THEORY OF MACHINES (TOM) MEEG 206 Theory of Machines (TOM) Lecture – 01 Basic Concepts Terms and Definitions THEORY OF MACHINES (TOM) MEEG 206 • Machines • Devices used to alter, transmit and direct forces to accomplish a specific objective. • transforms the available energy into some useful work e.g. Chain saw • Mechanisms • Mechanical portion of machine containing systems of links that transmits motion and forces from a power source to desired output. THEORY OF MACHINES (TOM) MEEG 206 Kinematic analysis Kinematic analysis provides insight into significant design questions: •What is the significance of the length of the legs that support the platform? • Is it necessary for the support legs to cross and be connected at their midspan, or is it better to arrange the so that they cross closer to the platform? • How far must the cylinder extend to raise the platform 8 in? Dynamic force analysis of the platform could provide insight into another set of important design questions: •What capacity (maximum force) is required of the hydraulic cylinder? •Is the platform free of any tendency to tip over? •What cross-sectional size and material are required of the support legs so they don’t fail? Terms and Definitions THEORY OF MACHINES (TOM) MEEG 206 • Links • Rigid structure connected with other links to transmit motion and forces. • E.g. connecting rod, piston, crankshaft • Flexible links are ignored during kinematic analysis but must be included during dynamic force analysis. • Joints • Movable connection between links that allows relative motion between links. • Primary joints are revolute and sliding joints THEORY OF MACHINES (TOM) MEEG 206 Types of joints On the basis of number of links • Binary joint • When two links are connected at same joint • Ternary joint • When three links are connected at same joint • Quaternary joint • When four links are connected at same joint On the basis of nature of contact • Higher Pair • Line or Point Contact (Example: Gear, Cam mesh) • Lower Pair: • Area Contact (Example: Slider) THEORY OF MACHINES (TOM) MEEG 206 Joints THEORY OF MACHINES (TOM) MEEG 206 Kinematic drawing • The representation of machine drawing in skeleton form so that the links and joints that influence the motion of the mechanism are shown. Machine drawing Kinematic drawing THEORY OF MACHINES (TOM) MEEG 206 Symbols Used in Kinematic Diagrams (Links) THEORY OF MACHINES (TOM) MEEG 206 Symbols Used in Kinematic Diagrams (Links) THEORY OF MACHINES (TOM) MEEG 206 Kinematic Diagram of Machine Figure 1.9 shows a shear that is used to cut and trim electronic circuit board laminates. Draw a kinematic diagram. Kinematic Diagram of Machine THEORY OF MACHINES (TOM) MEEG 206 Figure 1.11 shows a pair of vise grips. Draw a kinematic diagram. THEORY OF MACHINES (TOM) MEEG 206 Kinematic Diagram of Machine THEORY OF MACHINES (TOM) MEEG 206 Kinematic Diagram of Machine THEORY OF MACHINES (TOM) MEEG 206 Kinematic Diagram of Machine THEORY OF MACHINES (TOM) MEEG 206 Kinematic Diagram of Machine THEORY OF MACHINES (TOM) MEEG 206 Kinematic Diagram of Machine Gruebler’s Equation THEORY OF MACHINES (TOM) MEEG 206 Degree of Freedom (DOF) = 3(n - 1) - 2jp - jh Where: n = Number