Vibration and stability of a circular plate of unidirectionally varying thickness

An analysis is presented for the vibration and stability of an elastically restrained circular plate of unidirectionally varying thickness subjected to an in-plane force. For this purpose, the transverse deflection of a circular plate of variable thickness is written in a series of the deflection functions of a uniform circular plate without the action of a force. The dynamical energies of the plate are evaluated analytically, and the frequency equation of the plate is derived by the Ritz method. The analysis is applied to circular plates of unidirectionally tapered or stepped thickness; the natural frequencies and the divergence loads are calculated numerically, and the effects of the varying thickness and edge conditions on the vibration and stability are studied.     

Dynamic stability of thermoelastic coupling moving plate subjected to follower force

The dynamic characteristics and stability of the moving thermoelastic coupling rectangular plate subjected to uniformly distributed tangential follower force are investigated. Based on the heat conduction equation containing the thermoelastic coupling term and the thin plate theory, the thermoelastic coupling differential equation of motion of the rectangular plate under the action of uniformly distributed tangential follower force is established. Dimensionless complex frequencies of the moving thermoelastic coupling rectangular plate with four edges simply supported, two opposite edges simply supported and other two edges clamped are calculated by the differential quadrature method. The effects of the dimensionless thermoelastic coupling factor and dimensionless moving speed on the stability and critical load of the moving plate are analyzed. The results show that the divergence loads of the first order mode increase with the increase of the dimensionless thermoelastic coupling factor, and decrease with increasing the dimensionless moving speed.     

Dynamic stability of linearly varying thickness viscoelastic rectangular plate with crack and subjected to tangential follower force

Based on the two-dimensional viscoelastic differential constitutive relation and the thin plate theory, the differential equations of linearly varying thickness viscoelastic plate with crack and subjected to uniformly distributed tangential follower force in the Laplace domain are established, and the expression of the additional rotation induced by the crack is given. The complex eigenvalue equations of linearly varying thickness viscoelastic plate constituted by elastic behavior in dilatation and the Kelvin-Voigt laws for distortion with crack and under the action of uniformly distributed tangential follower force are obtained by the differential quadrature method. The generalized eigenvalue under different boundary conditions is calculated, and the curves of real parts and imaginary parts of the first three order dimensionless complex frequencies versus uniformly distributed tangential follower force are obtained. The effects of the aspect ratio, the thickness ratio, the crack parameters and the dimensionless delay time on the dynamic stability of the viscoelastic plates are analyzed.     

VIBRATION AND STABILITY OF ANNULAR PLATES IN NON-LINEAR CREEP CONDITIONS

This paper presents a study on the influence of the physical non-linearity of material in creep conditions on vibration and stability of non-uniform annular plates subjected to a follower force radially distributed at the outer edge. The stability analysis requires first the calculation of a membrane stress distribution and then the application of the kinetic stability criterion, making use of small superposed vibrations. The differential equations of the membrane and vibration states have been integrated by means of the transfer matrix method which allowed the determination of the relationships between the real and imaginary parts of the complex frequency and compressive force (characteristic curves). The results have been presented in numerous figures.     

Vibration and stability of a circular cylindrical shell subjected to a torque

An analysis is presented for the vibration and stability of a circular cylindrical shell subjected to a torque. The displacements of a circular shell are written in a series of beam eigenfunctions satisfying the boundary conditions. The kinetic and strain energies of the shell are evaluated analytically, and the frequency eauation of the shell is derived by the Ritz method. The method is applied to circular cylindrical shells under two types of boundary conditions at the edges; the natural frequencies and the divergence torques are calculated numerically, and the effects of the thickness ratio, length ratio and edge conditions on the vibration and stability are studied.     

Vibration and stability of a variable thickness annular plate subjected to a torque

The free vibration and stability of a variable thickness annular plate subjected to a torque are analyzed by the Ritz method. For this purpose, the transverse deflection of an annular plate is written in a series of the deflection functions of a uniform thickness annular plate without the action of a torque. The kinetic and strain energies of the plate are evaluated analytically and the frequency equation of the plate is derived by the conditions for a stationary value of the Lagrange functional. The present method is applied to annular plates with two types of radial thickness variation, power law and exponential, and the natural frequencies (the frequency parameters) and the divergence torques are calculated numerically, from which the effects of the varying thickness, inner/outer radii ratio and edge conditions are studied.     

