Perturbation method in the problem of large deflections of circular plates with nonuniform thickness

In this paper the perturbation method about two parameters is applied to the problem of large deflection of a cricular plate with exponentially varying thickness under uniform pressure. An asymptotic solution up to the third-order is derived. In comparison with the exact solutions in special cases, the asymptotic solution shows a precise accuracy.     

The perturbation parameter in the problem of large deflection of clamped circular plates

In the problems of large deflection of clamped circular plates under uniformly distributed loads, various perturbation parameters relating to load, deflection, slope of deflection, membrane force, etc, are studied. For a general perturbation parameter, the variational principle is used for the solution of such a problem. The applicable range of these perturbation parameters are studied in detail. In the case of uniformly loaded plate, perturbation parameter relating to central deflection seems to be the best among all others. The method of determination of perturbation solution by means of variational principle can be used to treat a variety of problems, including the large deflection problems under combine loads. This paper is completed under the guidance of Prof. Chien Wei-zang.     

Experimental and theoretical study on large amplitude vibrations of clamped rubber plates

In this paper, the large amplitude forced vibrations of thin rectangular plates made of different types of rubbers are investigated both experimentally and theoretically. The excitation is provided by a concentrated transversal harmonic load. Clamped boundary conditions at the edges are considered, while rotary inertia, geometric imperfections and shear deformation are neglected since they are negligible for the studied cases. The von Kármán nonlinear strain-displacement relationships are used in the theoretical study; the viscoelastic behaviour of the material is modelled using the Kelvin-Voigt model, which introduces nonlinear damping. An equivalent viscous damping model has also been created for comparison. In-plane pre-loads applied during the assembly of the plate to the frame are taken into account. In the experimental study, two rubber plates with different material and thicknesses have been considered; a silicone plate and a neoprene plate. The plates have been fixed to a heavy rectangular metal frame with an initial stretching. The large amplitude vibrations of the plates in the spectral neighbourhood of the first resonance have been measured at various harmonic force levels. A laser Doppler vibrometer has been used to measure the plate response. Maximum vibration amplitude larger than three times the thickness of the plate has been achieved, corresponding to a hardening type nonlinear response. Experimental frequency-response curves have been very satisfactorily compared to numerical results. Results show that the identified retardation time increases when the excitation level is increased, similar to the equivalent viscous damping but to a lesser extent due to its nonlinear nature. The nonlinearity introduced by the Kelvin-Voigt viscoelasticity model is found to be not sufficient to capture the dissipation present in the rubber plates during large amplitude vibrations.     

Free vibrations of ribbed cylindrical shells with local axisymmetric deflections

The paper proposes a method to calculate the natural frequencies of ribbed cylindrical shells with local axisymmetric deflections. The influence of initial imperfections of two types on the minimum frequencies of vibration is analyzed Translated from Prikladnaya Mekhanika, Vol. 44, No. 9, pp. 63–72, September 2008.     

THE METHOD OF THE RECIPROCAL THEOREM OF FORCED VIBRATION FOR THE ELASTIC THIN RECTANGULAR PLATES(Ⅰ)-RECTANGULAR PLATES WITH FOUR CLAMPED EDGES AND WITH THREE CLAMPED EDGES

In this paper the method of the reciprocal theorem (MRT) is extended to solve the steady state responses of rectangular plates under harmonic disturbing forces. A series of the closed solutions of rectangular plates with various boundary conditions are given and the tables and figures which have practical value are provided. MRT is a simple, convenient and general method for solving the steady state responses of rectangular plates under various harmonic disturbing forces. The paper contains three parts: (I) rectangular plates with four clamped edges and with three clamped edges; (II) rectangular plates with two adjacent clamped edges; (III) cantilever plates. We are going to publish them one after another.     

On the elastostatics of thin periodic plates with large deflections

Thin periodic plates with large (i.e. of the order of plate thickness) deflections are considered. In this note the tolerance and the asymptotic models of these plates are presented. As an example of applications, these models are used to analyse a bending of periodic plates under various loadings.     

Free vibrations of a partially clamped trapezoidal plate

The natural frequencies and mode shapes of partially clamped trapezoidal cantilever plates having various swept-back angles are examined numerically and experimentally by a finite-element computer program, SAP6, and holographic interferometry, respectively. A good agreement is found between the numerical and the experimental results. The influence of the clamped position, the free-width ratio, and the swept-back angle on the vibratory behavior of the trapezoidal plates is investigated.Paper was presented at the 1990 SEM Spring Conference on Experimental Mechanics held in Albuquerque, NM on June 3–6.     

