PROJECT MANAGEMENT & DESIGN II MAE 435 Rocket Project Final Report April 23rd, 2015 Group Members: Nathan Akers Class Supervisors: Dr. Sebastian Bawab Chad Bagley Dr. Colin Britcher Paul Campbell Mr. Michael Polanco Charles Juenger Brian Mahan Chris Skiba Michael Weber Michael Wermer Project Advisor: Dr. Thomas Alberts i Contents Table of Figures ............................................................................................................................. iii Nomenclature .................................................................................................................................. v Abstract ......................................................................................................................................... vii 1. Introduction ............................................................................................................................. 1 2. Literature Review .................................................................................................................... 1 3. Methods ................................................................................................................................... 4 3.1 Structural Design .............................................................................................................. 4 3.2 Fuel and Motor Design ..................................................................................................... 7 3.3 Nozzle Design ................................................................................................................ 10 3.4 Avionics Bay Design ...................................................................................................... 13 3.5 CFD Analysis ................................................................................................................. 14 3.6 Finite Element Analysis ................................................................................................. 15 3.7 Vibration Analysis.......................................................................................................... 16 3.8 Snatch Force Analysis .................................................................................................... 17 3.9 Ejection Charge Analysis ............................................................................................... 18 3.10 Rocket Manufacture ....................................................................................................... 18 3.11 Fuel Manufacture ........................................................................................................... 19 3.12 Motor Burn Testing ........................................................................................................ 21 ii 3.13 Wind Tunnel Testing...................................................................................................... 23 3.14 Launch Testing ............................................................................................................... 25 4. Results ................................................................................................................................... 25 5. Discussion .............................................................................................................................. 32 6. Appendix ............................................................................................................................... 36 6.1 Rocket Drawings ................................................................................................................. 36 6.2 Pictures ................................................................................................................................ 44 6.3 Tables and Plots .................................................................................................................. 47 6.4 Fuel and Combustion Chamber Calculations ...................................................................... 52 6.5 Gantt Chart .......................................................................................................................... 57 6.6 Budget ................................................................................................................................. 58 7. References ............................................................................................................................. 59 iii Table of Figures Figure 1: Rocket Components…………………....………………………...………...…………...5 Figure 2: Rocket Assembly in Inches………………………………………………...…………...6 Figure 3: Avionics Bay Components…………………………………………………………….36 Figure 4: Rocket Upper Body Components…………………………...………………...……….37 Figure 5: Rocket Lower Body Components.………..…………………………………...………38 Figure 6: Rocket Motor Components.…………………………………………………...………39 Figure 7: Propellant Published Values.………………...…………...…………….……...………47 Figure 8: Combustion Chamber Interior.………………………………………………...………..8 Figure 9: Kn vs Web Regression.…………………...…………………………….……...……..…9 Figure 10: Chamber Pressure vs Time……….…….……………………...………...…......…….10 Figure 11: Combustion Chamber with Converging-Diverging Nozzle……….…….…...………10 Figure 12: Velocity Contour Plot.……………………………………………..…….…...………12 Figure 13: Temperature Contour Plot.………………………………………………………...…13 Figure 14: Dual Deployment Wiring Diagram.…………………………...…..………...……….40 Figure 15: Avionics Bay Concept Sketch.………………………..…..………..………...………41 Figure 16: 3D Model of Avionics Bay Interior.……………...………………..………...………42 Figure 17: Coefficient of Drag vs Mach.………………………….…………..………...……….15 Figure 18: Vibration Analysis Input Parameters.…….………………………..………...………17 Figure 19: Amplitude vs Frequency Input Parameters.………………………..………...………17 Figure 20: Rocket Test Stand.……………………..…………………………..…….…...………23 Figure 21: Rocket Test Stand Components.……………………….…………..………...………43 iv Figure 22: Subsonic Wind Tunnel Diagram.………,………...……...………..…….…...………24 Figure 23: Wind Tunnel Mounted Model.………………...…………………..…….…...………24 Figure 24: Thrust vs Time.…...………………………………………………..…….…...………26 Figure 25: Predicted Rocket Motor Performance.………………………..…...…….…...………47 Figure 26: Predicted Rocket Flight Performance.……………………………..………....………48 Figure 27: Rocket Assembly Predicted Displacement ……………………………...…...………27 Figure 28: Avionics Bay Predicted Displacement……………………………..………...………28 Figure 29: Vibration Analysis Results.………………………………………...………...………49 Figure 30: Avionics Bay Deformation at a Frequency of 926.775 cycles/s…..………...….……29 Figure 31: Frequency vs Standard Deviation Maximum Displacement…………..…….…...…..29 Figure 32: Rocket Failure…….………………………………………………..………...………44 Figure 33: Thrust Force vs Time.………………………….…………………..………...………30 Figure 34: Recovered Combustion Chamber Pieces.……………………...…..………...………44 Figure 35: Design and Burst Pressures for Rocket Motor Casing..………..…..………...………50 Figure 36: Fin Vortices.……...………………………………………………..………...……….31 Figure 37: Rocket Wake……...………………………………………………..………...………45 Figure 38: Rocket Nose Cone Boundary Layer…………………...…………..….……...………46 Figure 39: Test Flight Data..…………………………….……………………..………...