A Refined Shear Deformation Theory of Bending of Isotropic Plates*

ABSTRACT

A nonlinear, in-plane displacement assumption is proposed, based on an undetermined variation df/dz of transverse shear strains through the plate thickness. A second-order ordinary differential equation for f(z) and two surface conditions, as well as a set of eighth-order partial differential equations and four associated boundary conditions, are derived from the principle of minimum potential energy. Coupling exists between the partial and ordinary differential equations. In the homogeneous solutions for the former, in addition to an interior solution contribution, there exist two edge-zone solution contributions, one of which induces self-equilibrated (in the thickness direction) boundary stresses. Three examples are calculated using the present theory. The last gives the stress couple and maximum-stress concentration factors at the free edge of a circular hole in a large bent plate. Numerical results for the examples are compared with those given by three-dimensional elasticity theory and several two-dimensional theories. It is found that the present theory can accurately predict nonlinear variations of in-plane stresses through the thickness of a plate.     

A Fully Symbolic Model of Multibody Systems Containing Flexible Plates

The modelling of flexible elements in mechanical systems has been investigated via several methods issuing from both the field of multi-body dynamics and the area of structural mechanics and vibration theory. As regards the multibody approach, recursive formulations in relative coordinates are quite suitable and efficient for a large variety of applications. Such a formalism is developed here for a general multibody system containing flexible plates and in such a way that its full symbolic generation is possible within the ROBOTRAN program [1].     

一种新的岩石非线性黏弹塑性蠕变模型研究

为了克服传统元件组合模型不能描述岩石蠕变过程中非线性特征的缺陷,首先根据加速蠕变阶段的应变和应变率随蠕变时间急剧增大的特点,建立黏塑性应变与蠕变时间的指数函数关系并提出非线性黏塑性体.将该非线性黏塑性体与广义Burgers蠕变模型串联,建立可以描述岩石全蠕变过程的非线性黏弹塑性蠕变模型,根据叠加原理得到一维应力状态下的轴向蠕变方程.然后基于塑性力学理论指出岩石三维蠕变本构方程建立过程中的不足之处,并给出非线性黏弹塑性蠕变模型合理的三维蠕变方程.最后采用不同应力水平下砂岩轴向蠕变试验对模型合理性进行验证,结果表明:拟合曲线与试验曲线吻合度较高,所建蠕变模型能够很好地描述砂岩在不同应力水平下的蠕变变形规律,尤其对加速蠕变阶段的非线性特征描述效果很好,验证了模型的合理性.     

Deformation of Piecewise-Homogeneous Plates

Forced planar vibrations of a rectangular plate consisting of two rectangular subdomains are studied. The mechanical properties of the subdomains are essentially different. The boundary conditions for the subdomains may also be different. It is shown that the more the difference between the properties of the subdomains, the closer to a certain limit is the first resonance frequency of the plate. It is established that under certain boundary conditions, the displacements in each of the subdomains may have their own specific features     

A Nonlinear Vibration Absorber for Flexible Structures

An approach for implementing an active nonlinear vibration absorber for flexible structures is presented. The technique exploits the saturation phenomenon exhibited by multidegree-of-freedom systems with quadratic nonlinearities possessing two-to-one autoparametric resonances. The strategy consists of introducing second-order controllers and coupling each of them with the plant through a sensor and an actuator, where both the feedback and control signals are quadratic. Once the structure is forced near its resonances, the oscillatory response is suppressed through the saturation phenomenon. We present theoretical and experimental results of the application of the proposed vibration absorber. The structure consists of a cantilever beam, the feedback signal is generated by a strain gage, and the actuation is achieved through piezoceramic patches. The equations of motion are developed and analyzed through perturbation techniques and numerical simulation. Then, the strategy is tested by assembling the controllers in electronic components and suppressing the vibrations of the first and second modes of two beams.     

Dynamic Plastic Deformation of Curvilinear Plates

A complete analysis is made of the dynamic plastic deformation of hinged or fixed plates with an arbitrary smooth contour. It is shown that the deformation of plates can significantly be dependent on the level of the dynamic load. Cases with low, medium, high, and ultrahigh levels of dynamic loading are distinguished and, for each of them, systems of resolving equations are derived and procedures for their solution are described. Some general properties of the solutions are revealed. An example is considered in which analytical formulas for the residual deflection are derived and limit passages to classical dynamic problems for a circular plate and a bar are analyzed     

Complex Parametric Vibrations of Flexible Rectangular Plates

In this paper we consider parametric oscillations of flexible plates within the model of von Kármán equations. First we propose the general iterational method to find solutions to even more general problem governed by the von Kármán–Vlasov–Mushtari equations. In the language of physics the found solutions define stress–strain state of flexible shallow shell with a bounded convex space R ² and with sufficiently smooth boundary . The new variational formulation of the problem has been proposed and his validity and application has been discussed using precise mathematical treatment. Then, using the earlier introduced theoretical results, an effective algorithm has been applied to convert problem of finding solutions to hybrid type partial differential equations of von Kármán form to that of the ordinary differential (ODEs) and algebraic (AEs) equations. Mechanisms of transition to chaos of deterministic systems with infinite number of degrees of freedom are presented. Comparison of mechanisms of transition to chaos with known ones is performed. The following cases of longitudinal loads of different sign are investigated: parametric load acting along X direction only, and parametric load acting in both directions X and Y with the same amplitude and frequency.     

