Issue 58

M. Utzeri et alii, Frattura ed Integrità Strutturale, 58 (2021) 254-271; DOI: 10.3221/IGF-ESIS.58.19

Note that the higher order terms are positive, meaning that the curvature increases (with respect to the linear case) for increasing amplitudes. This is the motivation for the hardening behaviour due to the geometric stiffness nonlinearity. The expression of the kinetic energy of the system then becomes:

   



2

      

  

  

  

3 1

3

5

y

3 ' y

'







    d

2

 E mL y  '

'



 

 y y

d

  

k

2

2 8

 

 

0

0

   

   

2

   

   

  

3

5

y

3 ' y

'





    2 y

'

'

 



 y y

d

(11)

2 8

0



    



  j y 

2

  2 y

4



y

' 1 '

'

Likewise, the potential energy becomes:

2

 

  

2

4

EI

y

3 ' y

'

1



2 ' ' 1 y





  

E

d

(12)

p

L

2

2 8

0

The vibration motion is assumed to have linear modes. The vibration of the beam with a concentrate intermediate mass is governed by the equation

          ¨ '''' 1 0 y m y EI 

(13)

Assuming a solution of Eqn.(13) in the form

  

    i t e

 , Φ

y t

(14)

Substituting the Eqn.(14) into Eqn.(13) gives

 2

m





  

         Φ







(15)

Φ

''''

1

0

EI

and the boundary condition at  0 x and  1 x are             Φ 0 0 Φ 0 ' 0 Φ 1 '' 0 Φ 1 ''' 0 EI EI

(16)

Taking the Laplace transformation of Eqn. (15), using the boundary conditions (16) and then taking the inverse Laplace transformation yields the eigenfunction of the beam with an intermediate concentrated mass

  Φ 0 ''

  Φ 0 '''









  

















2

3

















k

kL

kL

k

kL

kL

Φ

cosh

cos

sinh

sin

2

2

(17)

 

   

Φ

























2 kL H





 

 

  

kL

kL

sinh

sin

2

  2

4 k m

0 Φ '' and

0 Φ ''' can be determined imposing the boundary condition in

where

. Hence the unknown constants

EI

0 Φ ''' into Eqn. (17), and then letting    to eliminate    Φ , gives the characteristic

0 Φ '' and

Eqn.(17). Substituting

equation

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