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Structural Dynamics Concepts And Applications Solutions Manual 1st Edition Busby

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Chapter 2

2.1 What are the dimensions of the correlation function R x and the spectral density S x if the random function X (t ) is (A) Displacement (in.) (B) Acceleration ( in./sec 2 ) (C) Force (lb.) (D) Stress (psi)

Solution

R x   

1 T x(t ) x (t   )d T T

S x ( ) 

2 1 * 1 xT  i   2T 2T



xT (t )eit dt

Part

Function

R x  

S x ( )

A

Displacement (in.)

in.2

in.2  sec

B

Acceleration ( in./sec 2 )

in.2 / sec 4

in.2 / sec 3

C

Force (lb.)

lb.2

lb.2  sec

D

Stress (psi)

lb.2 / in.4

lb.2  sec/ in.4

2

2.2. What are the dimensions of the probability function Fx and the probability density function f x if the random function X (t ) is (A) Displacement (in.) (B) Acceleration ( in./sec 2 ) (C) Force (lb.) (D) Stress (psi) Solution Fx  



Part

f ( x ) dx

Function

fx 

2 2 1 e  ( x  x%) /2 c 2 c

Fx

fx


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A

Displacement (in.)

Dimensionless

1 / in

B

Acceleration ( in./sec 2 )

Dimensionless

sec 2 / in.

C

Force (lb.)

Dimensionless

1/lbs.

D

Stress (psi)

Dimensionless

in.2 /lbs

2.3. Define a stationary random process. Solution A stationary random process is defined by the correlations function of a random variable being constant regardless of the time at which it is evaluated, so long as the period over which each evaluation is taken is unchanged in duration.

2.4. Consider a stationary ergodic process X (t ) ; express E  X  , E  X 2  , and R x   as the time averages. Solution For a stationary ergodic process X (t ) we have in terms of time averages

1 T x(t )dt T  2T  T 1 T 2 E  X 2 (t )   lim x (t )dt T  2T T E  X (t )  lim

1 T  2T

R x    E  X (t ) X (t   )   lim

T

T

x (t ) x (t   )dt

2.5. Prove the relations between the spectral density S   and the correlation function R   :

R   

1 2

S   





S  ei d 

R  e  i d


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Solution

1 2T

R x    E  X (t ) X (t   )   lim

T 



T

T

x (t ) x (t   )dt

  1   i 1  e d  x (t )x (t   )dt  d  x (t )x (t   )e i ( t  ) eit dt       2T 2T  1  it  i ( t  )  x (t )e dt  x (t   )e d  2T 

R  e i d 

Since X * (i )   x ( t ) e it dt 



R  e  i d 

and

X * ( i ) 



x ( t   )e  i ( t  ) d

1 X *(i ) X *( i ) 2T

Take the limit of both sides as T   

lim  R  e i d  lim

T  

T 

1 2 X *(i ) X *( i ) 2T

lim R  e  i d  S ( )

 T 

S ( ) 



F R( ) and

Thus, S ( ) 

R( ) 

1 2

R  e  i d

R( ) 



F

S ( )

1

S   ei d 

t

2.6. Assuming that the input-output relation is y ( t )    ( t   ) x (  ) d  . Show that 

R y   





 

     R x       d  d and S y     * (i ) S x   2

Solution Given y (t ) we know that the correlation function is

R y    lim

T 

where

1 2T

T

T

y (t ) y (t   )dt  lim

T 

1 2T

T

T





dt  x(t   ) ( )d   x(t     ) ( )d


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t

t







y  t     (t   )x ( ) d    x (t   ) ( )d    x (t   ) ( ) d  y t     

t 



x (t     ) ( ) d   x (t     ) ( ) d 

R y    lim  d   d ( ) ( ) T  

1 T  2T lim

T

T



1  x(t   ) x(t     )dt 2T 

x(t   )x(t     )dt  R x      

t    t1

and t      t2 so t2  t1 =     

therefore R y   





 

     R x       d  d



  

S ( )   R  e  i d  

 

     R x       e  i d d  d







  d  d   d e  i      R x       







  d  d   d ei e  i e  i (     )     R x       e  i 







     e  i d      ei d   R x       e i (     ) d

  * (i ) * (i ) S x ( ) therefore S y     * (i ) S x   2

2.7. Derive the expression for the average number of crossings of the value x   assuming that the joint distribution function f ( ,  ) of X (t ) and X&(t ) is (A) two-dimensional normal (B) arbitrary

Solution (a)

The joint distribution function f ( ,  ) of X (t ) and X&(t ) is given by


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 2

2 

 2   1 2C 2C 2 f ( ,  )  e  1 2  2 C1C2

where C12  E  X 2  and C22  E  X&2  The probability that   X    d and   X&   d  is f ( ,  )d d  . The duration of the passage from  to   d is d /  . Thus the passages form  to   d at a velocity  per unit time is

 f ( ,  )d  d  f ( ,  ) d d      f ( ,  )d   d d  The passages from  to   d at any rate is equal to the sum of the passages at each value of  , or

N  



1  f ( ,  ) d   2

 2

2 

   2C 2  2C 2   C1C2 e  1 2  d  

2.8. The spectral density of the loading (input) is given in the form shown in Figure 2.29

Figure 2.29Loading input

Find the spectral density’s of the outputs (displacement, bending moment, etc.) of the systems described in problem 1(a) of Chapter 1.

Solution From Problem 3(a) of Chapter 1 we know that

1 1   U*   or  *(i )  2  115, 200  1.3  115, 200  1.3 2  


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S y   * (i ) Sx 2

At   100

 *(i100) 

1  9.785 106  9.8 106 115, 200  1.3(100)2

 * (i100)   9.8  106   9.6 1011 so S y   9.6  1011  S x 2

2

Also, the mean square value of u ( x, t ) is E (u 2 ) 

1 2



S x ( ) d  

1 2

100

101

(1)d  

1 2

1001

100

(1)d  

1 

2.9. A random function X (t ) has its spectral density as in Problem 2.8. Find the mean square value of X (t ) . Find the average number of crossings of the values x  0 , x  1 , and x  10 .

Solution From Problem 8 we have E (u 2 ) 

1 

For x  0 1/ 2

  2  1  0  S x ( )d   N0     S ( )d    0 x  For x  1

1 10, 000 100  rad/sec  1 


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N  N 0 e

2 2 C12

where C12  E (u 2 ) 

1 

N1 

100  (1)2 / 2 100  / 2 e  e rad/sec  

N1 

100  (10)2 / 2 100 50 e  e rad/sec  

For x  10

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Structural Dynamics Concepts And Applications Solutions Manual 1st Edition Busby  

Structural Dynamics Concepts And Applications Solutions Manual 1st Edition Busby. SOLUTIONS MANUAL for Structural Dynamics Concepts and Appl...

Structural Dynamics Concepts And Applications Solutions Manual 1st Edition Busby  

Structural Dynamics Concepts And Applications Solutions Manual 1st Edition Busby. SOLUTIONS MANUAL for Structural Dynamics Concepts and Appl...

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