【作者】 杜国君；

【导师】 刘宏民；

【作者基本信息】 燕山大学 ， 工程力学， 2004， 博士

【摘要】 本论文致力于研究弹性圆板和夹层圆板的非线性自由振动。重点讨论静载荷作用下板的非线性自由振动特性，将修正迭代法应用到所讨论的问题中，得到了求解这类非线性振动问题的新的解析方法。首先，讨论了弹性圆板的非线性振动问题，给出了静载荷作用下柔韧圆板以及阶梯圆形和环形薄板非线性振动的基本方程。对于静载荷作用下柔韧圆板非线性振动问题，按假设的时间模态函数，导出了该问题的非线性耦合的代数和微分特征方程组，利用修正迭代法求出了该方程组的近似解析解，得到柔韧圆板振动的幅频—载荷特征关系及非线性振子“漂移”随载荷和振幅的变化关系，详细讨论了各种边界条件下静载荷对其振动性态的影响。按上述方法，导出了阶梯变厚度圆形和环形薄板非线性振动问题的二阶修正迭代解，并给出了数值计算结果，对振动特性与板的几何参数和振幅之间的关系进行了详细讨论。由哈密顿原理导出了夹层圆板非线性振动的基本方程，并且给出了表板很薄情况下的简化形式。利用修正迭代法求解该非线性微分方程，得到了问题的近似解析解，讨论了夹层板参数对振动特性的影响。在此基础上，对夹层圆板非线性振动问题做了进一步的研究，讨论了具有滑动固定边界条件并计及表板抗弯刚度的夹层圆板轴对称非线性自由振动问题，由于该问题的控制方程属边界层型，求解有一定的困难，本文将修正迭代法推广应用于该问题的求解，得到了很好的结果，并将所得结果和忽略表板抗弯刚度情况的结果进行了对比，分析了表板抗弯刚度对振动特性的影响。在前述工作的基础上，讨论了静载荷作用下夹层圆板的非线性振动问题。利用能量原理导出了该问题控制方程的变分形式。基于时间模态假设和变分法，将挠度和应力函数设为时间和空间函数的分离形式，时间函数取谐函数，空间函数未知。将假定的模态函数代入本问题的变分方程，导出了空间模态的控制方程和求解非线性振子“漂移”的代数方程组。按修正迭代法求出了空间模态的渐近表达式，导出了均布载荷作用下夹层圆板的幅频-载荷特征关系，给出了静载荷对非线性振频和“漂移”影响的数值计算结果。更多还原

【Abstract】 The dissertation is devoted to studying the nonlinear free vibration of elastic circular plate and circular sandwich plate. The attention is mainly focused on the nonlinear free vibration characteristics of those plates. By using modified iteration method, some new analytic methods are arrived at. Firstly, the nonlinear vibration problem of elastic circular plate is discussed, the nonlinear vibration fundamental equations of flexible circular plate, circular and annular thin plate with multistepped thickness are given under static loading. For the nonlinear vibration of flexible circular plate under static loading, following an assumed time mode approach suggested, a set of nonlinear coupling algebraic and differential eigenvalue equations are established. The approximately analytic solution is derived by use of modified iteration method. The relations of amplitude frequency-loading, and nonlinear vibration “drift” with loading and amplitude are derived, the influences of static loading on vibration characteristics are discussed under all kinds of boundary conditions. Following the method mentioned above, second-order modified iteration solutions of nonlinear vibration problem on circular and annular thin plate with multistepped thickness are obtained. Some examples are given to show the influences of geometric parameter on amplitude. By means of Hamilton principle, the fundamental equations of nonlinear vibration for circular sandwich plate are derived. In most cases, the face plates of sandwich plate are very thin, consequently, the simplified fundamental equations are given for this case. Modified iteration method is employed to solve the nonlinear differential equation, the approximately analytic solution is used to analyzed the influences of sandwich parameter on vibration characteristics. Based on the studies above, further study on nonlinear vibration of circular sandwich plate is accomplished, the nonlinear axisymmetric free vibration of circular sandwich plate considering the face plate rigidity with clamped but free to slip boundary condition are discussed in detail. Because those control equations belong to boundary layer style, the solution is more difficult, modified iteration method is generalized to solve those equations, and fine solutions are obtained. The influences of face plate bending rigidity on vibration characteristics are discussed by comparing the solutions considering the face plate <WP=6>bending rigidity with those solutions ignoring the face bending rigidity. Based on studies above, the nonlinear vibration problem of circular sandwich plate under static loading is studied, energy principle is employed to obtain variation control equation. Based on time mode hypothesis and variation method, the deflection and stress function are assumed as the separate form of time and space function, the time function employs the simple harmonic, the space function is unknown, substituting the mode function into variation equation of problem, the control equation of space mode and a set of algebraic equations used to solve nonlinear vibration “drift” are obtained, and the expression of space mode is obtained by using modified iteration method, the relation of amplitude frequency-loading of circular sandwich plate under uniformed loading is derived. The influences of static loading on nonlinear vibration frequency and “drift” are discussed.更多还原