A general conclusion regarding the large amplitude flexural vibration of beams and plates.
01 Jan 1973-
TL;DR: In this paper, the modal equation is derived for the large amplitude flexural vibration of flat plates, and it is shown that the nonlinearity associated with this type is of the hardening type, that is, the frequency increases with amplitude.
Abstract: Use is made of the two coupled partial differential equations that govern the large amplitude flexural vibration of flat plates. On the basis of an assumed vibration mode taken to be separable in the space coordinates and time, the modal equation is obtained. In the case of a one-term solution, it is shown that the modal equation is the same as the dynamic equation of a mass-spring combination where the restoring force of the nonlinear spring is a cubic and an odd function of the displacement. Since the modal equation is a nonlinear, ordinary differential equation with positive coefficients, it is shown that the nonlinearity associated with the large amplitude flexural vibration of beams and plates is of the hardening type, that is, the frequency increases with amplitude.
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TL;DR: In this article, a high-order triangular membrane finite element is combined with a fully conforming triangular plate bending element to solve the geometrically nonlinear problems of plates where the membrane and flexural behaviors are coupled and the effect of the inplane boundary conditions is as significant as the flexural boundary conditions.
Abstract: A high-order triangular membrane finite element is combined with a fully conforming triangular plate bending element to solve the geometrically nonlinear problems of plates where the membrane and flexural behaviors are coupled and the effect of the inplane boundary conditions is as significant as the flexural boundary conditions. Each of the three orthogonal displacement components is represented by a two-dimensional polynomial of the same quintic order with no bias against one another giving the element a total of 54 degrees of freedom. The nonlinear stiffness matrices are formulated and a Newton-Raphson iteration procedure is used. Examples include the analyses of plane stresses of a parabolically loaded square plate, large deflections of a square plate under lateral pressure, postbuckling of a square plate, linear free vibration of a buckled rectangular plate, large amplitude free vibration of a square plate with and without inplane stresses. Various flexural and inplane boundary conditions are considered. Results are compared with those obtained by alternative finite element methods, analytical approximate methods, and an experiment. Physical interpretations of the results and explanations of the discrepancies among various solutions are provided. The results indicate that the present development is capable of accurately solving a wide variety of geometrically nonlinear plate problems.
59 citations
TL;DR: In this paper, a finite element formulation for analyzing large amplitude free flexural vibrations of elastic plates of arbitrary shape is developed for analyzing stress distributions in the plates, deflection shape and nonlinear frequencies.
26 citations
TL;DR: In this article, the authors derived general conclusions regarding the non-linear vibration of structural components like curved beams, rings and thin shells from the study of two specific examples, the circular ring and shallow spherical shell, and showed that with careful judgment in the use of mode shapes of one or more terms, the resulting modal equations help one to appreciate much better the physics of the problem.
18 citations
TL;DR: In this article, the modal equations applicable for the large amplitude flexural vibrations of plates and shells are obtained by the Lagrange's method, which can easily be specialised to obtain the corresponding equations for beams and rings.
Abstract: In a simple and straight forward manner the modal equations applicable for the large amplitude flexural vibrations of plates and shells are obtained by the Lagrange's method. These equations can easily be specialised to obtain the corresponding equations applicable for beams and rings. The basic nature of the modal equations for beams and plates on the one hand and rings and shells on the other hand are shown to exhibit hard and soft spring characteristics, respectively.
9 citations
TL;DR: In this article, a brief history of the analysis of large amplitude vibration is given, and nonlinearities exhibited by beams, plates, rings, and shells undergoing large amplitude vibrations are discussed.
Abstract: First, a brief history of the development of the analyses of large amplitude vibration is given. Second, nonlinearities exhibited by beams, plates, rings, and shells undergoing large amplitude vibrations are discussed. It is shown that, in case of large amplitude vibra tions, the modal equations for multilayered beams and plates are qualitatively the same as the modal equation for shells.
4 citations