基于高阶变形理论的夹层板的弯曲与振动

考虑面板和夹芯的面内刚度和横向剪切刚度以及抗弯刚度,考虑了高阶剪切变形,根据横向剪应变分布情况给出横向剪切转角的位移函数,基于哈密尔顿原理,推导了基于高阶变形理论、适用于软、硬夹芯情况夹层板的基本方程.作为算例,以四边简支条件下的夹层板的弯曲与振动,在不同的面板与芯层的弹性模量比和厚度比下进行了计算,并与Reissner理论、Hoff理论以及邓宗白基于Reissner理论的修正模型的计算结果进行了对比.与前述理论与方法相比,论文方法考虑因素更为全面,对夹层板的适用范围更为广泛,计算结果更为精确.针对Nastran软件计算夹层板的振动问题,对其适用范围作了简要分析.     

Wavelet-Based Analysis of Parametric Vibrations of Flexible Plates

Wavelet-based methods are reviewed, and their advantages in comparison to standard approaches are outlined. Then, a wavelet analysis is performed to investigate parametric vibrations of flexible plates under a sinusoidal load. In particular, a scenario leading from regular motion to chaos is analyzed     

Optimization of Dynamic Plastic Deformation of Plates with a Complex Contour

A method for studying dynamic deformation of ideal rigid–plastic plates with a complex contour on a viscoelastic foundation is proposed. The method allows one to optimize the process of pulsed forming. The optimization parameters are the amplitude of the pulsed load, viscoelastic damping coefficients of the foundation, the surface density of the plate material, and the shape and supporting conditions of the edges. Numerical examples of simply– and doubly–connected plates are given. It is shown that different combinations of the control parameters of the process can ensure the same final shape of the plate formed.     

考虑非线性变形的航天器柔性附件建模方法

针对航天器的柔性附件,将其简化为常见的大范围运动空间柔性梁结构。根据柔性体的非线性变形原理,考虑了弯曲和扭转的非线性因素的影响,在柔性梁的三个变形方向上均考虑了变形的二次耦合项。利用有限元方法进行离散并用Lagrange方程建立了非线性变形模式下的动力学方程,包含了较为完全的刚度矩阵和各种耦合项。对一大范围运动的空间梁进行了仿真计算,表明在运动速度较大并且基座具有大范围运动时,一次耦合模型与本文模型有一定差异,且初始变形对两种模型也会产生影响。     

Inelastic Deformation of Flexible Spherical Shells with Two Circular Openings

Elastoplastic analysis of thin-walled spherical shells with two identical circular openings is carried out with allowance for finite deflections. The shells are made of an isotropic homogeneous material and subjected to internal pressure of known intensity. The distributions of stresses (strains or displacements) along the contours of the openings and in the zone of their concentration are studied by solving doubly nonlinear boundary-value problems. The solution obtained is compared with the solutions that account for only physical nonlinearity (plastic deformations) and only geometrical nonlinearity (finite deflections) and with a numerical solution of the linearly elastic problem. The stress–strain state near the two openings is analyzed depending on the distance between the openings and the nonlinear factors accounted for     

带剪切型压电激励器的板的高阶剪切变形理论

利用压电材料固有的正，逆压电效应可以对结构变形和振动进行控制。与外加电场与极化方向平行于板厚度的压电材料的拉伸作动机制相比，外加电场与极化方向垂直的压电材料的剪切作动机制可以在作动器内产生较小的应力，从而降低作动器边界产生分层破坏的危险。本文对于压电材料的剪切作动机制进行研究，应用三阶剪切变形理论建立带剪切型压电激励器的智能层合板模型。采用哈密顿原理导出带剪切型压电激励器的层合板的控制方程。采用空间法得到了各种边界条件组合条件下板的解析解。数值算例对一三层板采用高阶和一阶剪切变形理论进行计算，结果表明两种理论所得的变形曲线很相似。但对于厚度剪切型激励器而言，由于激励器是引起板的剪切变形，而高阶剪切变形理论比一阶剪切变形理论能更好地反映结构的剪切应变能，因此高阶剪切变形理论可以提供板变形的更为精确的解。因此，对于厚度剪切型激励器，剪切变形理论的选取对于板变形结果的好坏有重要的作用。     

Refined Stress Analysis of Flexible Multilayered Shells of Revolution with Orthotropic Layers of Variable Thickness

A refined theory of flexible layered shells with orthotropic layers of variable thickness is considered. The theory assumes that each layer has a local rotation angle due to lateral shear. This makes it possible to derive equations whose order does not depend on the number of layers. The basic equations and calculation results are presented for a three-layer orthotropic toroidal shell under axisymmetric loading     

横观各向同性热弹性梁的精化理论

本文从横观各向同性梁的二维问题出发,研究了横观各向同性热弹性梁的精化理论。首先,在不作任何预先假设的条件下,利用横观各向同性热弹性理论和Lur’e算子函数,获得了由梁中线上的物理量表示的位移场和应力场。对热弹性梁上下表面承受非齐次边界条件的情况,推导出梁的近似控制微分方程。再舍去温度项,则横观各向同性热弹性梁的精化理论退化为横观各向同性梁的精化理论。     

折叠式柔性结构振动主动控制的研究

建立了具有刚性运动基的折叠式柔性结构振动主动控制实验系统，采用带实时预测误差修正的预测控制算法进行了这类结构振动主动控制的实验研究。实验结果表明，在实验建模的基础上设计预测控制器对折叠式柔性结构进行振动主动控制，能够有效地抑制柔性结构的振动和提高刚性基姿态定向精度。