Dynamics of Open and Closed Belt Conveyor Systems Incorporating Multiple Drives Peter Robinson PhD Thesis i ii Dynamics of Open and Closed Belt Conveyor Systems Incorporating Multiple Drives A thesis submitted in fulfilment of the requirements For the award of the degree of Doctor of Philosophy from THE UNIVERSITY OF NEWCASTLE By Peter William Alexander Robinson BEng (Hons) BSc (Physics) February 2016 iii iv Statement of Originality The thesis contains no material which has been accepted for the award of any other degree or diploma in any university or other tertiary institution and, to the best of my knowledge and belief, contains no material previously published or written by another person, except where due reference has been made in the text. I give consent to the final version of my thesis being made available worldwide when deposited in the University’s Digital Repository**, subject to the provisions of the Copyright Act 1968. **Unless an Embargo has been approved for a determined period. Peter Robinson 5th February, 2016 v For my Family vi Acknowledgements The production of this thesis would not have been possible without the help and guidance of many people and organisations. Firstly, I express my sincerest thanks to my supervisor A. Prof. Craig Wheeler, from the Centre for Bulk Solids and Particulate Technologies, in the University of Newcastle. The advice, encouragement and direction from Prof. Wheeler was crucial to the success of this thesis. Thank you. I’d also like to thank Dr Chris Wensrich, from the University of Newcastle. His mathematical knowledge was invaluable to the production of an accurate dynamic model. Acknowledgements are also given to all associate and technical staff at TUNRA Bulk Solids Handling Associates. Particular thanks to: - Mr Michael Carr, for his assistance with experimental work. - Mr Leo McFadden, Bill Rose, Tony Salmon and Paul Whitworth, for the invaluable assistance given designing, constructing and instrumenting test rigs. - Mrs Jayne O’Shea for her help retrieving material DMA data. Lastly, I would like to thank my friends and family. In particular my wife, Jessica Robinson. Thank you for your patience, support and understanding over the last 5 years, this would not have been possible without you. vii Table of Contents Abstract .................................................................................................................................. xii Nomenclature........................................................................................................................ xiii 1 Introduction ...................................................................................................................... 1 1.1. A comparison between Pouch and Troughed Conveying Systems .............................. 2 1.1.1 Belt .................................................................................................................... 3 1.1.2 Drive Method .................................................................................................... 4 1.1.3 Loading and Unloading ...................................................................................... 5 1.1.4 Routing Flexibility .............................................................................................. 6 1.2 Multiple Drive Systems on Conveyor Belts ................................................................. 6 1.2.1 1.3 2 3 Effect on Belt Tension ........................................................................................ 8 Thesis Statement and Outline .................................................................................... 9 Indentation Rolling Resistance ........................................................................................ 12 2.1 Modelling the Belt using a Generalised Maxwell Model ........................................... 13 2.2 Dynamic Mechanical Analysis and Master Curve Creation ....................................... 14 2.3 Indentation Rolling Resistance ................................................................................. 17 2.4 Experimental Setup ................................................................................................. 22 2.5 Findings ................................................................................................................... 25 2.5.1 Dependence on Idler Roll Diameter ................................................................. 25 2.5.2 Dependence on Cross Radius ........................................................................... 26 2.5.3 Comparison between Experimental and Simulated Results .............................. 27 2.5.4 Effect of Maxwell Elements .............................................................................. 31 2.6 Strain Dependency .................................................................................................. 33 2.7 Discussion ............................................................................................................... 38 2.8 Conclusions ............................................................................................................. 39 Drive Traction ................................................................................................................. 40 viii 3.1 Point Contact Drive Systems ....................................................................................41 3.2 General Incipient Sliding ..........................................................................................42 3.3 Tractive Rolling ........................................................................................................46 3.4 Experimental Validation ...........................................................................................47 3.4.1 Experimental Theory ........................................................................................48 3.4.2 Friction Experiments ........................................................................................50 3.4.3 Traction Experimental Setup ............................................................................51 3.4.4 Initial Results....................................................................................................54 3.4.5 Final Results .....................................................................................................59 3.5 4 5 The Transition to Tractive Rolling .............................................................................63 3.5.1 Force Dependent Traction ................................................................................64 3.5.2 Indentation Dependent Traction ......................................................................65 Troughed Conveyor Dynamics .........................................................................................68 4.1 Introduction.............................................................................................................69 4.2 Modelling the Conveyor Belt using Finite Elements ..................................................70 4.2.1 The Belt............................................................................................................70 4.2.2 Belt Sag ............................................................................................................71 4.2.3 Building a Mathematical solution for a Finite Element Model ...........................74 4.2.4 The Counterweight ..........................................................................................78 4.2.5 Lumped Masses ...............................................................................................84 4.2.6 Resistances to Motion ......................................................................................85 4.3 Velocity Prescribed Starting Procedures ...................................................................99 4.4 Belt Loading ...........................................................................................................101 4.5 Determination of the Elastic Modulus of a Conveyor Belt.......................................103 4.6 Conclusions............................................................................................................105 Case Study – Onsite Testing ...........................................................................................106 5.1 Conveyor Layout ....................................................................................................106 ix 5.2 Conveyor Parameters ............................................................................................ 108 5.3 Starting of an Elastic Conveyor Belt ....................................................................... 110 5.3.1 Drive Torque during Starting .......................................................................... 111 5.3.2 Counterweight Movement during Starting ..................................................... 114 5.3.3 Belt Velocity during Starting........................................................................... 116 5.4 5.4.1 Drive Torque during Stopping ........................................................................ 119 5.4.2 Counterweight Movement during Stopping ................................................... 122 5.4.3 Belt Velocity during Stopping ......................................................................... 124 5.5 Aborted Starting and Emergency Stopping Procedures of an Elastic Conveyor Belt 126 5.5.1 Drive Torque during Emergency Stopping Procedures .................................... 127 5.5.2 Counterweight Movement during Emergency Stopping Procedures ............... 128 5.5.3 Belt Velocity during Emergency Stopping Procedures..................................... 130 5.6 6 Stopping an Elastic Conveyor Belt .......................................................................... 119 Discussion ............................................................................................................. 131 Pouch Conveyor Dynamics ............................................................................................ 133 6.1 Wavefront Propagation through a Point Contact Drive System .............................. 134 6.2 Experimental Validation ........................................................................................ 135 6.3 Incorporating Wave Propagation in a Finite Element Model .................................. 141 6.3.1 Wave Propagation Constant .......................................................................... 141 6.3.2 Modelling the Suspended Mass of a Pouch System ........................................ 142 6.4 Simulating Wave Propagation for an Impulse Response ......................................... 149 6.5 Transient Behaviour in Pouch Conveyors Incorporating Multiple Drive Systems..... 153 6.5.1 Concurrent Starting ....................................................................................... 153 6.5.2 Sequenced Starting – Controlling the Peak Acceleration ................................ 157 6.5.3 Sequenced Starting – Controlling the Breakaway Friction ............................... 164 6.6 The Scalability of Pouch Conveyors Incorporating Multiple Drive Units .................. 168 6.7 Conclusion............................................................................................................. 170 x 7 Design Aspects, Conclusions and Future Work ...............................................................171 7.1 Design Aspects.......................................................................................................171 7.1.1 Main Resistances ...........................................................................................172 7.1.2 Drive System ..................................................................................................173 7.1.3 Dynamic Response during Starting and Stopping ............................................174 7.2 Summary ...............................................................................................................175 7.3 Future Work ..........................................................................................................177 7.3.1 Static Friction and Sag Take-up .......................................................................177 7.3.2 The Inclusion of an Inertial Frame of Reference within the Dynamic Model ....178 References ............................................................................................................................180 xi Abstract The incorporation of conveyor systems throughout industry has seen an increase in demand for systems that exceed the specification of conventional conveyors. This coupled with the demand to convey bulk materials over larger distances, at higher speeds and efficiencies, requires the development of a versatile design approach. This thesis explores the design aspects associated with modern pouch conveying systems, and how they vary, and can be adapted from theories used with conventional troughed conveyors. In particular, the indentation rolling resistance (IRR) is explored in detail, as this can account for up to 60% of the drag forces of a system. This is the drag force that arises due to an asymmetric pressure distribution as the idler roll shell indents the bottom cover of the belt. The potential idler roll arrangements for a generic pouch conveying system are analysed, and compared with experimental values. In addition to this, the drive traction attainable from suitable drive stations is analysed. Troughed conveyors typically wrap the conveyor belt around a large drive pulley, generating large amounts of traction. Given the layout of pouch conveying systems, a different approach is required, at multiple locations. As such, pouch conveyors are typically driven through simply supported drive stations, with small areas of contact with the belt. The useable traction from these point contact drives is considered. These theories are then united and applied to a dynamic package capable of handling multiple conveyor designs. This package utilises Finite Element Modelling (FEM) to model the viscoelastic nature of the system, based on the distributed drag forces, and inputs of the conveyor. Lastly, to qualify this theory, experimental analysis is conducted on an on-site installation, and compared with the theoretical results. xii Nomenclature 𝑎 Leading edge of the idler-belt contact region 𝑚 𝑎𝑐 Carry side idler roll spacing 𝑚 𝑎𝑟 Return side idler roll spacing 𝑚 𝐴 Array used to define the behaviour of the nodal elements - 𝐴𝑏 Cross sectional area of the belt 𝑚2 𝑏 Trailing edge of the idler-belt contact region 𝑚 𝐵 Belt width 𝑚 Array used to define the boundary conditions of each state - 𝐵𝐶 𝑐 Half-width of the idler-belt contact region 𝑐𝑏 Speed of sound in a conveyor belt 𝑑𝑙𝑠 Lip seal diameter 𝑚 𝑑𝑚 Mean diameter 𝑚 𝐸 Elastic Modulus 𝑃𝑎 𝐸′ Storage Modulus 𝑃𝑎 𝐸′′ Loss Modulus 𝑃𝑎 𝐸𝑏 Elastic modulus of a conveyor belt 𝑃𝑎 𝐸𝑒𝑓𝑓 Effective elastic modulus 𝑃𝑎 𝐹𝐼𝑅𝑅 Drag force due to indentation rolling resistance 𝑁 Normal Force 𝑁 Drag force due to bearing resistances 𝑁 Drag force due to belt skew 𝑁 Drag force due to special resistances within the system 𝑁 𝐹𝑑𝑟𝑎𝑔 Total drag force 𝑁 𝐹𝑑𝑟𝑖𝑣𝑒 Drive force produced by motors 𝑁 𝐹𝑓𝑙𝑒𝑥 Drag force due to belt and bulk solid flexure 𝑁 𝐹𝑔 Force due to gravity 𝑁 𝐹𝑖 Belt tension corresponding to node 𝑖 𝑁 𝐹𝑙𝑠 Drag force due to the lip seal 𝑁 𝐹𝑠 Shear force between the upper and suspended pouch 𝑁 𝐹𝑁 𝐹𝑅𝑖𝑚𝑑𝑟𝑎𝑔 𝐹𝑆𝑘𝑒𝑤 𝐹𝑆𝑝𝑒𝑐𝑖𝑎𝑙 𝑚 𝑚⁄ 𝑠 sections xiii 𝐺 Shear modulus 𝑃𝑎 ℎ Bottom cover thickness 𝑚 𝐻 Damping Coefficient 𝐼 Identity matrix 𝐼𝑝 Moment of inertia of the counterweight pulley 𝑘𝑔𝑚 2 𝐾 Spring constant 𝑁⁄ 𝑚 K𝑓 Foundation Modulus 𝐾𝑠 Static sag ratio - Height of the pouch from the lower contact of the drive 𝑚 𝑙 𝑁𝑠⁄ 𝑚 - 𝑃𝑎 wheel to the centre of gravity of the suspended mass 𝐿 Idler roll spacing 𝑚 𝐿∗ Additional length due to sag 𝑚 𝐿𝑒 Effective length of each finite element 𝑚 𝐿𝑖,𝑗 Width of the labyrinth seal 𝑚 Reduced mass 𝑘𝑔 𝑀 Nodal mass 𝑘𝑔 𝑀1 Nodal mass of a carry side element 𝑘𝑔 𝑀11 Nodal mass of the upper pouch element surrounding the 𝑘𝑔 𝑚𝑟𝑒𝑑 drive system 𝑀12 Nodal mass of the suspended pouch element surrounding 𝑘𝑔 the drive system 𝑀2 Nodal mass of a return side element 𝑘𝑔 𝑀𝑐 Mass of the counterweight 𝑘𝑔 Drag moment resulting from the labyrinth seal 𝑁𝑚 Mass of the counterweight pulley 𝑘𝑔 𝑀𝑙𝑎𝑏 𝑀𝑝 𝑛 Speed of rotation 𝑟𝑝𝑚 𝑁 Number of nodes - 𝑁𝑒 Number of Elements - 𝑝 Pressure 𝑃𝑎 𝑃 Normal load 𝑁 𝑃𝑒 Number of Parameters - 𝑞 Uniformly distributed load 𝑁⁄ 𝑚 xiv 𝑘𝑔⁄ 𝑚 𝑞𝑏 Belt mass per unit length 𝑞′ Traction generated within the slip region of contact 𝑁 𝑞′′ Traction generated within the stick region of contact 𝑁 𝑄 Tractive force 𝑁 𝑄𝑡 Conveyor throughput 𝑅 Radius of the counterweight pulley 𝑚 𝑅1 Primary radius of an idler roll shell 𝑚 𝑅2 Cross radius of an idler roll shell 𝑚 𝑅3 Radius of belt curvature 𝑚 Shear distribution 𝑃𝑎 𝑡 Thickness of the belt section 𝑚 𝑇 Temperature °𝐶 𝑇𝑏 Tension 𝑁 𝑇1 Tight side tension adjacent to the drive system 𝑁 𝑇2 Slack side tension adjacent to the drive system 𝑁 𝑇𝑑 Drive Torque 𝑣 Velocity 𝑆(𝑥) 𝑣𝑑𝑟𝑖𝑣𝑒 Velocity of the drive 𝑡⁄ ℎ𝑟 𝑁𝑚 𝑚⁄ 𝑠 𝑚⁄ 𝑠 𝑤 Indentation of a rigid idler roll into a belt 𝑚 𝑤𝑑 Drive wheel width in contact with the belt 𝑚 𝑥 Displacement 𝑚 𝑥𝑐 Displacement of the counterweight 𝑚 𝑥𝑖 Displacement of the belt corresponding to node 𝑖 𝑚 𝑋̂ State vector - 𝑧 Indentation profile 𝑚 𝑍0 Peak indentation 𝑚 𝛼𝑅 Reflection coefficient - 𝛼𝑇 Transmission coefficient - 𝛾 Shear strain - 𝛿 Peak belt sag 𝑚 𝜀 Strain xv 𝑠 −1 𝜀̇ Strain rate 𝜀∗ Total axial strain - 𝜀1 Apparent axial strain - 𝜀𝑠 Strain due to belt sag - 𝜂 Damping Coefficient 𝑁𝑠⁄ 𝑚 𝜃 Angle of wrap 𝑟𝑎𝑑 𝜃𝑐 Rotation of the counterweight pulley 𝑟𝑎𝑑 𝜇 Coefficient of Friction - Effective coefficient of friction - 𝜈 Poisson’s ratio - 𝜌 Density 𝜍 The ratio between the leading and trailing edge lengths 𝜇𝑒𝑓𝑓 𝑘𝑔⁄ 𝑚3 - within the contact region 𝜎 Stress 𝑃𝑎 𝜎̇ Stress rate 𝑃𝑎⁄ 𝑠 𝜏 Shear force 𝑃𝑎 𝜏𝑖 The wave period of the individual Maxwell elements 𝜐𝑑 Dynamic viscosity 𝑃𝑎 𝑠 𝜐𝑘 Kinematic viscosity 𝑃𝑎 𝑠 𝜔 Angular frequency Ω Angular velocity 𝑟𝑎𝑑⁄ 𝑠 𝑟𝑎𝑑⁄ 𝑠 𝑠 xvi 1 Introduction Globally, belt conveyors have secured their place in the mining industry, as one of the most favoured, and economical methods of bulk material transport. Their design and versatility allows them to handle materials of various particle sizes, at high speeds and throughputs, over confronting terrain. This makes them the primary choice for bulk material transportation, especially over terrain with little or no infrastructure. Conventional troughed conveyors are the most common form of installation. Their open trough design allows large particles, or clumps, to be conveyed relatively easily. The past century has seen demand for belt conveyors rise significantly. The growing need for efficient systems to convey material over more trying and diverse routes has stimulated significant research in this area, beyond the scope of conventional conveying systems. Two common areas of research associated with such systems are dust suppression, and the ability to convey up steep inclines. Pouch conveying systems have emerged as an improved solution to this problem. These conveyors are designed to enclose the material within the belt, creating a closed system. This 1 limits the exposure of the bulk solid to the air, reducing dust emissions and losses. It also generates internal friction within the material, allowing it to be conveyed up steeper angles. This is discussed further in Section 1.1. 1.1. A comparison between Pouch and Troughed Conveying Systems Pouch conveying systems were designed with a view to build on the limitations of open troughed conveyors. They consist of a flat belt, moulded into a tear drop cross section, to house the bulk material. An example of this cross section is depicted in Figure 1.1. Other pouch designs utilise a similar construction; a flat belt with modified edges to form a support structure. Clamping Idler Roll Idler Roll Conveyor Belt Bulk Solid Figure 1.1 Cross section of (a) Enerka-Becker pouch conveyor, (b) Sicon pouch conveyor, (c) Innovative Conveyor System. This cross section introduces several design concepts that differ significantly from troughed systems. Below, the primary differences between troughed and pouch conveying systems are outlined. 2 1.1.1 Belt Reinforcement In essence, the belt used in each system is based around similar principles. A fabric or steel reinforced carcass is encased in a highly resistant polymer or rubber blend to convey the bulk material. The principle difference between the two systems is the location of the carcass. Troughed conveyors utilise a unilateral or bilateral core throughout the entire belt. This is primarily due to the high tensile forces encountered with a large drive spacing. Pouch conveyors, as discussed in Section 1.1.2, aren’t exposed to such high tensile forces, and thus the reinforcement is often not required throughout the whole belt. Primarily, a form of reinforcement will be placed in the supporting member of the belt, in order to withstand the tensile load throughout the belt. Depending on the design intent of the system, reinforcement may also be required in the pouch section of the belt: either to assist with the axial loads in the belt, or to assist with the support of the bulk material. Shape and Particulate Behaviour The principle difference between the two systems, as outlined above, is the method of conveying the bulk material. The bulk material is conveyed on top of a troughed conveyor belt, compared to being encased within the conveyor belt in a pouch system, as depicted above in Figure 1.1. The encapsulation of the bulk solid ensures losses due to dust in transport are kept to a minimum. This means materials with a high fines content can potentially be conveyed via belt. It also enables moisture sensitive materials to be conveyed, at potentially high velocities. An added advantage to this is the internal friction generated within the material during loading (discussed in section 1.1.3). This friction allows steeper inclines to be negotiated, compared with its troughed counterpart. Some pouch systems claim to have the ability to convey up to 80 degrees [18]. Commonly, troughed systems are limited to approximately 20 degrees. This is due to material fall-back and loss. 3 Troughed conveyors are known in industry to have the greatest ability to handle large and odd shaped particles. Theoretically speaking, the particle size limitations of a troughed belt are governed by the terrain and speed of the system, as well as the belt width. For pouch conveyors, particle size bears a large impact on the operation of the conveyor. As they are an enclosed system, excessively sized particles present upon loading, will generate a significant closing force, risking belt or structural rupture. As such, the limiting particle size of a pouch conveyor is approximately 35-50% of the width of the pouch [18]. This is dependent on the pouch width, as well as the running configuration of the system. An added consequence of this is the possible throughputs associated with each system. Being an open, reinforced belt, a troughed system has much higher capabilities than a similar belt in a pouch configuration. 1.1.2 Drive Method The design of a pouch conveyor, as well as the required flexibility of the belt, limits the tensile capabilities of the belt structure. As such, excessive drive spacing is not practical. From this, an alternative method of applying power to the belt was required. Troughed systems have the ability to transfer enormous amounts of the power to the belt. This arises from a reinforced belt wrapped around a large drive pulley coated with high friction lagging. The belt is compressed onto the pulley surface through the internal tension in the belt, created by the counterweight, load and friction. Pouch conveyors instead use a point-contact drive to create traction. These drive wheels are indented into the belt surface via gravity, or through the use of a spring. Comparatively, the amount of traction achievable by this method is significantly less than the large drive pulley employed in a troughed system, resulting in more drive stations required. The drive arrangement of the Enerka-Becker pouch conveyor is depicted below in Figure 1.2. The implications of multiple drive stations are discussed in section 1.2. 4 Figure 1.2 Drive wheel arrangement of the Enerka-Becker System [35]. 1.1.3 Loading and Unloading Conventional conveyors are easily loaded and unloaded, owing to the nature of the open belt. Pouch conveyors however require the insertion of the material into the pouch. In order for this to be achieved, the belt needs to be loaded while in an open state. If the belt was loaded while completely open, risks would arise during closing of the belt, owing to shear between the particles, and the risk of overloading. Instead, pouch conveyors are typically loaded while partially open. The top of the pouch is open just enough for material to flow in via a chute. This minimises the amount of closing afterward, as well as the risk of belt rupture. This also allows a higher degree of filling (up to 80%). The unloading of a pouch occurs in a similar manner to a troughed system. Typically, the pouch is opened out flat, and travels around a head drum, discharging the material. Once the material has been discharged, the belt is once again folded back into an upright pouch, for travel along the return side of the conveyor. This conceptually allows material to be conveyed along the return side also. It also eliminates the needs for scraper blades and belt cleaners, as no idler rolls or running equipment comes into contact with the carrying surface of the belt. This is contrary to troughed systems, where the carrying surface becomes the underside of the belt along the return side. This means that any residual material on the belt requires cleaning; otherwise it will wear the idler roll surface and deteriorate bearing performance. Most excess material simply falls to the ground below. 5 1.1.4 Routing Flexibility The ability to negotiate horizontal and vertical curves is something that severely limits troughed installations. The width of the belt, along with the structural elasticity of the belt makes tight horizontal curves near impossible. Likewise, vertical curves are hindered by belt lift-off in concave cases, and excessive strain in convex situations. Most problems that arise are due to the belt being constrained by its own self weight. The support structure of pouch conveyors constrain the movement of the belt, in order to keep the pouch closed. As such, belt lift-off and excessive sag are a lot less of an issue during curves. The cross section and stiffness has also changed. The second moment of area of a troughed system is much more than the pouch counterpart, resulting in easier bending. As mentioned above, the pouch component often contains minimal or no reinforcement, making it more elastic. The result of all this is the ability to negotiate tight curves. The Innovative Conveying System [39], as well as the Sicon pouch conveying system [4], claim to be able to negotiate curves as low as 1.5m radius due to this. This is dependent on the size and loading state of the system. 1.2 Multiple Drive Systems on Conveyor Belts The addition of extra drive stations along a belt reduces the tension distribution in the system, negates the need for oversized belts and infrastructure, and provides an effective way to control belt dynamics and other transients. The method of applying this additional drive power has long been researched. The added infrastructure required to discharge the material from the belt, wrap the belt around a drive station, then reload the material onto the belt was deemed excessive. This configuration is commonly referred to as a ‘tripper’ drive, as depicted in Figure 1.3. As such, designs were researched that allowed the bulk material to remain on the belt. 6 Figure 1.3 Different methods of distributing drive power along a conveyor [35]. As can be seen in Figure 1.3, several methods were devised to add drive power along a conventional belt conveyor system. In addition to the tripper drive discussed previously, other conceptions involve using other drive belts to drive the loaded belt. The booster belt idea seen in (b) is a smaller dedicated conveyor belt placed inside the main infrastructure. Its purpose is only to apply drive power. The power strip design seen in (c) runs almost the full length of the system. This strip, in addition to transferring drive power to the loaded conveyor, bears the tensile load of the system. A cross section of the power strip design is shown below in Figure 1.4. As can be seen, the steel reinforced power strip is designed to sit at the base of the trough, with gravity maintaining a tractive contact between the two surfaces. This allows a more flexible fabric belt to convey the material, with better troughing characteristics. Figure 1.4 The Power-strip drive system [54]. 7 Due to their configuration, pouch conveying systems don’t have the ability to utilise a boosterbelt drive system, without considerable modification and infrastructure. As such, as mentioned above, point contact drives are a more suited approach. Point contact drive systems generate traction by indenting a rigid drive wheel into the viscoelastic surface of the conveyor. This approach allows wheels of various diameters, widths and materials to be used on a flat or curved belt surface. 1.2.1 Effect on Belt Tension The distribution of drive power along the length of the conveyor belt brings significant improvements to the tension gradient within a system. As expected, multiple drive systems reduce the overall tension in the system within simpler design limits. This is shown below in Figure 1.5. Figure 1.5 The effect of multiple drive stations on the tension distribution [35]. Smaller tensions in a system allow several components to be minimalized, reducing costs. The belt, in particular, isn’t required to withstand such a high load, and can thus be made of a lighter construction. Similarly, idler rolls and support components can be reduced. One of the biggest impacts of distributed drive power, is the ability to place drives at critical locations. Typically, on a conventional system, a conveyor travelling uphill has a drive at the 8 conveyor’s head. This is done such that a counterweight can be placed at the start of the return side, ensuring the slack side tension doesn’t drop below a certain value. This would not be possible, or practical with a tail drive. Likewise, a downhill conveyor typically has a drive at the tail end of the conveyor, at its highest point. For similar reasons, this ensures adequate operation of the counterweight. It also allows the motor assembly to be used as a brake also. Typically, however, conveyors are laid over hilly terrain consisting of many ups and downs. The ability to place drives, virtually at any point on the system, means they can be placed where they are needed, in any multiple, at any spacing. The use of multiple drives does not mean multiple take-up systems are also required. The function of a counterweight (or take-up) is to set a benchmark tension in the system, preventing excessive sag. It is typically placed next to a drive system on a troughed conveyor belt, so that the slack side tension of the drive can be maintained. This type of drive system relies on the tension in the belt to create friction at the belt-pulley interface. Point contact drives on the other hand, don’t rely on belt tension as a normal force. Most drive arrangements use a combination of the suspended weight of the belt and bulk solid, as well as a clamping spring, or similar. The effect of multiple drive systems on the dynamics of a pouch system is discussed extensively in Chapter 6. 1.3 Thesis Statement and Outline The statement of this thesis is as follows: “To investigate the dynamic implications of pouch conveyor systems incorporating multiple drive stations, associated design aspects, and how they differ from conventional troughed conveyors.” From this, the following work presents various models, and associated simulation programs, in an attempt to improve the design capabilities of complex conveyor systems, as they emerge into industry. An outline of the thesis is presented below. Chapter 2 investigates the indentation rolling resistance (IRR) associated with a generic pouch conveyor. In order to fully understand the transient behaviour of a conveying system, a comprehensive understanding of the main resistances in the system is required. IRR can 9 account for up to 60% of the total power usage in a system. IRR arises due to an asymmetric pressure distribution as a moving belt is indented by a rigid idler roll. This section uses a generalised Maxwell model to predict the viscoelastic behaviour of the belt, and compares it with experimental results from an in-house testing facility. Chapter 3 investigates the interface between the drive wheel and the belt surface, using theory developed in Chapter 2. Using this, a model is developed that predicts the tractive force attainable from various drive arrangements. The breakaway traction during start-up is modelled, and compared with experimental results from an MTS machine. This is then extrapolated to investigate the rolling traction of a drive system. That is, the traction available to accelerate the belt, once it is in motion. Chapter 4 builds a model to predict the transient behaviour of troughed conveyors. This model incorporates the effects of belt sag in the system, as well as the inertia of all moving components. In this section, several more resistances are assessed, and how they affect the dynamics of the belt. The standard method for determining the elastic modulus of the belt is also presented. Chapter 5 involves experimental validation of the dynamic theory presented in Chapter 4. Tests were performed on a 64m fabric installation, under a range of different starting and stopping procedures, for various loading conditions. Emergency procedures such as aborted starts and emergency power fails were also conducted. These were subsequently simulated, and the results compared. In each instance, there was a close correlation between the drive torque, counterweight movement and velocity profile. Chapter 6 analyses the dynamic differences between troughed and pouch conveying systems. Experiments are conducted to determine the effect of a point contact drive station on a wavefront propagating through the pouch. This effect is subsequently incorporated in the dynamics model of Chapter 4. Using the modified model, simulations are conducted to confirm the experimental results. That is, a propagating wavefront in a pouch conveyor has the ability to travel through a point contact drive system, introducing an interference pattern between subsequent drive stations. From here, the starting procedures of multiple drive stations are optimised. 