The so called archetypal smooth and discontinuous (SD) oscillator with irrational nonlinearity is a simple mass-spring system constrained to a straight line by a geometrical parameter $\alpha$ which is the dimensionless distance to the fixed point. The typical phenomenon of this oscillator is the dynamics transition from smooth to discontinuous depending on the smooth changing of the geometrical parameter, which has attracted a quite mount of investigations on the complex nonlinear behaviours of the SD oscillator since it was firstly proposed and appeared in Physical Review E74 (4)(2006)046218. The present work provides a comprehensive review of state-of-the-art researches on the SD oscillator begining with the complex nonlinear dynamics under the smooth and discontinuous cases, including the fundamental dynamical characteristics of the unperturbed system, perturbed bifurcations, chaotic motions and also the coexistence of multiple atrractors. Then, the work lists several extended oscillators with irrational type of nonlinear restoring forces based upon the SD oscillator. Finally, the work details the applications of the SD oscillator in engineering fields especially in vibration isolation and energy harvesting by means of the features of negative stiffness and the multiple stabilities. This review work shows the importance of irrational nonlinear restoring forces controlled by geometrical parameters in the engineering structures designing. The concluding remarks suggest further promising directions, such as the dynamics near local and global bifurcation with high co-dimension caused by the increasing geometrical parameters, the construction of universal unfoldings and irrational elliptic function for the situation of multiple geometrical parameters, the design of novel model with geometrical nonlinearity for engineering application and the improvement of experimental method for oscillators with irrational nonlinearity.

具有非理性非线性的所谓原型光滑和不连续 (SD) 振荡器是一个简单的质量弹簧系统，受几何参数约束为一条直线$\alpha$这是到固定点的无量纲距离。该振荡器的典型现象是根据几何参数的平滑变化从光滑到不连续的动力学转变，自首次提出并出现在物理评论 E74(4)(2006)046218。目前的工作从光滑和不连续情况下的复杂非线性动力学开始，对 SD 振荡器的最新研究进行了全面回顾，包括未扰动系统的基本动力学特性、扰动分岔、混沌运动以及多个吸引子的共存。然后，该工作列出了几个基于 SD 振荡器的具有非理性类型非线性恢复力的扩展振荡器。最后，该工作详细介绍了SD振荡器在工程领域的应用，特别是通过负刚度和多重稳定性的特点在隔振和能量收集方面的应用。这项审查工作表明了由几何参数控制的非理性非线性恢复力在工程结构设计中的重要性。结束语提出了进一步的有希望的方向，例如几何参数增加引起的高共维的局部和全局分叉附近的动力学，多几何参数情况下的通用展开和无理椭圆函数的构建，新颖的设计用于工程应用的几何非线性模型和非有理非线性振荡器实验方法的改进。