Analytical dynamic displacement response of rigid pavements to moving concentrated and line loads

This paper presents the formulation of a plate of infinite dimensions (without boundary conditions) on an elastic foundation, subjected to a moving concentrated and line load of constant amplitude and speed, using a triple Fourier transform. The solution is carried out integration by residues. A closed-form solution of displacement field has been obtained for a moving load with subsonic, transonic and supersonic speeds. It is found that the maximum response of the slab occurs beneath the moving load and travels with the load at the same speed. It is also shown that a critical speed exists. If the moving load travels at critical speed, slab displacement becomes infinite in amplitude.     

弹性基础无限大板对移动荷载的响应

用积分变换方法研究了无限大板在移运的任意点源荷载作用下的一般解．发现对于运动的平稳随机荷载，板的动力响应为非平稳随机过程     

Rigid Plastic Plates under a Concentrated Moving Load∗

A rigid-plastic rectangular plate subjected to a moving transverse load is analyzed. The load is too large for the plate to be supported under static conditions. The present study indicates that there is a critical value for the moving load, above which the crossing cannot be made, irrespective of its speed. The relation between the moving load and its speed is given for values less than ciritical, as well as the distribution of displacement for a simply supported rectangular plate.     

Effect of decoupled fluid loading on nonlinear vibration of a flat plate

Numerical simulations of nonlinear responses of a flat plate subject to decoupled fluid loading are carried out. Under clamped boundary conditions and subject to forced vibration at its natural frequency corresponding to the (5,1) mode, the various response modes of the plate are determined. It is found that increasing the excitation amplitude, the response changed from periodic to chaotic. In addition, the fluid-wall shear stresses are found to change the response from linear to nonlinear and vice versa depending on their magnitudes. When a static pressure load is combined with fluid-wall shear stresses and low excitation amplitude, the resulting response was chaotic.     

Radiation effects on the thermal boundary layer flow over a moving plate with convective boundary condition

The steady laminar boundary layer flow over a moving plate in a moving fluid with convective surface boundary condition and in the presence of thermal radiation is investigated in this paper. Under certain conditions, the present problem reduces to the classical Blasius and Sakiadis problems. The effects of radiation and convective parameters on the thermal field are thoroughly examined and discussed. Dual solutions are found to exist when the plate and the fluid move in the opposite directions.     

静载荷对夹层圆板超谐波共振的影响

研究静载荷作用下夹层圆板的超谐波共振问题.基于Hoff型夹层板理论,给出了静载荷作用下夹层圆板的非线性动力学方程.应用Galerkin法推导了静载荷作用下夹层圆板的轴对称非线性振动方程.运用多尺度法分别对系统的三次超谐波问题和二次超谐波问题进行了求解,并依据Lyapunov稳定性理论得到了系统稳态运动的稳定性判据.通过...     

Finite-Element Dynamic Analysis of a Rotating Shaft with or without Nonlinear Boundary Conditions Subject to a Moving Load

A C ⁰ continuity isoparametricfinite-element formulation is presented for the dynamic analysis of arotating or nonrotating beam with or without nonlinear boundaryconditions subject to a moving load. The nonlinear end conditions arisefrom nonlinear rolling bearings (both the nonlinear stiffness andclearance(s) are accounted for) supporting a rotating shaft. The shaftfinite-element model includes shear deformation, rotary inertia, elasticbending, and gyroscopic effect. Lagrange's equations are employed toderive system equations of motion which, in turn, are decoupled usingmodal analysis expressed in the normal coordinate representation. Theanalyses are implemented in the finite-element program DAMRO 1.Dynamic deflections under the moving load of rotating and nonrotatingsimply supported shafts are compared with those obtained using exactsolutions and other published methods and a typical coincidence isobtained. Samples of the results, in both the time and frequencydomains, of a rotating shaft incorporating ball bearings are presentedfor different values of the bearing clearance. And the results show thatsystems incorporating ball bearings with tight (zero) clearance have thesmallest amplitude-smoothest profile dynamic deflections. Moreover, fora system with bearing clearance, the vibration spectra of the shaftresponse under a moving load show modulation of the system naturalfrequencies by a combination of shaft rotational and bearing cagefrequencies. However, for a simply supported rotating shaft, the firstnatural frequency in bending dominates the response spectrum. The paperpresents the first finite-element formulation for the dynamic analysisof a rotating shaft with or without nonlinear boundary conditions underthe action of a moving load.     

