Investigation of the Free Vibrations of Bar Elements with Variable Parameters Using the Direct Integration Method

The vibration analysis of bar element systems with continuous variable parameters using the direct integration method is proposed. The method is based on analytical solutions of the appropriate differential equations. By using this method, the problem of free flexural vibrations of a straight bar with arbitrary, continuously varying transverse stiffness and mass per unit length is solved. Formulas in analytical form are derived for dynamic parameters of bar structures, and the procedure for their numerical implementation is presented. Free vibrations of a truncated cone and chimney with a variable wall thickness are studied.