Stick–slip chaos in a mechanical oscillator with dry friction

61 , 768 (1995)] can be expressed as a nonautonomous constraint differential equation owing to the static friction force. The object is constrained to the surface of a moving belt by a static friction force from when it sticks to the surface until the force on the object exceeds the maximal static friction force. We derive a 1D Poincaré return map from the constrained mechanical system, and prove numerically that this 1D map has an absolutely continuous invariant measure and a positive Lyapunov exponent, providing strong evidence for chaos....................................................................................................................


Introduction
In recent years, mechanical systems with dry friction that generate stick-slip chaos have been extensively studied. We investigate the forced mechanical system proposed by Yoshitake et al. [1], which generates stick-slip vibrations, finding that it is governed by a constraint differential equation, and we prove numerically that the dynamics has a positive Lyapunov exponent, which is strong evidence of chaos.
The phenomenon of stick-slip vibration is often encountered in nonsmooth mechanical systems with dry friction [2,3,9], such as seismic isolation systems [4], disk brakes [5], and dynamic absorbers [6]. Stick-slip oscillations generate complex bifurcations and chaos and have been the subject of intense research [1][2][3]5,8,9,[11][12][13]. In general, however, rigorously proving that mechanical systems generate chaotic behavior is difficult because dynamics with a single degree of freedom is 2D, i.e., the dimension of the mechanical dynamics tends to be higher.
The simple stick-slip mechanical system proposed by Yoshitake et al. is self-exciting and chaosgenerating, and the governing equation can be represented by a 2D, nonautonomous, nonsmooth differential equation [1]. In this system, the object is constrained to the surface of a moving belt due to static friction causing it to stick to the belt. They investigated the emergence of chaos by calculating the maximum Lyapunov exponent numerically. By contrast, several rigorous studies have considered chaos in electric circuits with 1D Poincaré return maps [14][15][16]. Specifically, they have discussed constrained dynamical circuits where one of the parameters diverges. Many fruitful discussions have yielded strong evidence of chaos in 1D maps [17][18][19]. For example, sufficient conditions for ergodicity, invariant measures, and the existence of scramble sets in 1D maps have been found by Li and Yorke [17], Lasota and Yorke [18], and Li and Yorke [19], respectively.
This study investigates the forced stick-slip-generating mechanical system proposed by Yoshitake et al. [1]. They analyzed its periodically forced dynamics through the conventional technique of using a stroboscopic section as a Poincaré section, so their Poincaré map was 2D. However, we have realized that their proposed dynamics is represented by a constrained dynamical equation that can be expressed as a piecewise 1D forced dynamical equation when the oscillator is governed by a stick state. We find that the structure of this constraint equation is similar to those for the electric circuits discussed in Refs. [14][15][16], and we derive a 1D Poincaré return map for the forced mechanical oscillator. We then analyze this strictly defined 1D Poincaré return map and show numerically that it has an invariant measure that is absolutely continuous with respect to the Lebesgue measure [18], is ergodic [17], and has a positive Lyapunov exponent. To the best of our knowledge, this study is the first attempt to investigate the emergence of chaos in mechanical dynamics exactly.

Stick-slip vibration dynamics from a mechanical system
In this section, we introduce the physical system discussed in this study. The forced self-excited mechanical system shown in Fig. 1 generates stick-slip vibrations via dry friction with a single degree of freedom, and was first presented in Ref. [1].
The mechanical system consists of a stiff body of mass m that is connected to a spring with spring constant k. The body rests on a belt moving with constant velocity V and experiences both a constant vertical force P and a periodic horizontal external force F 0 cos(ωt). In addition, there is an emergent nonlinear dry friction force between the body and the moving belt surface, which can be represented as where g is the acceleration due to gravity and V r is the relative speed between the body and the surface. The friction coefficient μ(V r ) is expressed as where γ 0 , γ 1 , and γ 3 are positive constants, with the parameter γ 0 corresponding to the maximal static friction force. The dry friction force represented by Eq. (2) can be understood as follows: once the body has stuck to the belt's surface, it remains fixed to it until the absolute force on the body exceeds the maximal static friction force, i.e., either γ 0 (mg + P) or −γ 0 (mg + P). Note that physical considerations mean that γ 3 must be positive because the belt's surface friction consumes energy. By contrast, −γ 1 can be negative because the belt is moving: it behaves like a negative resistance in an LC oscillator and increases the oscillations undergone by the object on the belt. Because of the −γ 1 V r term in μ(V r ), the object can thus become self-excited via a Hopf bifurcation in the absence of perturbations. Let the body be located at the point u. The dynamical equation can then be written as where the normalized equation is expressed by where Self-excited oscillations emerge for appropriately chosen parameter values via Hopf bifurcations.

Constraint dynamics representing the mechanical oscillator
Next, we examine the structure of the dynamics expressed by Eqs. (5) and (6). Figure 3 shows the characteristics of the normalized dry friction force f 0 (y) represented by Eq. (6)  absolute friction force of A 0 . From this, we can see that the variable y is constrained to be one when the body is stuck to the belt's surface by friction. The dynamics resembles that in Refs. [14][15][16], which considered electric circuits with completely saturated circuit elements. Given this, the forced stick-slip dynamics can be rewritten as follows:  Equations (7)-(9) correspond to Eqs. (5) and (6)  Throughout this study, we set A 0 = 1.5, A 1 = 1.5, A 3 = 0.45, and F = 0.35, and choose ν as the bifurcation parameter. Note that Transition II to Eq. (7) does not take place if F is small, meaning that y is always one or less than one. In the following discussion, we concentrate on this case where the solution with y > 1 (Eq. (7)) does not occur. The periodic and chaotic attractors shown in Fig. 4 were obtained numerically from Eqs. (8) and (9) with Transitions III and IV.

