台灣師範大學機電科技學系 C. R. Yang, NTNU MT -1- Chapter 14 Random Vibration 14.

台灣師範大學機電科技學系

C. R. Yang, NTNU MT

Chapter 14Random Vibration

C. R. Yang, NTNU MT

Chapter Outline

14.1 Introduction14.2 Random Variables and Random Processes14.3 Probability Distribution14.4 Mean Value and Standard Deviation14.5 Joint Probability Distribution of Several Random Variables14.6 Correlation Functions of a Random Process14.7 Stationary Random Process14.8 Gaussian Random Process14.9 Fourier Analysis14.10 Power Spectral Density14.11 Wide-Band and Narrow-Band Processes14.12 Response of a Single DOF system14.13 Response Due to Stationary Random Excitations14.14 Response of a Multi-DOF System

C. R. Yang, NTNU MT

14.1Introduction

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14.1 Introduction

• Random processes has parameters that cannot be precisely predicted.

• E.g. pressure fluctuation on the surface of a flying aircraft

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14.2Random Variables and Random Processes

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14.2 Random Variables and Random Processes

• Any quantity whose magnitude cannot be precisely predicted is known as a random variable (R.V)

• Experiments conducted to find the value of the random variable will give an outcome that is not a function of any parameter

• If n experiments are conducted, the n outcomes form the sample space of the random variable.

• Random processes produces outcomes that is a function of some parameters.

• If n experiments are conducted, the n sample functions form the ensemble of the random variable.

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14.3Probability Distribution

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14.3 Probability Distribution

• Consider a random variable x.

limfunction on distributiy Probabilit

toequalor smaller ofnumber theis

,, as available are of valuesalexperiment

valuespecified some is

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• Consider a random time function as shown:

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xdxpxP

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14.4Mean Value and Standard Deviation

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14.4 Mean Value and Standard Deviation

• Expected value of f(x) =μf

• The positive square root of σ(x) is the standard deviation of x.

22__2__

______

of Variance

xxdxxpxxxxE

dxxpxxxExxf

dxxxpxxExxf

dxxpxfxfxfE

C. R. Yang, NTNU MT

Example 14.1Probabilistic Characteristics of Eccentricity of a Rotor

The eccentricity of a rotor (x), due to manufacturing errors, is found to have the following distribution

where k is a constant. Find the mean, standard deviation and the mean square value of the eccentricity and the probability of realizing x less than or equal to 2mm.

elsewhere ,0

mm5x0 ,2kxxp

C. R. Yang, NTNU MT

Example 14.1Probabilistic Characteristics of Eccentricity of a RotorSolution

Normalize the probability density function:

Mean value of x:

Standard deviation of x:

3 i.e. 1

xkdxkxdxxp

mm75.34

xkxdxxpx

mm9682.0

9375.075.35

kxdxkx

dxxpxxxxdxxpxx

C. R. Yang, NTNU MT

Example 14.1Probabilistic Characteristics of Eccentricity of a RotorSolution

The mean square value of x is

064.0125

dxxkdxxpx

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14.5Joint Probability Distribution of Several RV

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14.5 Joint Probability Distribution of Several RV

• Joint behavior of 2 or more RV is determined by joint probability distribution function

• Joint pdf of single RV is called univariate distributions

• Joint pdf of 2 RVs is called bivariate distributions

• Joint pdf of more than one RV is called multivariate distributions

• Bivariate density function of RV x1 and x2:

222211112121 ,Prob, dxxxxdxxxxdxdxxxp

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• Joint pdf of x1 and x2:

• Marginal density functions:

1, 2121 dxdxxxp

221121

,Prob,x

xxdxdxxp

xxxxxxP

-dyyxpyp

dyyxpxp

C. R. Yang, NTNU MT

• Variances of x and y:

dyypyyE

dxxpxxE

dxdyyxpdxdyyxyp

dxdyyxxpdxdyyxxyp

dxdyyxpyxxy

dxdyyxpyx

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• Correlation coefficient between x and y:

