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Burn Baby Burn Gina Covarrubias John Dankanich Kristin Gates Jon Mah Mike Schreiner Selim Solmaz Executive Summary The Burn Baby Burn satellite system is a university built satellite. The manufacturing limitations of placing a thruster on the center of mass of a spacecraft lead to pointing errors. The primary mission of the satellite is to perform analysis on the removal of this pointing error through a two burn thrusting scheme. The secondary is mission to de-orbit the spacecraft while testing a trapezoidal thrusting scheme to remove pointing errors. The spacecraft is compatible with all available secondary payload launch systems. The satellite will have a circular orbit at a 350 km altitude and an inclination greater than 30o. The overall mass of the spacecraft is 24.85 kilograms and it has dimensions of 30 cm X 30 cm X 35 cm. The cost of production is estimated to be $48,000. 2 Table of Contents I. Mission Statement and Objectives 4 II. Concept of Operations 5 III. Major Design Requirements 6 IV. Spacecraft System Overview 9 V. Launch Vehicle Integration 13 VI. Orbit Selection 14 VII. Power Subsystem 15 VIII. Attitude Determination and Controls Subsystem 18 IX. Communication and Data Handling 37 X. Thermal Subsystem 42 XI. Structure and Mechanisms 47 XII. Propulsion 48 XIII. Summary of Spacecraft 53 XIV. References 54 XV. Appendix 55 A. QFD 55 B. ADCS 56 C. Power 62 D. Communications and Data Handling 64 E. Thermal 66 F. Structure 68 G. Propulsion 69 3 I. Mission Statement and Objectives Mission Statement Due to possible center of mass or thruster offset in axially spinning-thrusting spacecraft, the resulting angular momentum vector is not aligned with the intended flight path. A solution to this problem is to use a “two-burn scheme” during thrusting. This burn scheme will eliminate the average angular momentum bias, and cause the velocity vector to point along the axis of rotation. Spacecraft customers will benefit through decreased propellant costs and an increased mission life. Mission Objectives The primary objective is to test effectiveness of the “ Two-burn scheme” in an axially spinning- thrusting spacecraft with an intended thruster offset. The mission will also have secondary objectives to test the trapezoidal thrust profile using a solid rocket motor and to de-orbit the spacecraft. An additional secondary objective is to perform an outreach experiment. The outreach experiment will be limited to the available mass and volume after performing the university mission. Last, the satellite will have mirrored surfaced to aid in visualization of the craft. 4 II. Concept of Operations Figure 1: Illustration of the spacecraft mission. 1.Detach from launch vehicle 10. AOS 2.Initial acquisition of signal (AOS) 11. Downlink experiment #2 data 3.Command to Spin-Up •Telemetry accompanies data 4.Downlink telemetry (verify rates are within •Verify data is as expected limits) •After downlink, s/c goes to beacon mode 5.Perform experiment #1 (after performing 12. AOS experiment, s/c goes to beacon mode) 13. Perform experiment #3 (after performing 6.AOS experiment, s/c goes to beacon mode) 7.Downlink experiment #1 data 14. AOS •Telemetry accompanies data 15. Downlink experiment #3 data •Verify data is as expected •Telemetry accompanies data •After downlink, s/c goes to beacon mode •Verify data is as expected 8.AOS/downlink telemetry •After downlink, s/c goes to beacon mode 9.Perform experiment #2 (after performing 16. End of Life experiment, s/c goes to beacon mode) 5 III. Major Design Requirements The primary requirement of the mission is to design a spin stabilized spacecraft platform, at a minimal cost, to test the “Two-Burn” and “Trapezoidal” trusting schemes for velocity pointing control, which was developed by Prof. James M. Longuski, et al. of Purdue University, West Lafayette. The spacecraft shall have thrusters placed longitudinally along the spin axis with some intentional thruster offset. The reasons for this are explained along with The Two-Burn scheme in detail in the following chapters and information on the trapezoidal scheme can be found in US patent No: 6,332,592 B1. The major design requirements of the mission are explained below for each specific subsystem, and /or mission phase: 1. Orbit In order to maximize the chances of launching as secondary payload, the mission shall be designed for LEO (Low Earth Orbit). The spacecraft shall be designed to perform various pointing error correction techniques and transmit the data collected to the Purdue University ground station. The orbit shall have an inclination capable of direct communication with the Purdue ground station. The orbit shall have a minimum of 3 passes for communication each day. 3. Launch Vehicle The satellite shall be designed to fit any launch vehicle available (i.e. Delta, Shuttle Hitchhiker, Pegasus/OSP, and Minotaur), as a secondary payload. Due to the varying constraints of the different launch vehicles, the satellite shall be designed for the maximum allowable case. The Delta rocket imposes the greatest limits on the spacecraft dimensions, while the Pegasus will impose the largest structural constraints. The spacecraft shall have maximum dimensions of 30 cm X 30 cm X 35 cm and maximum mass of 40 kg, so the spacecraft shall fit in the Delta launch vehicle as a secondary payload. 4. Mission Life The satellite shall be designed for a one-week life in orbit. The experiments will take approximately three days, allowing for a redundancy of four days. 5. Propulsion The satellite shall have a cold gas thruster capable of providing the thrust profiles required to perform a test of the single and two burn maneuvers. The total velocity change for the cold gas thruster shall be 6 m/s. The satellite shall also have a solid rocket motor to de-orbit the spacecraft and that is capable of providing the thrust profile required to perform the trapezoidal burn maneuver. The total velocity change for the solid rocket motor shall be 50 m/s. The cold gas motor shall provide approximately 1 Newton of thrust to perform the burn maneuvers. Two arrays of small solid rocket thrusters shall be attached to the side of the spacecraft to provide the initial spin-up. The total velocity change for the spin-up thrusters is 0.5 m/s. Both the Cold-Gas thruster and the solid rocket motor shall have an intentional offset of 5 cm. 6. Attitude Determination & Control 6 The spacecraft shall have multiple sun sensors for precise attitude determination. The angular velocities shall be determined by rate gyros. The sun sensors shall be used in conjunction with gyros for reference correction. Accelerometers shall be used to detect acceleration changes of the spacecraft, which shall be used to determine the velocity history of the spacecraft during the experiments. The spacecraft shall spin about the maximum moment of inertia axis at a rate of 25 rpm. Two arrays of small solid thrusters that are placed about the spin axis shall provide the initial spin in case that the launch vehicle is incapable of providing it. The spacecraft shall not have an active attitude control system aside from the axially placed main cold-gas thruster (along the spin axis) and the SRM on the opposite face. Any nutation caused by the offset thrusters shall be damped out using passive nutation dampers in less than three hours. 7. Power The power requirement for the spacecraft shall have a maximum peak power requirement of 40 watts. The power shall be provided by primary D-cells with an operating voltage of 24 volts. The batteries’ state of charge shall be monitored throughout the mission. 8. Thermal Control The thermal control system shall have a thermal margin of 5°, meaning that temperatures of 5° higher than the lower temperature limit and 5° below the upper temperature limit will be maintained. Coating or insulating their outer surfaces shall control temperature of compartments, and conventional electronic equipment. 9. Telecommunications and Data Handling The goal shall be to use the ground station available at Purdue University, through a direct downlink from the spacecraft. Commands shall be up-linked to the spacecraft via a terminal node controller, which shall format the data for transmission. The primary use of the downlink shall be to transmit the attitude data, temperatures of different subsystems, and battery state of charge to the ground station during each pass. The uplink frequency shall be in the two-meter wavelength band (~150 MHz), and the downlink frequency shall be in the 70-centimeter wavelength band (~440 MHz). The exact frequencies shall be as allotted. The transmitter, receiver, and terminal node controller shall use existing, off-the-shelf technology. Communications shall use the A.X. 25 amateur radio protocol to package and transmit the data at a rate of 1200 bits per second. The antennae shall utilize two pairs of monopole antennae (one pair for transmit and one pair for receive), which shall be mounted on opposing corners of the spacecraft. The data handling shall use a processor, a data module, several analog-to-digital converters, a decoder, and a digital databus. The data shall be buffered in the memory module as the experiments are performed and then, upon ground command, down-linked data to the Purdue University ground station. 10. Structures A prime requirement for the structure is that it shall be designed to withstand dynamic loads present during testing, launch phases, and zero gravity environment. The main bulk of the structure shall consist of a durable, lightweight, cheap material that is relatively easy to manufacture. 7 8 IV. Spacecraft System Overview The Burn-Baby-Burn spacecraft is a spin-stabilized satellite that is spun about its z-axis. Figure IV-1 shows the spacecraft with it body fixed coordinate system. Figure IV-1: External view of spacecraft with body fixed coordinates. The spacecraft layout is shown in figure 2, 3, and 4. Figures 2 and 3 show the system without the batteries, while figure 5 shows the layout with the batteries included. The stability of the spacecraft requires Iz to be the maximum moment of inertia. Therefore the layout is designed to provide a maximum moment of inertia in the ‘z’ direction. Figure IV-2: Internal layout of the spacecraft. 9 All of the components except the inner battery packs are mounted directly to the walls of the spacecraft. The main cold-gas thruster and the solid rocket motor are intentionally offset five centimeters to cause a significant point error. Last, the CPU is a comprehensive computer package that includes the processor, data storage, transmitter, receiver, terminal node controller, and power regulator. Figure IV-3: Layout of the spacecraft components. Figure IV-4: Layout of spacecraft with dimensions. 