33.2 Structural Dynamic Modification of Finite Element Model

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The finite element method is an important tool to obtain numerical and computational solutions to

problems in structural vibration analysis. By applying the finite element method to a structure, a discrete

analysis model to idealize the continuum can be obtained. The finite equation of vibrations of a structure

in the global coordinate system is

Mq€ þ Cq_ þ Kq ¼ Q ð33:1Þ

where M; K; and C are the mass, stiffness, and damping matrices, respectively, q€ ; q_ ; and q are the

acceleration, velocity, and displacement vectors, respectively, and Q is the external load vector.

Neglecting the damping force and external load vector, Equation 33.1 becomes

Mq€ þ Kq ¼ 0 ð33:2Þ

This is the natural vibration equation for the structure. Its solution (the natural vibration) is harmonic

(see Chapter 30), and is given by

q ¼ u cosðvt 2 wÞ ð33:3Þ

where u is modal vector, and v the natural frequency of the system. Substituting Equation 33.3 into

Equation 33.2, the eigenproblem of structural vibration can be obtained as

Ku ¼ lMu ð33:4Þ

where l ðl ¼ v2Þ denote the eigenvalues of the system.

33-2 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

In structural vibration analysis, the natural frequencies and the corresponding modal vectors can

be obtained by solving the eigenproblem (Equation 33.4). The solutions for n eigenvalues and

corresponding eigenvectors satisfy

KU ¼ MUL ð33:5Þ

where U; which is called the modal matrix, is an ðn £ nÞ matrix with its columns equal to the n

eigenvectors, and L is an ðn £ nÞ diagonal matrix consisting of the corresponding eigenvalues as the

diagonal elements; specifically

U ¼ ½u1; u2; …; un􀀉 ð33:6Þ

L ¼ diagðliÞ; i ¼ 1; 2; …; n ð33:7Þ

An important relation for eigenvectors is that of M orthogonality and K orthogonality; that is, we have

uT

i Muj ¼ dij ð33:8Þ

uT

i Kuj ¼ lidij ð33:9Þ

where dij is the Kronecker delta. For n eigenpairs, Equation 33.8 and Equation 33.9 can be written as

UTMU ¼ I ð33:10Þ

UTKU ¼ L ð33:11Þ

Since the modal vectors are independent, an arbitrary displacement vector, u; can be expressed as a

linear combination of ui; i ¼ 1; 2; …; n; that is

u ¼

Xn

r¼1

cr ur ¼ UC ð33:12Þ

where cr is a constant. Each constant cr can be determined by

cr ¼ uT

r Mu; r ¼ 1; 2; …; n ð13:13Þ

This is known as the expansion theorem.

Suppose the physical parameter of a given structure is given a small modification. This will cause a

small change in the matrices K0 and M0; that is

M ¼ M0 þ 1M1; K ¼ K0 þ 1K1 ð33:14Þ

where 1 is a small parameter, K0 and M0 are the original mass and stiffness matrices, respectively, and

1M1 and 1K1 are the corresponding modifications. It is obvious that if M0 and K0 are symmetric, the

matrices M1 and K1 are also symmetric.

If 1M1 and 1K1 are small, the changes of eigenvalues and eigenvectors of the structure are also small.

According to the matrix perturbation theory, the eigensolutions of Equation 33.4 can be expressed in the

form of a power series in 1; thus

ui ¼ u0i þ 1u1i þ 12u2i þ · · · ð33:15Þ

li ¼ l0i þ 1l1i þ 12l2i þ · · · ð33:16Þ

where u0i and l0i are the eigensolutions of the original structure, l1i and l2i are the first- and the secondorder

perturbations of the eigenvalues, and u1i and u2i are the first- and the second-order perturbation of

the eigenvectors.

Since the eigensolutions of the original structure, u0i and l0i; are known, only the first- and the

second-order perturbations of the eigensolutions are required without solving Equation 33.4.

Structural Dynamic Modification and Sensitivity Analysis 33-3

© 2005 by Taylor & Francis Group, LLC

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