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Analysis of Propellant Slosh Dynamics and Generation of an Equivalent Mechanical Model for Use in Preliminary Voyager Autopilot Design Studies

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Analysis of Propellant Slosh Dynamics and Generation of an Equivalent Mechanical Model for Use in Preliminary Voyager Autopilot Design Studies NATIONAL AERONAUTICS AND SPACE ADMlNlSTRATlON Technical Memorandum No. 33-306 Analysis of Propellant Slosh Dynamics and Generation of an Equivalent Mechanical Model for Use in Preliminary Voyager Autopilot Design Studies Thomas A. Lance Approved by: R. J. Beale,Y Acting Manager Voyager Propulsion Section JET PROPULSION LABORATORY CALIFORNIA INSTITUTE OF TECHNOLOGY’ PASADENA. CALIFORNIA December 1, 1966 Copyright 0 1966 Jet Propulsion Laboratory California Institute of Technology Prepared Under Contract No. NAS 7-1 00 National Aeronautics & Space Administration JPL TECHNICAL MEMORANDUM NO. 33-306 . ~~-*~~~~NG?3 CONTENTS 1. Introduction .................. 1 I1 . Analysis ................... 2 A. Statement of the Problem ............. 2 B . Regime of Validity of Analysis ............ 2 C . Construction of the Fluid Dynamic Solution ........ 3 D . Constructionof anEquivalent MechanicalModel ...... 4 E . Generation of Slosh Model for Typical Voyager Missions .... 6 F . Discussion of Results .............. 6 111. Conclusions ..................15 Nomenclature ..................15 Appendix A . Derivation of Equations for Fluid Dynamics in a Cylindrical Tank and Generation of an Equivalent Mechanical Model ....17 I . Freesloshing .................17 I1. Small-Amplitude Forced Oscillations ..........24 I11 . Equivalent Mechanical Model ............27 Appendix B . Additional Data .............34 I . “Maximum” Planetary Vehicle ............34 I1. Orbit Insertion with Capsule Separated .........43 Bibliography ...................46 FIGURES 1 . Cylindrical tank approximation of LEMDS propellant tank .... 2 2 . Equivalent spring-mass system for propellant slosh in a cylindrical tank .............. 5 3 . Velocity increment. midcourse maneuver ......... 6 4 . Spacecraft mass (midcourse maneuver) .......... 7 5 . Fuel and oxidizer mass (midcourse maneuver)........ 7 6 . Fuel and oxidizer tank fluid depth (midcourse maneuver) .... 7 111 JPL TECHNICAL MEMORANDUM NO . 33-306 FIGURES (Cont'dl 7. Fuel and oxidizer tank natural slosh frequencies (midcoursemaneuver) ...............7 8 . Fixed and slosh fuel mass (midcourse maneuver) ....... 8 9. Fixed and slosh oxidizer mass (midcourse maneuver) ..... 8 10 . Fixed and slosh fuel mass depth (midcourse maneuver) ..... 8 1 1 . Fixed and slosh oxidizer mass depth (midcourse maneuver).... 8 12. Velocity increment. retromaneuver ........... 9 13. Spacecraft mass (retromaneuver) ........... 9 14. Fuel and oxidizer mass (retromaneuver) .........10 15. Fuel and oxidizer tank fluid depth (retromaneuverl ...... 10 16. Fuel and oxidizer tank natural slosh frequencies (retromaneuver) ................ 10 17 . Fixed and slosh fuel mass (retromaneuver)......... 10 18 . Fixed and slosh oxidizer mass (retromaneuver) ....... 11 19. Fixed and slosh fuel mass depth (refromaneuverl .......11 20 . Fixed and slosh oxidizer mass depth (retromaneuver) .....11 21 . Velocity increment, orbit trim maneuver .........12 22. Spacecraft mass (orbit trim maneuver) ..........12 23 . Fuel and oxidizer mass (orbit trim maneuver) ........12 24 . Fuel and oxidizer tank fluid depth (orbit trim maneuver)..... 13 25 . Fuel and oxidizer tank natural slosh frequencies (orbit trim maneuver) ............... 13 26. Fixedand slosh fuel mass (orbit trim maneuver) .......13 27. Fixed and slosh oxidizer mass (orbit trim maneuver) ......14 28 . Fixed and slosh fuel mass depth (orbit trim maneuver) .....14 29 . Fixed and slosh oxidizer mass depth (orbit trim maneuver) ....14 A-1 . Circular cylindrical tank of radius a, partially filled to an arbitrary depth h .............17 A-2 . Surface wave in cylindrical tank ............18 A-3 . Geometry of moment calculation for cylindrical tank ......29 A-4 . Equivalent spring-mass system ............ 31 B-1 . Velocity increment for midcourse maneuver. "maximum" planetary vehicle ............ 34 B-2 . Spacecraft mass. "maximum" planetary vehicle (midcourse maneuver)............... 34 iv JPL TECHNICAL MEMORANDUM NO. 33-306 FIGURES (Cont’dl B-3. Fuel and oxidizer mass, “maximum” planetary vehicle (midcourse maneuver). 34 B-4. Fuel and oxidizer tank fluid depth, ’~maximumO planetary vehicle (midcourse maneuver) . 35 B-5. Fuel and oxidizer tank natural slosh frequenciesr3naximum” planetary vehicle (midcourse maneuver) . 35 B-6. Fixed and slosh fuel mass, “maximum” planetary vehicle (midcourse maneuver) . 35 B-7. Fixed and slosh oxidizer mass, “maximum” planetary vehicle (midcourse maneuver) . 36 B-8. Fixed and slosh fuel mass depth, “maximum” planetary vehicle (midcourse maneuver) . 36 B-9. Fixed and slosh oxidizer mass depth, “maximum” planetary vehicle (midcourse maneuver) . 