轴向运动导电薄板磁弹性耦合动力学理论模型

针对磁场环境中轴向运动导电薄板的动力学理论建模问题进行研究,得到较为完备的磁弹性耦合振动基本方程及相应的补充关系式。在考虑几何非线性效应下,给出薄板运动的动能、应变能以及外力虚功的表达式。应用哈密顿变分原理,推得磁场中轴向运动薄板的非线性磁弹性耦合振动方程,并得到力和位移满足的边界条件。基于麦克斯威尔电磁场方程,并考虑相应的电磁本构关系和电磁边界条件,推得任意磁场环境中轴向运动导电薄板满足的电动力学方程和所受电磁力表达式。分别针对纵向磁场环境、横向磁场环境、条形板等具体情形,给出了振动方程、电动力学方程和电磁力的简化形式。所得结果,可为此类问题的进一步求解和分析提供理论参考。     

Reseach on stability of laminated composite plate under nonlinear aerodynamic load

In this paper, we propose an finite element approach based on classical plate theory to investigate the dynamic stability of a layered composite plate subject to nonlinear aerodynamic load. This study considers the influence of temperature, nonlinear geometry, and nonlinear aerodynamic load on composite plate structures simultaneously. Specifically, the present work conduct comparison the results of the critical pressure value between the nonlinear aerodynamic load and the linear aerodynamic load, thereby pointing out some necessary cases which must consider the nonlinearity of aerodynamic load for calculating the aerospace structures. We determine the critical pressure value and vibrational amplitude response of the plate by means of calculation. The outcomes of our calculations can be useful in designing and repairing body shells and wings of aircraft equipment.     

Transient responses of beams and plates subject to travelling load. Miscellaneous results

The steady-state response of a free and infinite Timoshenko beam is specified analytically in terms of non-dimensional displacements and stresses. The beam is supposed loaded by a travelling concentrated force or a moving step load. By a validated explicit numerical calculation, it is shown how a load travelling on a beam at constant velocity, from defined time and abscissa, generates a response which evolves towards the steady-state solution for a part, and towards a quantified transient solution for another part. Asymptotic values are given for the transient displacements and stresses according to the time and the speed of the loading. The solution is also found for a plate subject to a pressure, which spreads respecting the cylindrical symmetry. It is possible to identify in the response a part which follows the pressure front, and which is comparable with the steady-state response of a beam, and another transient part, which generates displacements and stresses with a much less oscillating character. An asymptotic solution is also presented for the plate.The whole series of the results makes it possible to better understand qualitatively the beginning of the transient response of a beam or of a plate to a moving load, and also makes it possible to estimate the stresses and displacements without needing specialised numerical codes.     

Application of active piezoelectric patches in controlling the dynamic response of a thin rectangular plate under a moving mass

The governing differential equation of motion for an undamped thin rectangular plate with a number of bonded piezoelectric patches on its surface and arbitrary boundary conditions is derived using Hamilton’s principle. A moving mass traveling on an arbitrary trajectory acts as an external excitation for the system. The effect of the moving mass inertia is considered using all the out-of-plane translational acceleration components. The method of eigenfunction expansion is used to transform the equation of motion into a number of coupled ordinary differential equations. A classical closed-loop optimal control algorithm is employed to suppress the dynamic response of the system, determining the required voltage of each piezoactuator at any time interval. In a numerical example for a simply supported square plate under two different loading paths, the effect of the mass velocity and mass weight of the moving load on the dynamic behavior of the uncontrolled system is investigated. The results show that, depending on the path of the moving mass, the inertia effect is very important, causing different behaviors of the system. In addition, the number of vibrational modes involved in determining the dynamic response of the system is crucial. The inertia effect is more important for an orbiting mass loading case compared to the case in which the moving mass is traversing the plate on a straight line. A number of equally spaced piezo patches are used on the lower surface of the plate to control the displacement of the center point of the plate. The implemented control mechanism proves to be very efficient in suppressing the near resonant dynamic response of the system, requiring fairly low levels of voltage for each patch. Increasing the area of the employed piezo patches would reduce the required maximum voltage for controlling the response of the system.     

在水槽中的加速平板问题

本文利用Lagrange方法分析在水槽中平板水平地向静止流体运动的问题,非线性自由面条件能精确满足,所得的结果大致与[2]符合。     

Three dimensional solution of thick rectangular simply supported plates under a moving load

Dynamic behavior of continuous systems such as beams and plates, under a moving load is an important engineering subject. In this paper, 3D elasticity equations are solved by use of the displacement potential functions and the exact solution of a simply supported thick rectangular plate under moving load is presented. For this purpose, the governing equations in terms of displacements, Navier’s equations, are converted to two linear partial differential equations of forth and second order using displacement potential functions. Then the governing equations in terms of the potential functions are solved using the separation of variables and Laplace integral transform, satisfying exact initial and boundary conditions. In order to validate the present approach, the obtained results of this study are compared with the results of the classical theory of plates for thin and existing solutions for moderately thick plates. Also, it is observed that the speed of a moving load has an important effect on the dynamic response of plate.     

Stochastic nonlinear bending response of laminated composite plates with system randomness under lateral pressure and thermal loading

In this paper, the effect of sensitivity of randomness in system parameters on the nonlinear transverse central deflection response of laminated composite plates subjected to transverse uniform lateral pressure and thermal loading is examined. System parameters such as the lamina material properties, expansion of thermal coefficients, lamina plate thickness and lateral load are modelled as basic random variables. A higher order shear deformation theory in the von-Karman sense is used to model the system behavior of the laminated plate. A direct iterative-based C ⁰ nonlinear finite element method in conjunction with the first-order perturbation technique developed by the authors is extended for thermal problem to obtain the second-order response statistics, i.e., mean and variance of the nonlinear transverse deflection of the plate. Typical numerical results of composite plates with temperature independent and dependent material properties subjected to uniform temperature and combination of uniform and transverse temperature are obtained for various combinations of geometric parameters, uniform lateral pressures, staking sequences and boundary conditions. The results have been compared with those available in the literature and an independent Monte Carlo simulation.     