………51 v Nomenclature r Propellant Burn Rate P0 Combustion Chamber Pressure Kn Propellant Burn Area to Nozzle Throat Area Ratio p Propellant Density Pexit Nozzle Exit Pressure Patm Atmospheri c Pressure M throat Nozzle Throat Mach Number Me Nozzle Exit Mach Number T F Thrust V exit m Mass Flow Rate A* Nozzle Throat Area * Nozzle Exit Velocity D De Nozzle Throat Diameter Nozzle Throat Diameter R Gas Constant T0 Combustion Chamber Temperatur e k Ratio of Specific Heats m m noz Mass Flow Rate X* Throat Position osi ci Outer Surface Inhibited (0 inhibited, 1 exposed) Core Surface Inhibited (0 inhibited, 1 exposed) ei dio Ends Surface Inhibited (0 inhibited, 1 exposed) Initial Core Diameter do lg o Initial Grain Diameter Initial Grain Length n Ab Number of Grains Propellant Burning Area At Propellant Throat Area two dto Initial Grain Web Thickness Propellant Initial Throat Diameter a Burn Rate Coefficien t at P0 G Kv Erosive Burning Factor Propellant Erosive Burning Velocity Coefficien t dc Combustion Chamber Diameter vi Aduct Flow Area G* Erosive Burning Threshold (G * 6) i Initial Time Step j v grain Subsequent Time Step Grain Volume v free Free Volume in Combustion Chamber m grain Grain Mass m gen Mass Generation of Combustion Products m sto Mass Storage Rate of Products in Combustion Chamber mass sto Mass of Combustion Products Remaining in Chamber sto Density of Combustion Production in Combustion Chamber Fty Yield Strength (ksi) Ftu Ult imate Tensile Strength (ksi) E Modulus of Elasticity (ksi) Poisson Ratio D0 Combustion Chamber Outside Diamter t Combustion Chamber Wa ll Thickness SD Combustion Chamber Design Safety Factor Fty / Ftu B Combustion Chamber Burst Factor PD Combustion Chamber Design Pressure (ksi) PU Combustion Chamber Burst Pressure (ksi) AE Nozzle Exit Area Cf Thrust Coefficien t nox Isentropic Efficiency of the Nozzle t Time Interval It Average Impulse Over Given Time Interval vii Abstract The goal of the project was to design a high-powered rocket capable of deploying a cosmic ray detector at an apogee of 10,000 feet. Detailed designs of the airframe, propulsion system, aerodynamic fins, avionics, and recovery system was developed using computer aided drafting software (CAD). Structural stability was determined through finite element analysis (FEA). The propulsion system is powered by a Potassium-Nitrate (KNO3) - Dextrose (C6H12O6) solid propellant. The expected thrust for the fuel was calculated and an optimized nozzle was designed to accommodate the combustion chamber conditions. Aerodynamic analysis was performed through modeled computational fluid dynamics (CFD) to determine coefficient of drag versus Mach number. Scaled rocket subsonic wind tunnel testing was performed to determine the accuracy of the CFD analysis. The avionics system autonomously records in-flight data and controls the deployment of the payload, the drogue and main parachutes, ensuring safe recovery after flight. Theoretical vibration analysis was performed to minimize determine if there were any harmful effects to the electronics. The analysese and testing performed allowed the team to extract the necessary information to validate the design and ensure a safe and reliable flight. 1 1. Introduction Rockets are a rapid means of payload delivery. Rocket engines operate at high thrusts, mass flow rates, temperatures, and pressures, with added complexity from multiple staging. The fuel is often very volatile which can lead to explosions if the wrong part, operating close to the design limit, fails. This is why rockets are designed based upon the payload. The requirements of this project are to design a high-powered rocket capable of deploying a five pound cosmic ray detector at 10,000 feet. These design requirements are the same as the SEDS (Students for the Exploration and Development of Space) 2014 Rocketry Competition Rules. However, the 2015 competition goals changed dramatically from the previous year’s goal, by not requiring a payload. The decision was made to not participate in the 2015 competition, and the rocket was built using the 2014 goal because the primary customer, the local SEDS chapter, is building the cosmic ray detector to be deployed. The rocket design was divided into four key areas necessary to satisfy the goal of the project. These included: structures, propulsions, aerodynamics, and avionics. 2. Literature Review A high-powered rocket is any rocket weighing more than 53 oz. and using a motor with more than 160 Newton-seconds total impulse and no more than 40,960 Newton-seconds or a having average thrust of 80 Newtons. CAD software is used to create an accurate visual representation of the rocket and its components. The representation is used to perform analysis that determines design feasibility prior to construction [1]. Finite Element Analysis (FEA) is a method for determining stress and strain properties on complex solid bodies due to applied forces [2-4]. FEA is used on the rocket assembly to design for structural integrity and to verify material 2 selection [5]. The analysis is needed to determine if the rocket design is capable of successful flight. Every new rocket design needs to be tested for structural integrity. Rocket motors are thrust engines which operate by generating a pressure and momentum thrust. This is achieved by combusting fuel inside a pressure vessel and expanding the combustion products through a nozzle. This is typically accomplished in small-scale rockets by using solid rocket fuels. Commercially available “high-power” hobby rocket motors are typically made of Ammonium Perchlorate, HTPB and Aluminum, which is similar to the formulations used in rockets designed for space flight as well as missiles. These formulations, commonly referred to as APCP (Ammonium Perchlorate Composite Propellant), provide a high specific impulse relative to other solid formulations. Specific Impulse is a key concept in the design and of a rocket motor because it is used to describe the efficiency of a fuel based on its weight. Specific Impulse is the ratio of the amount of thrust a fuel is able to produce to the weight flow of the propellants [6]. One drawback to using an APCP is the cost to produce; the material cost is high and the manufacturing process can be difficult. An alternative to APCP which is commonly used in homemade rockets is Potassium Nitrate-Sugar propellant, sometimes referred to as “RCandy”. This formula is easy to manufacture using readily available materials. The largest drawback to using these “R-Candy” formulations is the relatively low specific impulse they produce; meaning to reach a given altitude, more fuel is required. These formulations can be modified using different sugars such as sucrose, sorbitol, or dextrose, which can impact the material properties of the casted propellant and influence the ease of manufacturing and the reliability of the motor itself. To achieve maximum propulsive efficiency, a rocket motor must sustain a high flow of heavy particles and elevated combustion temperatures while maximizing the speed of gases 3 through the nozzle exhaust such that the exit pressure is equal to the atmospheric pressure [7, 8]. Exponentially increasing temperatures in the combustion chamber can cause rapid deformation of the materials of the nozzle and chamber [7]. Combustion gas expansion induces high internal pressures within the combustion chamber, which further exacerbates component deformation. Gases that travel through the nozzle carry solid fuel particles, which cause ablation to the throat section of the nozzle and agglomeration in the nozzle cone [9-11]. This results in decreased efficiency by altering the nozzle inner dimensions. Additionally, heat transfer in the chamber must be assessed to determine the effects on deformation of the combustion chamber walls and to select components such as O-rings that can withstand the heat[12]. The inability to accommodate these operating conditions results in performance inefficiencies as well as potential component failure. Determination of a nozzle design, which accommodates maximum operating conditions, will increase performance and longevity while reducing cost. Aerodynamics is examined to reduce drag and shock effects. Shapes of nose cones and fins have been examined in an effort to reduce drag and shock effects [13-18]. Nose cone designs showed that increased ratios of nose cone length to body length decreased drag, and reduction of the angle of the nose cone resulted in a smaller shock transition between the nose and the rocket body during supersonic flight [14, 15]. Fin designs were primarily focused on the flight stability, but as the size and bluntness of the leading edge increased, the drag and shock effects increased [13, 16-18]. While the basic aerodynamics of fins and nose cones are known, the final fin designs and nose cone designs are modified for every rocket based upon mass, and length of the rocket in order to move the center of pressure aft of the center of mass. This ensures flight stability