Bending of uniformly loaded rectangular plates with two adjacent edges clamped, one edge simply supported and the other edge free

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Multi-mode parametric excitation of a simply supported plate under time-varying and non-uniform edge loading

In this study, multi-mode parametric excitation of a simply supported plate under time-varying and non-uniform edge loading is modeled and the solution is found. Equations for multi-mode parametric excitation of a simply supported plate are derived using stress distributions within the plate as well as on the edges, considering both the effects of non-uniform edge loading and the non-linearity caused by the large deflection. The multi-mode equations are coupled by first-order linear terms, even in the case of simply supported boundary conditions, due to the non-uniform edge loading. The perturbation solutions of two-mode parametric excitation are examined by the method of multiple scales. For the edge loading, which consists of a uniform term as well as a non-uniform one, equations could be coupled or de-coupled by parametric excitation terms, and the numbers and values of the resonance frequencies of the parametric excitations could also differ, depending on whether the non-uniform term of the edge loading is time-varying or not. In addition to the resonant frequencies of the case when only the uniform term of the edge loading is time-varying, there are additional combination resonances at the vicinity of the sum of two natural frequencies of each mode when the non-uniform term of the non-uniform edge loading is time-varying.     

四边简支正交各向异性蜂窝型夹层板固有频率计算

该文以四边简支的方形蜂窝矩形夹层板为例,在经典夹层板理论的基础上,运用离散结构形式的运动控制方程和线性微分算子的可交换性,给出了一种把具有蜂窝型夹心的夹层板的包含三个广义位移的控制方程组化为,仅包含一个广义位移函数的单一方程的简单方法,并给出了四边简支蜂窝型夹层板的固有频率的精确解。研究结果对蜂窝夹层板的结构设计和工程应用具有指导意义。     

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.     

简支梁大挠度弹性后屈曲载荷与变形的优化算法

针对简支梁结构大挠度后屈曲载荷与变形的计算问题,本文提出了一种直接求解其后屈曲载荷和变形的优化算法。在简支梁处于大挠度屈曲平衡状态下,将梁结构划分为有限子段,以待求后屈曲载荷为设计变量,根据起点的边界条件和每个子段满足的弯矩变形公式,累积计算出其他各个节点的坐标,以得到的终点坐标满足的边界条件构建目标函数模型。在此基础上,通过MATLAB编制优化程序分析了两个典型算例,并将理论结果与相关软件的计算结果进行对比,从而证明了本文算法的正确性。本文算法求解过程简单、快速,具有一定的实用性,为变截面结构大挠度弹性屈曲稳定性问题的研究提供了参考。     

THE BENDING OF THIN RECTANGULAR PLATES WITH MIXED SUPPORTED SEGMENTS OF STRAIGHT EDGES

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A further study on the non-linear dynamics of simply supported pipes conveying pulsating fluid

In this paper, the non-linear dynamics of simply supported pipes conveying pulsating fluid is further investigated, by considering the effect of motion constraints modeled as cubic springs. The partial differential equation, after transformed into a set of ordinary differential equations (ODEs) using the Galerkin method with N=2, is solved by a fourth order Runge-Kutta scheme. Attention is concentrated on the possible motions of the system with a higher mean flow velocity. Phase portraits, bifurcation diagrams and power spectrum diagrams are presented, showing some interesting and sometimes unexpected results. The analytical model is found to exhibit rich and variegated dynamical behaviors that include quasi-periodic and chaotic motions. The route to chaos is shown to be via period-doubling bifurcations. Finally, the cumulative effect of two non-linearities on the dynamics of the system is discussed.     

Double-mode modeling of chaotic and bifurcation dynamics for a simply supported rectangular plate in large deflection

This paper presents a new approach to characterize the conditions that can possibly lead to chaotic motion for a simply supported large deflection rectangular plate by utilizing the criteria of the fractal dimension and the maximum Lyapunov exponent. The governing partial differential equation of the simply supported rectangular plate is first derived and simplified to a set of two ordinary differential equations by the Galerkin method. Several different features including Fourier spectra, state-space plot, Poinca?e map and bifurcation diagram are then numerically computed by using a double-mode approach. These features are used to characterize the dynamic behavior of the plate subjected to various excitation conditions. Numerical examples are presented to verify the validity of the conditions that lead to chaotic motion and the effectiveness of the proposed modeling approach. The numerical results indicate that large deflection motion of a rectangular plate possesses many bifurcation points, two different chaotic motions and some jump phenomena under various lateral loading. The results of numerical simulation indicate that the computed bifurcation points can lead to either a transcritical bifurcation or a pitchfork bifurcation for the motion of a large deflection rectangular plate. Meanwhile, the points of pitchfork bifurcation can gradually lead to chaotic motion in some specific loading conditions. The modeling result thus obtained by using the method proposed in this paper can be employed to predict the instability induced by the dynamics of a large deflection plate.     

On new symplectic approach for exact bending solutions of moderately thick rectangular plates with two opposite edges simply supported

The symplectic geometry method is introduced for exact bending solutions of moderately thick rectangular plates with two opposite edges simply supported. The basic equations for the plates are first transferred into Hamilton canonical equations. The whole state variables are then separated. Using the method of eigenfunction expansion in the symplectic geometry, typical examples for plates with selected boundary conditions are solved and exact bending solutions obtained. Since only the basic elasticity equations of the plates are used, this method eliminates the need to pre-determine the deformation function and is hence more reasonable than conventional methods. Numerical results were presented to demonstrate the validity and accuracy of this approach as compared to those reported in other literatures.     