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14.6Correlation Functions of a Random Process

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14.6 Correlation Functions of a Random Process

• Form products of RV x1, x2, …

• Average the products over the set of all possibilities to obtain a sequence of functions:

• These functions are called correlation functions

on... so and

321321321

212121

xxxEtxtxtxEtttK

xxEtxtxEttK

C. R. Yang, NTNU MT

• E[x1x2] is also known as the autocorrelation function, designated as

R(t1,t2)

• Experimentally we can find R(t1,t2) by multiplying x(i)(t1) and x(i)(t2)

and averaging over the ensemble:

21212121 ,, dxdxxxpxxttR

ii txtxn

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14.7Stationary Random Process

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14.7 Stationary Random Process

• Probability distribution remain invariant under shift of time scale

• Pdf p(x1) becomes universal density function p(x) independent of time

• Joint density function p(x1,x2) becomes p(t,t+τ)

• Expected value of stationary random processes

• Autocorrelation function depend only on the separation time τ where τ=t2-t1

tttxEtxE any for 11

tRtxtxExxEttR any for , 2121

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• R(0)=E[x2]

• If the process has zero mean and is extremely irregular as shown, R(τ) will be small.

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• If x(t)≈x(t+τ), R(τ) will be constant.

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• If x(t) is stationary, its mean and standard deviations will be independent of t: txtxtxEtxE and

,1 Since

:tcoefficienn Correlatio

txEtxEtxtxE

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• R(τ) is an even function of τ.

• When τ∞, ρ0, R(τ∞)μ2

• A typical autocorrelation function is shown:

RtxtxEtxtxER

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• Ergodic Process

We can obtain all the probability info from a single sample function and assume it applies to the entire ensemble.

x(i)(t) represents the temporal average of x(t)

dttxtxT

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14.8Gaussian Random Process

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14.8 Gaussian Random Process

• Most commonly used distribution for modeling physical random processes

• The forms of its probability distribution are invariant wrt linear operations

• Standard normal variable:

1 zexp

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• The graph of a Gaussian probability density function is as shown:

dxectx

dxectxc

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• Some typical values are shown below:

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14.9Fourier Analysis

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14.9 Fourier Analysis

• Fourier Series

Any periodic function x(t) of period τ can be express as a complex Fourier series

Multiply both side with e-imω0t and integrating:

where 00

tinnectx

dttmnitmnc

dtecdtetx

sincos

C. R. Yang, NTNU MT

• Fourier Series

x(t) can be expressed as a sum of infinite number of harmonics

Difference between any 2 consecutive frequencies:

If x(t) is real, the integrand of cn is the complex conjugate of that of

dtetxc tin

*nn cc

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• Fourier Series

Mean square value of x(t):

dtececc

dteccec

dtecdttxtx

2_______

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Example 14.2Complex Fourier Series Expansion

Find the complex Fourier series expansion of the functions shown below:

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Example 14.2Complex Fourier Series ExpansionSolution

tscoefficienFourier

2 and 2 where

dtetxc

tintin

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tintin

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The equation can be reduced to

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Note that

,6,4,2 ,0

,5,3,1 ,24

,6,4,2 ,1

,5,3,1 ,1

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Frequency spectrum is as shown:

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• Fourier Integral

A non periodic function as shown can be treated as a periodic function with τ∞

where 00

tinnectx

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As τ∞,

dtetxcX

dtetxdtetxc

lim Define

limlim2

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Integral Fourier Transform pair

Mean square value of x(t):

dtetxX

cccdttx

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This is known as Parseval’s formula for nonperiodic functions.

Xdttxtx

dXcXc nn

as and ,2

2_______

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Example 14.3Fourier Transform of a Triangular Pulse

Find the Fourier transform of the triangular pulse shown below.