by creating a restoring moment if the flight angle of attack is small. If the angle of attack becomes too large the rocket will become unstable and tumble, but the angle of attack will only 4 become too large if there is a motor failure or there are sudden large gusts of wind. Highpowered rockets can only be launched in wind less than 20 miles per hour. The avionics bay of a model rocket is typically positioned in the center section of the rocket, between the forward nose cone and rear motor section. The avionics bay must be designed to protect the rocket’s sensitive electronics, both from in-flight forces and impact from landing. The National Aeronautics and Space Administration (NASA) identifies four in-flight forces: weight, thrust, lift and drag [19]. The avionics bay must be designed to properly house and support the sensitive electronics, and mitigate the damaging effects from these in-flight forces. A rapid shock is an additional force the rocket could experience upon landing. The purpose of the parachute recovery system of the rocket is to lessen this shock load. The electronics mounting and structural design of the avionics bay must account for both the in-flight forces and the shock load upon landing. The maximum in-flight forces and shock load are unique to every rocket because every rocket has a different thrust which affects the in-flight forces and a different mass which affects the shock load. 3. Methods 3.1 Structural Design A 3D model of the rocket was created using Autodesk Inventor (Autodesk Inc., San Rafael, CA) and was developed by designing part assemblies of the different rocket components. These components (Figure 1) were then joined together to produce the main rocket assembly (Figure 2). 5 Figure 1: Rocket Components The first part assembly designed was the avionics bay (Figure 3), which houses the electrical equipment and connects the top and lower portions of the rocket. The ends are fitted with two circular end caps and are secured to the tubular housing by two threaded rods with accompanying fastener hardware. Eyebolts are fitted to the center of the two end caps by fasteners. These bolts serve as an anchor point for the shock chords after in-flight rocket separation has occurred. The nose cone (Figure 4) was designed with a hollow cavity surrounded by the nose cone walls. A circular end cap with a U-bolt and the accompanying fasteners are fitted to the end of nose cone. The U-bolt serves as another anchor point for the top shock chord. Connecting the nose cone and the avionics bay is the upper rocket body (Figure 4). 6 The lower rocket body is a tube fitted with six fins (Figure 5). The engine assembly is made up of a nozzle, outer casing, resin liner and end cap (Figure 6). This is attached to the lower rocket body using six circular motor mounts attached to inner walls of the bottom rocket body. Connecting the avionics bay to the nose cone are Kevlar® shock chords. Connecting the nose cone to the upper body and the avionic bay to the lower body are three shear pins for each connection. The avionics bay is connected to the upper rocket body using three riveted joints. Figure 2: Rocket Assembly in Inches 7 3.2 Fuel and Motor Design The rocket is powered by an “experimental” fuel, meaning that the motor utilizes a homemade propellant formulation in lieu of a commercially available motor. The formulation of the fuel was taken from Richard Nakka’s Experimental Rocketry Website [20]. The need to select a pre-investigated formulation was necessitated by the compressed budget and timeline of this project as there were not enough resources available to develop a unique formulation of APCP which could function safely and reliably. The propellant selected for this project was a mixture of 65% by mass Potassium Nitrate (KNO3) and 35% by mass Dextrose (C6H12O6) [21]. C6H12O6 + 3.31 KNO3 2.116 CO2 + 2.300 CO + 4.512 H2O + 1.424 H2 + 1.655 N2 + 1.585 K2CO3 + 0.133 KOH (Equation 1) (theoretical combustion reaction at 68atm [21]) This formulation has been tested extensively and is commonly used by high-power rocketry hobbyists. This relatively inexpensive formula has been proven to be safe and reliable while being relatively easily manufactured, and produces results within a reasonable margin of predicted results calculated by using the published values (Figure 7). The preliminary motor design has been modified using the Microsoft Excel (Microsoft Corporation, Redmond, WA) spreadsheet (“SRM_2014”) published by Richard Nakka [22]. This spreadsheet functions in three phases – calculating Kn and throat area, calculating chamber pressure and burn time and finally calculating thrust and impulse. Each of these functions occurs in an iterative process using a linear regression of the fuel from the inner core to the outer surface to calculate the next parameter. The process then continues using the data obtained from the previous iteration until 8 the regression of fuel is complete. This process is detailed step by step in the Appendix 6.5, in order of the operations as they occur. A target maximum operating pressure within the combustion chamber (Figure 8) of 1000 pounds per square inch (psia) was based on the final as built nozzle diameter of 0.625”. The motor casing was constructed from 3.0” outside diameter, 6061-T6 Aluminum tubing with a wall thickness of 0.125”, giving an inside diameter of 2.75”. The predicted maximum operating pressure of 1027psia gave a conservative factor of safety of 3.001. This conservative value was selected because of the need to re-use the motor casing multiple times and the relatively lowimpact of the casing weight on the rocket performance and design. Figure 8: Combustion Chamber Interior 9 The size of the motor was scaled to deliver the rocket as close to an apogee of 10,000 feet as possible, without exceeding that limit. The operating pressure is achieved by modifying the Kn value, which is the ratio of burning surface area of the propellant to the area at the throat of the nozzle [23]. This was done by selecting a commercially available propellant casting mold and modifying the nozzle diameter to meet the required Kn value. The KNDX propellant burns progressively in from the center core, causing the exposed surface area to increase (Figure 9), to a maximum Kn value of 342, resulting in a max pressure in the chamber of 1027psia (Figure 10). Before ignition, the “Kn” value is 276; this is favorable because a higher Kn value allows for more reliable ignition. Ignition tests using black powder charge igniter have shown that the motor should ignite completely and reliably. SRM 2014.XLS Kn vs Web Regression 400 25 350 250 Kn 15 200 10 150 100 5 50 0 0 0 2 4 6 8 10 12 14 16 18 20 22 24 26 Web Regression (mm) Kn Kn max Kn min Kn avg web thickness 342 276 320 Figure 9: Kn vs Web Regression web thickness 20 300 10 Pmax = t burn = t thrust = 1027 1.774 1.849 psi s. s. Figure 10: Chamber Pressure vs Time 3.3 Nozzle Design The nozzle (Figure 11) was assumed to be isentropic, and compressible flow equations were used to determine the thermodynamic properties at the subsonic, sonic, and supersonic regions. Figure 11: Combustion Chamber with Converging-Diverging Nozzle 11 The properties at each station and the constraint of three inches on the converging section were used to determine the dimensions of the nozzle. To reach maximum propulsive efficiency, the exit gases must be expanded through the diverging section of the nozzle so that Pexit equals Patm. The mass flow rate in the nozzle was determined by (Equation 2) where g -1 2g RTc é Pe ù g Ve = ê1- ú g -1 ë Pc û (Equation 3). Using the mass flow rate and exit velocity, the throat diameter (Equation 4) and 𝑋 ∗ is a function of the ratio was determined using the equation g +1 æ 2 ö 2(g -1) of specific heats X * (g ) = g ç (Equation 5). Next, the nozzle exit diameter was found ÷ è g +1 ø g +1 é ù 2 êæ Pc ö 2(g -1) ú by first finding the Mach number at the exit by M e = -1ú (Equation 6). ç ÷ g -1 êè Pe ø êë úû Substituting the Mach number into the equation for area ratios provided the diameter