10 The mass budget for the entire spacecraft is shown in Table 1. The individual subsystems mass budgets re shown in Tables 2 – 6. The goal for the spacecraft is to have a total mass under forty kilograms. The total mass of the spacecraft is currently 28.5 kg. This mass gives an 11.5 kg (29 %) margin. Because the majority of the components have already been selected, they do not require a significant mass margin. The margin will primarily be consumed by the estimated components in addition to a possible outreach experiment. Power 9.56 kg 38 % Structure 5.75 kg 23 % Propulsion 5.42 kg 22 % Attitude Determination 1.40 kg 6% Communication and Data 2.72 kg 11 % Total 24.85 100 % Margin 15.15 38 % Table IV-1: Overall system mass budget. Batteries *** 7.89 kg Frame* 1.39 kg Regulator/Converter* 0.42 kg Shell* 3.36 kg Wiring* 1.25 kg Fasteners** 1 kg Total 9.56 kg Insulation** 0.0042 kg Total 5.75 kg Table IV-2: Power mass Budget. Table IV-3: Structure mass budget. * Calculated ** Estimated *** Actual 11 Tank* 1.76 kg Transmitter* 0.5 kg Plumbing* 0.69 kg Receiver* 0.8 kg Spin-up Motors** 1 kg Antennae* 0.5 kg De-orbit SRM* 1.55 kg Processor*** 0.4 kg Nozzle* 0.012 kg Terminal Node Con.* 0.32 kg Propellant (gas)* 0.41 kg Data Storage* 0.5 kg Total 5.42 kg Total 2.72 kg Table IV- 4: Propulsion mass budget. Table IV-5: Com. and data mass budget. Gyros*** 0.06 kg Sun Sensor*** 0.31 kg Accelerometers*** 0.03 kg Nutation Damper** 1 kg Total 1.4 kg Table IV-6: AD & C mass budget. * Calculated ** Estimated *** Actual Ix = 0.32 kg / m Iy = 0.36 kg / m Iz = 0.50 kg / m Table IV-7. Mass moment of inertias for the spacecraft. 12 V. Launch Vehicle Integration 13 VI. Orbit Selection To meet the mission requirement of flexibility in launch vehicle, the spacecraft must be able to fly in any available orbits. However, for design considerations, a specific orbit must be chosen for calculations. The orbit inclination is determined by the communications subsystem. The spacecraft needs to have the ability to communicate with the Purdue University ground station at least twice per day for a minimum duration of 8.4 minutes. The minimum inclination for communications is then 30o. The altitude of the orbit is determined by the mission life, the communications system, and the size of the de-orbit motor. Our spacecraft can be in a low orbit because the mission life is very short. Because the size and mass of the de-orbit engine increase with altitude, the minimum altitude that allows fours days of communications before re- entry was chosen. Four days provides enough time for all of the data to be collected and transmitted to the ground station before the loss of the satellite. Therefore the altitude for the mission was chosen to be 350 km. Figure VI-1: Ground tracks of the orbit selected. The orbit for the mission is a circular orbit with an altitude of 350 km and an inclination of 40o, which will pass directly over the Purdue University ground station. This is the orbit is chosen for calculations, however; any inclination over 30o is acceptable. 14 VII. Power Subsystem The power subsystem limits the lifetime of the satellite. Under optimal conditions the mission can be completed in less than three days, so the mission lifetime requirement is one week for redundancy. Because of the short mission life, primary cells are a feasible option. A trade study was carried out to decide if solar cells or primary batteries would be best suited for the mission. The three methods of power generation considered were primary batteries, solar cells, and the combination of primary batteries with solar cells. Table VII-1 shows the results of the trade study. Primary cells are very limiting on the lifetime of the satellite and consume a large fraction of the satellite volume. The use of solar cells can increase the duration of the mission and also greatly reduce the amount of internal volume needed for the power subsystem. Using a combination of solar cells and primary batteries provide both an extended lifetime and an additional level of redundancy. The over all decision to use only primary cells and is to minimize mass and especially the cost. In order to accurately compare the various options rough calculations were carried out and are shown in Appendix C Batteries Solar Cells Batteries and Solar Simplicity ++ + - Mass ++ + - Cost ++ - - Redundancy - + ++ Volume - ++ + .Table VII-1: Trade study for choosing method of power generation. The power requirements for each subsystem as well as the total spacecraft are shown in table VIII-2. The power requirements are divided into the different mode of operation. Using the power required for each mode of operation with the duration of each mode provides the number of watt-hours the primary cells must provide. There is a margin shown in the table for a margin in both power for each mode of operation and the duration of each segment of the mission. The margin for the power consumed in each mode is approximately 10%. There is also a duration margin of 100% for the insertion, testing, and transmitting modes of operations in addition to 20% margin in the duration of the standby mode. 15 Insertion Testing Standby Transmitting Spin Up Motors Capacitor Cold Gas Thruster 1 De-Orbit SRM Capacitor Sun Sensor 0.05 0.05 0.05 0.05 Gyros 2 Accelerometers 1 Processor 3 6 3 6 Terminal Node 1 1 1 1 Controller Transmitter 7.5 Receiver 3 3 3 3 Power Losses 1.41 2.81 1.41 3.51 Total (Watts) 8.46 16.86 8.46 21.06 Power Margin 1 2 1 2 11 % Duration (Hours) 0.5 0.5 166 1 Duration Margin 0.5 0.5 36 1 23 % Total (W-hrs) 9.46 18.86 1910.92 46.12 1985 Table VII-2: Power budget in various modes of operation. After calculating how much power the primary cells must provide, the type of primary cells must be selected. Based on a maximum mass allocation of 25% (10 kg) for the power system, the primary cells needed to have a specific energy density greater than 200 W-hr / kg. This specific energy density limited the primary battery selection to Lithium Thionyl Chloride, Lithium Sulfur Dioxide, and Lithium Monoflouride. Table VII-3 shows the methodology of choosing which primary cell to use. Overall, the Lithium Sulfur Dioxide batteries are optimal for this mission. Lithium Thionyl Lithium Sulfur Lithium Chloride Dioxide Monoflouride Optimal Duration -- ++ + Discharge Curve + ++ + Used in Space + + + Cost + ++ + Specific Energy ++ + + Density Table VII-3: Trade study for choosing primary battery. Lithium Sulfur Dioxide primary cells have been used in space and are commonly used in military hardened applications. They have low internal resistance, and a temperature range from –60o to 80oC. The cell types vary with specific energy densities ranging from approximately 130 – 350 W-hr / kg. The cell chosen for the mission has a specific energy density of approximately 250 W-hr / kg. The mass budget of the power system is shown in table VII-4. 16 Batteries *** 7.89 kg Regulator/Converter* 0.42 kg Wiring* 1.25 kg Total 9.56 kg Table VII-4: Power mass budget. *Calculated *** exact The volume of the power consumption is also a concern for the mission. Lithium Sulfur Dioxide units can vary in the energy densities as well. The energy density is proportional to the cost of the cell. Because the objective in any space mission is to minimize the cost, the readily available D-cell was chosen. While the D-cell does not have the highest energy density available (0.4 W-hr / cm3), it is the most cost effective method of power generation. Because of the low energy density, the mission can only have a maximum duration of 12 days. The entire power system cost is approximately $2000. 17 VIII. Attitude Determination and Controls Subsystem 1) Major Requirements The requirements on the AD & C subsystem change during mission, especially if the satellite is performing multiple independent tasks. Usually the payload requirements at each specific mission phase dictate the AD & C subsystem requirements for that phase. The requirement on the AD & C subsystem at each distinct mission segment is called a control mode. Therefore, control modes of a satellite divide the mission into segments, according to specific attitude determination and control requirements. Our mission analysis points out four different control modes listed below. Detailed explanations for the each specific control mode can be found in the requirements section below. 1.1) Control Modes: Control Modes Explanation 1 ) Spin-Up Mode - Acquisition of Stability through spin-up maneuver after releasing from launch vehicle 2 ) Normal Mode - Orbit determination phase by tracking from Purdue ground station 3 ) Thrusting Mode - 2 Thrusting experiments for evaluating the “two-burn” thrusting scheme - 1 thrusting for evaluating the “Ramp-up” thrusting scheme while de-orbiting 4 ) Safe Mode - Stand-by mode in case of a major malfunction Table VIII-1: Control modes. 1.2) Performance Requirements: Specific requirements for each control mode were determined by the customer attributes and the minimal cost consideration. In other words, each specific requirement is sufficient enough to assess the validity of the thrusting experiment properly at a minimum cost. 1.2.1) Spin-Up Mode : This mode is the phase just after leaving the launch vehicle in orbit. Initially, the spacecraft will be unstable prior to spinning. Two pairs of 4-array small solid side thrusters will achieve spin-up with the aimed rate of 25 rpm. This angular rate has no significance, as long as it is within the range of 20 ~ 30 rpm. This flexibility is due to the robustness of the “two-burn” and “ramp-up” thrusting schemes which provide significant pointing error reduction, theoretically, for any given spin rate. However, due to the limitations on thruster on-off timing at high spin rates (we can’t turn-on and turn-off the thruster very quickly as it takes some time to reach a specific thrust value), it is not practical to apply this thrusting scheme at very- high spin rates. Fortunately, most spacecraft are spun around 20 ~ 30 rpm since higher spin rates cause excessive structural loads on the spacecraft. Thus we selected a nominal value of 25 rpm as the spin rate for our spacecraft, as it is practical for our experiment while employing a commonly used spin rate. In case of a malfunction in some of the side thruster units, the remaining thrusters will still be enough to spin the s/c within the desired range since we have 8 separate side thruster units. At the end of burnout of the side thrusters, the spacecraft will be spinning with a possible nutation about the maximum moment of inertia axis (resulting from uncertainties with initial motion and possible side thruster misalignments etc.). The excessive nutation will be damped out using a passive nutation damper. Attitude Determination Requirements: 18 - All attitudes (meaning that any random attitude shall be sensed) - Spin rate between 10 ~ 60 rpm - Accuracy: Spin rate must be sensed within 0.01 rad/s accuracy Attitude Control Requirements: - Accuracy: Not important - Range: Spin rate between 20 ~ 30 rpm o - Any nutation bigger than 0.1 shall be damped after spin-up maneuver using a nutation damper - Settling