# Analysis of Discontinuous Dynamical Behaviors of a Friction-Induced Oscillator with an Elliptic Control Law

Authors: Jinjun Fan; Ping Liu; Tianyi Liu; Shan Xue; Zhaoxia Yang
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

This paper develops the passability conditions of flow to the discontinuous boundary and the sticking or sliding and grazing conditions to the separation boundary in the discontinuous dynamical system of a friction-induced oscillator with an elliptic control law and the friction force acting on the massM through the analysis of the corresponding vector fields and G-functions. The periodic motions of such a discontinuous system are predicted analytically through the mapping structure. Finally, the numerical simulations are given to illustrate the analytical results of motion for a better understanding of physics of motion in the mass-spring-damper oscillator.


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## 1. Introduction

Discontinuous dynamical systems extensively exist in mechanics and engineering, such as turbine blades, dry friction, and impact processes. Luo and Rapp [1, 2] investigated different periodic motions in a periodic force discontinuous system. The switching boundary in the aforementioned papers was supposed to be an inclined line or a parabolic curve and the friction force acting on the oscillator was not considered. In this paper, the motions in the discontinuous dynamical system of a friction-induced oscillator with an elliptic control law will be investigated.In mechanical engineering, the friction contact between two surfaces of two bodies is an important connection and friction phenomenon widely exists. In recent years, much research effort in science and engineering has focussed on nonsmooth dynamical systems. This problem can go back to the 30s of last century. In 1930, Hartog [3] investigated the nonstick periodic motion of the forced linear oscillator with Coulomb and viscous damping. In 1960, Levitan [4] proved the existence of periodic motions in a friction oscillator with the periodically driven base. Discontinuous systems are usually described by ordinary differential equations with discontinuous right-hand sides. Filippov presented the basic concepts for such discontinuous differential equations and mainly discussed the existence and uniqueness of solutions for discontinuous dynamical systems in [5] in 1964 and gave the systematical summarization for such discontinuous differential equations in [6] in 1988. Since then, many authors have used and tried to extend the Filippov’s theory to investigate the flow passability and motion complexity in discontinuous dynamical systems. For instance, in 1995, Popp et al. [7] investigated dynamical behavior of a friction oscillator with simultaneous self-excitation and external excitation. In 1996, Oestreich et al. [8] studied the bifurcation and stability analysis for a nonsmooth friction oscillator. In 1997, Kunze et al. [9] employed the KAM theory to analyze the periodic motions for a forced oscillator with a jump of restoring force. In 2003, Awrejcewicz and Olejnik [10] studied stick-slip dynamics of a two-degree-of-freedom system. In 2003 and 2012, Cid and Sanchez [11] and Jacquemard and Teixeira [12] used an approximation procedure and the method of lower and upper solution to obtain the existence conditions of typical periodic solutions for some nonautonomous second-order differential equations with a jump discontinuity. In 2001 and 2006, Kupper and Moritz [13] and Zou et al. [14] reported the possibility of Hopf bifurcations in planar discontinuous dynamical systems from the aspects of stability change of an equilibrium point. In 2014, Pascal [15] discussed a system composed of two masses connected by linear springs: one of the masses is in contact with a rough surface and the other is also subjected to a harmonic external force. Several periodic orbits were obtained in closed form, and symmetry in space and time was proved for some of these periodic solutions. For more discussion about discontinuous system, refer to [16–32].On the other hand, the determination of periodic orbits along the switching boundary is very important in the nonsmooth control theory. In 2003, Bernardo et al. [33] showed that sliding bifurcations play an important role in analyzing the dynamics of dry friction oscillator. In 2004, Galvanetto [34] presented several detailed examples to address the fact that the occurrence of sliding bifurcation does not always correspond to the change in the number of periodic orbits at the bifurcation point. In 2005 and 2008, Kowalczyk et al. [35, 36] were concerned with the extension to the case of codimension 2 degenerate sliding bifurcations, numerical continuation, and analytical investigations of sliding bifurcations in Filippov systems. In 2010, Guardia et al. [37] analytically studied the sliding bifurcations of periodic orbits in a dry friction oscillator.Recently, new development in discontinuous dynamical systems, such as the motion switchability for stick and sliding and grazing motions, has been found. In 2005, Luo [38, 39] developed a theory for nonsmooth dynamical systems on connectable domains and the mapping dynamics of periodic motions for a piecewise linear system under a periodic excitation. In 2005, 2006 and 2008, Luo [40–42] introduced the concepts of imaginary, sink, and source flows in nonsmooth dynamical systems and developed a general theory for the local singularity of a flow to the discontinuous boundary and a theory for flow switchability in discontinuous dynamical systems. In 2006, Luo and Gegg [43] investigated periodic motions in an oscillator moving on a periodically vibrating belt with dry friction. In 2009, Gegg et