33.4 Design Sensitivity Analysis of Structural Vibration Modes

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In the optimization of structural analysis, the design sensitivity analysis of eigenvalues and eigenvectors

plays an essential role. The designer can use this information directly in an interactive computer-aided

design procedure as a valuable guide. Significant work has been done in this area (Haug et al., 1985;

Adelmen and Haftka, 1986; Chen and Pan, 1986; Wang, 1991).

33.4.1 Direct Differential Method for Sensitivity Analysis

Design sensitivity analysis of eigenvalues and eigenvectors will reveal how the changes in some design

parameters in the system affect the dynamic characteristics of the structure.

Let li;j and ui;j denote the sensitivity of the eigenvalue, li; and the eigenvector, ui; respectively, with

respect to the design variables bj ðj ¼ 1; 2; …; LÞ; and let K;j and M;j denote the derivative of the stiffness

and mass matrices, respectively, with respect to bj: The design sensitivity of the eigenvalue is

li;j ¼ uT

i ðK;j 2 liM;jÞui ð33:25Þ

The sensitivity of the eigenvector, ui;j; can be expressed as the following series:

ui;j ¼

Xn

s¼1

cijsus ð33:26Þ

where

cijs ¼

1

li 2 ls

uT

s ðK;j 2 liM;jÞui; i – s; i; s ¼ 1; 2; …; n; i – s ð33:27Þ

ciji ¼ 2

1

2

uT

i M;jui ð33:28Þ

33.4.2 Perturbation Sensitivity Analysis

Let DK and DM denote the increments of the stiffness and the mass matrices resulting from an

incremental change of the design variable, Dbj; and let Dli and Dui denote the corresponding

perturbations of the eigenvalue and eigenvector, respectively. The direct differential method of design

sensitivity analysis of vibration modes can now be put into perturbation form, approximately as

li;j ¼

Dli

Dbj ð33:29Þ

ui;j ¼

Dui

Dbj ð33:30Þ

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© 2005 by Taylor & Francis Group, LLC

where Dli and Dui can be evaluated by the perturbation formulas presented in this chapter. In practical

analysis, the design variables could be the cross-sectional area of the truss members, bending moment of

inertia, equivalent torsional moment of inertia of a beam, the thickness of a plate, or other variable. In

some complex structures, a mass, mr ; may be placed at a node point and moving in the direction of the

rth DoF, or an elastic support with spring stiffness, Kr ; may be placed at a certain node point. It is also

possible that an elastic connector of stiffness, Kj; might exist between two components. They can also be

considered as design variables. In finite element analysis, DK and DM are known to be the sum of the

element increments, DKe and DMe; thus

DK ¼

X

e

DKe ð33:31Þ

DM ¼

X

e

DMe ð33:32Þ

Hence, the sensitivity formulas of vibration modes as given above can be transformed into the finite

element perturbation form (Chen and Pan, 1986)

li;j ¼

1

Dbj

X

e

u􀀊 T

i ðDKe 2 liDMeÞu􀀊 i ð33:33Þ

and

ui;j ¼

1

Dbj

X

e

Xn

s¼1

s–i

1

li 2 ls

u􀀊 T

s ðDKe 2 liDMeÞu􀀊 ius 2

1

2

u􀀊 T

i DMe u􀀊 iui

0

BBB@

1

CCCAð33:34Þ

In these formulas, the overbar signifies that the eigenvector concerned contains only the components

needed for the eth finite element. It is important to observe that, in Equation 33.33 and Equation 33.34,

calculations are done on the element basis, and as a result, the calculations are greatly simplified.