for g +1 ö æ 2(g -1) é ù æ ö æ ö 4ç 1 2 g -1 2 ÷ the exit De = A* ç M e ÷ú êç ÷ç1+ ÷ (Equation 7). A simulation of the nozzle øû p ç M e ëè g +1 øè 2 ÷ è ø was run at the determined area and pressure ratios using the CD Nozzle Simulator (engApplets, Blacksburg, VA). Using Simulation CFD, (Autodesk Inc., San Rafael, CA) a CFD analysis was performed on the nozzle with the inlet conditions found in the fuel analysis. The assumptions were a fluid of air, compressible flow, sea level standard air conditions, steady flow, and no heat transfer. CFD analysis was run on the nozzle to supply velocity (Figure 12) and temperature (Figure 13) contour plots of operating conditions. This was used to reveal shockwaves that may occur in the 12 diverging section of the nozzle, with the understanding that this does not provide conclusive proof. Basic CFD packages such as this run blending algorithms that may smooth out smaller shocks. The effects of stress on the nozzle were modeled using FEA with a special attention was paid to the stress concentrations around the screws attaching the nozzle to the combustion chamber. Also, having an approximation for the heat through the nozzle walls helps select the proper materials for the two sets of O-rings at the front and rear of the motor. Hand calculations of the stress concentration also provided additional validations for the models. It was important to build a motor system that would handle the predicted operating conditions for several launches and still be light enough to allow for maximum payload weight. Figure 12: Velocity Contour Plot 13 Figure 13: Temperature Contour Plot 3.4 Avionics Bay Design The avionics system consists of a redundant, dual-deployment altimeter system capable of ejecting the drogue and main parachutes. The wiring diagram for the dual-deployment altimeters is seen in the appendix (Figure 14). The dual-deployment altimeters are programmed to deploy two separate parachute ejection charges that will allow failure of the rocket airframe shear pins. The ejection charges and airframe shear pins have been sized using theoretical analysis, common rocketry calculators [24] and experimental bench testing. The first ejection charge causes shear pin failure of the rocket’s forward airframe, releasing the drogue parachute and cosmic ray detector payload. The drogue parachute is deployed at the rocket’s apogee and it is used to decelerate and stabilize the descent of the rocket. The second ejection charge causes shear pin failure of the rocket’s rear airframe and deploys the main parachute at approximately 800 feet above the ground. 14 To verify the structural and electrical stability of the avionics bay, vibration analysis and snatch force analysis were conducted. The vibration analysis was conducted to verify the displacement due to structural oscillations would not produce damaging forces in the avionics bay causing failure of the electronics and associated mechanisms. Snatch force analysis was performed to ensure the avionics bay would not fail during the main parachute deployment. During main parachute deployment, the parachute ejects from the rocket, but remains connected by a shock chord attached to an eye bolt mounted to the avionics bay end cap. When the parachute canopy inflates, the shock chord is jerked under the force from the opening parachute. At this stage, the avionics bay is acting as a key structural element and is exposed to this “snatching force.” electronics, electrical mounting structures, parachutes, shock cords, and rocket airframe shear pins have been ordered. A conceptual sketch has been developed for the arrangement of the avionics bay section of the rocket and is in the appendix (Figure 15). A 3dimensional model of the avionics bay is in appendix (Figure 16). 3.5 CFD Analysis Simulation CFD (Autodesk Inc., San Rafael, CA). was chosen to perform aerodynamics analyses on the rocket. The assumptions for the flow to be used in the CFD analyses were a fluid of air, compressible flow, sea level standard air conditions, steady flow, and no heat transfer. The inputs were Mach numbers from 0.1 to 0.8 in step sizes of 0.1. The output criteria to judge the different designs were coefficient of drag and center of pressure. In order for the design to be stable the center of pressure had to be below the center of gravity. The Coefficient of Drag vs Mach number are plotted below (Figure 17). 15 Cd vs Mach 0.4 0.35 0.3 Cd 0.25 0.2 0.15 0.1 0.05 0 0 0.2 0.4 0.6 0.8 1 Mach Figure 17: Coefficient of Drag vs Mach 3.6 Finite Element Analysis The FEA was carried out through the application of NASTRAN In-Cad (Autodesk Inc., San Rafael, CA). Finite element analysis was completed on the rocket. The rocket was analyzed with the max thrust of 475 pounds applied to the rocket motor. It was opposed by the force of gravity and a drag force of 88.84 pounds. A standard mesh was developed and then refined to ensure the success of the analysis. FEA analysis revealed that the rocket had a safety factor of 1.8 at peak thrust and peak drag. Finite element analysis was also completed on the avionics bay. The opening of the main parachute exerts a high force on the avionics bay. A maximum force of 129 pounds was calculated by the avionics team. FEA analysis revealed that during the deploying of the parachute, there was a safety factor of 1.33. 16 3.7 Vibration Analysis The completed avionics bay was modeled using Inventor (Autodesk Inc., San Rafael, CA). This model was imported into Autodesk Simulation Mechanical (Autodesk Inc., San Rafael, CA) where random vibration analysis was conducted. Modal analysis was first employed by placing a 5 pound weight at the contact point in the Inventor model where the forward rocket mass would be resting on the avionics bay. Using the default 5 frequency inputs (Figure 18), the results generated a cumulative mass excited in all directions far below the required minimum of 80% mass usage [25]. The number of frequency inputs was increased to 25 before the analysis examined more than 80% of the cumulative mass. These 25 modal analysis frequency results were then loaded into the random vibration analysis tool in Autodesk Simulation Mechanical. Inputs (Figure 19) were required for frequency and power spectral density, and the inputs were chosen based on the typical power spectral density input for a model rocket [26]. The theoretical results obtained from the random vibration analysis produce a displacement result based on the root mean square response. The root mean square response is the square root of the mean of the squares of a sample. The results can demonstrate an expected displacement and deformation resulting from the inputs of the 5 pound mass load, 25 frequency points, and power spectral density inputs. These key values are combined through the modal and random vibration simulation to provide this deterministic output. 17 Frequency (Hz) 10 100 1000 2000 1 2 3 4 . Amplitude (g2/Hz) 0.002 0.04 0.04 0.02 Figure 18: Vibration Analysis Input Parameters Amplitude (g2/Hz) Vibration Testing Input Parameters 0.05 0.04 0.03 0.02 0.01 0 0 500 1000 1500 2000 2500 Frequency (Hz) Figure 19: Amplitude vs Frequency Input Parameters 3.8 Snatch Force Analysis In order to perform accurate FEA on the avionics bay, snatch force calculations were performed. At parachute inflation, an upward force is generated which “jerks” on the shock cord. This force is directed to the eye bolt mounted to the end cap of the avionics bay. The maximum force (Fmax) generated is the parachute suspension-line tension. The governing equation uses units of pounds-force and is outlined below [27]. 2𝑊𝑉𝑖 𝐹𝑚𝑎𝑥 = ( 𝑔𝑇 [1 − 𝑉𝑓𝑖𝑛𝑎𝑙 𝑉𝑖 ]) + 2𝑊 (Equation 8) W=weight of the rocket, 𝑉𝑖 =fall rate at the beginning of inflation, g=gravitational constant, T=duration of the inflation sequence, and 𝑉𝑓𝑖𝑛𝑎𝑙 =fall rate at the end of inflation. A conservative weight of 30 pounds was used for the rocket. The gravitational constant is in English units and equates to 32.17 ft/sec2. An initial velocity of 90 ft/sec was 18 used which is the maximum velocity that is sustained with the drogue parachute open. The final velocity will not be slower than 15 ft/sec. Finally, the inflation time will not exceed 2 seconds, if the parachute is properly folded and inserted into the rocket airframe. 