非齐次边界条件下轴向运动梁的非线性振动

轴向运动系统的横向非线性振动一直是国内外研究的热点课题之一.目前相关研究大都是针对齐次边界条件的.但是在工程实际中,非齐次边界条件更为常见,而针对非齐次边界条件的研究相对较少.为深入研究非齐次边界条件对轴向运动系统横向非线性振动的影响,本文以轴向变速运动黏弹性Euler梁为例,引入由黏弹性引起的非齐次边界条件,同时还引入由轴向加速度引起的径向变化张力,建立梁横向振动的积分-偏微分型运动方程,并导出了相应的非齐次边界条件.采用直接多尺度法分析了梁的次谐波参数共振.由可解性条件得到了梁的稳态响应,并根据Routh-Hurvitz判据确定了系统稳态响应的稳定性.通过数值例子讨论了黏弹性系数,轴向运动速度,轴向速度脉动幅值和非线性系数对幅频响应的影响,并详细对比分析了非齐次边界条件和齐次边界条件对幅频响应的影响.结果表明:随着黏弹性系数的增大,非齐次边界条件下的零解失稳区域和稳态响应幅值比齐次边界条件下的失稳区域和幅值大,非齐次边界条件对高阶次谐波参数共振的影响更加显著.最后,引入微分求积法来验证直接多尺度法的近似解结果.      

The bifurcation analysis on the circular functionally graded plate with combination resonances

The bifurcation and chaos of a clamped circular functionally graded plate is investigated. Considered the geometrically nonlinear relations and the temperature-dependent properties of the materials, the nonlinear partial differential equations of FGM plate subjected to transverse harmonic excitation and thermal load are derived. The Duffing nonlinear forced vibration equation is deduced by using Galerkin method and a multiscale method is used to obtain the bifurcation equation. According to singularity theory, the universal unfolding problem of the bifurcation equation is studied and the bifurcation diagrams are plotted under some conditions for unfolding parameters. Numerical simulation of the dynamic bifurcations of the FGM plate is carried out. The influence of the period doubling bifurcation and chaotic motion with the change of an external excitation are discussed.     

An investigation of the static and dynamic behavior of electrically actuated rectangular microplates

We present an investigation of the static and dynamic behavior of the nonlinear von-Karman plates when actuated by the nonlinear electrostatic forces. The investigation is based on a reduced order model developed using the Galerkin method, which rely on modeshapes and in-plane shape functions extracted using a finite element method. In this study, a fully clamped microplate is considered. We investigate the static behavior and the effect of different non-dimensional design parameters. The static results are validated by comparison with the results calculated by a finite element model. The forced-vibration response of the plate is then investigated when the plate is excited by a harmonic AC load superimposed to a DC load. The dynamic behavior is examined near the primary and secondary (superharmonic and subharmonic) resonances. The microplate shows a strong hardening behavior due to the cubic nonlinearity of mid-plane stretching. However, the behavior switches to softening as the DC load is increased. Finally, near-square plates are studied to understand the effect of geometric imperfections of microplates.     

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.     

薄板几何非线性弯曲分析的深度能量法

发展了一种增量形式的深度能量法求解薄板几何非线性弯曲问题。根据最小势能原理和Von-Karman非线性理论,构建以薄板势能为驱动的增量式深度神经网络模型。首先用网格离散薄板求解域,通过Python读取网格数据计算Hammer积分点,并以此作为训练集代入网络模型预测板的弯曲位移,再将荷载分成一系列的荷载增量,每个增量步中计算薄板势能作为神经网络的损失函数,以最小化势能为目标,结合Adam优化算法更新网络模型参数,待势能取驻值后再继续下一个荷载步的计算。本文求解了不同形状、不同边界条件下薄板的几何非线性弯曲问题,并将计算结果与文献解或有限元Abaqus解进行对比,研究表明,本文方法在求解薄板的几何非线性弯曲问题上具备有效性和准确性,且增量式的神经网络模型能够减小计算内存,有效提高计算效率和模型的稳定性。     

Three Dimensional Vibration Transmission of a Plate on a Layer of Porous Medium Due to a Moving Load

Three-dimensional (3D) transmission of vibration in an infinite elastic thin plate on a layer of poroviscoelastic medium, due to a harmonic, rectangular moving load, is investigated theoretically based on Biot’s theory. The material of the medium is idealized as a uniform, fully saturated poroviscoelastic layer on bedrock. By introducing four scalar potential functions and Helmholtz decomposition theorem, analytical solutions of stress, displacement, and pore pressure with and without thin plate are derived using Fourier transform technique. Numerical results are obtained with the help of inverse Fourier transform and are used to analyze the influence of load velocity, porosity, permeability, relative stiffness of plate versus ground, and the thickness of plate on the vibration. Furthermore, the results are compared with the available dynamic response results of a non-moving load on a layer of viscoelastic material.