常数激励与简谐激励联合作用下Duffing系统的非线性振动

This paper focuses on the basic dynamical characteristics of a Duffing system under the combination of constant excitation and harmonic excitation. The Harmonic Balance method was employed to solve the motion equation of the system. The amplitude-frequency relationship was obtained, and the backbone curve and the stability of the obtained periodic solution were analyzed as well. The amplitude-frequency curves and the backbone curves were used to show the main dynamical characteristics of the system. These dynamical characteristics affected by the constant excitation and the amplitude of the harmonic excitation were discussed significantly. In the vibration response of the system, it was shown that when the excitation frequency increases the constant term changes synchronous with the amplitude of the harmonic component, but towards an opposite direction. Nevertheless, the backbone curves for them both bend slightly to the left at the first stage and bend rightward after that. As a result, in some parameter regions, the system may have at most five periodic solutions, three of them are stable and the other two are unstable. By increasing the constant excitation, an effect of "stiffness enhancement" is presenting in the system, but it is accompanied by a more significant "stiffness softening" characteristic. Consequently, for a certain harmonic excitation, the increasing of the constant excitation may change the amplitude-frequency curve from a pure soft spring characteristic to a coexistence of soft and hard spring characteristics, and even to a pure hard spring characteristic finally. In a larger scale, however, the influence of the constant excitation to the backbone curve is mainly reflected at the down part. In other word, with the increase of the harmonic excitation, the effect of the constant excitation on the excitation frequency becomes weak, the backbone curves for different values of constant excitation tend to be similar.