R.E. Rossi

A note on transverse vibrations of a Timoshenko beam of non-uniform thickness clamped at one end and carrying a concentrated mass at the other

The lower natural frequencies of vibration of the structural system described the title have been determined and are presented for a significant range of values of the governing mechanical and geometric parameters for two types of configurations of structural interest: (a) discontinuous variation of the thickness and (b) continuous, linear variation. When the ratio of the radius of gyration of the cross-section to the beam length is very small, say 10−4, the frequency values coincide with the results corresponding to the Bernouilli-Euler classical theory of vibrating beams and which are available in the open literature. Some considerations are presented regarding the dynamic structural efficiency of the systems under study when the variation of the fundamental frequency of vibration is analyzed as a function of the ratio of the concentrated end mass to the total beam mass.

Vibration frequencies for a uniform beam with one end spring-hinged and subjected to a translational restraint at the other end

Abstract The purpose of the present study is to deal with the free vibration of a beam hinged at one end by a rotational spring and subjected to the restraining action of a translational spring at the other end. The eigenfrequencies for the fundamental mode are presented for different values of the parameters K R L / EI and K T L 3 / EI , where KR and K T are the stiffness constants of the springs.

In-plane vibrations of non-circular arcs of non-uniform cross-section

Abstract The title problem is solved by using polynomial co-ordinate functions and the Ritz method to generate the frequency equation. Symmetric and unsymmetric structural systems are considered. The agreement with results available in the literature or with eigenvalues determined by means of a finite element code is good from an engineering viewpoint.

Free vibrations of beams of bilinearly varying thickness

Free vibrations of Bernoulli beams of bilinearly varying thickness are studied using: (a) the optimized Rayleigh-Ritz method, (b) the differential quadrature technique and (c) the finite element approach. It is shown that the fundamental frequencies of vibration of the structural elements as determined by the methodologies (a) and (c) are in excellent agreement while rather considerable discrepancies are found when using the differential quadrature technique. The values of the second and third natural frequency coefficients determined via the finite element method are also presented.

The method of differential quadrature and its application to the approximate solution of ocean engineering problems

Abstract The method of differential quadrature developed by Bellman in the 1970s has only recently been applied in the solution of technically important problems. It consists, essentially, of the approximate solution of the differential system by means of a polynomial-collocation approach at a finite number of points selected by the analyst. The present paper reports some numerical experiments on vibrations, buckling and stress problems. An inherent advantage of the approach is that it has basic simplicity, is easily programmable, etc. Ocean engineers will find it very convenient when dealing with a large variety of solid mechanics, fluid flow and heat transfer problems.

Vibrations of a rectangular orthotropic plate with a free edge: a comparison of analytical and numerical results

Abstract Transverse vibrations of a rectangular orthotropic plate with a free edge while the others are supported according to four different combinations of boundary conditions are considered in the present paper. The presence of a concentrated mass is also taken into account. In view of the difficulty of satisfying exactly the natural boundary conditions at the free edge, the fundamental frequency of vibration is determined by means of the optimized Rayleigh–Ritz method and a pseudo-Fourier expansion which contains, in its argument, the optimization parameters γi. The results are in excellent agreement with those obtained by means of a finite element algorithmic procedure.

Forced vibrations of rectangular plates subjected to harmonic loading distributed over a rectangular subdomain

Quite frequently naval and ocean structural designers confront the problem of a plate or slab which supports a motor or engine which excites, dynamically, the structural element. Very often the excitation is treated as a concentrated force but, obviously, it acts over a finite area. The present study deals with the title problem for several combinations of boundary conditions by means of a variational approach and also using the finite element method. Dynamic displacement and bending moment amplitudes are determined.

Analysis of a cable-like system suddenly stopped at one end

The problem investigated is that of a spring and Hookean bar system, with a mass attached at the other end, moving axially with a constant velocity, Vo. A dynamic stress field results if the free end of the spring is suddenly stopped. Determination of these dynamic stresses has been accomplished by following a one dimensional wave approach. The model adopted constitutes a first order approximation to a type of cable system commonly used in ocean engineering applications. A severe sudden loading condition results when the situation previously described arises. The problem is also interesting from an applied mathematics viewpoint since it requires expansion of the displacement function in a series of non-orthogonal functions.

Vibrations of rings of variable cross section

Abstract The present paper deals with the determination of the lower natural frequencies of vibration of rings of variable cross-sectional area using three approximate schemes: • —using polynomial coordinate functions in the angular coordinate in order to approximate the fundamental mode shape • —expanding the ring response in terms of a sinusoidal truncated series • —by means of a finite element algorithm. When using the first two procedures the Ritz method is applied in order to obtain the frequency equation. In general very good agreement is obtained between the eigenvalues predicted by the three approaches.

Dynamic analysis of a simplified bone model during the process of fracture healing

The dynamic analysis of fracture healing is tackled numerically by means of a bone model which uses the finite element method. The model is of non-uniform cross-sectional area and moment of inertia. Shear and rotatory inertia are taken into account. Considerable variation of the upper natural frequencies is observed as the healing process progresses. The practical implications, as well as present limitations, of the technique are examined.