Parametric resonance in nonlinear vibrations of string under harmonic heating

•Geometric nonlinearity and Joule´s heating act opposite each other on string oscillation.•Single mode approximation results in Mathieu–Duffing equation.•Oscillator presents Mathieu´s type instabilities when proper nonlinearity is omitted.•Jump phenomenon is observed in sweeping the frequency of driving force.•String oscillates at the modulation frequency in a small only interval of resonant frequency. In this paper, vibrations of thin stretched strings carrying an alternating electric current in a non-uniform magnetic field are described by nonlinear equations. Within the frame of a simplified model, we studied the combined effect of geometric nonlinearity and Joule heating acting opposite to each other. An equation including Joule heating only shows unlimited growth in oscillation amplitude near resonant frequencies. Nevertheless, a single mode approximation resulting in Mathieu–Duffing´s equation shows a double resonance with bounded oscillation amplitude. At zero external force, the response frequency of steady-state oscillations is equal to parametric modulation frequency in an interval near the resonant frequency; otherwise, the response frequency equals the natural frequency of the oscillator.