10 Chapter 7 discusses the applications of the following research, along with relevant conclusions, and recommendations for future work. 11 2 Indentation Rolling Resistance In order to perform accurate predictions of the belt dynamic behaviour, the drag forces that arise from the way the belt interacts with the supporting structure must be understood. These motion resistances account for the majority of the main resistances, and if managed effectively, efficiency and productivity can be maximised. Indentation rolling resistance is a drag force that arises due to an asymmetric indentation pressure as a conveyor belt passes over an idler roll. This pressure profile results from hysteretic behaviour of the bottom cover of the belt, owing to the reduced relaxation time of the bottom cover. Hager and Hintz [13] postulated this loss can contribute up to 60% of the total power usage. Given the magnitude of this loss mechanism, several authors (including Jonkers [22], Spaans [46], Rudolphi & Reicks [42] and Lodewijks [24]) have proposed predictive models to estimate the extent of this loss in conventional conveying systems. Each model possesses different benefits, each with their own validity. These models apply to cylindrical idler rolls indented into a flat conveyor belt, and as such need modification to apply to the complex support systems existent in pouch conveyors. Nuttall [35] modified the theory presented by Lodewijks to predict the losses within the Enerka-Becker pouch conveyor system. 12 However as stated earlier, modern pouch conveyors utilise varying support configurations. Thus, a model needs to be developed that can simulate convex, flat and concave belt profiles, indented by flat and spherical (convex) idler rolls. It is common practice to model the indentation of the idler roll into the bottom cover of the conveyor by utilising a one-dimensional stress/strain analysis. In this model, inter-fibre relations are omitted, thus each fibre makes an independent contribution to the overall stress, as shear between the fibres is neglected. This simplification is done (opposed to a twodimensional model) in an attempt to minimise computational power, while still achieving accurate results. 2.1 Modelling the Belt using a Generalised Maxwell Model The cover material of a conveyor belt is a viscoelastic material, meaning its behaviour is dependent on the load and strain applied to the belt, as well as the temperature and strain rate. From above, Lodewijks [24] utilised a 3-parameter Maxwell model, which has been shown to provide reasonable predictions for viscoelastic materials. As conveyor belt manufacturers aim to improve the bottom cover material composition, in an attempt to minimise indentation rolling resistance, many polymers used are extremely temperature and strain rate dependent, and thus require a highly sensitive model. As will be shown, the 3parameter model yields a limited approximation, especially given the frequency of indentation on a belt conveyor may exceed 1000 Hz. The Generalised Maxwell Model, sometimes called the Maxwell-Wiechert model, utilises a series of springs and dampers in parallel, as seen in Figure 2.1. This model, a generic form of Maxwell’s 3-parameter model, allows the loading contributions from varying polymer lengths in the material to be accounted for. This is particularly important for polymer blends, which are increasingly common in conveyor belts. For the purpose of this research, the number of elements (𝑁𝑒 ) of the Maxwell model will denote the number of links, while the number of parameters (𝑃𝑒 ) will refer to the total number of springs and dashpots. Thus: 13 𝑃𝑒 = 2𝑁𝑒 + 1 (2.1) Figure 2.1 Generalised Maxwell Model. 2.2 Dynamic Mechanical Analysis and Master Curve Creation Dynamic Mechanical Analysis (DMA) is a technique used to characterise the properties of viscoelastic materials. A sinusoidal stress is applied via a machine (see Figure 2.2) to a sample of the material at varying temperatures, frequencies and strain rates, across a specified strain region. From this, many properties can be obtained; the two most common and useful however are the Storage (E’) and Loss (E”) Modulus. These two parameters define the materials ability to dissipate and store energy. From the model above in Figure 2.1, the materials capacity to store energy corresponds to the elastic portion of the material, whereas the energy dissipation is due to the viscous dampers. Figure 2.2 DMA Machine RSA-G2 and associated clamps [48]. 14 Shown below in Figure 2.3, is an example of the raw data extracted from a DMA test. Each plot corresponds to a certain temperature. Embedded in the storage and loss data, is the materials dependence on temperature. According to the Time-temperature Superposition Principle, temperature and frequency data can be related through the use of a master curve. A master curve is a consolidation of the raw data, into a single plot, extended over a large frequency range. This allows frequency information well in excess of the capabilities of the machine to be extrapolated. This process is outlined below. 1.000E9 1.000E8 1.000E8 1.000E7 1.000E7 1.000E6 E' (Pa) 1.000E9 1.000E6 0.1000 E'' (Pa) 1.000E10 1.000 frequency (Hz) 1.000E5 10.00 Figure 2.3 Graph of E', E" vs frequency [23]. To construct a master curve, an operating temperature is chosen as a basis for setting the frequency scale. Using this curve, the curves corresponding to one increment in temperature above and below would be shifted until the best fit for a uniform curve is created. To ensure continuity, a log-log scale must be used, as depicted in Figure 2.4. The process is then repeated for all curves with the end result shown in Figure 2.5. 15 1200 Storage Moduli, E' (MPa) 1000 800 T1 T3 600 T4 400 T2 200 T5 0 0.0001 0.01 1 100 10000 Frequency (Hz) Figure 2.4 Creation of a master curve [23]. 10 9 8 Log [E' (Pa)] Log [E'' (Pa)] 9 8 7 7 6 6 -4-3-2-1 0 1 2 5 3 4 5 6 7 8 9 10 11 12 13 Log [frequency (Hz)] Figure 2.5 Constructed master curve [23]. As can be seen, mechanical properties can be ascertained for frequencies well in excess of the capabilities of any testing procedure. Depending on the conveyor, the frequency of indentation of a belt passing over an idler roll is typically of the order of 50-100 Hz, per unit of belt velocity (m/s). Lighter indentation loads would result in a higher frequency, as would higher belt velocities. For the Generalised Maxwell Model, the storage (E’) and loss (E”) moduli are related to the mechanical elements of the model via: 𝑁 ′( 𝐸 𝜔) = 𝐸0 + ∑ 𝐸𝑖 𝑖=1 𝜔2 𝜏𝑖 2 1 + 𝜔 2 𝜏𝑖 2 (2.2) 16 𝑁 "( 𝐸 𝜔) = ∑ 𝐸𝑖 𝑖=1 𝜏𝑖 = 𝜔𝜏𝑖 1 + 𝜔 2 𝜏𝑖 2 𝜂𝑖 𝐸𝑖 (2.3) (2.4) Where Ei and ηi correspond to the values of the individual springs and dampers respectively, and 𝜏𝑖 corresponds to the wave periods of the individual dissipative elements [42]. From the created curve, equation (2.2), (or equation (2.3)) is fitted to the data using a function minimisation technique, to obtain the respective Maxwell parameters. 2.3 Indentation Rolling Resistance It is becoming increasingly common for pouch conveyor systems to contain curved running surfaces in contact with the idler rolls. This serves multiple purposes. The Enerka-Becker system (Figure 2.6 (b)), utilises a convex belt surface, to prevent jamming of the belt between support idler rolls. The ICS system (Figure 2.6(e)), alternatively, uses convex (spherical) idler rolls to support a concave belt section. In addition to this, curved surfaces allow the predictability of wear concentration, and the use of sacrificial parts, to minimise maintenance costs and downtime. As such, conventional methods used to calculate rolling resistance need to be modified. Multiple models have been developed to predict this friction, and the model presented below builds on the work of several authors, including Rudolphi & Reicks [42], and Nuttall [35]. Each model involves portraying the viscoelastic belt as a one-dimensional Winkler foundation, indented by a rigid idler roll. In addition to these models, curved belts and spherical idler rolls are investigated, as depicted in Figure 2.6. Previous tests have indicated that current bottom cover materials display a high dependence on temperature and load rate. To accommodate this, a high order Maxwell model is used in the approximation. For simplicity, a similar notation to Nuttall and Rudolphi & Reicks is used. This is illustrated below in Figure 2.7. 17 (a) (b) (c) (d) (e) Figure 2.6 Possible support combinations of pouch conveyors. 18 Figure 2.7 Schematic of the indentation profile. For small indentations (z<<h), it is possible to approximate the indentation depth as a parabolic function (Lodewijks [24]). Assuming that Z0 is the peak indentation, occurring at the centreline of the idler roll, the surrounding indentation can be estimated according to: 𝑤(𝑥, 𝑦) = 𝑍0 − 𝑥2 𝑦2 𝑦2 − + 2𝑅1 2𝑅2 2𝑅3 (2.5) In the above equation, it should be noted that the radius R3 is positive for a concave running surface, and negative for a convex surface. Similarly, this equation can be converted to a standard troughed configuration by setting R2 and R3 to infinity. From the generalised Maxwell model (Figure 2.1), it is evident that the total stress in the system can be written as the sum of the stress in each individual strand: 𝜎 = 𝜎0 + ∑ 𝜎𝑖 (2.6) 𝜎𝑖 = 𝜎𝐸𝑖 + 𝜎𝜂𝑖 (2.7) 𝜎0 = 𝐸0 𝜀 (2.8) 19 𝜎𝐸𝑖 = 𝐸𝑖 𝜀𝐸 𝜎𝜂𝑖 = 𝜂𝑖 (2.9) 𝑑𝜀𝜂 𝑑𝑡 (2.10) where 𝜀𝐸 and 𝜀𝜂 correspond to the strains in the individual springs and dashpots respectively. By taking the time derivatives of the individual strains: 𝜀𝑖̇ = 𝜀𝐸̇ + 𝜀𝜂̇ (2.11) 𝜀𝐸̇ = 𝜎𝐸𝑖̇ 𝐸𝑖 (2.12) 𝜀𝜂̇ = 𝜎𝜂𝑖 𝜂𝑖 (2.13) 𝜎𝑖̇ + 𝜎𝑖 𝐸𝑖 = 𝐸𝑖 𝜀̇ 𝜂𝑖 (2.14) The resulting ordinary differential equation relates the stress and strain in an individual element. Solving this, noting that 𝜀 = 𝑤(𝑥,𝑦) ℎ , the stress distribution over the contact region can be determined: 𝜎(𝑥, 𝑦) = 𝑎2 𝐸0 𝑥 𝑥 (1 − ) (1 + ) 2𝑅1 ℎ 𝑎 𝑎 𝑁 𝑎 𝑥 𝑎2 𝐸𝑖 𝑣𝜏𝑖 𝑣𝜏𝑖 𝑥 − (1− ) ∑ [(1 + ) (1 − 𝑒 𝑣𝜏𝑖 𝑎 ) − (1 − )] + 𝑅1 ℎ 𝑎 𝑎 𝑎 (2.15) 𝑖=1 In the above equation, 𝑎 is a function of 𝑦, and is determined from equation (2.5) by setting the indentation to zero, with 𝑥 equal to 𝑎: 𝑤(𝑎, 𝑦) = 0 → 𝑎 = √2𝑅1 𝑍0 + 𝑅1 𝑦 2 ( 1 1 − ) 𝑅3 𝑅2 (2.16) 20 From this, we can determine the overall normal load applied by the belt, the opposing moment, and thus the drag force. 𝑐 𝑎(𝑦) 𝐹𝑁 = 2 ∫ ∫ 𝜎(𝑥, 𝑦)𝑑𝑥𝑑𝑦 (2.17) 0 −𝑏(𝑦) 𝑐 𝑎(𝑦) 𝑀 = 2 ∫ ∫ 𝑥𝜎 (𝑥, 𝑦)𝑑𝑥𝑑𝑦 (2.18) 0 −𝑏(𝑦) 𝐹𝑑𝑟𝑎𝑔 = 𝑀 𝑅1 (2.19) From the above equations, it is possible to determine an explicit solution for a given 𝑦 value, by expressing the ratio 𝜍 = 𝑏 𝑎 [42]. From this, and equation (2.15), 𝑏 can be determined. For simplicity however, a computational approach was used to solve for the stress distribution between 𝑥 = −𝑎(𝑦) and 𝑥 = 𝑎(𝑦), and any negative data between 𝑥 = −𝑎(𝑦) and 𝑥 = −𝑏(𝑦) discarded. This method exploits the fact that |𝑎(𝑦)| > |𝑏(𝑦)|. For troughed conveyors, the indentation force is a function of the belt and bulk material weight on the idler, as well as the flexure force to trough the belt. Because of this, May et al [25] formulated a solution relying on the vertical load on the idler roll, contrary to the formula defined above which specifies an indentation depth. Using a specified depth, a convergence algorithm can be used to arrive at a defined weight. The most efficient process is dependent on the configuration of the idler stations. The carry side of pouch conveyors generally employs a belt suspended by idler rolls; however in areas where belt opening is an issue, clamping rollers may be employed to ensure the belt remains shut. This is shown in the ICS system in Figure 1.1. These idler stations may be used in turnover areas of the belt after discharge or along the return side where the weight of the empty belt is not sufficient to ensure adequate closure. In places where clamping idler rolls affect the normal load on the running idler rolls, a more appropriate approach would be to determine the indentation on each idler roll, and calculate the rolling resistance from this. 21 2.4 Experimental Setup In order to validate the theoretical results, experimental testing was conducted on a laboratory test facility, specifically designed to accurately measure the various drag forces associated with idler roll motion [27]. This facility has the ability to investigate various idler rolls of different diameters and materials, at different loads, speeds and temperatures. Details of the setup are shown below in Figure 2.8 and Figure 2.9. Figure 2.8 TUNRA In-house IRR test facility [27]. Figure 2.9 Detail of IRR measurement apparatus [Error! Reference source not found.]. As can be seen, the idler rolls are mounted on a knife edge surface, which exhibits minimal rotational resistance. This is designed to purely support the weight of the idler roll, whereas the attached torque load-beam prevents the shaft from rotating. The force required to stop the shaft rotating is due to the bearing rim-drag, and is subtracted from the measured results. In addition to this, hold down idler rolls are shown either side of the test apparatus. These are used to control the amount of sag experienced by the centre roller (typically 1-2%). From this, hydraulic cylinders on the tail drum allow the belt tension, and therefore the downward force on the test idler roll, to be controlled. 22 In order to test situations indicated in Figure 2.6, profiled idler rolls were created to represent case (c) in Figure 2.6. With the exception of case (a), testing all other configurations would require major modifications. This was done by machining idler rolls with corrugated shells to simulate spherical idler rolls. This is depicted in Figure 2.10. Figure 2.10 Profiled idlers used for testing. In order to investigate how diameter and cross-radius (R2 -Figure 2.7) affected measurements, 6 idler rolls were made, with the following specifications: 23 Table 2.1 Description of test idler rolls. Diameter (mm) Cross-radius (mm) Number of profiles Number of profiles in (total) contact with belt Ø75 R20 24 17 Ø 125 R10 40 28 Ø 125 R15 30 21 Ø 125 R20 24 17 Ø 125 R25 20 14 Ø 175 R20 24 17 As can be seen from Figure 2.10, not all profiles are in contact with the belt. This is because the machine has the capacity to handle belts up to 600mm, whilst the belt used for testing was 400mm wide. In order to account for this, a versatile design was achieved which accounted for belt stray while maintaining a constant number of profiles in contact with the belt, for a variety of belt widths. The aforementioned idler rolls were tested at a variety of loads and velocities, as specified below: Table 2.2 Test schedule. Variable Alterations Velocity 2, 4, 6, 8, 10 m/s Load* 900, 1250, 1750, 2250 N Temperature 30 °C Belt Sag 1% * The load values shown relate to certain positions of the hydraulic cylinder used to tension the belt, and thus increase the load. Due to temperature fluctuations, creep and material inconsistencies, the actual load varied with each test, as shown in the results. As noted by Wheeler and Munzenberger [48], as an idler roll indents a steel cord belt, the pressure created is not limited to the bottom cover, but instead enters the insulation rubber surrounding the cords. From this, Wheeler and Munzenberger predicted that the apparent bottom cover thickness is equal to the actual bottom cover thickness, plus half of the steel cord diameter. The belt used during these experiments was a steel cord belt; with Ø5 mm 24 cords and a 6 mm bottom cover. From this, the apparent thickness of the bottom cover is 8.5 mm. This is the value used in the simulations. 2.5 Findings The range of experiments and simulations conducted enabled a comprehensive analysis to be performed of spherical idler rolls indenting a flat belt. Below, an analysis of how the indentation rolling resistance is affected by velocity, load, wheel diameter and cross radius will be discussed, as well as the accuracy of the theoretical modelling techniques. 2.5.1 Dependence on Idler Roll Diameter For conventional troughed systems, idler roll diameters are typically within the range of Ø125 mm to Ø225 mm. This is based on a compromise between the indentation rolling resistance, manufacturing costs and the rotational inertia of the idler rolls. As can be seen in Figure 2.11, larger diameter idler rolls exhibit lower rolling resistance due to the increased contact area between the belt and idler roll. This distributes the load over a larger area resulting in lower stresses, and allows a more symmetrical pressure distribution. The corresponding cross sectional area (csa) of the contact is also depicted in Figure 2.11. This is formulated based on the stress profile presented in equation (2.15). Based on this, larger diameter idler rolls would appear more suited for conveyor installations. 2 300 1.8 250 Contact Area (mm2) IRR per Profile (N) 1.6 1.4 1.2 1 0.8 0.6 0.4 Ø75 mm Ø125 mm Ø175 mm 0.2 0 0 50 100 150 200 150 100 50 0 0 Normal Force (N) Ø75 mm Ø125 mm Ø175 mm 50 100 150 Normal Force (N) Figure 2.11 The influence of normal force on the indentation rolling resistance (left) and contact area (right) for various idler roll diameters. 25 In contrast to this however, manufacturing cost and rotational inertia must also be considered. Obviously, cost per idler roll impacts the overall cost of a system, however inertia of the idler roll also affects the required drive power from a dynamics standpoint. As shown in Chapter 4, the inertia of an idler roll corresponds to a reduced mass which must be overcome at start-up, and braked during stopping. It can also have drastic dynamic effects during acceleration and deceleration of the belt. 2.5.2 Dependence on Cross Radius The cross radius of an idler roll bears less impact on the overall efficiency of a system compared to the idler diameter. Instead, it is more of a design preference to suit the supporting structure of the conveyor. Nevertheless, the impact it has on the indentation rolling resistance was still analysed. 2 350 1.8 300 Contact Area (mm2) IRR per Profile (N) 1.6 1.4 1.2 1 0.8 0.6 0.4 Rad10 mm Rad15 mm Rad20 mm Rad25 mm 0.2 0 0 50 100 150 200 250 200 150 100 Rad10 mm Rad15 mm Rad20 mm Rad25 mm 50 0 0 Normal Force (N) 50 100 150 200 Normal Force (N) Figure 2.12 The influence of normal force on the indentation rolling resistance (left) and contact area (right) for various cross radii. 4 As can be seen from Figure 2.12, the resistance is proportional to 𝐹 3 , with the cross radius having little impact. In theory, this is not unexpected. As the cross radius increases, Figure 2.12 shows the cross-sectional area (CSA) also increases, as expected. As this happens however, the ratio of area on either side of the idler roll central axis (y-axis) remains constant. This means that the pressure is simply redistributed, with no change to the retarding moment on the idler roll. The material indentation and relaxation time remain unchanged. 26 In comparison, a decrease of the major diameter of an idler roll, and thus the contact area will see an increase in the overall resistance. This is because the contact area is redistributed along the x-axis, altering the location of the leading edge a(y). A larger diameter idler roll will allow the bottom cover to indent and relax slower than a smaller idler roll, resulting in a less skewed pressure distribution, and thus, a smaller retarding moment. 2.5.3 Comparison between Experimental and Simulated Results For belt conveying systems, the ability to accurately predict the indentation rolling resistance is critical to accurately predict the belt tensions and size associated components. As discussed earlier, many theories have been formulated, each with their own attributes, and unfortunately, downfalls. The following graph depicts a comparison between the theory outlined above, and the results achieved experimentally. These simulations were performed using a 20 element Maxwell model (41 parameters). The results below depict data sets corresponding to a 2000, 3000, 4000 and 5000 N/m load across the idler roll, which in turn was converted to a normal force on each individual profile. 2.5 Theory 2000 N/m Load Exp 2000 N/m Load Theory 3000 N/m Load Exp 3000 N/m Load Theory 4000 N/m Load Exp 4000 N/m Load Theory 5000 N/m Load Exp 5000 N/m Load IRR per Profile (N) 2 1.5 1 0.5 0 0 2 4 6 8 10 12 Velocity (m/s) Figure 2.13 Comparison of experimental and predicted results for Ø75xR20. 27 2.5 Theory 2000 N/m Load Exp 2000 N/m Load Theory 3000 N/m Load Exp 3000 N/m Load Theory 4000 N/m Load Exp 4000 N/m Load Theory 5000 N/m Load Exp 5000 N/m Load IRR per Profile (N) 2 1.5 1 0.5 0 0 2 4 6 8 10 12 Velocity (m/s) Figure 2.14 Comparison of experimental and predicted results for Ø125xR10. 2.5 Theory 2000 N/m Load Exp 2000 N/m Load Theory 3000 N/m Load Exp 3000 N/m Load Theory 4000 N/m Load Exp 4000 N/m Load Theory 5000 N/m Load Exp 5000 N/m Load IRR per Profile (N) 2 1.5 1 0.5 0 0 2 4 6 8 10 12 Velocity (m/s) Figure 2.15 Comparison of experimental and predicted results for Ø125xR15. 28 2.5 Theory 2000 N/m Load Exp 2000 N/m Load Theory 3000 N/m Load Exp 3000 N/m Load Theory 4000 N/m Load Exp 4000 N/m Load Theory 5000 N/m Load Exp 5000 N/m Load IRR per Profile (N) 2 1.5 1 0.5 0 0 2 4 6 8 10 12 Velocity (m/s) Figure 2.16 Comparison of experimental and predicted results for Ø125xR20. 2.5 Theory 2000 N/m Load Exp 2000 N/m Load Theory 3000 N/m Load Exp 3000 N/m Load Theory 4000 N/m Load Exp 4000 N/m Load Theory 5000 N/m Load Exp 5000 N/m Load IRR per Profile (N) 2 1.5 1 0.5 0 0 2 4 6 8 10 12 Velocity (m/s) Figure 2.17 Comparison of experimental and predicted results for Ø125xR25. 29 2.5 Theory 2000 N/m Load Exp 2000 N/m Load Theory 3000 N/m Load Exp 3000 N/m Load Theory 4000 N/m Load Exp 4000 N/m Load Theory 5000 N/m Load Exp 5000 N/m Load IRR per Profile (N) 2 1.5 1 0.5 0 0 2 4 6 8 10 12 Velocity (m/s) Figure 2.18 Comparison of experimental and predicted results for Ø175xR20. As can be seen, there is a consistent shortfall across the results. The simulations of the Ø75xR20 idler roll perform the worst, with an average shortfall of approximately 40%. The 125mm diameter results improved. The error for this idler roll was 34%, 31%, 38% and 36%, corresponding to the cross radii of 10, 15, 20 and 25 mm respectively. The largest idler roll was an improvement again. This 175mm diameter idler roll had an average deviation of 23% from the experimental results. These findings are evident in most one-dimensional Winkler foundation models attempting to simulate this behaviour. In addition to this: a. The use of a one-dimensional model, as stated earlier, does not account for shear between the polymer chains. The models produced by May et al [25] and Hunter [17] treats the cover material as a two-dimensional. The existence of shear between polymers would encourage a non-uniform strain distribution throughout the cover material. If this were the case, depending on the applied load and the thickness of the bottom cover, the strain profile throughout the cover would taper off at a certain depth. This would introduce another ‘apparent cover thickness’, over which the strain is distributed. Rudolphi et al [43] states the inaccuracies of a one-dimensional model are approximately 50% of the predicted value (or a third of the experimental value). This corresponds with the recorded error of 23-40%. 30 b. The test data, as expected, provided inconsistent results. It was noticed that while every measure was taken to ensure balanced rotation of the idler roll, minor material inaccuracies caused a continual change in the rolling resistance, which in turn caused a cyclic load on the measuring apparatus. The belt splice also induced a region of high rolling resistance which needed to be removed from the data. c. Friction between the idler roll and belt is very difficult to include in a theoretical model without developing a full FEM simulation. As the idler roll creates an increase in tension along the belt due to its drag force, this will create regions of stick and slip in the contact region, in a similar way to that described in Chapter 3. From Figure 2.7, it is evident that the belt is indented quicker than it relaxes. This will result in the majority of the slip occurring next to the trailing edge, thus opposing the motion. d. The model presented is based around a flat indentation of a viscoelastic backing, whereas the experimental tests were performed at a sag ratio of 1%. Incorporating sag in the model would increase the relative contact area, and thus the drag force on the idler. e. The DMA tests to obtain the rubber properties are performed at a constant strain setting. Typically between 0.1% and 2% values. This is discussed in section 2.6. 2.5.4 Effect of Maxwell Elements As discussed earlier in the Chapter, the Maxwell parameters used to describe the viscoelastic behaviour of rubber, are obtained through fitting a Prony series to a master curve of the data. For most software packages that perform this, the user may input the number of elements the fit is to, bearing in mind that a more detailed fit results in more time consuming calculations. In order to test the effect of this fit on the simulated results, data was determined that corresponded to a 3, 5, 10, 15 and 20 element Maxwell model (7, 11, 21, 31 and 41 parameters respectively). The simulation results are shown below in Figure 2.19. The results shown correspond to a Ø75xR20 idler roll, with a distributed load of 4000 N/m across the idler roll. 31 2 3 elements 5 elements 10 elements 15 elements 20 elements Exp 1.8 IRR per Profile (N) 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0 2 4 6 8 10 12 Velocity (m/s) Figure 2.19 Influence of material data on indentation rolling resistance accuracy. As can be seen above, the accuracy, although still different to the experimental data, is greatly improved with a more accurate predictive model. It can also be seen that the lower order models appear to have difficulty simulating data at low belt speeds. This can be explained from Figure 2.20. N=1 N=1 N=5 N=2 N=3 Figure 2.20 Plots of the various Prony series overlayed on the raw data. As can be seen, low order models embody a poor representation of the material properties. As the order increases, so does the accuracy, as well as the frequency range over which the data is accurate for. Figure 2.21 below shows the accuracy of the predicted results in relation to the r-square value of the master curve fit. 32 1 2 100 80 0.6 60 0.4 40 0.2 20 %Error 0.8 2 R Value of Curve Fit R Value % Error 0 0 5 10 15 20 0 25 Number of Maxwell Parameters Figure 2.21 Accuracy of master curve fit. 2.6 Strain Dependency As mentioned in Section 2.5.3, several factors contribute to the inaccuracy of this model. One of which, is the inability of the model to accommodate variations in strain. As stated above in Section 2.2, a DMA test is performed at a constant strain, however in reality, this is rarely the case as most situations involve varying strain levels in a material. For instance, an idler roll being indented into a belt cover will experience low strain at the edges, and a maximum at the centreline of the idler roll. To account for this, a strain dependency is added to the model. A strain dependent DMA test is conducted, that determines the influence of the strain level, on the corresponding storage and loss moduli. By adjusting the storage and loss moduli accordingly, and refitting the master curve and Prony series, it is possible to determine a more accurate estimation of the stresses in that element. The strain sweep for the cover material used in these calculations is shown below. As can be seen, the original test data was performed at 0.05% strain (corresponding to multiplier factors of 1). 33 1.2 Storage Modulus Loss Modulus Multiplying Factor 1 0.8 0.6 0.4 0.2 0 0 2 4 6 8 10 12 14 16 18 20 Strain (%) Figure 2.22 Effect of strain on Storage and Loss modulus. Using this data, the computational package is adjusted to calculate the strain at every point in the indentation profile. From this, an individual set of Maxwell parameters is determined, corresponding to each point, and used to determine the stress in that particular element. The results depicted below present non-strain dependent (NSD), strain-dependent (SD) and experimental (Exp) results. 2.5 NSD 2000 N/m Load Exp 2000 N/m Load SD 2000 N/m Load NSD 3000 N/m Load Exp 3000 N/m Load SD 3000 N/m Load NSD 4000 N/m Load Exp 4000 N/m Load SD 4000 N/m Load NSD 5000 N/m Load Exp 5000 N/m Load SD 5000 N/m Load IRR per Profile (N) 2 1.5 1 0.5 0 0 2 4 6 8 10 12 Velocity (m/s) Figure 2.23 Comparison between normal, strain dependent and experimental results for Ø75xR20. 34 2.5 NSD 2000 N/m Load Exp 2000 N/m Load SD 2000 N/m Load NSD 3000 N/m Load Exp 3000 N/m Load SD 3000 N/m Load NSD 4000 N/m Load Exp 4000 N/m Load SD 4000 N/m Load NSD 5000 N/m Load Exp 5000 N/m Load SD 5000 N/m Load IRR per Profile (N) 2 1.5 1 0.5 0 0 2 4 6 8 10 12 Velocity (m/s) Figure 2.24 Comparison between normal, strain dependent and experimental results for Ø125xR10. 2.5 NSD 2000 N/m Load Exp 2000 N/m Load SD 2000 N/m Load NSD 3000 N/m Load Exp 3000 N/m Load SD 3000 N/m Load NSD 4000 N/m Load Exp 4000 N/m Load SD 4000 N/m Load NSD 5000 N/m Load Exp 5000 N/m Load SD 5000 N/m Load IRR per Profile (N) 2 1.5 1 0.5 0 0 2 4 6 8 10 12 Velocity (m/s) Figure 2.25 Comparison between normal, strain dependent and experimental results for Ø125xR15. 35 2.5 NSD 2000 N/m Load Exp 2000 N/m Load SD 2000 N/m Load NSD 3000 N/m Load Exp 3000 N/m Load SD 3000 N/m Load NSD 4000 N/m Load Exp 4000 N/m Load SD 4000 N/m Load NSD 5000 N/m Load Exp 5000 N/m Load SD 5000 N/m Load IRR per Profile (N) 2 1.5 1 0.5 0 0 2 4 6 8 10 12 Velocity (m/s) Figure 2.26 Comparison between normal, strain dependent and experimental results for Ø125xR20. 2.5 NSD 2000 N/m Load Exp 2000 N/m Load SD 2000 N/m Load NSD 3000 N/m Load Exp 3000 N/m Load SD 3000 N/m Load NSD 4000 N/m Load Exp 4000 N/m Load SD 4000 N/m Load NSD 5000 N/m Load Exp 5000 N/m Load SD 5000 N/m Load IRR per Profile (N) 2 1.5 1 0.5 0 0 2 4 6 8 10 12 Velocity (m/s) Figure 2.27 Comparison between normal, strain dependent and experimental results for Ø125xR25. 36 2.5 NSD 2000 N/m Load Exp 2000 N/m Load SD 2000 N/m Load NSD 3000 N/m Load Exp 3000 N/m Load SD 3000 N/m Load NSD 4000 N/m Load Exp 4000 N/m Load SD 4000 N/m Load NSD 5000 N/m Load Exp 5000 N/m Load SD 5000 N/m Load IRR per Profile (N) 2 1.5 1 0.5 0 0 2 4 6 8 10 12 Velocity (m/s) Figure 2.28 Comparison between normal, strain dependent and experimental results for Ø175xR20. As can be seen from the above results, there is a consistent improvement through the addition of strain dependency to the material data. These changes are summarised below in Table 2.3, where an average improvement of 5.7% occurs, with the inclusion of strain dependent material properties, in the theoretical model. Table 2.3 Difference between idler roll results with and without strain dependency. Error without strain Error with Strain dependency Dependency Ø75xR20 40.4% 34.8% 5.6% Ø125xR10 34.0% 28.3% 5.7% Ø125xR15 30.8% 25.5% 5.3% Ø125xR20 37.9% 32.8% 5.1% Ø125xR25 35.5% 30.4% 5.1% Ø175xR20 23.2% 16.0% 7.2% Idler roll Average Improvement Improvement 5.7% 37 2.7 Discussion The model presented predicts the indentation rolling resistance drag force, due to an asymmetric pressure distribution, which occurs as a moving belt is indented by a rigid idler roll. This function is solely dependent on the parameters of the belt cover material, the geometry of the indenting idler roll as well as the indenting load and belt velocity. The results of the experimental indentation rolling resistance tests showed a consistent shortfall in the theoretical prediction of the drag force. This shortfall ranged between 23% and 40%. It was shown that part of the deficiency arises due to the inability of the model to accommodate strain variance. When included in this model, an average improvement of 5.7% was recorded, reducing the average deviation to 28.3% of the experimental value. The remaining error is the result of approximation techniques, and the limitations of the model. As mentioned earlier, the respective Maxwell parameters are formed through the fitting of a Prony series to the master curve data, produced from the raw DMA results. The construction of the master curve is, in itself, an approximation, as is the Prony series that is subsequently fitted. These techniques are subject to varying minimisation techniques, and thus accumulate error in the process. Some recent authors [40] utilise the raw DMA data directly into the simulation. Furthermore, this displays the inelasticity of a one-dimensional model. Shear between the polymer chains within the belt cover would exacerbate the impact stress during loading, increasing the resultant drag force on the idler roll. Friction between the idler roll surface and belt cover would also resultant in energy loss in the system. The relative sag value is also of importance. As mentioned in section 2.5.3, this model fails to account for wrap around produced by the sag of the conveyor belt between idler rolls. The actual amount of sag in a system varies consistently during operation. It is affected by load, tension and belt stiffness to name a few. In most systems, it is desirable to limit sag to approximately 2% of the idler spacing. This is to minimise belt and bulk solid flexure. A sag value of 2% equates to a wrap angle of 2.3 degrees. Using this theory to predict indentation rolling resistance losses of cylindrical idlers, O’shea [36] found similar conclusions. O’Shea [36] compared various rubber compounds, under 38 similar experimental conditions, to determine the validity of numerous models. The models investigated were Jonkers [22], Spaans [46], QC-N [40] and Lodewijks [24]. In addition to these, O’Shea also compared the theory presented herein (for cylindrical idler rolls), based on the model of Rudolphi & Reicks [42]. The results depicted above also show velocity only has a minor influence on indentation rolling resistance. The experimental results depict an increase of approximately 10% from the data taken at 1 m/s, compared to 5 m/s. This is contrary to the load dependence, which shows an increase in resistance, approximately proportional to the increase in load. 2.8 Conclusions An in-depth investigation was conducted, on the appropriateness of using a one-dimensional Winkler foundation model to predict the indentation rolling resistance that occurs in complex idler roll configurations and geometries, existent in current pouch conveying configurations. This involved the modification of an existing Maxwell model to accommodate complex support systems, and experimental validation of the theory. While this model, fails to predict various phenomena occurring in the idler-belt contact area, the model proved to provide reasonably accurate predictions. Upon the inclusion of strain dependency in the model, an average improvement of 5.7% was seen across the results. This reduced the initial error observed from 33.6% to 27.9%. 