3.9 Ejection Charge Analysis Ejection charge and shear pin sizing was performed, first using theoretical analysis and commonly available calculators used in model rocketry [23]. Data obtained from these calculators was used as a baseline to size the ejection charges and shear pins for experimental testing. Bench tests were performed on the assembled rocket testing the rocket’s separating force. Proper ejection charge sizing ensures failure of the rocket airframe shear pins during flight, allowing safe separation of the airframe and deployment of parachutes. Theoretical ejection charge sizing analysis was conducted to determine the predicted amount of black powder (BP) needed for proper rocket separation. 𝑉 𝐴𝑚𝑜𝑢𝑛𝑡 𝑜𝑓 𝑏𝑙𝑎𝑐𝑘𝑝𝑜𝑤𝑒𝑟 (𝑔𝑟𝑎𝑚𝑠) = 𝑑𝑃 𝑇 (Equation 9). 𝑅 dP=pressure inside rocket during separation is assumed to be 15 psi, V=the volume of the section under consideration, R=gas constant, and T=absolute temperature during ejection. After the ejection charges were calculated bench testing was performed to verify the theoretical ejection charge mass was sized appropriately. 3.10 Rocket Manufacture The rocket components were fabricated from G12 filament wound glass and arrived from the manufacturer in accordance with designated specifications. The manufacturer also cut the fin slits in the lower body tube in accordance with their specified positions. West System epoxy was used as the fastening agent for assembling the different rocket components. This epoxy system was employed by mixing three parts of the West System Epoxy resin with one part of the 19 West System Hardener. The motor mount was the first piece of the bottom portion to be assembled and was epoxied into the bottom rocket body. Next the fins were inserted into their corresponding holes and glued flush against the motor mounting tube. A thickening agent was mixed in with the epoxy and fillets were created between the fins and the outside of the rocket body. The motor retention ring was the final piece to be installed on to the bottom of the lower rocket body via bolts. The nose cone was assembled by joining the cone to a bulkhead via a bolt with accompanying nuts. An eyebolt was attached to this bulkhead by a corresponding nut. The avionics bay was assembled by gluing a centering ring in the middle of the avionics body tube. The last step was to drill holes in the top body tube, the lower body tube, and the avionics tube for the necessary connecting hardware. The combustion chamber and nozzle designs were submitted to the ODU machine shop for manufacture. The steel for the nozzle was provided by the machine shop. 3.11 Fuel Manufacture Several manufacturing investigations took place before actual fuel manufacturing began. This consisted of cooking an inert fuel where the Potassium Nitrate oxidizer was replaced with Sodium Chloride in the same mass ratios as the fuel mixture. These tests revealed that in order to make a homogenous mixture which could be molded easily, it is necessary to heat the mixture at a higher temperature than what is minimally necessary to cause melting of the Dextrose. The sample batch which was made at a lower temperature did not melt fully and was difficult to work with when melting occurred. During these tests, the mixture was exposed to an open flame which showed that the dextrose fuel was resistant to overheating. The dextrose tended to decompose, rather than burn in the presence of excessive heat. Tests involving a dry mixture of the Potassium Nitrate and Dextrose confirmed this result, even in the presence of the oxidizer. When exposed 20 to an open flame, prolonged expose to the flame source was necessary to obtain ignition of the mixture. This is desirable from a safety standpoint while manufacturing fuel to know that there is a large margin between melting and ignition of the fuel. It was even shown that when the fuel mixture was put directly onto the heating element, ignition did not occur, rather, the Dextrose caramelized and decomposed. The process of manufacturing the fuel required the two components to be thoroughly mixed together and then heated until the dextrose melts, homogenously binding the two products together. The particles were mixed together while over heat and stirred constantly throughout heating to ensure a thorough and homogenous mixture. Initial fuel batches consisted of Potassium Nitrate and Dextrose particles which were finely milled in a blender before heating; these batches were highly susceptible to caramelization and burned significantly more slowly and less energetically than batches which were made with dry ingredients which were not milled before heating. Although this is counterintuitive to the common design practice to make the dry particle grain sizes as small as possible to ensure a more efficient burn, the higher rate of dextrose caramelization had a larger negative impact on burning rate. Later grains which used the dry ingredients in their native state, which was already fairly fine, showed almost no signs of caramelization during heating, were much easier to melt homogenously, and the final product was much lighter in color than earlier batches. A comparison test of samples from each manufacturing group showed that the unmodified grains burned much faster than the grains which were milled before heating. To manufacture the fuel, the Potassium Nitrate-Dextrose dry mixture was heated between 145-150°C. The Dextrose begins to melt at 145°C and caramelize at 157°C so it was important to precisely control the temperature; caramelization of the sugar negatively impacts the performance of the propellant. The fuel mixture was heated in a heavy 21 bottom metal pot over a hot plate and stirred throughout heating with a silicon spatula. The constant stirring was necessary to prevent hot spots and premature caramelization of the Dextrose which would negatively impact fuel performance. After the mixture was fully melted and homogeneous, the fuel was packed into a 2.493” inside-diameter, cylindrical casting tube, placed around a cylindrical core with a diameter of 0.6875” and then secured in an air-tight bag and left to cool and solidify. To form the motor, four of these castings were made at a length of 5.0”; each of these castings is known as a grain. The completed motor is comprised of four grains, totaling 20” long with an outside diameter of 2.493”, and a core diameter of 0.6875”, for a total volume of 97.6in3 and theoretical fuel mass of 5.817 lb. A key manufacturing concern was eliminating as many air gaps in the fuel mixture as possible; any air gaps in the fuel can cause a rapid increase in burning area of the fuel and therefore cause a spike in pressure. Before motor fires, the grains were weighed to ensure they were within 95% of their theoretical density. Tests of the fuel showed that the fuel mixture gave off tremendous amounts of smoke but burned rather slowly but data for this formulation shows that the mixture burns faster at higher pressures. Ignition with open flame was inefficient and could not get the entire fuel sample to ignite rapidly which was necessary for use in a motor. A comparison test showed that a similar size fuel sample will burn more completely and rapidly when ignited with black powder. Black powder is the ideal igniter for our motor because achieving target operating pressure at ignition improves the efficiency of the motor. 