14台灣師範大學機電科技學系

(1)

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

-2-

Chapter Outline

14.1 Introduction

14.2 Random Variables and Random Processes 14.3 Probability Distribution

14.4 Mean Value and Standard Deviation

14.5 Joint Probability Distribution of Several Random Variables 14.6 Correlation Functions of a Random Process

14.7 Stationary Random Process 14.8 Gaussian Random Process 14.9 Fourier Analysis

14.10Power Spectral Density

14.11Wide-Band and Narrow-Band Processes 14.12Response of a Single DOF system

14.13Response Due to Stationary Random Excitations 14.14Response of a Multi-DOF System

14.1 Introduction

14.1 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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-5-

14.2 Random Variables and Random Processes

14.2

-6-

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.

14.3

-7-

14.3 Probability Distribution

-8-

14.3 Probability Distribution

• Consider a random variable x.

 

  n

n x P

x x

n

x x x x

n x

n n x x

n i

n

~

2 1

~

lim function

on distributi y

Probabilit

to equal or smaller of

number the is

, , as available are

of values al experiment

value specified some

is

Prob













(3)

-9-

14.3 Probability Distribution

• Consider a random time function as shown:

   

 

   

 

    1

Prob

0 Prob

lim 1 1 Prob

i

~ i







































P t

x

P t

x t t x P

t t x t x

n i

i

-10-

14.3 Probability Distribution

       

   

    1

lim











 







 

















x d x p P

x d x p x P

x x P x x P dx

x x dP

p

n

14.4 Mean Value and Standard Deviation

14.4 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.

         

     

 

^__² ^__ ²

__ 2 __ 2

2

2 __

2 2 2

__

______

of Variance

, If

2



 







 







 

 











 



 

 





















x x dx x p x x x

x E

x

dx x p x x x E x x f

dx x xp x x E x x f

dx x p x f x f x f E

x x x f





(4)

-13-

14.4 Mean Value and Standard Deviation

Example 14.1

Probabilistic 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.

      

 0 , elsewhere mm 5 x 0

2

, x kx p

-14-

14.4 Mean Value and Standard Deviation

Example 14.1

Probabilistic Characteristics of Eccentricity of a Rotor Solution

Normalize the probability density function:

Mean value of x:

Standard deviation of x:

 

125

3 i.e.

3 1 i.e.

1

5

0 5 3

0

2   

 



 





^ x k

k dx kx dx x p

 

3.75mm 4

5

0 5 4

0  

 



 





^p^x^xdx ^k ^x

x

       

     

mm 9682 . 0

9375 . 0 75 . 5 3 3125 5

2

2 2

5

0 5 5 2

0 4

5 0

2 5 2

0 2 2











 



 

 



 



 

















x x

k x x

k x dx kx

dx x p x x x x dx x p x x



-15-

14.4 Mean Value and Standard Deviation

Example 14.1

Probabilistic Characteristics of Eccentricity of a Rotor Solution

The mean square value of x is

   

064 . 125 0

8 3

2 Prob

mm 5 15

3125

2

0 3 2 0

2 0 2 2 ___

2



 



 



 









 



 

 

k x

dx x k dx x p x

k x

-16-

14.5 Joint Probability Distribution of Several RV

14.5

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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 x1and x2:



x1,x2



dx1dx2 Prob



x1 x1 x1 dx1,x2 x2 x2 dx2



p       

-18-

14.5 Joint Probability Distribution of Several RV

• Joint pdf of x1and x2:

• Marginal density functions:

 

 

^





p x

₁

, x

₂

dx

₁

dx

₂

 1

   

 

 

 

   













1 2

2 1 2 1

2 2 1 1 2

1

,

, Prob ,

x -

x

p x x d x d x

x x x x x

x P

   

    









-

p x y dy

y p

dy y x p x p

, ,

14.5 Joint Probability Distribution of Several RV

• Variances of x and y:

 

     

 

      

























dy y p y y

E

dx x p x x

E

y y

y

x x

x

2 2 2









   

 

     

   

 

_x _y ^x ^y

x

y y x x y

y x

y x xy

xy E

dxdy y x p dxdy

y x yp

dxdy y x xp dxdy

y x xyp

dxdy y x p y

x xy

dxdy y x p y x

y x E





























   

 

















































, ,

,

14.5 Joint Probability Distribution of Several RV

• Correlation coefficient between x and y:

1 1  





xy y x

xy xy





 

(6)

-21-

14.6 Correlation Functions of a Random Process

14.6

-22-

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

, , ,

3 2 1 3 2 1 3

2 1

2 1 2 1 2

1

x x x E t x t x t x E t t t K

x x E t x t x E t t K



-23-

14.6 Correlation Functions of a Random Process

• 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:

   

^

 