39 3 Drive Traction In order for a conveyor system to move, traction must be generated through a friction based contact at a drive station. In conventional systems, the belt is wrapped around a large drive pulley, with the belt tension generating the required normal force for traction. Given the complex arrangement of pouch systems, a drive system must take a different form. In current pouch systems, two varying techniques are employed: - A profiled corner drum is used to drive the belt (see Figure 3.1(a)). - Drive wheels that indent into the running surface of the belt are used to generate the required traction (see Figure 3.1(b)). Figure 3.1 (a) Corner drive unit of the Sicon System [24], (b) Drive wheel arrangement of the Enerka-Becker System [6]. 40 As can be seen from Figure 3.1, a drive drum requires a belt capable of negotiating particularly tight horizontal curves, potentially when filled. It also imposes the limitation that the system must incorporate curves in order to function. Conversely, a drive wheel indented into the belt to create a point contact may be located at any place along the belt, without causing any increased wear issues. From this, an in depth analysis has been undertaken to study the possible drive configurations for a point contact drive. 3.1 Point Contact Drive Systems A point contact drive system indents a driven wheel into the cover of the belt. This wheel has the ability to form a part of the supporting structure. In comparison to drum drive systems used in conventional conveyors, point contact drives cannot compare to the generated traction without creating significant belt wear issues, through high indentations subjected to high frequency cyclic loading. From this perspective, multiple small units are preferred. Nuttall [35] offered an approach to quantify the traction generated from the drive systems of the Enerka-Becker system, which utilises a brush model to analyse the shear induced in the contact region. Due to the viscoelastic nature of the belt cover, Nuttall notices an apparent velocity difference between the belt and drive wheel, as the brushes used to model the belt deform under a tractive force (see Figure 3.2). Figure 3.2 Brush model used by Nuttall [35]. 41 Nuttall designates this velocity difference the term, slip (δ), calculated by: δ= vb − ωd r |vb | (3.1) where: - vb is the velocity of the belt - ωd is the rotational velocity of the drive - r is the radius of the drive wheel From this, Nuttall develops an accurate theory to predict the drive traction, based on the apparent slip between the belt and drive wheel. From this model however, it becomes evident that in order for traction to be transferred to the belt, there must be a velocity difference between the drive wheel and belt. This is not the case during breakaway at start-up, and is only applicable during acceleration of the belt from a non-zero starting velocity. From this, it becomes clear that in order to analyse the interaction between the drive wheel and the belt completely, the problem must be separated into two categories: - Incipient sliding: this ensures the wheels have sufficient traction to overcome the start-up friction held in the conveyor infrastructure. This analysis is only valid for a belt velocity of zero. - Tractive Rolling: this investigates the contact region for a belt driven by a rolling cylinder, thus a non-zero belt velocity. 3.2 General Incipient Sliding As a drive wheel begins to apply traction to a conveyor belt, the contact interaction must be wholly understood, in order to ensure the drives are a sufficient size. The breakaway torque demands a considerable amount of traction, given static resistances throughout the system need to be overcome. The highest torque demand on the drive is dependent on the starting profile, loading scenario and topography. In addition to this, the velocity dependence of the main resistances also has a significant impact. 42 When a tractive force is applied to an interaction between an elastic body (belt) and rigid surface (idler roll), regions of stick and slip develop. This is true for section 3.3 also, with the primary difference being the location of the respective zones. This section builds on the theory presented by Johnson [20], which investigates the sliding interaction between similar cylinders, and develops a theory that describes the interaction for a constant normal force before relative sliding occurs between the belt and idler roll. When an elastic body is indented by a rigid body with a normal force 𝑃, the maximum allowable tractive force 𝑄, is given by the friction limit of the interaction, according to Coulombic mechanics: 𝑄 = 𝜇𝑃 (3.2) where 𝜇 is the coefficient of friction between the two materials. At this point however, complete sliding occurs in the contact, resulting in high amounts of wear on the surfaces. An illustration of the problem is given in Figure 3.3. Figure 3.3 Depiction of the stick and slip zones induced in a sliding interaction [20]. As can be seen from Figure 3.3, a tractive force applied to a contact induces regions of slip and stick. These are indicated through the curves 𝑞′ and 𝑞′′ respectively, combining to yield curve B. The stick regions exist due to the normal force being sufficient to exceed the tractive force. 43 The slip zones arise due to the tailing normal force towards the edge of the contact, in addition to the tractive gradient that arises across the contact. The inexistence of this gradient would create a discontinuous tension along the contact, as well as the belt. The analysis of the stick and slip zones in an elastic contact, was first investigated by Cattaneo [1], and subsequently by Mindlin [26]. It was proposed that each zone contributed to the overall traction in a different manner. Thus, each zone may be investigated individually, before being superimposed upon each other. Firstly, in order to quantify the sliding traction, as shown in Equation (3.2), the pressure distribution across the contact must be understood. While complex methods are used to calculate indentation rolling resistance, for computational reasons, a simpler approach is taken, as adopted by Johnson [20]. A comparison between the two methods is undertaken in section 3.4.5. Johnson proposes a simple pressure distribution across the contact, which follows a parabolic nature. This is in contrast to Hertz theory, which offers a circular representation. Bearing in mind this is a static assessment, no phenomena that arise due to rolling contact are evident. This will be observed in section 3.3. Following a Hookean approach, for a belt moving in the xcoordinate, the pressure observed is based on the following: 𝑝(𝑥, 𝑦) = 𝐾𝑓 𝑤(𝑥, 𝑦) ℎ (3.3) where: - K f is the modulus of the foundation (belt) - h is the thickness of the belt cover - 𝑤(𝑥, 𝑦) is the depth profile of the contact The indentation is governed by Equation (3.4): 𝑤(𝑥, 𝑦) = 𝑍0 − 𝑥2 𝑦2 𝑦2 − + 2𝑅1 2𝑅2 2𝑅3 (3.4) Where - Z0 is the peak indentation located at (0,0) - R1 is the primary radius of the drive wheel 44 - R2 is the secondary (cross) radius of the drive wheel - R3 is the radius of curvature of the belt surface As can be seen, this pressure is dependent on the indentation. Depending on the setup of the drive station (the presence of clamping rollers), an indentation or force based approach can prove beneficial. Given this is a static calculation, the reactive force arising from a given indentation can be determined through integration. From this, the tangential traction can be determined: 𝑞′(𝑥) = 𝜇 𝐾𝑓 𝑥2 𝑦2 𝑦2 [𝑍0 − ] − + ℎ 2𝑅1 2𝑅2 2𝑅3 (3.5) From Equation (3.4), the half-length of the contact region, a, can be determined: 𝑎(𝑦)2 = 2𝑅1 [𝑍0 − 𝑦2 𝑦2 ] + 2𝑅2 2𝑅3 (3.6) This yields: 𝑞′(𝑥) = 𝜇 𝐾𝑓 [𝑎(𝑦)2 − 𝑥 2 ] 2ℎ𝑅1 (3.7) This traction represents the tractive limit of the interaction, and would occur during complete sliding. At tractive forces less than this limit however, stick zones reduce the applicable traction. This, noted as a stick traction distribution, is subtracted from the tangential traction limit, as the traction generated throughout a stick zone reduces, the further from the boundary it is observed. This is given by: 𝑞′′(𝑥) = −𝜇 𝐾𝑓 [𝑐(𝑦)2 − 𝑥 2 ] 2ℎ𝑅1 (3.8) As can be seen, Equation (3.8) is simply a scaled representation of Equation (3.5). It should be noted that the units of 𝑞′(𝑥) and 𝑞′′(𝑥) are force per unit contact width. From this, the total tractive force is given as: 𝑄 = 𝑞′(𝑥) + 𝑞′′(𝑥) = 𝜇 𝐾𝑓 𝐾𝑓 [𝑎(𝑦)2 − 𝑥 2 ] − 𝜇 [𝑐(𝑦)2 − 𝑥 2 ] 2ℎ𝑅1 2ℎ𝑅1 (3.9) 45 3.3 Tractive Rolling In addition to the theory presented above in section 3.2, once the belt begins to move, the behaviour within the contact area changes. Instead of the stick zone being centrally located within the contact area, it shifts to be tangential with the leading edge of the interface. This change arises due to the mechanics of the interaction. Johnson [20] notes that the direction of slip must oppose the traction in a slip zone. If this is the case, a slip zone located at the leading edge would violate this principal. Johnson went on to prove this hypothesis. The shape of the stick region is one of discussion. For a cylinder rolling on a flat belt, the stick and slip zones are rectangular, in conjunction with the shape of the contact. When spherical interactions are introduced however, as described by Equation (3.4), there is debate over the internal shape of the stick zone. Johnson [21] postulated the shape of the region should be consistent with the shape of the overall interaction; thus, an ellipse. Haines and Ollerton [14] on the other hand suggested a lemon shaped region, in full contact with the leading boundary. The difference between the two theories is illustrated below in Figure 3.4. In this figure, an elliptical contact region is placed under a longitudinal traction force Qx. Figure 3.4 Stick and slip zones within an elliptical contact under at tractive force. Dashed line: elliptical stick zone [21], Dash-dot line: Lemon stick zone [14]. 46 In a similar method to Section 3.2, the tractive rolling solution is obtained through the superposition of two rectangular (or elliptical) contact areas. Once rolling begins however, the viscoelastic properties of the belt become more vital in the model. As discussed above in Section 3.2, Johnson uses a static pressure distribution, in comparison to the viscoelastic model presented by Rudolphi & Reicks [42]. For sake of comparison, simulations were performed using both pressure distributions. 3.4 Experimental Validation In order to verify the above relation, an apparatus was designed and built that pulled a belt sample between two static drive wheels, until the traction limit was obtained. A schematic of the setup is shown in Figure 3.5. The assembly is designed to accept belt samples up to 60 mm wide, and drive wheels up to 300 mm in diameter. In order to pull the belt sample through the apparatus, the setup was placed in an MTS (Mechanical Testing and Simulation) machine, which provides a controlled, uniaxial tension. This allows accurate control and recording of multiple parameters simultaneously. Figure 3.5 Schematic of the drive traction test assembly [47]. 47 3.4.1 Experimental Theory Testing was done on cylindrical idler rolls indenting a flat belt, as is the case with many clamping roller installations. Given this, the only radius in the drive configuration is that of the drive wheel. From this, the above equations can be simplified. Belt indentation: w(x) = Z0 − x2 2R 1 (3.10) Pressure Distribution: p(x) = K𝑓 x2 [Z0 − ] h 2R 1 (3.11) The normal force is then determined by integrating over the contact area. From Equation (3.10), the half-length of the contact zone, a, can be determined: a = √2R 1 Z0 (3.12) If we denote the width of the belt to be B, and note this integration can be simplified through symmetry, the normal force is found: a F = B ∫ p(x)dx (3.13) −a F = 2B K𝑓 a x2 ∫ [Z0 − ] dx h 0 2R 1 (3.14) K𝑓 a3 [Z ] F = 2B a− h 0 6R 1 (3.15) Substituting in a from Equation (3.12) 3⁄ 2 K𝑓 [2R 1 Z0 ] F = 2B [Z0 √2R 1 Z0 − h 6R 1 ] (3.16) 48 For this arrangement, it is also possible to numerically calculate the portion of stick and slip zones present at a certain tractive force. Starting with Equations (3.5) and (3.10), we can determine the tangential traction distribution: q′(x) = μ K𝑓 x2 [Z0 − ] h 2R 1 (3.17) By rearranging Equation (3.12), the indentation depth Z0 can be expressed in terms of a: Z0 = q′(x) = μ a2 2R 1 (3.18) K𝑓 a2 x2 [ ] − h 2R 1 2R 1 (3.19) K𝑓 [a2 − x 2 ] 2hR 1 (3.20) q′(x) = μ If we denote the half-length of the stick region to be c, the traction distribution within this region is given by: q"(x) = −μ K𝑓 [c 2 − x 2 ] 2hR 1 (3.21) The total tractive force is found through the superposition of these two zones. a c −a −c Q = ∫ q′ (x)dx + ∫ q"(x)dx Q=μ a c K𝑓 [∫ [a2 − x 2 ]dx − ∫ [c 2 − x 2 ]dx] 2hR 1 −a −c Q= 2 μK𝑓 3 [a − c 3 ] 3 hR 1 (3.22) (3.23) (3.24) From this, the amount of slip within the interaction can be determined, for a given traction force. 49 3.4.2 Friction Experiments The foundation of all traction calculations is the coefficient of friction between the two relevant materials. Simple mechanics dictates the existence of a static and dynamic friction factor, however, given the viscoelasticity involved, and the nature of stick and slip zones within the contact, an apparent static coefficient cannot be ascertained, as sliding begins with the application of a small traction. From this, the only definitive value is that of the dynamic portion. This correlates to the theory outlined above, as the available traction is a portion of the sliding traction, depending on the extent of the slip zones within the contact. Using this, it is necessary to perform tests to determine the coefficient of friction between the belt sample, and the various lagging materials. A friction test is performed by pulling a sample piece of material across another surface, under different loads (see Figure 3.6). To prevent a bow-wave occurring, a small chamfer was created on the leading edge. In this figure: - The shear tester is a machine designed to pull or push a sample relative to another surface, and detect small changes in force using a stiff load cell. - The mounting bracket not only serves as a platform to place the weights on, but also ensures the lagging sample remains flat on the belt sample. The trailing edge of the bracket is bent downwards to encapsulate the lagging sample. - The lagging sample is a flat piece of lagging cut to the appropriate size of the mounting bracket. - This experiment placed 200kg of weights on the platform in increments of 20kg. Figure 3.6 Friction test setup. 50 From this, the results were plotted; a trend line was fitted with the intercept set to zero, and the average gradient taken as the coefficient of friction for that sample. Steel Rubber 1500 Pull Force (N) Pull Force (N) 1500 1000 500 0 0 500 1000 1500 2000 1000 500 0 0 2500 500 Normal Force (N) Ceramic 1500 2000 2500 2000 2500 1500 Pull Force (N) Pull Force (N) 1500 1000 500 0 0 1000 Normal Force (N) Polyurethane 500 1000 1500 2000 1000 2500 500 0 0 500 Normal Force (N) 1000 1500 Normal Force (N) Figure 3.7 Friction test results. The average friction value for each material is given in Table 3.1. Lagging is used on drive wheels to increase the coefficient of friction with the belt, as seen in Table 3.1. Small nodules protruding from the ceramic tiles are used to increase the friction coefficient by indenting the rubber surface. Discussed in section 3.4.4, it is seen that the comparative pressure profiles (curved versus flat) between the friction tests above and the experimental results below is significantly different, resulting in an altered coefficient of friction. Table 3.1 Coefficients of friction. Material Coefficient of Friction Steel 0.60 Rubber 0.70 Ceramic 0.65 Polyurethane 0.62 3.4.3 Traction Experimental Setup The experimental setup depicted in Figure 3.5 is a specifically designed facility designed to investigate the breakaway traction of a point-contact drive system on a belt conveyor. In order to adequately quantify the attainable traction, a series of tests was performed on wheels of varying diameters and surfaces, as outlined below. 51 Table 3.2 List of wheels used in traction experiments. Wheel Diameter (mm) Surface Ø100 Machined Steel (STL) Ø 150 Machined Steel (STL) Ø 200 Machined Steel (STL) Ø 250 Machined Steel (STL) Ø 270 Ceramic Tile (CER) Ø 270 Polyurethane (POLY) Ø 270 Natural Rubber (RUB) Figure 3.8 Drive traction test wheels: Steel (left), Polyurethane (top right), Rubber (middle right) and Ceramic (bottom right). A belt sample was created using two sections of flat conveyor belt, and gluing the top covers together. As pouch systems are typically driven along the bottom of the belt, this leaves the bottom covers exposed to the clamping wheels. In order to model the breakaway (start-up) of a conveyor, the belt was pulled at a constant speed through the wheels. This speed must be slow enough to prevent viscoelastic 52 phenomena entering the experiments, while adequately maintaining a sliding contact. Through trial and error, a speed of 5 mm/min was determined. This speed was utilised in the above friction tests. Figure 3.9 Traction test rig in the MTS machine. Figure 3.10 Belt sample clamped between two Ø150 mm steel wheels. 53 As mentioned earlier, drive wheels can contact the belt through either a force or indentation governed contact. This system uses indentation to produce a required normal force, and is thus a combination of both arrangements. Typically, a force dependant setup uses the weight of the belt and material along with spring loaded idler rolls to maintain traction. Given the current capabilities of pouch systems, a generalisation was made that this force would be limited to 200kg per wheel. Naturally, this is wholly dependent on the belt, material and idler roll spacing. From this, indentation forces were tested between 300N and 1900N, in 200N increments. Throughout each test, the normal load, pull load and stretch of the belt was monitored. 3.4.4 Initial Results Upon correlation of the obtained results, a trend was discovered that altered the coefficient of friction in various scenarios. As mentioned earlier, the limiting tractive value is given by Equation (3.2). However upon analysis, it was noted that altering the normal load, and thus the indentation into the belt, also altered the apparent coefficient of friction. This is shown in Figure 3.11. As can be seen, the coefficient is reduced by approximately 20% over the course of the testing. Coefficient of Friction 1.5 D100 STL D150 STL D200 STL D250 STL D250 PU D250 RUB D250 CER 1 0.5 0 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Normal Load (N) Figure 3.11 Influence of normal load on friction coefficient. Coulombic friction is defined for a planar contact, and is thus limited in describing the indented contact between a rigid cylinder and elastic plane. This also explains the observed friction 54 coefficients for the ceramic lagging being higher than the friction test. The ceramic lagging includes small nodes on each tile that extrude outwards. As a friction test is performed on a flat plane, the load is evenly distributed over a large number of nodes. However, for a curved contact, the load is localised to a few nodes, which in turn indent the belt, increasing the friction coefficient. Comparatively, both the friction test and traction test were performed at a maximum 200kg load; however the relative contact area between each test, along with the curved surface sees a severe increase in pressure result in the tractive test. An explanation of this phenomena can be found through the work of J. Golden [11, 10], and Galin et al [9]. Golden [11] states that when contact between elastic bodies exhibits curved surfaces, the resultant frictional force is the sum of two components. The first being the frictional shear over the entire contact region; the usual result. The second, affected by curvature and load, is negative in nature. This reduction in friction is a direct consequence of the asymmetrical shear distribution induced in the contact area as the surfaces slide relative to one another. This skewed distribution gives rise to a force that aids the motion. Figure 3.12 A viscoelastic layer indented by a rigid cylinder, subjected to a tangential load. If we consider the layout shown above in Figure 3.12, with the indentation w(x) given by equation (3.10), then the equation to be solved is given by (3.25): b k1 P(X ′ )dX ′ w ′ (x) = − ∮ ′ − k 2 S(X) π X −X (3.25) a 55 2(1 − ν2 ) E (3.26) (1 + ν)(1 − 2ν) E (3.27) k1 = k2 = where: - X ∈ [a,b] from Figure 3.12 - S(X) represents the shear distribution - P(X) denotes the pressure profile due to load F - E and ν is the elastic modulus and poisons ratio respectively, of the foundation A complete derivation to equation (3.25) is found in Golden’s publications [10, 11]. This equation describes the indentation of the material as a function of the wheel geometry, pressure distribution and shear applied to the surface. From this, Golden [11] offers an indepth derivation of an effective friction coefficient that arises due to the asymmetrical contact area. This is given by: 4 μeff = μ + ( d1 γ) 3 Lk1 d1 = ( ) ( 2πθ 1 − θ)R 1⁄ 2 γ = 1 − 2θ θ= 1 k1 tan−1 ( ) π g g = −k 2 μ (3.28) (3.29) (3.30) (3.31) (3.32) Care should be observed in the use of these equations however. Limitations exist due to the limits of Poisson’s ratio. Conventional materials (excluding complex composites and auxetic materials) have a maximum ratio of 0.5. This limit exists due to the fundamental definition. Taking this limit, boundaries can be imposed on the above equations, aiding their solution. For instance: Equation (3.26): Equation (3.27): k1 = k2 = 2(1 − ν2 ) > 0 for 0 ≤ ν ≤ 0.5 E (1 + ν)(1 − 2ν) > 0 for 0 ≤ ν ≤ 0.5 E 56 Using this, g<0, and thus k1 ⁄g < 0. The next condition lies in the denominator of equation (3.29). For a real solution, the value in brackets must be positive. Thus: θ(1 − θ) ≥ 0 (3.33) 0≤θ≤1 (3.34) This condition holds such that: By definition in equation (3.31), this means: 0 ≤ tan−1 ( k1 )≤π g (3.35) k k1 g g By definition, tan−1 ( 1 ) cannot lie within the above limits, for a negative value of . Thus, a more appropriate definition would be: θ= 1 k1 [ tan−1 ( ) + π] π g (3.36) This conforms to condition (3.34) above. Using the above relations, the effect of curvature on the friction coefficient was determined. The observed and theoretical results are presented below in Figure 3.13 and Figure 3.14. 0.8 D100 STL Exp D100 STL Theory D150 STL Exp D150 STL Theory D200 STL Exp D200 STL Theory D250 STL Exp D250 STL Theory Coefficient of Friction 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Normal Load (N) Figure 3.13 The effect of normal load on the coefficient of friction for steel pulleys. 57 1.4 Ø270 PU Exp Ø270 PU Theory Ø270 RUB Exp Ø270 RUB Theory Ø270 CER Exp Ø270 CER Theory Coefficient of Friction 1.2 1 0.8 0.6 0.4 0.2 0 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Normal Load (N) Figure 3.14 The effect of normal load on the coefficient of friction for lagged pulleys. As can be seen from the above figures, the experimental findings for the steel and lagged wheels vary dramatically. The steel wheel experiments form a reasonable representation of the theoretical model. A similar trend is followed, although it is noticeable that the model is limited at lower normal loads. This is most likely due to viscoelastic effects within the contact. At higher loads however, each wheel agreed well with the predictive theory. In contrast, with the exception of the rubber, the lagged pulley results differ significantly from the theoretical calculations. This is due to the surface protrusions of the lagging, indenting the belt cover at high loads. The rubber and polyurethane lagging consisted of a chevron pattern extending past the lagged surface, indenting the belt. The rubber pattern was designed in a way to maintain reasonable contact with the belt surface, however the polyurethane layout caused sharp indentations around the edge of the protrusion. Similarly, the ceramic lagging consisted of small tiles with sharp edges, and small nodules on the surface designed to aid traction. The high pressures within this contact (small contact area, high load) accentuated the effect of these protrusions. This effect was not evident in the flat friction tests performed earlier. These effects introduce a microscopic element to the above model, far beyond its capabilities. It is possible that the microscopic effects can be minimised by increasing the diameter of the drive wheel (and thus increasing the contact area and distributing the inaccuracies over a greater area). 58 3.4.5 Final Results The shape dependence of the friction coefficient has proven to have a significant impact on the results. Also noticeable during the tests was a reduction in the normal force. This is caused by the necking of the trailing edge when placed under tension, and is a direct result of an indentation based system, in comparison to a force dependent drive. Given this, a reduced normal load was used in the theoretical predictions, to ensure accuracy with the experimental results. The apparent normal load for each test is given in Table 3.3. As can be seen, in most cases a reduction was noticeable. For some instances with the lagged pulleys, the surface texture induced an additional normal force, causing the load to exceed the pre-set limit. Due to this, in addition to the high coefficients of friction discovered in Figure 3.14, some tests were beyond the capabilities of the apparatus for the lagged pulleys, and as such, only the lower loads were examined. Table 3.3 Summary of the observed normal loads during each traction test. Observed Normal Load (N) Ø 100 Ø 150 Ø 200 Ø 250 Ø 270 Ø270 Ø270 STL STL STL STL POLY RUB CER 300 285 285 270 275 310 260 280 500 470 470 470 450 470 450 475 700 670 660 650 645 655 640 700 900 860 855 840 830 840 800 900 1100 1055 1035 1020 1020 1030 970 1100 1300 1245 1225 1200 1210 - - 1320 1500 1435 1400 1380 1390 - - - 1700 1625 1590 1570 1580 - - - 1900 1800 1770 1770 1760 - - - Set Normal Force (N) Using the formulas presented above, the theoretical sliding limit was determined for each loading case. The incipient sliding limit was analysed using both pressure distributions as given by Johnson [20] and Rudolphi & Reicks [42]. The results are presented below. 59 300N Exp 300N Johnson 300N Rudolphi 700N Exp 700N Johnson 700N Rudolphi 1100N Exp 1100N Johnson 1100N Rudolphi 1500N Exp 1500N Johnson 1500N Rudolphi 1900N Exp 1900N Johnson 1900N Rudolphi Pull Force (N) 500 400 300 200 600 400 300 200 100 0 0 500N Exp 500N Johnson 500N Rudolphi 900N Exp 900N Johnson 900N Rudolphi 1300N Exp 1300N Johnson 1300N Rudolphi 1700N Exp 1700N Johnson 1700N Rudolphi 500 Pull Force (N) 600 100 50 0 0 100 Slip (%) 50 100 Slip (%) Figure 3.15 Incipient sliding results for a 100mm diameter steel wheel. 300N Exp 300N Johnson 300N Rudolphi 700N Exp 700N Johnson 700N Rudolphi 1100N Exp 1100N Johnson 1100N Rudolphi 1500N Exp 1500N Johnson 1500N Rudolphi 1900N Exp 1900N Johnson 1900N Rudolphi 700 Pull Force (N) 600 500 400 300 800 600 500 400 300 200 200 100 100 0 0 50 500N Exp 500N Johnson 500N Rudolphi 900N Exp 900N Johnson 900N Rudolphi 1300N Exp 1300N Johnson 1300N Rudolphi 1700N Exp 1700N Johnson 1700N Rudolphi 700 Pull Force (N) 800 0 0 100 Slip (%) 50 100 Slip (%) Figure 3.16 Incipient sliding results for a 150mm diameter steel wheel. 300N Exp 300N Johnson 300N Rudolphi 700N Exp 700N Johnson 700N Rudolphi 1100N Exp 1100N Johnson 1100N Rudolphi 1500N Exp 1500N Johnson 1500N Rudolphi 1900N Exp 1900N Johnson 1900N Rudolphi 700 Pull Force (N) 600 500 400 300 800 600 500 400 300 200 200 100 100 0 0 50 Slip (%) 100 500N Exp 500N Johnson 500N Rudolphi 900N Exp 900N Johnson 900N Rudolphi 1300N Exp 1300N Johnson 1300N Rudolphi 1700N Exp 1700N Johnson 1700N Rudolphi 700 Pull Force (N) 800 0 0 50 100 Slip (%) Figure 3.17 Incipient sliding results for a 200mm diameter steel wheel. 60 300N Exp 300N Johnson 300N Rudolphi 700N Exp 700N Johnson 700N Rudolphi 1100N Exp 1100N Johnson 1100N Rudolphi 1500N Exp 1500N Johnson 1500N Rudolphi 1900N Exp 1900N Johnson 1900N Rudolphi 700 Pull Force (N) 600 500 400 300 800 600 500 400 300 200 200 100 100 0 0 50 500N Exp 500N Johnson 500N Rudolphi 900N Exp 900N Johnson 900N Rudolphi 1300N Exp 1300N Johnson 1300N Rudolphi 1700N Exp 1700N Johnson 1700N Rudolphi 700 Pull Force (N) 800 0 0 100 Slip (%) 50 100 Slip (%) Figure 3.18 Incipient sliding results for a 250mm diameter steel wheel. 300N Exp 300N Johnson 300N Rudolphi 700N Exp 700N Johnson 700N Rudolphi 1100N Exp 1100N Johnson 1100N Rudolphi 800 Pull Force (N) 700 600 500 400 900 700 600 500 400 300 300 200 200 100 100 0 0 50 500N Exp 500N Johnson 500N Rudolphi 900N Exp 900N Johnson 900N Rudolphi 800 Pull Force (N) 900 0 0 100 Slip (%) 50 100 Slip (%) Figure 3.19 Incipient sliding results for a 250mm diameter polyurethane wheel. 300N Exp 300N Johnson 300N Rudolphi 700N Exp 700N Johnson 700N Rudolphi 1100N Exp 1100N Johnson 1100N Rudolphi 600 Pull Force (N) 500 400 300 700 500 400 300 200 200 100 100 0 0 50 Slip (%) 100 500N Exp 500N Johnson 500N Rudolphi 900N Exp 900N Johnson 900N Rudolphi 600 Pull Force (N) 700 0 0 50 100 Slip (%) Figure 3.20 Incipient sliding results for a 250mm diameter rubber wheel. 61 300N Exp 300N Johnson 300N Rudolphi 700N Exp 700N Johnson 700N Rudolphi 1100N Exp 1100N Johnson 1100N Rudolphi 1200 Pull Force (N) 1000 800 600 1400 1000 800 600 400 400 200 200 0 0 50 Slip (%) 100 500N Exp 500N Johnson 500N Rudolphi 900N Exp 900N Johnson 900N Rudolphi 1300N Exp 1300N Johnson 1300N Rudolphi 1200 Pull Force (N) 1400 0 0 50 100 Slip (%) Figure 3.21 Incipient sliding results for a 250mm diameter ceramic wheel. As can be seen in the results above, the theory presented in this chapter represents an accurate prediction for the breakaway sequence of a conveyor. It is observed however that the primary limitation with this theory is the surface profile, and corresponding area of contact of the drive wheel. The smooth surfaced wheels (steel and rubber) show accurate predictions of the experimental results. For lagged pulleys involving surface protrusions within the contact area (ceramic and polyurethane), the experimental findings well exceeded the theoretical predictions. As noted previously, this is due to the localised loading on these protrusions. For an experimental situation involving a large contact area, the loading will be distributed over more nodules, thus better reflecting the results of the friction test presented in 3.4.2. When comparing the theoretical models of Johnson [20] and Rudolphi and Reicks [42], it can be seen that the full viscoelastic model of Rudolphi and Reicks bears a closer resemblance. This is evident in the steeper initial gradient of the curves. As discussed above however, this model is much more computationally demanding. A reasonable approximation may be obtained using Johnson’s model. These findings also investigated the dependence of wheel diameter on the results. As can be seen from the results depicted below in Figure 3.22, there is an appreciable difference between the pull forces associated with each wheel diameter. It is evident that the largest variation occurs for the Ø100mm wheel, while only a finite difference is noticeable between the rest. This is explainable by the increase in the effective coefficient of friction given in Figure 3.23. As can be seen, the friction coefficient increases sharply between the Ø100 and Ø150 62 mm wheels, before tapering off for the larger wheels. This indicates the larger wheels would be more suited for use in drive systems. This would also be true for surface wear on the driven surface. A larger drive wheel would result in a smaller indentation, with the load distributed over a larger surface area, resulting in less stress on the surface. 