Blelloch, Paul

Coppolino, Bob

Stewart, Eric

NASA Technical Reports Server

© The Aerospace Corporation 2010
© ATA Engineering, Inc., 2018
Improving Accuracy of Structural Dynamic 
Modifications Using Augmented Mode 
Residual Vectors
Paul Blelloch, ATA Engineering, Inc.
Bob Coppolino, Measurement Analysis Corporation
Eric Stewart, NASA Marshall Space Flight Center
Spacecraft and Launch Vehicles Dynamic Environments Workshop
June 26–28, 2018
https://ntrs.nasa.gov/search.jsp?R=20180005538 2019-08-31T15:00:32+00:00Z
2Overview
• Review of structural dynamic modification (SDM) 
method for reduced order representation of parameter 
changes on dynamic results
• Review of Augmented Mode Residual Vector (AMRV) 
extension to SDM
• Application to launch vehicle component modes
• Application to SLS integrated modal test (IMT) modes
• References
– Coppolino, R. “Sensitivity method for uncertainty and reconciliation analysis,” 
Proceedings of the 29th IMAC, Jacksonville, FL, 2011
– Coppolino, R. “FEM sensitivity technique for dynamic response uncertainty 
analysis,” Proceedings of the 30th IMAC, Garden Grove, CA, 2012
– Coppolino, R., “Methodologies for Verification and Validation of Space Launch 
System (SLS) Structural Dynamic Models, NASA/CR-2018-219800, Jan. 2018 
3Structural Dynamic Modification (SDM) Used to 
Approximate Parameter Changes in Dynamic Results
• Consider a “parameterized” FEM where mass and stiffness matrix can 
be expressed as follows:
– 𝑀 = 𝑀0 + σ𝑝𝑖 Δ𝑀𝑖 , 𝐾 = 𝐾0 + σ𝑝𝑖 Δ𝐾𝑖
– Parameters can be anything that result in linear change in mass or stiffness
• e.g. material density, Young’s modulus, Spring stiffness, etc.
• SDM reduces problem to Modal Space
– ෡𝑀 = 𝐼 + σ𝑝𝑖 Φ
𝑇 Δ𝑀𝑖 Φ , ෡𝐾 = Λ + σ𝑝𝑖 Φ
𝑇 Δ𝐾𝑖 Φ
෡𝐾 Ψ = ෡𝑀 Ψ ෩Λ , ෩Φ = Φ Ψ - Matrices size of # modes
– Φ 𝑇 Δ𝑀𝑖 Φ and Φ
𝑇 Δ𝐾𝑖 Φ are not diagonal so SDM captures effect of 
parameters on coupling modes (changes mode shapes as well as 
frequencies)
– Fundamental assumption is that modified mode shapes are a linear 
combination Ψ of nominal mode shapes – carry along extra modes
– Excellent approximation for moderate (± 10% - ± 20% parameter changes)
4Lots of Applications for SDM
• In vocabulary of uncertainty quantification we call this a “surrogate 
model” – i.e. a reduced model that approximates full model
• Model Updating
– Model updating programs use SDM to find a set of parameters that 
minimize difference between measured and predicted modes (shape 
and frequency)
– Can run 10,000’s of different parameter values in a genetic algorithm
• Uncertainty Quantification
– Allows for rapid Monte Carlo evaluations of parameter uncertainty
– 10,000’s of uncertain parameters can be run in seconds or minutes
• Design Studies
– Rapidly evaluate effect of many parameters on dynamic response
– Structural optimization
• Best Model Estimate (BME)
– Prepare a “library” of parameterized models prior to a modal test in order 
to be in position to rapidly select best model when data is available
5Augmented Mode Residual Vectors (AMRVs)
Address Biggest Draw Back of SDM
(limitation to moderate parameter variations)
• Limitation comes from assumption that perturbed modes are 
linear combination of nominal modes
• Poor assumption for large parameter variations 
• What if we included modes for large parameter variations in our 
nominal mode set?