time for nutation shall be less than 3 hours 1.2.2) Normal Mode : This is the phase in between thrusting maneuvers. Spacecraft will be spinning without nutation. Attitude of the spacecraft will be monitored at this phase. Ground station will provide the orbit tracking. The spacecraft will be waiting for any command uplink and data downlink. Thus, antennas should provide enough coverage for communication at this phase. Attitude is random and inertially fixed due to spinning. Random attitude means that we are not too concerned about the orientation of the satellite since any orientation will allow us to perform our mission. Attitude Determination Requirements: - Inertially fixed due to spin - All attitudes (orientation + spin rate) o - Attitude will be monitored within 0.5 accuracy - Orbit will be tracked from the ground station Attitude Control Requirements: - None 1.2.3) Thrusting Mode : This is the control mode to be used during the thrusting experiments. The initial and final orientations of the spacecraft shall be determined precisely. Velocity changes shall be monitored during experiments. Each experiment shall be approximately 3 minutes long (thrusting experiments are not very sensitive to small variations in burn times). Attitude Determination Requirements: - Spin rate must be sensed within ~0.01 rad/s precision. - Attitude must be sensed within ~0.5 deg accuracy (Before and after thrusting maneuver) - Accelerations must be sensed with 0.01 m/s2 accuracy during thrusting Attitude Control Requirements: - None. - The effect of the nutation damper must be insignificant during thrusting (for 3 min). This is a reasonable assumption although the nutation damper will be operational during the experiments. Since the nutation is expected to be eliminated in less than 3 hours, we can overlook the effect of the nutation damper for a three minute period. 1.2.4) Safe (Stand-by) Mode: 19 Safe mode is the operating condition when there is a major malfunction like a thruster problem, spin-up booster malfunction etc. The spacecraft should cut down the energy consumption to minimum and wait for commands from the ground station. Telemetry system should function and must be kept at full power in order to receive commands. Attitude information will be sensed and stored for downlink if possible. Attitude Determination Requirements: - If spinning, spin rate must be measured and stored - Attitude shall be sensed and stored if possible - Orbit shall be tracked from the ground station Attitude Control Requirements: - None. 2) Orbit and Disturbance Environment In order to maximize the possibilities of launching as a secondary payload in any launch vehicle (which is the main driver for determining the envelope of the satellite), the LEO was selected as the mission orbit. Most of the Earth orbiting satellites are placed and lunched in the LEO, so that choosing this orbit regime will maximize the possibilities of getting a ride into the orbit. The orbit selected has a mean altitude of 350 km with a nominal inclination of 40o. The altitude is somewhat smaller than usual LEO orbits due to the de-orbiting requirement. The selected orbit parameters were used in the calculation of disturbance environment torque calculations. Typical values for solar and magnetic exposures, aerodynamic and gravitational disturbances for this specific orbit were assumed for the calculations, details of which are given in the Appendix-B. The disturbance torques considered are: gravity gradient, solar pressure, magnetic and aerodynamic disturbances of the Earth. The results of the calculations are, in summary: Disturbance Environment Disturbance Torque (N.m) 3.53380 × 10 -7 Gravity Gradient 7.65525 × 10 -8 Magnetic 3.77534 × 10 -5 Solar Pressure 7.03584 × 10 -6 Aerodynamic Table VIII-2: Disturbance torques. As the results point out, the disturbance torques are very small and insignificant compared to the stiffness of the spacecraft. Therefore, for the length of the mission and the possible disturbance environment we do not need to have an active attitude control & compensation system since our experiments do not require an active control other than the initial spin-up. The nutation frequencies due to the disturbance torques about the two smaller principal axes of the spacecraft were found to be; ω ni |x = 39.0625rpm ω ni | y = 34.7222rpm for the fixed spin rate of 25 rpm. The details of the calculations are given in Appendix B. 3) Attitude Control Strategy As the experimental approach requires (this is an external requirement), the only spacecraft control will be passive spin stabilization. This is realistic since, any spinning-thrusting vehicle has an inherent gyroscopic stiffness and therefore attitude control systems are not activated without slowing down (if not stopping) the spin of the spacecraft. Thus, the spacecraft is going to be a single spinner (pure spin) with inertially fixed attitude in the LEO (Low Earth Orbit) with no attitude control. 20 During the trade studies, several options ranging from a 3-axis stabilized platform to a variable spin control enabling platform and several combinations of other possible options were considered. However due to the nature of the mission, having any control on the spinning platform requires a high degree of sophistication that is beyond the reach of the mission mass, size limitations and the desired low cost requirement. The decision of using no attitude control was chosen because of great simplicity and cost effectiveness. The inertially fixed attitude due to spinning can also be used to determine the thrusting direction, albeit in a limited sense, by specifying the time of thrusting for a specific location in orbit. Although this will not give complete flexibility, it will be sufficient to achieve our mission goals. The de-orbiting requirement will be achieved by the same control idea, again setting the time of initiation of the burn for the solid rocket booster with the aim of decreasing the relative spacecraft velocity and possibly pushing it towards the atmosphere for orbit decay and eventual reentry. 4) Selection & Sizing of ADCS Hardware In light of above requirements, calculations and design decisions, the attitude determination equipment was selected so that all the selected components are compatible with the accuracy requirements set in “Major Requirements” section above. The specific hardware selected for the mission consists of 2 sun sensors, a 3-axial accelerometer, a rate gyro and 2 nutation dampers. Nutation dampers will be of custom design, and the compliance with the nutation damping requirements will be determined during testing The specific information about each selected component is as follows: 4.1) Sun Sensors ( × 2) : Vendor: TNO TPD Space Instrumentation Part : Sun Acquisition Sensor (SAS) • Field of view: Hemispherical, typically +/- 97 degrees about bore sight. • Accuracy: Better than +/- 0.5 degrees on bore sight for GEO missions under all environmental conditions and for the whole mission lifetime. Albedo will degrade the accuracy in LEO. • Power consumption: No input power required. • Electrical output: In current mode 0 - 30 mA. In voltage mode 0 - 200 mV. Output can be of individual detectors or of combinations of detectors (balance, sum). • Operating temperature: Typically -100 C to +100 C. • Mass/dimensions: Mass : 0.155 kg Dimensions: 110 x 110 x 28 mm without connector, alignment cube, grounding stud or specific baffling. • Reliability: Depend strongly on output arrangement (single cell or combination output) and philosophy with regard to redundancy; in SAS for GEO application outputs are redundant; in SAS for LEO application only single-cell type of output is redundant; failure probability for single cell voltage output 2.4 x 10-4 worst case (+100 degrees C) per year mission duration. • Qualification status: Fully qualified and flight proven sensor. A technical drawing and a photo of the part are given on the next page. 21 Figure VIII – 1: Sun Sensor. Figure VIII –2: Sun Sensor. 22 4.2) Gyroscope ( × 1) : Vendor: B E I -T E C H., I N C. SYSTRON DONNER INERTIAL DIVISION Part : Model QRS11Micromachined Angular Rate Sensor 23 Figure VIII-3: Gyroscope. 24 Figure VIII-4: Gyroscope assembly drawing. 25 4.3) Accelerometer ( × 1) : Vendor: PCB PIEZOTRONICS Part : 356B07 Low-Noise Triaxial ICP® Accelerometer Figure VIII-5: Accelerometer . 26 Figure VIII-6: Accelerometer assembly drawing 4.4) Nutation Damper ( × 2) : A spacecraft undergoes periodic motion if it is disturbed from a stable equilibrium position. For a pure spin-stabilized spacecraft, this periodic motion is rotational and is known as nutation. Nutation occurs as a result of control and environment torques, separation from the launch vehicle or, as in our experiment, it may result from offset thrusters. The problem of nutation damping is that of aligning the nominal spin axis with the angular momentum vector by dissipating the excess rotational kinetic energy associated with the nutation motion. Also it should be noted that nutation damping is only possible when the spacecraft is spinning about the maximum principle moment of inertia axis, as in our design. Nutation motion can be damped by passive and active devices. A passive damper is one which does not require attitude sensing, is driven by the motion itself, and dissipates rotational kinetic energy. The frequency of the damper is intentionally kept near or equal to the rigid body frequency so that it significantly affects the motion of the spacecraft. Nutation damping plays a significant role in this mission, since the validity of the thrusting experiments will depend highly on the initial attitude of the spacecraft. Any possible nutation prior to the thrusting experiments will cause additional thruster misalignment to the existing intentional offset, and will reduce the validity of the experiment. Available options for passive nutation dampers and their characteristics are shown below; Damper Energy Dissipation Characteristics Type Mechanism Pendulum fluid friction sturdy, long life Eddy Current eddy currents delicate, high energy dissipation rate, variable damping constant Ball-in- rolling and fluid sturdy, long life, remains tuned for different Tube friction spin rates Viscous Ring fluid friction Simple construction, long life Table VIII-3: Nutation damper types. A simple comparison of the available options is given below: Eddy Ball-in- Viscous Pendulum Current Tube Ring Simplicity - - + + safety + + + + 28 support multiple spin rates - + + + price + - + - Table VIII-4: Trade study for nutation dampers. According to table VIII-4, the ball-in-tube nutation damper was selected for our spacecraft. Nutation dampers are custom built components and designs tend to change from one spacecraft to another. A ball-in-tube damper consists of a closed, curved tube in which a ball is allowed to roll freely. The damping caused by rolling friction may be augmented by viscous damping if the tube is filled