Dynamic stiffness matrix of a conical bar using the Rayleigh-Love theory with applications

Abstract

Based on the Rayleigh-Love theory, the dynamic stiffness matrix of a conical bar in longitudinal vibration is developed for the investigation of free vibration and response characteristics of such bars and their assemblies. First the governing differential equation of motion in free longitudinal vibration of a conical bar using the Rayleigh-Love theory which accounts for the inertia effects due to transverse or lateral deformations is derived by applying Hamilton's principle. Next, for harmonic oscillation, the governing differential equation is recast in the form of Legendre's equation, providing a series solution connected by integration constants. The expressions for the amplitudes of displacements and forces are then obtained by means of the series solution. Finally, the frequency dependent dynamic stiffness matrix is formulated by relating the amplitudes of forces to those of the corresponding displacements at the ends of the conical bar and thereby eliminating the integration constants. As an established solution technique, the Wittrick-Williams algorithm is applied to the resulting dynamic stiffness matrix when computing the natural frequencies and mode shapes of some illustrative examples. The theory is also applied to investigate the response of a cantilever conical Rayleigh-Love bar with a harmonically varying load applied at the tip. The results computed from the Rayleigh-Love model based dynamic stiffness theory are compared and contrasted with those computed from conventional classical theory with significant conclusions drawn.