Sliding Mode Control for Two-Degree-Of-Freedom Fractional Zener Oscillator

Fractional-order derivatives provide a powerful tool for the characterization of mechanical properties of viscoelastic materials. A fractional oscillator refers to mechanical model of viscoelastically damped structures, of which the viscoelastic damping is described by constitutive equations involving fractional-order derivatives. This paper proposes active control of vibration in a two-degree-of-freedom fractional Zener oscillator utilizing sliding mode technique. Firstly, with a state transformation, the fractional differential equations of motion are equivalently transformed into a relatively simple form. Meanwhile, a virtual fractional oscillator is generated, which is further used to analyze the original oscillator. Then, the stored energy in the two fractional derivative terms is derived based on the diffusive model of fractional integrator. Thus, the total mechanical energy in the virtual oscillator is determined as the sum of the kinetic energy, the potential energy and the fractional energy. Furthermore, sliding mode control of vibration in the fractional Zener oscillator is designed, of which the Lyapunov function is chosen as the total mechanical energy. Finally, numerical simulations are conducted to validate the effectiveness of the proposed controllers.