Free flexural axisymmetric vibration of a clamped circular sandwich plate with a functionally graded core

In a companion article, a model for a clamped circular unimorph piezoelectric plate has been developed for the purpose of analyzing the influence of geometric design parameters and electrode configuration on the amount of electrical energy that can be harvested from an applied pressure source. It has been shown that the ratio of layer thickness (piezoelectric layer to substrate layer) and electrode pattern have a significant effect on energy conversion for harvesting. Specifically, the theoretical analysis shows that regrouping of the electrodes (i.e., segmenting the electrode into a specific pattern) can lead to optimized energy harvesting in a clamped circular plate structure. This article provides experimental validation of these results. In this article, three circular plate piezoelectric energy generators (PEGs), one unmodified and two regrouped unimorph PEGs, were used to support the analysis.

The effects of large vibration amplitudes on the axisymmetric mode shapes and natural frequencies of clamped thin isotropic circular plates. Part I: iterative and explicit analytical solution for non-linear transverse vibrations

Large deformations and failure of clamped circular steel plates under uniform impulsive loads using various phenomenological damage models

Based on von Karman plate theory, axisymmetric geometrically nonlinear governing equations of simple power-law type functionally graded material (FGM) circular plates are derived. Material properties are assumed to be temperature-independent, and thermal or/and mechanical loads are considered in the derivation. Considering clamped boundary conditions, the dimensionless critical (lowest) buckling temperature (temperature difference) 14.684 of the systems is obtained by analyzing linear-eigenvalue problem. With this constant and an analytical formula proposed in this paper, accurate dimensional critical (lowest) buckling temperature (temperature difference) of any specified clamped FGM circular plate can be easily calculated. Moreover, two-point boundary value problem posed by the governing equations and the clamped boundary conditions is solved using the shooting method. Thermal buckling response under thermal load and geometrically nonlinear mechanical behaviors under thermomechanical load of the system are discussed. When the temperature difference of the upper and lower surfaces of FGM plates exists, the solution of one-dimension Fourier heat conduction equation is an infinite series, which depicts a nonlinear temperature field (NLTF) along the thickness direction of plates. The effects of the initial terms of the series on the accuracy of solutions are examined. The results reveal it is necessary to take enough the initial terms of the series even for thinner FGM plates. Typical load–deflection curves of the clamped FGM circular plates with different temperature fields (along the thickness) are presented. For a given temperature field, the clamped FGM circular plates under transverse uniform load are a hard stiffness nonlinear system. When the values of thermal and mechanical load and their loading sequence meet certain combinations, the clamped circular plate will inevitably undergo secondary buckling (snap-through buckling).

Frequency equation for the in-plane vibration of a clamped circular plate

Response simulations of clamped circular steel plates under uniform impulse and effects of axisymmetric stiffener configurations

The investigation is concerned with the axisymmetric non-linear vibrations of a thin composite circular plate carrying a concentric rigid mass. Hamilton's principle is utilized to derive the von Karman equations and the associated boundary conditions. Harmonic vibrations are assumed and the time variable is eliminated by a Kantorovich averaging method. Numerical solutions are obtainable by introducing the related initial value problem. Responses of the free and forced vibrations of the plate-mass system are obtained. Basic Differential Equations and Approximate Analyses Consider a flat circular plate having an outer radius a, constant thickness h, and an attached concentric rigid mass, M . The radius of the rigid mass is b and equals th: inner radius of the plate. The geometry is shown in Figure 1. The plate material is assumed to be elastic and homogeneous, but the material is made of composites which behave as a cylindrically orthotropic plate. By Hamilton's principle, it can be shown that the equations of motion for a finite amplitude axisymmetric vibration of a circular plate with a concentric rigid mass in nondimensional form are,