of Link jp = Number of Lower Pair Joint jh = Number of Higher Pair Joint Note: Number of DOF implies that the number of driver required to get desired motion for the mechanism is equal to DOF. THEORY OF MACHINES (TOM) MEEG 206 Calculation of DOF Figure 1.14 shows a toggle clamp. Draw a kinematic diagram, using the clamping jaw and the handle as points of interest. Also compute the degrees of freedom for the clamp. n = 4, jp = 4 pins, jh = 0 DOF = 3(n - 1) - 2jp - jh = 3(4 - 1) - 2(4) - 0 = 1 THEORY OF MACHINES (TOM) MEEG 206 Calculation of DOF Figure 1.16 shows a beverage can crusher used to reduce the size of cans for easier storage prior to recycling. Draw a kinematic diagram, using the end of the handle as a point of interest. Also compute the degrees of freedom for the device. n = 4, jp = (3 pins + 1 slider) = 4, jh = 0 DOF = 3(n - 1) - 2jp - jh = 3(4 - 1) - 2(4) - 0 = 1 THEORY OF MACHINES (TOM) MEEG 206 Calculation of DOF Figure 1.18 shows another device that can be used to shear material. Draw a kinematic diagram, using the end of the handle and the cutting edge as points of interest. Also, compute the degrees of freedom for the shear press. n = 3, jp = 2 pins, jh = 1 gear DOF = 3(n - 1) - 2jp - jh = 3(3 - 1) - 2(2) - 1 = 1 THEORY OF MACHINES (TOM) MEEG 206 Calculation of DOF Figure 1.21 shows an outrigger foot to stabilize a utility truck. Draw a kinematic diagram, using the bottom of the stabilizing foot as a point of interest. Also compute the degrees of freedom. n = 4, jp = (3 pins + 1 slider), jh = 0 DOF = 3(n - 1) - 2jp - jh = 3(4 - 1) - 2(4) - 0 = 1 THEORY OF MACHINES (TOM) MEEG 206 Calculation of DOF Figure 1.26 presents a lift table used to adjust the working height of different objects. Draw a kinematic diagram and compute the degrees of freedom. n = 6, jp = (5 pins + 2 slider), jh = 0 DOF = 3(n - 1) - 2jp - jh = 3(6 - 1) - 2(7) - 0 = 1 THEORY OF MACHINES (TOM) MEEG 206 Calculation of DOF Figure 1.29 shows a mechanical press used to exert large forces to insert a small part into a larger one. Draw a kinematic diagram, using the end of the handle as a point of interest. Also compute the degrees of freedom. n = 6, jp = (6 pins + 1 slider), jh = 0 DOF = 3(n - 1) - 2jp - jh = 3(6 - 1) - 2(7) - 0 = 1 THEORY OF MACHINES (TOM) MEEG 206 Calculation of DOF Number of Driver and DOF for desired motion C THEORY OF MACHINES (TOM) MEEG 206 C 2 2 3 B A 1 4 D C C' D 1 A 4 1 5 B D 1 A DOF = 1 3 Number of Driver required = 1, 2 To get2' determined motion3' B 3 E D' 4' 4 1 5 E DOF = 2 Number of Driver required = 2, To get determined motion 4 THEORY OF MACHINES (TOM) MEEG 206 C 2 3 B A 1 4 DOF=3(n-1)-2PL-Ph=33 D -24 =1 THEORY OF MACHINES (TOM) MEEG 206 C 3 2 B D 1 A 4 1 5 E DOF=3(n-1)-2PL-Ph=34 -2 5 =2 4 THEORY OF MACHINES (TOM) MEEG 206 3 DOF=3(n-1)-2PL-Ph=32 - 22 -1=1 2 1 THEORY OF MACHINES (TOM) MEEG 206 Conditions for a Mechanism to have a Determined Motion Independent parameters are supplied only by the driving links. Since the driving links are always connected to the frame by lower pairs, one driving link (together with the corresponding lower pair) will provide only one independent parameter. Therefore, the condition for a mechanism to have