Flexural and torsional non-linear free vibrations of beams using a p-version finite element

A p-version beam finite element with hierarchic basis functions and which may experience longitudinal, torsional and bending deformations in any plane is employed to investigate the geometrically non-linear vibrations of beams. Clamped–clamped, isotropic and elastic beams of circular cross section are analysed. The geometrical non-linearity is taken into account by considering a simplified version of Green’s strain tensor. The stiffness matrix and the consistent mass matrix are derived using the principles of d’Alembert and of the virtual work. The harmonic balance method is employed to map the equations of motion to the frequency domain and the resulting algebraic non-linear system of equations is solved by a continuation method. Assuming a Fourier series where the constant term and the first three harmonics are considered it is concluded that internal resonances appear both in bending and torsion.