Free Vibration Of Non-uniform Beams/rods With Variable Cross-section By Finite Difference Method

Generally, non-uniform beam is widely used in engineering. It is difficult to obtain the analytical solution because of the variable coefficients of the differential governing equations. Numerical methods can be used to get the approximate solution. On the basis of the Euler-Bernoulli beam theory, free vibration of the non-uniform beam with variable cross section, subjected to axial loads and rested on the elastic foundation, is analyzed by the finite difference method. Natural frequency and mode shapes are given. The effects of various parameters on the response of the vibration are discussed. The main contents of this thesis are as follow:1. By using the central difference scheme, a discrete linear algebraic equation, with penta-diagonal coefficient matrices is obtained from an arbitrary fourth-order ordinary differential equation. Then by using the recurrence method, the recurrence formula of the linear algebraic equations is obtained. Based on the recurrence formula, only five one-dimensional arrays are used to store the coefficient matrix in the numerical calculations. Therefore, it has the advantage of being easy to program, using the least storage space and having high computing speed. The program of difference recurrence solution in this thesis can be used to solve all kinds of fourth-order ordinary differential equations with two-point boundary conditions.2. Based on the Euler-Bernoulli beam theory, the free vibration of the non-uniform beams with the variable cross section, subjected to axial pressure and rested on the elastic foundation, is studied by the recurrence formula of the finite difference method. In the analysis, the cross-section and elasticity modulus of beam are assumed to be continuous change in the axial direction. The free vibration problem of the fixed-fixed and pined-pined beam, with varying circular cross-section, subjected to the axial pressure and rested on the elastic foundation, is researched respectively as an computing example. The non-dimensional frequencies and the corresponding vibration modes of uniform cross-section beam with no axial pressure and elastic foundation by this method are very close to the classical analytical solutions. The effects of variable parameters of cross-section and elasticity modulus, the stiffness of the elastic foundation, axial pressure and the boundary conditions on the natural frequency are discussed. The numerical results show that, the frequency decreases with the increase in the value of the axial pressure, and increases with the increment of the parameter of elastic foundation. The natural frequency increases with increasing the parameter of cross-section and the material parameter.3. The transverse vibration of the non-uniform column which carrying a concentrated rigid mass at the free end is studied. The effects of the inertial force of the mass, as well as the axial pressure generated by the mass and weight of the beam on the natural frequency are discussed. Using the finite difference method, two point boundary problem of fourth order ordinary differential equations is solved. The numerical results of natural frequency of variable cross-section column are given. The influence of the varying cross-section and the axial pressure generated by gravity on the natural frequency is investigated. The results show that the frequency increases with increasing parameter of the cross-section, and decreases with increase of the axial pressure.