Multi-scale analysis for dynamic stability of an axially accelerating viscoelastic beam subjected to combination parametric resonance

The analytical–numerical approach has been adopted to investigate the nonlinear planner response of an axially accelerating beam with the coexistence of additive-type combination parametric resonance and internal resonance. This study includes geometric nonlinearity developed due to the stretching of the neutral layer, longitudinally varying tension, harmonically fluctuating speed, material, and modal dampings. For the suitable value of the system parameters, the second natural frequency of the moving system is approximately equal to three times of first mode, consequently, three-to-one internal resonance activates for a specific range of mean axial speed. The perturbation method of multiple time scales is adopted to solve the beams governing integro-partial differential equation motion with associated end conditions, resulting in complex variable modulation equations that control amplitude and phase modulation. The continuation algorithm technique is used to compute these modulation equations to study the impact of various control parameters, such as internal frequency detuning parameter, variable speed, pulley stiffness parameter, and axial stiffness through the frequency and amplitude response curves. Trivial state stability plots are also presented to illustrate the impact of material and external dampings on the stability of the system. The findings of this analysis are unique and still need to be addressed in the literature.