Measuring Net CG Motion

Transcription

1 Measuring Net CG Motion Paul Blelloch ATA Engineering, Inc. San Diego, CA (858) Marcos Franceschini and Martin Arregui INVAP SC S.C. de Bariloche, Argentina June 2012 The Aerospace Corporation 2010 The Aerospace Corporation 2012

2 Measuring net CG Motion What is net CG motion? Why is it important? How do we calculate it analytically? How might we approximate it from test measurements? Guyan expansion Least squares Example INVAP ARSAT-1 Sine Vibration Test Article 2

3 Net CG motion is mass weighted average motion of all DOF Net CG motion is not motion of a point nominally at the CG of a structure Deformation of the structure will shift CG relative to nominal point No one point on a structure follows the net CG motion Net CG motion can always be written as a linear combination of the motion of all DOF with mass in the structure t Net CG motion is the most commonly used output from a coupled loads analysis (CLA) Satisfies F=ma F is the interface forces/moments iff the only input is coming from base Does not apply to an acoustic test Used to compare against static G-loads Net CG motion is often referred to as net load 3

4 Net CG Motion Can Be Calculated from Craig-Bampton Model Net interface force Convert to Acceleration Transform to CG This step almost always performed as part of CLA (DMAPs galore) 4

5 Often Need to Recover net CG motion of internal components As FEMs become more detailed most components represented with detailed models (not lumped masses) In this case net CG motion is developed as mass weighted average of all DOF in component FEM Easily implemented in DMAP by using PARTN set to identify component DOF, or by using breakout FEM of component Equations can be converted to MPC for easy recovery 5

6 How do We Recover net CG Motion from Test? Limited number of measurements are available Typically 100 s for S/C and a few for a component Cannot exactly measured net CG motion with a finite number of sensors If we can measure net CG motion we can response limit Since net CG acceleration is directly related to interface force (for base shake) net CG acceleration can be used to force limit Poor choice would be to place accel at location that is close to nominal CG location Very unlikely to give good results (except in case where structure is rigid) More general problem is how to estimate a linear combination of all DOF in a model given a relatively small number of measurements Equally applicable to measuring optical performance Or measuring resultant interface forces 6

7 Using Guyan Expansion to Recover net CG Motion Use Guyan Reduced mass matrix to recover net CG motion Equivalent to using Guyan shapes to expand measured response up to all DOF in FEM Must have enough measured DOF to restrain rigid body motion or M0 is not invertible or 7

8 Least Squares Offers More Direct Solution Review of least squares Approximately solve b=ax where number of equations (length of b) ) > number of unknowns (length of x) Least squares solution minimizes 2-norm of error (min b-ax 2 ) Solution is x=(a T A) -1 A T b (x=a\b \ in Matlab) Or regularized LS is x=(a T A+γI) -1 A T y minimizes b-ax 2 +γ x 2 Increasing γ decreases size of coefficients (x) and results in better conditioning. Trick is to find γ such that b-ax 2 is not degraded. In this case x are unknown coefficients multiplying measured DOF, A are transfer functions from input to measured DOF and b are exact net CG responses Stack real and imaginary parts to match both Stack as many transfer functions as desired to ensure good solution 8

9 Setting up Problem as Least Squares Solution Would like to match response for a base shake and also a free-free force input Would like to recover 6 sets of coefficients (3 to recover translational motion and 3 to recover rotational motion) 6 # Measurements Re ( HCGbaseX ) Re( H AbaseX ) Im( HCGbaseX ) ( ) Im( H AbaseX ) Re HCGbaseY ( ) Re H AbaseY ( ) Im HCGbaseY Im( H AbaseY ) Re( H ) CGbaseZ Re( H ) AbaseZ Im( H ) Im( H ) CGbaseZ b = ( ) Re HCGfreeX ( HCGfreeX ) Re( H ) CGfreeY ( ) ( ) Im HCGfreeY Re H CGfreeZ ( ) Im HCGfreeZ AbaseZ A = ( ) Re H AfreeX Im Im( H AfreeX ) Re( H ) AfreeY ( ) Im H AfreeY Re( H ) AfreeZ ( ) Im H AfreeZ x = 6 [ x x x x x x ] CGx CGy 6x # Frequency Lines CGz CGrx CGry CGrz # Measurements 9

10 INVAP ARSAT-1 FEM Sine Vibe Configuration Used as Example ARSAT-1 first of three commercial communication satellites being developed by INVAP for coverage of the southern hemisphere Managed by AR-SAT S.A. INVAP (located in Bariloche, Argentina) has previous experience with SAC-A, SAC-B, SAC-C and SAC-D/Aquarius satellites ARSAT-1 be launched on Ariane or Soyuz Launch Vehicles ~3,000 Kg total weight (3m x 2m x 2m) 10