Experimental-theoretical correlations of impulsively loaded clamped circular plates

Clamped circular plates are impulsively loaded with sheet explosive, and the resulting large-defection response is monitored using a high-speed streak camera, and dynamic-strain measurements. Dynamic and final-plate deflections as well as strain-time histories of various locations on the plates are compared to deflections and strains obtained with the elastic-plastic structural computer program DEPROSS. It is shown that DEPROSS adequately computes the dynamic response of this highly nonlinear biaxial-stress problem.     

Design of heterogeneous clamped circular plates with specified fundamental natural frequency

The titled problem is solved by the aid of the semi-inverse formulation. At the first glance, a surprising demand is posed: Design a radially heterogeneous polar orthotropic circular plate whose fundamental mode shape coalesces with the static deflection of a homogeneous circular plate of the same radius and thickness under uniformly distributed load. Compatible polynomial variations of the radial and circumferential flexural rigidities are introduced. It turns out that these are infinite amount of polar orthotropic circular plates that solve the posed semi-inverse problem, depending on the free parameter represented by one of the coefficients in variation of flexural rigidities. This allows to provide an unique solution to the titled problem: there is a single plate that possesses the specified fundamental natural frequency.     

Primary resonance excitation of electrically actuated clamped circular plates

We investigate the response of an electrically actuated clamped circular plate to a primary resonance excitation of its first axisymmetric mode using an analytical reduced-order model (macromodel). We discuss the inf luence of the number of modes retained in the discretization on the predicted solutions. The reduced-order model, which is a system of coupled nonlinear ordinary-dif ferential equations, accounts for general residual stress and strain hardening and allows for general material and geometric design variables. Our reduced-order model is robust up to the pull-in instability and is general enough to be an ef fective design tool for capacitive micromachined ultrasonic transducers.     

Bending of simply supported rectangular plates with clamped portions along arbitrary sections of the edges

Conclusion A method of handling such a rectangular plate having special mixed boundary conditions was presented and also its reliability was investigated. The labour required for computation amounts to a substantial quantity and thus an approximate method seems to be desirable. However, such a difficulty could be solved through the use of a high speed digital computer at present. Actually, the author has taken advantage of such a devise.In conclusion, the assistance of Mr. S. Hatano and Mr. H. Okamura on numerical computation and experiments is gratefully acknowledged.     

Response of circular clamped plates to square-wave stress pulses

An account is given of an experimental investigation of the reeponse of clamped circular mild-steel plates of various thicknesses subjected to rectangular stress pulses over a small circular region. The stress pulses were transmitted to the plates through a 1/2-in.-diam shock bar and the strain-time responses of the plates were measured. The stress-wave interactions between the bar and the plates were measured for a number of thicknesses and the effect of the applied stress on the extent of the plastic deformation was determined. It was found that the elastic response was accurately predicted by the theory of Sneddon and the plastic response behaved according to a simple modification of this theory. The interaction between the stress pulse and plates of various thickness was theoretically predicted and found to be in excellent agreement with experimental measurements. The final plate deflections were theoretically predicted using a rigid viscoplastic theory and was in substantial agreement with the data. From this theory, the data were analyzed to determine the visco-plastic constant or relaxation time of the material. It is proposed that this testing arrangement is a suitable and convenient method for determining dynamic yield properties under biaxial-loading conditions.     

Airy stress function for describing the non-linear effect of simply supported plates with movable edges

The general form of the solution of the Airy function for the stress distributions that describe the non-linear effect developed from the large deflection of simply supported plates with movable edges are found by superposition of the Airy functions, which satisfy the large deflection condition and the boundary conditions of the edges. Each term of the Airy function consists of a particular solution and a homogeneous one. The particular solution satisfying the large deflection condition is classified into six cases, depending on the combinations of the modal numbers of the comparison functions. The corresponding homogeneous solution is found to make each Airy function satisfy the boundary condition by using the Fourier series method. The solution is applied to the non-linear analysis of the deflection of the simply supported plates with movable edges under transverse loading, and is verified by comparison with other investigation.