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Example 14.3Fourier Transform of a Triangular PulseSolution

otherwise ,0

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sincos

sincos2

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14.10Power Spectral Density

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14.10 Power Spectral Density

• Power spectral density S(ω) is the Fourier transform of R(τ)/2π

• If the mean is zero,

• R(0) is the average energy

dSRx 02

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• S(-ω)=S(ω)

• Only positive frequencies are counted in an equivalent one-sided spectrum Wx(f)

2 dffWdSxE x

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dffWdS

C. R. Yang, NTNU MT

14.11Wide-Band and Narrow-Band Processes

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14.11 Wide-Band and Narrow-Band Processes

• Wide-band random process:

• E.g. pressure fluctuations on surface of rocket

• Narrow-band random process:

• A process whose power spectral density is constant over a frequency range is called white noise.

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• Ideal white noise – band of frequencies is infinitely wide

• Band-limited white noise – band of frequencies has finite cut off frequencies.

• Mean square value is the total area under the spectrum: 2S0(ω2 –

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Example 14.4Autocorrelation and Mean Square Value of a Stationary Process

The power spectral density of a stationary random process x(t) is shown below. Find its autocorrelation function and the mean square value.

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Example 14.4Autocorrelation and Mean Square Value of a Stationary ProcessSolution

We have

sinsin2

cos2cos2

dSdSR xx

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Example 14.4Autocorrelation and Mean Square Value of a Stationary ProcessSolution

Mean Square Value 12002 22

SdSdSxE x

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14.12Response of a Single DOF System

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14.12 Response of a Single DOF System

2,,, where

C. R. Yang, NTNU MT

• Impulse Response Approach

Let the forcing function be a series of impulses of varying magnitude as shown:

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y(t)=h(t-τ) is the impulse response function

Total response can be found by superposing the responses.

Response to total excitation:

tdthxty

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• Frequency Response Approach

Transient function:

, H(ω) is the complex frequency response function.

Total response of the system:

titi eHtyetx ~~

deXtx ti

deXHtxHty

C. R. Yang, NTNU MT

Since h(t- )=0 when t< or >t,

Change the variable from to θ=t- ,

Both the superposition integral and the Fourier integral can be used to find system response

dthxty

dhtxty

dtethHdeHth

dtetdtetxX

deHXthty

C. R. Yang, NTNU MT

14.13Response Due to Stationary Random Excitations

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14.13 Response Due to Stationary Random Excitations

• When excitation is a stationary random process, the response is also a stationary random process

dhtxEtyE

dhtxty

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Autocorrelation

212121

222111

ddhhtxtxE

tytyER

ddhhtxtx

dhtxdhtxtyty

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Power Spectral Density

212121

C. R. Yang, NTNU MT

dehdehS

C. R. Yang, NTNU MT

H(-ω) is the complex conjugate of H(ω).

Mean Square Response:

xy SHS2

ddhhRRyE

2111212

C. R. Yang, NTNU MT

Example 14.5Mean Square Value of Response

A single DOF system is subjected to a force whose spectral density is a white noise Sx(ω)=S0. Find the following:

a)Complex frequency response function of the systemb)Power spectral density of the responsec)Mean square value of the response

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Example 14.5Mean Square Value of ResponseSolution

a)Substitute input as eiωt and corresponding response as y(t)=H(ω)eiωt

eeHkicm

txkyycymtiti

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Example 14.5Mean Square Value of ResponseSolution

b) We have

c) Mean square value

kicmSSHS xy

dSyE y

C. R. Yang, NTNU MT

Example 14.6Design of the Columns of a Building

A single-storey building is modeled by 4 identical columns of Young’s modulus E and height h and a rigid floor of weight W. The columns act as cantilevers fixed at the ground. The damping in the structure can be approximated by a constant spectrum S0. If each column has a

tubular cross section with mean diameter d and wall thickness t=d/10, find the mean diameter of the columns such that the standard deviation of the displacement of the floor relative to the ground does not exceed a specified value δ.

