Transverse vibrations of rectangular orthotropic plates with one or two free edges while the remaining are elastically restrained against rotation

A survey of the literature reveals that the title problem has not been previously solved. The fundamentals frequency of transverse vibration of the orthotropic structural element is determined by expanding the displacement amplitude function in terms of simple polynomial coordinate functions which satisfy identically the essential boundary conditions and approximately some of the natural edge requirements.The Ritz method is then used to generate the frequency equations.     

Transverse vibrations of quadrilateral and triangular plates with one edge elastically restrained against rotation while the remaining are free

The present paper deals with an analytical and experimental investigation of the title problem when a concentrated mass is rigidly attached to the plate. Very simple yet quite accurate engineering frequency equations are derived.     

Free vibrations of rectangular plates elastically restrained against rotation with circular or square free openings

The fundamental frequency of vibration of plates elastically restrained against rotation along the edges and with centrally located circular or square holes is determined by means of the optimized Rayleigh's method. Independent solutions are also obtained using the finite element method.A comparison with experimental values is also presented in the case where the plate is rigidly clamped along the four edges.     

Analytical and experimental investigation on vibrating rectangular plates with two opposite free edges and internal supports while the remaining are elastically restrained against rotation

This paper deals with the determination of an approximate fundamental frequency equation for a rectangular plate with two edges (y = 0, b) elastically restrained against rotation while the edges x=0, a are free but internal supports parallel to the free edges are present in the structural system. The algorithmic procedure can be easily implemented on a micro computer or even a hand, programmable calculator.Good engineering agreement with experimental results is shown to exist in the case of a plate with edges y=0,b rigidly clamped.     

Transverse vibrations of circular annular plates with edges elastically restrained against rotation, used in acoustic underwater transducers

Simply supported or clamped boundary conditions are rather ideal situations difficult to satisfy from a physical viewpoint. This paper considers a more “moderate” restriction: the case of edges elastically restrained against rotation for which no exact solution appears in the open literature. Eigenvalues corresponding to a wide range of the intervening geometric and mechanical parameters are determined. Good agreement is obtained with frequency coefficients determined two decades ago by means of a variational method. Obviously the problem is of basic interest in many ocean engineering applications: from the design of certain underwater acoustic transducers to pump and compressor elements passing through the design of naval vehicles and ocean structures.     

Forced vibrations of a beam elastically restrained against rotation and carrying a spring–mass system

An exact solution for the title problem is obtained in closed form fashion in the case of a Bernoulli–Euler beam. It is assumed that the exciting force is applied to the mass which is elastically mounted on the beam. The mathematical model constitutes a first order approximation to a motor or engine elastically mounted on a structural element. The operation of the machine generates a transverse, sinusoidally varying force. The problem is of basic interest in mechanical, naval and ocean engineering systems from the point of view of the determination of dynamic displacements and stresses; sound radiation calculations, etc. The present problem arose in connection with the mounting of an engine on a structural beam in a small naval vessel and when excessive vibrational level was noted. This study was undertaken in order to understand the physical problem and to correct the mechanical situation     

Forced vibrations of a beam elastically restrained against rotation at the ends in the case of a concentrated load

The title problem is solved using Bernouilli's classical theory and by means of the separation of variables technique.Numerical data is presented for the simply supported beam, simply supported - clamped and clamped - clamped situations for the case of a concentrated load which varies sinusoidally with time.Certainly no claim of originality is made by the authors since the problem is of a classical nature but it hoped that the rather complete tabulation of displacements, bending moments and shear forces amplitudes included in the present study will be of practical use to ocean and naval engineers.     

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

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 γ_i. The results are in excellent agreement with those obtained by means of a finite element algorithmic procedure.     

Forced transverse vibrations of a simply supported rectangular orthotropic plate in the case where the force acts over a plate subdomain

The present study deals with the exact solution of the title problem in the case where a uniformly distributed p₀ cos ωt-type force acts over a rectangular portion of the plate. The problem is of interest in naval and ocean engineering systems where a motor or an engine mounted on a plate or slab induces a dynamic excitation over a finite area. Mechanical designers commonly consider the problem from the point of view of a concentrated dynamic or static force. On the other hand, modelling the structural system as an orthotropic element is of considerable interest in ocean and naval design in view of the ever-increasing use of composite materials and also of the anisotropic characteristics often induced by metallurgical processes.     

Vibrations of a rectangular plate with a free edge in the case of discontinuously varying thickness

Transverse vibrations of a rectangular plate with three edges elastically restrained against rotation while the fourth is free are studied using (a) an analytical approach based on the optimized Rayleigh-Ritz method and (b) the finite element method. The study was motivated by the necessity of finding out how the lower frequencies of vibration were affected by introducing a step discontinuity in the plate thickness. This variation in the plate thickness was practiced in order to obtain a reduction in the total plate weight.     

Free vibrations of a cantilever beam with a spring–mass system attached to the free end

An exact solution for the title problem is obtained using the Bernoulli–Euler theory of beam vibrations. Natural frequencies are obtained for a wide range of the intervening physical parameters. The problem is of interest in naval and ocean engineering systems since in order to avoid dangerous resonance conditions the designer must be able to predict natural frequencies of the overall mechanical system: structure–motor and its elastic mounting.     

Forced vibrations of a clamped-free beam with a mass at the free end with an external periodic disturbance acting on the mass with applications in ships’ structures

An exact, analytical solution is obtained for the title problem which constitutes a classical one although no solution is available in well known textbooks and handbooks normally used by the structural engineer in several fields of technology: ocean and naval engineering, aerospace applications, etc. The authors performed this study motivated by a situation where excessive displacements were noticed in a structural element carrying a relatively small motor at the free end and placed at the engine room of a naval vessel. The Bernoulli-Euler model has been employed.