600 D100 D150 D200 D250 Pull Force (N) 500 400 300 200 100 0 0 10 20 30 40 50 60 70 80 90 100 Slip (%) Figure 3.22 Comparison of steel wheels for a 1300N normal load. 0.5 0.48 Coefficient of Friction 0.46 0.44 0.42 0.4 0.38 0.36 0.34 0.32 0.3 100 150 200 250 Wheel Diameter (mm) Figure 3.23 Effective coefficient of friction for steel wheels. 3.5 The Transition to Tractive Rolling Once the belt starts moving, the location of the stick zone migrates to be tangential with the leading edge. Any small amount of traction applied to a moving belt will induce some portion of slip at the trailing edge. This will then propagate through the contact area as the tractive force is increased, until it meets the leading edge, at which point complete sliding occurs. 63 Given this, a drive sequence must ensure the drive contact is adequate, at all stages during the starting and stopping procedures. For this to occur, one must consider a velocity dependent stress profile, which will in turn induce a load dependent coefficient of friction. For illustrative purposes, let’s assume a starting configuration as follows: Table 3.4 Test parameters for Tractive Rolling. Drive Wheel Diameter Ø200 mm Drive Wheel Width 50 mm Normal Load (dead weight) 2000 N Tractive Force 1000 N Coefficient of Friction 0.7 Operating velocity 5 m/s Belt Type Fabric Bottom Cover Thickness 6.5 mm Based on the theory presented above, two possible situations may arise during starting. 3.5.1 Force Dependent Traction Force dependent traction is a situation where a constant normal load is applied to the drive wheel, as would be the case if traction is governed by the weight of the belt and bulk material, plus any additional spring loads by design of the system. From this, two outcomes may occur resulting from the different pressure distributions given by Johnson [20] and Rudolphi & Reicks [42]. The stress profile presented by Johnson [20] incorporates no velocity dependent behaviour, and as such the contact region is expected to remain constant. Comparing with the theory presented by Rudolphi & Reicks [42], which includes viscoelasticity, changes are expected to occur. As this is a force dependent situation, it is assumed that the indentation of the idler roll is free to change as required. According to viscoelastic theory, as the belt begins to move, an apparent hardening of the belt cover will occur, reducing the indentation. As there is no 64 change to the normal load, the friction coefficient between the two surfaces will remain constant; however the magnitude of this friction factor is open to the theory presented in Section 3.4.4. For the purposes of this comparison, the influence of normal load on the friction factor is not considered. It is however investigated in Section 3.5.2 below. The findings are presented below in Figure 3.24. 2.5 50 45 40 35 1.5 Slip (%) Indentation (mm) 2 1 30 25 20 15 0.5 0 0 10 5 Johnson Rudolphi 1 2 3 4 5 Velocity (m/s) 0 0 Johnson Rudolphi 1 2 3 4 5 Velocity (m/s) Figure 3.24 Force dependent starting - Analysis of the indentation depth (left) and slip portion (right) in the drive contact. From the above figure, it can be seen that while both theories are similar, each has their limitations. It is evident that the static stress profile presented by Johnson provides an adequate explanation of the overall behaviour of the system, however is limited if any depth analysis is required. As can be seen Rudolphi & Reicks [42] predicts that once the belt begins moving, the indentation reduces dramatically, and continues to do so over the length of the starting procedure. Although the indentation is reduced, the portion of slip in the system remains reasonably constant. Johnson predicts approximately 47% slip, compared to 35% given by Rudolphi & Reicks. This is evident in section 3.4.5, as given by the initial gradient of the slip profile. 3.5.2 Indentation Dependent Traction In circumstances where light loads are being conveyed, it isn’t practical to rely on the dead weight of the belt and material to provide traction for the system. To overcome this, an 65 additional spring load is added to the system to increase the tractive load on the drive. This would thus conform to the results presented in Section 3.5.1. For comparison however, a study was performed on the use of a fixed indentation to provide the necessary normal load. In a similar manner to that above, the theories presented by Johnson and Rudolphi & Reicks are utilised to predict the behaviour within the contact region. The results are presented below in Figure 3.25. 3500 50 45 3000 40 35 Slip (%) Normal Load (N) 2500 2000 1500 30 25 20 15 1000 10 500 0 0 5 Johnson Rudolphi 1 2 3 4 5 Velocity (m/s) 0 0 Johnson Rudolphi 1 2 3 4 5 Velocity (m/s) Figure 3.25 Indentation dependent starting - Analysis of the normal load (left) and slip portion (right) in the drive contact. Given the indentation restrictions imposed on the contact, viscoelasticity now plays a major role in the behaviour of the contact. As can be seen, Johnson’s results mirror those presented for a force dependent situation, as expected. It is clear however, from Rudolphi & Reick’s predictions that the normal load on the clamping idler roll as the belt begins to move increases significantly. In this situation, the normal load increases 45% from when the belt is stationary, to moving at 5 m/s. This in turn reduces the portion of slip in the contact, from 34% to 21%. This reduction would see a significant reduction in the wear of the belt over the lifetime of the installation. The limitation of the above results however, is the absence of dependence of the normal load, on the coefficient of friction. As shown in Section 3.4.4 however, the friction coefficient reduces with an increase of the normal load. Thus, using the normal loads presented in Figure 66 3.25, an effective friction coefficient was determined, and used to analyse the slip portion within the contact. The results are presented below in Figure 3.26. 100 Coefficient of Friction % Slip 90 Coefficient of Friction 0.43 0.42 80 0.41 70 0.4 60 0.39 50 0.38 40 0.37 30 0.36 20 0.35 10 0.34 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Slip (%) 0.44 0 5 Velocity (m/s) Figure 3.26 The effect of normal load on the coefficient of friction and portion of Slip. Introducing load dependence on the friction coefficient appears to change the results drastically. So much so, that 100% slip occurs during breakaway, meaning in theory that the starting procedure would not begin. This is a result of an apparent friction coefficient of 0.438, with a normal load of 2000 N, resulting in a maximum attainable traction of 876 N. From here, the friction coefficient continues to decline, however the increase in normal force within the contact sees the amount of slip gradually decay within acceptable margins. This process has outlined the need for clarity and understanding during these calculations, as it is clear that a brief analysis could result in controversial behaviour. It also shows that if load dependence was included in the earlier findings in Section 3.5.1, breakaway may not have occurred. Subsequently, this process highlights the risks of a fixed indentation for a drive system. Material inconsistencies along the length of the belt would induce varied normal loads, enforcing the need for a lower slip portion using this method. It would also see a higher rate of wear on the system, especially at high running velocities. From this, it is evident that a load dependent design is the preferred solution, with the addition of clamping springs if necessary. 67 4 Troughed Conveyor Dynamics A belt conveyor, due to its distributed mass, is a high inertia system. During the starting and stopping schedules of the conveyor, this inertia induces transient effects that, if not sufficiently accounted for, can cause significant damage to the belt, drives or supporting infrastructure. As a conveyor belt exhibits viscoelastic properties and distributed drag forces, the acceleration along the belt is not instantaneous. As such, a wave is produced, beginning at the source of the velocity change; for instance a drive station, or brake. In the case of head drive starting, a tension wave propagates from the drive, along the carry side of the belt, whereas a compression wave travels along the return side. Where a counterweight is placed adjacent to the drive on the return side, this absorbs the majority of the compression wave. If no counterweight exists, as is the case with a tripper drive, or a tail end take-up system, a standing wave is produced as the waves interact. For small installations, this behaviour is less obvious due to the smaller mass of the system, and as such can be easily accounted for through a safety factor. For longer installations however, or those with severe starting or stopping requirements, dynamics play an increasingly important role in the optimum design and installation of a belt conveyor. 68 4.1 Introduction In industry, technological advancements coupled with global demand have seen conveyor belts integrated throughout many industrial processes. As such, the scale and productivity of these machines are pushing the limitations of many installations, beyond their original design specification. Due to this, the overall need to quantify the power demand on a system is higher than ever. The overall power consumption in a system is a function of multiple variables, as well as many that are beyond reasonable control, such as temperature. In order to calculate the gross tension distribution in a system, design standards such as DIN 22101 are used. These however assume a rigid and inextensible configuration along the belt, which behaves according to Newtonian mechanics. For static operation, or small installations with minimal transients, this procedure has been proven to give acceptable results. On larger systems however, transient operation induces shock waves that are beyond the capabilities of elastic mechanics. A moving shockwave on a belt exists in the form of a region of high and low tension. These tension gradients can induce several problems including: Rupture of the belt Excessive displacement of the counterweight system Belt tracking issues Excessive belt sag Lift off in vertical curves Damage to supporting infrastructure For conventional systems, multiple theories have been developed to estimate the belt dynamics during starting and stopping [16, 24, 28], each with their own merit. Harrison [16] modifies the traditional wave equation to represent the stress variations that propagate as sound waves in a conveyor, whereas Lodewijks [24] and Nordell [28] utilise a Finite Element Model (FEM) to approximate the behaviour. 69 4.2 Modelling the Conveyor Belt using Finite Elements The Finite Element Method (FEM) breaks a system down into a series of fundamental components. In this instance, the basis consist of point masses, representing the mass of the belt and bulk material, connected with a series of springs and dampeners, which correspond to the elastic and viscoelastic properties of the belt. In many cases, the incurred resistances can also be modelled in a similar manner. Through modelling a system using finite elements, the user maintains a high degree of control over the construction of the system. Depending on the length of each element, various behavioural or drag characteristics may be included for each idler roll set if required. One step further can be taken, breaking the belt down even further, to investigate the transverse behaviour of the system; i.e. belt flap, as shown by Lodewijks [24]. Lodewijks [24] states that reasonable longitudinal transients can be obtained through an element size between 10-250m, whereas transverse effects can be noted for 0.1-0.25m. Obviously the element length is heavily dependent on the overall length, as well as the computational power. FEM models allow drag forces to be placed at the actual location, instead of being distributed over the length of the belt, as is the case with design standards such as DIN 22101. 4.2.1 The Belt In a similar way to that presented in Chapter 2, a Maxwell model has proven to be a solid representation for viscoelastic materials. In order to completely model the belt however, the model needs to be modified to incorporate the reinforced carcass. Typically, a conveyor incorporates a carcass comprising of layered fabric, or parallel steel cords. These are designed to withstand the tensile loads applied to the belt. These can simply be modelled using a spring, as they behave in an elastic manner. In reality, these materials would include a small amount of dampening; however as will be shown, the intrinsic dampening of the conveyor belt is negligible with reference to the dampening interactions with the infrastructure (belt flexure, rolling resistance etc.). This results in a model shown below in Figure 4.1. 70 Figure 4.1 Finite element model of the conveyor belt. From the above figure, it is evident that the lower part of the model (K1, K2, and C1) corresponds to Maxwell’s representation of the belt covers, whereas the top strand reflects the carcass. It is possible to reduce this by combining the springs K0 and K1, however for simplicity, they are kept independent. Combining these two springs would result in the model utilised by Nordell [28]. Also seen in Figure 4.1, a third element lies in series with the model of the belt. This spring, K 3, originally investigated by Nordell [28], corresponds to the sag of the belt between idler roll sets. This is discussed below. 4.2.2 Belt Sag Troughed installations that experience a high degree of belt sag will behave in a different way to a taut conveyor. Instances where this may occur include inclined or heavily loaded systems, or during an emergency stop. The resulting regions of low tension in the belt obstruct a transient wave propagating due to the difference in acoustic velocity. From this, when the wave encounters the region of low tension, the wavefront will self-steepen, causing the faster portion of the wave to overtake the slower portion. Ellis and Miller [8] compare this scenario to waves breaking on a beach in shallow water. This obstruction must be accounted for in the Finite Element Model. 71 Figure 4.2 Belt sag between Idler roll sets. The static sag of a tensioned belt is depicted above in Figure 4.2. It should be noted that this investigation into belt sag bears no correlation to belt flexure; the drag force due to belt and material flexure between idler rolls. This is purely an investigation into the effect of belt sag on the propagation of transient stresses. In order to quantify the belt sag δ, the belt is compared to a simply supported string, in which case catenary mechanics apply. This states: 𝛿= 𝑞𝐿2 = 𝐾𝑠 𝐿 8𝑇𝑏 (4.1) Where: - δ is the mid-span sag - q is a uniformly distributed load along the length of the belt (weight of the belt and bulk solid) - L is the idler roll spacing - 𝑇𝑏 is the tension in the belt - Ks is the static sag ratio (δ/L) The change in belt modulus is found through the additional strain in the belt due to sag. For a static scenario belonging to the coordinate system described in Figure 4.2, the basic equation for catenary sag is: 𝐿 𝑞 (𝑥 − 2) 𝑇𝑏 𝑦(𝑥) = (cosh ( ) − 1) − 𝛿 𝑞 𝑇𝑏 (4.2) From this, the length of a curve is given by: 72 𝑏 𝐿∗ = 𝐿 + ∆𝐿 = ∫ √1 + ( 𝑎 𝑑𝑦 2 ) 𝑑𝑥 𝑑𝑥 (4.3) 1 𝑞 (𝑥 − 2 𝐿) 𝑑𝑦 = sinh ( ) 𝑑𝑥 𝑇𝑏 (4.4) 1 𝑞 (𝑥 − 2 𝐿) 𝐿 = ∫ √1 + sinh2 ( ) 𝑑𝑥 𝑇𝑏 0 (4.5) Thus: ∗ 𝐿 𝐿 𝐿∗ = ∫ 𝑐𝑜𝑠ℎ ( 1 𝑞 (𝑥 − 2 𝐿) 0 𝐿∗ = 𝑇𝑏 ) 𝑑𝑥 2𝑇𝑏 𝑞𝐿 𝑠𝑖𝑛ℎ ( ) 𝑞 2𝑇𝑏 (4.6) (4.7) From this, the additional strain due to sag is: 2𝑇𝑏 𝑞𝐿 ( ) ∆𝐿 𝑞 𝑠𝑖𝑛ℎ 2𝑇𝑏 𝜀𝑠 = = −1 𝐿 𝐿 (4.8) Through an alternative derivation using approximations, Lodewijks [24] was able to show that: 2𝑇𝑏 𝑞𝐿 ( ) 1 𝑞𝐿 2 8 2 𝑞 𝑠𝑖𝑛ℎ 2𝑇𝑏 𝜀𝑠 = −1≈ ( ) = 𝐾𝑠 𝐿 24 𝑇𝑏 3 (4.9) From this, the belt is subjected to a higher strain than that imposed by the resistances to the belts motion. The simplest way to include this in a finite element model is through an apparent reduction in the Elastic Modulus. For a linearly elastic belt with cross sectional area, 𝐴𝑏 , and modulus Eb, the total axial strain is given by: 𝜀 ∗ = 𝜀1 + 𝜀𝑠 (4.10) From this, the apparent axial deformation is 𝜀1 : 73 𝜀1 ≈ 𝜀1 = 𝜀 ∗ − 𝜀𝑠 (4.11) 𝑇𝑏 1 𝑞𝐿 2 − ( ) 𝐸𝑏 𝐴𝑏 24 𝑇𝑏 (4.12) From here, the effective Elastic Modulus, Eeff can be obtained through differentiation of the axial strain, with respect to the belt tension: 1 𝐸𝑒𝑓𝑓 = 𝐴𝑏 1 𝐸𝑒𝑓𝑓 = 𝑑𝜀1 1 𝐴𝑏 (𝑞𝐿)2 = + 𝑑𝑇𝑏 𝐸𝑏 12 𝑇𝑏 3 (4.13) 1 16𝐸𝑏 𝐴𝑏 2 (1 + 𝐾𝑠 ) 𝐸𝑏 3𝑇𝑏 (4.14) 𝐸𝑏 𝐸 1 + 𝐸𝑏 𝑠 (4.15) 𝑇𝑏 3 3 𝑇𝑏 = 2 (𝑞𝐿) 𝐴𝑏 16 𝐴𝑏 𝐾𝑠 2 (4.16) Several authors [8, 24] rewrite this as: 𝐸𝑒𝑓𝑓 = Where 𝐸𝑠 = 12 This reduction in modulus predominantly affects areas of low tension, such as vertical curves during high loading scenarios. 4.2.3 Building a Mathematical solution for a Finite Element Model Given the above model, a solution method must now be ascertained. Software packages capable of modelling such a system are in existence, however in order to maintain the highest degree of user control, a mathematical solution is chosen. For this, a set of generalised formulae are derived to describe the system. A graphical representation of the system is shown in Figure 4.3. 74 Figure 4.3 Detailed finite element model of a conveyor belt. Figure 4.4 Mathematical representation of the finite element model. In order to derive a general expression for the element depicted in Figure 4.1, the contribution of each strand of the element must be analysed and then combined to determine the overall behaviour of the element. The respective forces experienced by each element are shown in Figure 4.4. From this, we denote: 𝑥2 − 𝑥1 = ∆𝑎 + ∆𝑑 = ∆ (4.17) If we denote the global force in the element to be F, Hookean mechanics also dictates: ∆𝑎 = ∆𝑏 + ∆𝑐 (4.18) 𝐾2 ∆𝑏 = 𝐶1 ∆̇𝑐 (4.19) 𝐾3 ∆𝑑 = 𝐾2 ∆𝑏 + 𝐾1 ∆𝑎 + 𝐾0 ∆𝑎 = 𝐹 (4.20) Beginning with Equation (4.20), substituting in Equation (4.19), yields 𝐹 = 𝐾0 ∆𝑎 + 𝐾1 ∆𝑎 + 𝐶1 ∆̇𝑐 (4.21) From Equation (4.18), we can see that: 75 Thus ∆̇𝑐 = ∆̇𝑎 − ∆̇𝑏 (4.22) 𝐹 = 𝐾0 ∆𝑎 + 𝐾1 ∆𝑎 + 𝐶1 [∆̇𝑎 − ∆̇𝑏] (4.23) Also from Equation (4.20), we can derive the expression for ∆̇𝑏: ∆̇𝑏 = 𝐾3 ∆̇𝑑 − 𝐾0 ∆̇𝑎 − 𝐾1 ∆̇𝑎 𝐾2 (4.24) Substituting this into Equation (4.23), and noting that: and ∆𝑎 = ∆ − ∆𝑑 (4.25) ∆̇𝑎 = ∆̇ − ∆̇𝑑 (4.26) 𝐹 = 𝐾0 [∆ − ∆𝑑] + 𝐾1 [∆ − ∆𝑑] + 𝐶1 [∆̇ − ∆̇𝑑] yields − 𝐶1 [ 𝐾3 ∆̇𝑑 − 𝐾0 [∆̇ − ∆̇𝑑] − 𝐾1 [∆̇ − ∆̇𝑑] ] 𝐾2 (4.27) As shown in Equation (4.20), the overall force in the element can be described by the extension of spring K3. Thus: and ∆𝑑 = 𝐹 𝐾3 (4.28) ∆̇𝑑 = 𝐹̇ 𝐾3 (4.29) This is done on the assumption that the spring constant does not alter with time. As was shown in the previous section, it changes with tension, however remains constant with time. Substituting this into Equation (4.27), yields the behaviour of the element, as a function of the movement of the two endpoints: 𝐹 = 𝐾0 [∆ − 𝐹 𝐹 𝐹̇ ] + 𝐾1 [∆ − ] + 𝐶1 [∆̇ − ] 𝐾3 𝐾3 𝐾3 𝐹̇ 𝐹̇ 𝐹̇ 𝐾3 𝐾 − 𝐾0 [∆̇ − 𝐾 ] − 𝐾1 [∆̇ − 𝐾 ] 3 3 3 ] − 𝐶1 [ 𝐾2 (4.30) It can be seen that this is an Ordinary Differential Equation (ODE) in F(t). Converting this to characteristic form yields: 76 𝐹̇ + 𝐽𝐹 + 𝑁∆̇ + 𝑃∆= 0 𝐽= where 𝐾2 𝐾0 + 𝐾1 + 𝐾3 [ ] 𝐶1 𝐾0 + 𝐾1 + 𝐾2 + 𝐾3 𝐾0 + 𝐾1 + 𝐾2 ] 𝑁 = −𝐾3 [ 𝐾0 + 𝐾1 + 𝐾2 + 𝐾3 𝑃= −𝐾2 𝐾3 𝐾0 + 𝐾1 [ ] 𝐶1 𝐾0 + 𝐾1 + 𝐾2 + 𝐾3 (4.31) (4.32) (4.33) (4.34) From this, we create a Free Body Diagram (FBD) showing individual forces on each belt element. The majority of the elements will be subjected to similar loads, which allows repetition. This is shown in Figure 4.5. Figure 4.5 Free body diagram of a belt element. As depicted in the model above, in addition to the belt forces (𝐹𝑖 , 𝐹𝑖−1 ) acting between nodes, several other forces exist. Prior to defining in detail these drag forces, the construction of the differential equation will be presented, which will subsequently be used in the drag force description. Firstly however, an ODE is presented to represent Figure 4.5. Summing forces in the 𝑥 direction: ∑ 𝐹𝑥 = 𝑀𝑥𝑖̈ = 𝐹𝐷𝑟𝑖𝑣𝑒 + 𝐹𝑖 − 𝐹𝑖−1 − 𝐹𝑔 − 𝐹𝐹𝑙𝑒𝑥 − 𝐹𝐼𝑅𝑅 − 𝐹𝑅𝑖𝑚𝑑𝑟𝑎𝑔 − 𝐹𝑆𝑘𝑒𝑤 (4.35) − 𝐹𝑆𝑝𝑒𝑐𝑖𝑎𝑙 77 Through rearranging, and noting 𝑣𝑖 = 𝑑𝑥𝑖 𝑑𝑡 (4.36) We achieve: 𝑑𝑣𝑖 𝐹𝑖−1 𝐹𝑖 1 + − = [𝐹𝐷𝑟𝑖𝑣𝑒 − 𝐹𝐷𝑟𝑎𝑔 ] 𝑑𝑡 𝑀 𝑀 𝑀 (4.37) 𝐹𝐷𝑟𝑎𝑔 = 𝐹𝐹𝑙𝑒𝑥 + 𝐹𝐼𝑅𝑅 + 𝐹𝑅𝑖𝑚𝑑𝑟𝑎𝑔 + 𝐹𝑆𝑘𝑒𝑤 + 𝐹𝑆𝑝𝑒𝑐𝑖𝑎𝑙 (4.38) Where The above equations explain the behaviour as an incomplete system, as it fails to incorporate the counterweight. Because of this, it is necessary to describe the counterweight, prior to developing the differential equation of the complete model. 4.2.4 The Counterweight To finalise the model, boundary conditions must be imposed on the outermost limits of the model. Commonly, installations place the counterweight adjacent to the main drive system, and as such, some authors [24] incorporate the drive and counterweight system as one node, as the length between them is negligible compared to the overall length of the system. For installations such as these, the effect of transients impacting the drive is often simplified. As a wave passes through the take-up system (gravity), the displacement of the take-up changes to accommodate the variations in tension. This sees the majority of the wavefront inverted, and reflected. Also, given the rigidity of large drive systems, as well as the large area of contact between the drive drum and the belt, any transients that propagate through the counterweight, do not pass through the drive. This behaviour depends on the layout of the conveyor, as well as the formulation of the model. As this dissertation is aimed at the transient behaviour in conveyors incorporating multiple drive systems, to prevent any potential inaccuracies from this assumption, each drive station and counterweight will be modelled separately, with the potential for wavefront propagation. From this, the FEM model is shown below. 78 Figure 4.6 Static diagram of counterweight. As can be seen from Figure 4.3: - A conveyor belt is broken down into N elements. The nodes 1 → 𝑁2 correspond to the return side of the belt, incorporating the counterweight and tail pulley. The nodes 𝑁 +1→𝑁 2 - relate to the carry side, and include the head pulley. The drive location is moveable, and can be located anywhere along the belt. It does not need to be located at the head pulley. - The counterweight moves only in the vertical direction. Its location along the belt is moveable, and for design purposes, is located at 𝑥𝑐 . - This model utilises N displacements, N velocities, and N forces. For the purposes of this investigation, counterweight inertia is included. Including the counterweight in the model is shown in Figure 4.6. From here, a set of equation is derived that predict the behaviour of the counterweight, dependent on the tensions throughout the rest of the belt. It should be noted that the proceeding equations differentiate between the gross mass of the counterweight, and the mass of the pulley, as depicted in Figure 4.6. This is designed to separate the inertia of the counterweight pulley from the rest of the system. By summing Forces in the y-direction: 𝑥𝑐̈ + 1 1 𝐹𝑐−1 + 𝐹 =𝑔 𝑀𝑐 + 𝑀𝑝 𝑀𝑐 + 𝑀𝑝 𝑐 (4.39) By summing Torques about the pulley centroid: 𝜃𝑐̈ + 𝑅 𝑅 𝐹𝑐−1 − 𝐹𝑐 = 0 𝐼𝑝 𝐼𝑝 (4.40) 79 where R and Ip correspond to the radius and moment of inertia of the pulley respectively. Lastly, a differential equation can be derived from Equation (4.31). This equation references ∆ and ∆̇ as the extension and rate of change of extension (or contraction respectively) between the adjacent nodes. From this, we can define the relative extensions as: ∆𝑥𝑐−1→𝑐 ∆= 𝑥𝑐 − 𝑥𝑐−1 + 𝑅𝜃 (4.41) ∆𝑥𝑐→𝑐+1 ∆= 𝑥𝑐 + 𝑥𝑐+1 − 𝑅𝜃 (4.42) Alternatively, winch counterweights can also be used to provide sufficient tension throughout the system. This method employs a mechanical tensioning device (typically hydraulic or electric) to displace the take-up pulley a fixed amount, stretching the belt and raising the base tension of the system. This is done to provide sufficient tension during starting, and to release this tension during normal operation, in an attempt to reduce wear and operating costs. In order to include a winch take-up in the model, restrictions are placed on the counterweight travel 𝑥𝑐 , in equations (4.41) and (4.42). A force controlled system would be utilise a similar representation to a gravity take-up, with alterations based around any feedback systems that may be in place. Alternatively, if the winch is displacement controlled, this displacement is assigned to 𝑥𝑐 . The preceding set of equations can now be used to form a differential equation for the belt behaviour. The formulation of an FEM model should quantify the belts displacement, velocity and tension. These three parameters provide sufficient information for an appropriate conveyor design. From this, we specify a state vector: 80 𝑥1 𝑥2 ↓ 𝑥𝑁 𝜃𝑐 𝑣1 𝑣2 𝑋̂ = ↓ 𝑣𝑁 𝜃𝑐̇ 𝐹1 𝐹2 ↓ [𝐹𝑁 ] (4.43) As can be seen from Equation (4.43), the vector 𝑋̂ represents the displacement and velocities of all nodes (including the angular motion of the counterweight pulley), as well as the respective forces between these nodes. From this we can state that: 𝑑𝑋̂ + 𝐴𝑋̂ = 𝐵𝐶 𝑑𝑡 (4.44) In this definition, A corresponds to a system of (3𝑁 + 2) × (3𝑁 + 2) coefficients, as defined by equations (4.31), (4.36), (4.37), (4.39) and (4.40), whereas BC is a vector describing the boundary conditions of the system. For ease, it is simpler to consider the matrix A, as an assembly of 3x3 sub-matrices, each describing the dependency of each variable, on one another. For instance, consider: 𝐴11 𝐴 = [𝐴21 𝐴31 𝐴12 𝐴22 𝐴32 𝐴13 𝐵𝐶1 𝐴23 ] and 𝐵𝐶 = [𝐵𝐶2 ] 𝐵𝐶3 𝐴33 (4.45) Here, row 1 describes the effect of displacement, velocity and force on the rate of change of 𝑑𝑣 displacement (𝑑𝑥 ), whereas rows 2 and 3 describe the rate of change of velocity ( 𝑑𝑡 ) and force 𝑑𝑡 𝑑𝐹 ( 𝑑𝑡 ) respectively. From this, it is simplest to define each sub-matrix individually. From equation (4.36), which simply defines velocity, we can see that: 𝑑𝑥𝑖 − 𝑣𝑖 = 0 𝑑𝑡 (4.46) By the definition provided in equation (4.44), we can see that the sub-matrices correspond to: 𝐴11 = 𝐴13 = 0 (4.47) 81 𝐴12 = −𝐼 (NxN Identity Matrix) and (4.48) It can also be seen that 𝐵𝐶1 = 0. This form also holds for the counterweight pulley angular displacement. The velocities in the second row of matrix A are drawn from equations (4.37) and (4.39) for the linear velocities, and equation (4.40) for the angular velocity of the counterweight pulley. From these, and for arguments sake placing the counterweight at node 1, directly after the head drive located at node N, we see that the sub-matrices break down to: 𝐴21 = 𝐴22 = 0 1 𝑀𝑐 + 𝑀𝑝 1 𝑀2 0 and 𝐴23 = [ 0 1 𝑀2 1 𝑀2 − (4.49) 0 0 0 0 0 1 𝑀𝑐 + 𝑀𝑝 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 𝑀2 1 𝑀2 − 1 𝑀2 1 𝑀1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 − 𝑅 𝐼𝑝 − 1 𝑀1 1 𝑀1 − 1 𝑀1 1 𝑀1 − 1 𝑀1 1 𝑀1 − 0 (4.50) 0 1 𝑀1 𝑅 𝐼𝑝 ] − This depicts an 8 element FEM model, with the 9th row corresponding to the inertia of the counterweight pulley. Typically, 8 elements is insufficient for a comprehensive analysis (as dynamics typically play a larger role in long conveyors), and as such is only for graphical representation. It should also be noted that the counterweight velocity described in row 1 will have a slightly different form if the counterweight is moved along the return side of the belt. If this is the case, it will follow a similar form to the rest of the matrix, along the diagonal, with positive values of 𝑀 1 𝑐 +𝑀𝑝 𝑅 only. The corresponding values of 𝐼 will also be moved to lie directly 𝑝 82 underneath. The first row will thus read, 𝐴23−11 = − 1 𝑀2 and 𝐴23−1𝑁 = 1 𝑀2 . It should also be observed that the nodal masses are divided into the carry and return components of the belt. The carry side masses (M1) are a function of the weight of the belt and bulk material, whereas the return side (M2) only consist of the belt mass. From the equations, it can also be seen that: 𝑔 1 [𝐹 − 𝐹𝐷𝑟𝑎𝑔 ] 𝑀2 𝐷𝑟𝑖𝑣𝑒 1 [𝐹 − 𝐹𝐷𝑟𝑎𝑔 ] 𝑀2 𝐷𝑟𝑖𝑣𝑒 1 [𝐹 − 𝐹𝐷𝑟𝑎𝑔 ] 𝑀2 𝐷𝑟𝑖𝑣𝑒 1 [𝐹 𝐵𝐶2 = − 𝐹𝐷𝑟𝑎𝑔 ] 𝑀1 𝐷𝑟𝑖𝑣𝑒 1 [𝐹 − 𝐹𝐷𝑟𝑎𝑔 ] 𝑀1 𝐷𝑟𝑖𝑣𝑒 1 [𝐹 − 𝐹𝐷𝑟𝑎𝑔 ] 𝑀1 𝐷𝑟𝑖𝑣𝑒 1 [𝐹 − 𝐹𝐷𝑟𝑎𝑔 ] 𝑀1 𝐷𝑟𝑖𝑣𝑒 [ ] 0 (4.51) From above, a drive station can be placed at any node along the return or carry side of the belt. Depending on the layout of the system also, the values of 𝐹𝐷𝑟𝑎𝑔 will also vary along the belt. This is discussed in detail in Section 4.2.6. Lastly, the 3rd row of matrix A denotes the interconnecting forces between the nodes. These are described by equations (4.31) - (4.34), with modifications from equations (4.41) and (4.42) which describe the counterweight tension. Breaking these equations down into the individual components dependent on displacement, velocity and force yields: 83 𝐴31 and 𝐴32 𝑃 0 0 0 = 0 0 0 [𝑃 𝑁 0 0 0 = 0 0 0 [𝑁 𝑃 −𝑃 0 0 0 0 0 0 0 𝑃 −𝑃 0 0 0 0 0 0 0 𝑃 −𝑃 0 0 0 0 0 0 0 𝑃 −𝑃 0 0 0 0 0 0 0 𝑃 −𝑃 0 0 0 0 0 0 0 𝑃 −𝑃 0 0 0 0 0 0 0 𝑃 −𝑃 𝑁 −𝑁 0 0 0 0 0 0 0 𝑁 −𝑁 0 0 0 0 0 0 0 𝑁 −𝑁 0 0 0 0 0 0 0 𝑁 −𝑁 0 0 0 0 0 0 0 𝑁 −𝑁 0 0 0 0 0 0 0 𝑁 −𝑁 0 0 0 0 0 0 0 𝑁 −𝑁 Similarly, −𝑃𝑅 0 0 0 0 0 0 𝑃𝑅 ] −𝑁𝑅 0 0 0 0 0 0 𝑁𝑅 ] (4.52) (4.53) 𝐴33 = 𝐽𝐼 (4.54) 𝐵𝐶3 = 0 (4.55) Given this set of equations, the various drive and drag forces depicted in Figure 4.5 will now be discussed. 4.2.5 Lumped Masses In a dynamic simulation, the mass of each finite element is beyond the physical mass of the system. It is easy to assume that the mass of each node is given as: Carry: 1 𝑄𝑡 𝑀1 = 𝐿𝑒 [ + 𝑞𝑏 ] 3.6 𝑣 (4.56) Return: 𝑀2 = 𝑞𝑏 𝐿𝑒 (4.57) Where - 𝑄𝑡 is the throughput (t/hr) - 𝑣 is the belt velocity (m/s) - 𝑞𝑏 is the belt mass (kg/m) - 𝐿𝑒 is the effective length of each finite element (m) While equations (4.56)-(4.57) accurately describe the static weight of each element, they omit the inertia of all rotating components within the element. This includes idler rolls, flywheels 84 connected to the drive system, pulleys, brake drums etc. Thus, it is crucial to include a reduced mass of these components in dynamic calculations, however these have no bearing on gravitational loads. It should be recognised that the inertia of the counterweight pulley is previously included in equation (4.40). The principal of reduced mass, is a method of translating the inertia of components rotating at different radii, to a single mass moving at a linear velocity. By definition, the moment of inertia 𝐼, is given by: 𝐼 = ∑ 𝑚𝑖 𝑅𝑖 2 (4.58) If the moment of inertia of a component is known, the reduced mass of all consolidated components is thus: 𝑚𝑟𝑒𝑑 = 𝐼 𝑅2 (4.59) Equation (4.59) requires further discussion. It is common for rotating components to have the moment of inertia specified. This is due to the increasing functionality of modelling programs. This incorporates the mass distributed over the total radius of the component. To translate this to a line mass however, it must be done so at the required radius of the line mass. In the case of idler rolls, as we are interested in the load at the belt line, the radius will be that of the idler roll. For a flywheel attached to a head drum however, if the flywheel is larger than the drum, we are interested in the load at the belt line, and thus use the radius of the head drum (not the flywheel) in the calculations. 4.2.6 Resistances to Motion Self-weight 𝑭𝒈 : Very rarely will a conveyor be completely flat. Inclines and declines induce varying tensions in the belt, due to the elevation and loading of each section. From this, Newtonian mechanics enables the additional connecting tensions to be determined, due to the angle of elevation. Thus, from Figure 4.5: 𝐹𝑔 = 𝑀𝑔 sin 𝜃 (4.60) 85 where 𝑀 corresponds to the physical mass of the element (not rotational masses, as discussed earlier). The inclusion of the gravitational force allows the static tension to be determined in the belt. Flexure 𝑭𝑭𝒍𝒆𝒙 : As a conveyor belt travels unsupported between successive idler roll sets, the belt will sag between the idler rolls. It will also attempt to return to its unstressed state (flat). This behaviour is exacerbated with the addition of bulk material. Thus, in order for the belt to pass over the subsequent idler roll set, a compressive force is required to re-form the belt into its dynamic shape. This force is dependent on the loading and geometry of the belt, and idler roll arrangement, however its dependence is centred on the axial tension in the belt. As such, it is common to express the flexure as a function of the tension in the belt. When fitting this relationship to the model, it is important to consider the velocity dependence. No drag will be experienced during stationary operation, and this will gradually ramp up to full value as the velocity increases. Thus, it is crucial that this ramp be approximated during simulations. In order to simulate this, a ‘ramp factor’ is included, portraying the flexure as a full-weight percentage, dependent on velocity. The factor was designed to ramp to full value, over a specified velocity range, and remains constant thereafter. A continuous curve fit was used, to prevent discontinuities during simulations. The ramp function was determined as: 𝑅 (𝑣) = 4𝑣 [1 − |𝑣 |𝑛 𝑣 𝑛 + 𝑥𝑓 ]+[ 𝑛 |𝑣 |𝑛 |𝑣|𝑛 + 𝑥𝑓 𝑛 ] (4.61) This function is depicted in Figure 4.7. The power n is used to determine the acuteness of the corner. From this, the resultant drag force is 𝐹𝐹𝑙𝑒𝑥 = 𝑅 (𝑣)𝐹𝐹𝑙𝑒𝑥 (𝑇) (4.62) 86 The direction of this force is governed by the velocity of the belt. This ramp factor is also used to describe the velocity dependence of other drag forces, as shown below. This curve corresponds to 𝑛 = 20, 𝑥𝑓 = 0.1, and is continuous. 1.2 Ramp Percentage 1 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Velocity (m/s) Figure 4.7 Ramp function to simulate flexure increase with velocity. Indentation Rolling Resistance, 𝑭𝑰𝑹𝑹 : The drag force induced by a belt indented by a roller, is discussed extensively in Chapter 2. It is possible to include the full Maxwell model in a differential solver, however this would require significant computational power and time. Thus, a simpler approach is taken. As can be seen in Figure 4.8 for a given load, the rolling resistance typically remains relatively constant over the range of operating velocities. From Chapter 2, this is dependent on the viscoelastic properties of the bottom cover of the conveyor belt. Like belt flexure however, indentation drag bears some velocity dependence at low speed. In order to fully comprehend this behaviour, the effect of velocity on IRR is determined from the Maxwell model. This is depicted below in Figure 4.8. 