with a liquid. The ends of the tube may have energy-absorbing bumpers. The damper behaves like a centrifugal pendulum and its frequency of vibration is directly proportional to the spin rate of the body on which it is mounted. Hence if such a damper is tuned initially, it remains tuned for different spin rates. For our spacecraft we are going to have 2 ball-in-tube nutation dampers, so that we comply with the requirements. The flexibility of the antennas will also contribute to the energy damping so that our nutation settling time will be less than 3 hours. The dimensions of the damper are shown on the figure below: Figure VIII-7: Ball-in-tube nutation damper. The tube will be filled with a viscous fluid (such as engine oil), which will cause energy damping due to friction as the ball reciprocates in the tube. During manufacturing 29 phase the nutation damping capability of the device will measured with simple tests and a suitable viscous fluid will be selected to achieve the design requirements. The total mass of a single unit was estimated to be 0.5 kg. Burn Thrusting Scheme A very good way of providing directional stability for spacecraft and rockets is to spin them about their maximum or minimum principal axes. We know from dynamic analysis of rigid bodies that the angular momentum of a spinning rigid body will remain constant unless acted upon an external torque. Due to production tolerances, small errors in the thruster location and direction are inevitable. Therefore an axially thrusting-spinning spacecraft or rocket will experience unwanted transverse torques during the thrusting maneuver1, as shown in figure VIII-8. In the Figure VIII-8: Thrusting Problem example configuration thruster offset causes a body fixed torque in to the page. We know that such a torque will distort the angular momentum vector in inertial coordinates and cause it to trace a circular path as it is shown in figure VIII-9. The average angular momentum bias angle ρ is measured from the vertical and in the YZ plane, as shown in the reference1 and it is shown that the ∆V pointing errors occur along the axis set by ρ in axially thrusting spin-stabilized spacecraft and rockets. In the figure, Ho shows the initial position of the angular momentum vector and the H vector is the angular momentum during the thrusting maneuver. 1 Longuski J.M., T. Kia, W.G. Breckenridge, “Annihilation of Angular Momentum Bias During Thrusting and Spinning-up Maneuvers,” The Journal of the Astronautical Sciences , Vol. 37, No.4, October-December 1989, pp.443-450. 30 Figure VIII-9 The Angular Momentum and Velocity Pointing Bias During Thrusting A remedy for the problem lies in using a two-burn scheme as proposed1 .And indeed, this method is the simplest and probably most effective way of achieving a solution, provided that we have an on-off type pulse thruster. With the conclusion that that the velocity pointing error will occur along the direction set by the angular momentum bias, then the basic idea lies behind eliminating the angular momentum bias. We know that for the case of spinning and axially thrusting spacecraft with the presence of thruster offset, there is no way of eliminating the angular momentum bias except highly sophisticated controllers. However, with the two-burn scheme proposed by Longuski et al.1 it is possible to eliminate the time average of the angular momentum bias. The idea is to shift the center of the circle that is traced by the angular momentum vector to the origin of the inertial axis system as shown in figure 3 below. Figure VIII-10: Initial and Final Paths of the Angular Momentum Vector in the Two- Burn Scheme Uncompensated angular momentum vector moves on the depicted “initial angular momentum path”. When the angular momentum vector comes to the point A in the figure VIII-10, along the solid line, the thruster is turned off (coasting) and consequently the angular momentum vector stops moving since there is no external torques acting on the spacecraft. The A point corresponds to a 60o rotation of the spacecraft and the time required to arrive this point can be found simply from the relation t b = π / 3Ω , where, tb denotes 1st burn time and Ω is the spin rate. We note that if we define θ as the spacecraft rotation angle, the time relation of it is simply θ = Ω ⋅ t After the first ignition, the thruster is kept off for a period of “coast time” t c = π / 3Ω and after that it is ignited again for the rest of the maneuver. In the end this causes the angular momentum vector to fall in the track of the final path shown in dashed lines, 31 which has an average angular momentum bias of 0 degrees. The resulting behavior of the velocity pointing error and angular momentum path can be obtained numerically using the designed spacecraft parameters (MOI, thrust and an intentional offset value of 5 cm). The promise of the two-burn scheme is evident from the plots given for numerical simulations. Simulations were performed for 120 seconds (2 minutes). 32 Figure VIII-11: Simulation of Angular Momentum Path and Velocity Path without the Two-Burn Scheme Figure VIII-12: Non-dimensional velocity time history (the last plot is the dimensional z- velocity) without the Two-Burn Scheme 33 Figure VIII-13: Dimensional time history of the position vector without the Two-Burn Scheme 34 Figure VIII-14: Simulation of Angular Momentum Path and Velocity Path with the Two- Burn Scheme 35 Figure VIII-15: Non-dimensional velocity time history (the last plot is the dimensional z- velocity) with the Two-Burn Scheme Figure VIII-16: Dimensional time history of the position vector with the Two-Burn Scheme According to the numerical results, the spacecraft departed about 0.45 m laterally during 2 minute thrusting using the two-burn scheme as compared to the approximately 5 m lateral departure of the single-burn case. So, these are some of the results that we want to demonstrate with our experiments. 36 IX. Communication and Data Handling 1) Control Modes 1.1) Transmit Mode: This mode is commanded when an experiment has been performed and the data is ready to be down-linked. The processor shall be commanded to choose from which of the data for the three experiments shall be down-linked. Along with the data, the battery state of charge values and temp sensor data shall also be down-linked before and after data transmission. During this time, the receiver shall also be on. This will allow commands to be uplinked if problems occur in the transmission. 1.2) Normal mode: This mode is the default mode after acquisition of signal. The transmitter shall be off during this time, and the receiver shall be on. 1.3) Acquisition/contingency mode: This mode serves as both the initial signal acquisition mode and the contingency mode. The spacecraft transmits a beacon signal allowing the ground station to track it before sending first command. This is also a low power mode, which will still allow the ground station to track the spacecraft. 2) Link Budget The design requirements assume that the altitude, h, of the satellite is 350 km at an inclination of 40 degrees. From orbit calculations (see Appendix), the total time of the longest pass is 13 minutes, which occurs at least twice a day. The requirement to down-link data was found to be 1200bits/sec. Each experiment lasts 120 seconds. Each sample of accelerometer data will require at most 1 byte per axis (3 bytes total per sample); likewise with the gyro data. Using a sampling period of .01 seconds, the total amount of data for each experiment is 576,000 bits. Including a margin of 5%, the total increases to 604,800 bits per experiment. Using the data rate of 1200bps, it will take approximately 8.4 minutes of continuous transmission to down-link 1 experiment. Total on-board data storage for three experiments is then 1,814,400 bits (226.8 Kbytes). Table 1 shows the link budget parameters for the satellite. Some of the equations used are located in the Appendix. Units Uplink downlink Frequency MHz 150 435 transmitter power Watts 50 2 37 transmitter power dBW 16.99 3.01 transmitter line loss dB -1 -1 transmit antenna length meters 0.6667 0.1667 transmit antenna gain dBi 14 2 equiv. isotropic radiated power dBW 29.99 4.01 Propagation path length km 350 350 space loss dB -146.85 -156.101 propagation & polarization loss dB -0.3 -0.3 receive antenna length meters 0.5 2 receive antenna gain dBi 2 14 system noise temperature K 614 221 data rate bps 400 1200 Eb/No dB 59.537716 35.97326 C/No dB-Hz 85.558316 66.76508 bit error rate 1.00E-05 1.00E-05 required Eb/No dB 2.5 2.5 Implementation loss dB -2 -2 Margin dB 55.037716 31.47326 Table IX-1: Link Budget 3) Hardware Sizing, Commands, & Requirements Much of the hardware chosen was based on the fact that they were used on previous successful spacecraft. Figure 1 shows the top-level block diagram for the telemetry, communications, command and data handling subsystem. All the power consumption and mass data is located in the budget section of this document. 38 Figure IX-1: Block diagram for telemetry, communications, command, and data handling subsystem The processor samples the values of temperature sensors, sun sensors, and batteries to be made available for telemetry, and samples the values of the gyro and accelerometer, which is stored in the data module. Table 2 shows the command list that will be defined in the processor’s ROM. Command Mode Description 1. Spin-up normal All sensors are on. Processor thrusters sends command to fire spin- up thrusters. 2. Perform normal All sensors are on. Processor experiment #1 sends command to fire cold gas thruster for experiment #1 3. Perform normal All sensors are on. Processor experiment #2 sends command to fire cold gas thruster for experiment #2 4. Perform normal All sensors are on. Processor experiment #3 sends command to fire cold gas thruster for experiment #3 5. Downlink data transmit All sensors are on. Processor from experiment routes data from experiment #1 #1 and downlinks with telemetry. 6. Downlink data transmit All sensors are on. Processor from experiment routes data from experiment #2 #2 and downlinks with 39 telemetry. 7. Downlink data transmit All sensors are on. Processor from experiment routes data from experiment #3 #3 and downlinks with telemetry. 8. Abort downlink acquisition/contingency All sensors are on. Transmitter quits sending data, and beacon is transmitted. 9. Command to acquisition/contingency Temp sensors and battery Power- voltage monitors are safe/overheat streaming data to processor. mode All other sensors are off. Once downlink telemetry command is issued, temp sensor and battery state of charge data will be made available through telemetry. 