3.12 Motor Burn Testing A test stand was constructed to measure motor thrust (Figure 20, Figure 21). The purpose of the test stand was to capture the motor thrust and direct it to a single point where it could be measured accurately with a load cell. It was built with a 2 foot W10 x 22 I beam as the base. 3 22 square 3/16 inch thick steel plates held the motor in place. A 3.125 inch hole was bored in each of the plates to accommodate the motor. The load cell was bolted to a ½ steel plate at the end of the stand. The 3 plates aligned the motor so that the front of the motor would be directed into the load cell, allowing freedom of movement towards the load cell, while ensuring that the motor stayed secured during a test. The test stand then had 4 holes drilled into it and it was securely bolted to a pickup truck bed. The truck bed was used since it provided a stable, heavy, mobile platform. The steel bed also provided some shielding for the testing personnel in the case of a catastrophic failure. The data was then transmitted into the cab of the truck where it was read by a computer. The load cell used was an Interface 1010AF 1000 lb load cell which was fixed to the test stand as a means of measuring the thrust. A cable was connected to the load cell which attached it to a National Instruments data acquisition system and a power amplifier. The amplifier provided the load cell with 15 volts of electricity. The data acquisition system was connected to a computer through a USB cable. LabView was installed on the computer along with the appropriate drivers to recognize the data acquisition system. A program was set up on LabView to record the voltage change across the load cell and convert it to pound force. This program was also set up to record a point of a data per millisecond. 23 Figure 20: Rocket Test Stand 3.13 Wind Tunnel Testing Wind tunnel testing was conducted in the subsonic wind tunnel (Figure 22) to obtain a visualization of the air moving along the rocket body. In order to use the subsonic wind tunnel, an attachment was machined to connect to the stand which held the model in the tunnel. The attachment was made from a block of aluminum. Three holes were drilled 1 inch apart into the top of block. These holes were made so the attachment could be screwed into the testing stand. A 0.5625 inch hole was drilled through the aluminum for the holding rod to slide into. This rod was held tightly in place by 2 more screws drilled into the side of the block. Glued onto the rod was a long wooden shaft which slid inside the back of the model (Figure 23). 24 Figure 22: Subsonic Wind Tunnel Diagram The model, connected to the test stand, was then placed inside the subsonic wind tunnel. Using a laser wall and a smoke rod, flow visualization was done to show the different vortices created by the fins as well as the flow around the body and nosecone. Figure 23: Wind Tunnel Mounted Model 25 3.14 Launch Testing Launching the constructed rocket required a TRIPOLI level 1 & 2 rocketry certification. The level 1 certification was attained by building another rocket for launching with a motor having an impulse of less than 640 N-s. Level 2 entailed taking a written test and launching the same rocket with a motor with an impulse of between 640 and 5120 N-s. This certification was completed in one day moving the project closer to completion and a successful launch. The original plan was to launch the team rocket at a SEVRA research motor event the next day. However, the event was cancelled and the next one would not be until after graduation. It was decided to take the opportunity to test launch the team rocket using a commercial motor. The motor used was manufactured by Aerotech and had a total impulse of 1762 N-s and an average thrust of 805 N. The test launch took place on after the certification launches making three successful launches in one afternoon. 4. Results A rocket design was completed based upon commercially available rocket designs. These include the nose cone, avionics bay, rocket outer body, motor mount, fins and engine (Figure 1). Based upon the design the rocket was manufactured. The motor was designed to produce a thrust for 1.849 seconds (Figure 10) with an average thrust of 475 lbf (Figure 24). This configuration will deliver a total impulse of 770.8lbf – seconds (Figure 25). 26 SRM 2014.XLS Thrust vs time 500 450 400 Thrust (lbf) 350 300 250 200 150 100 50 0 0.0 0.5 1.0 1.5 2.0 Time (sec.) F max = F avg = t thrust = 475 417 1.849 lbf lbf sec. Figure 24: Thrust vs Time With an estimated empty-weight for the rocket assembly of 17 pounds and estimated peak subsonic drag coefficient of 0.321 (Figure 17), this motor configuration will take the rocket to a predicted altitude of 10757 feet in 26.0 seconds, with a max velocity of 1128 feet/sec (Figure 26). From the CFD analysis the center of pressure for the rocket was found to be 87.473 inches from the nose cone, and the center of gravity was approximately 57.3935 inches from the nose cone. This makes the rocket stable during flight because the center of pressure is below the center of gravity. Finite element analysis was completed on the rocket assembly (Figure 27.) The rocket was analyzed with the max thrust of 475 pound applied to the rocket motor. The force of gravity and a drag force of 88.84 pounds opposed it. A standard mesh was developed and then refined to 27 ensure the success of the analysis. FEA analysis revealed that the rocket had a safety factor of 1.8 at peak thrust and peak drag. Figure 27: Rocket Assembly Predicted Displacement Finite element analysis was also completed on the avionics bay (Figure 28). The opening of the main parachute exerts a high force on the avionics bay. The avionics team calculated a maximum force of 129 pounds. FEA revealed that during the deploying of the parachute, there was a safety factor of 1.33. 28 Figure 28: Avionics Bay Predicted Displacement The vibration analysis displacement results showed the deformation in the normal direction to the axially applied load, which is the direction the avionics bay bends during frequency oscillations. These results (Figure 29) presented that the maximum displacement of the part under loading would not exceed 0.01738060 inches displacement during the frequency values being considered. An example of the graphical output at one frequency is shown in figure 30. The results also demonstrated the average displacement value to be 0.00635975 inches. From the truly random nature of the results, some were inclusive as to their effect on the avionics bay during a simulated rocket flight ascent. At the very low default frequency input values, the displacement was generated to be an excessively high value. Looking at the expected natural frequency range of approximately 100-1000 Hz, the results demonstrate a very low expected displacement based on the location of the load and the input frequency values. These values range from 0.00064418 inches to 0.01738060 inches of total displacement. 29 Figure 30: Avionics Bay Deformation at a Frequency of 926.775 cycles/s Figure 31 pictures the truly non-linear nature of random vibration analysis. The theory behind random vibration analysis is that there is no linear relationship between frequency and the oscillations produced from the vibrations. Frequency vs Std Dev Max Displacement Standard Dev Max Displacement 0.02000000 0.01500000 0.01000000 0.00500000 0.00000000 0.00E+00 2.00E+02 4.00E+02 6.00E+02 8.00E+02 1.00E+03 -0.00500000 Frequency (Hz) Figure 31: Frequency vs Standard Deviation Maximum Displacement Using the parachute force equation and analysis values, the value of force during the main parachute deployment was determined to be a maximum of 129.94 pounds force (578 30 Newtons) applied directly to the eye bolt on the avionics bay end cap. This did not exceed the yield stress of the eyebolts. Bench testing revealed the theoretical black powder amounts were insufficient for proper rocket separation. After several ejection tests it was determined that 2 grams (30.8 grains) of BP was required for the forward section of the rocket. The burn test resulted in a burst combustion chamber. After ignition, there was a slight delay before exhaust was observed, after which thrust occurred for approximately 0.4 seconds. At the 0.4 second mark, an abrupt pressure change occurred, which caused catastrophic failure of the motor (Figure 32). During failure the force exerted on the load cell measuring the thrust of the rocket exceeded the maximum rating of the load cell (Figure 