Natural vibration of arbitrary spatially curved rectangular rods with pre-twisted angles

For the natural vibration of rods of pre-twisted rectangular cross-sections with its centerline spatially curved and twisted arbitrarily, a system of non-dimensional ordinary differential equations is obtained in three translation displacements and three rotational angles of the rod. The eigenvalue problem is solved for the natural frequencies of various pre-twisted angles of the rectangular cross-section area, aspect ratios, slenderness ratio, end conditions and functions of varying cross-section area along the rods by using a numerical method of Runge-Kutta integration. The eigenvalues are determined by the vanishing of a rank-six determinant whose column vectors are obtained at the end of six independent numerical integrations. The differential equations of the adjoint operator system for the eigenvalue problem with its associated boundary conditions are also derived and solved for the eigenvalues to check with that of the vibration system. It is found that the effect of increasing pre-twisted angle of the rod is that the lowest natural frequency of an arbitrary curved rod will asymptotically tend to be a constant, in spite of slenderness ratio, aspect ratio of the rectangular cross-section, E/G ratio, types of end conditions, or the functions of the cross-section area along the rod.