On asymptotic solutions to the non-linear vibrations of curved elements
About: This article is published in Journal of Sound and Vibration.The article was published on 1978-06-08. It has received 5 citation(s) till now. The article focuses on the topic(s): Method of matched asymptotic expansions & Asymptotic analysis.
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TL;DR: In this paper, a non-linear equation of the free motion of a heavy elastic cable about a deformed initial configuration is developed, which is obtained via a Galerkin procedure, an approximate solution is pursued through a perturbation method.
Abstract: 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.
164 citations
TL;DR: Partial differential equations of motion suitable to study moderately large free oscillations of an clastic suspended cable arc are obtained in this paper, where 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.
Abstract: 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.
77 citations
TL;DR: In this article, the nonlinear free flexural vibrations of isotropic/laminated orthotropic straight/curved beams have been studied using a cubic B-spline shear flexible curved element, based on the field consistency principle.
Abstract: 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.
22 citations
TL;DR: In this article, 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, which includes the influence of shear deformation which is represented through trigonometric function and it accounts for in-plane and rotary inertia effects.
Abstract: 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.
16 citations
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References
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01 Feb 1970
TL;DR: In this article, a general theory of elastic stability is presented, augmented by an investigation of the buckled structure in the immediate neighborhood of the bifurcation point, which explains why some structures, such as a flat plate supported along its edges and subjected to thrust in its plane, are capable of carrying loads considerably above the buckling load, while other structures, e.g., an axially loaded cylindrical shell, collapse at loads far below the theoretical critical load.
Abstract: : A general theory of elastic stability is presented. In contrast to previous works in the field, the present analysis is augmented by an investigation of the behavior of the buckled structure in the immediate neighborhood of the bifurcation point. This investigation explains why some structures, e.g., a flat plate supported along its edges and subjected to thrust in its plane, are capable of carrying loads considerably above the buckling load, while other structures, e.g., an axially loaded cylindrical shell, collapse at loads far below the theoretical critical load.
883 citations
TL;DR: In this paper, it was shown that the vibration of an extensible bar, carrying no transverse load and having the ends fixed at the supports, causes an axial tensile force with a period equal to the half-period of the vibration.
Abstract: It can be shown that the vibration of an extensible bar, carrying no transverse load and having the ends fixed at the supports, causes an axial tensile force with a period equal to the half-period of the vibration of the bar. This force modifies the process of the vibration to a nonlinear one and produces an increase of the frequency of vibration according to the increase of the amplitude.
508 citations
97 citations
TL;DR: For the case of similar flows in the plane of symmetry of an inclined axisymmetric body with zero streamwise pressure gradient and insulated walls, the following conditions prevail: e\ = 1, e» = r(x), KI = 0, ft = 0 and dgr/d^ = 0.
Abstract: It is noted that Eqs. (22, 24, and 25) are essentially those obtained by Beckwith 1 except for the coefficient constants. Finally, for the case of similar flows in the plane of symmetry of an inclined axisymmetric body with zero streamwise pressure gradient and insulated walls, the following conditions prevail: e\ = 1, e» = r(x), KI = 0, ft = 0, and dgr/d^ = 0. In order to transform the resulting equations into a familiar form, we first differentiate Eq. (14) with respect to f *, which is defined as rf * = f. Then, with the aid of the following definitions:
58 citations
TL;DR: In this article, an approach to nonlinear free vibrations of elastic structures is developed with the aid of Hamilton's principle and a perturbation procedure, analogous to the theory of initial postbuckling behavior due to Koiter.
Abstract: An approach to nonlinear free vibrations of elastic structures is developed with the aid of Hamilton's principle and a perturbation procedure. The theory is analogous to the theory of initial postbuckling behavior due to Koiter. It provides information regarding the first order effects of finite displacements upon the frequency, period and dynamic stresses arising in the free, undamped vibration of structures. Attention is restricted to structures which are linearly elastic. The theory is illustrated by application to the free vibration of beams and rectangular plates.
55 citations