This technical note presents a numerical study on the stability of single degree of freedom (SDOF) systems with asymmetric bi-linear hysteretic restoring force, subjected to earthquake excitations. The aim is to report: (a) the existence of an unstable behavior in the response of such systems, under a specific ground motion, given small modifications of the yielding conditions of the hysteresis model, and (b) the introduction of a novel three-dimensional graphic visualization of the problem. The modifications of the yielding conditions were introduced via the symmetry-breaking produced by very small variations of the static equilibrium position of the system, equivalent to having an initial position and restoring force different from zero and symmetric yielding. The concise study comprises of nonlinear dynamic analyses of three system cases, one of them with symmetric (reference) and two with asymmetric yielding conditions. The results show that the system presented a stable response and severe ratcheting toward the weakest yielding direction for the symmetric and asymmetric cases, respectively. Differences as large as $\pm 2800$% between the asymmetric and reference cases were obtained for the residual displacement of the systems, due to variations as small as $\pm 7$% in the static-equilibrium position, and consequent $\pm 7$% variations of the positive/negative yielding displacements and forces. In turn, negligible variations of the velocity between the three cases were predicted. To conclude, the paper introduces novel three-dimensional representations of the solution-curve and of the hysteresis cycles of the systems, deepening the discussion on the identified bifurcation. The 3D hysteresis curve, in particular, can be of much use for seismic engineering and mechanical studies, either numerical or experimental, as it allows visualizing the sequence of events in the hysteresis plots in a much clearer fashion compared to the traditional two-dimensional counterparts.

本技术说明提供了关于具有非对称双线性滞后恢复力的单自由度（SDOF）系统在地震激励下的稳定性的数值研究。目的是报告：（a）在磁滞模型的屈服条件稍作修改的情况下，在特定的地面运动下，此类系统在响应中存在不稳定行为；以及（b）引入了新颖的三种问题的三维图形可视化。屈服条件的修改是通过系统静态平衡位置的很小变化产生的对称破坏来引入的，等效于初始位置和恢复力不同于零屈服和对称屈服。简洁的研究包括对三种系统情况的非线性动力学分析，其中一个具有对称（参考），两个具有非对称屈服条件。结果表明，对于对称和非对称情况，该系统分别表现出稳定的响应和向最弱屈服方向的严重棘轮。差异最大$\pm 2800$对于系统的残余位移，由于变化小至 $\pm 7$％在静态平衡位置，因此 $\pm 7$正/负屈服位移和力的变化百分比。反过来，可以预测这三种情况之间速度的变化可以忽略不计。总之，本文介绍了系统的解曲线和滞后循环的新颖的三维表示，加深了对确定的分叉的讨论。3D磁滞曲线尤其可用于数值或实验的地震工程和力学研究，因为与传统的二维对应物相比，它可以更清晰地可视化磁滞图中的事件序列。