Topics in Modal Analysis, Volume 7 pp 743-769 | Cite as

# Expansion of Nonlinear System Response Using Linear Transformation Matrices from Reduced Component Model Representations

## Abstract

Finite element models of structural systems have become increasingly complicated as they are used for many types of applications. Because a highly detailed model is not required for every purpose, much work has been presented in generating reduced-order models (ROMs) that accurately reflect the dynamic characteristics of the system. Recent work has shown that these ROMs can be used to compute both the linear and nonlinear dynamic response of the structure at the reduced space without loss of accuracy due to the reduction procedures.

Although computationally beneficial to calculate dynamic response at drastically reduced space, there are significant advantages in expanding the response out to the full-field solution. Much work has been focused on the expansion of linear dynamic solutions for improved visualization. This work builds on previous efforts and demonstrates that the transformation matrix generated from the linear model can be used to accurately expand the nonlinear dynamic response; this is true for component as well as system models. Both analytical and experimental cases are examined to demonstrate the accuracy of this technique.

### Keywords

Nonlinear analysis Efficient computation Model reduction Test-verified results Dynamic response### Nomenclature

### Symbols: Matrix

- [M]
Analytical Mass Matrix

- [K]
Analytical Stiffness Matrix

- [C]
Physical Damping Matrix

- [U]
Analytical Modal Matrix

- \([\bar{\mathrm{M}}]\)
Diagonal Modal Mass Matrix

- \([\bar{\mathrm{K}}]\)
Diagonal Modal Stiffness Matrix

- \([\bar{\mathrm{C}}]\)
Diagonal Modal Damping Matrix

- [T]
TransformationMatrix

- [E]
Experimental Modal Vectors

- [I]
Identity Matrix

### Vector

- {e}
Experimental Mode Shape

- {p}
Modal Displacement

- {F}
Force

- {u}
Analytical Mode Shape

- {x}
Physical Displacement

- \(\{\dot{\mathrm{x} } \}\)
Physical Velocity

- \(\{\ddot{\mathrm{x} } \}\)
Physical Acceleration

- {t}
Time Vector

### Subscript

- 1
State 1

- 2
State 2

- 12
State 1–2

- i
Row i

- j
Column j

- n
Full Set of Finite Element DOF

- a
Reduced Set of DOF

- d
Deleted (Omitted) Set of DOF

- U
SEREP

### Superscript

- T
Transpose

- g
Generalized Inverse

- k
kth Degree of Freedom

- − 1
Standard Inverse

- A
Component A

- B
Component B

- AB
System AB

### Variable

*Δ*tTime Step

- ζ
Modal Damping

### Acronyms

- ADOF
Reduced Degrees of Freedom

- DOF
Degrees of Freedom

- ERMT
Equivalent Reduced Model Technique

- FEM
Finite Element Model

- MAC
Modal Assurance Criterion

- MMRT
Modal Modification Response Technique

- POC
Pseudo-Orthogonality Check

- SEREP
System Equivalent Reduction Expansion Process

- TRAC
Time Response Assurance Criterion

## Notes

### Acknowledgements

Some of the work presented herein was partially funded by Air Force Research Laboratory Award No. FA8651-10-1-0009 “Development of Dynamic Response Modeling Techniques for Linear Modal Components”. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the particular funding agency. The authors are grateful for the support obtained.

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