Issue 70

S.K. Shandiz et alii, Frattura ed Integrità Strutturale, 70 (2024) 24-54; DOI: 10.3221/IGF-ESIS.70.02

f n

2



T       c k k sk k k sk k

2 [ ] 





diag

(

)

(37)

K

v

bb

j

mL



k

1

l

f n

2



  

T





sk k c r k r

(

)

(38)

F

R v wk

bb

sk k k

m L



k

1

l

0 0

   

  



v c r k r c r k r             v v c r k r 3 1 3 1   4 2 4 2 2 3 2 3 0 t t 

 

(39)

F

vv

    1 k



T

2 



( ) k x

( ) k x

( ) x



(40)

n k

 

 

 

( ) ( ) x x

( ) ( ) x x

( ) ( ) x x

     

     

1 1

1 2

1 3

2 1

2 2

2 3



(41)

Φ







n 

n 

n 

1 ( ) x

2 ( ) x

3 ( ) x

0 0 0 0 0 0

0 0 0 0 0 0

        

        

        

        

c

k

0 0

0 0

3

3





,

,

(42)

c

k

sk

sk

c

k

0

0

0

0

4

4

0 0 0

0 0 0

c

k

0 0

0 0

t

t

2

2

In the above equations, m l , E , I ,  j , and  j are the per-unit-length mass, modulus of elasticity, the flexural moment of inertia, the j th undamped natural frequency, and damping ratio, respectively. M v , C v and K v are the vehicle matrices in Eqn. (3). The natural frequency of the beam is given as:

2

j EI L m 

      

(43)

j 

l

The TT vibration is then achieved from both methods by implementing the parameters specified in Tab. 1 for beam and TT. Two approaches were used to determine the vertical displacement of the trailer ( d 5 ) at different speeds (see Fig. 5a). A comparison of modal and finite element results is made to explore the influence of the number of elements on the accuracy of the FEM results, where 30 vibration modes are evaluated in the case of modal analysis (see Fig. 5b). It should be noted that when the front axle reaches the end of the beam, the TT stops moving.

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