Effect of non-linearity in free large oscillations of a shallow catenary

Non-linear effects in free large amplitude oscillations of a shallow catenary are analyzed by representing the motion in terms of the well-known linear eigenmodes. The motion is comprised of a fundamental frequency and its harmonics. For each harmonic, including the fundamental, the mode shape is synthesized from the linear eigenmodes. The equation of motion is then cast as a non-linear eigenproblem to determine the fundamental frequency and the amplitudes of the contributory linear eigenmodes at each harmonic. Numerical results confirm the previously reported shift of the fundamental frequency due to non-linear effects as amplitude increases. The shift is found to have both tension- and non-tension-dependent components. For given structural assumptions, there may be different solutions to the non-linear equation of motion, differing significantly in mode shape and waveform.