Víctor H. Cortínez

Numerical experiments on the determination of natural frequencies of transverse vibrations of rectangular plates of non-uniform thickness

Abstract A comparison of natural frequencies of transverse vibration of thin, rectangular plates of non-uniform thickness with different combinations of boundary conditions is presented in this paper, as obtained by using the following methodologies: (1) the Rayleigh-Ritz method with characteristic orthogonal polynomial shape functions; (2) the Rayleigh-Ritz method with a shape function which includes two exponents that are to be determined by minimizing the fundamental frequency coefficient; (3) the optimized Kantorovich method which has been proposed rather recently; (4) the finite element method. The results are presented in tabular form and discussed.

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.

Analysis of vibrating rectangular plates of discontinuously varying thickness by means of the Kantorovich extended method

Abstract The title problem is solved by using a polynomial co-ordinate function in the x -direction, where the discontinuity in the thickness variation occurs, and which contains an exponential parameter γ. Use of a variational approach allows for the determination of a differential equation in the y -direction which contains the fundamental eigenvalue of the problem. Since the eigenvalue is a function of the optimization parameter γ, by minimizing it with respect to γ one finds an optimized value of the fundamental frequency coefficient.

Vibrations of non-homogeneous rectangular membranes

Abstract An exact solution for the title problem is obtained for the case where the two subdomains of the non-homogeneous membrane are of rectangular shape. It is shown that the fundamental frequency coefficient obtained by means (a) of the classical Kantorovish method and (b) the optimized Kantorovich method are in very good agreement with the result predicted by the exact method.

Transverse vibrations of circular plates and membranes with intermediate supports

The present paper deals with an approximate solution of the title problem in the case of (a) a concentric circular support and (b) a secant support. Fundamental frequency coefficients are determined by means of the Rayleigh minimization procedure; the modal shapes been approximated by simple polynomial co-ordinate functions. In the case of a circular plate with a secant support an independent solution is obtained by means of a finite element code and very good agreement is shown to exist.

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.

Further optimization of the Kantorovich method when applied to vibrations problems

Abstract It is shown in the present study that, in general, the accuracy of the Kantorovich method can be improved considerably by including an exponential optimization parameter, γ, and a multiplier factor, ξ, in the part of the expression giving the solution which is chosen a priori when determining eigenvalues in a vibrations problem.

WHIRLING OF FLEXIBLE SHAFTS WITH INTERMEDIATE SUPPORTS

Abstract The present paper deals with an approximate solution of the title problem using the Rayleight-Schmidt procedure. It is shown that rotating shafts of non-uniform cross section can be analyzed in a straightforward manner. Experimental and analytical results are obtained for the case of uniform shafts with intermediate supports, and good engineering agreement is shown to exist.

Optimization of the Kohn–Kato enclosure theorem: Application to vibrations problems

Finding close upper and lower bounds to eigenvalues in vibrations and elastic stability problems constitutes a highly desired situation for the applied mathematician or research engineer since their knowledge allows for a clear evaluation of the error involved when determining natural frequencies or buckling loads. Among the methods which allow for the determination of upper and lower bounds one must certainly cite the Kohn–Kato bounding technique. The present paper deals with a simple yet quite substantial optimization of the original approach by Kohn and Kato. It is shown that by inclusion of an exponential undetermined parameter in the coordinate function it is possible to minimize the upper bound and maximize the lower bound, improving considerably the ‘‘closeness’’ of the bounds, especially in the case of higher eigenvalues.

In-plane vibrations of a circumferential arch elastically restrained against rotation at one end and with an intermediate support

Abstract The determination of the fundamental frequency of in-plane vibration of the structural system described in the title is tackled using three different approaches: (a) an optimized Rayleigh-Ritz method; (b) a modified Dunkerleys method; (c) the finite element formulation. The effect of a concentrated mass placed at the free end of the arch structure is also taken into account. It is concluded that, in view of the simplicity and accurate results provided by the first approach, one could use it in more complex situations where an exact solution seems out of the question (non-uniform cross section, presence of elastically mounted masses, etc.).