87 Indentation Rolling Resistance (N)) 20 18 16 14 12 10 8 6 4 780N 1265N 1775N 2245N 2 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Velocity (m/s) Figure 4.8 Rolling resistance as a function of velocity for various loads. This plot builds on the data portrayed in Figures 2.23-2.28 in Chapter 2, and corresponds to a Ø125 mm idler roll, consisting of 17 profiles, subjected to total indenting load of 780N, 1265N, 1775N and 2245N. As can be seen, the breakaway rolling resistance (as 𝑣 → 0+ ) represents approximately 90% of the drag when compared to the full running velocity of 5 m/s. From this, it can be concluded that while zero rolling resistance exists with zero velocity, a significant jump occurs as the belt begins to move. This allows a linear approximation to be formulated: 𝐹𝐼𝑅𝑅 = [0.02𝑣 + 0.9]𝐹𝐼𝑅𝑅 (5𝑚/𝑠) for |𝑣| > 0 (4.63) This observed trend is supported by data received from the test facility presented in chapter 2. Rim drag, 𝑭𝑹𝒊𝒎 𝒅𝒓𝒂𝒈 : Rim drag is a term used to describe a conveyor idler rolls resistance to motion. This drag arises due to the friction of the rolling elements in the bearings, the friction of the contact lip seals, and the viscous drag of the lubricant. Quantifying this force is particularly important on long conveyor installations, as these can typically contain approximately one thousand idler rolls per kilometre of belt [51]. A theoretical model is presented below, based on the work of Wheeler [51,53] to predict the individual contributions of the friction categories listed above. 88 Viscous Drag due to the Labyrinth Seal As conveyors are typically located in open, dusty environments, they are heavily sealed to prevent contaminants, such as dust and water entering the housing. This is done via a grease packed labyrinth seal. The seal, as shown in Figure 4.9, creates a viscous shear moment opposing the direction of motion. Labyrinth seals comprise of a part attached to the rotating assembly, as well as one attached to the fixed components. The alignment of the seal typically falls into one of two configurations: axial or radial alignment. This is depicted below in Figure 4.9. Figure 4.9 Labyrinth seal configurations [51]. As can be seen, Newtonian Mechanics dictates a shear force arises between the two surfaces separated by a viscous fluid. For simplicity, the lubrication is assumed to be a Newtonian fluid, and the end discs are neglected. The magnitude of the drag force is dependent on the rotational velocity of the bearing, lubricant viscosity, as well as seal configuration. The viscosity is further dependent upon the operating temperature and contamination. For an axial arrangement, Wheeler [53] derives the viscous drag from the labyrinth seal as: 89 𝑀𝑙𝑎𝑏 = 4𝜋𝜐𝑑 Ω 𝐿1,2 𝑅1 2 + 𝑅 2 [1 − ( 1 ) ] 𝑅2 [ 𝐿3,4 𝑅3 2 𝑅 2 [1 − ( 3 ) ] 𝑅4 + ⋯+ 𝐿2𝑚−1,2𝑚 𝑅2𝑚−1 2 2 𝑅 [1 − ( 2𝑚−1 ) ] 𝑅2𝑚 ] (4.64) Where 𝜐𝑑 is the dynamic viscosity, Ω is the angular velocity of the idler roll, 𝐿2𝑚−1,2𝑚 describes the separation between the 2𝑚 − 1 and 2𝑚 labyrinth seals, with corresponding radii of 𝑅2𝑚−1 and 𝑅2𝑚 respectively. Similarly, the radial configuration obeys: 𝑀𝑙𝑎𝑏 = 𝜋𝜐𝑑 Ω 𝑅2 4 𝑅1 4 𝑅4 4 𝑅3 4 𝑅2𝑚 4 𝑅2𝑚−1 4 [( − )+( − )+ ⋯+( − )] 2 𝐿1,2 𝐿1,2 𝐿3,4 𝐿3,4 𝐿2𝑚−1,2𝑚 𝐿2𝑚−1,2𝑚 (4.65) Friction due to Roller Bearings Conveyor idler rolls typically operate at relatively low operating speeds, in comparison to their design specification. The lubricant in the bearing thus acts as a load sharing element, preventing metal-metal contact. Palmgren [38] states that the total resistive moment is defined as the sum of the free (no load) and load dependent components. Palmgren defines the no-load friction moment as: 2⁄ 3 𝑑𝑚 3 𝑀0 = 0.1𝑓0 (𝑣𝑛) provided 𝑣𝑛 ≥ 2000 (4.66) where 𝑓0 is a design specified factor, typically between 1.5 and 2 for single row deep groove ball bearings, 𝜐𝑘 is the kinematic viscosity of the lubricant, 𝑛 is the speed of rotation and 𝑑𝑚 is the mean diameter of the bearing. Note the kinematic viscosity is related to the dynamic viscosity 𝜐𝑑 through the density 𝜌: 𝜐𝑑 = 𝜐𝑘 𝜌 (4.67) Palmgren specifies the load dependent friction according to: 𝑀1 = 𝑓1 𝐹𝑟 𝑑𝑚 (4.68) where 𝐹𝑟 describes the radial component of the bearing load, and 𝑓1 is defined below: 0.6𝐹𝑟 0.55 ] 𝑓1 = 𝑓 [ 𝐶0 (4.69) 90 The factor 𝑓=0.0008 for 62 series deep groove ball bearings, or 𝑓=0.0009 for series 63. 𝐶0 is the static load rating as specified by the manufacturer. From this, the total rolling friction is given by: 𝑀𝑏𝑟𝑔 = 𝑀0 + 𝑀1 (4.70) Resistance due to the Lip Seal: The outer lip seal provides the outermost defence to contamination, whereas the inner lip seal contains the lubrication within the bearing. For this, Wheeler [51] derived an approximation based on data provided by SKF [45]. This is given by: 𝐹𝑙𝑠 = 0.3𝑒 25𝑑𝑙𝑠 𝐷𝜋 (4.71) Where 𝑑𝑙𝑠 is the diameter of the lip seal. It should be noted, that given the large array of lip seal configurations in operation, equation (4.71) purely represents an approximation. Based on these formulations, an idler roll of diameter D, supported by 2 bearings, undergoes a total resistance of: 𝐹𝑅𝑖𝑚𝑑𝑟𝑎𝑔 = 4[𝑀𝑙𝑎𝑏 + 𝑀𝑏𝑟𝑔 ] + 𝐹𝑙𝑠 𝐷 (4.72) Using these relations, Wheeler and Madden [51] determined experimental values for the labyrinth and bearing resistances using a custom designed facility capable of operating at various belt speeds and temperatures. As can be seen below in Figure 4.10, the theoretical predictions compared closely to the experimental data, particularly at temperatures beyond 10°C. 91 Figure 4.10 Experimental testing of bearing and seal resistance over a range of belt speeds and grease temperatures [51]. The preceding equations form a complex regime of friction determination that may not always be suitable in some situations. Bearing manufacturers [45] include either a friction value based on standard conditions, or a mechanism to calculate the friction. SKF [45] offers a friction calculator, which utilises the formula presented prior. For most conveyor installations, it is not practical to calculate the friction of each idler roll based upon the individual operating conditions. To simplify this process, experimental tests are often conducted according to DIN 22112 (2010-12), which specifies a set of standard operating conditions as shown below: Table 4.1 Standard operating conditions as specified in DIN 22112. Variable Value Load 250 N Rotational Velocity 650 rpm Temperature 20 °C From this test data, it is possible to calculate the bearing resistance using the above equations, for a range of operating conditions. The only limitation of these equations is the temperature of the bearing during operation. While this variable is dependent on many factors, such as internal friction and lubrication, the primary influence is the ambient temperature of the environment. 92 To investigate the dependence of this, a series of experimental tests were performed, designed to investigate the labyrinth and bearing temperatures, at a range of ambient temperatures, for two separate lubricants. This analysis will provide some insight into the temperature used when determining the lubricant viscosity. Two idler rolls were used which incorporated the same bearing type, from different manufacturers. These are listed below: Table 4.2 Bearings used in Idler roll tests. Bearing Manufacturer Shielding Lubricant 6309 Z NTN Single Shielded Alviana S2 6309 NSK None NS7S In addition to the bearings listed above, the corresponding labyrinth seals were analysed, in order to determine the effect of bearing temperature on the seal temperature, as bearings are the primary heat source during operation. Each labyrinth seal was physically similar, consisting of six axially aligned seals, including the outer seals. Tests were performed at the abovementioned load and rotational velocity. To investigate ambient temperature dependence on the operating temperature, temperatures were varied between 0 °C and 50 °C, in 10 °C increments. To ensure a uniform temperature throughout the idler roll and its components prior to testing, both idler rolls were acclimatised for a minimum of 18 hours beforehand. Thermal scans were taken to confirm a consistent temperature throughout. Each idler roll was then run for a minimum of two hours, enabling all thermal transients to decay, and a steady operating temperature to be achieved. Thermal scans were then repeated displaying the thermal distribution across the idler roll. The operating temperature of each bearing and seal was recorded, and subsequently used to determine the temperature rise. 93 70 Temperature Rise (°C)) 60 50 40 30 20 6309 Bearing 6309 Lip Seal 6309Z Bearing 6309Z Lip Seal 10 0 0 10 20 30 40 50 60 70 Ambient Temperature (°C) Figure 4.11 Operating temperature vs ambient temperature. Depicted in Figure 4.11, it is evident the operating temperature of any component is limited by the ambient temperature (dashed line). This is further shown in Figure 4.12 displaying the temperature rise. A noticeable decay function is evident, asymptotically bounded by the ambient temperature. 50 6309 Bearing 6309 Lip Seal 6309Z Bearing 6309Z Lip Seal 45 Temperature Rise (°C)) 40 35 30 25 20 15 10 5 0 0 5 10 15 20 25 30 35 40 45 50 Ambient Temperature (°C) Figure 4.12 Temperature rise vs ambient temperature. From the above data, it is evident that the shielded bearing performs better at lower temperatures. This is due to the shield acting as a heat sink for the bearing. The high conductivity of the metal efficiently draws heat out of the grease, while circulating it within the bearing. For temperatures above 20 °C however, both bearings and lip seals are thermally comparable. 94 Figure 4.13 Thermal scans of the open NSK-6309 (left) and shielded NTN-6309Z (right) operating at 0°C. The effect of the shield is evident in Figure 4.13. The outer surface of the shield is considerably cooler, owing to its low specific heat capacity. The internal temperature of the bearing is just visible through the small gap between the shield and the inner race. For both idler rolls, a distinct trend is evident between the temperature of the lip seal, and the bearing it encapsulates. Given friction in the bearing is the primary source of heat, the temperature of the lip seal should remain lower than the bearing. For both installations, it was discovered that the bearing temperature is, on average, 60% hotter than the corresponding lip seal. Skew, 𝑭𝑺𝒌𝒆𝒘 : The misalignment between the belt and idler rolls is typically known as idler roll skew. It denotes the sliding contact induced when the roller is not exactly perpendicular to the belts motion. Typically, even when every physical measure is taken to ensure alignment, some amount of skew exists, or develops over time. In addition to alignment issues from installation, structural movement over the length of the system is inevitable to some degree, depending on the terrain, mounting style, temperature, footing size etc. The Conveyor Equipment Manufacturers Association (CEMA) provides a reasonable method of incorporating skew in power calculations, based on the installation method and accuracy. The following section is based on that presented in CEMA [3]. For troughed installations, CEMA [3] divides idler roll misalignment into 3 categories: 95 - Misalignment due to the installation not being perpendicular to the velocity of the belt in the horizontal plane, ∆𝐴𝑒𝑖 - Manufacturing inaccuracies, ∆𝐴𝑒𝑚 and - Idler tilt in the vertical plane, ∆𝐴𝑒𝑡 . From this, CEMA [3] thus defines the effective misalignment, ∆𝐴𝑒 , as the sum of all factors: ∆𝐴𝑒 = ∆𝐴𝑒𝑖 + ∆𝐴𝑒𝑚 + ∆𝐴𝑒𝑡 (4.73) For practical purposes, the installation alignment is given as a function of operating conditions. This eliminates the need for estimates. From this, the following environments are defined: Table 4.3 Installation misalignment values as given by CEMA [3]. ∆𝐴 (mm) ∆𝑎 (deg) 6 0.25 Structural Mounting Conditions Idler rolls installed on a permanent steel structure using specialised aligning procedures. 13 0.5 Idler rolls installed on a permanent steel structure with no aligning procedures. 19 0.75 Idler rolls installed on independent footings. 38 1.5 Idler rolls installed in difficult conditions, such as unstable footings, underground or suspended from the roof. From this, the total installation misalignment is defined as: ∆𝐴𝑒𝑖 = ∆𝐴 ∆𝑎 + tan ( ) 𝐴𝑠 2 2 (4.74) with As being the distance between the bolt holes mounting the idler roll. Please note, that to avoid confusion, some of the above notation has been altered from the original text, CEMA [3]. The manufacturing misalignment is introduced through the specified tolerance of the structure. As such, reasonable effort is required to deduce the value of ∆𝐴𝑒𝑚 . If actual values are not available however, CEMA recommends an approximation of 2.5mm. Idler tilt induces a sliding force on the wing idler rolls of a troughed system, as shown in Figure 4.14 (a). It is sometimes used intentionally to aid belt tracking, at the expense of additional running costs. From this, the misalignment due to idler tilt is defined as: 96 ∆𝐴𝑒𝑡 = 2𝐴𝑤𝑐 𝑎 tan(𝛽) 𝐴 𝐴𝑎 𝑖𝑡 (4.75) Where - A is the cross sectional area of the load on the belt, as described by: 𝐴 (𝑚 2 ) = 𝑡ℎ𝑟𝑜𝑢𝑔ℎ𝑝𝑢𝑡 (𝑡⁄ℎ𝑟) 1 × 3600 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 (𝑚⁄ ) × 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 (𝑡⁄ 3 ) 𝑠 𝑚 (4.76) - 𝛽 is the troughing angle (degrees) - 𝑎𝑖𝑡 is the idler roll misalignment in mm/mm. This can also be described as the tangent of the tilt angle, as depicted in Figure 4.14 (a) - 𝐴𝑎 is the idler roll mounting bolt centres (m) - 𝐴𝑤𝑐 is the cross section of the load above the wing idler rolls. It is defined by: 𝜑𝑐 sin 𝜑𝑐 cos 𝜑𝑐 − ) + (𝑏𝑤𝑚𝑐 2 sin 𝛽 cos 𝛽)] 2 2 𝑏𝑐 𝜑𝑐 = sin−1 ( ) 2𝑟𝑠𝑐ℎ 𝐴𝑤𝑐 = 𝑏𝑤 2 × [𝑟𝑠𝑐ℎ 2 × ( (4.77) (4.78) The variables above are portrayed in Figure 4.14. The dimensions 𝑏𝑤 , 𝑏𝑐 and 𝑏𝑤𝑚𝑐 define the portion of the belt in contact with the specific idler rolls and the material, and are thus dimensionless, and less than one. Figure 4.14 (a) Idler tilt [3] (b) Belt and material cross section showing nomenclature [3]. The relations shown in equations (4.75)-(4.78) that describe idler tilt are a new edition to the seventh edition of CEMA [3], and are thus incomparable to certain operating conditions. The previous edition of CEMA [2], however states that for precision made framework, installed using proper aligning techniques, the drag force due to idler tilt is negligible. 97 Now that the effective misalignment due to the above three factors is defined, the tension loss due to this can now be explained. From equation (4.73), the average misalignment (as a ratio of the bolt hole spacing As) can be defined: 𝑎𝑖𝑚 = ∆𝐴𝑒 𝐴𝑠 (4.79) This is subsequently used to determine a design factor, based on the sliding friction induced by the calculated misalignment: 𝐶𝑖𝑚 = 𝐶𝑏𝑖 × 𝑎𝑖𝑚 (4.80) Where 𝐶𝑏𝑖 represents the sliding coefficient of friction between the belt and idler roll material. From here we calculate the tension loss over a length L, under a gravitational load W b from the belt, and Wm due to the material. ∆𝑇𝑖𝑚𝑛 = 𝐶𝑖𝑚 𝐿(𝑊𝑏 + 𝑊𝑚 )𝑅𝑟𝑖𝑚 (4.81) The factor Rrim is defined as a modifying factor, introduced in the 7th edition of CEMA, to account for various installation conditions. Although ineffectively defined, it is noted that for standard installations, 𝑅𝑟𝑖𝑚 = 1. Special Resistances, 𝑭𝑺𝒑𝒆𝒄𝒊𝒂𝒍 : Certain sections of a conveyor will include various processes which induce an additional drag force at that location. These forces are typically location dependant, and generally don’t apply to the whole length of the belt. Certain examples include: - Belt scrapers that clean the belt - The force required to accelerate the bulk material on the belt during loading. - Skirt plate resistances in the loading zone. - Seized idler rolls. Many special resistances can be quantified by theories presented in publications such as CEMA [3]. Given the nature of transients during starting or stopping, these forces won’t typically affect the behaviour of the belt, however an accurate steady state model will require the inclusion of these. 98 Drive Force, 𝑭𝑫𝒓𝒊𝒗𝒆 The parameter 𝐹𝐷𝑟𝑖𝑣𝑒 describes a force or velocity specification, located at the node corresponding to a drive station. Through this description, a torque or velocity dependent start-up procedure may be specified. A torque start-up can easily be produced by applying a drive force to the node corresponding to the drive station. A velocity controlled start-up on the other hand requires some minor modifications to the above equations presented in Section 4.2.3. A velocity controlled start obeys the following form: 𝑣𝐷𝑟𝑖𝑣𝑒 = 𝑑𝑥𝐷𝑟𝑖𝑣𝑒 = 𝑣(𝑡) 𝑑𝑡 𝑑𝑣𝐷𝑟𝑖𝑣𝑒 = 𝑣′(𝑡) 𝑑𝑡 (4.82) (4.83) In order to include this in matrix A, it must be done so using accelerations rather than velocities. If this were done using velocities (i.e. row 1 of matrix A), the coupling between displacement and velocity would be broken, thus rendering the results useless. As such, the row corresponding to the acceleration of the drive in matrices 𝐴21 , 𝐴22 and 𝐴23 must be zeroed, and the boundary condition in 𝐵𝐶2 used to define the acceleration of the drive, as described in equation (4.83). 4.3 Velocity Prescribed Starting Procedures The starting sequence of a conveyor dictates the stresses induced on the infrastructure, and as such governs their behaviour. Naturally, a longer starting procedure will generate less strain on the system, however this is not always economically viable. The length and severity of a starting procedure is not standardised, and as such open to discussion. Manufacturers of drive systems will often recommend a sequence involving constant acceleration, in order to reduce the acceleration force. However the jerk in the system, defined as the derivative of the acceleration profile, will induce high stresses at the beginning 99 and end of the starting procedure. As such, several authors [15, 30] have suggested an S-shape start-up. Based on their relevant Finite Element Models, Harrison [15] and Nordell [30] define similar starting characteristics that were discovered to exhibit minimal stress on the system. For a running belt velocity of 𝑣𝑏 , and starting procedure of length 𝑇, Harrison prescribes a sinusoidal velocity profile, as given by: 𝑣𝐷𝑟𝑖𝑣𝑒 (𝑡) = 𝑣𝑏 𝜋𝑡 (1 − cos ( )) 0 ≤ 𝑡 ≤ 𝑇 2 𝑇 (4.84) Conversely, Nordell specifies a parabolic start-up: 2𝑡 2 𝑇 𝑣𝐷𝑟𝑖𝑣𝑒 (𝑡) = 𝑣𝑏 ( 𝑇2 ) for 0 ≤ 𝑡 ≤ 2 𝑣𝐷𝑟𝑖𝑣𝑒 (𝑡) = 𝑣𝑏 (−1 + 4𝑡 𝑇 − 2𝑡 2 𝑇2 ) for 𝑇 2 (4.85) ≤𝑡≤𝑇 (4.86) These are compared below in Figure 4.15. As can be seen, both profiles are similar, with Nordell exhibiting a slightly steeper acceleration. 1 0.9 0.8 0.7 V/Vb 0.6 0.5 0.4 0.3 0.2 0.1 0 0 Harrison Nordell 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t/T Figure 4.15 S-type starting procedures as given by Harrison [15] and Nordell [30]. In addition to this, it is becoming increasingly common for conveyors to utilise inspection periods upon start-up, or regions of low velocity. These periods are a region of constant low velocity, which allow the system to overcome breakaway resistances. Visual inspections of the belt can also be performed, making emergency stops less stressful on the system if required. Lodewijks [24] notes that utilisation of an inspection period can reduce the peak belt tension 100 by 15% (Lodewijks defines them as “rest periods”), as the static transients from breakaway are given time to decay. The length of the start-up procedure is a topic that has long been debated over. For torque controlled starting, Singh [44] states that the length of the start-up should be at least five times the time required for a stress wave to travel along the unloaded return side of the belt, from head to tail. Thus, if we define the speed of sound in the empty belt as 𝑐𝑏 : 𝑇≥ 5𝐿 𝑐𝑏 (4.87) In the absence of calculated estimates, a conservative rule of thumb is given as ‘one minute per kilometre of conveyor length’. Typically, the speed of sound in a conveyor varies dependent on the reinforcing material. Fabric conveyors transmit elastic waves at between 500 ms-1 and 1500 ms-1 for an empty belt, while waves in a steel cord belt can propagate at up to 4300 ms-1 [15]. From this, we can modify Singh’s equation: 𝑇≥ 30𝐿 260𝐿 → 𝑇≥ 𝑐𝑏 𝑐𝑏 (4.88) This would create unnecessarily long starting procedures. From this, it is evident that dynamic analysis of a system will provide an appropriate starting procedure. 4.4 Belt Loading The behaviour of a conveyor during starting and stopping is, as shown above, reliant on the distributed mass of the system. From this, it is prudent to, where possible, start and stop the system in an unloaded state. Practicality however dictates that this will not always be the case. Emergency stops, breakdowns and/or irregular loading means the system can theoretically be required to start with any given loading scenario. Thus it is necessary to account for this in the dynamic simulations. In order to comprehensively predict the behaviour of the belt, 5 different loading scenarios are considered, as depicted in Figure 4.16: (a) Empty – no material on the belt. (b) Full – fully loaded belt. 101 (c) Inclines Only Loaded – this load distribution places full load on the inclined sections of the system, whereas the flats and declines are empty. (d) Declines Only Loaded – Declines only are loaded. This loading scenario will require the least amount of power, as it can be regenerative. (e) Worst Case – Similar to the Inclines Only scenario, load is also placed on the flat sections of the belt, in addition to the inclines. This distribution will place the highest demand on the drive during start-up, if there is zero power regeneration from declines. (a) (b) (c) (d) (e) Figure 4.16 Conveyor belt loading scenarios As will be shown in Chapter 5, the various starting scenarios are important for two reasons. Firstly, as mentioned above, the start-up procedure, infrastructure and belt must all be designed to accommodate the worst case scenarios. Conversely, the minimum tension in the belt must remain above a prescribed value, as governed by sag and drive design. The tension drop across a drive drum must remain above a critical value otherwise slip will occur. If we denote the high tension running onto the drive, T1, and the low tension coming out of the drive T2, the ratio of the two must conform to: 𝑇1 ≤ 𝑒 𝜇𝜃 𝑇2 (4.89) Where 𝜇 is the coefficient of friction between the drive surface and the belt, and 𝜃 is the angle of wrap around the drum. It is for this reason many drive configurations utilise snub pulleys 102 which allow a higher wrap angle. Once the minimum tension in the belt is determined, the counterweight can be designed to modify this value. 4.5 Determination of the Elastic Modulus of a Conveyor Belt The Modulus of Elasticity is a dominant parameter when assessing the elastic behaviour of a material. For elastic materials, it is determined via a stress-strain test. Given the composite and viscoelastic nature of conveyor belts however, a different approach is required. Adopting the procedure given in ISO 9856 [19], the dynamic elastic modulus was analysed on 3 different samples of belting, cut from the same species of conveyor. Figure 4.17 Dynamic tensile test for the modulus of elasticity. This standard proposes a procedure utilising cyclic tension tests between 2% and 10% of the minimum breaking load of the belt, after consolidated with a 0.5% dead load. The sample dimensions are to be 50mm wide, at least 300mm long between clamps, and sourced from various locations of the belt. The covers were removed from the top and bottom. The belt in question is a 16mm thick, PN250, 4 ply fabric belt. From this, it is seen that the required loads are 0.5% DW=250N, 2% MBL=1000N and 10% MBL= 5000N. A picture of the test is shown above, incorporating the extensometer measuring the strain in the sample. Once the covers were removed, the carcass measured 7mm thick. The samples were stressed 200 times, ensuring all viscoelastic effects had dissipated. A comparison between the first and last cycle of each sample is shown below in Figure 4.18. As 103 can be seen, there is a distinct difference between the behaviour during the initial loading of the belt and the last cycle. Given this, the last cycles are used to determine the elastic modulus of the sample, and then averaged to find that of the belt. Effects due to hysteresis are ignored. 6 5 Load (kN) 4 3 2 Sample 1 - 1st Sample 1 - 200th Sample 2 - 1st Sample 2 - 200th Sample 3 - 1st Sample 3 - 200th 1 0 0 0.002 0.004 0.006 0.008 0.01 0.012 Strain (mm/mm) Figure 4.18 The first and last cycle of the three belt samples. From the data displayed in Figure 4.18, the elastic modulus of each sample was determined: Table 4.4 Summary of belt elastic properties. Sample Carcass Modulus 1 2275 MPa 2 1899 MPa 3 2519 MPa Average 2231 MPa As can be seen, the modulus varies significantly across the width of the belt. This would be affected by an inconsistent tension distribution of the fabric weft during the rubber vulcanisation. It reinforces the need to test multiple samples from different regions of the belt. The above moduli were calculated according to Hooke’s Law. Comparatively, ISO 9856 dictates an elastic modulus be presented in Newtons per millimetre width of test piece. This is the gradient of the curve when strain is plotted against force per mm width of belting. This is the standard form as given by belt manufacturers. Thus, a conversion is required: 104 𝐸(𝑁⁄𝑚𝑚) = Alternatively 𝐸(𝑀𝑃𝑎 ) 𝐴(𝑚𝑚 2 ) 𝐵(𝑚𝑚) 𝑁⁄ 𝐸 (𝑀𝑃𝑎 ) 𝐸 ( 𝑚𝑚⁄𝑝𝑙𝑦) = 𝐴(𝑚𝑚 2 ) 𝑃𝐵(𝑚𝑚) (4.90) (4.91) Where P is the number of plies, A is the area of the carcass and B is the width of the belt. Using this, the above samples give: Table 4.5 Summary of belt elastic properties. Sample Carcass Modulus 1 3980 N/mm/ply 2 3324 N/mm/ply 3 4408 N/mm/ply Average 3904 N/mm/ply 4.6 Conclusions The theory presented above forms a flexible framework for the dynamic analysis of troughed conveyor systems. It incorporates all major drag forces, with the potential for the inclusion of minor resistances also. The accuracy of the model is investigated in Chapter 5, when it is compared to an onsite test facility. 105 5 Case Study – Onsite Testing As previously mentioned, the starting and stopping of a conveyor system is not an instantaneous reaction, owing to the viscoelastic nature of the belt, as well as the distributed mass and drag forces along the system. As such, a change in acceleration of a drive system causes stress waves to propagate outwards from the point of origin. In order to predict this behaviour, Chapter 4 saw a theoretical approach developed. This theory is compared to site measurements in this Chapter. In Section 5.3, various conveyor starting procedures are investigated, followed by the associated stopping procedures in Section 5.4. From this, emergency stops and aborted starts are modelled in Section 5.5. 5.1 Conveyor Layout The conveyor system used for this analysis is an onsite conveyor specifically designed for monitoring and control. The layout consists of 2 conveyors of similar length, aligned parallel to form a recirculating system, with hoppers and/or transfer chutes connecting each end. With design in mind, a fabric belt was used on one system, while the other utilises a steel reinforced belt. As conveyor dynamics are directly dependent on the elastic modulus of the belting, for investigative purposes, only the fabric belt is analysed. 106 The layout of the system, depicted in Figure 5.1, represents a constant incline, with a midpoint return drive and gravity take-up. Lconv h ar ac 𝜔 𝑀𝑐 Figure 5.1 Conveyor schematic. Figure 5.2 Photo of the test conveyor. 107 As stated in Chapter 4, the dynamics of a conveyor system are more predominant in long belts, or belts under severe load or operating conditions. Albeit a small system of 64m, the fabric belt construction allows relative comparisons to be made with a steel reinforced belt 6-8 times its length. This is due to the difference in wave speed between the belts. The system is loaded via a feeding conveyor, at the base of a storage bin. For practical purposes, the conveyed material in this instance is coarse river sand, given its lack of dust and high flowability. The actual composition of the bulk material is irrelevant, as only the loaded mass is of concern. In order to comprehensively understand the dynamics of this system, the starting and stopping characteristics of an empty, half loaded and loaded belt are examined. In addition to this, aborted starts and emergency stops are also analysed. 5.2 Conveyor Parameters From here it is necessary to quantify the specific parameters used for a dynamic simulation. In particular, idler roll parameters, belt characteristics, drag forces, drive specifics and the counterweight arrangement. These are given below. Table 5.1 Conveyor properties. Parameter Value Conveyor Length 64 m Conveyor Elevation 4.4 m Table 5.2 Carry idler roll properties. Parameter Value Idler roll spacing 1.07 m Configuration 3-idler troughed Idler roll diameter Ø100 mm Troughing angle 30° Reduced mass 6.45 kg/idler 108 Table 5.3 Return idler roll properties. Parameter Value Idler roll spacing 2.9 m Configuration 1-idler flat Idler roll diameter Ø100 mm Reduced mass 15.15 kg/idler Parameter Value Belt Class PN 150 Number of Plies 2 Belt thickness 7 mm Top cover thickness 3 mm Bottom Cover thickness 1.5 mm Carcass thickness 2.5 mm Belt mass 4.43 kg/m Table 5.4 Belt properties. Maxwell Parameters 𝐸0 1.15 GPa 𝐸1 9.155 MPa 𝐸2 12.84 MPa H 35990 Ns/m Parameter Value Pulley Diameter Ø430 mm Number of Drives 1 Drive Location Return Drive Inertia 50 kgm2 Reduced Mass 1081.7 kg Table 5.5 Drive properties. 109 Table 5.6 Counterweight properties. Parameter Value Counterweight Mass 927 kg Pulley Mass 110 kg Pulley Diameter Ø430 mm In order to quantify the drag forces of the system, the running drive torque was determined and calibrated to return the associated drag forces. These values were not known from prior data. As such, the velocity dependence of these forces could be ascertained, however the tension dependence could not. While a direct correlation between the running torque is evident, no calibration was done during starting. The torque profile during starting is a direct output, based on the behaviour of the belt and material. 5.3 Starting of an Elastic Conveyor Belt The starting procedure of a conveyor system dictates the stress on the structure. As shown in Chapter 4, the control of the drive system governs the peak stress, as well as the propagation of acceleration waves throughout the belt. The theory presented in Chapter 4, suggest a starting time of approximately 8s is suited to this installation. This represents a soft start. In order to exaggerate the dynamics of the system however, a starting time of 4.5s is chosen as a hard start, whereas a 9 s procedure represents a conservative start. A ramp and S-type profile is utilised for each start. These starting profiles are depicted below in Figure 5.3. 4.5s Ramp 4.5s S-Start 2.5 2 Velocity (m/s) Velocity (m/s) 2.5 1.5 1 0.5 0 0 2 4 6 8 2 1.5 1 0.5 0 0 10 2 Time (s) 2 2 1.5 1 0.5 4 8 10 8 10 9s S-Start 2.5 Velocity (m/s) Velocity (m/s) 9s Ramp 2 6 Time (s) 2.5 0 0 4 6 8 10 1.5 1 0.5 0 0 2 Time (s) 4 6 Time (s) Figure 5.3 Starting profiles. 110 Upon first inspection, the ramp function will induce points of discontinuity in the starting procedure, resulting in additional dynamics along the belt. The S-start however, as represented by Nordell (28) and Harrison (16), controls the acceleration. Beginning with a soft start, the peak acceleration is achieved half way through the procedure, before tailing off to the chosen velocity. Typically, a conveyor is started in an empty state. This induces less wear on the components, and requires less power input. In certain situations however, it is not practical to empty the belt before stopping; for instance, an emergency stop. This will see material left on the belt, and will thus require starting. As such, simulations were performed for an empty, half full, and fully loaded belt. The corresponding tonnages for the loaded case were 90 t/hr at half capacity, and 180 t/hr fully loaded. 5.3.1 Drive Torque during Starting The limits of any starting procedure are those imposed by the installed drive system and gearbox. As such, the resulting torque must be within reasonable limits, to prevent drive or belt failure. In this instance, the drive torque was measured through strain gauges attached directly to the drive shaft of the motor. The measured and simulated torques during start-up is depicted below for the various loading cases. 