A new perspective on static bifurcations in the presence of viscoelasticity

This manuscript explores the effect of viscoelasticity on static bifurcations: such as pitchfork, saddle-node, and transcritical bifurcations, of a single-degree-of-freedom mechanical oscillator. The viscoelastic behavior is modeled via a differential form, where the extra degree of freedom represents the internal force provided by the viscoelastic element. The governing equations are derived from a simplified lumped parameter model consisting of a rigid rod incorporating a viscoelastic element and subjected to axial and transverse forces at the free end, in addition to an external time-varying moment applied to the rod. In order to study the effect of viscoelasticity on bifurcation diagrams, the equations of motion are non-dimensionalized. Next, a review of static bifurcations in the absence of viscoelasticity is conducted, followed by an examination of the effect of viscoelasticity on the bifurcation diagrams. Finally, this paper investigates the effects of viscoelasticity on the transient behavior of the oscillator. Results show that the Deborah number, which measures the ratio of the viscoelastic time constant to the natural periodic time of the system, controls the duration of time needed to maintain oscillations around an unstable point before jumping to a stable equilibrium point.