7.3 Vibration Analysis

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According to the vibration characteristics to be extracted, vibration analysis can be categorized into the

following two types: natural vibration analysis, including modal analysis (see Chapter 1 and Chapter 3),

and (forced) response analysis (see Chapter 2). Natural vibration analysis can extract natural vibration

frequencies and the associated mode shapes, which is a matrix eigenvalue problem (see Chapter 3), and can

result from a finite element discretization. The response analysis refers to the calculation of the response,

which can be displacements, strain, or stress, when the system is subjected to time-varying excitation forces.

The response analysis can be further divided into any one a combination of harmonic response analysis,

transient response analysis, and response spectrum analysis, depending on the nature of excitation forces.

7.3.1 Natural Vibration

As noted in previous chapters, the natural vibration frequencies (or simply natural frequencies) and the

associated mode shapes of a vibrating system are independent of excitation forces. In other words, they are

intrinsic characteristics of the vibration problem. Therefore, they constitute an important part of vibration

theory and vibration engineering. When vibration engineers specify design requirements in terms of

vibration, they normally do so by restricting natural frequencies, and sometimes restricting mode shapes

as well. For instance, in order to enhance the passenger comfort, vehicle designers have to ensure that the

first few natural frequencies of the vehicle are not within a certain range; in order to avoid vibration

resonance, the natural frequencies of a transmission shaft should be designed not to be identical or even

close to the rotating speeds of the shaft; in order to effectively control vibration, vibration sensors and

actuators have to be located at those places where the dominant mode shapes have large displacements.

Newton’s law

Mu€ ðtÞ ¼ 2F

Hooke’s Law

F ¼ KuðtÞ

Hamilton’s principle

d

ðt2

t1 ðT 2 V Þdt þ

ðt2

t1

dW dt ¼ 0

Finite element equation without damping

Mu€ þ Ku ¼ F

M ¼

X

e

LTMeL; K ¼

X

e

LTKeL; F ¼

X

e

LTFe

Guyan reduction scheme

QTMQu€ 1 þ QTKQu1 ¼ F

Q ¼

I

2K21

22 K21

" #

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© 2005 by Taylor & Francis Group, LLC

From a mathematical standpoint, the natural vibration analysis of a multi-DoF system requires the

solution of a matrix eigenvalue problem. According to the theory of second-order ordinary differential

equations, the solution of Equation 7.34, when F ¼ 0; can be expressed as u ¼ v eivt : By substituting

u ¼ v eivt into Equation 7.34 and letting F ¼ 0; one can obtain

v2Mv ¼ Kv ð7:45Þ

Equation 7.45 represents a generalized matrix eigenvalue problem. For an N-dimensional matrix pair

ðM; KÞ; there exist N pairs of solutions ðvi; viÞ; v2i

Mv ¼ Kvi ði ¼ 1; 2; …; NÞ, where vi and vi are called

the ith natural frequency and the associated ith mode shape, respectively.

The numerical methods for solving the matrix eigenvalue problem given by Equation 7.45 have been

well developed with the symmetric and sparse features of ðM; KÞ being fully considered. Those that have

been used by commercial finite element software packages include the power method, the subspace

iteration method, the LR method, the QR method, the Givens method, the Householder method, and the

Lanczos method.

When conducting vibration modeling, modelers need to understand how idealization and

simplification will affect the resulting natural frequencies and the associated mode shapes. Idealization

and simplification cause a difference between the actual mass matrix M and the resulting mass matrix Mr

ðM ¼ Mr þ DMÞ; and a difference between the actual stiffness matrix K and the resulting stiffness matrix

Kr ðK ¼ Kr þ DKÞ: Rayleigh’s quotient [9,11] can be used to determine the effect of DK and DM on a

particular natural frequency. Rayleigh’s quotient is defined as

RðxÞ ¼

xTKx

xTMx ð7:46Þ

Note that Rayleigh’s quotient RðxÞ becomes the square of the ith natural vibration frequency, RðxÞ ¼ v2i