a determined motion is: 1. DOF1 2. DOF= number of the driving links. THEORY OF MACHINES (TOM) MEEG 206 DOF=3(n-1)-2PL-Ph =34 -26=0 DOF=3(n-1)-2PL-Ph =32 -23=0 2 1 2 1 C 5 3 D A 3 E 4 If DOF=0, it is a truss (Structure). F THEORY OF MACHINES (TOM) MEEG 206 If the number of driving links is greater than the DOF of the mechanisms, the kinematic chain cannot move due to the conflict between the input forces. In extreme cases, the weakest link in the chain may be broken. C 2 3 B A 1 4 C D THEORY OF MACHINES (TOM) MEEG 206 If the number of driving links is less than the DOF of the mechanisms, some driven links will not have determined motion. C 3 2 B DOF=3(n-1)-2PL-Ph=34 -2 5 =2 D 1 A 4 1 5 E 4 THEORY OF MACHINES (TOM) MEEG 206 Compound Hinge (Multiple Joint) If the axes of two or more revolutes coincide, they constitute a compound hinge. 2 11 3 The number of revolutes in a compound hinge is equal to one less than the number of links joined at that hinge. (a) (a) 11 22 33 (b) (b) THEORY OF MACHINES (TOM) MEEG 206 Note: In many cases, compound hinge does not show clearly. Attention must be paid to it. THEORY OF MACHINES (TOM) MEEG 206 Note: In many cases, compound hinge does not show clearly. Attention must be paid to it. 4 3 1 1 2 2 3 3 3 2 1 2 1 THEORY OF MACHINES (TOM) MEEG 206 All machine elements belonging to the same link in the kinematic diagram must be connected firmly by welding symbols. THEORY OF MACHINES (TOM) MEEG 206 THEORY OF MACHINES (TOM) MEEG 206 Is there compound hinge here? 3 2 1 THEORY OF MACHINES (TOM) MEEG 206 Calculate the DOF of the gear-linkage mechanism. In the mechanism, there are three moving gears(1, 2, and 3) and two moving rods (4 and 5). 2 4 1 B A E C F 5 D 6 3 THEORY OF MACHINES (TOM) MEEG 206 C is a compound hinge of links 2, 4, 5 D is a compound hinge of links 3, 5, 6. DOF=3(n-1)-2PL-Ph=35 -26 -2=1 2 4 1 B A E C F 5 D 6 3 Passive DOF(Local DOF) B THEORY OF MACHINES (TOM) MEEG 206 B 2 1 O 2 1 A A O 3 3 DOF=3(n-1)-2PL-Ph=33-23-1=2 DOF=3(n-1)-2PL-Ph=32 -22 -1=1 33-231=2 THEORY OF MACHINES (TOM) MEEG 206 Passive DOF B B 2 1 O 2 1 A A O 3 3 The DOF which does not change the output motion of the mechanism is called a passive DOF. Any passive DOF should be deleted by welding the roller to the follower before the calculation of the DOF of mechanism. THEORY OF MACHINES (TOM) MEEG 206 Passive DOF DOF=3(n-1)-2PL-Ph=32 -22 -1=1 33-23-1=2 THEORY OF MACHINES (TOM) MEEG 206 Redundant Constraints (void constraint) have exactly the same kinematic function as other constraints. Redundant constraints must not be counted during the calculation. (1)When two links are connected by more than one parallel sliding pair, only one sliding pair can be counted during the F calculation, others are redundant constraints and must not be counted. THEORY OF MACHINES (TOM) MEEG 206 For the plate cam with translating flat-faced follower, if there are two parallel sliding pairs between the follower and the frame, then one of the guide-ways is a redundant constraint DOF=3(n-1)-2PL-Ph =32 -22 -1=1 32-23-1= -1 THEORY OF MACHINES (TOM) MEEG 206 (2) When two links are connected by more than one revolute pair whose axes coincide, A A' DOF=3(n-1)-2PL-Ph=31 -21=1 