Parametric bifurcation of a viscoelastic column subject to axial harmonic force and time-delayed control

• We study a viscoelastic column subject to parametric excitation whose damping is governed by fractional derivatives.
• We find the multiple bifurcation solutions, jump phenomenon and saddle-node analytically.
• Nonlinear fractional derivative and time delay are investigated.
• We find that the fractional order, time delay and material modulus ratio can increase the stability.
• New criteria of stability determination for fractional order derivatives are established.

We investigate the steady state response of a simply supported viscoelastic column subject to axial harmonic excitation. The viscoelastic material is modeled in fractional derivative Kelvin sense. The equation of motion is derived and discretized by the Galerkin approximation resulting in a generalized Mathieu–Duffing equation with time delay. Bifurcations in parametric excitation can be eliminated by appropriate feedback gain and time delay. The bifurcating behavior for various fractional orders and material ratios are also investigated. New criteria of stability determination are established. Based on the Runge–Kutta method, numerical results are obtained and compared with analytical solutions for verification.