The magneto-elastic subharmonic resonance of current-conducting thin plate in magnetic filed

Based on Maxwell equations and corresponding electromagnetic constitutive relations, the electrodynamic equations and electromagnetic force expressions of a current-conducting thin plate in electromagnetic field are deduced. Nonlinear magneto-elastic vibration equations of the thin plate are given. In addition, nonlinear subharmonic resonances of the thin plate with two opposite sides simply supported which is under the mechanic live loads and in constant transverse magnetic field are studied. The corresponding vibration differential equation of Duffing type is deduced by the Galerkin method. The method of multiple scales is used to solve the equation, and the frequency-response equation of the system in steady motion under subharmonic responses is obtained, and the stability of solution is analyzed. According to the Liapunov stability theory, the critical conditions of stability are obtained. By the numerical calculation, the curves of resonance amplitude changing with the detuning parameter, the excitation amplitude and the magnetic intensity and corresponding state planes are obtained. The existing regions of nontrivial solutions and the changing law of stable and unstable solutions are analyzed. The time history response plots, the phase charts and the Poincare mapping charts are plotted. And the effect of the magnetic intensity on the system is discussed, and some complex dynamic performances as period-doubling motion and quasi-period motion are analyzed.     

NON-LINEAR VIBRATION AND THERMAL BUCKLING OF AN ORTHOTROPIC ANNULAR PLATE WITH A CENTRIC RIGID MASS

A computational analysis of the non-linear vibration and thermal post-buckling of a heated orthotropic annular plate with a central rigid mass is examined for the cases of immovably hinged as well as clamped constraint conditions of the outer edge. First, based on von Karman's plate theory and Hamilton's principles, the governing equations, in terms of the displacements of the middle plane, of the problem are derived. Then, upon assuming that harmonic responses of the system exist, the non-linear partial differential equations are converted into the corresponding non-linear ordinary differential equations through elimination of the time variable by using the Kantorovich time-averaging method. Finally, by applying a shooting method, the fundamental responses of the non-linear vibration and thermal post-buckling of the plate are numerically obtained. For some prescribed values of the parameters, such as the material rigidity ratio, temperature rise and so on, the curves of the fundamental frequency versus specified amplitude and the thermal post-buckled equilibrium paths of the plate are numerically presented.     

Vibration and stability of a non-uniform Timoshenko beam subjected to a follower force

An analysis is presented for the vibration and stability of a non-uniform Timoshenko beam subjected to a tangential follower force distributed over the center line by use of the transfer matrix approach. For this purpose, the governing equations of a beam are written in a coupled set of first-order differential equations by using the transfer matrix of the beam. Once the matrix has been determined by numerical integration of the equations, the eigenvalues of vibration and the critical flutter loads are obtained. The method is applied to beams with linearly, parabolically and exponentially varying depths, subjected to a concentrated, uniformly distributed or linearly distributed follower force, and the natural frequencies and flutter loads are calculated numerically, from which the effects of the varying cross-section, slenderness ration, follower force and the stiffness of the supports on them are studied.     

Vibration and stability of elasticity supported circular plates under conservative and non-conservative loads

This paper deals with the vibration and stability of circular plates elastically restrained against translation and rotation along concentric intermediate circles as well as at the outer edge. The plate is subjected to a horizontal or tangential radial load at the periphery. Both axisymmetric and non-axisymmetric vibrations are considered. It is assumed that the plate is thin, elastic, and isotropic. The solutions are obtained in terms of Bessel functions for both the complete and annular plate sections. The effect of the support stiffnesses on the plate natural frequency and the divergence and flutter instability loads are studied in detail.     

Scattering by cavities of arbitrary shape in an infinite plate and associated vibration problems

This paper presents a solution for the displacement of a uniform elastic thin plate with an arbitrary cavity, modelled using the biharmonic plate equation. The problem is formulated as a system of boundary integral equations by factorizing the biharmonic equation, with the unknown boundary values expanded in terms of a Fourier series. At the edge of the cavity we consider free-edge, simply-supported and clamped boundary conditions. Methods to suppress ill-conditioning which occurs at certain frequencies are discussed, and the combined boundary integral equation method is implemented to control this problem. A connection is made between the problem of an infinite plate with an arbitrary cavity and the vibration problem of an arbitrarily shaped finite plate, using the jump discontinuity present in single-layer distributions at the boundary. The first few frequencies and modes of displacement are computed for circular and elliptic cavities, which provide a check on our numerics, and results for the displacement of an infinite plate are given for four specific cavity geometries and various boundary conditions.     

Vibration analysis of a circular arch with variable cross-section using differential transformation and generalized differential quadrature

Vibration analysis of circular arches is an important subject in mechanics due to its various applications. In particular, circular arches with variable cross-section have been widely used to satisfy modern architectural and structural requirements. Recently, the generalized differential quadrature method (GDQM) and differential transformation method (DTM) were proposed by Shu and Zhou, respectively. In this study, GDQM and DTM are applied to vibration analysis of circular arches with variable cross-section. The governing equation of motion is derived and the non-dimensional natural frequencies are obtained for various boundary conditions. The concepts of differential transformation and generalized differential quadrature are briefly introduced. The results obtained by these methods are compared with previously published works. GDQM and DTM showed fast convergence, accuracy and validity in solving the vibration problem for circular arches with variable cross-sections.     

