Transverse vibrations of a non-uniform rectangular plate on an elastic foundation

Free transverse vibrations of an isotropic rectangular plate of variable thickness resting on an elastic foundation has been studied on the basis of classical plate theory. The fourth-order differential equation governing the motion is solved by using the quintic spline interpolation technique. Characteristic equations for plates of exponentially varying thickness have been obtained for three combinations of boundary conditions at the edges. Frequencies, mode shapes and moments have been computed for different values of the taper constant and the foundation moduli for the first three modes of vibration.     

Transverse vibrations of circular plates of varying thickness with non-uniform edge constraints

Free vibrations of circular plates varying in thickness and with flexible edge supports have been studied by several investigators for the restricted case when the supports are represented by uniformly distributed springs of constant stiffness.In the present study an approximate method is presented for dealing with supports possessing rotational flexibility which varies arbitrarily around the boundary.The method consists in representing the varying stiffness in terms of a Fourier expansion in the polar angle and approximately expressing the displacement function using a summation of polynomial co-ordinate functions which exactly satisfies only the essential boundary condition. The Ritz method is then applied in order to obtain the frequency determinant. The method can be easily extended to the forced vibrations case.     

Approximate study of the free vibrations of a cantilever anisotropic plate carrying a concentrated mass

This study is concerned with the vibration analysis of a cantilevered rectangular anisotropic plate when a concentrated mass is rigidly attached to its center point. Based on the classical theory of anisotropic plates, the Ritz method is employed to perform the analysis. The deflection of the plate is approximated by a set of beam functions in each principal coordinate direction. The influence of the mass magnitude on the natural frequencies and modal shapes of vibration is studied for a boron-epoxy plate and also in the case of a generic anisotropic material. The classical Ritz method with beam functions as the spatial approximation proved to be a suitable procedure to solve a problem of this analytical complexity.     

A nonlocal plate model incorporating interatomic potentials for vibrations of graphene with arbitrary edge conditions

This article presents analytical explicit frequency expressions for investigating the vibrations of single-layer graphene sheets (SLGSs). The interatomic potential is incorporated into a nonlocal continuum plate model through establishing a linkage between the strain energy density induced in the continuum and nonlocal plate constitutive relations. The model which is independent of scattered value of Young's modulus is then applied and explicit frequency formulas for the SLGSs with different edge conditions are derived using static deflection function of the nanoplate under uniformly distributed load. The reliability of the present formulation is verified by the results obtained by the molecular dynamics (MD) simulations and other research workers. The formulas are of a simple short form enabling quick and accurate evaluation of the frequency of the SLGSs and also simple calibration of scale coefficient by the use of MD simulations results.     

Transverse vibrations of circular plates with stepped thickness

The study of the dynamic behaviour of circular plates with stepped thickness is of interest in view of their use in the construction of high frequency transducers. A simple analytical approach which allows for the prediction of their natural frequencies is proposed in the present Note.     

Transverse waves in a relativistic rigid body

We present relativistic elasticity as a scalar field theory. We apply it to rigid bodies, i.e., relativistic bodies with a nonlinear elastic law and a definite longitudinal wave velocity _l equal to the light velocity,c. We obtain the transverse wave equation with a definite velocity _t , and the relation between _l , _t , and the Poisson coefficient is the classical one. This is an indication that we have the relativistic extension of a classical Hooke elastic law.     

Non-linear vibrations of a flat plate with initial stresses

Non-linear free vibrations of a simply supported rectangular elastic plate are examined, by using stress equations of free flexural motions of plates with moderately large amplitudes derived by Herrmann. A modal expansion is used for the normal displacement that satisfies the boundary conditions exactly, but the in-plane displacements are satisfied approximately by an averaging technique. Galerkin technique is used to reduce the problem to a system of coupled non-linear ordinary differential equations for the modal amplitudes. These nonlinear differential equations are solved for arbitrary initial conditions by using the multiple-time-scaling technique. Explicit values of the coefficients that appear in the forementioned Galerkin system of equations are given, in terms of non-dimensional parameters characterizing the plate geometry and material properties, for a four-mode case, for which results for specific initial conditions are presented. A comparison of the results with those obtained in previous studies of the problem is presented. In addition, effects of prescribed edge loadings are examined for the four-mode case.     

Forced vibrations of a rectangular plate with non-uniform thickness

The dependence on frequency of the maximum deflection and surface stresses of a simply supported rectangular plate subjected to a uniformly distributed sinusoidal excitation is discussed and simple formulae are proposed for estimating the deflection and surface stresses. The thickness of the plate varies linearly in one direction parallel to a side of the plate.     

Free vibrations of a plate with an inner support

The title problem is tackled by using a simple polynomial co-ordinate function and the Rayleigh-Schmidt method. It is assumed that the inner support is parallel to the free edge. When the support coincides with the free edge the frequency equation degenerates properly into the case of a simply supported edge. Numerical results are presented for the situation where two opposite edges are simply supported and the edge parallel to the free edge is either clamped or simply supported.     

Transverse rod vibrations under a short-term longitudinal impact

Transverse vibrations of a thin rod caused by a short-term longitudinal impact are considered. After the impact, a system of compression-tension waves, which causes transverse vibrations, appears in the rod. Parametric resonance, which leads to an unbounded rise in the amplitude of transverse vibrations, is investigated in the linear approximation. With a nonlinear approach, beats appear in the resonance vicinity, in which the mutual exchange of the energy of longitudinal and transverse vibrations occurs. The influence of viscoelastic resistance forces is investigated.     

Transverse nonlinear vibrations of a circular spinning disk with a varying rotating speed

We analyze the transverse nonlinear vibrations of a rotating flexible disk subjected to a rotating point force with a periodically varying rotating speed. Based on Hamilton’s principle, the nonlinear governing equations of motion (coupled equations among the radial, tangential and transverse displacements) are derived for the rotating flexible disk. When the in-plane inertia is ignored and a stress function is introduced, the three nonlinearly coupled partial differential equations are reduced to two nonlinearly coupled partial differential equations. According to Galerkin’s approach, a four-degree-of-freedom nonlinear system governing the weakly split resonant modes is derived. The resonant case considered here is 1:1:2:2 internal resonance and a critical speed resonance. The primary parametric resonance for the first-order sin and cos modes and the fundamental parametric resonance for the second-order sin and cos modes are also considered. The method of multiple scales is used to obtain a set of eight-dimensional nonlinear averaged equations. Based on the averaged equations, using numerical simulations, the influence of different parameters on the nonlinear vibrations of the spinning disk is detected. It is concluded that there exist complicated nonlinear behaviors including the periodic, period-n and multi-pulse type chaotic motions for the spinning disk with a varying rotating speed. It is also found that among all parameters, the damping and excitation have great influence on the nonlinear responses of the spinning disk with a varying rotating speed.     

Transverse vibrations of plates with stepped thickness over a concentric circular region

This paper deals with the determination of the fundamental frequency of vibration of: (a) rectangular, (b) regular polygonal and (c) circular plates with stepped thickness over a concentric, circular subdomain of the plates. The problems are solved in a unified fashion by adopting simple polynomial co-ordinate functions and making use of the Ritz method to generate the frequency determinant.