Stability in parametric resonance of an axially moving beam constituted by fractional order material

The stability of an axially moving beam constituted by fractional order material under parametric resonances is investigated. The governing equation is derived from Newton's second law and the fractional derivative Kelvin constitutive relationship. The time-dependent axial speed is assumed to vary harmonically about a constant mean velocity. The resulting principal parametric resonances and summation resonances are investigated by the multi-scale method. It is found that instabilities occur when the frequency of axial speed fluctuations is close to two times the natural frequency of the beam or when the frequency is close to the sum of any two natural frequencies. Moreover, Numerical results show that the larger fractional order and the viscoelastic coefficient lead to the larger instability threshold of speed fluctuation for a given detuning parameter. The regular axially moving beam displays a higher stability than the beam constituted by fractional order material.