Periodic motions with impact chatters in an impact Duffing oscillator (Q6555026)

The authors consider a periodically forced Duffing oscillator with one-sidewall constraint. This problem belongs to the class of piecewise continuous dynamical systems. Denote the displacement of the Duffing oscillator by \(x(t)\). The nonlinear spring is described by the expresion \(k_1 x + k_2x^3\), and the linear damping is \(d\dot{x}\) (\(d\) is a number), where \(\dot{x} = dx/dt\) is the velocity. The external force is periodic function \(Q \cos{\Omega t}\), where \(Q\) and \(\Omega \) are excitation amplitude and frequency, respectively. The gap between the Duffing oscillator and a rigid wall is \(\triangle \). The impact Duffing oscillator is governed by the ODE\N\[\N m\ddot{x}+d\dot{x}+k_1x+k_2x^3= Q\cos{\Omega t}, \ \ x\in (-\infty ,\triangle ),\tag{a}\N\]\Nwhere \(\dot{x}^{+}=-e\dot{x}^{-}\) at \(x=\triangle\). Here \(\dot{x}^{-}\) and \(\dot{x}^{+}\) are the velocities between the oscillator and wall before and after impact, respectively. The parameter \(e\) is the coefficient of restitution. If one is interested in the phase space to analyze the the solutions of the equation under consideration, then the equation \((a)\) can be transformed into the system\N\[\N\begin{array}{l} \dot{x}=y, \\\N\dot{y}=Q_0\cos{\Omega t}-\delta y- \alpha x-\beta x^3, \end{array}\tag{b}\N\]\Nfor \(x\in (-\infty ,\triangle )\). In addition the impact relation and parameters are\N\[\N\begin{array}{l} y^{+}=-cy^{-} \ \mbox{at} \ x=\triangle , \\\N\delta =d/m, \ \alpha =k_1/m, \ \beta = k_2/m, \ Q_0=Q/m. \end{array}\tag{c}\N\]\NAfter letting \(\mathbf{x}=(x,y)^T\), \(\mathbf{F}=(F_1,F_2)^T\) the system takes simpler form. The two boundary functions for impact and stuck motions are defined by \(\varphi_{1\infty }= x-\triangle \) and \(\varphi_{12}=y\). Through the boundary functions, the open domains for impact and stuck motions are defined as \(\Omega_1=\{(x,y)|\varphi_{1\infty }= x-\triangle < 0\}\), \(\Omega_2=\{(x,y)|\varphi_{12}= y = 0,\, x=\triangle \}\), and the corresponding boundaries are defined by \(\partial\Omega_{1\infty}= \{(x,y)|\varphi_{1\infty }= x-\triangle = 0, \, y\ne 0\}\), \(\partial\Omega_{12}= \partial\Omega_{21}= \{(x,y)|\varphi_{12}= y = 0, \, x=\triangle \}\). Note that on some domain \(\Omega_{\sigma }\) (\(\sigma =1,2\)) the considered dynamical system for the impact Duffing oscillator can be described by the system of ODE (in vector form) \(\dot{\mathbf{x}}^{(\sigma )}= \mathbf{F}^{(\sigma )} (\mathbf{x}^{(\sigma )},t)\). The authors present a semi-analytical method for existence of periodic motions in the problem under consideration. The stability and grazing bifurcations of such periodic motions are studied. Some analytical conditions for motion grazing at the boundary are developed. The impact Duffing oscillator is discretized implicitly, and the corresponding implicit mapping is discussed. A specific impact periodic motion for the impact Duffing oscillator is considered. The bifurcation trees of impact chatter periodic motions are analyzed. The grazing and period-doubling bifurcations are obtained. The impact chatter periodic motions with and without grazing are considered for illustration of impact periodic motion complexity in the impact oscillator.