Exact Solution Of Free Vibration And Buckling Of Non-uniform Members And The Sturm-Liouville Problem Using Dynamic Stiffness Method

For free vibration analysis of skeletal structures, dynamic stiffness method (DSM) is an exact method. In this method, the exact dynamic stiffness matrices are generated by using exact shape functions which satisfy the control differential equation, and then the corresponding transcendental eigenvalue problem is solved exactly. Compared with usual finite element method (FEM), only one element is needed for each member and the refinement of elements is usually unnecessary, which considerablely reduces the number of degrees of freedom with exact solution being remained.While the Wittrick-Williams algorithm is a powerful mean dealing with the transcendental eigenvalue problems, the recent recursive second order method proposed by Yuan Si et al can efficiently and reliably produce accurate solutions for both eigenvalues and mode vectors. In this paper, DSM is applied to the exact solution of free vibration of non-uniform members and further to the Sturm-Liouville (SL) problem. The main ideas are: (1) the member is divieded into a number of sub-members so that the calculation of the number of fixed-end frequencies below a given value is not necessary when using the Wittrick-Williams algorithm, and (2) the exact stiffness and mass matrices are computed by using a standard adaptive linear ordinary differential equation (ODE) solver.The main work of this thesis is as follows:The DSM is applied to free vibration problems of both axial and flexural vibration of non-uniform members. For any particular order of eigensolutions, both frequency and mode vector can be exactly obtained.The above method is extended to the free vibration of non-uniform Timoshenko beams and elastic buckling of non-uniform rods. Both the frequency (buckling load) and mode vector (buckling mode vector) can be obtained exactly for any order of .frequency and mode vector.The above method is further applied to general second- and forth-order SL eigenvalue problems, which is again very successful.Theoretical analysis and numerical examples given in this paper show that the proposed method can effectively solve the free vibration of non-uniform members (including Timoshenko beams), as well as buckling problems. In addition, when extended to general second- and fourth-order SL problems, the method performs excellently in terms of the efficiency, precision and stability.