In this paper, a dynamical model of a spring-mass system with a single degree of freedom is proposed, which can be designed as a nonlinear supporting system for GVT of large scale aircraft and vibration isolation owing to stable-quasi-zero-stiffness (SQZS). The SQZS structure is constructed by a positive stiffness component and a pair of inclined linear springs providing negative stiffness, which is typical for an irrational restoring force due to geometrical configuration. The unperturbed analysis demonstrates complex equilibrium bifurcations and stabilities for this peculiar system, based upon which parameter optimizations are performed for SQZS and the maximum interval of low frequency. Furthermore, the dynamics analysis of the perturbed system near the optimized parameters reveals complicated behaviour of KAM structures, period doubling, chaos crisis, coexistence of multiple solutions, intermittency chaos, chaos saddle; etc. All those presented herein provide a better understanding for the complicated dynamics of SQZS nonlinear system.