31-22= -1 THEORY OF MACHINES (TOM) MEEG 206 (3) When two links are connected by more than one higher pair whose common normals passing through the points of contact coincide, DOF=3(n-1)-2PL-Ph =32 -22-1=1 3 2 -22-2=0 A 1 n O 3 2 n B THEORY OF MACHINES (TOM) MEEG 206 (4) When the distance between two points on two links remains constant during the motion of the mechanism, adding one link and two revolutes with their centers at these two points will create a redundant constraint. DOF=3(n-1)-2PL-Ph=33 -24=1 34 -26=0 2 B C E 1 A AB 4 CD 3 F BC AD D BE AF 1 A C 2 B 5 4 F 3 D CD THEORY OF MACHINES (TOM) MEEG 206 THEORY OF MACHINES (TOM) MEEG 206 Note: (1) Redundant constraints can improve the rigidity of a mechanism, improve the force condition in links, etc. and are widely used. (2) Deleting the redundant constraints during the calculation of DOF does not mean that the redundant constraints should be omitted from real mechanisms. (3) All redundant constraints require some special dimensions. Therefore, attention should be paid to manufacturing accuracy when any redundant constraint is used. 2 THEORY OF MACHINES (TOM) MEEG 206 B 1 A C E 1 3 4 A F F CD AB EFDOF=3(n-1)-2P EF CD AB-24=1. L-Ph=33 1 A E 2 B C 5 F 4 C 5 4 D E 2 B 3 D DOF=3(n-1)-2PL-Ph=33 -24=0. truss!! It is a 3 D THEORY OF MACHINES (TOM) MEEG 206 C 1 A F B 3 D 4 5 E 2 C 1 A F DOF=3(n-1)-2PL-Ph =33 -24=1 3 D 4 5 E DOF=3(n-1)-2PL-Ph =34 -26=0 It is a truss!! THEORY OF MACHINES (TOM) MEEG 206 Calculate the DOF of the mechanism shown below. THEORY OF MACHINES (TOM) MEEG 206 Solution: (1)The spring should not be counted. (2) C is a compound hinge of links 2, 3, 4. (3) The roller has a passive DOF. (4) There are two parallel sliding pairs E and E between the frame 8 and the slider 6. One of the sliding pairs is redundant. C 2 B 1 3 D A E 4 E' 6 G F 7 O 5 8 THEORY OF MACHINES (TOM) MEEG 206 After the mechanism is redrawn, DOF=3n-2PL-Ph=37 -29 -1=2. Needs two drivers. C 2 B 4 3 1 E' D A 6 E 8 5 4 3 1 A 7 O C 2 B G F E D G F 6 7 O 5 8 THEORY OF MACHINES (TOM) MEEG 206 THEORY OF MACHINES (TOM) MEEG 206 Calculate the DOF of the mechanism. THEORY OF MACHINES (TOM) MEEG 206 (1) C is a compound hinge of links 4, 5, 7. (2) A is a compound hinge of links 1, 2, 7. (3) The roller has a passive DOF. After the roller is welded with the link 6, there is still a revolute pair between links 5 and 6. DOF=3n-2PL-Ph=36 -28 -1=1 B 1 6 A 2 D 5 C 3 E 4 7 F THEORY OF MACHINES (TOM) MEEG 206 AB EC FD. Calculate the DOF of the mechanism. H C D B G E F K A = This implies parallel and equal THEORY OF MACHINES (TOM) MEEG 206 Link EC, revolute E, and revolute C create a redundant constraint. remains 6 moving links. H C D 4 2 B G 3 5 E F K 6 7 1 A THEORY OF MACHINES (TOM) MEEG 206 D is a compound hinge of links 2, 3, 4. H C D 4 2 B G 3 5 E F K 6 7 1 A THEORY OF MACHINES (TOM) MEEG 206 The roller has a passive DOF. There is only one higher pair between links 4 and 5. H C D 4 2 B G 3 5 E F K 6 7 1 A THEORY OF MACHINES (TOM) MEEG 206 There is only one higher pair between cam 6 and link 5. H C D 4 2 B G 3 5 E F K 6 7 1 A THEORY OF MACHINES (TOM) MEEG 206 DOF=3n-2PL-Ph=36 -27 -2=2. Needs two drivers. H C D 4 2 B G 3 5 E F K 6 7 1 A