36 B-10. Velocity increment for retromaneuver, “maximum” planetary vehicle . 37 B-11. Spacecraft mass, “maximum” planetary vehicle (retromaneuver) . 37 B- 12. Fuel and oxidizer mass, ‘haximum” planetary vehicle (retromaneuver) . 37 B-13. Fuel and oxidizer tank fluid depth, “maximum” planetary vehicle (retromaneuverl . 38 B-14. Fuel and oxidizer tank natural slosh frequencies, “maximum” planetary vehicle (retromaneuverl . 38 B-15. Fixed and slosh fuel mass, “maximum” planetary vehicle (retromaneuver) . 38 B-16. Fixed and slosh oxidizer mass, “maximum” planetary vehicle (retromaneuver) . 39 B-17. Fixed and slosh fuel mass depth, “maximum” planetary vehicle (retromaneuver) . 39 B-18. Fixed and slosh oxidizer mass depth, “maximum” planetary vehicle (retromaneuver) . 39 B-19. Velocity increment for orbit trim maneuver, “maximum” planetary vehicle . 40 B-20. Spacecraft mass, “maximum” planetary vehicle (orbit trim maneuver) . 40 B-21. Fuel and oxidizer mass, “maximum” planetary vehicle (orbit trim maneuver). 40 V JPL TECHNICAL MEMORANDUM NO. 33-306 FIGURES (Cont’d 1 B-22. Fuel and oxidizer tank fluid depth, “maximum” planetary vehicle (orbit trim maneuver) . 41 8-23. Fuel and oxidizer tank natural slosh frequencies (orbit trim maneuver) . 41 8-24. Fixed and slosh fuel mass, “maximum” planetary vehicle (orbit trim maneuver) . 41 B-25. Fixed and slosh oxidizer mass, ”maximum” planetary vehicle (orbit trim maneuver). 42 B-26. Fixed and slosh fuel mass depth, ”maximum” planetary vehicle (orbit trim maneuver). 42 8-27. Fixed and slosh oxidizer mass depth, “maximum” planetary vehicle (orbit trim maneuver). 42 B-28. Velocity increment, orbit insertion with capsule separated . 43 8-29. Spacecraft mass, orbit insertion with capsule separated . 43 8-30. Fuel and oxidizer mass, orbit insertion with capsule separated. 43 B-31. Fuel and oxidizer tank fuel depth, orbit insertion with capsule separated . 44 B-32. Fuel and oxidizer tank natural slosh frequencies, orbit insertion with capsule separated . 44 B-33. Fixed and slosh fuel mass, orbit insertion with capsule separated . 44 B-34. Fixed and slosh oxidizer mass, orbit insertion with capsule separated . 45 8-35. Fixed and slosh fuel mass depth, orbit insertion with capsule separated . 45 B-36. Fixed and slosh oxidizer mass depth, orbit insertion with capsule separated . 45 VI JPL TECHNICAL MEMORANDUM NO. 33-306 ABSTRACT The linearized dynamic equations of propellant motion in the regime dominated by gravity force have been developed for a cylindrical tank approximation of the Lunar Excursion Module Descent Propulsion System ( LEMDS ) propellant tanks. An equivalent mechanical model consisting of a fixed mass and an infinite series of springs and masses has been generated. A computer program has been developed to calculate the parameters required to describe the equivalent mechanical model for an assumed spacecraft flying an assumed mission profile. These parameters are presented as a function of time for three mission profiles and can be used for preliminary analysis of Voyager closed-loop autopilot designs. Transfer functions for the propellant dynamics are also presented. 1. INTRODUCTION Propellant sloshing is a potential source of spacecraft Since current Voyager spacecraft designs envisage disturbances that may adversely affect the stability of liquid propulsion systems with propellant loads of up to the attitude control system during both powered and 60% of the total planetary vehicle mass, sloshing can be unpowered phases of flight. expected to be a potential source of disturbances. Sloshing refers to the free surface oscillations of a fluid Various theoretical and analytical investigations have in a partially filled tank. These oscillations result from shown that sloshing waves can be large enough to pro- lateral and longitudinal displacements or angular motions duce forces whose magnitude and phase can cause insta- of the spacecraft. The magnitude of the free surface oscil- bilities in closed-loop control systems unless more liquid lations, and hence the forces and moments acting on the damping than that provided by the wiping action of the vehicle as a result of propellant motion, depends upon propellant on the tank walls is made available. The addi- several factors. The most significant parameters that in- tion of baffles to the tanks will provide the additional fluence propellant slosh are the following: damping, if required, and limit the resultant slosh to an acceptable value. 1. Tank geometry Analysis of the coupled response of the spacecraft atti- 2. Propellant properties tude control system and propellant free surface oscilla- tions requires either an equivalent mechanical model for 3. Slosh damping the fluid system or a transfer function for the forces and 4. Depth of propellant in the tank moments produced by the propellant motion in response to an input excitation. The effort reported
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