4.5s Ramp 4.5s S-start 600 Theory Experimental 400 200 0 0 5 10 15 Torque (Nm) Torque (Nm) 600 Theory Experimental 400 200 0 0 20 5 Time (s) 9s Ramp 20 600 Theory Experimental 400 200 5 10 15 20 Torque (Nm) Torque (Nm) 15 9s S-start 600 0 0 10 Time (s) Theory Experimental 400 200 0 0 5 Time (s) 10 15 20 Time (s) Figure 5.4 Starting torques for an empty belt. 111 4.5s Ramp 4.5s S-start 1200 Theory Experimental 1000 800 600 400 200 0 0 5 10 15 Torque (Nm) Torque (Nm) 1200 800 600 400 200 0 0 20 Theory Experimental 1000 5 Time (s) 9s Ramp 20 1200 Theory Experimental 1000 800 600 400 200 5 10 15 Torque (Nm) Torque (Nm) 15 9s S-start 1200 0 0 10 Time (s) 800 600 400 200 0 0 20 Theory Experimental 1000 5 Time (s) 10 15 20 Time (s) Figure 5.5 Starting torques for a half loaded belt. 4.5s Ramp 4.5s S-start 2000 Theory Experimental 1500 1000 500 0 0 5 10 15 Torque (Nm) Torque (Nm) 2000 1000 500 0 0 20 Theory Experimental 1500 5 Time (s) 9s Ramp 20 2000 Theory Experimental 1500 1000 500 5 10 15 20 Torque (Nm) Torque (Nm) 15 9s S-start 2000 0 0 10 Time (s) Theory Experimental 1500 1000 500 0 0 5 Time (s) 10 15 20 Time (s) Figure 5.6 Starting torques for a fully loaded belt. As can be seen, the simulation of the empty belt, in comparison to the loaded case, takes longer to decay to steady state. This is not evident in the experimental results. In fact, all cases decay over approximately the same timeframe. The presence of this in the empty simulations is due to load independent resistances omitted from the simulation. The inclusion of indentation rolling resistance, belt and bulk material flexure, rim drag and skew in calculations 112 ensures the main resistances of the system are included. As discussed in Chapter 4, these forces are dependent on the normal force (load) acting downward on the idler roll. In reality, resistances exist which are load independent, and difficult to quantify. Such as belt cleaners, balance of idler rolls and pulleys, surface inconsistences of idler roll and pulleys etc. All of these impact the behaviour of the system. When drag forces are minimal, as with an empty case for a short system, these resistances are more predominant, leading to slight inaccuracies in the simulation. As can be seen, the decay time is similar for the half and fully loaded case, due to the insignificance of these forces. Based on this, the depth of the analysis must be tied to the length of the system. From the theory presented in section 4.2.6, the inclusion of localised resistances is simple, but would be negligible on a longer system. Upon further inspection, it can be seen that the starting torque for the loaded case is non-zero. This is a direct result of the resistance model depicted in Figure 4.7. Friction is a velocity dependent entity, and from this model decays to zero with zero velocity. This is a requirement for accurate settling of the system. The experimental conveyor however contains sufficient internal friction to prevent rollback of the system, and thus a drive brake is not installed. From above, the system only inclines at a constant rate of 4 degrees. In the theoretical model, the anti-rollback placed on the drive during the settling period causes the drive to brake the system. This imposes an initial torque on the drive correlating to the force required to hold the material on the incline. This behaviour is more accurately depicted through the stopping sequence in Section 5.4. As expected, it is seen that an empty belt requires no braking torque. Comparatively, the model accurately represents the torque during the starting period. The peak torque is similar, as well as the period of the acceleration waves. As discussed in chapter 4, the peak torque experienced by the drive does not correspond to the peak acceleration of the system. This would be the case for an inelastic system, however the propagation of elastic waves results in a support wave aiding the drive, once the acceleration has propagated throughout the entire system. As such, the peak torque is entirely dependent on the length and profile of the starting procedure, as well as the length of the system and modulus of the belt. The arrival of the support wave is represented by the first trough in the drive torque. 113 5.3.2 Counterweight Movement during Starting In addition to the drive torque, the design of the counterweight is of crucial importance. The counterweight’s function is to pretension the belt, to avoid excessive sag or lift-off during curves, and to maintain adequate slack side tension for the drive system to attain traction. In order to accomplish this, gravitational or winch counterweights are commonly used. This system utilises a gravitational system, providing a constant tension adjacent to the drive. As such, the counterweight must have a suitable amount of travel, based on the belts elastic properties. As outlined above, the counterweight of this system has a total mass of 1037 kg. This includes the pulley, bearings, frame and weights. Of this, 110 kg represents the steel pulley. Its movement is guided by tracking steel beams through a channel, and has a total travel available of 0.7m. The counterweight motion was measured using an infrared laser displacement sensor. The results are shown below. Consistent with the theory presented in chapter 4, a downward movement is given as a positive displacement. 4.5s S-start Theory Experimental 100 50 0 0 5 10 15 Displacement (mm) Displacement (mm) 4.5s Ramp 150 150 Theory Experimental 100 50 0 0 20 5 Time (s) Theory Experimental 100 50 5 10 Time (s) 15 20 9s S-start 15 20 Displacement (mm) Displacement (mm) 9s Ramp 150 0 0 10 Time (s) 150 Theory Experimental 100 50 0 0 5 10 15 20 Time (s) Figure 5.7 Counterweight displacement during starting of an empty belt. 114 4.5s S-start Theory Experimental 200 100 0 0 5 10 15 Displacement (mm) Displacement (mm) 4.5s Ramp 300 300 Theory Experimental 200 100 0 0 20 5 Time (s) Theory Experimental 200 100 5 10 15 20 9s S-start 15 Displacement (mm) Displacement (mm) 9s Ramp 300 0 0 10 Time (s) 20 300 Theory Experimental 200 100 0 0 5 Time (s) 10 15 20 Time (s) Figure 5.8 Counterweight displacement during starting of a half loaded belt. 4.5s S-start Theory Experimental 400 300 200 100 0 0 5 10 15 Displacement (mm) Displacement (mm) 4.5s Ramp 500 500 300 200 100 0 0 20 Theory Experimental 400 5 Time (s) Theory Experimental 300 200 100 0 0 5 10 Time (s) 15 20 9s S-start 15 20 Displacement (mm) Displacement (mm) 9s Ramp 500 400 10 Time (s) 500 Theory Experimental 400 300 200 100 0 0 5 10 15 20 Time (s) Figure 5.9 Counterweight displacement during starting of a fully loaded belt. 115 From the figures above, a distinct correlation is noticeable between the movement of the counterweight, and the drive torque depicted earlier. This is expected given the location of the counterweight adjacent to the drive system. It results from the behaviour of the support wave travelling through the belt. As the support wave reaches the drive station, tension throughout the belt is equalised, resulting in a decrease in drive torque, and thus an upward movement of the counterweight. Similarly, it can be seen that a non-zero starting position exists for the counterweight during loaded belt simulations. This is a direct result of the pretension placed on the system, as a result of the braking torque. As such, the actual starting position of the counterweight is irrelevant. Again, this is further explained in Section 5.4. Lastly, small discrepancies are evident in the running position of the experimental counterweight, during successive tests. While all measures were taken to ensure consistency between tests, being an outside system, the moisture content of the bulk material beared a small impact on the results. As such, the bulk density changed, and the feed rate was also affected. In addition to this, material inconsistencies were noticeable in the belt. Age has seen the belt creep, and consolidate unevenly, resulting in a varying elastic modulus throughout the system. 5.3.3 Belt Velocity during Starting To comprehensively display the effect of conveyor dynamics on a system, encoders were placed at various locations on the belt designed to measure the belt velocity: - Encoder 1 was placed on the drive pulley. This was done partly to ensure the system was obeying the specified starting characteristics, but more so to provide a baseline for comparison of the other encoders. - Encoder 2 was placed at the head of the conveyor. - Encoder 3 was placed at the tail of the conveyor The location of encoders 2 and 3 represent the transition between the carry and return side of the belt. As shown later in chapter 6, there is a distinct difference in behaviour between the 116 two sides of the belt, and these encoders allow this behaviour to be quantified. The results are graphed below. 4.5s Ramp 4.5s S-start 3 Velocity (m/s) Velocity (m/s) 3 2 1 Drive Head Tail 0 -1 0 5 10 15 2 1 -1 0 20 Drive Head Tail 0 5 10 Time (s) 9s Ramp 3 Velocity (m/s) Velocity (m/s) 20 9s S-start 3 2 1 Drive Head Tail 0 -1 0 15 Time (s) 5 10 15 2 1 -1 0 20 Drive Head Tail 0 5 10 Time (s) 15 20 Time (s) Figure 5.10 Belt velocity of an empty belt during starting. 4.5s Ramp 4.5s S-start 3 2 1 Drive Head Tail 0 -1 0 5 10 15 Velocity (m/s) Velocity (m/s) 3 2 1 -1 0 20 Drive Head Tail 0 5 Time (s) 9s Ramp 20 3 2 1 Drive Head Tail 0 5 10 Time (s) 15 20 Velocity (m/s) Velocity (m/s) 15 9s S-start 3 -1 0 10 Time (s) 2 1 Drive Head Tail 0 -1 0 5 10 15 20 Time (s) Figure 5.11 Belt velocity of a half loaded belt during starting. 117 4.5s Ramp 4.5s S-start 3 2 1 Drive Head Tail 0 -1 0 5 10 15 Velocity (m/s) Velocity (m/s) 3 2 1 -1 0 20 Drive Head Tail 0 5 Time (s) 9s Ramp 20 3 2 1 Drive Head Tail 0 5 10 15 Velocity (m/s) Velocity (m/s) 15 9s S-start 3 -1 0 10 Time (s) 20 Time (s) 2 1 Drive Head Tail 0 -1 0 5 10 15 20 Time (s) Figure 5.12 Belt velocity of a fully loaded belt during starting. As can be seen, the velocity profiles of the encoders display the definition of conveyor dynamics. A clear delay is visible, resulting from the elastic nature of the system subjected to inconsistent drag forces. This delay is increasingly noticeable in the half and fully loaded case, owing to the increased weight on the belt, and the reduced elastic modulus of the system. As can be seen above, the delay along the carry side of the belt during starting is significantly affected by the loading scenario. An empty belt responds quickly to an initial displacement, taking only 0.3s for the acceleration wave to propagate along the unloaded carry side. Half loading increases this to 0.65s, with a fully loaded belt requiring 1.1s. This increase in time is a direct result of the increase in drag forces, material weight and sag take-up required. In addition to these delays, the natural frequency of the system is also seen to reduce. This frequency may be measured once all starting loads are overcome, as the system decays to steady state. As such, an empty belt has a period of oscillation of 2.36 s, whereas a half and fully loaded belt require 2.99 s and 3.24s respectively. 118 5.4 Stopping an Elastic Conveyor Belt In many situations, the stopping of a conveyor system requires more design than the starting. In particular, downhill (regenerative) systems require appropriate stopping sequences to prevent overrun of the conveyor, material loss or excessive sag. As this system exhibits an incline, the material weight on the belt aids deceleration, however it remains critical that the stopping sequence be of sufficient length. In order to model the stopping behaviour, the starting sequences presented in section 5.3 were mirrored to determine the corresponding stopping time. These represent a velocity controlled stop. A power-fail, corresponding to an emergency stop or aborted start, is investigated in Section 5.5 below. As such, the controlled stops performed are depicted below in Figure 5.13. 4.5s S-Stop 2.5 2 2 Velocity (m/s) Velocity (m/s) 4.5s Ramp Stop 2.5 1.5 1 0.5 0 0 5 10 15 1.5 1 0.5 0 0 20 5 Time (s) 2 2 1.5 1 0.5 5 10 15 20 15 20 9s S-Stop 2.5 Velocity (m/s) Velocity (m/s) 9s Ramp Stop 2.5 0 0 10 Time (s) 15 20 1.5 1 0.5 0 0 Time (s) 5 10 Time (s) Figure 5.13 Stopping profiles. Given the length of the test system, and associated incline, a stopping procedure of 9s is longer than required, however it is simply modelled for symmetry. As can be seen, the stopping sequence is introduced after 5 seconds. No brake exists in the system or drive. 5.4.1 Drive Torque during Stopping In a similar fashion to that outlined in section 5.3.1, the motor torque during stopping is of crucial importance. It is generally desirable to stop a conveyor as quickly as practical, especially 119 if an emergency stop is required, thus the capabilities of the motor and infrastructure must be adhered to. The results are shown below. 4.5s Ramp 4.5s S-stop 400 Theory Experimental 200 0 -200 0 5 10 15 Torque (Nm) Torque (Nm) 400 Theory Experimental 200 0 -200 0 20 5 Time (s) 9s Ramp 20 400 Theory Experimental 200 0 5 10 15 Torque (Nm) Torque (Nm) 15 9s S-stop 400 -200 0 10 Time (s) Theory Experimental 200 0 -200 0 20 5 Time (s) 10 15 20 Time (s) Figure 5.14 Stopping torques for an empty belt. 4.5s Ramp 4.5s S-stop 600 Theory Experimental 400 200 0 -200 0 5 10 15 Torque (Nm) Torque (Nm) 600 400 200 0 -200 0 20 Theory Experimental 5 Time (s) 9s Ramp 20 600 Theory Experimental 400 200 0 5 10 15 20 Torque (Nm) Torque (Nm) 15 9s S-stop 600 -200 0 10 Time (s) Theory Experimental 400 200 0 -200 0 5 Time (s) 10 15 20 Time (s) Figure 5.15 Stopping torques for a half loaded belt. 120 4.5s Ramp 4.5s S-stop 1200 Theory Experimental 1000 800 600 400 200 Torque (Nm) Torque (Nm) 1200 0 -200 0 Theory Experimental 1000 800 600 400 200 0 5 10 15 -200 0 20 5 Time (s) 9s Ramp 20 1200 Theory Experimental 1000 800 600 400 200 0 Torque (Nm) Torque (Nm) 15 9s S-stop 1200 -200 0 10 Time (s) Theory Experimental 1000 800 600 400 200 0 5 10 15 20 -200 0 5 Time (s) 10 15 20 Time (s) Figure 5.16 Stopping torques for a fully loaded belt. From the results above, it is immediately noticeable that the drive torque for the loaded cases decays to a non-zero value, once the system has come to a halt. In the simulations, as mentioned above, this is due to the motor acting as a brake to prevent rollback of the belt and bulk material once the system stops. This non-zero value corresponds to the starting torques of Figures 5.4- above. The ability of this theory to model this behaviour is well presented in the empty case. As can be seen, rightly so, the torque on the motor decays to zero, as no residual load is required to prevent rollback. The system weight is the same on the carry and return side of the system. In order for the test system to return the torque to zero, as mentioned above, power to the motor is ceased, allowing the motor to free-spool, and back-track. This results in a fast relaxation. In the model, as this is a controlled stop, the velocity of the motor is prescribed. Thus, it remains fixed, once the system comes to a stop. This results in a residual load along the belt. In accordance with the resistance model depicted in Figure 4.7 however, this then allows the load to disperse throughout the system slowly, until equilibrium is again found. 121 With the exception of this, the theoretical model accurately predicts the torque on the drive motor during a controlled stop. The presence of a dynamic wave is less prominent during stops of these lengths. Given that the motor torque remains positive during stopping, it appears that a stopping time of 4.5s for a fully loaded belt is longer than required. This is further investigated in the power fail shutdowns presented in section 5.5. It also displays the influence of material weight on the incline during stopping. 5.4.2 Counterweight Movement during Stopping During stopping, the counterweight’s function reverses. Where a reverse torque is applied by the motor, the counterweight load becomes the tight side tension of the belt. If this load is insufficient, traction is not possible and slippage occurs, as it would do during starting. The upper limit of the counterweight’s travel is of concern during stopping. As shown below, the counterweight travels upwards as the belt stops, in an attempt to return to the unloaded, static position. In the graphs below, continuing with the trend outlined during starting, an upward movement by the counterweight is depicted by a negative change in y-direction. 4.5s S-stop Theory Experimental 40 20 0 -20 -40 0 5 10 15 Displacement (mm) Displacement (mm) 4.5s Ramp 60 60 20 0 -20 -40 0 20 Theory Experimental 40 5 Time (s) Theory Experimental 20 0 -20 -40 0 5 10 Time (s) 15 20 9s S-stop 15 20 Displacement (mm) Displacement (mm) 9s Ramp 60 40 10 Time (s) 60 Theory Experimental 40 20 0 -20 -40 0 5 10 15 20 Time (s) Figure 5.17 Counterweight displacement during stopping of an empty belt. 122 4.5s S-stop Theory Experimental 150 100 50 0 -50 0 5 10 15 Displacement (mm) Displacement (mm) 4.5s Ramp 200 200 100 50 0 -50 0 20 Theory Experimental 150 5 Time (s) Theory Experimental 100 50 0 -50 0 5 10 15 20 9s S-stop 15 Displacement (mm) Displacement (mm) 9s Ramp 200 150 10 Time (s) 20 200 Theory Experimental 150 100 50 0 -50 0 5 Time (s) 10 15 20 Time (s) Figure 5.18 Counterweight displacement during stopping of a half loaded belt. 4.5s Ramp 4.5s S-stop 300 Theory Experimental 200 100 0 -100 0 5 10 15 Displacement (mm) Displacement (mm) 300 200 100 0 -100 0 20 Theory Experimental 5 Time (s) 9s Ramp 20 300 Theory Experimental 200 100 0 5 10 Time (s) 15 20 Displacement (mm) Displacement (mm) 15 9s S-stop 300 -100 0 10 Time (s) Theory Experimental 200 100 0 -100 0 5 10 15 20 Time (s) Figure 5.19 Counterweight displacement during stopping of a fully loaded belt. 123 Consistent with the starting results above, the counterweight movement resembles the motor torque during stopping. In a similar manner, the relaxation of the drive can also be noticed, and its influence on the counterweight movement shown. As mentioned previously, the upward movement of the counterweight must be monitored, and the stopping procedure designed accordingly. This is to prevent the counterweight ‘topping out’ – hitting the stops in place to prevent excessive or destructive movement. If this were to happen, all elasticity of the system (counterweight travel) is removed, meaning that the full shock of the load must be absorbed by the belt and infrastructure. As will be shown later in section 5.5, this is particularly crucial during an emergency shutdown of the system. 5.4.3 Belt Velocity during Stopping The controlled stopping of this belt, as shown by the behaviour of the torque and counterweight movement exhibits little dynamic behaviour. This is a direct result of the inclination of the system, as well as the longer than required stopping procedures. This behaviour is explained by the velocity of the belt, as shown in the figures below. 4.5s Ramp Stop 4.5s S-stop 3 Drive Head Tail 2 1 0 -1 0 5 10 15 Velocity (m/s) Velocity (m/s) 3 2 1 0 -1 0 20 Drive Head Tail 5 10 Time (s) 9s Ramp Stop 3 Drive Head Tail 2 1 0 5 10 15 20 Velocity (m/s) Velocity (m/s) 20 9s S-stop 3 -1 0 15 Time (s) Drive Head Tail 2 1 0 -1 0 5 Time (s) 10 15 20 Time (s) Figure 5.20 Belt velocity of an empty belt during stopping. 124 4.5s Ramp Stop 4.5s S-stop 3 Drive Head Tail 2 1 0 -1 0 5 10 15 Velocity (m/s) Velocity (m/s) 3 2 1 0 -1 0 20 Drive Head Tail 5 Time (s) 9s Ramp Stop 20 3 Drive Head Tail 2 1 0 5 10 15 Velocity (m/s) Velocity (m/s) 15 9s S-stop 3 -1 0 10 Time (s) 2 1 0 -1 0 20 Drive Head Tail 5 Time (s) 10 15 20 Time (s) Figure 5.21 Belt velocity of a half loaded belt during starting. 4.5s Ramp Stop 4.5s S-stop 3 Drive Head Tail 2 1 0 -1 0 5 10 15 Velocity (m/s) Velocity (m/s) 3 2 1 0 -1 0 20 Drive Head Tail 5 Time (s) 9s Ramp Stop 20 3 Drive Head Tail 2 1 0 5 10 Time (s) 15 20 Velocity (m/s) Velocity (m/s) 15 9s S-stop 3 -1 0 10 Time (s) Drive Head Tail 2 1 0 -1 0 5 10 15 20 Time (s) Figure 5.22 Belt velocity of a fully loaded belt during stopping. 125 For the loaded cases, it can be seen that as the drive velocity approaches zero, a spike occurs. This is the motor relaxing back to a state of equilibrium, resulting in a zero drive torque. Although this spike represents a positive velocity (in the direction of travel), this is because encoders cannot differentiate between direction. They simply count pulses to gauge velocity. Visual inspection (as well as the counterweight movement and drive torque) however confirms this movement is in the reverse direction. Knowing this, the starting and finishing positions above can be understood. The stopping sequences shown above cause the counterweight and drive torque to decay to values close to those predicted through the theoretical simulation. It is only due to the motor relaxation that the end values are offset. 5.5 Aborted Starting and Emergency Stopping Procedures of an Elastic Conveyor Belt A critical test of a system’s design comes with the application of an emergency stop or aborted start procedure. These typically occur as a result of a failure or incident, meaning the conveyor system needs to be shut down immediately. In long systems, an emergency stop is often associated with a power failure at the motor, and sometimes a break applied at an appropriate location. As this test system conveys uphill, an emergency stop simply sees the power cut to the motor, and the system comes to rest under its own weight and friction. From this, three scenarios are considered, based on two loading states - Aborted starts are performed based on a 6s ramp and s-start up procedure. This is where the emergency stop (power failure) is applied, approximately half-way through the starting procedure, at the time of peak acceleration. Initial tests based on the starting time of 4.5s presented above, displayed a potential risk to the system, given the severity of the procedure, especially once material is applied to the belt. As such, a 6s starting time was chosen. - Emergency stops are performed from the full running velocity. Each of these tests were performed in an empty and loaded state. In a similar manner to the tests performed above, conditions were chosen to exaggerate the dynamic nature of the 126 system, while working within acceptable limits. As such, initial tests with a fully loaded belt (throughput 180 t/hr), saw the counterweight reach its travel extremities during an aborted start. From this, the throughput was lowered to 140 t/hr, to keep the counterweight within the limits of the structure. 5.5.1 Drive Torque during Emergency Stopping Procedures The torque on the drive shaft during an aborted start is governed by the time at which the emergency procedure is enacted. Timed precisely with the arrival of the support wave, and the effects on the drive torque will be minimal. If however the timing corresponds to a high torque instance (peak acceleration, sag take-up etc.), the torque reversal during equalisation may have permanent effects. As can be seen, the timing of the aborted start for the empty tests corresponds to a peak in the drive torque. As such, when power is cut from the motor, the drive over equalises to a torque approximately 60% of the peak during starting. Comparatively in the loaded case, it is seen that the power fail occurs as the support wave is approaching the drive, and thus the over correction is much less; approximately 20% of the peak value. 6s Ramp Aborted Start 6s S Aborted Start 600 Theory Experimental 400 200 0 -200 -400 0 5 10 15 20 Time (s) Torque (Nm) Torque (Nm) 600 Theory Experimental 400 200 0 -200 -400 0 5 10 15 20 Time (s) Emergency Stop Torque (Nm) 400 Theory Experimental 200 0 -200 0 5 10 15 20 Time (s) Figure 5.23 Drive torque during emergency stopping of an empty belt. 127 As can be seen from both simulations, the theoretical approach represents an accurate model to the recorded data. The decay time of all simulations is close to that observed. The theoretical data tends to overshoot the experimental results, once the power is removed however the trend in the data remains consistent. Like the loaded results presented earlier, the loaded torque begins and ends at a value other than zero, for reasons presented in Section 5.4.2. As this test is for a different throughput however, no correlation exists with the previous results. 6s Ramp Aborted Start 6s S Aborted Start 1500 Theory Experimental 1000 500 0 -500 0 5 10 15 20 Time (s) Torque (Nm) Torque (Nm) 1500 Theory Experimental 1000 500 0 -500 0 5 10 15 20 Time (s) Emergency Stop Torque (Nm) 1500 Theory Experimental 1000 500 0 -500 0 5 10 15 20 Time (s) Figure 5.24 Drive torque during emergency stopping of a loaded belt. It is also noticeable that the running position of the emergency stop (prior to power loss) differs slightly from the predicted value. As mentioned above, this is due to variations in the moisture content of the bulk material, as well as inconsistencies in the elastic modulus of the belt. 5.5.2 Counterweight Movement during Emergency Stopping Procedures Following a similar trend to above, the counterweight movement, again, bears a strong resemblance to the torque on the drive shaft. As can be seen, when the emergency stop begins, the immediate relaxation of the drive wheel induces a sharp upwards movement in the counterweight. 128 6s S Aborted Start Displacement (mm) Displacement (mm) 6s Ramp Aborted Start 200 Theory Experimental 100 0 -100 -200 0 5 10 15 200 100 0 -100 -200 0 20 Theory Experimental 5 Time (s) 10 15 20 Time (s) Displacement (mm) Emergency Stop 100 Theory Experimental 50 0 -50 -100 0 5 10 15 20 Time (s) Figure 5.25 Counterweight displacement during emergency stopping of an empty belt. 6s Ramp Aborted Start 6s S Aborted Start 600 Theory Experimental 400 200 0 -200 0 5 10 15 20 Time (s) Displacement (mm) Displacement (mm) 600 Theory Experimental 400 200 0 -200 0 5 10 15 20 Time (s) Emergency Stop Displacement (mm) 300 Theory Experimental 200 100 0 -100 -200 0 5 10 15 20 Time (s) Figure 5.26 Counterweight displacement during emergency stopping of a loaded belt. 129 The loaded case, depicted above in Figure 5.26, requires explanation. The two aborted starts begin and end at a point above the experimental data, with no overlay observed. This differs from the stopping data of Section 5.4.2, in that the relaxation of the drive is inherent in the emergency stop. As such, the start and end point is irrelevant, and only the relative movement is of concern. From this, it can be seen that the theoretical profiles accurately model the experimental results. 5.5.3 Belt Velocity during Emergency Stopping Procedures The experimental velocity profiles, shown below, bear a significant difference to the stopping profiles depicted in Section 5.4.3. This is due to the system coming to a stop of its own accord, under the influence of the distributed friction along the conveyor. 6s Ramp Aborted Start 6s S Aborted Start 2 Drive Head Tail 1 0 -1 0 5 10 15 20 Time (s) Velocity (m/s) Velocity (m/s) 2 Drive Head Tail 1 0 -1 0 5 10 15 20 Time (s) Emergency Stop Velocity (m/s) 3 Drive Head Tail 2 1 0 -1 0 5 10 15 20 Time (s) Figure 5.27 Belt velocity during emergency stopping of an empty belt. 130 6s Ramp Aborted Start 6s S Aborted Start 2 Drive Head Tail 1 0 -1 0 5 10 15 20 Velocity (m/s) Velocity (m/s) 2 Drive Head Tail 1 0 -1 0 Time (s) 5 10 15 20 Time (s) Emergency Stop Velocity (m/s) 3 Drive Head Tail 2 1 0 -1 0 5 10 15 20 Time (s) Figure 5.28 Belt velocity during emergency stopping of a loaded belt. This means several things: - Deceleration by a distributed force (friction) severely reduces dynamic waves throughout the system, as opposed to deceleration via an external stimulus. - Small dynamic waves are observable, as a result of the instantaneous relaxation of the drive system. - No backwards travel is evident, once the system stops. From this, unfortunately no correlation can be made as to the suitability of an S-type starting procedure, when compared to a ramp procedure, from an emergency stop stand-point. It is noticeable that the emergency stop procedure was not engaged simultaneously, as this was a manual function. During each test, the procedure was engaged at approximately half-way through the starting sequence, corresponding to half of the running velocity, and the peak acceleration in an S-type start. As this could not be achieved, no correlations may be made. 5.6 Discussion The simulations presented above bear a good resemblance to the experimental data attained. It does however reveal some shortfalls of the theory, along with potential improvements. The 131 majority of which are centred around the inclusion and cross correlation between drag forces, and belt sag. In reality, each drag force is dependent on a superposition of complex factors. In addition to velocity and belt tension utilised in this model, temperature, age, counterweight mass (for belt sag), tracking and friction between the belt and idler roll, amongst others play quite a definitive role. While this model presents an accurate approximation, the static breakaway upon starting requires further insight, as discussed in 7.3.1. Newtonian mechanics indicates that the breakaway friction in a system exceeds the dynamic friction as the system runs. In addition to this, sag take-up in a system must also be accounted for. As a dynamic wave travels along a static belt, sag is going to reduce, slowing the propagation of the dynamic wave. This is, in turn, dependent on the static friction of the system and its components. 132 6 Pouch Conveyor Dynamics The previous chapters have focussed predominantly on the transition between the conventional troughed systems, and an individual component of a pouch configuration. Primarily, this is focussed on the interaction between a rigid wheel and the belt surface. The final analysis involves the transient coupling between the drive wheel and belt cover. The connection between a drive system and belt cover provides the necessary boundary conditions imposed on a dynamic model. As mentioned in Chapter 4, elastic wave propagation is limited through drive systems in a troughed system. The reason for this impediment lies within the configuration of the drive system: A large contact area exists between the belt and drive pulley, subjected to a high normal load. This creates an area of high friction opposing the relative motion of the belt with the pulley surface. With particular reference to long conveyor systems, drive units must be adequately sized. This means high mass components which are impervious to sudden movement. In addition to this, gearbox designs usually prevent motion disruption from the shaft end. 133 The counterweight is designed to absorb tension disruptions, while maintaining a constant base tension. This means that a wavefront is reflected and inverted due to a counterweight. The above points describe a drive’s inability to accommodate sudden changes in motion. Having said this however, all components behave elastically, ensuring some transmission. This transmission can be modelled through the use of torsional springs and dampeners, and applied to the model outlined in Chapter 4, however the design of such a model is dependent on the drive system and head layout. For pouch conveyors, the analysis of the drive contact presented in Chapter 3 shows the small nature of the required systems. This in turn means multiple units are often required. From this, it is evident that such a system is missing the magnitude and design which restricts wave propagation, as outlined above. Also seeing as the drive connects to an unloaded belt section, it is postulated that the suspended mass offers an additional degree of freedom, creating a high portion of transmission through the drive. Troughed systems that require multiple drive units are often a significant length. Thus the location of these units is typically greater than 50% of the belt length apart. Because of this, a dampened wavefront that manages to propagate through a drive station is unlikely to reach the subsequent drive station. Given the small amount of achievable drive traction in point contact drives, pouch conveyor systems are typically closely spaced. This introduces the concept of interference between a primary wave, generated from a drive stations change in velocity and a secondary wave which has passed through an adjacent drive station. This would create a standing wave between drive stations which would need to be accurately monitored to prevent failure. 6.1 Wavefront Propagation through a Point Contact Drive System In order to account for the transmission of stress waves in a dynamic model, the reason for the propagation must be addressed. Initial concepts outlined the addition of a second strand to the FEM model presented in Chapter 4, as shown in Figure 6.1. This allows the separation of the drive and carry strand of the pouch to be analysed separately. 134 Figure 6.1 Two strand FEM model. This approach would suit systems with separate belt components. From the above figure, it can be seen that the elastic properties of both the tensile members of the system, connected to the drive, may be modelled separately to the load bearing portion of the belt. Often pouch conveyors contain reinforcement in the supporting component of the belt, in order to handle the tensile loads on the belt, whereas the pouch may or may not have reinforcement. Any reinforcement may lay in the transverse direction also to assist with carrying the load, which bears no effect on the longitudinal behaviour. The issue with this design is the lack of consistency between pouch systems. The coupling between the upper and lower structure exists in many forms. The ICS system uses bolted steel sections to connect two sections, at intervals of approximately 100mm, whereas the Sicon and Enerka Becker Systems are completely vulcanised. This break in the ICS coupling would induce discontinuities in the wavefront propagation, which are difficult to predict. From this, a simpler approach was taken. Given the assumption transmission occurs, any transmission and reflection would be modelled through a coefficient of transmission 𝛼 𝑇 , and coefficient of reflection 𝛼𝑅 . For an elastic collision, it can be assumed: 𝛼 𝑇 + 𝛼𝑅 = 1 (6.1) An elastic collision is a reasonable assumption, as the damping through a drive system would be negligible in comparison to that of the overall system. 