Derivation of the Poincaré return map
This section discusses the derivation of the Poincaré return map. Yoshitake et al. [1] investigated a periodically forced mechanical system using the conventional method of analyzing the dynamics of Eq. (5) by defining the stroboscopic Poincaré map. Although this reduces the dynamics to a degenerate equation when the object is stuck to the belt's surface, they did not take full advantage of their dynamics' simplicity. Because the dynamics is expressed as a constraint equation, we can construct a 1D Poincaré return map. The oscillations can generate two types of attractors: stick-slip and slip-only vibrations. Our numerical results show that chaos is generated by stick-slip vibrations, and these vibrations can be analyzed precisely using the 1D Poincaré return map. We therefore analyze the stick-slip vibrations generated by Eq. (5), or Eqs. (8) and (9) with Transitions III and IV, in this study. The 1D Poincaré return map can be constructed as follows. First, we define the half-plane r ∈ R 2 and curve 1 ∈ R 1 as (10) Figure 5 shows the geometric structure of the vector fields. Here, the solution is constrained to the plane r corresponding to the belt's surface y = 1 by the static friction force, and Transition IV occurs at a curve 1 where the static friction force is maximal.
We define the solution as where the parameter set λ includes A 0 , A 1 , A 3 , F, and ν.
Let us now consider solutions that are initially located on 1 , as denoted by the solid circle in Fig. 5. When the parameter values are chosen so as to generate stick-slip vibrations, all solutions that leave 1 eventually return back to 1 again. Now, we derive the 1D Poincaré return map. Let ϕ(τ ) and φ(τ ) be solutions with initial point τ = τ 0 on 1 , meaning that they satisfy Note that x 0 is a function of τ 0 . A solution that leaves 1 strikes the plane r at (τ r , x r , y r (= 1)), marked by the solid square in Fig. 5. It is then constrained to r and strikes 1 again at (τ 1 , x 1 , y 1 ), marked by the solid triangle in Fig. 5. Note that y 1 = 1 and x 1 = A 0 + F cos(ντ 1 ).
The 1D Poincaré return map can be represented explicitly as follows. Let us consider a projection h −1 r1 from 1 = [0, 1) to 1 , namely In addition, let us consider the following mapping from an initial point (τ 0 , x 0 , y 0 ) on 1 to the point (τ r , x r , y r ) on r where the solution that leaves the initial point returns to r : The projection h r can be defined as The mapping T 1 from a point on r to the point on 1 where a solution that leaves the solid square point in Fig. 5 returns to the solid triangle can be written as where τ 1 is the time when the solution reaches the solid triangle in Fig. 5. Since x 1 is a function of τ 1 , we can define the projection h 1 as Using Eqs. (13)-(17), the 1D Poincaré return map can now be expressed as where T (θ) can be defined naturally as The stick-slip vibration behavior can now be explained using this Poincaré return map. In our calculations, T was actually obtained numerically by solving Eqs. (8) and (9) with Transitions III and IV using the Runge-Kutta method. Figure 6 shows a bifurcation diagram for T in terms of the single bifurcation parameter ν. Nonsmooth bifurcations can be observed, which are caused by the fact that the dry friction force is not smooth. In addition, a nonsmooth bifurcation transition from a periodic to a chaotic solution can be observed at around ν 0.605.  Fig. 7(a.2). In this study, however, we concentrate on demonstrating chaos numerically by finding a positive Lyapunov exponent for the ν = 0.602 map (Fig. 7(c)).  Fig. 7(c.2). Let us denote the right-hand discontinuous point of θ as θ 0 , as shown in Fig. 8(b). The bottom-left and bottom-right plots in Fig. 8(b) show the trajectories that depart from θ = 0.845 > θ 0 and θ = 0.86 < θ 0 , respectively. The trajectory leaving θ = 0.845 is doubly twisted in the x-y plane. Because the object has mass m and the vector fields are continuous,  the following relationships hold in the vicinity of the discontinuous point θ = θ 0 :

Bifurcation analysis of the stick-slip vibrations using the Poincaré return map
where T n represents T applied n times. It is clear from Fig. 8 becomes an invariant interval of T . Let us consider T 2 on J with ν = 0.602, as shown in Fig. 9. From this, it is clear that the following inequality holds: Therefore, when combined with a result in Ref. [18], we find that T 2 (θ) has an absolutely continuous invariant measure. This numerical result indicates that T has a positive Lyapunov exponent, which is strong evidence for chaos.

Conclusion
In this paper, we have investigated stick-slip vibrations generated in a mechanical system with dry friction. Based on the dynamical model of a mechanical oscillator presented by Yoshitake et al., we have derived a 1D Poincaré return map. Our numerical results show that the absolute differential coefficient of the composite map consisting of the return map applied twice was greater than one in the invariant interval. This numerical result indicates a positive Lyapunov exponent, and hence provides strong evidence for chaos. To the best of our knowledge, this study is the first attempt to demonstrate strong evidence for chaos in a mechanical system.