10. Command to acquisition/contingency Temperature sensors are Low temperature streaming data to processor. mode All other sensors are offOnce downlink telemetry command is issued, temp sensor data will be made available through telemetry 11. Command to normal All sensors commanded on. Normal mode Transmitter is shut off. 12. Downlink transmit Chosen sensors are on. Telemetry Battery state of charge, temperature sensor data, gyro data, accelerometer data and sun sensor data are made available through telemetry. Table IX-2: Command List Hardware Choice: A.) Processor/data module: The processor/data module was chosen from the Citizen Explorer space- qualified board. The CoreModule 3SXi board built by Ampro Computers, Inc. utilizes a 25MHz 386SX-compatible CPU. The 4 Megabyte surface- mounted memory module supplies more than enough storage for the 3 experiments (226.8 Kbytes). Operating temperatures are 0º-70ºC 40 standard or -40º-85º C specially ordered. Figure 2 shows a picture of the board. Figure IX-2: Processor board with data module B.) Quarter-wavelength monopole antenna: Monopole, omni-directional antennae were chosen since there is no active attitude control. One pair of antennae is used for transmitting (69 cm wavelength), and one is used for receiving (2 meter wavelength). Both of the frequencies for transmitting and receiving were chosen because of the requirement that the satellite be able to communicate with the Purdue University ground station. Each pair of antennae shall be mounted on opposing corners of the satellite. The antennae shall be made of unpainted tape measures that are cut to a quarter of the wavelength. This follows the design of PCsat built by the Navy. C.) Terminal node controller (TNC), transmitter, and receiver: These were chosen also based on the design of PCsat. The terminal node controller, the KWM-1200plus (see Figure3), built by Kantronics, shall be used to package the data per the AX.25 amateur radio protocol. It is capable of the required 1200bps data rate for down-linking data. The transmitter (Figure 4) and receiver (Figure 5) are also off-the-shelf amateur radio parts built by Hamtronics. They were chosen based on the supportable frequencies (435MHz down-link, 150 MHz up-link). 41 Figure IX-3: KWM-1200plus Terminal Node Controller Figure IX-4: Hamtronics T304 Transmitter board Figure IX-5: Hamtronics R304 Receiver board X. Thermal Control Subsystem X.1 Overview The thermal control subsystem (TCS) is an integral part of every spacecraft. It's purpose is to maintain all the components of a spacecraft within their respective temperature limits. There are several different sources of thermal energy acting on a spacecraft; solar radiation, albedo, earth emitted infrared, and heat generated by onboard equipment. Therefore, the thermal control subsystem is different for every spacecraft. In general, there are two types of TCS, passive and active. A passive system relies on conductive and radiative heat paths and has no moving parts or electrical power input. An active system is used in addition to the passive system when passive system is not adequate, for example, on manned missions. Active systems rely on pumps, thermostats, and heaters, use moving parts, and require electrical power. Based on the calculations contained in this section, it is concluded that soley insulation is needed in the satellite’s thermal control design, resulting in a passive thermal system. The satellite’s outer surface (0.6 m2) insulation will consist of 0.5 mil aluminized Kapton/ITO and thin nets of Dacron. The resulting equilibrium temperature during eclipse, will now be within the satellite’s temperature limits. 42 Type Kapton/ITO 0.5 mil aluminized Absorptivity 0.34 Emissivity 0.55 Area Density 7 g/m2 Area 0.6 m2 Total Weight .0042 kg Total Cost $30 Table X-1: Thermal Control Design Specifications Fig.(X-1) below, gives an overview of the design process for the thermal control system. 43 Figure X-1: Design Process for Thermal Control System X.2 Temperature Ranges The following table shows the operating temperature ranges for all of the satellite’s components. 44 Category Description Tmin (°C) Tmax (°C) Spacecraft Internal Units Worst case envelope 7 50 Telecommunications Payload Units -10 50 Onboard Computer -10 50 Telemetry & Command Units -10 50 Electrical Power Batteries Power Control Unit Attitude Control Sun Sensors -100 100 Propulsion Tanks, filters, valves, lines 7 55 Thrusters 7 55 Harness Spacecraft internal -15 55 Thermal Control Multilayer Insulation (MLI) -160 250 Structures Nonalignment critical -45 65 Antennas TT&C -65 95 Table X-2: Temperature Operating Ranges for Satellite Components X.3 Preliminary Thermal Performance Thermal analysis, found in Table X-3 below, was performed based on the spherical satellite analysis outlined in SMAD III. Equations used, to calculate the values, are listed in Appendix E. Figure X-2: Thermal Radiation Environment 45 No. Item Symbol Value Units Source Comments 1 Surface A 0.6 m2 satellite satellite is Area .3x.3x.35 m3 2 Diameter of D 0.437 m Eq. (E-1) sphere which has equal surface area 3 Max. power QW 12 W satellite given produced 4 Min. power QW 9 W satellite given produced 5 Altitude H 350 km given 6 Radiaus of RE 6378 km given Earth 7 Angular ρ 1.247 rad Eq.(E-2) radius of Earth 8 Albedo Ka 0.978 -- Eq. (E-3) correction 9 Max. Earth qI 258 W⋅m-2 Fig. (X-2) Use for worst- IR emission case hot at surface 10 Min. Earth qI 216 W⋅m-2 Fig. (X-2) Use for worst- IR emission case cold at surface 11 Direct solar GS 1418 W⋅m-2 Fig. (X-2) Use max. value flux 12 Albedo a 0.35 % Fig. (X-2) Use max. value 13 Emissivity ε 0.92 -- White Paint 14 Absorptivity α 0.25 -- White Paint 15 Worst case TMAX -10 °C Eq. (E-5) hot temp. 16 Worst case TMIN -73 °C Eq. (E-6) cold temp. 17 Upper temp. TU 45 °C Table X-1 Assume 5°C limit thermal margin 18 Lower TL 12 °C Table X-1 Assume 5°C temp. limit thermal margin Table X-3: Preliminary Thermal Performance Estimates The values found in Table X-3 for TMAX and TMIN appear to be significantly lower than the expected temperatures seen by the satellite. It is concluded that the spherical model 46 analysis of the satellite is inaccurate for such a small satellite so close the Earth. As a solution to this discrepancy, historical TMAX and TMIN data taken from the FalconSat2 (an existing satellite with relatively the same size and altitude as this satellite) will be used to complete a more accurate analysis. Therefore, TU = 45o C TMAX = 12 o C ü ï from FalconSat2 ý TL = 12 o C TMIN = −3 C ï o þ It is seen that the worst case hot temperature (12°C) lies within the upper and lower temperature limits of the satellite, while the worst-case cold temperature (-3°C) falls below the lowest allowable operating temperature. It will, therefore, be necessary to use a thermal control system in order to maintain an equilibrium temperature of at least 12°C, during eclipse. X.4 Choosing Coating/Insulation Material It must first be determined how much energy needs to be dissipated. Solving the energy balance, Eq.(E-4), for heat dissipated yields: Qdis = −120.6 W Therefore, 120.6 W must be dissipated in order to achieve thermal equilibrium. Consider a satellite which must dissipate heat. Qdis can be broken down into the sum of Q of the insulation and Q of the radiator. Assuming no radiator, the coating emissivity required to maintain the satellite at an equilibrium temperature of 12°C (lower temperature limit) is: ε = 0.55 The following table gives the thermal properties for various types of coating material: Material Solar Absorptivity Infrared Emissivity α ε White Apoxy 0.248 0.924 Black Paint 0.975 0.874 Aluminized Teflon 0.163 0.8 Silvered Teflon 0.08 0.66 Kapton/ITO 0.5 mm 0.34 0.55 aluminized Aluminum tape 0.12 0.06 Table X-4: Radiation Properties 47 Aluminized Kapton (0.5 mm) possesses the exact required emissivity value, and will therefore be chosen as the insulation material. This will yield an equilibrium temperature of 12°C during eclipse, which is an acceptable value for the satellite. The aluminized Kapton will cover the entire outer surface of the satellite, and will therefore have a total mass of 4.2 grams. A conservative cost estimate for the coating is $30. 48 XI. Structure and Mechanisms 49 XII. Propulsion Cold Gas Thruster (CG) a. Major Requirements - The requirements as stated in the systems requirement document (SRD) are the cold gas thruster providing approximately 1.0 Newton of thrust for a single burn and two-burn maneuver, which combined have a total velocity change of 6.0 meters/second. Other requirements that became factor are the size of tank, because the spacecraft was space limited. b. Concept of Operation - After spacecraft is spun up using the spin-up thrusters data will be sent to the ground to gather necessary information on attitude prior to the first burn. After this collection the computer will send an electronic signal to the on/off valve to open for 120 seconds then shut send another signal to close the on/off valve. After data has been sent for the burn and data has been sent prior to the next burn after the spacecraft has been allowed to stabilize. Again the computer will send a signal to open the on/off valve, close the on/off valve for the already mentioned first part of the two burn scheme then open again for the rest of the burn then close again. This completes the operation of the (CG) thruster and any leftover propellant in tank could be used to further de-orbit spacecraft after the solid rocket motor fires. Gas N2 Gas (kg) 0.412 Tank Sphere Tank (kg) 1.76 Tank Material D6aC Steel Pressure Regulator (kg) 0.23 Overall Diameter (cm) 12.52 On/Off Valve (kg) 0.46 Thickness (cm) 0.49 Nozzle (kg) 0.012 Piping (kg) 0.00146 Pressure Regulator TESCOM Total (kg) 2.87546 Part Number BB-13PL3KEB2 Diameter (cm) 4.42 Table VII-2: Cold Gas Component Mass Height (cm) 5.72 On/Off Valve Metal W ork Pneumatic Part Number PIV22I0SNC Diameter (cm) 3.45 Length (cm) 3.00 Nozzle Cone Nozzle Material 2219-Al Length (cm) 2.00 Thickness (cm) 0.50 Cone Angle (deg) 5.25 Throat Diameter (cm) 0.08 Exit Diameter (cm) 0.48 Piping Copper Total Length (cm) 1.0 Diameter (cm) 0.3175 Table VII-1: Cold Gas Component Dimensions 50 Figure VII-1: Cold Gas Diagram System c. Component Selection and Sizing - Table VII-1 gives dimensions of all components while table VII-2 gives mass of all components and Figure VII-1 gives the schematic of the cold gas system. The selection of nitrogen gas was determined because of its low reactivity, safe, performance, and zero freezing due to water droplets. The design and use of nitrogen provides 0.95 Newton of thrust, which is within 5% of the 1.0 Newton thrust desired in the SRD. The tank material was determined by comparing various materials and consulting how much room was available in the spacecraft for the tank. The D6aC material provided the smallest case thickness and mass for the tank. The pressure regulator was chosen for its ability to handle the pressures; ability to set pressure drop through system so after testing can adjust for pressure drops, and being small. Figure VII-2 shows the appearance and workings of the pressure regulator courtesy of Tescom Corporation. 