Fractional dynamic of two-blocks model for earthquake induced by periodic stress perturbations

In this paper, the resonance behavior of a spring-block model with fractional-order derivative under periodic stress perturbation is investigated. Using the harmonic balance method, we derive the frequency-response equations for the system consisting of two blocks linked by a linear spring. The results have shown that the fractional-order derivative and perturbation parameter can affect the dynamical properties of fault rock, which is characterized by the equivalent linear damping coefficient and the equivalent linear stiffness coefficient. The frequency-response curve displays the resonance peaks and one anti-resonance. The effects of parameters $q,{\beta }_0,{\epsilon }_0,{\beta }_1$ and ${\epsilon }_1$ on the resonance and anti-resonance periods and the response amplitudes at the resonance frequency are analyzed. The shear stress response shows that the system accumulates a lot of energy at the resonance frequency. This accumulation can lead to the destabilization of the fault system. The blocks move without accumulating energy at the anti-resonance frequency. This can lead to the stabilization of the fault system.