;

when x ¼ vi: Thus, Rayleigh’s quotient can be expressed as

v2i

þ Dv2i

¼ ðvT

i þ DvT

i ÞðKr þ DKÞðvi þ DviÞ

ðvT

i þ DvT

i ÞðMr þ DMÞðvi þ DviÞ ð7:47Þ

where Dv2i

and Dvi are the increase of the ith natural frequency and the variation of the ith mode shape,

respectively, induced by DK and DM: Because of the fact that RðxÞ reaches the stationary value when x is

equal to the eigenvector vi; Equation 7.47 can be simplified as [1,4]

Dv2i

¼ vT

i ðDK 2 DMÞvi ð7:48Þ

Equation 7.48 indicates that an increase in stiffness leads to a rise in a natural frequency, but an increase

in mass causes a decrease in a natural frequency, as is intuitively clear.

7.3.2 Harmonic Response

Harmonic response analysis determines the response of a vibration system (model) to harmonic

excitation forces. A typical output is a plot showing response (usually displacement of a certain DoF)

versus frequency. This plot indicates how the response at a certain DoF, as a function of excitation

frequency, changes with excitation frequency. The harmonic response can also be used to calculate the

response to a general periodic excitation force, if it can be satisfactorily approximated by a summation of

its major harmonic components.

Consider a harmonic excitation force, FðtÞ ¼ F0 eivt : Substituting it into Equation 7.34, we have

Mu€ ðtÞ þ KuðtÞ ¼ eivt F0 ð7:49Þ

According to the theory of differential equations, its steady solution can be written as uðtÞ ¼ eivt U: After

substitution of uðtÞ ¼ eivt U into Equation 7.49, one obtains

ð2v2M þ KÞU ¼ F0 ð7:50Þ

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© 2005 by Taylor & Francis Group, LLC

Harmonic response analysis will solve Equation 7.50 for U against v: There are many numerical methods

available for solving Equation 7.50. The most efficient one is the modal superposition method.

In the modal superposition method, the response is expressed as a linear combination given by

uðtÞ ¼

Xj

i¼1

viu_

iðtÞ ¼ Fu_ðtÞ ð7:51Þ

where F ¼ ½v1; v2; …; vj􀀉 is a modal matrix that contains the dominant mode shapes,

and u_TðtÞ ¼ ½u_

1ðtÞ; u_

2ðtÞ; …; u_

jðtÞ􀀉 are called modal coordinates. Substituting Equation 7.51 into

Equation 7.49 and premultiplying the result by FT; we obtain

u_€ ðtÞ þ Lu_ðtÞ ¼ FT eivt F0 ð7:52Þ

Note that the modal matrix F has already been normalized against the mass matrix ðFTMF ¼ IÞ;

and that L ¼ diagðv1; v2; …; vjÞ:

Equation 7.52 represents a set of decoupled modal equations with a much smaller dimension

than Equation 7.49. After solving Equation 7.52 for u_ðtÞ and transforming u_ðtÞ back to uðtÞ through

Equation 7.51, we obtain uðtÞ:

7.3.3 Transient Response

Transient response analysis (sometimes called time-history analysis) determines the dynamic response of

a structure under the action of time-varying excitation. Excitation forces are explicitly defined in the time

domain. The computed response usually includes the time-varying displacements, accelerations, strains,

and stresses. Consider Equation 7.34 in its general form

Mu€ ðtÞ þ KuðtÞ ¼ FðtÞ

uð0Þ ¼ u0; u_ ð0Þ ¼ u_ 0

(

ð7:53Þ

where FðtÞ is the excitation force, u0 is the initial displacement, and u_ 0 is the initial velocity. As in the

harmonic response analysis, Equation 7.53 can be solved by the modal superposition method.

