Heat Conduction and Its Related Interdisciplinary Areas: Self-Excited Oscillations in a Thermomechanical Elastic Sheet

We present a minimal theoretical model for self-sustained oscillations of a thin elastic sheet on a hot plate, induced by thermomechanical coupling. As the plate temperature increases, the sheet’s static deflection becomes unstable via a Hopf bifurcation at a critical temperature T C , giving rise to spontaneous periodic motion. Linear stability analysis yields analytical expressions for the critical oscillation temperature T C and the oscillation period at onset. Numerical simulations of the nonlinear equations confirm the bifurcation and reveal how key parameters (stiffness, thermal softening, thermal coupling, etc.) govern the oscillation amplitude and waveform. Finally, we demonstrate that the self-oscillating sheet can perform mechanical work as a heat engine, and we compare its performance to the Carnot efficiency limit. This work provides design principles for thermally driven self-oscillators with potential applications in soft robotics, adaptive structures, and thermal energy harvesting.