Free vibration of nonlocal piezoelectric nanoplates under various boundary conditions

This paper investigates the thermo-electro-mechanical vibration of the rectangular piezoelectric nanoplate under various boundary conditions based on the nonlocal theory and the Mindlin plate theory. It is assumed that the piezoelectric nanoplate is subjected to a biaxial force, an external electric voltage and a uniform temperature rise. The Hamilton's principle is employed to derive the governing equations and boundary conditions, which are then discretized by using the differential quadrature (DQ) method to determine the natural frequencies and mode shapes. The detailed parametric study is conducted to examine the effect of the nonlocal parameter, thermo-electro-mechanical loadings, boundary conditions, aspect ratio and side-to-thickness ratio on the vibration behaviors.     

变厚度弹性圆薄板问题

一.引言 变厚度的圆薄板常常在机器零件的设计中遇到,例如蒸汽涡输机和活塞的膜片便是。 用平板小挠度理论求解变厚度平板问题并不是容易的事,因此到目前为止,这类的实际问题已解决的为数并不多。     

Stability of transverse vibration of a circular plate subjected to a periodically varying torque

The paper is concerned with transverse vibration of a circular plate subjected to periodically varying torsion. The instability regions are found and plotted for the plate under a tangent uniform load and a tangent concentrated force.     

An investigation into the vibration analysis of a plate with a surface crack of variable angular orientation

An analytical approach is presented for the forced vibration analysis of a plate containing a surface crack of variable angular orientation, based on three different boundary conditions. The method is based on classical plate theory. Firstly, the equation of motion is derived for the plate containing the angled surface crack with respect to one side of the plate and subjected to transverse harmonic excitation. The crack formulation representing the angled surface crack is based on a simplified line-spring model. Then, by employing the Berger formulation, the derived governing equation of motion of the cracked plate model is transformed into a cubic nonlinear system. The nonlinear behaviour of the cracked plate model is thus investigated from the amplitude–frequency equation by use of the multiple scales perturbation method. For both cracked square and rectangular plate models, the influence of the boundary conditions, the crack orientation angle, crack length, and location of the point load is demonstrated. It is found that the vibration characteristics and nonlinear characteristics of the plate structure can be greatly affected by the orientation of the crack in the plate. Finally the validity of the developed model is shown through comparison of the results with experimental work.     

Vibration and stability of a free circular plate subjected to non-conservative loading

This paper is concerned with the stability and vibration of a completely free circular plate subjected to a non-conservative edge loading. The eigencurves and mode shapes of the plate are obtained for various values of the non-conservativeness parameter. Numerical results are presented for the asymmetrical mode shapes of the plate. Interesting conclusions are drawn from these results some of which are verified analytically.     

Axisymmetric vibrations of linearly tapered annular plates under an in-plane force

Analysis and numerical results are presented for the axisymmetric vibrations of circular annular plates with linear variation in thickness under the action of a hydrostatic in-plane force on the basis of the classical theory of plates. The governing differential equation with variable coefficients has been solved by Chebyshev collocation technique. The effect of inplane force on the natural frequencies of vibration has been investigated for two different boundary conditions and for different radii ratio and taper constant. Transverse displacements, moments and the critical buckling loads in compression with thickness variation have also been computed for the first two modes.     

Vibrations of circular plates of variable thickness under an inplane force

The natural frequencies of a circular plate of variable thickness under the action of an inplane force are discussed on the basis of the classical theory of plates. The governing differential equation of motion is solved by the method of Frobenius. Frequency parameters of clamped as well as simply supported plates in the first two modes of vibration are computed for various values of a taper parameter, β, and the inplane force, both for linear and parabolic variations of thickness.     

Free vibration analysis of the completely free rectangular plate by the method of superposition

While the subject of free vibration analysis of the completely free rectangular plate has a history which goes back nearly two centuries it remains a fact that most theoretical solutions to this classical problem are considered to be at best approximate in nature. This is because of the difficulties which have been encountered in trying to obtain solutions which satisfy the free edge conditions as well as the governing differential equation. In a new approach to this problem, by using the method of superposition, it is shown that solutions which satisfy identically the differential equation and which satisfy the boundary conditions with any desired degree of accuracy are obtained. Eigenvalues of four digit accuracy are provided for a wide range of plate aspect ratios and modal shapes. Exact delineation is made between the three families of modes which are characteristic of this plate vibration problem. Accurate modal shapes are provided for the response of completely free square plates.