Continuum non-linear equations of free motion of a heavy elastic cable about a deformed initial configuration are developed. Referring to an assumed mode technique one ordinary equation for the cable planar motion is obtained via a Galerkin procedure, an approximate solution of which is pursued through a perturbation method. Suitable nondimensional results are presented for the vibrations in the first symmetric mode with different values of the cable properties. Which procedure is the proper one to account consistently for the non-linear kinematical relations of the cable in one ordinary equation of motion is discussed.

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Planar non-linear free vibrations of an elastic cable

Parametric analysis of large amplitude free vibrations of a suspended cable

In this paper, the large amplitude free flexural vibration characteristics of fairly thick and thin functionally graded graphene platelets reinforced porous curved composite beams are investigated using finite element approach. The formulation includes the influence of shear deformation which is represented through trigonometric function and it accounts for in-plane and rotary inertia effects. The geometric non-linearity introducing von Karman’s assumptions is considered. The non-linear governing equations obtained based on Lagrange’s equations of motion are solved employing the direct iteration technique. The variation of non-linear frequency with amplitudes is brought out considering different parameters such as slenderness ratio of the beam, curved beam included angle, distribution pattern of porosity and graphene platelets, graphene platelet geometry and boundary conditions. The present study reveals the redistribution of vibrating mode shape at certain amplitude of vibration depending on geometric and material parameters of the curved composite beam. Also, the degree of hardening behaviour increases with the weight fraction and aspect ratio of graphene platelet. The rate of change of nonlinear behaviour depends on the level of amplitude of vibration, shallowness and slenderness ratio of the curved beam.

Large amplitude free flexural vibrations of functionally graded graphene platelets reinforced porous composite curved beams using finite element based on trigonometric shear deformation theory

Using a cubic B-spline shear flexible curved element, based on the field consistency principle, the nonlinear free flexural vibrations of isotropic/laminated orthotropic straight/curved beams have been studied. The nonlinear governing equations are solved by employing Newmark's numerical integration scheme coupled with modified Newton-Raphson iteration technique. Amplitude-frequency relationships are obtained from the non-linear dynamic response history. Detailed numerical results are presented for various parameters for isotropic and laminated orthotropic beams. The present study brings out the type of non-linearity associated with the curved beams and its dependency on the interaction of curvature with initial amplitude of the beams.

G. Prathap

Papers

On asymptotic solutions to the non-linear vibrations of curved elements