The influence of bending and shear stiffness and rotational inertia in vibrations of cables: An analytical approach

Abstract The main objective of this work is to analyze the influence of bending and shear stiffness, and rotational inertia in the natural frequencies of overhead transmission line conductors, and to compare the results with a vibrating string, where only geometrical stiffness is considered. Five formulations, based on Bernoulli’s and Timoshenko’s beam theories, taking into account the effects of geometrical, bending and shear stiffness and rotational inertia are considered here. They are also based on the assumption that the cable is inextensible. The equation of motion of the vibrating cable is developed analytically, assuming small strains. The Newton–Raphson method is employed for the solution of the nonlinear equations. A numerical example is presented and the results are compared with solutions obtained from the technical literature.

[1]  B. Posiadała,et al.  Free vibrations of uniform Timoshenko beams with attachments , 1997 .

[2]  Jong-Shyong Wu,et al.  A lumped-mass TMM for free vibration analysis of a multi-step Timoshenko beam carrying eccentric lumped masses with rotary inertias , 2007 .

[4]  R. E. Rossi,et al.  Free vibrations of Timoshenko beams, carrying elastically mounted, concentrated masses , 1993 .

[5]  Jong-Shyong Wu,et al.  Free vibration analysis of a Timoshenko beam carrying multiple spring–mass systems by using the numerical assembly technique , 2001 .

[6]  M. J. Maurizi,et al.  Natural frequencies of the beam-mass system: Comparison of the two fundamental theories of beam vibrations , 1991 .

[7]  Federico Cheli,et al.  Aeolian vibrations of overhead transmission lines: computation in turbulence conditions , 1993 .

[8]  I. Elishakoff,et al.  Application of the Krein's method for determination of natural frequencies of periodically supported beam based on simplified Bresse-Timoshenko equations , 1987 .

[9]  H M Irvine,et al.  STATICS OF SUSPENDED CABLES , 1975 .

[10]  Haim Abramovich,et al.  Vibration of a uniform cantilever Timoshenko beam with translational and rotational springs and with a tip mass , 1992 .

[11]  I. Elishakoff,et al.  Influence of shear deformation and rotary inertia on vibration frequencies via Love's equations , 1990 .

[12]  K. D. Do,et al.  Boundary control of transverse motion of marine risers with actuator dynamics , 2008 .

[13]  Hsien-Yuan Lin.  On the natural frequencies and mode shapes of a multispan Timoshenko beam carrying a number of various concentrated elements , 2009 .