Stability Of Hysteretic System

Dynamical system is a mathematical model of a large number natural phenomenon whose state varies with time, in which the nonlinear dynamical system is the most interesting but also difficult part. Nonlinear system is different from linear system, for in its state equation the time derivative of the state vector is not a linear function of the state vector, which leads to many different properties. Furthermore, hysteretic systems is a special class of nonlinear system, and this paper firstly answer what the difference is in the state function, and then study its property such as stability. As Bouc-Wen model is a widely used hysteretic system, this paper use it as an example to study the phenomena of the hysteretic system, and Emphatically analyze the stability of it under condition of excitation.Main contents are included as follows:Some numerical studies of the Bouc-Wen model are firstly made, which reflect some special properties: the dynamical ratcheting occurs when the system is excited by drifted input or dual-frequency input with certain phase difference, and whether it is bounded is decided by the parameters of the system; hysteretic system can be oscillated when it is excited by certain dual-frequency input, and the oscillation period can be much larger than the natural vibration period.; And the response of hysteretic system under static load with noise is then studied, and some interesting conclusion is discoverd.A mathematical definition of the stability of the Bouc-Wen model is then made, and whether it is stable is proved. As hysteretic system is a special class of nonlinear system, its stability cannot be conveniently studied by traditional methods, so it is necessary to give a new method.In the end of this paper, artificial seismic ground motion is made to study if a hysteretic system can have a large residual deformation. A comparison with linear system is made. Though the linear system cannot have a residual deformation, it can have a large displacement under certain artificial seismic ground motion. The seismic ground motion is constructed by wavelet with the vanishing moment 2 to avoid integral drift.