33). The explosion destroyed the load cell. Remnants of the motor assembly were found at the testing site and have been photographed (Figure 34) and measured in an attempt to ascertain the failure method. A cursory investigation indicated a burst failure of the aluminum chamber. The predicted burst pressure for this assembly was 3936 psi (Figure 35). Figure 33: Thrust Force vs Time 31 The wind tunnel testing showed the fluid flow along the rocket. The drag was shown directly by seeing the vortices created by the fins (Figure 36). The wake caused by these vortices was also seen behind the rocket (Figure 37). When viewing the flow over the nosecone, the airflow was shown to ride right along the boundary layer on the surface of the rocket (Figure 38). Overall, the wind tunnel proved to be a useful tool to visualize something that couldn’t otherwise have been seen. Figure 36: Fin Vortices The test launch using a commercial motor was partially successful. After analyzing the flight data extracted from the altimeters, it appears the back-up altimeter did not operate as intended. The back-up altimeter never calibrated itself properly prior to launch. That agrees with why 2 of the 4 charges were expelled (drogue and main, from the primary altimeter only). From the plot in the appendix (Figure 39), the descent velocity curve is constant which indicates it was the main parachute that deployed at apogee. However, the plot also indicates that there was a "main" event at the preset 800 ft. There were two events that took place, agreeing with the 32 expelled ejection charges (2 of 4). The primary altimeter expelled 2 charges, one at apogee and the other at 800 ft (as programmed). The main charge was expelled at apogee and the drogue event fired at 800 ft. The drogue chute should have deployed at apogee and the main chute should have deployed at 800 ft. The 800 ft charge failed to shear the pins in the forward cone. Even though the drogue parachute did not deploy, the rocket was recovered safely because the main parachute deployed. The maximum altitude reached was 4407 ft, the maximum velocity reached was 583 ft/s and the total flight time was 3.75 minutes. The ascent was 16 seconds and the descent was 3.5 minutes with an average velocity of 22 ft/s. 5. Discussion The purpose of this project was to design high-powered rocket capable of carrying and deploying a cosmic ray detector at 10,000 feet. This was done by designing the structure, propulsions, aerodynamics, and avionics sections of the rocket. The results of the analyses showed that the rocket design was structurally sound and aerodynamically stable. FEA predicted the rocket would withstand the expected body forces. CFD analysis showed the rocket was aerodynamically stable. The vibration analysis displacement results showed the avionics bay design is acceptable. It was not expected to experience the damaging vibrations leading to failure. The snatch force analysis predicted a force that did not exceed the structural limitations of the bolts when a FEA was conducted using the expected force. The nozzle and combustion chamber were designed to handle the predicted pressure, thrust, and heat expected from analysis of the fuel. The lowest factor of safety was the aluminum combustion chamber with a burst safety factor of 3. Additional calculations predicted the flight performance of the rocket. These calculations used the average thrust, weight of the rocket, weight of the propellant, mass flow rate, and 33 maximum coefficient of drag to predict a maximum altitude of 10757 feet, which would have met the purpose of the project. At 10000 feet, the ejection charge analysis calculated the necessary force to fail the shear pins and safely eject both the parachute and the cosmic ray detector. However, there were limitations in the analyses and flight predictions. The CFD analysis was run assuming sea level standard conditions which will not be the case as the rocket’s altitude changes. The sea level standard conditions resulted in higher predicted coefficients of drag than occurred during flight. Bench testing of the ejection charges proved the calculations undersized the charge. For the FEA, the drag was assumed to be a point force acting in the middle of the body, while in reality, drag affects most of the outer surface of the rocket. FEA was also conducted at maximum thrust and maximum drag. In reality, maximum thrust and drag do not occur at the same time so this was a conservative estimate. Unfortunately, testing only confirmed some of the results of the analyses. Launch testing was done with a commercial motor to show that the rocket was capable of flight and returning safely. The rocket succeeded in doing this but not in the way expected. The backup altimeter never ran through its prelaunch setup and was still in idle mode. This was not noticed so the flight was conducted without the backup altimeter. Additionally, at apogee the nose cone failed to separate as designed despite the ejection charge igniting. Instead the shear pins connecting the avionics bay to the lower rocket body failed which resulted in the main parachute deploying. A possible reason for this is that the forward nose cone stuck when the charge went off creating a tension force which sheared the pins connecting the avionics bay to the lower rocket body. This would be easy to fix in any subsequent launches by using graphite chalk at all connections to ensure no binding of parts and by verifying that both altimeters have initialized. 34 The test burn of the experimental motor to measure the thrust of the rocket motor resulted in a catastrophic failure from a sudden over pressurization which exceeded the burst pressure of the aluminum combustion chamber. Such pressure jumps are typically caused by a rapid increase of the burning area of the propellant. Possible causes of this are air gaps trapped in the propellant during manufacturing or an over-powered igniter causing more of the fuel to ignite than just the inner core surface. Additionally, the black powder ignition charge could have fractured the fuel, so that a fragment broke off of the grains and plugged the nozzle. This could have also resulted in the sudden increase in pressure. Contact was made with Richard Nakka, an expert on model rocketry, to get a second opinion, and his theories agreed with ours. Based on a review of the report you kindly provided, your design looks sound. Rocketry is a very unforgiveable beast and sometimes the smallest of non-perfect details are pounced upon and used to punish us...this happens to the best of us, even professional motors often CATO or behave badly on the first try. I've had my share of ruptured motors taunt me over the years. A CATO such as yours could only (IMHO) be a result of either: 1) severe propellant fracturing due to an overpowered igniter 2) significant disbonding of the propellant from the casting tube My bet is on the latter. I have had exactly the same thing happen. KNDX is kinda tricky to work with in this regard, as it is cast hot, 35 then cools and shrinks, and will often debond while cooling. Sorbitol propellant is superior as it does not shrink this way...it retains its flexible nature well past cooling. With KNDX, I use clamping pressure to prevent the propellant from shrinking radially (this is done with springs) [28]. Although catastrophic failures are not uncommon in rocketry, it is highly disappointing to not achieve the intended results. Because of the motor failure, limited time, and budget, it is impossible to complete a full launch with the experimental motor. Recommendations to future designers would be to start will smaller scale motors and work up both in size and power. Arrangements had been made to work with a SEVRA (Sout Eastern Virginia Rocketry Association) member who is experienced in the design and manufacturing of APCP motors. However, an experienced user of sugar-powered motors could not be found to assist with this project. Although there are extensive online resources for this topic, it is no substitute for in person assistance. It would be recommended that in the future, arrangements be made to work with a person who has practical experience in the design, manufacturing and testing of rocket motors. 