Hard-Spring Bistability and Effect of System Parameters in a Two-Degree-of-Freedom Vibration System with Damping Modeled by a Fractional Derivative

The harmonic balance coupled with the continuation algorithm is a well-known technique to follow the periodic response of dynamical system when a control parameter is varied. However, deriving the algebraic system containing the Fourier coefficients can be a highly cumbersome procedure, therefore this paper introduces polynomial homotopy continuation technique to investigate the steady state bifurcation of a two-degree-of-freedom system including quadratic and cubic nonlinearities subjected to external and parametric excitation forces under a nonlinear absorber. The fractional derivative damping is considered to examine the effects of different fractional order, linear and nonlinear damping coefficients on the steady response. By means of polynomial homotopy continuation, all the possible steady state solutions are derived analytically, i.e. without numerical integration. Coexisting periodic solutions, saddle-node bifurcation and various effects of fractional damping on the steady state response are found and investigated. It is shown that the fractional derivative order and damping coefficient change the bifurcating curves qualitatively and eliminate the saddle-node bifurcation during resonance. Moreover, the system response depicts bigger and bigger region of hard-spring bistability with increasing fractional derivative order, but the region of hard-spring bistability of steady response becomes gradually small and then disappears when we increase the linear and nonlinear damping coefficients. In addition, the analytical results are verified by comparison with the numerical integration ones, it can be found that the present approximate resonance responses are in good agreement with numerical ones.