Vibrational Behaviour of Plane Frame Structures Composed of Prismatic and Tapered Sections

Abstract

The dynamic analysis of prismatic structures is further developed to include structures with members of different forms of taper. The mass, static stiffness and dynamic stiffness matrices are formulated for two assumed displacement functions — polynomial and quasi-exact, the latter giving an exact solution for prismatic structures. Both functions give an exact solution with finer subdivisions of the elements. The methods of subdivision are compared for their effectiveness. The formulation of the property matrices, for both functions in both prismatic and tapered sections, are fully documented and are proved to be valid in the analyses. The solution methods described give natural frequencies, the modal shapes and the analysis of dynamic response. The matrix iteration methods are developed to solve the linear eigensystems which are derived from the polynomial expressions. Nonlinear eigensystems, developed from the quasi-exact function, are studied by means of the count algorithm. This algorithm identifies a root with the concept of the Sturm sequence and the isolation of the singularity. It also serves as a powerful tool in dealing with abnormalities. The behaviour of plane frame structures, both of prismatic and tapered section, is studied with a wide variety of examples. Certain special features are noted: the convergence tests; the extensional mode in flexural vibration; the discontinuities in sectional properties; the optimisation of structures and the half-structuring analysis at the plane of symmetry. The analytical results obtained are supported by experimental evidence.