36 6. Appendix 6.1 Rocket Drawings Figure 3: Avionics Bay Components 37 Figure 4: Rocket Upper Body Components 38 Figure 5: Rocket Lower Body Components 39 Figure 6: Rocket Motor Components 40 Figure 14: Dual Deployment Wiring Diagram 41 Figure 15: Avionics Bay Concept Sketch 42 Figure 16: 3D Model of Avionics Bay Interior 43 Figure 21: Rocket Test Stand Components 44 6.2 Pictures Figure 32: Rocket Failure Figure 34: Recovered Combustion Chamber Pieces 45 Figure 37: Rocket Wake 46 Figure 38: Rocket Nose Cone Boundary Layer 47 6.3 Tables and Plots Isp Isp Parameter Specific Impulse, ideal Specific Impulse, measured 137 Units sec. sec. 164 C* Characteristic exhaust velocity, theoretical 2993 (912) ft/s (m/s) C* Characteristic exhaust velocity, measured 2922 (891) ft/s (m/s) To Combustion temperature, theoretical @1000 psia 1437 (1710) deg Celsius (K) Density, ideal Density, as cast Mass fraction of condensed-phase products 1.879 1.859 gram/cu.cm. gram/cu.cm. - Ratio of specific heats Effective molecular wt. of exhaust products 1.043 plateau ro r Burn rate behavior Burn rate @ 1 atm. Burn rate @ 1000 psia 0.509 in/sec in/sec Tcr Auto-ignition temperature > 300 deg. C. X k M 0.425 g/mole 42.39 0.084 Figure 7: Propellant Published Values ODU Rocket Team Motor utilizing KNDX propellant. Grain mass Total impulse Average thrust Thrust time Specific Impulse Motor Classification 2.639 5.817 3428.6 770.8 1854.0 416.8 1.849 132.5 L kg. lb. N-sec. lb-sec. N. lb. sec. sec. 1854 Figure 25: Predicted Rocket Motor Performance 48 FLIGHT PERFORMANCE ESTIMATOR FOR HOBBY ROCKETS (valid for subsonic rockets only) EzAlt.xls (MS Excel 97) Version: 1.2 Date: Feb. 2007 Instructions: Enter data in blue text (boxed cells). Title Motor average thrust Motor total impulse Motor propellant weight Rocket dead weight Rocket diameter (max) Rocket drag coefficient Motor thrust time Motor classification Rocket avg. flight weight F= It = mp = mr = D= Cd = t= mra = Acceleration (average) a= or a = Peak altitude (zero drag) Time to peak altitude (zero drag) Max velocity (zero drag) Burnout altitude (zero drag) z2 = t2 = v1 = z1 = Drag Influence number Peak altitude reduction factor Time to peak reduction factor Max velocity reduction factor Burnout altitude reduction factor N= f1 = f2 = f3 = f4 = example rocket 416.8 lb. 770.8 lb-sec. 5.817 lb. 17.000 lb. 4 inch 0.351 Input data 1.849 sec. K 19.909 lb. 674 feet/sec 2 20.9 g's 22967 38.7 1186 1097 feet sec. feet/sec. feet Ideal (no drag resistance) 614 (valid range between 0 and 900) 0.468 0.621 Drag reduction factors 0.951 0.973 Peak altitude Time to peak altitude Max velocity Z peak = 10757 feet t peak = 24.0 sec. V m ax = 1128 feet/sec. or V m ax = 769 MPH Burnout altitude Z bo = 1067 feet Warning: Rocket is supersonic, results may be invalid Figure 26: Predicted Rocket Flight Performance Predicted (with drag) 49 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Frequency (Hertz) 1.36E-04 1.98E-04 5.53E-04 7.51E-04 7.72E-04 7.76E-04 2.08E-03 1.95E+02 2.10E+02 3.04E+02 3.39E+02 3.52E+02 3.80E+02 4.16E+02 4.30E+02 4.92E+02 5.05E+02 5.07E+02 5.29E+02 5.85E+02 6.71E+02 7.26E+02 7.77E+02 7.96E+02 9.27E+02 Period (Seconds) 7.34E+03 5.05E+03 1.81E+03 1.33E+03 1.30E+03 1.29E+03 4.80E+02 5.13E-03 4.77E-03 3.29E-03 2.95E-03 2.84E-03 2.63E-03 2.41E-03 2.33E-03 2.03E-03 1.98E-03 1.97E-03 1.89E-03 1.71E-03 1.49E-03 1.38E-03 1.29E-03 1.26E-03 1.08E-03 Standard Dev Max Displacement (in) 1079000.00000000 1248900.00000000 41394.60000000 230371.00000000 496343.00000000 344531.00000000 54998.20000000 0.01226130 0.01433770 0.00061118 0.00480574 0.00123981 0.00102288 0.01021890 0.00121483 0.00196697 0.01101720 0.00847359 0.00254671 0.00030500 0.01738060 0.01475140 0.00964377 0.00203377 0.00064418 Figure 29: Vibration Analysis Results 50 Design and Burst Pressures for Rocket Motor Casing [ Input data in blue text, English or (SI) units] Casing Dimensions and Design Factors Do = t= SD = 3 in. (mm) 0.125 in. (mm) 3.001 Diameter, outside wall thickness Design Safety factor Material Properties Fty = Ftu = E= = B= 37 ksi (MPa) 42 ksi (MPa) 9.9 Msi (MPa) 0.33 0.881 1.277 Yield Strength Ultimate Strength Modulus of Elasticity Poisson Ratio Fty/Ftu Burst factor Design and Burst Pressures PD = PU = SU = 1027 psi (kPa) 3936 psi (kPa) 3.83 Design pressure Burst pressure Burst Safety Factor Elastic Deformation under Pressure * DD = Dc = 0.00312 in. (m.) 0.00980 in. (m.) Change in casing diameter, at P D Change in casing circumference, at P D Figure 35: Design and Burst Pressures for Rocket Motor Casing 51 Figure 39: Test Flight Data 52 6.4 Fuel and Combustion Chamber Calculations Given data for specified fuel mixture KNDX 𝑇0 = 1710 𝐾 Adiabatic flame temperature obtained from GUIPEP/PROPEP software 𝑇 = 1625 𝐾 𝑓𝑜𝑟 𝜂𝑐 = 0.95 (fine grain propellant) 𝑀 = 42.39 𝑘𝑔 𝑘𝑚𝑜𝑙 𝑘 = 1.1308 𝑅 = 196.1 𝑐∗ = √ 𝐽 𝑘𝑔 ∗ 𝐾 𝑅 𝑇0 𝑚 = 889 𝑘+1 2 𝑘−1 𝑠 𝑘( ) 𝑘+1 Kn Calculations - Empirically Determined Data by R. Nakka a b c d e KNDX 43.500 0.24168 50.484 -9.9115 163.80 22.224 -0.2240 5095.0 -2460.4 453.48 -35.532 1.0175 calc Kn -1070 313 363 Range incr 150-400 psi 450-850 psi 900-1300 psi 𝐾𝑛 = 𝑎 + 𝑏 ∗ 𝑃 + 𝑐 ∗ 𝑃2 + 𝑑 ∗ 𝑃3 + 𝑒 ∗ 𝑃4 Where P is Pressure in MPa Therefore, for target pressure of 1050 psi (7.24 MPa), Kn = 363 Burn Rate Calculations - Empirically Determined Data by R. Nakka KNDX a n Pressure, psia psi, in/sec Pressure, Mpa 14.7 to 113 0.0160 0.619 0.100 to 0.779 113 to 373 0.3105 -0.009 0.779 to 2.572 373 to 860 0.0049 0.688 2.572 to 5.930 860 to 1233 1.4155 -0.148 5.930 to 8.502 1233 to 1625 0.0209 0.442 8.502 to 11.20 a n Mpa, mm/sec 8.875 0.619 7.553 -0.009 3.841 0.688 17.20 -0.148 4.775 0.442 53 Calculation of Kn and Throat Area – Assume linear web regression 𝑥𝑖𝑛𝑐 = 𝑥 𝑖𝑛𝑐𝑟𝑒𝑚𝑒𝑛𝑡 (mm) 𝑑 = 𝑑𝑖𝑜 + 𝑐𝑖 ∗ (2𝑥𝑖𝑛𝑐 ) (mm) 𝐷 = 𝑑𝑜 − 𝑜𝑠𝑖 ∗ (2𝑥𝑖𝑛𝑐 ) (mm) 𝐿 = 𝑙𝑔𝑜 − 𝑒𝑖 ∗ (2 ∗ 𝑛 ∗ 𝑥𝑖𝑛𝑐 ) (mm) 𝑡𝑤𝑒𝑏 = 𝐷 − 𝑑 (mm) 𝜋 𝐴𝑏𝑒 = 𝑒𝑖 ∗ 2 ∗ 𝑛 ∗ 4 ∗ (𝐷2 − 𝑑 2 ) (mm2) 𝐴𝑏𝑐 = 𝑐𝑖 ∗ 𝜋 ∗ 𝑑 ∗ 𝐿 (mm2) 𝐴𝑏𝑠 = 𝑜𝑠𝑖 ∗ 𝜋 ∗ 𝐷 ∗ 𝐿 (mm2) 𝐴𝑏 = 𝐴𝑏𝑐 + 𝐴𝑏𝑒 + 𝐴𝑏𝑠 (mm2) 𝑑𝑡𝑜 = 𝑚𝑎𝑥(𝐴𝑏 )/(𝐾𝑛 @ 𝑃 = 𝑃𝑚𝑎𝑥 ) 𝜋 𝐴𝑡 = 4 ∗ (𝑑𝑡𝑜 ∗ 𝐾𝑛 = 𝐴𝑏 𝐴𝑡 (𝑡𝑤𝑜−𝑡𝑤𝑒𝑏 ) 2 𝑡𝑤𝑜 ) (mm2) (dimensionless) 54 Calculation of Pressure and Burn Time In the following set of calculations only n=pressure exponent at Po 𝑥𝑖𝑛𝑐𝑝 = 𝑥 𝑖𝑛𝑐𝑟𝑒𝑚𝑒𝑛𝑡 𝑓𝑜𝑟 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 (mm) 𝑑 = 𝑑𝑖𝑜 + 𝑐𝑖 ∗ (2𝑥𝑖𝑛𝑐 ) (mm) 𝐷 = 𝑑𝑜 − 𝑜𝑠𝑖 ∗ (2𝑥𝑖𝑛𝑐 ) (mm) 𝐿 = 𝑙𝑔𝑜 − 𝑒𝑖 ∗ (2 ∗ 𝑛 ∗ 𝑥𝑖𝑛𝑐 ) (mm) 𝑡𝑤𝑒𝑏 = 𝐷 − 𝑑 (mm) 𝜋 𝐴𝑡 = 4 ∗ (𝑑𝑡𝑜 ∗ 𝐴∗ = (𝑡𝑤𝑜−𝑡𝑤𝑒𝑏 ) 2 ) (mm2) 𝑡𝑤𝑜 𝐴𝑡 (m2) 10002 𝜋 𝜋 𝐴𝐷𝑈𝐶𝑇 = 4 ∗ 𝑑𝑐 2 − 4 ∗ (𝐷2 − 𝑑 2 ) (mm2) 𝐺 = 𝐺 ∗ − 𝐴𝐷𝑈𝐶𝑇 𝑟 = (1 + 𝑘𝑣 ∗ 𝐺) ∗ 𝑎 ∗ 𝑃0𝑛 (mm/s) 𝑡= 𝑥𝑖𝑛𝑐𝑝 (s) 𝑟+𝑡𝑖 𝜋 𝑉𝑔𝑟𝑎𝑖𝑛 = 4 ∗ (𝐷2 − 𝑑 2 ) ∗ 𝐿 (mm3) 𝑉𝑔𝑟𝑎𝑖𝑛 = 𝑉𝑔𝑟𝑎𝑖𝑛 10003 (m3) 𝑣𝑐 𝑉𝑓𝑟𝑒𝑒 = 10003 −𝑉 𝑔𝑟𝑎𝑖𝑛 (m3) 𝑚𝑔𝑟𝑎𝑖𝑛 = 𝜌𝑔𝑟𝑎𝑖𝑛 ∗ 𝑉𝑔𝑟𝑎𝑖𝑛 /10002 (kg) 𝑚̇𝑔𝑒𝑛 = 𝑚̇𝑛𝑜𝑧 = 𝑚𝑔𝑟𝑎𝑖𝑛,𝑖 −𝑚𝑔𝑟𝑎𝑖𝑛,𝑗 𝑡𝑗 −𝑡𝑖 (𝑃0 −𝑃𝑎𝑡𝑚 )∗1,000,000∗𝐴∗ √𝑅𝑇 (kg/s) 2 ∗ √𝑘 ∗ (𝑘+1) 𝑘+1 2 𝑘−1 (kg/s) 𝑚̇𝑠𝑡𝑜 = 𝑚̇𝑔𝑒𝑛 − 𝑚̇𝑛𝑜𝑧 (kg/s) 𝑚𝑎𝑠𝑠𝑠𝑡𝑜,𝑗 = 𝑚̇𝑠𝑡𝑜,𝑗 ∗ (𝑡𝑗 − 𝑡𝑖 ) + 𝑚𝑎𝑠𝑠𝑠𝑡𝑜,𝑖 (kg) ρprod = 𝑚𝑎𝑠𝑠𝑠𝑡𝑜 𝑉𝑓𝑟𝑒𝑒 (kg/m3) 𝑃0 = ρprod ∗ 𝑅 ∗ 𝑇 + 𝑃𝑎𝑡𝑚 ∗ 1,000,000 (Pa absolute) 𝑃 0 𝑃0 = 1,000,000 (MPa absolute) 𝑃0 = 𝑃0 − 𝑃𝑎𝑡𝑚 (MPa gage) 55 Calculation of Thrust and Impulse 𝑃0 = ρprod ∗ 𝑅 ∗ 𝑇0 + 𝑃𝑎𝑡𝑚 ∗ 1,000,000 (Pa absolute) 𝐴𝑡 𝐴∗ = 10002 (m2) 𝜋 𝐴𝑡 = 4 ∗ (𝑑𝑡𝑜 ∗ (𝑡𝑤𝑜−𝑡𝑤𝑒𝑏 ) 2 𝑡𝑤𝑜 ) (mm2) 𝐴𝑒 ⁄𝐴 = expansion ratio 𝑡 𝑃𝑒 = [ 𝑃0 𝑘 (𝑘−1) 𝑘−1 (1+ ) 2∗𝑀2 𝑒 ] 𝑤ℎ𝑒𝑛 𝑃𝑒 ≥ 𝑃𝑎𝑡𝑚 , 𝑒𝑙𝑠𝑒 𝑃𝑒 = 𝑃𝑎𝑡𝑚 (Pa absolute) −1 1 𝑘+1 𝐴𝑒 𝑘+1 , 𝑜𝑝𝑡 = ( ) 𝐴𝑡 2 [ 1 𝑃𝑒 𝑘 𝑘−1 𝑃𝑒 𝑘 𝑘+1 ( ) √( ) (1 − ( ) 𝑃0 𝑘−1 𝑃0 ] 𝑘+1 𝑘−1 𝑘 2𝑘 2 2 𝑘−1 𝑃𝑒 𝐶𝐹 = 𝜂𝑛𝑜𝑧 √ ∗( ) ∗ (1 − ( ) 𝑘−1 𝑘+1 𝑃0 )+ (𝑃𝑒 − 𝑃𝑎𝑡𝑚 ∗ 1,000,000)𝐴𝑒 𝑃0 𝐴∗ 𝐹 = 𝐶𝐹 ∗ 𝐴∗ ∗ 𝑃0 (N) 𝑡= 𝐼𝑡 = ) 𝐹𝑗 −𝐹𝑖 2 𝑥𝑖𝑛𝑐𝑝 𝑟+𝑡𝑖 (sec) ∗ (𝑡𝑗 − 𝑡𝑖 ) (N-sec) 56 Design of Motor Chamber – Design and Burst Pressure Fty Ftu β= 𝐵 = 𝐴 ∗ β4 + B ∗ β3 + C ∗ β2 + D ∗ β + E 𝑃𝐷 = 2 𝑃𝑈 = 2 Burst Factor Polynomial Coefficients: A = 9.5833 B= -33.528 C= 44.929 D = -28.479 E= 8.6475 𝑡 𝐹𝑡𝑦 𝐷𝑜 𝑆𝐷 𝑡 𝐹𝑡𝑦 𝐵 𝐷𝑜 57 6.5 Gantt Chart 58 6.6 Budget 59 7. 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Available e-mail: Message: Based on a review of the report…