Study on dynamical behaviors of the spring-pendulum system with an irrational and fractional nonlinear restoring force

A novel model comprised of a lumped mass linked with a pair of inclined elastic stiff springs is proposed, which can be regarded as a smooth and discontinuous oscillator under constant excitation, i.e., CSD (originally introduced in Chin. Phys. Lett. 29, 0847061-4 (2012)), vibrating vertically and laterally. Of particular concern is the influence of the parameters on its steady-state response. Neglecting its lateral vibration, the system is a single-degree-of-freedom system, i.e., the CSD oscillator, whose amplitude-frequency response curves were studied by using average method and elliptical integral. The third-order approximation form of the two-degree-of-freedom system was introduced and the amplitude-frequency response curves were obtained. By simulating the original system and the approximation one using the Matlab software, we obtained phase portraits, Poincaré sections, bifurcations and maximum Lyapunov exponents of the two systems. And the practicality of the approximation system was certified by comparing the characteristics of bifurcations and chaos of the two systems, which can offer theoretical foundations for practical engineering.