几种典型板考虑初始荷载效应的基频近似解

基于两组板考虑初始荷载效应的动力控制微分方程:一般形式的动力控制微分方程和极坐标形式的动力控制微分方程，运用Galerkin（伽辽金)法求解得到了简支矩形板、固支矩形板、简支等边三角形板、固支椭圆形板、简支圆形板和固支圆形板6种典型板考虑初始荷载效应的自由振动基频（第一阶频率）近似解．通过与相关文献提出的有限元法计算结果对比，验证了公式的正确性．基频近似解表达式简单明了，物理意义明确，清楚地说明了初始荷载及相关因素对板自由振动基频的影响，直观地说明了板的初始荷载效应这一概念．计算分析表明:初始荷载的存在增加了板的弯曲刚度，提高了板的自振频率．这种初始荷载效应对频率的影响主要受初始荷载大小、跨厚比及边界条件等因素的影响．在计算分析和设计中应考虑并重视这种初始荷载效应对板计算分析带来的影响.

Approximate Fundamental Frequency Solutions Under Initial Load Effect for 6 Typical Plates

Based on two sets of dynamic equilibrium differential equations for plates under initial load effect, which were respectively expressed as general and polar coordinate forms to fit different boundary conditions. The approximate solutions of fundamental frequencies under initial load effect for the simply supported rectangular plate, the clamped rectangular plate, the simply supported equilateral triangular plate, the clamped elliptic plate, the clamped circular plate and the simply supported circular plate, were derived with the Galerkin method. These approximate solutions were verified with the finite element method under initial load effect, which clearly illustrated the initial load effect and corresponding factors that influence the plates’ fundamental frequencies. Initial load effect on fundamental frequencies of the above 6 typical plates was analyzed with these solutions. Due to initial load effect, bending stiffnesses of the plates increased, and their fundamental frequencies rose. The key physical factors governing the initial load effect on the plates are the initial load magnitude，the ratio of span to thickness and the boundary conditions, etc. The bigger the initial loads and the smaller the bending stiffnesses of the plates are, the higher the initial load effect on the fundamental frequencies is. This initial load effect is obvious and should not be neglected in the design and analysis of plates.