PSI - Issue 58

Victor Rizov et al. / Procedia Structural Integrity 58 (2024) 150–156 V. Rizov / Structural Integrity Procedia 00 (2019) 000–000

153

4

a A    ( ) ( )    A

N

dA

,

(12)

M

z dA 1

,

(13)

a

where N is the axial force, M is the bending moment, A is the cross-section area. The MatLab is used to solve (12) and (13). In frame members, 2 3 B B and 3 4 B B , the curvature and the coordinate of neutral axis are determined in a similar way. We obtain the damping energy by integrating the damping energy density, u  . The latter is determined by formula (14).

1

1



 1 41  



n

n H n 



C

u  

C a

.

(14)

1

  C C C n

We give an account of the influence of the temperature, T , on the damping by changing

C H . For this purpose,

we apply the following relations (Narisawa (1987)):

t      T

  

T



  H t C

H

0 0

,

(15)

CT

T

1 

0





 C T T C T T    q P

   lg

,

(16)

 q

T

R

C H at a reference temperature, q T is a constant. The rest of the constants involved in

0 CT H

where

is the value of

8.86  P C ,

101.6  R C and

/( ) 1 1 0 0    T T

(15) and (16) are

(Narisawa (1987)).

We apply formula (17) for deriving the damping energy, U  , in the frame.

2 3 B B B B U U U U     1 2

,

(17)

3 4

B B

where the subscripts, 1 2 BB , 2 3 B B and 3 4 B B , refer to the corresponding frame member (Fig. 1). Formula (18) is used to determine the damping energy in frame member, 1 2 BB .

U B B V B B     ) ( 1 2 1 2

udV

,

(18)

where V is the volume of this frame member. The damping energies in frame members,

2 3 B B and 3 4 B B , are found-out by formulas (19) and (20), i.e.

U 2 3     V B B B B ) ( 2 3

u dV B B 2 3

,

(19)