Comparison of CMS, Krylov and Balanced Truncation Based Model Reduction from a Mechanical Application Engineer’s Perspective

Conference paper
Part of the Conference Proceedings of the Society for Experimental Mechanics Series book series (CPSEMS)

Abstract

Component Mode Synthesis (CMS) is a well known and established method for order reduction of Finite Element (FE) models. One advantage of CMS is a clear physical interpretability and another, more practical one, is the availability in common FE packages. In the last years a lot of research has been done, in order to adapt reduction methods, which are based on Krylov subspaces and balanced truncation for FE models. Several recent publications denote mode based reduction methods, like CMS, as out-dated while the latter ones are so called ‘modern methods’. For a mechanical application engineer the question arises, whether these methods are really so advantageous, that the reliable CMS should be exchanged against one of the two other methods.

This paper is devoted to a numerical and qualitative comparison of these three methods with respect to each other. The contribution starts with an introduction, where the ‘mechanical application engineer’s perspective’ is explained in terms of requirements and boundary conditions of the reduction process. Next, all three approaches will be briefly outlined and representative literature will be cited. In the next chapter, all three methods will be demonstrated by a simple three mass example, so that some basic characteristics can easily be seen and first conclusions can be drawn. In the subsequent section the different methods will be applied to simple FE structures and the quality of the reduced models will be examined. Finally a conclusion will be drawn whether one of the three methods is clear better (or worse) with respect to the other ones.

Keywords

Rubber Cyan 

Nomenclature

M

Stiffness matrix

K

Stiffness matrix

B, B1

Input matrix

C, C1

Output matrix

E, A

System matrices for LTI system

q

State space vector

\( {{ \tilde{\mathbf M}}} \)

Quantity M in reduced model

u

Force vector

x

Nodal DOF vector of FE model

y

Output vector

n

Number of DOF of FE model

u

Number of external forces

y

Number of outputs

m

Number of DOF in reduced model

p

Number of considered modes

s

Number of static trial vectors

V, W

Subspace for reduction

VD

Modes gained by an eigenvalue problem

VS

Static deformation shapes

xB

Boundary DOF

xI

Inner DOF

b

Number of boundary DOF

s

Complex laplace variable

H(s)

Transfer function matrix

f

Frequency

ωi

Eigenfrequency of the i-th mode

ω*

Eigenfrequency limit

\( {{\mathbf T}}_j^{{{s_0}}} \)

j-th moment around s0

WC

Controllability Gramian matrix

WO

Observability Gramian matrix

t

Time

T

Particular time instant

σi

Hankel Singular Value

r

Number of considered trial vectors

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Copyright information

© Springer Science+Buisness Media, LLC 2012

Authors and Affiliations

  1. 1.University of Applied Sciences Upper AustriaWelsAustria

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