11 Consider Star Tracker from INVAP ARSAT-1 Spacecraft as Example Goal is to measure net CG motion of ST from sine vibration test ARSAT-1 FEM has ~440,000 nodes and elements Matrix representations ti of key components such as tanks Preliminary correlation to modal test results is excellent First consider net CG motion of Star Tracker Weighs ~ 9 kg - FEM has ~7,000 nodes and elements ST Galileo South component Two cases considered 1. Triax measurements at 4 locations 2. Add triax to measure lumped mass representing 50% of weight 11

12 Increased number of measurements improves solution (typical response) Y Response for Y Input 10 1 Measuring only lumped mass degrades solution Measuring four external locations better Acceler eration (m/s 2 ) 10 0 Adding lumped mass gives best solution (,1001Y+) Exact 4 (,1002Y+) triaxes (,1003Y+) 5 triaxes (,1004Y+) (, Y+) Lumped mass Frequency (Hz) All measurements work VERY well to 60 Hz (1 st component mode) 12

13 Now Consider Measurement of Full S/C Net CG Motion Total of 248 accelerometers available from sine test Distributed widely, but not with intention of measuring net CG motion Guyan method does not do well above 60 Hz 0-90 X net CG Frequency Response Function for X Input Phase Approximate solution diverging from exact Acceleration (m/s 2 ) Exact (, X+) Guyan (, X+) Frequency (Hz) 13

14 Least Squares Solution Nails Response 0 X net CG Response for X Input Frequency Response Function -90 Phase ration (m/s 2 ) Acceler Exact (, X+) Least Squares (,30106Y+) Frequency (Hz) Not surprising i since LS directly minimizes i i error in this transfer function 14

15 0.30 LS Coefficients are quite different from Guyan Coefficients (smaller on the most part) 1st 50 Coefficients in X-Direction Guyan Least Squares

16 How Robust is Solution? Not surprising that Least Squares does excellent job of matching response for analytical transfer function But what if model used for LS is different from real structure? What if mass distribution is different from model? To check robustness randomly vary mass properties of FEM Scale every mass entry by normally distributed random variable with mean of 1.0 and standard deviation of 0.1 ~ 1/3 of mass values vary by > 10% ~ 5% of mass values vary by > 20% Some mass values vary by > 30% Mass values include ρ on material cards, NSM on property cards, lumped masses, and DMIG mass matrices (1000 s of random variations) Overall mass decreased by ~5% (largely due to random decreases in tank masses) Modes shifted significantly (99 modes vs. 104 < 100 Hz) Coefficients calculated from nominal model, but applied to modified model and compared with exact netcg response of modified model 16

17 Least Squares Solution Fairly Robust 0-90 X net CG Response for X Input Frequency Response Function Phase Accele eration (m/s 2 ) (,1001X+) Exact (, X+) Guyan (,30106Y+) Least Squares Frequency (Hz) LS Coefficients from nominal model still work fairly well on modified model 17

18 Least Squares Solution Fairly Robust (Y direction) 0 Y net CG Response for Y Input Frequency Response Function -90 Phase Accele eration (m/s 2 ) (,1001Y+) Exact (, Y+) Guyan (,30106Y+) Least Squares Frequency (Hz) LS Coefficients from nominal model still work fairly well on modified model 18

19 Least Squares Solution Fairly Robust (Z direction) 0 Z net CG Response for Z Input Frequency Response Function -90 Phase Accele eration (m/s 2 ) 10 0 Exact (,1001Z+) (, Z+) Guyan (,30106Y+) Least Squares Frequency (Hz) LS Coefficients from nominal model still work fairly well on modified model 19

20 Open Questions Would Least Square results for full S/C apply to component net CG calculation? Would least square solution be more accurate/robust than Guyan Could I do better with some other expansion method (e.g., modal)? To recover component net CG motion would I want to use just measurements on components or on full S/C? Intuition suggests that net CG motion of component should be function of only component measurements But blending more measurements could improve accuracy Which would be more robust? If I can effectively measure interface forces/moments as a linear combination of accelerometer measurements, is this a good idea? How would such a measurement compare with direct measurement of interface forces? How good would this method be at measuring optical performance? 20

21 Summary of Measuring net CG Motion Net CG motion (net load) very useful output to compare with static G- loads for S/C and components Can be used to response limit to design load Is equivalent to measuring resultant interface forces Can be developed from CB model (includes terms from modes) Standard process in CLA Can be developed directly from mass matrix using DMAP Is effectively mass weighted average of all DOF in model Can be approximated from limited it measurements Guyan approach straight forward (used in ARSAT-1 sine vibration test) Least squares appears to be more accurate and robust Use regularization to give smoother answer 21