A new method for the non-linear deflection analysis of an infinite beam resting on a non-linear elastic foundation

The aim of this paper is to develop a new method of analyzing the non-linear deflection behavior of an infinite beam on a non-linear elastic foundation. Non-linear beam problems have traditionally been dealt with by semi-analytical approaches that involve small perturbations or by numerical methods, such as the non-linear finite element method. In this paper, in contrast, a transformed non-linear integral equation that governs non-linear beam deflection behavior is formulated to develop a new method for non-linear solutions. The proposed method requires an iteration to solve non-linear problems, but is fairly simple and straightforward to apply. It also converges quickly, whereas traditional non-linear solution procedures are generally quite complex in application. Mathematical analysis of the proposed method is performed. In addition, illustrative examples are presented to demonstrate the validity of the method developed in the present study.     

A complete symplectic eigenfunction expansion for the elastic thin plate with simply supported edges

The eigenvalue problem for the Hamiltonian operator associated with the mathematical model for the deflection of a thin elastic plate is investigated. First, the problem for a rectangular plate with simply supported edges is solved directly. Then, the completeness of the eigenfunctions is proved, thereby demonstrating the feasibility of using separation of variables to solve the problem. Finally, the general solution is obtained by using the proved expansion theorem.     

弹性约束边界圆环板面内自由振动的二维弹性解

基于二维线弹性理论,应用哈密顿原理导出弹性约束边界圆环板面内自由振动的控制微分方程。采用微分求积法(DQM)数值研究了弹性约束边界圆环板面内自由振动的频率特性。通过设置弹性刚度系数为0或∞,问题退化为四种典型边界圆环板的面内自由振动,与已有文献的计算数值结果进行比较,证实本文的分析求解方法行之有效。最后全面考虑了圆环板边界条件、几何系数及刚度系数对自振频率的影响。     

A semi-analytical approach for the non-linear large deflection analysis of laminated rectangular plates under general out-of-plane loading

A semi-analytical approach for the geometrically non-linear analysis of rectangular laminated plates with general inplane and out-of-plane boundary conditions under a general distribution of out-of-plane loads is developed. The analysis is based on the elastic thin plate theory with geometrically non-linear von Kármán strains. The solution of the non-linear partial differential equations is reduced to an iterative sequential solution of non-linear ordinary differential equations using the multi-term extended Kantorovich method. The efficiency, accuracy, and convergence of the proposed method are examined through a comparison with other semi-analytical methods and with finite element analyses. The capabilities of the approach and its applicability to the non-linear large deflection analysis of plate structures are demonstrated through various numerical examples. Emphasis is placed on combinations of lamination, boundary, and loading conditions that cannot be analyzed using alternative semi-analytical methods.     

Bending problem of rectangular plates of unidirectional nonhomogeneity and variable thickness with two opposite edges simply supported under arbitrary distributed loads

Using the step reduction method^[1,2] suggested by the first author of this paper, we investigate the problem indicated in the title and obtain the stepped approximate solutions. As an example, the case of a square plate of linearly varying thickness with four edges simply supported under linearly distributed loads is calculated. The obtained results agree well with those given in [3] and thus the exactness of the new method is verified.Projects Suppotred by the Science and Technic Fund of the National Education Committee of the People's Republic of China.Although this paper has published in Journal of Lanzhou University, Special Number of Mechanics, No. 1, (1979) (in Chinese), it republishes here in order to correct some mistakes due to calculation.     

On the effective stiffness of plates made of hyperelastic materials with initial stresses

Within the framework of the direct approach to the plate theory we consider the infinitesimal deformations of a plate made of hyperelastic materials taking into account the non-homogeneously distributed initial stresses. Here we consider the plate as a material surface with 5 degrees of freedom (3 translations and 2 rotations). Starting from the equations of the non-linear elastic body and describing the small deformations superposed on the finite deformation we present the two-dimensional constitutive equations for a plate. The influence of initial stresses in the bulk material on the plate behavior is considered.     

Perturbation analyses for the postbuckling of simply supported rectangular plates under uniaxial compression

In this paper, applving perturbation method to von Kármán nonlinear large deflection equations of plates by taking deflection as perturbation parameter, the posibuckling behavior of simply supported rectangular plates under uniaxial compresion is investigated. Two types of in-plane boundary conditions are now considered and the effects of initial imperfections are also studied. It is found that the theoretical results are in good agreement with experiments. The method suggested in this paper which has not been found in previous papers is rather simple and easy for the postbuckling analysis of rectangular plates.     

Probability density function for stochastic response of non-linear oscillation system under random excitation

Probability density function (PDF) of stochastic responses is a critical topic in uncertainty analysis. In this paper, orthogonal decomposition technique was extended to discuss non-stationary response of non-linear oscillator under random excitation. The PDF of stochastic reponses is represented by a set of standardized multivariable orthogonal polynomials. According to the Galerkin scheme, the original problem, which has to solve the Fokker-Planck-Kolmogorov (FPK) equation, was converted to a first-order linear ordinary differential equation, in terms of unknown time-dependent coefficients. Then, stationary and non-stationary PDFs of uncertainty responses were obtained. In numerical examples, first-order and second-order non-linear systems exposed to the Gaussian white noise were considered. Finally, the accuracy of the proposed method was demonstrated through appropriate comparisons to Monte-Carlo simulation and analytical results.     

AN ANALYTICAL SOLUTION OF TRANSVERSE VIBRATION OF RECTANGULAR PLATES SIMPLY SUPPORTED AT TWO OPPOSITE EDGES WITH ARBITRARY NUMBER OF ELASTIC LINE SUPPORTS IN ONE WAY

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