Non-linear steady state vibration and dynamic snap through of shallow arch beams

The non-linear steady state vibration of shallow arch beams is studied by a finite element method based on the principle of virtual work. The spatial discretization is achieved by the usual finite element method and the steady state nodal displacements are expanded into a Fourier series. The harmonic balance method gives a set of non-linear algebraic equations in terms of the vibrating frequency and the Fourier coefficients of nodal displacements. The non-linear algebraic equations are solved by the Newtonian algorithm iteratively. The combined algorithm is called the incremental harmonic balance method. The importance of the conditions of completeness and balanceability is presented. Since the non-linearity is essentially softening, different orders of internal resonances between two modes can occur repeatedly. Isolated response curves are possible and are connected to the bifurcation of a particular excited mode.