6.2 Experimental Validation To adequately qualify the transmission and reflective coefficient outlined above, experimental testing is required. To derive a generic theory, the test setup must exclude any design specific 135 components which may affect the results. As such, a test assembly was constructed, designed to impact a scale pouch conveyor, and analyse the behaviour of the transients as they pass through the belt. The interaction of the waves with the drive station was also of concern. Drive Traction Test Rig Load cell and Tensioning system Spring Loaded Impact Plate Impact Mass Pouch Belt mounted on track wheels End Frame Clamped Fixed wheels Support Frame Figure 6.2 Model of the pouch test facility. The facility is shown in Figure 6.2. Key components of the setup are: A 5m, 600 mm PN150 fabric belt was folded into a pouch shape. This was supported by track wheels at 900mm spacing, allowing free longitudinal movement. Each end of the belt utilised stainless steel turnbuckles, capable of tensioning the belt to 600kg force. A load cell at the right hand end of the belt measures this load. The centre of the belt uses the drive traction facility presented in Chapter 3, to provide a variable clamping force to the belt surface. The left end of the belt is connected to an impact plate mounted on fixed springs. This plate, when struck, sends a significant vibration through the belt. In total, 7 accelerometers are placed along the belt, to measure the frequency and magnitude of the travelling vibrations. Their placement is shown in Figure 6.3. Figure 6.3 Placement of accelerometers on the pouch conveyor. 136 The accelerometers depicted in Figure 6.3 describe the location and magnitude of vibrations travelling through the belt. Accelerometer 1 and 2 are located to measure the initial impact on the belt. This forms a reference point, from which other readings can be based. Units 3 and 5 show the immediate loss through the drive station, while 4 displays the continuity of the wave through the suspended portion of the belt. Cells 6 and 7 define the full scale wave that propagates through the clamped drive wheel. This wave should be a combination of the waves experienced by positions 4 and 5. The end of the belt also forms a starting point for a reflected wave (if any), to be measured by units 4 and 5. The use of accelerometers 1 and 2 eliminates the need for a consistent input. As this process investigates the attenuation of a travelling wave through a belt, the input is irrelevant. All the same however, 3 different impact loads were used. The test schedule is as follows: Table 6.1 Pouch conveyor test schedule. Clamping Force on the Drive Wheels 0 N – 1800 N in 300 N increments. Belt Tension 300 kg for all clamping forces except 600 N and 1500 N. These clamping loads are tested at 100 kg – 600 kg in 100 kg increments. Impact Force 3 impact loads were tested. Fill Ratio of the belt A full schedule of tests were performed at fill ratios of 0% - 75% in 15% increments. 15% fill corresponded to 4.04 kg/m of material. The results set accumulated enabled an in-depth analysis of the wavefront transmission to be determined. An example results set is shown below in Figure 6.4. In this results set, the accelerometers are defined as follows: - Accelerometer 1: Bottom Start - Accelerometer 2: Top Start - Accelerometer 3: Top Middle Start - Accelerometer 4: Bottom Middle - Accelerometer 5: Top Middle End - Accelerometer 6: Top End - Accelerometer 7: Bottom End 137 Acc (m/s2) (a) 500 Bottom Start Top Start 250 0 -250 0.42 0.425 0.43 0.435 0.44 0.445 0.45 0.455 0.46 0.465 0.47 (b) Acc (m/s2) Time (s) 500 Top Middle Start Bottom Middle Top Middle End 250 0 -250 0.42 0.425 0.43 0.435 0.44 0.445 0.45 0.455 0.46 0.465 0.47 (c) Acc (m/s2) Time (s) 500 Top End Bottom End 250 0 -250 0.42 0.425 0.43 0.435 0.44 0.445 0.45 0.455 0.46 0.465 0.47 Time (s) Figure 6.4 Results set for 45% full belt, with a clamping force of 1500 N and tension of 100 kg. From the above results, several trends are noticeable. It is evident that the form of the wave remains consistent as it travels through the belt. Time causes the majority of the waveform transients that result from vibrations in the infrastructure during impact to decay. For the purposes of these results, the magnitude of the initial peak in acceleration is analysed, to prevent standing waves affecting the results. The location of the wavefront in the belt was also of interest. The initial impact is produced by striking the tensile member of the belt with a weight, to simulate an exaggerated impact from a drive station. As can be seen from Figure 6.4, the initial impact resides predominantly in the tensile (upper) region of the belt. Results show the impact in the lower section of the belt (below the drive station) represents approximately 73% of that inflicted on the upper section, for an unloaded belt. As expected, this result is dependent on the fill (and therefore the mass) of the pouch. As material was added to the belt, these results varied. As expected, a fuller belt exhibiting a higher inertia would be affected less from a similar magnitude input. This caused pouches with higher fill ratios to contain a lower portion of the impact. As this wave progresses through the belt, the behaviour becomes dependent on that of the pouch. 138 At a distance of 2.35m from the point of impact (represented by accelerometers 3-5), the pouch acceleration was now 150% that of the tensile member. This remained acceptably constant regardless of fill. By the time the wave reached the end of the belt (accelerometers 6-7), an empty pouch represented 255% of the upper section. In correlation with the material dependence at the start of the belt, a higher fill ratio resulted in a higher portion of the impact being contained in the pouch, up to 296% for a fill ratio of 75%. These results are depicted below in Figure 6.5. Wavefront ratio (Bottom/Top - %) 400 Start Ratio Middle Ratio End Ratio 350 300 250 200 150 100 50 0 0 10 20 30 40 50 60 70 80 Fill Ratio (%) Figure 6.5 The vertical movement of the wavefront from the impact location. The above ratios can be explained through the decay of the wavefront in the individual sections of the belt. On average, an unimpeded (no clamping) wavefront in the tensile member decayed 76% of its original value. Comparatively, a wave through a loaded pouch section decayed 28%. This is a direct result of the respective elasticity of each section. As discussed previously, rubber exhibits viscoelastic properties, resulting in a dampened, delayed effect along the belt. Conversely, the material in the pouch displays rigid elasticity in compression, causing the pouch to become the primary driving force in the system. This decay gives rise to the notion of reflected waves. The initial impact, depicted in Figure 6.4(a), shows transients resulting from vibration in the supporting structure of the belt. The two maximum peaks however are the initial acceleration of the mass hitting the belt end, and the belt recovering from the impact and adjusting back to its static position. As the wave travels along the belt however at a speed of 670 m/s (evident from Figure 6.4(a)-(c)), 139 dampening removes minor transients. By the time the wave reaches the middle and end sections of the belt, the initial impact is no longer the second largest peak. The relaxation movement still remains the highest acceleration, but a third delayed peak has arisen resulting from the interference pattern produced from reflected waves. This peak occurs 0.014 s after the initial disturbance, which corresponds to the time it takes for the initial disturbance to perform a full cycle of the system. Lastly evident in Figure 6.4 is the presence of propagation through the clamped drive wheel. The readings of accelerometers 3 and 5 show a distinct transmission of the impact across the drive-belt boundary. The determined transmission coefficients are graphed below in Figure 6.6 1 0% 15% 30% 45% 60% 75% Transmission Coefficient 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Clamping Force (N) Figure 6.6 The propagation of waves at various clamping forces. As can be seen, a distinct trend is visible between the portion of wave transmitted and the clamping force, as well as the fill ratio. The clamping force governs the resistance experienced by the oncoming wave, in addition to the cross section of the contact area. An increase in these factors results in a higher degree of reflection, as shown. The effect of fill ratio continues on from the location of the wavefront in the belt, discussed earlier. A higher fill ratio causes the majority of the momentum to be carried by the bulk material in the pouch of the conveyor. From this, little inertia is transmitted through the tensile member of the belt, which causes a higher effect from the clamped drive wheels. This in turn causes the apparent transmission coefficient to be lower; however, as depicted by Figure 6.5, the realistic transmission of momentum through the whole conveyor would in fact be higher. 140 The sole purpose of this set of experiments was to prove the existence of wave propagation through a drive system on a pouch conveyor, for which it has been successful. The actual amount of propagation is heavily dependent on the drive system, belt characteristics and bulk material density. 6.3 Incorporating Wave Propagation in a Finite Element Model The Finite Element Model presented in Chapter 4 presented respectable results when compared with the 64 m incline test facility, as shown in Chapter 5. From here, it is necessary to alter this definition to apply to pouch systems, and also define the behaviour of a wavefront surrounding the drive system. As shown in Section 6.2, a distinct wavefront is visible through the drive, owing to the majority of the momentum being held in the suspended mass of the conveyor. The inclusion of this factor is in itself a complex issue. Seeing as the majority of drive systems are velocity specified, it is very difficult to define the velocity of the node with drive attached, as a function of the wave behaviour at the adjacent nodes, as this would decouple the node from the drive sequence. From this, two options are apparent, each with their own merits and limitations. 6.3.1 Wave Propagation Constant A propagation constant, as defined in Equation (6.1), represents the portion of an oncoming wave that passes through a drive system, with the remainder reflected. This approach assumes a purely elastic collision, with zero losses. The incorporation of this factor into the model presented in Chapter 4 may be done through reliance between the nodes adjacent to a drive system. Thus, the behaviour of an adjacent node is that described through equation (4.37), plus a portion of the behaviour of the node on the opposite side of the drive system. This portion may be 𝛼 𝑇 or 𝛼𝑅 depending on the nature of the oncoming wavefront. 141 While this approach is simple in theory, its limitation is that the propagation constant must first be known. This means experimental testing is required prior to simulations. 6.3.2 Modelling the Suspended Mass of a Pouch System A more efficient approach is to include the suspended mass below the drive system. This mass represents the material in the pouch, and is governed by the behaviour of the drive node, as well as the nodes adjacent to the drive. As the independent behaviour of the upper and lower sections of the pouch are insignificant at areas other than the drive system, the full model presented in Figure 6.1 is superfluous. As such, the FEM model is altered around each respective drive system, as shown below. 𝑣𝑑𝑟𝑖𝑣𝑒 Fi-2 F(i-1)1 Fi1 Fi+1 𝑖1 M11 𝑖−1 𝑖+1 Fs M1 M1 F(i-1)2 𝑖2 Fi2 M12 Figure 6.7 FEM modification to incorporate wave propagation. As can be seen in Figure 6.7 above, an additional strand is added to the model, to allow bypass of the drive system by a wavefront. For definitive purposes, to continue with the notation from Chapter 4, denoting the carry side of the belt ‘chain’ 1, and the return side of the belt ‘chain’ 2, the additional strand will be designated ‘chain’ 12. From this, several components require definition: Nodes 𝒊𝟏 and 𝒊𝟐 The mass of the upper and lower section varies depending on the size of the drive system, and the amount of material in the belt. For the purposes of this model, M11 represents the mass of the belt in contact with the drive wheels. 142 𝑀11 = 2𝑤𝑑 𝑞 𝐿 𝐵 𝑏 (6.2) Where: - 𝑤𝑑 is the width of the drivewheel in contact with the belt - 𝐵 is the width of the belt when laid out flat - 𝑞𝑏 is the mass of the belt per metre - 𝐿 is the nodal separation For a velocity controlled starting procedure, M11 is redundant, as it is eliminated from the defining equations. For a torque start however, it must be included. Subsequently, M12 represents the remaining mass of the belt, as well as the bulk material. 𝑀12 = (1 − 2𝑤𝑑 ) 𝑞𝑏 𝐿 + 𝑞𝑚 𝐿 𝐵 (6.3) It should also be noted that: 𝑀11 + 𝑀12 = 𝑀1 (6.4) The mass associated with nodes 𝑖 − 1 and 𝑖 + 1 remains unchanged from the definition presented in Chapter 4. Graphically, they are extended to depict their rigidity in the vertical direction. Inter-nodal Forces 𝑭𝒊−𝟐 , 𝑭(𝒊−𝟏)𝟏, 𝑭𝒊𝟏 and 𝑭𝒊+𝟏 The forces connecting the nodes, follow a similar definition from Chapter 4, with one modification. Element k3 from Figure 4.1, which defines the belt sag between idler roll sets, may be neglected due to the increased stiffness in the vertical plane [35]. This principle also allows belt and material flexure to be excluded from the simulation. From this, we can redefine the respective coefficients from equation (4.31). The FEM model now obeys: 143 Figure 6.8 Modified FEM model without belt sag. Following a similar process to that depicted in Chapter 4, the force exhibited by the above element is given by: 𝐹̇ + 𝐵𝐹 + 𝐷∆̇ + 𝐺∆= 0 Where: 𝐵= 𝑘2 𝐶1 (6.5) (6.6) 𝐷 = −𝑘0 − 𝑘1 − 𝑘2 (6.7) 𝑘2 (𝑘 + 𝑘1 ) 𝐶1 0 (6.8) 𝐺=− When defining 𝑘0 , 𝑘1 , 𝑘2 and 𝐶1 , it is necessary to use the appropriate cross sectional area of the belt also. In this instance: 𝐴1 = 2𝑤𝑑 𝑡 (6.9) where t represents the thickness of the appropriate belt section (cover or reinforcement). Inter-nodal Forces 𝑭(𝒊−𝟏)𝟐 and 𝑭𝒊𝟐 To define the forces connecting the suspended mass to the nodes adjacent to the drive, a simpler approach to that presented in Figure 6.8 is sufficient. For generic purposes, a simple Kelvin-Voigt model incorporating a single spring and dashpot in parallel is appropriate. It may be argued, that for small oscillations, dampening may be neglected. This however will result in continual vibrations travelling through the belt from an oscillating mass. The model is shown below in Figure 6.9. 144 𝑘4 𝐻1 Figure 6.9 Kelvin-Voigt model used to connect the suspended mass. The associated values 𝑘4 and 𝐻1 represent the material properties of the pouch rubber. A pouch system produced from a flat belt will simply utilise the tensile and dampening properties of the belt. For pouch systems that incorporated corrugations to negotiate corners easier, these corrugations will affect the results. In this instance, a tensile test, as shown in Chapter 4 is required. As stated above, the correct cross-sectional area must be used when defining the relevant material properties. In correlation with above, the suspended section of the belt will have a cross sectional area: 𝐴2 = (𝐵 − 2𝑤𝑑 )𝑡 (6.10) Given the complex particular interaction amongst a bulk solid, there is no allowance for the influence of the bulk solid on the material properties. From this, the force in the element is given by: 𝐹 = 𝑘4 ∆ + 𝐻1 ∆̇ (6.11) Shear Force 𝑭𝒔 The force connecting the upper and lower strand of the drive station is due to shear between the two components. In order for the pouch to deviate with respect to a fixed drive, a rectifying force is introduced. To quantify this, some assumptions must be made: - There is a frictionless contact between the belt and bulk material. The bulk material bears no impact on the shear force. - The pouch is continually fixed to the drive member of the belt. 145 From this, starting with the definition of shear: 𝜏= 𝐹𝑠 = 𝛾𝐺 𝐴𝑠 (6.12) This formula defines a shear force 𝐹𝑠 , acting over an area 𝐴, resulting in an angular displacement of 𝛾, depending on the shear modulus 𝐺. Figure 6.10 Definition of shear If we define the distance between the bottom of the drive wheel contact, and the centre of gravity of the bulk material, 𝑙, for small angles, we can assume: ∆= 𝑙𝛾 (6.13) Where ∆ defines the linear displacement of node 𝑖1 relative to 𝑖2 . Redefining the shear modulus in terms of the elastic modulus, 𝐸, and poissons ratio, 𝜐: 𝐺= 𝐸 2(1 + 𝜐) (6.14) From this, we can define the shear force between the drive node, and suspended pouch. 𝐹𝑠 = where 𝐴𝑠 𝐸 1 Δ 𝑙 2(1 + 𝜐) (6.15) 𝐴𝑠 = 2𝑡𝐿 (6.16) This theory provides a reasonable representation of the shear force between the upper and lower section of the belt. Care should be taken however, when determining the elastic modulus of the material. Many reinforcement materials behave very differently in the transverse direction, when compared to their longitudinal properties. For instance, steel cables placed lengthways will have no influence on shear, as the pouch will shear in the material around the cables. Similarly, if fabric was used to reinforce the pouch, it is relatively weak in 146 shear, due to the alignment of the fibres. In many cases, the most resistance to shear arises due to the belt rubber alone, with very little impact from the reinforcement. Due to this, the elastic modulus of the rubber provides a reasonable approximation. If in doubt or complex reinforcement methods are utilised, a shear test should be conducted. Based on the above relations, the system of equations presented in section 4.2.3 can be modified to account for the suspended mass. Using the notation presented in Figure 6.7: Drive node 𝑖1 , 𝑑𝑣𝑖1 𝐹𝑖1 𝐹(𝑖−1)1 𝐹𝑠 = − − 𝑑𝑡 𝑀11 𝑀11 𝑀11 (6.17) Using Equation (6.15): 𝑑𝑣𝑖1 𝐹𝑖1 𝐹(𝑖−1)1 1 𝐴𝑠 𝐸 1 1 𝐴𝑠 𝐸 1 − + + ( ) 𝑥𝑖1 − ( )𝑥 = 0 𝑑𝑡 𝑀11 𝑀11 𝑀11 𝑙 2(1 + 𝜐) 𝑀11 𝑙 2(1 + 𝜐) 𝑖2 (6.18) Node 𝑖 − 1, 𝑑𝑣𝑖−1 𝐹(𝑖−1)1 𝐹𝑖−2 𝐹(𝑖−1)2 = − + 𝑑𝑡 𝑀1 𝑀1 𝑀1 (6.19) Using Equation (6.11): 𝑑𝑣𝑖−1 𝐹(𝑖−1)1 𝐹𝑖−2 𝑘4 𝑘4 𝐻1 𝐻1 − + + 𝑥 − 𝑥 + 𝑥̇ − 𝑥̇ = 0 𝑑𝑡 𝑀1 𝑀1 𝑀1 𝑖−1 𝑀1 𝑖2 𝑀1 𝑖−1 𝑀1 𝑖2 (6.20) Similarly for node 𝑖 + 1, 𝑑𝑣𝑖+1 𝐹𝑖+1 𝐹𝑖1 𝑘4 𝑘4 𝐻1 𝐻1 − + + 𝑥𝑖+1 − 𝑥𝑖2 + 𝑥̇ 𝑖+1 − 𝑥̇ = 0 𝑑𝑡 𝑀1 𝑀1 𝑀1 𝑀1 𝑀1 𝑀1 𝑖2 (6.21) The suspended node, 𝑖2 , is governed by: 𝐹(𝑖−1)2 𝑑𝑣𝑖2 𝐹𝑖2 𝐹𝑠 = + − 𝑑𝑡 𝑀12 𝑀12 𝑀12 (6.22) Through Equations (6.11) and (6.15): 𝑑𝑣𝑖2 𝑘4 𝑘4 1 𝐴𝑠 𝐸 1 1 𝐴𝑠 𝐸 1 − 𝑥𝑖+1 − 𝑥𝑖−1 − ( ) 𝑥𝑖1 + (2𝑘4 + )𝑥 𝑑𝑡 𝑀12 𝑀12 𝑀12 𝑙 2(1 + 𝜐) 𝑀12 𝑙 2(1 + 𝜐) 𝑖2 𝐻1 𝐻1 𝐻1 − 𝑥̇ 𝑖+1 − 𝑥̇ 𝑖−1 − 2 𝑥̇ = 0 𝑀12 𝑀12 𝑀12 𝑖2 (6.23) 147 In order to include these equations in the differential equation presented in Chapter 4, it is firstly necessary to include the displacement and velocity of the suspended mass in the state vector. Extending equation (4.43) yields: 𝑥1 ↓ 𝑥𝑁 𝜃𝑐 𝑣1 ↓ 𝑣𝑁 𝜃𝑐̇ ̂ 𝑋 = 𝐹1 ↓ 𝐹𝑁 𝑥𝑑1 ↓ 𝑥𝑑𝑁 𝑣𝑑1 ↓ [𝑣𝑑𝑁 ] (6.24) As can be seen from equation (6.24), the behaviour of the N suspended masses (corresponding to N drive stations) is defined after that of the rest of the system. This is to prevent confusion during the solve process. The addition of these elements however, also increases the size of Matrix A (equation (4.45)). Matrix A is now extended to a 5x5 configuration, as shown below: 𝐴11 𝐴21 𝐴 = 𝐴31 𝐴41 [𝐴51 𝐴12 𝐴22 𝐴32 𝐴42 𝐴52 𝐴13 𝐴23 𝐴33 𝐴43 𝐴53 𝐴14 𝐴24 𝐴34 𝐴44 𝐴54 𝐴15 𝐵𝐶1 𝐴25 𝐵𝐶2 𝐴35 and 𝐵𝐶 = 𝐵𝐶3 𝐴45 𝐵𝐶4 [𝐵𝐶5 ] 𝐴55 ] (6.25) Chapter 4, presents a formulation for the construction of matrices 𝐴12 , 𝐴23 , 𝐴31 , 𝐴32 and 𝐴33 . The adjusted cross sectional area presented in equations (6.9) and (6.10) induces a minor change in this structure, however the balance remains unchanged. Obviously, the definitions of 𝐽, 𝑁 and 𝑃 will be changed to comply with 𝐵, 𝐷 and 𝐺 in equations (6.6)-(6.8). In addition to this however, equations (6.17)-(6.23) induce non-zero entities in matrices 𝐴21 , 𝐴22 , 𝐴24 , 𝐴25 , 𝐴45 , 𝐴51 , 𝐴52 , 𝐴54 and 𝐴55 . Given these values are only used to define the 148 behaviour around a drive system, and do not occupy the entirety of these matrices, the construction of these matrices is not explored in this section. 6.4 Simulating Wave Propagation for an Impulse Response In order to verify the above theory, a simulation is set up to mimic the experiments performed in Section 6.2. To clearly see the results, the system is expanded to 1 km in length, with a material weight of 100 kg/m. The specifics of the simulation are detailed below. Table 6.2 Simulation properties. Conveyor Properties Conveyor Length 1000 m Number of elements in simulation 200 Material mass 100 kg/m Belt construction PN 1000, 4 ply belt. Carcass 4mm, with top and bottom covers 7.5mm and 4.5 mm respectively. Belt Width, B 1000 mm Pouch height, 𝑙 450 mm Rubber Properties Reinforcing Modulus, 𝐸0 2.23 GPa Rubber Modulus, 𝐸1 9.16 MPa Rubber Modulus, 𝐸2 12.8 MPa Rubber Dampening, 𝐻 35991 Ns/m Poisson’s Ratio 0.3 Drive Properties Primary Drive Location Head, N=100 Movement 0.2s Impulse to 1 m/s Secondary Drive Location Half-way along the carry side, N=75 Movement Fixed 149 Counterweight Properties Mass 20 T Type Gravity Location Immediately after the Primary drive, on the return side of the belt, N=1 This simulation serves to investigate the behaviour of transients throughout a pouch system. It is for this purpose that the conveyor is flat, as no gravitational effects are present. Also, as the system is stationary in response to an impulse, no velocity dependent effects are noticed. The propagation of the impulse is shown below in Figure 6.11 and Figure 6.12. Figure 6.11 Wave propagation due to a 0.1s impulse. Figure 6.12 Plan view of the wave propagation. 150 From this figure, several features of the system become apparent: 1) Consistent with previous simulations, the conveyor layout is split into the return side of the belt (0-1000m), and the carry side of the belt (1000-2000m). This simulation represents a loaded system. 2) A short impulse is evident at the head end of the belt (x=2000m). As this profile consists of both a sharp acceleration and deceleration, tension and compressive waves extend through the system. These waves are seen to propagate outward from the origin, gradually decaying due to internal (viscoelastic) resistances within the rubber, as well as motion resistances as discussed in Chapter 4. 3) A difference in wave velocity is evident between the return and carry sides of the belt. This can clearly be seen in the plan view shown in Figure 6.12. This is due to the lower mass and resistances on the return side, resulting in faster wave propagation. This inertial obstruction creates an interface for reflection of the wave. Thus, as can be seen, a portion of the wave is reflected back along the return side of the belt, while the remaining portion propagates up the carry side, at a slower velocity. 4) A stationary drive unit is placed at x=1500m. As can be seen, the drive exhibits no movement, while the wavefront passes through the location. This simulation shows a rapid decay as the wavefront approaches the drive system. This is best seen in Figure 6.11. This is not due to resistances within the model presented above, it exists due to the superposition of the incoming wave, and the leading edge of the reflected wave. In reality, constraining a portion of the belt, as is the case with a drive wheel, will induce a reflected wave of some magnitude. The leading edge of this wave will interfere with the remainder of the approaching wave, diminishing its amplitude. This can further be seen in Figure 6.14. Figure 6.14 displays the behaviour of the fixed drive node (with zero velocity), the suspended node, as well as the two nodes either side of the drive. This is depicted in Figure 6.14. As can be seen, there is a significant change in amplitude between two nodes before, and the node before (approaching node). This is a direct consequence of the reflected wave. Once the wave enters the drive system through the suspended node, no further superposition occurs. A consequence of this however is the observed portion seen to propagate through the drive system. For this simulation, the propagation constant, defined as the ratio of the wave amplitude either side of the drive, is seen to be 0.685. If this definition is based on 151 the node two before the drive however, this constant decays to 0.467. This, again, is because the amplitude of the wave immediately before the drive is a superposition of the incoming and reflected wavefronts. From this, a model that utilises a propagation constant to show transmission must account for superposition. Surrounding Drive Velocities 0.4 Suspended Node Fixed Drive Node After Approaching Node 2 After 2 Before 0.3 0.2 Velocity (m/s) 0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 Time (s) Figure 6.14 Wave behaviour surrounding a fixed drive system. Oncoming Wave Fixed Drive 2 Before Approaching Node 2 After Node After Suspended Node Figure 6.13 FEM modification to incorporate wave propagation. Where a higher degree of transmission is required, minor modifications may be required to the model presented in Section 6.3.2. ‘Locking’ the adjacent nodes; done by maximising K4 and minimising the reinforcement modulus in the drive rubber, results in a theoretical 100% transmission. As discussed above however, this is based on the amplitude immediately adjacent to the drive, and includes negative effects due to the reflection of the incoming wave. To circumvent this, two approaches are possible: - Reducing the number of nodes eliminates the resolution surrounding the drives. This prevents the simulation from detecting the reflected wave, increasing the apparent 152 propagation. This however has the adverse effect of reducing the overall resolution of the simulation. - Extending the dependence of the suspended nodes further than the adjacent nodes. In order to maintain an accurate model, it is recommended that the dependence on the surrounding nodes be adjusted accordingly, so as to not affect the resolution of the model. 6.5 Transient Behaviour in Pouch Conveyors Incorporating Multiple Drive Systems It was shown in Section 6.4 above, that the superposition of waves in a pouch conveyor can induce some erratic behaviour. The previous simulation was simply a stationary system consisting of a fixed and moving drive unit, as tested experimentally in Section 6.2, in order to verify a portion of propagation through a fixed drive system. As mentioned in Chapter 3 the reduced traction available to pouch conveyor drive systems results in more drive units placed along the length of the belt to generate the required traction. From this, it is necessary to start each system in a way to minimize the transient effects. Below, investigations are performed into 4 various starting methods: - Concurrent Starting: Each drive station is started instantaneously. - Basic Sequence: A sequenced starting procedure attempts to minimise transients - Complex Sequence: A refined adaptation of the basic approach. - Single Drive: An analysis is performed with a single head drive, for comparison. 6.5.1 Concurrent Starting In order to comprehend the optimum starting sequence of each drive unit, the same system as above is utilised, with a starting sequence applied to each drive. A flat system is used to eliminate loads due to gravity. Consider drive units, placed at 400m increments along the return and carry side of the belt. In addition to the drive placed at the head, this yields a drive station on the carry and return side of the belt at distances of 400m and 800m from the head drum, as depicted in Figure 6.15 . 153 Figure 6.15 Location of drive stations on 1km conveyor. Firstly, for comparative purposes, the starting of the above system is analysed using a direct start procedure for all drive systems. All systems begin instantaneously, with no correlation between the adjacent drives. This is done to analyse the transient interactions between drive stations, and to investigate the propagation of wavefronts through the system. Consider a 20 second starting procedure consisting of the following: 𝑣𝑑 = 0.05𝑣𝑚𝑎𝑥 𝑡 0≤𝑡≤2 𝑣𝑑 = 0.1𝑣𝑚𝑎𝑥 2≤𝑡≤4 0.9 𝑡−4 𝑣𝑚𝑎𝑥 [1 − cos (𝜋 ( ))] 2 16 4 ≤ 𝑡 ≤ 20 𝑣𝑑 = 0.1𝑣𝑚𝑎𝑥 + Starting Procedure of the Drive System 5.5 5 4.5 Velocity (m/s) 4 3.5 3 2.5 2 1.5 1 0.5 0 0 2 4 6 8 10 12 14 16 18 20 Time (s) Figure 6.16 Starting procedure of the drive system. As can be seen from Figure 6.16, the drive system ramps to 10% of the operating velocity over a period of 2 seconds. It then remains at this velocity for a further 2 seconds before entering a sinusoidal s-start curve, given by Harrison [16], for 16 seconds. The tension distribution along the belt throughout the starting procedure is given in Figure 6.17. 154 Figure 6.17 Belt forces during start-up. Concurrent starting of multiple drive systems essentially breaks down to starting 5x400m individual conveyors. Each belt section remains the same length, so barring the behaviour of the counterweight, each belt section represents a 400m conveyor. This introduces the concept of scalability, and questions if a conveyor is increased in length, and drives added such that the drive spacing remains the same, will the dynamics be the same? This theory is investigated in Section 6.6. The force profile depicted in Figure 6.17, shows severe transient behaviour during the starting process. This is a direct result of the superposition of wavefronts from adjacent drive stations. The system then dampens to steady state, with each belt section consisting of a high and low tension distribution, defining the T1 (tight side) and T2 (slack side) tensions of each drive. The peak tensions for each belt section are defined below in Table 6.3. Table 6.3 Tension extremities for concurrent starting. Station Maximum Tension Minimum Tension 0-400m 109.1 kN 92.1 kN 400-800m 104.1 kN 91.8 kN 800-1200m 118.8 kN 83.9 kN 1200-1600m 127.6 kN 68.3 kN 1600-2000m 120.2 kN 79.7 kN 155 The influence of the transients from surrounding drive stations is also affected by the reduced drag in the system. As shown in Chapter 4, belt flexure contributes a significant amount of drag to the system, and the removal of this facilitates dynamic behaviour. It can also be seen that apart from the initial acceleration to the inspection velocity, the peak tension during start-up corresponds approximately to the peak acceleration of the belt. This differs slightly from the results from Chapter 5, which showed a discrepancy between the peak acceleration and peak tension in the belt. This is a direct consequence of the time it takes for the tension wave to perform a complete lap of the system. Figure 6.18 Tractive force per drive station. The traction required by each motor is given by Figure 6.18. As can be seen, the transients in each element can actually facilitate the subsequent drive stations, resulting in a negative traction value. For a system that isn’t designed to accommodate these transients, the reverse cyclic loading on the motor and gearbox can have disastrous consequences. It is best to dampen these transients using sequenced starting as shown in Section 6.5.2. 156 Counterweight Displacement during Start-up 0.04 0.02 Displacement (m) 0 -0.02 -0.04 -0.06 -0.08 -0.1 0 5 10 15 20 25 Time (s) Figure 6.19 Counterweight displacement during start-up. The movement of the counterweight is shown above, in Figure 6.19. For concurrent starting, the motion of the counterweight is governed by the movement of drives D1 and D5. Drive D5 at the head of the conveyor, will displace the counterweight downward, until the acceleration wave from D1 reaches the counterweight. The peak acceleration of each drive system occurs at approximately 12s. This corresponds to the lowest position of the counterweight. It can be seen that the relative displacement of the counterweight during this starting procedure is relatively low. This is due to the unloaded state of the belt, as well as the concurrent starting procedure of the drives. This section of belting also lies on the return side of the conveyor, with fewer motion resistances. This low displacement can be used to explain the high order transients evident in Figure 6.17. Very little energy has been absorbed through the counterweight, resulting in un-damped stress waves throughout the rest of the system. This is further explained below. 6.5.2 Sequenced Starting – Controlling the Peak Acceleration In an attempt to minimise the number of stress waves propagating through the system, a sequenced start can be utilised to absorb the waves. In this process, the timing of each drive is such that the oncoming acceleration wave from a drive starting, is replicated by the subsequent drive. This ‘speed matching’ allows the oncoming wave to be absorbed, and in fact utilised to assist the next drive station. 157 For this two-stage starting procedure, the initial wave results from the breakaway of the conveyor as the belt accelerates to the inspection velocity. As discussed in Chapter 4, this corresponds to the greatest internal friction of the system. In a sequenced start, the order in which the drives are started is critical. The drive adjacent to the counterweight must begin the procedure, ideally located at the location of peak tension; D5 in this instance. The counterweight ensures a reasonably constant slack side tension is maintained, enabling traction, while the location ensures a positive tension distribution. As the stress wave propagates from D5 to D4, the starting procedure of D4 is timed to correspond to the impact of this wave, minimising the transients resulting from the D 5. Given this is a soft starting procedure (in comparison to the impulse depicted in Section 6.4), the majority of the compression wave attempting to propagate through the return side of the belt will be absorbed by the adjacent counterweight, resulting in downward movement. Given the inertia of the counterweight pulley, a small compression wave may propagate, but this is not enough to warrant the starting of D1. Utilising such a starting procedure must be done with caution. If all drive systems incorporate the same sequence, as portrayed in Figure 6.16, by the end of the starting sequence, each drive will have travelled a different distance, affecting the inter-nodal tensions between the drives. For instance, if D5 begins the sequence, followed by D4 through to D1, the resultant tension profile between each drive will be significantly higher to the running tensions depicted in Figure 6.17. This is owing to the sustained change in nodal separation during starting. The profile between drives D5 and D1, will remain the same, owing to the freedom of the counterweight, and consistent tension. Any change in nodal separation is absorbed by the counterweight, providing it does not exceed the counterweights travel. To compensate for this behaviour, the starting procedure of each drive must be adjusted such that the distance travelled by each drive is uniform by the end of the starting procedure. To begin analysing the impact of sequenced starting, we must first determine the time delay between adjacent drives. To find this, an impulse response is utilised in a similar manner to Section 6.4 above. The belt responds according to Figure 6.20: 158 Figure 6.20 Impulse response to 5 drive stations. Table 6.4 Wavefront travel time between drive stations. Location Travel Time (s) Wave Speed (m/s) 1→2 1.009 396.4 2→3 1.374 291.1 3→4 1.626 246.0 4→5 1.626 246.0 5→1 1.009 396.4 The times determined above depend on the construction of the system, and as discussed in Section 6.4, show the dependence of drag forces and material weight on the propagation speed of the wave. It is also evident that the time delay between drives 2 and 3 is a linear combination of the times between the drives on the carry and return side. From the above simulations, it is evident that the peak tension during starting lies shortly after the peak acceleration, with the time delay proportional to the length of the belt associated with the drive. Based on this, it is prudent to separate these occurrences as much as possible. From this, investigations are performed to analyse the impact of the s-start timing. 