Forced axisymmetric response of circular plates

Forced axisymmetric response of circular plates is considered on the basis of an improved theory which takes into account the effect of rotatory inertia and shear deformation. Explicit solutions are obtained for distributed, ring and concentrated impulsive loadings with half-sine, triangular, blast and rectangular pulse shapes. Natural frequencies are presented for the first 20 modes of four plate models with clamped and supported edges. Numerical results are presented to illustrate the differences between the improved and classical theories, and the effects of edge conditions, loading conditions, pulse shapes and pulse durations.     

Non-linear resonances in the forced responses of plates,part II: Asymmetric responses of circular plates

The dynamic analogue of the von Karman equations is used to study the forced response, including asymmetric vibrations and traveling waves, of a clamped circular plate subjected to harmonic excitations when the frequency of excitation is near one of the natural frequencies. The method of multiple scales, a perturbation technique, is used to solve the non-linear governing equations. The approach presented provides a great deal of insight into the nature of the non-linear forced resonant response. It is shown that in the absence of internal resonance (i.e., a combination of commensurable natural frequencies) or when the frequency of excitation is near one of the lower frequencies involved in the internal resonance, the steady state response can only have the form of a standing wave. However, when the frequency of excitation is near the highest frequency involved in the internal resonance it is possible for a traveling wave component of the highest mode to appear in the steady state response.     

Asymmetric non-linear forced vibrations of free-edge circular plates. Part II: experiments

This article is devoted to an experimental validation of a theoretical model presented in an earlier contribution by the same authors. The non-linear forced vibrations of circular plates, with the excitation frequency close to the natural frequency of an asymmetric mode, are investigated. The experimental set-up, which allows one to perform precise measurements of the vibration amplitudes of the two preferential configurations, is presented. Experimental resonance curves showing the amplitude and the phase of each configuration as functions of the driving frequency are compared to the theoretical ones, leading to a quantitative validation of the predictions given by the model. Finally, all the approximations used are systematically discussed, in order to show the scope and relevance of the approach.     

Response of circular plates to thermal impact

The thermally induced vibrations of simply supported circular plates are investigated analytically. The solution is composed of two parts; the first is obtained by neglecting the inertia term and the second represents the vibrating component which oscillates about the first, due to the effect of the inertia. One of the reasons for separating the solution in two parts is to obtain the complete solution expediently. But when this is represented in non-dimensional form, the quasi-static part has the particular desirable feature that it is independent of the strength of the thermal shock and thus will give an index for the displacement due to the temperature moment if (and only if) the inertia force has a small effect. The basic equation is governed by only one non-dimensional parameter B which contains the density, the thermal diffusivity of the material and the radius of the plate in its numerator, and the flexural rigidity and thickness of the plate in its denominator, all to various powers. B represents the strength of the thermal shock. As B increases the non-dimensional amplitude of the vibration becomes large and the natural frequency decreases.     

Natural frequencies of simply supported circular plates

Although the problem of finding the natural frequencies of free vibration of a simply supported circular plate has a straightforward solution, very few numerical results are available in the literature. In the present work accurate (six significant figure) non-dimensional frequency parameters (λ²) are given for all values of n + s ? 10, where n and s are the numbers of nodal diameters and internal nodal circles, respectively, and for Poisson's ratios 0, 0·1, …, 0·5. Simplified formulas for determining additional values of λ² for large s are derived by the use of asymptotic expansions.     

Vibrations of nearly annular and circular plates

A general procedure is presented to determine the natural frequencies and mode shapes of nearly annular and circular plates. A straightforward perturbation procedure is used with a transfer of the boundary conditions to obtain solutions given by simple expressions. As well as yielding good accuracy quantitatively, the method accurately predicts qualitative behavior. Numerical results are presented for clamped elliptical and square plates.     

Vibration analysis of circular segment shaped plates

Circular segment shaped plates are analyzed to determine their natural frequencies and mode shapes of vibration. The analysis is based on the finite element approach. The curved sided triangular plate bending element is used for solving the problem. The effect of variation of the size of the plate on the vibrational characteristics is studied and several important conclusions are made.     

Distributed piezoelectric modal sensors for circular plates

In this note, we deal with finding the shape of distributed piezoelectric modal sensors for circular plates with polar symmetric boundary conditions. The problem is treated by an optimization approach, where a binary function is used to model the design variable: the polarization profile of the piezoelectric layer. The numerical procedure proposed here allows us to find polarization profiles which take on two values only, i.e. either positive or negative polarization, that isolate particular vibration modes in the frequency domain.     

Periodic geometrically nonlinear free vibrations of circular plates

The geometrically nonlinear free vibrations of thin isotropic circular plates are investigated using a multi-degree-of-freedom model, which is based on thin plate theory and on Von Kármán's nonlinear strain-displacement relations. The middle plane in-plane displacements are included in the formulation and the common axisymmetry restriction is not imposed. The equations of motion are derived by the principle of the virtual work and an approximated model is achieved by assuming that the in-plane and transverse displacement fields are given by weighted series of spatial functions. These spatial functions are based on hierarchical sets of polynomials, which have been successfully used in p-version finite elements for beams and rectangular plates, and on trigonometric functions. Employing the harmonic balance method, the differential equations of motion are converted into a nonlinear algebraic form and then solved by a continuation method. Convergence with the number of shape functions and of harmonics is analysed. The numerical results obtained are presented and compared with available published results; it is shown that the hierarchical sets of functions provide good results with a small number of degrees of freedom. Internal resonances are found and the ensuing multimodal oscillations are described.     

Axisymmetric non-linear oscillations of isotropic layered circular plates

Non-linear equations of motion for isotropic layered circular plates are presented for axisymmetric motion. Further simplification is made by ignoring the in-plane and rotatory inertia terms. Explicit solutions are obtained for the forced and free oscillations. In this case it is found that the non-linearity is of hardening type. Numerical results are presented for the case of a two layered plate of aluminium and steel.     

Forced axisymmetric motion of circular,viscoelastic plates

Williams' method for forced motion of elastic systems is applied to circular, viscoelastic plates where the effects of rotatory inertia, transverse shear and time-dependent boundary conditions are included. The viscoelastic material is assumed to have a constant Poisson's ratio. A particular problem is solved for a symmetrically loaded, completely free plate. The material used is vulcanized rubber where the viscoelastic behavior in shear is used in specifying the material parameters of a three-element solid.     

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.