The analysis of dynamic instability and vibration motions of a pinned beam with transverse magnetic fields and thermal loads

Dynamic instability and transient vibrations of a pinned beam with transverse magnetic fields and thermal loads are studied. The magnetoelastic model, whose beam thickness and the deflection are very small compared with the length, is taken for analysis. Applying the Hamilton's principle, the equation of motion with damping factor is derived. The governing equation is reduced to the Mathieu equation by Galerkin's method with the assumed mode shape. The incremental harmonic balance (IHB) method is applied to analyze the dynamic instability. The amplitude versus time behavior of the system is investigated by using the Runge-Kutta method. The study shows that the instability and transient vibratory behaviors of the beam are influenced by the magnetic fields, thermal loads, and the frequencies of oscillation of the transverse magnetic field. The beat phenomenon and primary resonance are presented and discussed when the frequencies of the oscillating transverse magnetic field are close to the fundamental natural frequency of the system.