Random Processes - Wind Engineering - Lecture Slides, Slides of Environmental Law and Policy

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• Topics :
  • Concepts of deterministic and random processes stationarity, ergodicity
• Basic properties of a single random process mean, standard deviation, auto-correlation, spectral density
• Joint properties of two or more random processes correlation, covariance, cross spectral density, simple input-output relations

Refs. : J.S. Bendat and A.G. Piersol “Random data: analysis and measurement procedures” J. Wiley, 3rd^ ed, 2000.

D.E. Newland “Introduction to Random Vibrations, Spectral and Wavelet Analysis” Addison-Wesley 3rd^ ed. 1996

• random processes :
  • The probability density function describes the general distribution of the magnitude of the random process, but it gives no information on the time or frequency content of the process

fX(x)

time, t

x(t)

• Averaging and stationarity :
  • Sample records which are individual representations of the

underlying process

• Ensemble averaging : properties of the process are obtained by averaging over a collection or „ensemble‟ of sample records using values at corresponding times
• Time averaging :

properties are obtained by averaging over a single record in time

• Underlying process

• Mean value :
• The mean value,x , is the height of the rectangular area having the same area as that under the function x(t)

time, t

x(t)

x

T

 (^)  

T T 0

x(t)dt

T

x Lim

• Can also be defined as the first moment of the p.d.f. (ref. Lecture 3)

• Mean square value, variance, standard deviation :

variance,

 (^)  

T 0

2 T

2 x (t)dt

T

x Lim

standard deviation, x, is the square root of the variance

mean square value,

  2 2

σ x  x(t)x

(average of the square of the deviation of x(t) from the mean value,x)

time, t

x(t)

x

T

x

 ^  

T 0

2 T

x(t)-x dt

T

Lim

• Autocorrelation :
• The autocorrelation for a random process eventually decays to zero at large 

R(  )

Time lag, 

1

0

• The autocorrelation for a sinusoidal process (deterministic) is a cosine function which does not decay to zero

• Autocorrelation :
• The area under the normalized autocorrelation function for the fluctuating wind velocity measured at a point is a measure of the average time scale of the eddies being carried passed the measurement point, say T 1

R(  )

Time lag, 

1

0

• If we assume that the eddies are being swept passed at the mean velocity, U.T 1 is a measure of the average length scale of the eddies
• This is known as the „integral length scale‟, denoted by lu





(^1 ) T R()d 

• Spectral density :

Basic relationship (2) :

Where XT(n) is the Fourier Transform of the process x(t) taken over the time interval -T/2<t<+T/

The above relationship is the basis for the usual method of obtaining the spectral density of experimental data



 

  

2 x (^) T XT(n) T

2 S (n) Lim

Usually a Fast Fourier Transform (FFT) algorithm is used

• Spectral density :

Basic relationship (3) :

The spectral density is twice the Fourier Transform of the autocorrelation function

Inverse relationship :

Thus the spectral density and auto-correlation are closely linked - they basically provide the same information about the process x(t)

 

2 n

Sx (n) 2 x( )e dτ

  i ^ 

 

0 0 x

2 n ρx () Real Sx( )e dn S ( ) os(2n )dn

n i ^  n c

• Covariance :

 ^ ^  

T xy (^) T 0

x(t)-x.y(t)-y dt

T

c (0) x(t).y(t) Lim

• The covariance is the cross correlation function with the time delay, , set to zero

(Section 3.3.5 in “Wind loading of structures”)

Note that here x'(t) and y'(t) are used to denote the fluctuating parts of x(t) and y(t) (mean parts subtracted)

• Correlation coefficient :
  • The correlation coefficient, , is the covariance normalized by the standard deviations of x and y

When x and y are identical to each other, the value of  is + (full correlation)

When y(t)=x(t), the value of  is  1

In general,  1<  < +

σx .σy

x'(t).y' (t) ρ 

• Cross spectral density :

By analogy with the spectral density :

The cross spectral density is twice the Fourier Transform of the cross- correlation function for the processes x(t) and y(t)

The cross-spectral density (cross-spectrum) is a complex number :

Cxy(n) is the co(-incident) spectral density - (in phase) Qxy(n) is the quad (-rature) spectral density - (out of phase)



 

2 n

Sxy (n) 2 ( )e dτ

 i ^ 

cxy

Sxy ( n )  Cxy ( n ) iQxy ( n )

• Normalized co- spectral density :

It is effectively a correlation coefficient for fluctuations at frequency, n

Application : Excitation of resonant vibration of structures by

fluctuating wind forces

If x(t) and y(t) are local fluctuating forces acting at different parts of the structure, xy(n 1 ) describes how well the forces are correlated („synchronized‟) at the structural natural frequency, n 1

xy(n)

S n S n

C n

x y

xy