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

  • Tim Marinone
  • Louis Thibault
  • Peter Avitabile
Conference paper
Part of the Conference Proceedings of the Society for Experimental Mechanics Series book series (CPSEMS)

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

Δt

Time 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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Copyright information

© The Society for Experimental Mechanics 2014

Authors and Affiliations

  1. 1.Structural Dynamics and Acoustic Systems LaboratoryUniversity of Massachusetts LowellLowellUSA

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