Carlos P. Filipich

Vibrations of beams and plates carrying concentrated masses

The fundamental frequency of vibration of beams and plates elastically restrained against rotation at the supports and carrying finite masses is determined in the present paper using the optimized Rayleigh methodology. The analysis takes into account the effect of both translational and rotational inertias. In the case of simply supported vibrating plates a lower bound for the frequency coefficient is obtained by means of an extension of Dunkerleys method.

Static and dynamic behavior of circular plates of variable thickness elastically restrained along the edges

Abstract Simple polynomial approximations and a variational approach are used to solve a rather complex elasto-mechanics problem. It is assumed that the plate is elastically restrained against rotation and translation along the edge. The approach developed in the present paper allows for a unified solution of both free and forced vibration problems, the static situation being a special situation of the dynamic state.

Free vibrations of a spinning uniform beam with ends elastically restrained against rotation

The title problem is considered in the case where the spinning beam cross-section possesses only one axis of symmetry. An exact solution is obtained by means of an extension of Bauers approach, and with account taken of orthotropic properties of the beam supports. The effect of coupling with torsional modes is not taken into account in the present investigation.

In-plane vibrations of portal frames with end supports elastically restrained against rotation and translation

The title problem is solved by using classical beam theory. The general problem has not been treated previously in the technical literature and is quite important from a structural dynamics viewpoint since ideal boundary conditions—frictionless pins or clamps of infinite rigidity—cannot be met in realistic technological situations.

A note on the vibrations of rectangular plates of variable thickness with two opposite simply supported edges and very general boundary conditions on the other two

Abstract An approximate solution for the title problem is obtained by making use of the Galerkin method. The plate displacement function is approximated by means of a sinusoid multiplied by a polynomial. Translational and rotational flexibilities are taken into account at x = ±a/2. It is shown that the free edge situation (Kirchoffs boundary condition) can be treated as a special case by means of the approach developed herein. A simple algorithm which allows evaluation of the fundamental frequency of vibration is derived and rough estimates of amplitudes and stress resultants are also given when the plate is subjected to a p0cosωt-type excitation.

A variant of Rayleigh's method applied to Timoshenko beams embedded in a Winkler-Pasternak medium

This paper deals with the determination of the fundamental frequencies of Timoshenko beams in a Winkler-Pasternak medium by means of the variant of Rayleighs method which allows an optimization of the approximate modal functions through a non-integer exponential parameter. The case of a Timoshenko beam gives rise to a two-variable problem which constitutes an extension of the title method with respect to previous work. A static functional relationship between the two unknown functions is proposed. Numerical results are given for simply supported and clamped-clamped beams of uniform and variable cross section. Comparisons with available exact solutions were made, showing very good agreement.

In-plane vibrations of an elastically cantilevered circular arc with a tip mass

Abstract Upper and lower bounds are determined for the fundamental frequency of in-plane, transverse vibration of the structural system described in the title in the case of constant cross-section and moment of inertia. The upper bound is determined by approximating the fundamental mode shape with a polynomial co-ordinate function in the angular co-ordinate which includes an exponential optimization parameter. The fundamental frequency equation is generated by means of the Rayleigh-Ritz method and the resulting upper bound is minimized with respect to the previously mentioned exponential parameter. The lower bound for the frequency coefficient is obtained by means of an extension of Dunkerleys method. It is felt that the methodologies developed in the present study are especially useful in the case of variable cross-section of the arch structure, presence of internal supports, etc.

Vibration of orthotropic plates: discussion on the completeness of the solutions used in direct methods

Fil: Rosales, Marta Beatriz. Consejo Nacional de Investigaciones Cientificas y Tecnicas. Centro Cientifico Tecnologico Conicet - Bahia Blanca; Argentina. Universidad Nacional del Sur. Departamento de Ingenieria; Argentina

Uniform Convergence Series to Solve Nonlinear Partial Differential Equations: Application to Beam Dynamics

Extended trigonometric series of uniform convergence are proposed as a method to solve the nonlinear dynamic problemsgoverned by partial differential equations. In particular, the method isapplied to the solution of a uniform beam supported at its ends withnonlinear rotational springs and subjected to dynamic loads. The beam isassumed to be both material and geometrically linear and the end springs are of the Duffing type. The action may be a continuous load q = q(x, t) within a certain range and/or concentrated dynamic moments at the boundaries. The adopted solution satisfies the differential equation, the initial conditions, andthe nonlinear boundary conditions. It has been previously demonstrated that, due to the uniform convergence of the series, the method yieldsarbitrary precision results. An illustration example shows theefficiency of the method.

Alternative Approach for the Exact Solution of the Forced Vibrations of Beams

The present work is an extension of a tool vastly used by the authors to solve static boundary problems in one, two, and even three dimensions. It consists in a so-called generalized solution with special trigonometric Fourier functions to solve the equations of motion of beams. An important theorem that guarantees that the classic answer is attained through an alternative way is demonstrated. In other words, it is a variational methodology to solve differential equations in engineering. An example solved numerically completes the present proposal.