PSI - Issue 13

Tianyu Chen et al. / Procedia Structural Integrity 13 (2018) 613–618 T. Chen et al./ Structural Integrity Procedia 00 (2018) 000 – 000

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3

2.2. Beam ’s transverse deflection under general displacement excitation

  0 w t can be expanded into the summation of several functions

If the general applied displacement

n

1    j

 

 

0 w t

w t

.

(3)

j

0

with each having a finite number of linearly-independent derivatives (Grant, 1983), or of periodic functions, then shifting functions can be introduced to derive the transverse deflection of the beam by forcing homogenous conditions. Under the above conditions, the transverse deflection of the beam is of the form

n

 

  ,

  ,

   

w x t

fv w x t

F x w t



.

(4)

j

j

0

j

1



where   fv , w x t is the free vibration of the beam and   j F x are the shape functions. To demonstrate the process and facilitate the derivation, the applied displacement is taken here as the sum of one linear term and one harmonic term, that is,     2 0 sin w t vt H t    . The governing equation for vibration of an Euler-Bernoulli beam (Rao, 2007) is       4 , , 0 EIw x t Aw x t    . (5)

The transverse deflection is thus assumed to be of the form as

  2

  ,

  ,

 

 

w x t

sin w x t F x vt F x H t     .

(6)

fv

1

2

By combining Eqs. (5) and (6) and forcing homogenous conditions, the following differential equations are obtained:       4 fv fv , , 0 EIw x t Aw x t    ,     4 1 0 F x  ,       4 4 2 2 0 EIF x A F x     . (7)

  fv 0, 0 w t  ,

  fv , 0 w L t  ,

    1 fv 0, 0 w t  ,

    2 fv , 0 w L t  .

The boundary conditions for the first of Eqs. (7) are

Note that   fv , w x t is the free vibration of a fixed-pinned beam. The solution for the free vibration of an Euler-Bernoulli beam can be found by using the separation method (Rao, 2007; Blevins, 1979) as

 

  

B

 

  ,

  W x A t   cos

.

fv w x t

t

(8)

sin i





i 

i

i

i

i 

i

1



where   i W x is the i th normal mode given in

1 cos 

i 

i 

i 

i 

  

  

  

 

W x

x

x

x

x

cosh

sin

sinh







i 



.

(9)

 

i

AL L 

L

L

L