Substituting Equation 7.51 into Equation 7.53, premultiplying the result of the first equation by FT;

and premultiplying the result of the initial condition by FTM; we obtain

u_€ ðtÞ þ Lu_ ðtÞ ¼ FTFðtÞ

u_ ð0Þ ¼ FTMu0; u__ ð0Þ ¼ FTMu_ 0

(

ð7:54Þ

Equation 7.54 represents a set of decoupled modal equations, which can be solved by means of numerical

integration techniques. After solving Equation 7.54 for u_ðtÞ and transforming u_ðtÞ back to uðtÞ through

Equation 7.51, we can obtain uðtÞ:

To implement the numerical integration techniques, the overall time period being studied has to be

divided into a number of smaller time steps. If the time step is too large, portions of the response (such as

spikes) could be missed or truncated. On the other hand, if the time step is too small, the analysis will

take an excessively long time or even a prohibitive amount of time.

7.3.4 Response Spectrum

The excitation forces, resulting from earthquakes, winds, ocean waves, jet engine thrust, uneven roads,

and so on, do not have repeated patterns, for a variety of reasons, and thus it is difficult to describe them

using a deterministic time history. Such excitations are normally treated as random excitations. The

assumption that such excitation forces are random is recognition of our lack of knowledge of the detailed

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characteristics of the excitation forces. Some excitation forces, like those resulting from an uneven road,

could be measured to any desired accuracy, and thus they would become deterministic rather than

random. But it is not cost-effective and not convenient to do so. Therefore, engineers prefer to

characterize these excitation forces by a statistical description that can be easily measured on any

particular representative length of time history. Of the statistical descriptions, the autocorrelation

function and the power spectral density function are the most important. Denote f ðtÞ as a stationary

random excitation force, and Rf ðtÞ as its autocorrelation function and Sf ðvÞ as its power spectral density

function. Their relations [6] are

Rf ðtÞ ¼ lim

1

T

ðT

0

f ðtÞf ðt þ tÞdt ð7:55Þ

Sf ðvÞ ¼

ð1

21

Rf ðtÞe2ivt dt ð7:56Þ

The response spectrum analysis here calculates the power spectral density function of the response of

a vibration model to a random excitation force, described by its power spectral density function

(see Chapter 5). For the single DoF system given in Figure 7.1

mu€ ðtÞ þ kuðtÞ ¼ f ðtÞ ð7:57Þ

The power spectral density function of the response uðtÞ is given by

SuðvÞ ¼ lHðvÞl2Sf ðvÞ ð7:58Þ

where

HðvÞ ¼ ð2mv2 þ kÞ21 ð7:59Þ

is the frequency response function representing the natural vibration characteristic of the system.

In the case of the multiple-DoF system given by Equation 7.34, the random excitation force FðtÞ is a

column vector. For the sake of simplicity, we assume all of the components in the vector FðtÞ are

stationary and statistically independent. Accordingly, all of the components in the vector of the response

uðtÞ are stationary. Under this assumption, the power spectral density function of the response uðtÞ is

determined by

SuðvÞ ¼ HðvÞSf ðvÞHTðvÞ ð7:60Þ

where Sf ðvÞ is a diagonal matrix with its ith element as the power spectral density function of the

ith element in FðtÞ; and HðvÞ is the frequency response function matrix defined by

HðvÞ ¼ ð2Mv2 þ KÞ21 ð7:61Þ

From Equation 7.60 and Equation 7.61 one can see that the power spectral density function matrix of the

response is correlated to the power spectral density function matrix of the excitation force by means of

the frequency response matrix of the system HðvÞ: In commercial finite element software packages, HðvÞ

is often calculated by the truncated modal method in which HðvÞ is approximately expressed by the

dominant natural frequencies and the associated mode shapes, neglecting the contributions of the other

mode shapes to HðvÞ; as given below.

HðvÞ <

X

i

vivT

i

v2i

2 v2 ð7:62Þ

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