Parametric resonance in nonlinear vibrations of elastic stretched string under harmonic heating
The existence of parametric resonance and transient phenomena in nonlinear systems under the action of an external force are important characteristics of dynamical systems. Nonlinear vibrations of a thin stretched string, with an alternating electric current passing through, in a non-uniform magnetic field are described by complicated equations of motion. The general mathematical model involves modes coupling by means of the intrinsic and improper nonlinearities; besides, the string suffers Joule heating. The purpose of the work is to study the combined effect of the intrinsic (geometrical) nonlinearity and Joule heating on the elastic string oscillation in the frame of a simplified model. We use a combined analytical-numerical approach in studying the dynamics of the proposed model. First, we solve our model analytically by iterations; then we solve it numerically. Both analytical and numerical results show a good agreement almost everywhere except small intervals near resonant frequencies of different modes. It was found that numerical solutions show instabilities near resonant frequencies in contrast to that of the approximate analytical solutions by iterations. We explain those instabilities using the theory of Mathieu equations.