Issue 70

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

 

 

 



b





m

m

2

R w

g

0       1 1 2 b b

(26)

1

 

 

 



b





m

m

1

R w

g

0       2 1 2 b b

(27)

2

0 t t m m   (

)

R

g

(28)

w

3

1

d d 0 1 2 m b b b    1 2 

I

F d 1 3 w m  1

 

0 1

(29)

d 0 1 



b b  

I

m b

d

F d 2 4 w m  2

 

0 2 2

(30)

1 2

F d d 3 1 6 0 5 w t t m m    

(31)

Based on modal superposition and separation of variables, beam displacement can be represented as:

n

1    j

x q x 

( , ) x t

( ) ( )

(32)

w

j

j

where { ( ), 1,2, , } j q t j n   are the generalized coordinates, and n is the number of vibration modes. Vibration modes are considered to be of the form of solution a simply supported beam:

j x L 

 

  

j 

sin  

(33)

Combination of Eqns. (3) and (23) yield an interaction relation that contains modal components of the beam and physical components of the TT: 2 2 0 0 T T sk sk bb bb bb bb l l v vv sk v sk sk v mL mL                                                         c k C Φ Φ M M c Φ C k Φ c Φ     K q q q F d d d F v K (34) where, [ M bb ], [ C bb ], and [ K bb ] are the mass, damping, and stiffness matrices of the beam, respectively. In the aforementioned equations, F bb represents the forces that act on the beam due to TT, and F vv shows the forces that act on TT due to the beam vibrations. The sub-matrices of the above system are defined as follows [68,69]:

(35)

bb  M I

[ ]

f n

2



T   c k sk k



[2 ]  



diag

(

)

0

(36)

C

bb

j

j

m L



k

1

l

33