• We’ll call these shapes “augmented” modes
• We can’t exactly do that, because augmented modes aren’t 
orthogonal w.r.t. nominal modes
• They aren’t even necessarily independent
• Nastran already has machinery to turn static shapes into 
“residual vectors”
• Residual vectors the part of the static shapes that are orthogonal to the 
nominal modes and to each other
• Approach is to hijack Nastran residual vector calculation to turn 
augmented modes into residual vectors
• Hence augmented residual vectors (AMRVs)
6Calculating Augmented Mode Residual Vectors
• Start with a set of augmented mode shapes Ψ
• Can come from any number of modal solutions with large parameter 
changes (these are not static solutions)
• Orthogonalize augmented modes w.r.t. normal modes
• Ψ′ = 𝐼 − Φ(Φ𝑇𝑀) Ψ (i.e. Φ𝑇𝑀Ψ′ = 0)
• Calculate “coherence” since columns of Ψ′ are not independent
• መ𝐶 = Ψ′
𝑇
𝑀Ψ′
• Calculate singular values (neglect those below threshold)
• መ𝐶 = 𝑈𝑟 𝑈𝑛
Σ𝑟 0
0 Σ𝑛
𝑈𝑟 𝑈𝑛
𝑇
• Transform augmented modes to retained space
• ෡Ψ = Ψ′𝑈𝑟
• Pass ෡Ψ through Nastran residual vector calculation to turn into 
residual vectors (෡Ψ𝑇𝑀෡Ψ = I, ෡Ψ𝑇𝐾෡Ψ = Λ𝑟)
• Just a linear transformation
• Treated just like modes for rest of solution
• Number of AMRV’s typically << than number of augmented modes
7Implementing AMRVs in Nastran
1. Run any number of augmented modal solutions
• Save mode shapes in OP2 or OP4 format
• Augmented modes should represent “extreme” cases
• Don’t need to run augmented modes for every parameter – just 
those that will vary a lot
• Only requirement is that the DoF (G-set) be the same
2. Execute nominal modes with “trial_resvec” DMAP alter
• OP2/OP4 files from all augmented mode solutions ASSIGNed to 
INPUTT2/INPUTT4 units
• PARAM,TFILT sets threshold for retaining residual vectors
• Output is nominal modes plus augmented mode residual vectors
3. Nastran process works on its own, but is most useful when 
combined with SDM calculation (e.g. ATTUNE)
• Another DMAP alter does the SDM step for model updating
• Running SDM for model updating remains unchanged
81st Example is Launch Vehicle Component Modal Test 
• FEM reflects modal test configuration
• Small model runs quickly
• Modes to 50 Hz takes < 20 seconds on desktop
• Parameterized by 24 groups of “E” and 4 groups of 
“K” values (DESVARs)
• Total of 28 parameters to “tune”
• Assume that K’s can decrease by 90% but not increase
• Assume that E’s can increase or decrease by 50%
• 52 sets of augmented modes (2*24 for E + 4 for K)
• Augmented modes run to 50 Hz
• Total of 3,248 augmented modes
• Consider one case
1. 62 nominal modes to 50 Hz + 678 AMRVs based on singular 
value cutoff of 1E-6 (total of 740 ‘modes’ used for sensitivity 
calculation
9Nastran Run to Calculate AMRVs
ASSIGN INPUTT2=‘mode_v12_d01_p50.op2' UNIT=101
꞉
ASSIGN INPUTT2=‘mode_v12_d28_m90.op2' UNIT=152
SOL 200
INCLUDE 'trial_resvec_op2.dmap'
INCLUDE 'attune22.dmap’
CEND
DSAPRT(END=SENS) = ALL
ANALYSIS = MODES
METHOD=909
RESVEC = YES
BEGIN BULK
PARAM,VUNIT1,101
PARAM,VUNIT2,152
PARAM,TFILT,1.E-6
Assign 52 OP2 files from 
augmented modes runs
Include ‘trial_resvec_op2’ and 
ATTUNE (SDM) DMAP alters
Turn on residual vectors
1st and last unit number for OP2 files
Singular value threshold for retaining residual vectors
Standard modal sensitivity request
Note: OP2 files written using PARAM,POST,-1 only works for models 
without SPOINTs.  For models with SPOINTs you need to use DMAP to 
write mode shapes in OUTPUT4 format and use ‘trial_resvec’ DMAP alter
10
Selected Output from AMRV Run
R E A L   E I G E N V A L U E S
(BEFORE AUGMENTATION OF RESIDUAL VECTORS)
MODE    EXTRACTION      EIGENVALUE            RADIANS             CYCLES            GENERALIZED         GENERALIZED
NO.       ORDER                                                                       MASS              STIFFNESS
