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MODEL REDUCTION USING
GUYAN, IRS, AND DYNAMIC METHODS
Christopher C. Flanigan
Quartus Engineering Incorporated
San Diego, California
Model Reduction Using Guyan, IRS, and Dynamic Methods
AGENDA
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Background and introduction
Guyan reduction
IRS reduction
Dynamic reduction
Comparison of reduction methods
Mode shape expansion
Conclusions
Background and Introduction
MODAL SURVEY OFTEN PERFORMED
TO VERIFY FINITE ELEMENT MODEL
• Must be confident that structure will survive operating
environment
• Unrealistic to test flight structure to flight loads
• Alternate procedure
– Test structure under controlled conditions
– Correlate model to match test results
– Use test-correlated model to predict operating responses
• Modal survey performed to verify analysis model
– “Reality check”
Background and Introduction
TEST AND ANALYSIS DATA HAVE
DIFFERENT NUMBER OF DOF
• Finite element model (FEM)
– 10,000-1,000,000 DOF
• Test
– 50-500 accelerometers
• Compare test results to analysis predictions
Ortho  T M 
• Many other comparison techniques
– Cross-ortho, MAC, COMAC, CORTHOG, etc.
• Need a common basis for comparison
Background and Introduction
TEST-ANALYSIS MODEL (TAM)
PROVIDES BASIS FOR COMPARISON
• Test-analysis model (TAM)
– Mathematical reduction of finite element model
– Master DOF in TAM corresponds to accelerometer
• Transformation (condensation)
Kaa  TgaT Kgg Tga
Maa  TgaT Mgg Tga
• Many methods to select optimum accelerometer locations
• Many methods to perform reduction transformation
• Sensor locations and transformation method critical for
accurate TAM and test-analysis comparisons
Transformation Methods
GUYAN REDUCTION IS THE
INDUSTRY STANDARD METHOD
• Robert Guyan, Rockwell, 1965
– Pronounced “Goo-yawn”, not “Gie-yan”
• Implemented in many commercial software codes
– NASTRAN, I-DEAS, ANSYS, etc.
• Start with static equations of motion
K oo K oa  Uo  Po 
  
K

 ao Kaa  Ua  Pa 
• Assume forces at omitted DOF are negligible
Po  0
Transformation Methods
GUYAN REDUCTION IS A
SIMPLE METHOD TO IMPLEMENT
• Solve for motion at omitted DOF
Uo   Koo1 Koa Ua
• Rewrite static equations of motion
Uo   K oo 1 K oa 
 
 Ua
U
Iaa
 a 

• Transformation matrix for Guyan reduction
 K oo 1 K oa 
TGuyan  

I


aa
Transformation Methods
IRS REDUCTION ADDS
FIRST ORDER MASS CORRECTION
• Guyan neglects mass effects at omitted DOF
• IRS adds first order approximation of mass effects
GGuyan  GIRS 
TGuyan  

Iaa


GGuyan   Koo1 Koa


GIRS  Koo1 Moa  Moo GGuyan Maa 1 Kaa
Transformation Methods
DYNAMIC REDUCTION ALSO
ADDS MASS CORRECTION
• Start with eigenvalue equation
i
i
K oo K oa  o 
i Moo Moa  o 
   
K

  
K

M
M
 ao
 ao
aa   a 
aa   a 
• Replace eigenvalue with constant value L
 K oo  L Moo 1 K oa  L Moa 
TDyn Re d  

I


aa
• Equivalent to Guyan reduction if L = 0
Comparison of Reduction Methods
EACH REDUCTION METHOD HAS
STRENGTHS AND WEAKNESSES
• Guyan reduction
– Strengths
• Easy to implement
• Computationally efficient
• Widely available in commercial software (NASTRAN, etc.)
• Extensive use in pretest analysis and correlation
• Works well for many structures when good A-set selection
– Weaknesses
• Poor treatment of mass at omitted DOF
• Unacceptable accuracy for structures with high M/K
• Errors if poorly selected A-set
Comparison of Reduction Methods
EACH REDUCTION METHOD HAS
STRENGTHS AND WEAKNESSES
• IRS reduction
– Strengths
• Relatively easy to implement
– NASTRAN rigid format alter
• Computationally efficient
• Generally more accurate than Guyan reduction
– Weaknesses
• Not COTS available
• Inaccurate if poor A-set (Gordis, 1992)
• Limited industry experience
Comparison of Reduction Methods
EACH REDUCTION METHOD HAS
STRENGTHS AND WEAKNESSES
• Dynamic reduction
– Strengths
• Relatively easy to implement
– NASTRAN rigid format alter
• Computationally efficient
• Generally more accurate than Guyan reduction
– Weaknesses
• Not COTS available
• What is good choice for L?
• Limited industry experience
Comparison of Reduction Methods
SHAPE EXPANSION IS AN
ALTERNATIVE TO MATRIX REDUCTION
• Expand test mode shapes to FEM DOF
Ug  Tga Ua
• Expansion and reduction give same results if same
matrices used
• Dynamic expansion based on eigenvalue equation



oi   Koo  i Moo Koa  i Moa ai
• Computationally intensive, but computers are getting faster
all the time!
Conclusions
GUYAN, IRS, AND DYNAMIC REDUCTION
ASSIST TEST-ANALYSIS CORRELATION
• TAM provides basis for test-analysis comparisons
• Many transformation methods
– Guyan (static) reduction
– IRS reduction
– Dynamic reduction
• Each method has strengths and weaknesses
– Applicability to structures and models
– Availability and experience
– Robustness
• Expansion methods should also be considered