FEM在求解连续梁振动问题中的应用

-- 1. 问题描述利用有限元方法（FEM）对以下两个两单元问题进行求解，梁的横截面为矩形，其约束情况如图 1 所示。已知梁的几何尺寸和物理参数如下⑴ 几何尺寸：长度m，宽度m，厚度m；⑵ 物理参数：弹性模量GPa，泊松比，密度。 Figure 1．梁及其横截面示意图要求：(1) 至少划分五个节点（四个单元）；(2) 给出单元节点信息；(3) 给出单元刚度矩阵和质量矩阵；(4) 给出总刚度矩阵和总质量矩阵；(5) 求出梁各界固有频率及振型；(6) 将所得结果与理论值进行对比，验证方法的可行性。2. 有限元方法求解1. Divide the structure up into a number of elements of finite size.As shown in figure 2, we select 6 nodes equally spaced on the beam, thus the beam is divided into 5 elements with equal length and length of each element is m. 2. Associate with each node point a given number of degrees of freedom The number of each node and each element is presented in figure 2. Here, we take two degrees of freedom at each node because we have to involve the rotation in the calculation, namely the plane bending is considered. Figure 2 the diagram of the discrete elements3. Construct a set of functions such that each one gives a unit value for one degree of freedom and zero values for all the others From the Rayleigh-Ritz conditions, we note that the prescribed functions is required to be times differentiable, where is the order of the highest derivative appearing in the expression for the strain energy. Similarly, we can applies this principle to the choice of basis function here as well as the shape functions. Otherwise we would not be able to calculate the strain energy properly when we are using the finite element approximation method. For the beam bending problem, the strain energy is as follows 11\*MERGEFORMAT ()Here, is the elastic modulus of the beam material, refers to the inertia ...