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.

本手稿探讨了粘弹性对单自由度机械振荡器的静态分叉的影响：例如叉，鞍形节点和跨临界分叉。通过微分形式对粘弹性行为进行建模，其中额外的自由度表示由粘弹性元素提供的内力。该控制方程式是从简化的集总参数模型得出的，该模型包括一个刚性棒，该棒结合了粘弹性元件，并在自由端承受了轴向和横向力，此外还施加了一个随时间变化的外部力矩。为了研究粘弹性对分叉图的影响，对运动方程进行了无量纲化处理。接下来，对在没有粘弹性的情况下的静态分叉进行了回顾，然后检查粘弹性对分叉图的影响。最后，本文研究了粘弹性对振荡器瞬态行为的影响。结果表明，Deborah数测量了系统的粘弹性时间常数与自然周期的比值，控制了在跳至稳定平衡点之前维持不稳定点附近振荡所需的时间。