51 Fig VII-2: Functional Schematic of Pressure Regulator The on/off valve was selected due to its ability to turn off or on by electric signal, light, small, small power consumption of only 0.9 watts during operation, and able to get the job done. The nozzle was designed for simplicity and the limiting factor does not come from firing but from launch where it supports the rest of the system as the only connection it has to the rest of the spacecraft. The equations and method for solving the component selection and sizing problem can be found in the appendix. d. Propellant Budget - 1-Burn 2-Burn mp (kg) 0.199 0.199 mr (kg) 0.00656 0.00656 mt (kg) 0.206 0.206 ∆V (m/s) 3.58 3.58 SRD ∆V (m/s) 3 3 Margin (%) 19.34 19.34 Table VII-3: Cold Gas Thruster Propellant Budget e. Trade Studies/Comparisons - Gas mt (kg) rtank(cm) F (N) Isp (sec) ∆ V (m/s) He 0.18 10.74 0.98 169.09 7.38 Air 0.42 5.74 0.95 71.86 7.18 N2 0.41 5.77 0.95 72.81 7.16 CO2 0.46 5.25 0.92 62.87 6.96 Table VII-4: Propellant Gas Comparison Material ρ (kg/m^3) Ftu (GPa) tcs (cm) mtank (kg) D6aC Steel 7830 1.52 0.49 1.76 2219 Al 2800 0.41 1.82 2.86 Titanium 4460 1.23 0.61 1.26 Graphite 1550 1.34 0.56 0.40 Table VII-5: Tank Material Comparison for N2 Gas Solid Rocket Motor (SRM) a. Major Requirements - The solid rocket motor as stated by the SRD is required to provide approximately 43.3 m/s change of velocity while providing a trapezoidal thrust 52 scheme and able to fit within the space limitation. Obtaining a low maximum thrust is a secondary objective to accomplishing the above two stated objectives. b. Concept of Operations - After the two burn schemes and the spacecraft has stabilized an electronic signal from the computer shall let the capacitor discharge itself igniting the Pyrogen igniter. The Pyrogen igniter ignites the propellant grain and discharges the nozzle plug. The propellant burns in the trapezoidal scheme, as depicted by figure VII-3, due to the geometry of the casing until all propellant is gone and thus ending the SRM operation. Figure VII-3: Thrust Profile of SRM c. Component Selection and Sizing - Table VII-6 illustrates the SRM’s component dimensions and Table VII-7 displays the mass breakdown of the component and figure VII-4 shows the diagram of the SRM. 53 Propellant TP-H-3340 Propellant (kg) 1.059 Aluminum 18% Tank (kg) 0.192 Ammonium Perchlorate 71% Insulation (kg) 0.207 Hydroxy-terminated 11% Nozzle (kg) 0.028 Polybutadiene (HTPB) Igniter (kg) 0.057 Tank Cylinder w/ Plug (kg) 0.005 Hemispherical Total (kg) 1.548 End Caps Material D6aC Steel Table VII-7: Solid Rocket Motor Component Mass Overall Diameter (cm) 11.85 Length (cm) 7.06 Thickness (cm) 0.1 Insulation Ablative Material Carbon/Phenolic Thickness (cm) 0.62 Nozzle Cone Material (Body) D6aC Steel Material (Insert) Pyrolytic Graphite Length (cm) 0.21 Cone Angle (deg) 12.0 Throat Diameter (cm) 1.0 Exit Diameter (cm) 1.09 Plug Diameter (cm) 1.01 Table VII-6: Solid Rocket Motor Component Dimensions Figure VII-4: Diagram of SRM The propellant selection was chosen for its common use, low cost, and ability to be obtained by Purdue University. It had to have the propellant grain design of end burning, because only possible way to have a low thrust and be able to accomplish the task of the trapezoidal thrust scheme. The casing is to be made of D6aC steel, because of its high strength and common usage in solid rocket motors, such as the space shuttle solid rocket boosters. The insulation material will be made of carbon/phenolic, because of its relatively low erosion rate. The low erosion rate is needed along with minimum char build-up to provide the desired thrust and to prevent blockage of the throat during thrusting. The nozzle is to have D6aC steel as the body with pyrolytic graphite insert in the throat region due to the high temperatures and the necessity that the throat erosion be kept to a minimum so that the trapezoidal thrust is possible. The pyrogen igniter was selected because of minimum power input and common usage. Finally a nozzle plug is needed to prevent debris from entering the rocket while the cold gas thruster performing its experiments. d. Propellant Budget – 54 mp (kg) 1.059 mr (kg) 0.0 mt (kg) 1.059 ∆V (m/s) 43.3 SRD ∆V (m/s) 43.3 Margin (%) 0.0 Table VII-8: Solid Rocket Motor Propellant Budget e. Trade Studies/Comparisons - Material ρ (kg/m^3) Ftu (GPa) tcs (cm) mtank (kg) 2219-Al 2800 0.413 0.0321 0.0219 Titanium 4460 1.230 0.0108 0.0117 D6aC Steel 7830 1.520 0.0087 0.0166 4130 Steel 7830 0.862 0.0154 0.0293 Graphite 1550 1.343 0.0099 0.0037 Kevlar 1380 0.964 0.0138 0.0046 Fiberglass 1990 1.100 0.0121 0.0058 Table VII-9: Solid Rocket Motor Case Material Material Erosion Rate (cm/s) tinsul (cm) minsul (kg) Pyrolytic Graphite 0.05 0.232 0.119 Polycrystalline Graphite 0.10 0.464 0.188 3-D Carbon/Carbon 0.10 0.464 0.210 Carbon/Phenolic 0.18 0.835 0.288 Graphite/Phenolic 0.28 1.298 0.467 Silica/Phenolic 1.30 6.027 3.699 Glass/Phenolic 1.50 6.955 5.040 Paper/Phenolic 1.90 8.809 4.463 Table VII-10: Insulation for Solid Rocket Ablative Material 55 XIII. Summary of Spacecraft a) Requirement Compliance Design of the spacecraft and all the subsystems are based major design requirements given in section III of the report. The specific requirements on each subsystem level and the corresponding design decisions were explained in detail within subsystem overviews. In summary, our designed spacecraft complies with most of the subsystem level requirements and all of the major mission requirements, while allowing for some contingency margins. Compliance with some subsystem level requirements couldn’t be determined within the scope of the project objectives, as these require more sophisticated, in-depth study. Simplified analyses were used for such complicated decisions. b) Cost Estimation Power $ 2,000 ADCS $ 44,550 Communications and Data Handling $ 430 Thermal $ 30 Structure and Mechanisms $ 200 Propulsion $ 780 Total for Production $ 47,990 Launch 200,000 Total $ 247,990 Table XIII-1: Cost estimation assuming labor is at no cost. c) Areas Needing Further Design Consideration 1. Nutation Damper Testing. 2. Venting of Sulfur Dioxide during battery short. 3. Construction of SRM plug. 56 XIV. References [1] Longuski J.M., T. Kia, W.G. Breckenridge, “Annihilation of Angular Momentum Bias During Thrusting and Spinning-up Maneuvers,” The Journal of the Astronautical Sciences , Vol. 37, No.4, October-December 1989, pp.443-450. [2] Javorsek II, D. and J.M. Longuski, “Velocity pointing errors associated with spinning thrusting spacecraft,” Journal of Spacecraft and Rockets, Vol. 37, No. 3 , May-June 2000, pp.359-365 [3] Wertz, R. J., and W. J. Larson, Space Mission Analysis and Design, 3rd Ed., Kluwer Academic Publishers, Microcosm Press, El Segundo CA 1999. [4] Wertz, James R., Spacecraft Attitude Determination and Control, D. Reidel Publishing Company, Dordrecht, Holland,1984. [5] Chobotov, Vladimir A., Spacecraft Attitude Dynamics and Control, Krieger Publishing Company, Malabar-Florida, 1991. [6] Longuski, James M., AAE 507 Principles Of Dynamics, Class Notes, Fall 2001 semester, Purdue University, West Lafayette, 2001. [7] Greenwood, D.T., Principles of Dynamics, 2nd edition, Prentice-Hall,1988 [8] Humble, R. W., Henry, G. N., and Larson, W. J.1995. Space Propulsion Analysis and Design, 1st ed-Revised. New York: Primis Custom Publishing [9] Javorsek, Daniel II, and Longuski, J., "Velocity Pointing Errors Associated with Spinning Thrusting Spacecraft," Journal of Spacecraft and Rockets, Vol. 37, No.3, 2000, pp. 359-365. Internet References [10] http://www.hdssystems.com/LithiumBattery.htm [11] http://www.firefox-fx.com/ [12] http://www.tescom.com [13] http://www.globalspec.com/Frames?URL=%2FSpecSearch%2FMatchingSuppliers%3 FQID%3D1413807%26Comp%3D1412%26state%3Dspecsearchable [14] www.ampro.com [15] www.kantronics.com [16] www.hamtronics.com 57 Appendix A: QFD Ground Station Power consumption ` accelorometers Fuel quantity Angular Momentum History Comm Architecture -- Life ++ Loss of Velocity Spin Rate C.M. History S/C Size - ++ + Liftoff Mass ++ + - ++ + Do Not Delete Direction of improvement ( + or -) ê ê ê é ê é ê Do Not Delete Launch Mission Payload HOW Angular Momentum History IMPORTANCE WHAT Power consumption Comm Architecture Loss of Velocity accelorometers Ground Station Fuel quantity C.M. History Liftoff Mass Spin Rate S/C Size Export to Next Phase = * Life Mission Burn Test 20 9 9 9 9 9 9 9 3 Attitude Determination 20 9 9 9 9 1 3 Command and Data Handling 20 3 9 9 9 9 De-Orbit after mission 10 3 1 3 9 3 Cost Quick Assembly 5 1 3 1 Launch Flexibility 8 9 9 1 9 Low Cost 17 9 9 9 9 3 1 9 3 290.0 240.0 390.0 368.0 370.0 363.0 338.0 360.0 341.0 377.0 483.0 303.0 ABSOLUTE IMPORTANCE 11% 7% 6% 9% 9% 9% 9% 8% 9% 8% 9% 7% RELATIVE IMPORTANCE RANK 11 12 2 5 4 6 9 7 8 3 1 10 30X30X35 high accu 3 months constant reliable 20 rpm 40 kg 20 kg 50 W TARGET VALUE 58 Appendix B: ADCS Disturbance Environment Calculations 1) Gravity Gradient: Type of disturbance: Cyclic (since spacecraft is inertially fixed) Influenced by: - spacecraft orientation - orbital altitude Formula: 3µ Tg = I z − I y sin(2θ ) (1.1) 2R3 where, Tg = maximum gravity torque, R = orbit radius, Iz , Iy = spacecraft moment of inertias, µ = gravitational constant for Earth, θ = maximum deviation of the Z – axis from local vertical Parameters are: Altitude = 350 km Þ R = 350 km + REarth = 350 km + 6378km = 6728 km Þ R = 6728 km (1.2) Spacecraft moments of inertia were first calculated using a cuboid model given below for initial iterations. Later on the design process, using the actual size and the masses of selected hardware, the moment of inertias were calculated using solid modeling in the Unigraphics CAD Package. The moments of inertias were generated by the software, based on the location and masses of the each component. The resulting moments of inertias were found to be (in the principal directions shown above) I x = 0.2995 kg ⋅ m2 I y = 0.3488 kg ⋅ m2 I z = 0.3890 kg ⋅ m 2 59 Since the spacecraft is a single spinner, as dynamics of the motion points out, spinning about the maximum moment of inertia will be the only stable spinning motion. What’s more, having the maximum moment of inertia as big as possible compared to inertias about the other axes provides better stability. Our mass budget (most of the mass values were calculated using the actual component masses) indicates that we have approximately % 40 margin in our aimed mass budget. Thus in order to enhance the stability properties of the spacecraft we are going to add additional masses to increase Iz moment of inertia. Since we have a big margin, we can have as much as 0.4 kg.m2 increase in our Iz moment of inertia but to stay on the safe side we are going to assume a smaller increase in Iz moment with added masses at the corners. This will account for some uncertainties during actual manufacturing of the spacecraft. With the modifications, the principal moment of inertia values used in calculation are as follows I x = 0.32 kg ⋅ m 2 I y = 0.36 