Singular characteristics of nonlinear normal modes in a two degrees of freedom asymmetric systems with cubic nonlinearities

Nonlinear normal modes in a two degrees of freedom asymmetric system with cubic nonlinearities as singularity occurs in the system are studied, based on the invariant space in nonlinear normal modes and perturbation technique. Emphasis is placed on singular characteristics as the linear coupling between subsystems degenerated. For nonresonances, it is analytically presented that a single-mode motion and localization of vibrations occur in the system, and the degree of localization relates not only to the coupling stiffness between oscillators, but also to the asymmetric parameter. The parametric threshold value of localization is analytically given. For 1:1 resonance, there exist bifurcations of normal modes with nonlinearly coupling stiffness and asymmetric parameter varying. The bifurcating set on the parameter and bifurcating curves of normal modes are obtained.