A transfer matrix method is used to study free vibration characteristics of an axial-loaded Euler–Bernoulli beam with variable cross sections and multiple concentrated elements in the article. The differential equation for bending vibrations of the beam element is solved by the Frobenius method, and the solution is in power series form. Then, the transfer matrix method is applied to establish the state vector equation for both ends of the beam. Combined with boundary conditions, the frequency equation is obtained and expressed in a two-order determinant. The numerical results in this article are compared with those of the finite element method, which illustrates the accuracy of the method we proposed. The influence of the size of each concentrated elements and axial force on the natural frequency coefficients and the influence of the concentrated elements on the first critical buckling load are discussed.

本文采用传递矩阵法研究具有可变截面和多个集中单元的轴向加载欧拉-伯努利梁的自由振动特性。用Frobenius方法求解梁单元弯曲振动的微分方程，其解为幂级数形式。然后，采用传递矩阵法建立梁两端的状态矢量方程。结合边界条件，获得了频率方程，并用二阶行列式表示。将本文中的数值结果与有限元方法的数值结果进行了比较，这说明了我们提出的方法的准确性。