Acoustic metamaterial, which can prohibit effectively the elastic wave propagation in the bandgap frequency range, has broad application prospects in the vibration and noise reduction areas. The Lindstedt–Poincaré method was utilized to analyze the dispersion curves of nonlinear metamaterial with a pure Duffing oscillator. The first‐order perturbation solutions of acoustic and optical branches were obtained. Both the starting and cutoff frequencies of the bandgap are determined consequently. It was found that the soft/hard characteristics of pure Duffing oscillators could lead to the lower/upper movement of the starting and cutoff frequencies of the bandgap. By further researching the degraded linear system, the conclusion that actual nonlinear metamaterial bandgap region is wider than effective negative mass region is drawn and that both mass and stiffness ratio effect on the starting frequency is obtained. Effective positive mass can also lead to the vibration attenuation in bandgap. For nonlinear metamaterial, the translation effect of the external excitation amplitude on the bandgap range and the zero mass at the nonlinear bandgap cutoff frequency were discussed, and all above conclusion are identified by numerical analysis.

The stochastic P-bifurcation behavior of a bistable Van der Pol system with fractional time-delay feedback under Gaussian white noise excitation is studied. Firstly, based on the minimal mean square error principle, the fractional derivative term is found to be equivalent to the linear combination of damping force and restoring force, and the original system is further simplified to an equivalent integer order system. Secondly, the stationary Probability Density Function (PDF) of system amplitude is obtained by stochastic averaging, and the critical parametric conditions for stochastic P-bifurcation of system amplitude are determined according to the singularity theory. Finally, the types of stationary PDF curves of system amplitude are qualitatively analyzed by choosing the corresponding parameters in each area divided by the transition set curves. The consistency between the analytical solutions and Monte Carlo simulation results verifies the theoretical analysis in this paper.

Top tension riser is one of the most frequently used and vulnerable equipment in deep-sea petroleum engineering. With experiments and CFD simulations, researchers have found some hysteresis phenomenon and bistable behavior in vortex-induced vibration of the cylinders. However, there is still insufficient research on the bifurcation phenomenon and the multi-stable behavior of the riser vortex-induced vibration. In this study, based on the Van der Pol wake oscillator model, the vortex-induced vibration cross-flow governing equations of the riser fluid structure–coupled system in linear shear flow are established. Then the simplified multi-degrees-of-freedom model with 15-order modes is derived by the Galerkin method. Poincare map is utilized to obtain bifurcation diagram and identify the dynamic property of the top tension riser in a wide flow velocity range. Nonlinear dynamic behavior such as bistable behavior and even tristable behavior are discovered from the bifurcation diagram. Combined with the riser coupled system’s eigen analysis and the three-dimensional spectrum contour, it is found that there are various phenomena of mode interaction in the riser system, including 1:3 internal resonance phenomenon. Such study can provide reference and guide the design and optimization of riser structural parameters.