C. R. Yang, NTNU MT

Example 14.6Design of the Columns of a BuildingSolution

Model the building as a single DOF system.

4403 64

4 , iddIh

EIkgWm

C. R. Yang, NTNU MT

47592.003966.012

03966.08000

101,10With

tdtdtdtdtdtd

ddddddI

tddtdd

C. R. Yang, NTNU MT

When the base moves, equation of motion:

eHtzex

xmkzzczm

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iSdSzE

C. R. Yang, NTNU MT

47592.0

47592.0or

47592.0, Since

C. R. Yang, NTNU MT

14.14 Response of a Multi-DOF System

C. R. Yang, NTNU MT

• Equation of motion:

• Physical and generalized coordinates are related as:

nitQtqtqtq iiiiiii ,,2,1 ;2 2

jii tqXtxtqXtx

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• Physical and generalized forces are related as:

tFXtQtFXtQ

tNtfXtQ

1 where

, Assume

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• Mean square value of physical displacement:

2_______

dttHHT

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dttHHT

angles. phase eNeglect th

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• For stationary random process,

dttHHT

dSdttT

2_______

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• Mean square value of xi(t)

rii dSHH

NNXXtx

_______2

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• For lightly damped systems,

dHSdSH

_______2

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Example 14.7Response of a Building Frame Under an Earthquake

A 3-storey building is subjected to an earthquake. The ground acceleration during the earthquake can be assumed to be a stationary random process with a power spectral density S(y)=0.05(m2/s4)/(rad/s). Assuming a modal damping ratio of 0.02 in each mode, determine the mean square values of the responses of the various floors of the building frame under the earthquake.

C. R. Yang, NTNU MT

Example 14.7Response of a Building Frame Under an EarthquakeSolution

C. R. Yang, NTNU MT

Compute eigenvalues and eigenvectors using k=106N/m and m=1000kg

rad/s 0001.578025.1

rad/s 4368.392471.1

rad/s 0734.1444504.0

C. R. Yang, NTNU MT

01036.0

02330.0

01869.0

5544.0

2468.1

0000.15991.0

01869.0

01037.0

02331.0

8020.0

4450.0

0000.17370.0

02330.0

01869.0

01037.0

2470.2

8019.1

0000.13280.0

C. R. Yang, NTNU MT

Relative displacements of the floors: zi(t)=xi(t)-y(t), i=1,2,3

Equation of motion:

where [Z] denotes the modal matrix.

ymzkzczm

zkzcxm

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Assume damping ratio ζi = 0.02

Uncoupled equations of motion:

tytmmf

tftymtymtF

3,2,1 ;Q2

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Mean square values

35.5205235.01000

37.5205237.01000

36.5205236.01000

_______2

C. R. Yang, NTNU MT

Mean square values of relative displacements of various floors of the building frame:

2_______

m 00216455.0

m 00139957.0

m 00053132.0

C. R. Yang, NTNU MT

Example 14.8Probability of Relative Displacement Exceeding a Specified Value

Find the probability of the magnitude of the relative displacement of the various floors exceeding 1,2,3, and 4 standard deviations of the corresponding relative displacement for the building frame of Example 14.7

C. R. Yang, NTNU MT

Example 14.8Probability of Relative Displacement Exceeding a Specified Value Solution

Assume ground acceleration to be normally distributed random process with zero mean.

Relative displacements of various floors can also be assumed to be normally distributed.

C. R. Yang, NTNU MT

Example 14.8Probability of Relative Displacement Exceeding a Specified Value Solution

4for 00006.0

3for 00270.0

2for 04550.0

1for 31732.0

3,2,1 ;_______

台灣師範大學機電科技學系 C. R. Yang, NTNU MT -1- Chapter 14 Random Vibration 14.

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