 ₁ ₂ ₁ ₂ ₁ ₂ 2

1

, t x x p x , x dx dx

t R

  

^{ }

 

^{ }

 



ⁿ

i

t x t

n x t t R

1

2 1 2

1

, 1

-24-

14.6 Correlation Functions of a Random Process

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-25-

14.7 Stationary Random Process

14.7

-26-

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

  ^x   ^t ^E  ^x  ^t ^t   ^t

E

₁



₁

 for any

  t t E   x x E  x    t x t   R   t R

₁

,

₂



₁ ₂

     for any

14.7 Stationary Random Process

• R(0)=E[x²]

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

14.7 Stationary Random Process

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

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-29-

14.7 Stationary Random Process

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

E   x   t  E  x  t       and 

_x_{ }_t

 

_x_{ }_t__

 

 

     

 

   

         

 

² ²

2 2

2 2 2

2

2 2

, 1 Since i.e.

: t coefficien n

Correlatio









































 















 









 





 

R R

R

t x E t

x E t

x t x E

t x t x E

-30-

14.7 Stationary Random Process

• R(τ) is an even function of τ.

• When τ∞, ρ0, R(τ∞)μ²

• A typical autocorrelation function is shown:

    E  x    t x t      E  x    t x t      R     R

-31-

14.7 Stationary Random Process

• 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)

   

^{ }

 

    

^{ }

  

      

^{ }

 

^{ }

 









 



 











2 2 2

2

2 2

lim 1 lim 1 lim 1

T

i i T

T

T i T

T

T i T

dt t x t T x

t x t x R

dt t T x

t x x E

dt t T x

t x x E



-32-

14.8 Gaussian Random Process

14.8

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-33-

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:

 

2

2 1

2

1

^^_^{ } ^_^



^x

x x

x

e x

p

^





x

x z x



 

 

²¹ ²

2

1

^z

e x

p 

^



-34-

14.8 Gaussian Random Process

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

   

    _



 











 



c x c

c

z

dx e c

t x

dx e c

t x c

2 2

2

2 1

2 Prob 2

2 Prob 1

14.8 Gaussian Random Process

• Some typical values are shown below:

14.9 Fourier Analysis

14.9

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-37-

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-imω0tand integrating:

 

^

 2 _ ^

where

₀

0



 

^



n t in n

e c t x

 

^ ^

   

 

 





















- n

n

- n

t m n i n t

im

dt t m n i t m n c

dt e c dt

e t x

2

2 0 0

2 2 2

2

sin cos

0 0



 





-38-

14.9 Fourier Analysis

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 cnis the complex conjugate of that of c-n





 



²  2

1

^ 0





 ^x ^t ^e ^dt

c

_n ⁱⁿ ^t

 

₀ ₀ ₀

1

1 2 



 



_n_



_n

 n   n    

* n

n

c

c 

_

-39-

14.9 Fourier Analysis

Mean square value of x(t):

   

 



 

  

 































 



























 









  





 



  





 



 



n n n

n n

n n n

t in n t in n

n t in n n

t in n

n t in n

c c c

dt c c c

dt e c e c c

dt e c c e c

dt e c dt

t x t x

2 1

2 2 0

2

2 1

* 2 0 2

2

1

* 0

2 2

2

1 0 1

2 2

2 2 _______

2

1 2 1 1

1 1

0 0

0











 



-40-

14.9 Fourier Analysis

Example 14.2

Complex Fourier Series Expansion

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

(11)

-41-

14.9 Fourier Analysis

Example 14.2

Complex Fourier Series Expansion Solution

 



 



 



 

 

 



 

 















 

 



 



 

 







2 0 0

2 2

2 0

0 0

0

1 1 1

1

ts coefficien Fourier

2 and 2 where

0 2 , 1

2 0 , 1

 















 



dt a e A t dt a e A t

dt e t x c

a a a t A t

a t A t t x

t in t

in t in n

-42-

14.9 Fourier Analysis

Example 14.2

 

   

2

0 2 0

0 2

0 0

0

2 2 0

0 0

0 2 2

1 1 1

1

0 0

 













 



 





 



 

  

 

 

 



 

  

 

 





 





 





in in

e a e A

in A

in in e a e A

in c A

k kt dt e te

t in t

in

t in t

in n

kt kt

14.9 Fourier Analysis

Example 14.2

The equation can be reduced to

 

^

 

^



 





in in

n

e n in

a e A n in

a A

n e a e A n a A

in e A n a e A in c A







 



 

2 0 2 2

0 2

2 0 2 2

0 2

0 2 0 2 0

1 1

1 2 1

14.9 Fourier Analysis

Example 14.2

Note that





















 



















 



