Issue 76

A. Huynh-Thai et alii, Fracture and Structural Integrity, 76 (2026) 99-116; DOI: 10.3221/IGF-ESIS.76.07





* 4 2 E J + ρ J ω -N - ρ A ω =0 s s 2 2

(7)

The solutions of Eqn. (7) are given by:

   

 2





2 ρ J ω -N

2

*

2



* ρ J ω -N 4E J ρ A ω 2E J 

2



s

(8)

In this study, the cable ends are modeled as elastic support boundary conditions by applying stiffness values (like springs) denoted by k 1 and k 2 , respectively.         2 2 2 2 2 2 , , , EJ ±C ±k , ± ρ A =0 0 n u x t u x t u x t J u x t withx or L x x t x t                               (9) where: k n denotes the spring stiffness, with k 1 at left end ( x=0) and k 2 at the right one ( x=L), see Fig. 1. In this case, the bending moment at both ends is zero.   2 2 , EJ =0 0 u x t withx or L x    (10) If the boundary condition of both cable ends is simply supported [20], the viscoelastic damping coefficient is determined as follows

E

η



C

(11)

α

 

  

  











f

sin

cos

2

η

2

2

Complex modulus shall be rewritten [13] [14] as:

  α E ω =E+C i ω =E’+E    α *

(12)

The storage modulus ( E’ ), the real part of Eqn. (12), is directly measured from vibration signal, and the dissipative modulus ( E” ) which is the imaginary part, is indirectly determined as shown in Eqns. (13) and (14) [15].       * α α E'=Re E ω =E+C ω cos απ /2 (13)       * α α E =Im E ω =C ω sin απ /2  (14)

And the loss factor from cable material is also a function of frequency is derived:





α

 E' E+C ω cos απ /2  α α C ω sin απ /2

  E η ω = =

(15)



α

The spatial modes remain unchanged because the damping component only affects the frequency domain. For a nontrivial solution of C 2 and C 4 in boundary conditions, the determinant of their coefficient matrix is set equal to zero. Consequently, Eqn. (16) is obtained as follows:

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