kg ⋅ m 2 (1.3) I z = 0.50 kg ⋅ m 2 The other parameters in equation (1.1) are given next: θ = 45o (worst case) (1.4) µ = 3.986 × 1014 m3/s2 (1.5) Thus having defined all the parameters in equation (1.1) we can compute Tg as follows: 3 × (3.986 ×1014 m3 / s 2 ) Tg = 0.5 − 0.32kg ⋅ m 2 × sin(90o ) 2 × (6728 ×103 m)3 Þ Tg = 3.5338 ×10−7 N ⋅ m (1.6) 2) Solar Radiation: Type of disturbance: Cyclic (since spacecraft is inertially fixed) Influenced by: - spacecraft geometry - spacecraft surface reflectivity - center of gravity location Formula: Tsp = F (C ps − C g ) (2.1) Fs where, F= As (1 + q ) cos i (2.2) C The parameters are defined below: 60 Tsp = solar radiation pressure (torque), Fs = solar constant, (= 1367 W/m2) C = speed of light, (= 3 × 108 m/s) As = surface area, Csp = location of the center of solar pressure, Cg = center of gravity location, q = surface reflectance (ranges between 0~1), i = angle of incidence to the sun We set these parameters as follows for the worst-case conditions: As ≅ 0.14 m2 ( ≅ 0.3 2 × 0.35m2 - the worst possible case) q = 0.6 (semi reflective) cos i = 1 (i = 0 worst case) Csp - Cg ≅ 0.075 m (this is an approximation based in the fact that the spacecraft is highly symmetric and the color pattern and surface properties do not vary significantly) Thus from equation (2.1) and (2.2) we calculate as follows: 1367 W F= ( ) m 2 × 0.14m 2 × (1 + 0.6) × cos(0o ) 3 × 108 m (2.3) s Þ F = 1.0207 × 10−6 N Thus, Tsp = (1.0207 × 10−6 ) × (0.075m) (2.4) Þ Tsp = 7.65525 ×10−8 N ⋅ m 3) Magnetic Field: Type of disturbance: Cyclic (since spacecraft is inertially fixed) Influenced by: - orbit altitude - residual spacecraft magnetic dipole - orbit inclination Formula: Tm = D ⋅ B (3.1) where, D is the residual dipole of the vehicle in amp ⋅ turn ⋅ m2 ( A ⋅ m ) and, 2 2M B= (for polar orbit with i = 90o) R3 M B= 3 (for equatorial orbit with i = 0o) R 61 Since our inclination is i = 40o, by linear interpolation we find the corresponding number for our orbit: 13M B= (3.2) (Exact using linear interpolation) 9R3 In the above equation M is the magnetic moment of the Earth and measured as; M = 7.96 × 1015 [tesla ⋅ m2] and, R is the radius from dipole (Earth) center to spacecraft in [m] . Thus we calculate for R = 6728 km, and for D = 1 A ⋅ m2 (this is a common value for small-sized, uncompensated vehicle); æ 13 × (7.96 ×1015 tesla ⋅ m3 ) ö TM = (1A ⋅ m 2 ) × ç ÷ è 9 × (6728 ×103 m)3 ø Þ TM = 3.77534 ×10−5 N ⋅ m (3.3) 4) Aerodynamic Disturbance: Type of disturbance: Variable (since spacecraft is inertially fixed) Influenced by: - orbit altitude - spacecraft geometry - center of gravity location Formula: Ta = F ⋅ (C pa − Cg ) = F ⋅ L (4.1) where, F = 0.5 ⋅ é ρ ⋅ Cd ⋅ A ⋅ V 2 ù ë û (4.2) The parameters are defined below: Ta = aerodynamic torque, F = aerodynamic force, ρ = atmospheric density Cd = drag coefficient (usually between 2 ~ 2.5), A = exposed surface area, V = spacecraft velocity, Cpa = center of aerodynamic pressure, Cg = center of gravity We assign the following numbers to these parameters; ρ ≅ 6.98 × 10-12 kg/m3 (mean density at 350 km altitude), Cd ≅ 2.25 (usually between 2 and 2.5 – we take it to be constant at 2.25) A ≅ 0.14 m2 (worst possible case) Vmax ≅ 8000 m/s (this is calculated from the circular velocity at 700km altitude in µ 3.986 ×1014 m3 / s 2 orbit where Vcirc = = ≅ 7697m / s R 6728 ×103 m Since the spacecraft may not be in the circular orbit during the 62 experiments, we can take V ≅ 8000 m/s to compensate higher speeds of an elliptic orbit at perigee. ) Cpa-Cg ≅ 0.1 m (this is an approximation based in the fact that the spacecraft is geometrically symmetric) Thus from equations (4.4.1) and (4.4.2) we calculate as follows; F = 0.5 ⋅ é(6.98 ×10−12 kg / m3 ) ⋅ (2.25) ⋅ (0.14m 2 ) ⋅ (8000m / s)2 ù ë û Þ F = 7.03584 ×10−5 N Thus Ta=( 7.03584 × 10−5 N ) × (0.1m) Þ Ta = 7.03584 ×10−6 N ⋅ m (4.3) This concludes the calculation of major disturbances in the LEO orbit. Next we will try to give the idea of how significant are the disturbance torques on the motion and the attitude of the spacecraft in orbit. We are going to assume that spacecraft is spinning with 25 rpm about the maximum moment of inertia axis, and all the disturbance torques are acting on the same direction! (This is a much exaggerated assumption but this will give a fairly good idea how effective these torques are). Thus we sum all the disturbing torques; Tdisturbance = Tg + Tsp + TM + Ta Tdisturbance = 3.5338 × 10−7 + 7.65525 × 10−8 + 3.77534 × 10 −5 + 7.03584 × 10−6 Þ Tdisturbance ≅ 4.52192 × 10−5 N ⋅ m Thus we conclude that even if the torques were exaggerated greatly and assumed to be all in the same direction, the net effect is not very significant. Therefore, for the length of the mission and the possible disturbance environment we do not need to have an active attitude control & compensation system since our experiments do not require an active control other than the initial spin-up. As the last step in our calculations we are going to calculate the nutation frequency of the spacecraft. The relation between the inertial nutation frequency (wni) and the spacecraft rotation frequency is: Is ωni = ωs (4.4) IT where ws is the spin frequency and Is and IT are moments of inertia about the spin axis and transverse axis, respectively. Thus setting Is=Iz nutation frequencies for the transverse axes x and y are; Iz 0.5kg ⋅ m2 ωni | = ωz = × 25rpm = 39.0625rpm x Ix 0.32kg ⋅ m 2 63 Iz 0.5kg ⋅ m 2 ωni | = ωz = × 25rpm = 34.7222rpm y Iy 0.36kg ⋅ m 2 Thus, ωni | = 39.0625rpm (4.5) x ωni | = 34.7222rpm y We must make sure that the nutation damper to be selected is capable of eliminating nutation within the range of above numbers. ADCS Cost Estimation: The following table represent the actual and estimated component costs Unit Price Number Component ($) of parts Sun Sensor 20000 (estimated) 2 Gyroscope 2450 (actual) 1 Accelerometer 1100 (actual) 1 Nutation Damper 1000 (estimated) 2 Total = 44550 $ 64 Appendix C: Power In order to accurately compare the solar panels to the batteries, there are rough calculations to determine the mass and cost of solar panels. The equations are from reference three. Pe = power during eclipse Pd = power during daylight Psa = power provided by solar array PBOL = beginning of life power PEOL = end of life power Psun = 1367 W / m2 Po = ideal solar output per area Te = time in eclipse Td = time in daylight Xe = efficiency Xd = efficiency Msa = mass of panels Mcontrol = mass of power control unit Mreg = mass of regulator and converters Mwiring = mass of wiring Mtot = total mass of subsystem Msecondary = mass of rechargeable batteries Mdry = dry mass of the spacecraft = 40 kg C = capacity in W hrs Ld = life degradation Eff = solar conversion efficiency = 22% for Multijunction GaInP / GaAs Asa = area of solar array θ = angle of sun striking solar array For short mission, it is assumed that there is no degradation and that PBOL = PEOL. Equation 1 calculates the power the solar panel must provide. Equation 12 gives and total mass of a solar panel system and equation 13 gives an estimated cost of the system. The important results are shown in table C-1. It is important to note that the maximum production efficiency of the solar panels was used, which does not account for the losses due to thruster’s exhaust. Psa = [(Pe Te / Xe) + (Pd Td / Xd)] / Td (1) Po = Psun * Eff (2) PBOL = Po Id cos θ (3) Ld = (1 – degradation / yr)^satellite life (4) PEOL = PBOL Ld (5) 65 Asa = Psa / PEOL (6) Msa = 0.04 * Psa x 4 (7) Mcontrol = 0.02 * Psa (8) Mreg = 0.025 * Psa (9) Mwiring = 0.04 * Mdry (10) Msecondary = C / 45 (for NiH2) (11) Mtot = Msa + Mcontrol + Mreg +M wiring + Msecondary (12) Costsa = $800 - $3000 / watt (13) Psa 37 Watts Mtotal 12.3 kg Asa (outer surface area) 0.6 m2 Costsa $29,600 – $111,000 Table C-1: Important results for solar cell comparison. For the calculations of the primary batteries exact figured were found using available devices. The important results are shown in Table C-2: After comparing Table C-2 with C-1, primary cells are the better option for this particular mission. Also, in the generation of table VII -2, power losses were calculated with equations 14 and 15. Preg = 0.2 P (14) Pwiring = 0.02 P (15) Energy Density 0.4 W / cm3 Energy Specific Density 250 W / kg Cost regulator/converter and wiring $ 300 Cost per cell $ 17 66 Mtotal 8.3 kg Costtotal $ 2000 Table C-2: Important results for primary cell comparison. 67 Appendix D: Communications and Data Handling The Earth angular radius, ρ is calculated as follows, æ RE ö ο ρ = arcsinç ç R + h ÷ = 71.44 ÷ (1) è E ø where RE is the radius of the Earth. Also, the period, P, of the orbit is, P = 1.658669 × 10 −4 ⋅ (RE + h ) = 91.54 min (2) The maximum nadir angle, ηmax, is calculated from the following equation η max = arcsin (sin ρ cos ε min ) = 56.2 o (3) where εmin = minimum elevation that the ground station can view the satellite = 5deg. Next, the maximum Earth central angle, λmax, is λ max = 90 o − ε min − η max = 28.8 o (4) From this, the maximum range is defined as æ sin λ max ö Dmax = RE ç ç sin η ÷ = 1656.76km ÷ (5) è max ø Next, the minimum Earth central angle, λmin, is λ min = arcsin(sin lat pole sin lat gs + cos lat pole cos lat gs cos(∆long )) = 14.2 o (6) where latpole = 90º – inclination, and latgs = 40 deg. for the Purdue ground station. The mean motion, n, is −3 n ≅ 8681660.4 ⋅ a 2 = 15.73 rev (7) day where, a is the semi-major axis defined by, 2 a ≅ 331.24915 ⋅ P 3 (8) The total time in view, T, is calculated from, 68 æ P ö æ cos λ max ö T =ç ÷ arccosç ç cos λ ÷ = 13.0 min ÷ (9) è 180 ø è min ø The received energy-per-bit to noise-density (Eb/N0) was calculated from Eb = 10 log( Pt ) + L + Ll + Gt + Gr + 228.6 − 10 log Ts − 10 log R (10) N0 And the effective isotropic radiated power, EIRP, was calculated from EIRP = 10 log(Pt ) + Gt + Ll (11) Space losses, Ls, was calculated from the following equation ( ) Ls = 20 log 3 × 10 8 − 20 log(4π ) − 20 log(S ) − 20 log( f ) (12) where S = 350,000meters, and f is the frequency in Hz. 69 Appendix E: Thermal E.1 Equations for Table X-2: D, diameter of sphere which has equal surface area D= Aπ (E-1) ρ, angular radius of the Earth RE sin ρ = (E-2) H + RE Ka, albedo correction factor K a = 0.644 + 0.521ρ − 0.203 ρ 2 (E-3) E.2 Energy Balance and Required Emissivity Calculations It must first be determined how much energy needs to be dissipated, using the equilibrium energy balance equation: Qout = Qin + Qdis (E-6) Qout is the equilibrium energy, or the heat generated by the satellite, which is given in Table X-2 as 9W. During eclipse, the only heat input will be from the Earth infrared, which emits at 216 W/m2. This gives Qin = 129.6 W, where A = 0.6m2. Solving the energy balance for heat dissipated yields: Qdis = −120.6 W The positive value of Qdis confirms that the satellite is too cold and heat must be added. Consider a satellite which must dissipate heat. Qdis can be broken down as follows: Qdis = Qins + Q prad (E-7) where, Qins = Heat radiated into space by insulation coating 70 Qins = εAσT 4 (E-8) Q prad = Heat radiated into space by passive radiators Q prad = εAσηT 4 (E-9) It will first be determined whether or not the proper insulation will suffice to dissipate the required energy (Assume no radiator: Q prad = 0). Therefore, Qdis = Qins (E-10) Substituting Eq.