159 This sequenced start attempts to control the peak acceleration during starting. This is done by sequencing the starting of the s-procedure, while maintaining a simultaneous breakaway. In order to conform to a uniform displacement of each drive, the various stages of the starting procedure are integrated, as shown below in Figure 6.21. Start-up Curve 6 5 Velocity (m/s) 4 3 2 1 0 T0 T1 T2 Tf Ts Time (s) Figure 6.21 Start-up curve. Based on the above figure, integrating each individual procedure allows the s-curve to be customised to suit. The distance travelled from the above procedure is given as: 1 1 𝑥 = (𝑇1 − 𝑇0 ) + (𝑇2 − 𝑇1 ) + 0.55𝑣(𝑇𝑓 − 𝑇2 ) + 𝑣(𝑇𝑠 − 𝑇𝑓 ) 4 2 (6.26) where 𝑣 represents the full belt velocity (5 m/s). By definition above: 𝑇0 = 0 (6.27) 𝑇1 = 𝑇0 + 2 (6.28) 𝑇𝑠 = 20 𝑠 (6.29) For the drive sequence depicted in Figure 6.16, assigned to drive 5, a total of 45.5m is travelled upon start-up. Following the data presented in Table 6.4, the following properties are derived for the subsequent drive systems. This is depicted graphically in Figure 6.22. 160 Table 6.5 Starting characteristics of a complex start. Drive 𝑻𝟎 (s) 𝑻𝟏 (s) 𝑻𝟐 (s) 𝑻𝒇 (s) 𝑻𝒔 (s) 1 0 2 9.635 14.365 20 2 0 2 8.626 15.374 20 3 0 2 7.252 16.748 20 4 0 2 5.626 18.374 20 5 0 2 4 20 20 Starting Procedure of each Drive System 5 4.5 4 Velocity (m/s) 3.5 3 2.5 2 1.5 Drive 1 Drive 2 Drive 3 Drive 4 Drive 5 1 0.5 0 0 2 4 6 8 10 12 14 16 18 20 Time (s) Figure 6.22 Starting Procedure of each drive system. As can be seen from Figure 6.22, the acuteness of each starting procedure is increased, when compared to Figure 6.16. When compared to Figure 6.26 however, it is seen that the acceleration of each drive is lower, owing to the concurrent starting of each system during breakaway. All drives still maintain a constant displacement, and a 20s starting time. The forces on the belt are depicted in Figure 6.23, with the peak tensions displayed in Table 6.6. 161 Figure 6.23 Belt forces during start-up. Table 6.6 Tension extremities for a complex sequenced start. Station Maximum Tension Minimum Tension 0-400m 150.9 kN 89.2 kN 400-800m 136.4 kN 91.6 kN 800-1200m 160.7 kN 89.5 kN 1200-1600m 172.7 kN 77.9 kN 1600-2000m 161.6 kN 83.7 kN The above results represent an increase in control of the tension transients, compared with concurrent starting in Figure 6.17. Although higher tensions are experienced in each belt section, some transients remain. It is clear that these transients result from the breakaway start of each drive. Each stress wave is seen to bear a significant impact on each belt section, with some propagation and superposition of the waves also noticeable. The traction of each drive system is shown below in Figure 6.24. Similar to Figure 6.18, the oscillating waves in the system have a significant effect on the traction of the each drive. Oscillating tractions, and consequently, oscillating torques, result in considerable wear on the drive components, and should thus be avoided. 162 Figure 6.24 Tractive force per drive station. As can be seen from Figure 6.25, the counterweight displaces a great deal more, than during concurrent starting, depicted in Figure 6.19. The initial peak is the same, owing to the simultaneous breakaway. This peak, approximately 80 mm, represents half the amount D5 travels, before the tension wave from D1 equalises the counterweight, returning it to the zero position. After 4s however, D5 begins to accelerate, as D1 remains at the inspection velocity. This sees the counterweight move rapidly downward. The peak displacement is approximately 12.5 s into the starting procedure. This corresponds to the point where D1 begins to travel faster than D5, allowing for the delay for the wavefront to reach the counterweight. Counterweight Displacement during Start-up 0.5 0 Displacement (m) -0.5 -1 -1.5 -2 -2.5 -3 0 5 10 15 20 25 Time (s) Figure 6.25 Counterweight displacement during start-up. 163 6.5.3 Sequenced Starting – Controlling the Breakaway Friction Section 6.5.2 analysed the effect of sequencing the primary acceleration curve, in an attempt to control the transients within the system. While some control was evident, it highlighted the influence of the waves that arise during breakaway of the conveyor. As such, this simulation is aimed at controlling the transients through the proper sequencing of both breakaway, and the S-start, and analysing the resulting behaviour. Using equation (6.26), the starting characteristics can be determined, specifying the following constraint: 𝑇2 = 𝑇0 + 4 (6.30) This fixes the time between the breakaway of the drive, and the transition into the starting procedure, ensuring each starting procedure is delayed according to Table 6.4. Based on this, the starting parameters shown in Table 6.7 are determined. These are depicted graphically in Figure 6.26. Table 6.7 Starting parameters for sequenced starting. Drive 𝑻𝟎 (s) 𝑻𝟏 (s) 𝑻𝟐 (s) 𝑻𝒇 (s) 𝑻𝒔 (s) 1 5.635 7.635 9.635 13.113 20 2 4.626 6.626 8.626 14.346 20 3 3.252 5.252 7.252 16.026 20 4 1.626 3.626 5.626 18.013 20 5 0 2 4 20 20 Starting Procedure of each Drive System 6 5 Velocity (m/s) 4 3 2 Drive 1 Drive 2 Drive 3 Drive 4 Drive 5 1 0 0 2 4 6 8 10 12 14 16 18 20 Time (s) Figure 6.26 Starting procedure of each drive system. 164 As can be seen, the starting profiles closely resemble those shown in Figure 6.22. In this instance however, the S profiles of drives D1 to D4 are shortened, to accommodate the conservation of displacement. The belt forces during the above starting sequence, is given in Figure 6.27, with the corresponding tension peaks given in Table 6.8. Figure 6.27 Belt forces during start-up. Table 6.8 Tension extremities for a sequenced start. Station Maximum Tension Minimum Tension 0-400m 182.3 kN 74.3 kN 400-800m 151.2 kN 93.1 kN 800-1200m 169.3 kN 93.8 kN 1200-1600m 178.0 kN 88.0 kN 1600-2000m 175.2 kN 89.4 kN As can be seen from Figure 6.27, a much smoother start is experienced by each belt section. This is owed to the absorption of the primary acceleration wave during each start sequence. All belt sections achieve a single peak, with the exception of D1, owing to the movement of the counterweight. 165 As expected, drives D5 and D4 experience severe starting tensions, given they are accelerating a loaded belt. Similarly, drives D3 and D2 produce lower starting tensions given they are starting a semi loaded belt, and empty belt respectively. Drive D1 on the other hand, experiences the highest starting tension of all. This is partly due to the extreme starting procedure of the system, but is primarily due to the counterweight. Drive D1 serves to accelerate the full 400m of the belt, as well as ensuring the counterweight remains within reasonable limits, as drive D5 attempts to displace the counterweight down. No compressive force from D5 assists D1, as this is absorbed by the counterweight. It also serves to accelerate 200m of belt in front of the drive, using compression. It is the only drive responsible for starting 600m of belt. The relative displacement of each drive is also visible in Figure 6.27. During the initial acceleration of the s-shape procedure, there is an elongation present between each drive station. This is evident in the tension drop across the drive. Comparing Figure 6.27 to Figure 6.17 showing concurrent starting, it can be seen that the overall tension in the belt element is higher than the static load of the counterweight. In the instance where the displacement remains constant, a period of low tension is seen in front of each drive station. This causes the tension distribution to centre itself around the counterweight load, as this is a horizontal simulation. A higher average starting tension can have its benefits. Provided proper design techniques are utilised to prevent belt or infrastructure failure, these increases in belt tension can be put to good use. For a system with vertical curves, regions of low tension cause excessive sag, resulting in abnormal transients, and in worse cases, alignment issues and material loss. The proper placement and starting of drive stations can be used to increase belt tension appropriately during starting or stopping, and return it to its running value once the transients decay. The traction required by each drive to sustain this basic starting sequence is shown below in Figure 6.28. Comparing this with the tractions derived in Figure 6.18, several changes are noticeable. Firstly, it can be seen that the peak tension, associated with drive D5, is significantly higher than experienced during concurrent starting. This is due to the delayed compressive force from D2, resulting in D5 being responsible for the initial acceleration of the belt segment. This produces a peak traction of 85.2 kN, compared to the 22.1 kN from Figure 6.18. Evidently, 166 the assistance received from D4 reduces the traction, resulting in D5 essentially starting 200m of belting. Similarly, drives D2 and D3 bear the main load during concurrent starting, with this load fluctuating heavily, even progressing into negative figures. During a sequenced start, these forces are contained and reduced within acceptable limits. Evidently the propagation of acceleration waves through the drive systems cause the greatest fluctuation of traction. All drives subsequently decay to their respective operating tractions. Figure 6.28 Tractive force per drive station. Lastly, drive D1, remains stationary for 5.6s before starting. Meanwhile, D5 begins starting, resulting in counterweight movement, and a small compressive wave propagating towards D 1, as shown in Figure 6.27. This is also evident in Figure 6.29, showing the counterweight movement. The counterweight is designed to absorb any stretch in the conveyor system, preventing these areas of high and low tension from existing as transient waves. From this, the transients in the system during concurrent starting (Figure 6.17) can be partially explained through the relatively low displacement of the counterweight, shown in Figure 6.19. 167 Counterweight Displacement during Start-up 0.5 0 Displacement (m) -0.5 -1 -1.5 -2 -2.5 -3 -3.5 -4 0 5 10 15 20 25 Time (s) Figure 6.29 Counterweight displacement during start-up. It is also noticeable that the initial equalisation of the counterweight, depicted in Figure 6.25 no longer occurs. This results in a larger displacement, close to 4m. 6.6 The Scalability of Pouch Conveyors Incorporating Multiple Drive Units The use of multiple drive units queries the scalability of a system. For a velocity prescribed starting procedure, as shown above, the transients are typically contained within the belt element following the drive. The only exception to this is wave propagation due to the pouch, torsional components of the drive system or the counterweight. This concept induces the theory that long conveyors containing multiple drive systems, will essentially behave in the same way as a short conveyor containing one drive, so long as the spacing between drive units is maintained. Practically, this allows for the extension of pouch conveyor systems, should additional length be required beyond their design specification. In order to simulate this, a comparative study to the 1km flat system presented above is undertaken. In this system, the drives are spaced at 400m intervals along both the return and carry side of the belt. Based on this, a 2km and 4km system is studied, based on concurrent starting. For simplicity, only the steady state tensions are displayed, along with the peak tensions in each element. Note – the peak tension does not necessarily occur simultaneously along the belt. It is purely a representation of the maximum and minimum tension experienced at that point on the belt, with no temporal correlation. The results are displayed in Figure 6.30. 168 Upon first inspection, one noticeable difference becomes apparent. The drive spacing of 400m places a drive at the tail of both the 2km and 4km systems; however this is not the case with the 1km system. This intern reduces the belt force owing to only half of the material being conveyed. Apart from this however, all tension profiles are comparable. It is seen that the steady state tension of each system consists of a ramp function centred on the counterweight tension, as expected. 1.5 x 10 Tension Plots for a 1km System 5 Running Tensions Peak Tension Minimum Tension 1.4 Tension (N) 1.3 1.2 1.1 1 0.9 0.8 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Distance along the belt (m) 1.5 x 10 Tension Plots for a 2km System 5 Running Tensions Peak Tension Minimum Tension 1.4 Tension (N) 1.3 1.2 1.1 1 0.9 0.8 0.7 0 500 1000 1500 2000 2500 3000 3500 4000 Distance along the belt (m) 1.5 x 10 Tension Plots for a 4km System 5 Running Tensions Peak Tension Minimum Tension 1.4 Tension (N) 1.3 1.2 1.1 1 0.9 0.8 0.7 0 1000 2000 3000 4000 5000 6000 7000 8000 Distance along the belt (m) Figure 6.30 Tension plots for a 1km, 2km and 4km system. 169 For both systems with a drive placed at the tail, it is seen that the first drive on the carry side experiences considerably higher tensions upon starting, than the proceeding drive along the carry side. This occurrence is a direct result of the transmission and reflection of transient waves in the system. As shown in Figure 6.12, an interface exists at the tail of a conveyor owing to the difference in mass between the carry and return side of the belt. In addition to the drive located at the tail, this interface contains transients from surrounding drives, within this belt element. As these waves superimpose, a peak tension is observed. Overall, this simulation allows the possibility of belt scaling. From a design perspective, experimental testing on a scale model may give indicative results of a much larger system. In addition to this, in-situ systems have the ability to be extended easily, and, if done so properly, with minimal alteration to the dynamics of the system. 6.7 Conclusion This chapter has analysed three different starting sequences, and the corresponding belt behaviour, according to multiple drive units. Concurrent starting of all drives experienced the lowest belt tensions, and smallest counterweight movement. However this was at the cost of significant transients continually operating through the belt. Sequencing the S-start saw a slight improvement in the transient behaviour during starting. From this however, a consequence of delayed starting, saw higher belt forces and counterweight movement. When the entire starting procedure was sequenced (including the breakaway), this saw the elimination of the transients. With the exception of belt between D5 and D1, including the counterweight, no transients were seen to remain in the belt for more than one period. The only reason they remain between D5 and D1 is because the counterweight makes it impractical to absorb the oncoming wave. The sequencing of the starting procedure saw higher accelerations of the belt, resulting in higher tensions during starting. This increase in tension however had the bonus effect of keeping the traction of each drive within reasonable limits. With the exception of the transients due to the counterweight, all drives recorded positive (or slightly negative) tractions, quickly decaying to the running tractions soon after the starting sequence. 170 7 Design Aspects, Conclusions and Future Work The ever increasing demand for bulk material handling systems has seen belt conveyors change more in the past 20 years, than they have since their inception. In order to maximise efficiencies and productivity, more complex conveyor systems are being installed without adequate knowledge of the system. In order to rectify this, this Thesis is aimed at specific design aspects of pouch conveying systems, and their relevance when compared to conventional troughed conveyors. The following chapter provides an overview of how the theoretical models developed apply to the design of pouch conveyor systems. Following this, conclusions are drawn about the work contained herein, and recommendations for future work presented. 7.1 Design Aspects Pouch conveyors are typically used in small installations, however hold the potential for much larger uses. As such, throughout this thesis, several theories and simulation programs were 171 developed in an attempt to improve the design capabilities of generic pouch conveyors. This thesis separated the various design issues into 3 forms: the main resistances, drive system implications, and the subsequent dynamic response during starting and stopping. 7.1.1 Main Resistances The drag forces of a belt conveyor govern the overall efficiency of the system. In particular, they determine the size of drive and belt required for a particular installation. Due to this, each resistance has accrued significant amounts of research in an attempt to maximise the efficiency. In particular: indentation rolling resistance (IRR). This dissertation builds on the extensive research of others, in an attempt to modify the theories of troughed conveying systems, to a simply supported pouch conveyor. Taking the form of a generic design, several concepts were considered, as discussed in Chapter 2. As can be seen, both a force and indentation based approach is considered. For IRR, a force based system would typically be used. The option of an indentation based approach is more applicable to the drive traction, discussed later. From a design perspective, the indentation force is ordinarily comprised of three components. Firstly, the gravitational load on the idler roll from the belt and bulk material. Secondly, the tension distribution throughout curves must also be considered, and any additional loads as a result of this. Lastly, any force that is required to maintain closure of the belt must be allowed for. Care must be taken to consider the normal load only on the idler roll, not necessarily the vertical load. This is dependent on the arrangement of the pouch system, and the inclination angle of the idler rolls. From here, an iteration based model can be used to determine the indention rolling resistance for the given load. This approach is also applicable to pipe conveyors. With the exception of belt flexure, the remaining resistances of the system can largely be determined using existing theories applicable to troughed conveyors. This is due to the inherent similarities between the respective components. In many pouch configurations, the drag force due to belt and bulk solid flexure may be neglected, as discussed in section 6.3.2. This is a result of the reduced sag compared to troughed installations. 172 7.1.2 Drive System Using the resistances determined with this method, the belt and drive can be designed. Contrary to troughed installations which typically place a drive at the head or tail, a pouch system has the ability to employ virtually any drive location. As such, the permissible traction from the relevant drive wheel must be determined, based on the design of the belt support system. Building on the velocity dependent stress profile produced in Chapter 2, Chapter 3 offers a method to analyse the belt-wheel interface under different loading conditions. In order to overcome the static friction in the system, the breakaway torque is often the peak load during start-up. As such, the traction model is separated into two categories: incipient sliding, and tractive rolling. Incipient sliding refers to the permissible traction during the initial stages of start-up. As tension is accumulated in the belt, and the drive starts to move, sag is at its greatest and lubricants are most viscous, resulting in a demanding motor torque. As discussed in Section 3.2, as the applied torque increases, regions of stick and slip will develop within the interaction. The drives must be sized and spaced appropriately, to ensure the amount of slip within the contact doesn’t exceed an appropriate safety factor. Once the breakaway traction is determined for a particular design, the tractive rolling model is employed to ensure the design can accelerate the belt to the required velocity, within an appropriate timeframe. It should be noted that this timeframe will vary for each drive system, discussed below. The above procedure presents the designer with two values of traction. Incorporating an appropriate safety factor, the drive sizing, spacing and start-up procedure, can subsequently be determined. Depending on the layout of the system, the designer may choose to place drives at critical locations along the belt, as opposed to equal spacing. Areas of high tension during static operation can be determined using design standards such as DIN 22101, ISO 5048 or the CEMA Handbook. The tensile program discussed in Chapter 4 may also be used. These 173 methods give the added bonus of counterweight design. The designer may consequently choose the location of drives and counterweight, based on the tension distribution. 7.1.3 Dynamic Response during Starting and Stopping The system layout is now temporarily complete, pending a dynamic analysis. In order to perform this however, certain material and belt properties are required: - The counterweight pulley mass, given the mass of the counterweight required from the static analysis. - Idler roll diameter, spacing and reduced mass. - Elastic modulus of the belt carcass, as specified by ISO 9856. - DMA properties of the bottom cover of the belt. - Belt mass and bulk solid throughput. - Estimated drive pulley diameter. - Pouch configuration, as shown in Chapter 5 (if performing a pouch analysis). This information allows a preliminary dynamic analysis to be performed, with parameters modified should any results require adjustment. The pouch configuration determines an estimated propagation (and reflection) coefficient of the system during transient response. Lastly, the starting sequence and procedure for each drive station must be determined. As discussed extensively in Section 6.5, each drive station must be started in a way as to minimise the dynamic effects of the adjacent drives. To do this, the wave period between adjacent drives must be determined. This is done through a simulated impulse response, as outlined earlier. Knowing this, a preliminary starting procedure can be determined, such that the total distance travelled by each drive during the start-up period, is the same. This is crucial to preventing an uneven distribution of tension between drive systems. A dynamic simulation can now be performed, including stopping procedures, and the conveyor design modified where appropriate. In order to conduct a comprehensive system analysis, extreme cases must be considered, however unlikely. Section 4.4 discusses the 5 possible loading states when starting a conveyor. Ideally, the system will be empty upon starting and stopping, as this results in the least amount 174 of strain on the drive system. Under special circumstances however, uneven feed rates could result in one of the more difficult cases presented earlier; material only lying on the inclines. In this instance, the conveyor would need to be designed in order to overcome this high tractive state. Similarly, it would be required to stop in an adequate timeframe, if needed particularly with only the downhills loaded. 7.2 Summary This thesis offers an in-depth analysis of the major design considerations associated with pouch conveying systems. The thesis statement: “To investigate the dynamic implications of pouch conveyor systems incorporating multiple drive stations, associated design aspects, and how they differ from conventional troughed conveyors.” In order to comprehensively understand the transient behaviour of any belt system, the major resistances of the conveyor must be understood. As such, Chapter 2 saw a theoretical model developed to predict the indentation rolling resistance of a simply supported pouch conveyor system, which as stated, can be responsible for up to 60% of the total power usage. The model, based on the work of Rudolphi & Reicks [42] for troughed installations, modifies the indentation profile to suit a range of possible designs. When compared with experimental results, a consistent shortfall was noticed, of approximately 34%. Upon examination of this model, improvements were made to allow the model to incorporate strain variance in the indentation. This further improved the results by 5.7%, reducing the error to 28.3%. Further comparison, the work by O’Shea [36] defined the losses associated with the indirect integration of master curve data. Other inaccuracies arose from shear between the polymer strands, belt sag and friction between the belt and idler roll. The ability to transfer a tractive force to a pouch conveyor is perhaps the largest design difference compared to troughed systems. Point contact drives, utilised in pouch conveyors, generate much smaller amounts of traction, compared to their troughed counterparts, and therefore require more units. The amount of traction provided by this type of drive is the sole focus of Chapter 3. Utilising the pressure distribution presented in Chapter 2, a friction model 175 is developed to analyse the stick/slip portions of the contact region. This is subsequently compared to experimental ‘pull tests’, using ceramic, polyurethane, rubber and steel lagging. The comparison showed a direct correlation between the theory and experimental results. Of particular interest, was the observed reduction in the friction coefficient for higher normal loads. This phenomenon was later explained, through the work of Golden [11]. Chapter 4 saw the development of a dynamic software package, building on the work of Nordell et al [28] and Lodewijks [24]. This program, predicted the transient response of troughed systems, given the conveyor layout, idler roll parameters, belt and bulk material properties and drive configuration. Subsequently in Chapter 5, measurements were taken from a fabric site conveyor. These measurements, namely the drive torque, counterweight displacement and belt velocity, agreed well with the theoretical results Adapting this dynamic theory to pouch conveyors, raised several questions: - Does the suspended mass of the belt and bulk material affect the propagation of a transient wave through the belt? - Given the low contact area of a drive station with the belt, can a transient wave pass through a drive? If so, is there any superposition of waves from adjacent drives? - What is the optimum sequence to start multiple drive stations? In Chapter 5, impact tests were performed on a length of pouch conveyor, loaded at various rates. This was designed to confirm or refute the ability of a wave to pass through a drive station. The results showed a portion of the wave propagating through, as well as reflected from the contact surface. The degree of transmission was found to be dependent on the fill state of the pouch, as well as the indentation load of the drive system. Using this confirmation, the dynamic theory from Chapter 4 was modified surrounding the drive systems. A parallel chain was added to the model, suspended below a drive node, to represent the hanging mass. This mass was elastically bound to the adjacent nodes, and through shear to the drive node. Results from this model showed a distinct propagation, and reflected component of an oncoming wave. Knowing this, investigations were conducted into the starting sequence of multiple drive systems. By knowing the wave travel time between successive drive stations, it was shown that the optimum starting procedure begins with the 176 drive adjacent to the counterweight. Then, travelling against the velocity of the belt, each subsequent drive is started after a delay equal to the travel time of the wave. This allows the impact of the dynamic wave to assist the drive, being heavily absorbed in the process. The delayed starting of sequenced drive systems introduces another concern. If each drive station obeys the same velocity profile, a state of high tension will exist at the end of the procedure, given each drive has travelled a different distance. As such, the starting procedure of each system must be tailored so that each drive travels the same distance, at the end of the start time. This will result in the last drive experiencing the steepest acceleration profile. 7.3 Future Work The area of belt conveying will always require further research, in order to keep up with the increasing demand and expansion of the industry, as well as the newer complex conveying systems emerging. This author believes extensive research in the following areas is justified. 7.3.1 Static Friction and Sag Take-up The starting of a belt conveyor system is a function of the drag forces and terrain of the system, as well as the belt and bulk solid properties. This combination, as shown above provides a reasonable estimation of the dynamic response. As noted in Section 5.6 however, the breakaway case of a conveyor isn’t wholly understood. Upon the application of traction from a drive source, a chain reaction propagates throughout the system, until the whole system is in motion, as follows: - In the idler roll space adjacent to the drive station, tension will gradually increase. This will reduce the slack between the successive idler rolls, causing a load on the subsequent idler roll system. Once this load creates a tension gradient across this idler roll set, sufficient to overcome the static friction involved, it will begin to move. - Repeated in the next idler spacing. As can be seen, the internal friction of each idler roll set plays a significant role in the breakaway torque required by the motor, and consequently the initial movement of the counterweight. This internal friction is primarily due to the rim drag of the bearings, and the 177 static flexure of the belt and bulk solid, but is also dependent on friction between the idler roll and belt, as well as environmental conditions. The corollary is that the stopping process will have a similar effect. The deceleration of the belt will occur unevenly, as belt sections of different terrain and layout slow at different rates. The combination of these two processes results in an uneven distribution of tension and sag, accentuated through different loading, starting and stopping processes. As such, while the running properties of the system will remain consistent, the breakaway conditions, at this point in time, are difficult to predict, and simulate. The flexure of the belt and bulk solid during start-up is also worthy of research. The variances of flexure with tension and velocity are well understood, during steady state conditions [7, 53], however little is known of this interaction during breakaway and periods of acceleration. 7.3.2 The Inclusion of an Inertial Frame of Reference within the Dynamic Model The dynamic model presented in this thesis captures the transient behaviour of a conveyor system, under constant loading conditions. By this, it is assumed that the aggregate mass of the system is unchanged. As shown above, this has proven to provide accurate results for continually loaded or empty situations. Consider however, a system loaded according to Figure 4.16 (c)-(e). If upon starting a load only exists on the uphills of a conveyor, as the belt starts, this load will move and gradually exist on the decline, aiding the motion of the belt. Similarly, should a load only exist on a decline, as the conveyor begins to move, this load is transferred to the incline. This change in gravitational load of the system affects the transients in an unknown way. Similarly, should a conveyor be stopped completely full, loading will not restart until the conveyor is successfully back to its operating speed. This again results in a moving load in the system. The discharge of a conveyor induces a similar effect. The consequence of this behaviour is that extended simulations cannot be accurately performed, if they change the total mass of the system to a large extent. Thus, a moving 178 coordinate system should be integrated to account for this transition. The location of the drive would also need to be adjusted, as the material is theoretically discharged from the conveyor. 179 References 1. Cattaneo, C.: Sul Contatto di due corpi elastici: distribuzione locale degli sforzi, Rendiconti dell’ Accademia nazionale dei Lincei, 1938. 2. CEMA: Belt Conveyors for Bulk Materials, 6th Edition, 2005, pp 115-117. 3. 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