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

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.

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.

Partial differential equations of motion suitable to study moderately large free oscillations of an clastic suspended cable arc obtained. An integral procedure is used to eliminate the spatial dependence and to reduce the problem to one ordinary differential equation which shows quadratic and cubic nonlincarities. The frequency-amplitude relationship for symmetric and antisymmetric vibration modes is studied and a numerical investigation is performed to describe the nonlinear phenomenon in a large range of values of the cable sag-to-span ratio. Softening and hardening behaviour is evidenced dependent on both the cable properties and the amplitude of oscillation.

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.

On développe les équations non linéaires du mouvement llbre d'un câble élastique lourd autour d'une configuration initiale déformée. En se référant à une technique de mode supposé on obtient, avec une méthode de Galerkin, une équation ordinaire pour le mouvement plan du câble dont on recherche une solution approchée par une méthode de perturbation. On donne des résultats non dimensionnels appropriés pour les vibrations du premier mode symétrique avec differentes valeurs des propriétés du câble. On discute quelle est la procédure propre à expliquer logiquement les relations cinématiques non linéaires du câble au moyen d'une équation ordinaire du mouvement.

Nichtlineare Kontinuumsgleichungen der freien Bewegung eines schweren elastischen Kabels um eine verformte Ausgangsform werden entwickelt. Eine gewoehnliche Gleichung fuer die ebene Bewegung des Kabels wird mit Hilfe eines Galerkinschen Verfahrens aufgestellt, und eine Naeheŕungsloesung wird durch eine Perturbationsmethode erreicht. Fuer die Schwingungen mit der ersten, symmetrischen Schwingungsform und fuer verschiedene Werte der Kabeleigenschaften werden geeignete nichtdimensionale Ergebnisse dargestellt. Es wird untersucht, welches Verfahren zuverlaessig die nichtlinearen, kinematischen Beziehungen des Kables in einer gewoehnlichen Bewegungsgleichung gibt.