Single and multi-purpose optimization of vibrating Timoshenko shafts

Abstract

Optimization of transversely vibrating shafts with respect to eigenfrequencies, with constraints on the design variables, has been almost fully investigated in recent years - as far as the Bernoulli-Euler equation of motion is concerned. In the present paper, Timoshenko's equations are applied, i.e. the effects of shear and rotational energy are taken into account in the calculation of the transverse vibration modes and frequencies of the shaft. This is done via a finite element formulation. The sensitivity analysis of the transverse vibration frequencies is based on analytical differentation of the stiffness- and mass matrices. The computer program developed provides the user with an option to suppress the Timoshenko effects, such that the analysis can be carried out within the Bernoulli-Euler theory as well and then comparisons can be made. Torsional vibrations of shafts are also considered, computing the vibration modes and natural frequencies by means of a finite difference approach. Again, the sensitivity analysis is carried out analytically. Objective functions and behavioral constraints are selected from fundamental frequencies, higher order frequencies and differences between adjacent frequencies of the vibrational types considered. The cross-sectional area function of the shaft is used as the design variable and the total volume of structural material is assumed to be given, along with some sizing constraints. The shafts considered may be equipped with non-structural disks and flexible supports. Several examples are presented and some notable differences between optimized Timoshenko and Bernoulli-Euler shafts derived.