, 6 , 4 , 2 , 0

, 5 , 3 , 1 2 , 4

0 2,

, 6 , 4 , 2 , 1

, 5 , 3 , 1 , 1

0 , 1 or

2 2 2 0 2

n n n

A n

a A

A n c

n n

n e

e

n in in









(12)

-45-

14.9 Fourier Analysis

Example 14.2

Frequency spectrum is as shown:

-46-

14.9 Fourier Analysis

• Fourier Integral

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

 

^

^where ^

0

² _ ^

0



 

^



 n

t in n

e c t x

-47-

14.9 Fourier Analysis

As τ∞,

   

     

 

  























































 

 



 





 





 





















 







d e X

e c

e c t

x

dt e t x c X

dt e t x dt e t x c

t i n

t i n n

t i n

t i t

i n

2 1

2 1 lim 2

2 lim 2

lim

Define lim

lim

²

2

-48-

14.9 Fourier Analysis

Integral Fourier Transform pair

Mean square value of x(t):

   

    













dt e t x X

d e X t

x

t i



  2

1

 

   



 























 





 



 



n n n n

n n

n n n n

n

c c c

c

c c c dt t x



 

 



 





2 1

* 0

0

* 0 2 2

2 2

(13)

-49-

14.9 Fourier Analysis

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

   

     



 ^











 













 

X d

dt t x t

x

d X

c X

c

_n _n

2 lim 1

as and

,

2 2

2 2 _______

2

0

*

-50-

14.9 Fourier Analysis

Example 14.3

Fourier Transform of a Triangular Pulse

Find the Fourier transform of the triangular pulse shown below.

14.9 Fourier Analysis

Example 14.3

Fourier Transform of a Triangular Pulse Solution

 



 









 

 

  

 



 

  



 

 



 





 

   



 

 



0

1 1

1 otherwise

, 0

, 1

dt a e A t dt a e A t

dt a e A t X

a a t A t t x

t i t

i t i



14.9 Fourier Analysis

Example 14.3

 

   

 

 

 

 



 

 





 



 

  

 

 

 





 

 

 

  

 



 

  





 

 







2 2

2

0 2

0

0 0

2

1 1 1



 



 



a e A a e A a

A

t i i

e a e A i A

dt a e A t dt a e A t X

t i t

i

t a a i

t i

t i t

i

(14)

-53-

14.9 Fourier Analysis

Example 14.3

   

 

  



 



 













sin 2 cos 4

2 1

sin cos

sin 2 cos

2 2 2

a a

a A a

A

a i a a

A

a i a a

A a X A



 



 



 

-54-

14.10 Power Spectral Density

14.10

-55-

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

   

      





















 



 

d S x E R

d e S R

d e R S

i i

0

2

2 1

   

^

 

  



_x²

R 0 S d

-56-

14.10 Power Spectral Density

• S(-ω)=S(ω)

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

   

^

   

^

 

0

2

S d W f df

x

E  

_x

(15)

-57-

14.10 Power Spectral Density

   

         





 



x x

d S S d df S d f W

df f W d S

2 4 2

2 2



-58-

14.11 Wide-Band and Narrow-Band Processes

14.11 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.

14.11 Wide-Band and Narrow-Band Processes

• 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– ω1)

(16)

-61-

14.11 Wide-Band and Narrow-Band Processes

Example 14.4

Autocorrelation 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.

-62-

14.11 Wide-Band and Narrow-Band Processes

Example 14.4

Autocorrelation and Mean Square Value of a Stationary Process Solution

We have

   

 

 

 









 

 















sin 2 cos 2

4

sin 2 sin

1sin 2

cos 2 cos 2

2 1 2 1 0

1 2 0 0

0 0

2

1

2

1



 



 



 



 





^



S S S

d S

R_x _x

-63-

14.11 Wide-Band and Narrow-Band Processes

Example 14.4

Autocorrelation and Mean Square Value of a Stationary Process Solution

Mean Square Value

   

0 0



2 1



2

     2 

^

  2   





S d S d S

x

E

_x

-64-

14.12 Response of a Single DOF System

14.12

(17)

-65-

14.12 Response of a Single DOF System

 

   

mk c c

c m k m

t t F x

t x y y y

c c n

n n

2 , , ,

where

2 ²













-66-

14.12 Response of a Single DOF System

• Impulse Response Approach

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

14.12 Response of a Single DOF System

• Impulse Response Approach

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

Total response can be found by superposing the responses.

Response to total excitation: y

 

t 



^tx

  

ht



d

14.12 Response of a Single DOF System

• Frequency Response Approach Transient function:

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

Total response of the system:

 

t eⁱ^t y

 

t H

 

eⁱ^t

x  ^ 



^

~ , ~

If

 





^

 



 

 ^X^e ^d t

x ⁱ^t

2 1

         

   

 

      



 



 



 



X H Y

d e Y

d e X H

d e X H t x H t y

t i

















2 1