1         1        9.028311E+03        9.501742E+01        1.512249E+01        1.000000E+00        9.028311E+03
:
62        62        9.793706E+04        3.129490E+02        4.980738E+01        1.000000E+00        9.793706E+04
R E A L   E I G E N V A L U E S
MODE    EXTRACTION      EIGENVALUE            RADIANS             CYCLES            GENERALIZED         GENERALIZED
NO.       ORDER                                                                       MASS              STIFFNESS
1         1       -2.412605E-17        4.911828E-09        7.817417E-10        1.000000E+00       -2.412605E-17
:
3248      3248        6.016507E+00        2.452857E+00        3.903843E-01        1.000000E+00        6.016507E+00
^^^RETAINING          678 OF       3248 RESIDUAL TRIAL VECTORS  
^^^BASED ON A FILTER OF  1.000000E-06 
R E A L   E I G E N V A L U E S
(AFTER AUGMENTATION OF RESIDUAL VECTORS)
MODE    EXTRACTION      EIGENVALUE            RADIANS             CYCLES            GENERALIZED         GENERALIZED
NO.       ORDER                                                                       MASS              STIFFNESS
1         1        9.028311E+03        9.501742E+01        1.512249E+01        1.000000E+00        9.028311E+03
:
740       740        1.811352E+09        4.255998E+04        6.773631E+03        1.000000E+00        1.811352E+09
Nominal Modes to 50 Hz
3248 singular values from -2E-17 to 6.02
62 Normal modes + 678 Residual Vectors based on 1E-6 cutoff
11
Look to see how well Model Updating estimates 
changes
• Model updating software makes “random” changes in parameters based on 
matching test data
• Parameters allowed to move within range of augmented mode solutions
• Evaluate predicted vs. actual comparison to test data after a single iteration of 
model updating software
• Results indicate that predicted frequencies and shapes in model updating software 
matched the actual shapes calculated in Nastran very accurately
-40%
-35%
-30%
-25%
-20%
-15%
-10%
-5%
0%
5%
10%
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Fr
eq
ue
nc
y 
Er
ro
r
Mode Number
Frequency Errors for 216 Modes + 622 Residual Vectors 
Initial Predicted Actual
40%
50%
60%
70%
80%
90%
100%
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
C
ro
ss
-O
rt
ho
go
na
lit
y
Mode Number
Shape Errors for 216 Modes + 622 Residual Vectors 
Initial Predicted Actual
12
Computational Cost of AMRVs Can Be Significant
• Modal solution for each set of augmented 
modes (up to 2x number of parameters)
• Sensitivity solution with AMRVs increases cost from 
1 minute to 15 minutes
• Largest computational cost is መ𝐶 = Ψ′
𝑇
𝑀Ψ′
3%8%
40%
4%
12%
2%
31%
Nastran RunTime Breakdown Out of 859 Seconds for 
62 Normal Modes and 678 Residual Vectors
Calculate 62 Nominal Modes
Read in 3248 Augmented Modes
Calculate Coherence Matrix
Calculate Singular Values
Convert to 678 Residual Vectors
Mass Matrix Reduction
Sensitivity Matrices
13
Evaluation of Random Variations
• First case varied all 28 variables uniformly by ± 20%
• Second case varied 24 variables uniformly by ± 50%
• Spring variables varied -90% + 0%
• Run all 1000 solutions to 50 Hz and compare with sensitivity 
based solutions
• Solution with 62 normal modes to 50 Hz + 678 AMRVs (740 
modes total)
• Straight modes (62 to 50 Hz, 216 to 100 Hz, 740 to 200 Hz)
• Gather statistics on frequency error and mode shape fit for 
all modes for all models
• Total of 61,485 modes across 1,000 +/- 20% random models
• Total of 65,540 modes across 1,000 +/- 50% random models
• Modes matched using cross-orthogonality
• Mode shape error based on RSS cross-orthogonality for +/- 2% 
frequency band
14
AMRVs Not Really Necessary for 1000 ± 20% Variations
• With 216 modes to 100 Hz and no AMRVs 95% of 
