Dynamic analysis of modal shifting and mode jumping in thermally buckled plates

Both analytical and finite element investigations are performed for the various static and dynamic aspects of the mode jumping phenomenon of a simply-supported rectangular plate heated deeply into the post-buckling regime. For the analytical method, the von Kármán plate equation is reduced to a system of non-linear ODEs by expressing the transverse deflection as a series of linear buckling modes. The ODEs, combined with the non-linear algebraic constraint equations obtained from in-plane boundary conditions, are then solved numerically under the parametric variation of the temperature. The results are checked by the finite element method, where a hybrid static–dynamic scheme is implemented. The contribution of each assumed (buckling) mode component is studied systematically. Characterized by the strong geometrical non-linearity, the secondary bifurcation point of the thermally loaded plate with fixed in-plane boundary conditions occurs far beyond the primary buckling point, and the jump behavior cannot be predicted correctly without sufficient assumed modes. Stationary bifurcation analysis indicates that while the post-buckling deflection before mode jumping is composed of pure symmetric modes, additional pure antisymmetric modes will appear after the occurrence of the snapping and they play the role of destabilizing the equilibrium. Furthermore, by monitoring natural frequencies and modal shapes, we find that a mode shifting phenomenon (the exchanging of vibration modes) exists in the primary post-buckling regime. Breaking of the symmetry of the dynamic modes is also found. By introducing a linear temperature sweeping scheme, transient analysis is performed to capture the snapping phenomenon dynamically, which occurs with moderate heating ratio. Comparison between the analytic and finite element results shows good agreement.