Issue 65

V. Le-Ngoc et alii, Frattura ed Integrità Strutturale, 65 (2023) 300-319; DOI: 10.3221/IGF-ESIS.65.20

M ETHODOLOGY

Power spectral density of vibration ccording to Euler–Bernoulli beam theory, the governing differential equation of beams is as follows:                         2 2 2 2 2 2 , , , ( ) ( , ) u x t u x t u x t EJ x c f s t t x x t (1) where u(x,t) is the displacement response of the beam in location x at the time t , EJ is bending rigidity,  is linear density, c is damping coefficient, and f(s,t) is the external force in location s at time t . With singly supported beam, boundary and initial conditions are shown as follows: A

 (0, ) 0 ; u t

 ( , ) 0,

u l t

2

2





u x t

u x t

( , )

( , )





(2)

0;

0

2

2





x

x



 x l

x

0



u x t

( , )

 ( ,0) 0 ; u x



0;



t



t

0

The solution of beam response with length L can be found in general forms as follows:



   1 r

 ( ) ( ) r r x w t (3) in which  r (x) and w r (t) are r t h mode shape and r th generalized displacement of the beam, respectively. From Eq. (1) and Eq. (3), the differential equation of the beam in each generalized coordinate is shown as: ( , ) u x t

2

    ( ) r r r w t 



 w t r

w t

r f t

( ) 2

( )

( )

(4)

r r

with  r ,  r and f r (t) are natural frequency, damping ratio and generalized force of r th mode shape, respectively. This parameter is defined as

    2



 r

2 ( ) x

  

  2 r r

( ) x

EJ x

( )

(5)

2





x

x



L



 

   r rk

( ) ( )d x x x

(6)

r

k

0

L



  ( ) ( )d 2 c x x x

    r nr r rk

(7)

r

k

0

1

L

  

 r

r f t

( ) ( , ) x f x t dx

( )

(8)

0

r

with H r (  ) is the frequency response function (FRF)of r th mode shape, the vibration response of the beam can be found in the form as follows:

 

  ( )

    d

( )F ( )e i t

w t

H

(9)

r

r

r

302