modes are within 0.34% and 99% are within 0.61%
• With AMRVs 95% of modes are within 0.05% and 
99% are within 0.10%
• AMRVs improve situation, but having 99% of the 
modes well within 1% is probably close enough
15
AMRVs Really Helpful for ± 50% Variations
• Adding modes increases frequency 
accuracy very slowly
• Even with 740 modes to 200 Hz only 95% of modes 
within 2.32% and 99% within 4.08%
• Adding AMRVs dramatically improves 
frequency match
• With 740 modes almost all modes within 0.5% (95% 
within 0.24%, 99% within 0.55%)
• For this case errors acceptable with AMRVs 
but unacceptable without
62 Modes to 50 Hz 216 Modes to 100 Hz 740 Modes to 200 Hz
62 Modes + 678 AMRVs
16
Mode Shapes Also Match Better
(RSS in +/-2% Frequency Band)
• RSS match looks at all modes without ± 2% of frequency of best fit 
mode (deals with ‘rotated’ modes)
• Adding AMRVs improves RSS shape match
• Almost all modes better than 99.5% RSS cross-ortho
• Without AMRVs most results are still not too bad
• This is mostly because parameterized variation doesn’t move mode shapes 
around that much anyway
17
Observations from Component Modal Data
• For ± 20% changes, AMRVs aren’t really necessary
• Do improve shape match a little bit but probably not worth it
• AMRVs do a good job of predicting changes within the range of 
changes accounted for
• ± 50% changes in materials and -90% changes in springs for this test 
case
• AMRVs don’t necessary improve solution beyond range of 
parameter accounted for
• Recommended approach
• Don’t bother calculating AMRVs for parameters that vary within ± 20%
• For parameters that do vary more than ± 20% calculate AMRVs for 
max and min range
18
Second Case is SLS on Mobile Launcher (ML)
• Much larger model
• 53 parameters
• 44 E’s – allowed to vary ± 50%
• 9 interface K’s – allowed to vary ± 90%
• 106 augmented mode sets run in steps due 
to limitation in old version of Nastran
• Run augmented modes in groups through 
DMAP alter until all 106 sets accounted for
• Final AMRV run with sensitivities took about 2 
hours
• 56 nominal modes to 6 Hz + 213 AMRVs for 
total of 269 “modes”
• Ran 133 cases using DOE
• 106 cases for extremes
• 27 cases for intermediate values within extremes
• Consider all modes to 5 Hz
• Modes aligned using MAC with 132 DOF
19
Results for 133 ± 50%/90% Parameter Variations
• With just 56 modes to 6 Hz 95% of frequencies are within 1.73% and 
99% are within 3.76%
• With 269 modes to 12 Hz 95% of frequencies are within 1.38% and 
99% are within 2.98%
• With AMRVs 95% of frequencies are within 0.21% and 99% are 
within 0.92%
• For this case solutions without AMRVs are not acceptable, but 
solutions with AMRVs are
• Adding modes from 6 Hz to 12 Hz didn’t help much
56 Modes to 6 Hz 269 Modes to 12 Hz 56 Modes to 6 Hz + 213 AMRVs
20
Summary
• SDM is a very powerful way of generating a reduced order model 
to track effect of parameter variations on modal properties
• Method typically works well for ±10% to ±20% parameter variations 
using modes slightly beyond frequency range of interest
• Outside this range typically requires multiple steps or accepting 
significant errors relative to “exact” solution
• Adding more modes doesn’t help SDM very much
• Augmented Modal Residual Vectors (AMRVs) Extend Applicability 
of SDM to much larger parameter variations
• Order of magnitude variations are possible
• Cost in setting up runs and generating AMRVs is significant
• Recommendation is to use AMRVs judiciously for those variables 
that will undergo large variations
• Typical examples are springs at joints
© The Aerospace Corporation 2010
© ATA Engineering, Inc., 2018
Thank you


Improving Accuracy of Structural Dynamic Modifications Using Augmented Mode Residual Vectors

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