Steady state response of fractionally damped nonlinear viscoelastic arches by residue harmonic homotopy

The nonlinear single-mode dynamic behaviour of the viscoelastic arch whose damping is governed by fractional derivatives is considered. A set of ordinary differential equations is derived for the primary resonance and is solved by the residue harmonic homotopy to obtain all the steady state solutions. A parametric study is carried out to determine the influence of the viscoelastic damping characteristics of the material on the responses. Both the fractional order and material modulus ratio are effective in enhancing the stability of the structure. Multiple bifurcation solutions, jump phenomenon, saddle-node are observed analytically without numerical integration and are confirmed by numerical integration.

► We study a viscoelastic arch whose damping is governed by fractional derivatives. ► We find the multiple bifurcation solutions, jump phenomenon and saddle-node analytically. ► Nonlinear fractional derivative is investigated. ► The method used is residue harmonic homotopy. ► We find that both the fractional order and material modulus ratio can increase the stability.