In this paper, the thermoelastic transient behavior of a clamped circular plate composed of functionally graded material (FGM) is investigated. The material properties of the FGM circular plate are assumed to vary through the plate thickness according to a power law distribution of the volume fraction of constituent materials, except Poisson's ratio, which is assumed as constant. Based on the von Karman equation and classical theory of thin plates, the equation of motion for the FGM circular plate is derived by the Hamilton principle. The nonlinear governing equation is solved by the Galerkin method, along with Newmark's integration method, in an iterative manner. Numerical results reveal that the functional gradient index, ratio of thickness to radius, thermal and mechanical loads have significant effect on the thermoelastic transient behavior of the clamped FGM circular plate. The result presented herein may be used as a reference for solving other transient coupled problems of thermoelasticity.

A MEMS capacitive sensor is basically an electrostatic transducer and an analytical approach to evaluate the pull-in voltage associated with a clamped circular plate or a circular membrane due to a bias voltage is presented. The approach is based on a linearized uniform approximation of the nonlinear electrostatic force due to a bias voltage and the use of a 2D load-deflection model for the clamped plate or membrane. In particular, the large deflection of the clamped thin, circular and isotropic plate is investigated when a transverse loading and an initial in-plane tension load are applied. The transition from plate behavior to membrane behavior is analyzed. The edge zone region is explored and properties of this region are given. The resulting electrostatic pressure on the diaphragm of the MEMS capacitive sensor and the pull-in voltage are studied.

Response of thin walled transversely stiffened clamped circular plates under uniform impulsive loads

ABSTRACTComplex variable methods have been applied to isotropic and aelotropic plate problems by several authors. The notation used here is that of Stevenson(14). Dawoud(5) has expressed the continuity conditions between two differently loaded regions in terms of the complex potentials and the particular integrals for the two regions.The problem of a transverse load at any point of a clamped circular plate was solved by Clebsch(4), Michell(11), Melan(10) and Flügge(6). A series solution for the simply supported circular plate under the same load was given by Foeppel(8). Using Stevenson's tentative method Dawoud(5) applied complex potentials to solve the problem of an eccentric isolated load under certain boundary conditions. Applying Muskhelishvili's method, Washizu(15) obtained the same results for clamped and simply supported boundaries.It is easy to get solutions for a circular plate concentrically and uniformly loaded. For non-uniform loadings there are the solutions found by Sen (13) for certain distributions of normal thrust over the complete plate or over a concentric circle and the solution of Flügge (7) for a linearly varying load over the simply supported circular plate. The present author and Dawoud(3) obtained the solutions for a circular plate with the load over the complete plate or over a concentric circle, under a general boundary constraint including as special cases the usual clamped and hinged boundaries. Ghose (9) worked out the problem of a clamped circular plate when the load is uniformly distributed between two concentric circles and two radii. Schmidt (12) found the solution for a clamped circular plate uniformly loaded over an eccentric circle. The complex variable method was applied by the author and Dawoud(2) to obtain the solutions for a circular plate having an eccentric circular patch symmetrically loaded with respect to its centre under the general boundary condition mentioned before. The author (1) also found the solution for a linearly varying load over an eccentric circle under the same boundary condition. In this paper the power of the complex variable method is exhibited by rinding the appropriate complex potentials corresponding to the loadover an eccentric circular patch, where R, θ are measured from the centre of the patch and the common diameter of the plate and the patch. Since the two cases n = 0, 1 require special consideration and were dealt with separately (in (2) and (1) respectively), we see that this paper completes the solution of the problem of a circular plate with an eccentric circular patch symmetrically loaded with respect to the common diameter of the plate and patch, the load being in this case expressible in the form .For a clamped boundary the solution is obtained in finite terms.

A response prediction method for clamped circular plates subjected to repeated blast loading

Deformation and tearing of circular plates with varying support conditions under uniform impulsive loads

Full-range response of clamped ring-stiffened circular steel plates-comparisons between experiment and theory

Chapter 13 - Relationships for Inhomogeneous Plates