Non-linear axisymmetric vibration of orthotropic thin circular plates on elastic foundations

This study deals with the large amplitude axisymmetric free vibrations of cylindrically orthotropic thin circular plates resting on elastic foundations. Geometric non-linearity due to moderately large deflections has been included. Movable and immovable simply supported plates and immovable clamped plates resting on Winkler, Pasternak and non-linear Winkler foundations have been considered. The von Kármán type governing equations have been employed. Harmonic vibrations are assumed and the time t is eliminated by the Kantorovich averaging method. An orthogonal point collocation method is used for spatial discretization. Numerical results are presented for the linear natural frequency of the first axisymmetric mode and for the ratio of the non-linear period to the linear period of natural vibration. The effects of foundation parameters, the orthotropic parameter and the edge conditions on the non-linear vibration behaviour have been investigated.     

Non-linear dynamic behaviour of plates on elastic foundations

The large amplitude transverse vibration characteristics of a simply-supported isotropic equilateral triangular plate on elastic foundation have been investigated by following Berger's well-known approximate method in conjunction with a special type of co-ordinates known as tri-linear co-ordinates. A second order non-linear differential equation for the unknown time function has been obtained and solved, as usual, in terms of Jacobian elliptic functions. Results obtained from numerical calculations are presented graphically.     

Free vibration of circular footing on elastic foundations

The free vibration of a circular plate on an elastic foundation is analyzed by using Vlasov's two-parameter model. The natural frequencies of the system under axisymmetric conditions are determined. In the region of the plate, the general solution is represented by Bessel functions. Modified wave equations are used to predict the harmonic motion of the elastic foundation. Since the region outside of the plate is infinite, a reflected wave is not produced, thereby eliminating the need to consider a wave moving toward the plate. Finally, the effects of various parameters of the plate and the foundation on the natural frequencies system are discussed.     

弹性薄圆环的超声频径向振动及其等效电路研究

对各向同性弹性薄圆环的径向振动进行了研究,得出了其等效电路。在此基础上,推出了其径向振动的频率方程以及振子的共振频率的表达式,利用数值法得出了不同材料弹性薄圆环径向振动频率方程的一系列根。经过拟合,得出了频率方程的根与振子材料的泊松系数的拟合关系曲线,从而大大方便了径向振动薄圆环的工程设计及计算。     

Natural vibration of circular and annular thin plates by Hamiltonian approach

The present paper deals with the natural vibration of thin circular and annular plates using Hamiltonian approach. It is based on the conservation principle of mixed energy and is constructed in a new symplectic space. A set of Hamiltonian dual equations with derivatives with respect to the radial coordinate on one side of the equations and to the angular coordinate on the other side are obtained by using the variational principle of mixed energy. The separation of variables is employed to solve Hamiltonian dual equations of eigenvalue problem. Analytical frequency equations are obtained based on different cases of boundary conditions. The natural frequencies are the roots of the frequency equations and corresponding mode functions are in terms of the dual variables q₁(r, θ). Three basic edge-constraint cases for circular plates and nine edge-constraint cases for annular plates are calculated and the results are compared well with existing ones.     

Thermal postbuckling and vibration analyses of functionally graded plates

Thermal postbuckling and vibration behaviors of the functionally graded (FG) plate are investigated. The material properties of the FG plate are assumed to vary continuously through the thickness of the plate and as temperature with the nonlinearity. The nonlinear finite element equations based on the first-order shear deformation plate theory are formulated for the FG plate. The von Karman nonlinear strain–displacement relationship is used to account for the large deflection of the plate. The incremental form considering the initial displacement and initial stress is adopted for the nonlinear temperature-dependent material properties of the functionally graded material. The numerical result shows the characteristics of the thermal postbuckling and vibration of the FG plate in the pre- and post-buckled regions.     

Free vibration of polar orthotropic circular plates with elastic growth stresses and drying stresses

This study was performed to analyze effect of transverse growth stresses and drying stresses on natural frequencies of transverse free vibration of thin tree disk with free boundary condition. Using Rayleigh-Ritz method and finite element method, explicit closed form expression was derived for homogeneous orthotropic circular plates with internal stresses, which enables to predict both the effects of growth stresses, drying stresses, and combined growth and drying stresses on the natural frequency changes. This model can be used for predicting growth stresses from known natural frequencies. The same approach can also be undertaken to validate models visco-elastic drying stresses.     

Fundamental frequency of vibration of thin elastic plates elastically supported along internal segments

The title problem is approximately solved in a unified manner in the case of rectangular and circular plates elastically restrained against rotation along the boundary.Polynomial coordinate functions and a variational approach are used in order to obtain an approximate frequency equation. Judging from the excellent accuracy attained using the present methodology in the case where no internal supports are present, it is reasonable to expect that the results presented in this study will be sufficiently accurate for engineering purposes in the situations considered.     

Experiments on elastic cloaking in thin plates

Following a theoretical proposal [M. Farhat et al., Phys. Rev. Lett. 103, 024301 (2009)], we design, fabricate, and characterize a cloaking structure for elastic waves in 1?mm thin structured polymer plates. The cloak consists of 20 concentric rings of 16 different metamaterials, each being a tailored composite of polyvinyl chloride and polydimethylsiloxane. By using stroboscopic imaging with a camera from the direction normal to the plate, we record movies of the elastic waves for monochromatic plane-wave excitation. We observe good cloaking behavior for carrier frequencies in the range from 200 to 400?Hz (one octave), in good agreement with a complete continuum-mechanics numerical treatment. This system is thus ideally suited for demonstration experiments conveying the ideas of transformation optics.     

黏性介质中圆截面弹性细杆的平面振动

讨论圆截面弹性细杆在黏性介质中的平面振动. 基于Kirchhoff理论，以杆中心线的Frenet坐标系为参考系，建立其动力学方程，杆中心线为任意平面曲线时，其扭转振动与弯曲振动解耦. 讨论两端固定条件下任意形状杆的平面扭转振动，以及无扭转的轴向受压直杆和圆环杆的平面弯曲振动，导出其自由振动频率和阻尼系数. 证明空间域内压杆的Lyapunov稳定性和欧拉稳定性条件为时域内渐近稳定性的充分必要条件，或无阻尼压杆的稳定性必要条件. 圆环杆平衡恒满足渐近稳定性条件. 关键词： 弹性细杆 黏性介质 扭转振动 弯曲振动     

Non-linear oscillations of elastic orthotropic annular plates of variable thickness

A computational analysis of the non-linear oscillations of elastic orthotropic annular plates of variable thickness is presented. The non-linear boundary value problem is converted into a corresponding eigenvalue problem by using a Kantorovich time-averaging method. Then, by a Newton-Raphson iteration scheme in conjunction with the concept of analytical continuation, the solution to the non-linear oscillations of elastic orthotropic annular plates of variable thickness are obtained.     

Non-linear vibrations of elastic plates subjected to transient pressure loading

The large amplitude vibrations of elastic plates of arbitrary plan form subjected to transient pressure loading are analyzed in a relatively simple fashion by using the Berger method in conjunction with the iso-amplitude contour lines method. The analysis provides for both clamped and simply supported edge conditions. By way of illustration, the large amplitude response of elliptical plates under various types of dynamic loading, namely a step function, a sinusoidal pulse and an N-wave, is investigated and the results are presented graphically. Some comparison is made with previously obtained results for circular plates, as available in the literature.