Leung, Andrew Yee TakAndrew Yee TakLeungGe, T.2022-05-072022-05-071997https://repository.sfu.edu.hk/handle/sfu/3112For a strongly non-linear multi-degree-of-freedom system, in general, one cannot consider one mode at a time as in linear modal analysis. In the absence of external excitation, the natural vibration often involves more than one mode at a time resulting in quasi-periodic or multi-periodic (toroidal) vibration. The normal multi-mode in free vibration have been formulated by means of the action-angle transformation and the resulting ordinary differential equations embedded in partial differential equations. Final multi-periodic solutions have been obtained by extending the newly developed Toeplitz Jacobian matrix method with multi-periodic fast Fourier transforms.enjournal article10.1006/jsvi.1996.0830