Vibration and sensitivity analysis of structures with optimal design using transfer dynamic stiffness matrix

A method for exact vibration analysis of 3-D frame structures is developed in this dissertation. Given a 3-D frame consists of N prismatic members, the vibration analysis of each member involves the solution of 4 differential equations: these are axial vibration, torsional vibration and flexural vibration in 2 planes. So the vibration analysis of this frame involves the solution of 4 N differential equations. A method is developed in this dissertation to solve this problem. The solution is exact: that they satisfy the governing differential equation and all the boundaries and interconnection condition. Sensitivity analysis and optimal design based on the exact method are also developed.; The developed method is based on transfer dynamic stiffness matrix method (TDSM). The first step in TDSM is the solutions of the governing beam equations for free vibration based on Euler-Bernoulli or Timoshenko beam theory. These solutions are then expressed in a transfer matrix form for a 3-D beam element. The transfer matrix for each beam “element” is then rearranged in dynamic stiffness matrix that is called transfer dynamic stiffness matrix (TDSM) that relates beam end forces and displacements. The TDSM for each fame members are then assembled to yield global dynamic stiffness matrix. Applying boundary conditions to this global matrix yield exact frequency equation for frame vibration analysis. The exact natural frequencies can be found by using a golden section method based on Williams's equation. For each frequency, an eigenvector for displacement at the ends of beam elements can be computed, and the associated eigenfunction can be determined by the eigenvector and the dynamic shape function based on the Euler-Bernoulli and Timoshenko beam theories.; Eigensolution sensitivity analysis based on the exact frequency equation is also developed in this dissertation. Once the frequency sensitivity of structures is available, the minimum weight design with frequency constraint and its gradient can be performed. Several examples are presented to demonstrate the principles and algorithms and the results are compared and show good agreement with those computed by ANSYS or given in the references.