(E-8) into Eq.(E-10) and rearranging to solve for ε, gives: Qdis ε= (E-11) AσT 4 where, ε = emissivity A = surface area of spacecraft = 0.6 m2 W σ = Stefan-Boltzmann constant = 5.67 × 10 −8 m2K 4 T = min. absolute temperature of spacecraft = 12°C = 285.15 K Qdis = 120.6 W Eq.(E-11) yields a required emissivity of, ε = 0.55 in order to maintain the satellite within its lower temperature limit. 71 Appendix F: Structure Appendix G: Propulsion a. Cold Gas Thruster (CG) The steps in developing the cold gas thruster were first determining gases to compare against each other and finding the gas constant, R, ratio of specific heat, γ, and molecular mass, m, of the gases. Then calculate the acoustic velocity, ao, a o = γRTo (G-1a) where To is the stagnation temperature of the gas during operation assumed equal to 273.15 oK, because this is the average temperature the gas should be at during operation. Next the characteristic velocity, c*, was calculated. ao c∗ = γ +1 (G-2a) æ 2 ö 2(γ −1) ç γ + 1÷ γç ÷ è ø • The mass flow m was then calculated 1 γ +1 − é ù 2 • F êæ 2 öæ 2 ö ú γ −1 m= ∗ ç ÷ç ÷ (G-3a) c γ êç γ − 1 ÷ç γ + 1 ÷ ú è øè ø ú ê ë û where F is the thrust and equals 1.0 Newton from the SRD. The mass of propellant, mp, was calculated. • m p = m ∆T (G-4a) where ∆T is the burn time, which was assumed to be 300 seconds to provide a propellant margin since for the two burn schemes on 240 seconds is needed. The volume of the storage tank was then calculated m p RTo Vp = (G-5a) Pc and graphed for a range of pressures to determine what pressure would be needed in the tank for using the gas. 72 As the figure illustrates the tank volume decreases and most of the gases begin to converge as the tank pressure is increased. From graph values for the initial tank pressure, Pti, were chosen that would give a low tank volume where any more increase in pressure would provide negligible result in reducing the tank volume. Next expansion ratios were determined by use of the expansion-mach number relation equation for a range of mach numbers. γ +1 A 1 éæ 2 öæ γ - 1 2 öù 2(γ -1) ε= e = êç ÷ç1 + M e ÷ú (G-6a) A t M e ëç γ + 1 ÷è è ø 2 øû The pressure ratio was also determined by similar means γ Pe éæ γ - 1 2 ö ù (1-γ ) = ç1 + M e ÷ú (G-7a) Pc êèë 2 øû The specific impulse, Isp, was calculated 1 ∗ é γ +1 ì üù 2 γ −1 c γ êæ 2 öæ 2 ö γ −1 ï æ Pe ö ïú γ I sp = ç ÷ç ÷ í1 − ç÷ ý (G-8a) g o êç γ − 1 ÷ç γ + 1 ÷ ï ç Pc÷ ú êè ë øè ø î ø ïú è þû The specific impulse, equation (G-8a), was graphed against the expansion ration, equation (G-6a), so a choice for exit Mach number, pressure ratio, and specific impulse could be chosen. Next a range of throat and exit areas was determined for a range of regulated pressures, Pr. • m c∗ At = (G-9a) Pr A e = εA t (G- 10a) where ε is the chosen expansion ratio determined earlier for that specific gas. The diameter of the throat and exit were determined and then graphed against the range of regulated pressures 73 A D=2 (G- π 11a) The graph was used to determine the best regulated pressure, exit diameter, and throat diameter, which occurs when the throat and exit diameters are near each other and when an increase in the regulated pressure does not affect the diameter difference greatly. Do not want a large regulated pressure, because more residual propellant would be left in the tank. Having determined the critical components of the design it is possible to calculate the thrust of system. ì 1 ü ï é γ +1 ì γ −1 ü ù2 ï ï êæ 2 öæ 2 ö γ −1 ï1 − æ Pe ö γ ïú + (P A )ï F = λ íA t Pr γ êç ç ÷ ý ç γ − 1 ÷ç γ + 1 ÷ í ç P ÷ ÷ç ÷ ú e e ý (G- ï êè øè ø ï è c ø ïú ï ï ë î þû ï î þ 12a) where λ is the nozzle efficiency which is 99%, because the nozzle cone angle is so small. The mass flow is again recalculated by rearranging equation (G-9a) for the mass flow and solving. This is done so that a mass flow that is determined more from design considerations is used. The propellant mass was recalculated using equation (G-4a) for the same burn time. The volume of the tank taking in consideration for mass residue was then calculated m p RTo V= (G- (Pti − Pr ) 13a) The mass residue, mr, was then calculated by PV mr = r (G- RTo 14a) The total propellant mass is the sum of the mass residue and propellant mass. A spherical tank was chosen for the tank design due to its high strength capability and it is commonly used. The inner radius of tank was determined 1 æ 3V ö 3 ri = ç ÷ (G- è 4π ø 15a) The burst pressure was calculated Pb = f s (MEOP ) (G- 16a) where fs is the factor of safety, which equals 2.25. MEOP is the maximum expected operating pressure, which is calculated by using the ideal gas law and determining the pressure for the tank filled at a temperature of 335.93 oK for the highest probable temperature that the tank could while waiting for launch at the launch site. The thickness of the case was then determined 74 Pb ri t cs = (G- Ftu 17a) where Ftu is the ultimate tensile strength of case material. Next the mass of the case was determined m cs = ρ cs π(rcs − ri3 ) 4 3 (G- 3 18a) where ρcs is the density of the case material and rcs is sum of tcs plus ri. b. Solid Rocket Motor (SRM) In calculating the amount of propellant required to achieve the ∆V an iterative approach was used. The mass ratio, MR, was calculated ∆V g o I spv MR = e (G-1b) where ∆V is the change in velocity required, go equals 9.807 m/s2, and Ispv is the vacuum specific impulse that solid rocket motors have historically. Assumed on the low end of the range and chose Ispv = 200 sec. The mass of propellant was calculated λ p (MR − 1) m p = mi (G-2b) 1 − MR (1 − λ p ) where mi is the inert mass of the spacecraft and λp is the propellant mass fraction. The propellant mass fraction is what is iterated on so it is initially guessed. Having a value for the propellant mass and knowing the desired initial total mass of the spacecraft, mo, calculated the propellant mass fraction mp λp = (G-3b) mo If the guessed propellant mass fraction is no more than 0.00001 different than the propellant mass fraction calculated from equation (G-3b) then values are correct. If the value is more than 0.00001 different then use the new propellant mass fraction in equation (G-2b) and recalculate; continue this process till propellant mass fraction is within the limit. Now knowing the mass of the propellant can calculate the volume of the propellant. Vp = m p /ρ p (G-4b) where ρp is the density of the propellant mixture. The average mass flow was calculated by assuming steady state for the constant burn. • F m = max (G-5b) I spv g o where Fmax is the maximum thrust, which occurs during the constant burn process of the thrust scheme. The burn time was then calculated mp tb = • (G-6b) m 75 Once knowing the burn time it is possible to calculate the ramp up time, tr. The limitations on the ramp up time so that it has some effect comes from Javorsek and Longuski 2nπ trñ (G-7b) Ω where Ω is the spin rate and n is some integer greater than zero. Knowing how long the burns are can develop the propellant grain and thus the casing of the SRM. Using information on propellant for typical chamber pressures set the throat radius. Since their needs to be more propellant to ensure choked flow set the initial radius of propellant. With the ramp up time the burn surface area was calculated A b = πr (t ) (G-8b) where r(t) is the radius of the propellant as a function of time. Next the chamber pressure is calculated 1 æ aρ p A b c ∗ ö 1− n Pc = ç ÷ (G-9b) ç At ÷ è ø where a and n are constants determined by the propellant. Assumed St. Robert’s Law was applicable so the rate of burn was calculated rb = aPcn (G-10b) The web distance burned was found ω b = rb ∆T (G-11b) where ∆T is the step in time for the process. The height was calculated h new = h old + ω b (G- 12b) where hold was the height from the last time step; initial hold equals 0.0 cm. The new propellant radius was then calculated for the burn in height é æ æ r − r ö öù rnew = rold + h new ê tanç atanç c i ÷ ÷ú ç ç r t ÷÷ (G- ê è ë è bi r ø ø ú û 13b) where rc is the radius needed to obtain the constant thrust, ri is the initial propellant radius, and rbi is the initial burn rate. This is continued till tb is reached with the few minor changes. During the constant burn equation (G-13b) rnew = rold and during the ramp down equation (G-13b) remains the same expect that the plus sign is negative. The average volume of the tank was calculated from taking the burn surface area and height half way through each burn time. Vt = A bru h ru + A bc h c + A brd h rd (G- 14b) where the subscripts ru signify the ramp up phase and rd signify the ramp down phase. Then compared the volume of the tank to the volume of propellant. If not in agreement varied Fmax it till the tank volume converged with the propellant volume within 0.1 differences. The thrust coefficient was then found for the constant burn time, because this when greatest thrust occurs. 76 Fmax Cf v = • (G- c∗ m 15b) Using thrust coefficient tables determined the optimal expansion ratio by interpolation. The thrust was determined Fv = Cf v At Pc (G- 16b) The thickness of casing used equation (G-17a) except burst pressure used a factor of safety of 2.0 and MEOP equaled the chamber pressure at constant burn. The thickness of insulation was determined by • t insul = t b e f s (G- 19b) • where e is the insulations erosion rate, which is dependent on insulation material selected. The mass of the casing was found by m cs = ρ cs π(rcs − ri2 )(h (t b − t r )) 2 (G- 17b) where rcs equals rc plus tcs plus tinsul, ri equals rc plus tinsul, and h(tb-tr) is the height at the time right after the constant burn occurs. The mass of insulation is calculated in the same manner of equation (18.17b) except the density for case materials are the insulation material’s density and rcs is replaced by rti (rti = rc + tinsul) and ri is replaced with rc. The mass of the igniter was determined from an equation that correlates the propellant volume and free space in SRM to the mass of the igniter using a number of SRMs that have been actually used. 0.571 é æ 1 öù ç η − 1÷ú m ig = 0.0138êVp ç ÷ (G- ë è v øû 18b) where ηv is the volumetric loading efficiency, which is equal to 98% in this case. The nozzle mass was determined by using a correlation equation ( ) 0.917 é m p c ∗ ε 0.3 ù 1.2 m noz = 0.0000256 ê 0.8 0.6 ú (G- ê Pc t b (tan (θ cn )) ú 0.4 ë û 19b) where θcn is the nozzle cone angle, which equals 12 degrees so that a high nozzle efficiency is possible. Finally the nozzle length was determined by D − Dt Ln = e (G- 2tan (θ cn ) 20b) where De and Dt are the exit diameter and throat diameter, respectively. 77 78