Using the shorthand notations

le

i;j ¼

1

Dbj

u􀀊 T

i DKe 2 liDMe u􀀊 i ð33:35Þ

ue

i;j ¼

1

Dbj

Xn

s¼1

s–i

1

li 2 ls

u􀀊 T

s DKe 2 liDMe u􀀊 ius 2

1

2

u􀀊 T

i DMe u􀀊 iui

0

BBB@

1

CCCA

ð33:36Þ

Equation 33.33 and Equation 33.34 can be written as

li;j ¼

X

e

le

i;j ð33:37Þ

and

ui;j ¼

X

e

ue

i;j ð33:38Þ

where le

i;j and ue

i;j are the design sensitivity of the eth element for the eigenvalue li and the eigenvector ui;

respectively. Let us consider the following important cases.

For a concentrated mass, mr ; placed at a node point and moved in the direction of the rth DoF,

Equation 33.33 and Equation 33.34 become

li;r ¼

Dli

Dmr ¼ 2liu2

ir ð33:39Þ

Structural Dynamic Modification and Sensitivity Analysis 33-9

© 2005 by Taylor & Francis Group, LLC

and

ui;r ¼

Dui

Dmr ¼

Xn

s¼1

s–i

2li

li 2 ks

usr uir us 2

1

2

u2

ir ui ð33:40Þ

where uir is the rth element of the ith eigenvector ui:

For an elastic connector with stiffness Kj between the rth and the lth DoF of two components, Equation

33.33 and Equation 33.34 become

li;j ¼

Dli

Dkj ¼ ðuir 2 uilÞ2 ð33:41Þ

and

ui;j ¼

Dui

Dkj ¼

Xn

s¼1

s–i

1

li 2 ls ðusr uir 2 usluir 2 usr uil þ usluilÞus ð33:42Þ

For an elastic support with spring stiffness Kr placed in the direction of the rth DoF, Equations 33.33 and

Equation 33.34 become

li;r ¼

Dli

Dkr ¼ u2

ir ð33:43Þ

and

ui;r ¼

Dui

Dkr ¼

Xn

s¼1

s–i

1

li 2 ls

usr uir us ð33:44Þ

33.4.3 Numerical Example

The design sensitivity analysis of an automotive chassis is presented here as an illustration of the method.

Example 33.4

The finite element model of an automobile chassis consists of 39 beam elements involving 30 nodal

points and 180 DoF (Figure 33.4).

The design variables for the sensitivity analysis of eigenvalues in this example are the equivalent

torsional moment of inertia, J; and the bending moment of inertia, Iy ; of the beam element of

FIGURE 33.4 Finite element model of the automotive chassis for Example 33.4.

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© 2005 by Taylor & Francis Group, LLC

the structure. Results for the sensitivity of eigenvalues with respect to J and Iy are given in Table 33.5,

in which NE denotes the number of the element. Only the highest eight values are given, and they are

listed in descending order.

From Table 33.5, it is seen that for this particular chassis, the sensitivities of the first natural

frequency, le

1Iy ; are much smaller than le

1J : This indicates that there is very little effect of the change of

bending moment of inertia, Iy ; of the beams on the vibration of the chassis at its first natural frequency.

Thus, we can conclude that the first mode is a torsional mode. Similarly, the results indicate that the

third mode is also a torsional mode. On the other hand, the second and the fourth modes are

recognized to be bending modes. This information is very useful to the designer when deciding on a

change in the design. For example, if he wants to increase the first torsional frequency, the efficient way

is for him to increase the equivalent torsional moments of inertia of beam elements 15, 11, 19, 24, 6,

and so on.

It should be noted that only the first low-frequency modes are available and can be used as basis

vectors of eigenvector derivatives in Equation 33.26. However, modal truncation induces errors, and

the errors become significant if more high-frequency modes are truncated. An improvement to

truncated modal superposition representation of eigenvector derivatives is presented in the next

section.

33.4.4 Concluding Remarks

As can be seen from the numerical examples given above, the matrix perturbation method is an extremely

useful tool for fast reanalysis of a modified structure. It is widely used in a range of structural

modifications, such as the modification of various types of elements, local modification of structures,

sensitivity analysis of vibration modes, and so on. Therefore, matrix perturbation plays an important role

in dynamic analysis and optimization of structures.

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