Torsional vibration analysis of bars including secondary torsional shear deformation effect by the boundary element method

Abstract

In this paper a boundary element method (BEM) is developed for the torsional vibration problem of bars of arbitrary doubly symmetric constant cross section, taking into account the nonuniform warping and secondary torsional shear deformation effects (STSDE). The bar is subjected to arbitrarily distributed or concentrated dynamic torsional loading along its length, while its edges are subjected to the most general torsional and warping boundary conditions. Apart from the angle of twist, the primary angle of twist per unit length is considered as an additional 1-D degree of freedom in order to account for the STSDE in the equations of motion of the bar. The warping shear stress distribution and the pertinent secondary torsional rigidity are computed by satisfying local equilibrium considerations under dynamic conditions without adhering to assumptions of Thin Tube Theory (TTT). By employing a distributed mass model system accounting for rotatory and warping inertia, an initial boundary value and two boundary value problems with respect to the variable along the bar time-dependent 1-D kinematical components, to the primary and secondary warping functions, respectively, are formulated. The latter are solved employing a pure BE method, requiring exclusively boundary discretization of the bar׳s cross section. The numerical solution of the aforementioned initial boundary value problem is performed through a BE method leading to a system of differential equations with displacement only unknowns, which is solved using an efficient direct time integration technique. Additionally, for the free vibrations case, a generalized eigenvalue problem is formulated through a similar BE technique. The accuracy and reliability of the results is assessed by FEM solutions employing solid or shell modelling. Both open- and closed-shaped cross section bars are examined and the necessity to include nonuniform torsional and STSD effects in the dynamic analysis of bars is demonstrated.