al. [44] presented the analytical conditions for sliding and passable motions on the periodically time-varying boundary for a friction-induced oscillator through the relative force product. In 2011, Luo [45] systematically presented a new theory for flow barriers in discontinuous dynamical systems, which provides a theoretic base to further develop control theory and stability. As to applications of the aforementioned local singularity theory, Luo and Huang [46] used such theory to determine the flow switchability on the discontinuous boundary for the nonlinear, friction-induced, periodically forced oscillator in 2012. In 2015 and 2017, Zhang and Fu [25, 47, 48] studied the periodic motions, stick motions, grazing flows in an inclined impact oscillator, and the flow switchability of motions in a horizontal impact pair with dry friction. In 2016, Chen and Fan [49] presented the analytical conditions for the motion switchability in a double belt friction oscillator system with a periodic excitation. In 2018, Fan et al. [50] further investigated dynamics of such double belt friction oscillator system with a periodic excitation. In 2017 and 2018, Fan et al. [51, 52] studied a friction-induced oscillator with two degrees of freedom on a speed-varying traveling belt and discontinuous dynamical behaviors in a vibroimpact system with multiple constraints, respectively.In study of dynamical system, the switching control law also plays an important role. Some researchers have obtained some results by using the theory of discontinuous dynamical systems developed by Luo. In 2009, Luo and Rapp [1] investigated the flow switchability and periodic motions in a periodically forced, discontinuous dynamical system with switching control law of an inclined line. In the next year, Luo and Rapp [2] studied the motions and switchability of an oscillator in a periodically forced, discontinuous system with switching control law of a parabolic boundary. In 2015, Zheng and Fu [53] proposed the switched Van der Pol equation with impulsive effect as switched system and analyzed its features from a discontinuous point of view. In 2017, Fan et al. [54] studied discontinuous dynamics of a friction-induced oscillator with switching control law of a straight line. For the examples of impulsive control and Boolean control as switching control law, see [55–68]. It is worth noting that the switching control law of an elliptic boundary has an important application in practical problem, such as satellite rendezvous and formation flight. The elliptical controller is designed as a reference orbit for formation flight. By selecting the appropriate parameters to design the controller, the orbit of the satellite can be controlled so that it can conduct sliding or periodic motion in the reference orbit. This can effectively improve flight efficiency and save fuel. Compared with the circular reference orbit, the flight mode of elliptical reference orbit has the advantages which can reach the earth layer boundary region of space physics research, travel through specific spaces for longer periods of time, collect more information, and realize multipoint synchronous measurement by three-dimensional configuration and so on. Some scholars do a lot of work. For example, Senqupta [69] investigated elliptic rendezvous in the chaser satellite frame in 2012. In the same year, Chang et al. [70] studied the transfer of satellites between elliptic Keplerian orbits using Lyapunov stability theory. In 2013, Yan et al. [71] investigated pseudospectral feedback control for three-axis magnetic attitude stabilization in elliptic orbits. In 2012, Do [72] presented a design of cooperative controllers that force a group of N mobile agents with an elliptical shape and with limited sensing ranges to perform a desired formation.In this paper, the analytical conditions for motion switchability on the discontinuous boundaries in a periodically forced, discontinuous system with an elliptic control law and the friction force acting on the massM will be developed through the G-functions of the vector fields to the discontinuous boundaries. The rest of the paper is organized in the following manner. In Section 2, the physical model of a periodically forced, linear, friction-induced oscillator is introduced by using mechanical models. In Section 3, different domains and boundaries are defined due to the elliptic switching control law and friction discontinuities. The G-functions are introduced to discuss the motion switchability, and the analytical conditions for the switching conditions of the passable motions, stick or sliding motions, and grazing motions are derived mathematically in Section 4. In Section 5, the switching sets and two-dimensional mappings are defined to describe the complex motions in the friction-induced oscillator with an elliptic control law and the friction acting on the mass. Further, the periodic motions, such as sliding periodic motions and stick periodic motions, are analytically predicted. Numerical simulations are carried out to illustrate the analytical conditions of motion switchability for a better understanding of the complex dynamical behaviors in Section 6. Finally, Section 7 concludes the paper.


## 2. Mechanical Model

Consider a periodically forced, linear, friction-induced oscillator, as shown in Figure1(a). Denote displacement by x and time by t, and assume that the switching boundary in this discontinuous dynamic system is (1)x2a2+x˙2b2=1,where a>0, b>0, and x˙=dx/dt, as sketched in Figure 1(b). This elliptical switching boundary will allow the system to switch from one dynamic system to another when the oscillator meets certain conditions. Assume also that the oscillator consists of a mass M, a switching spring of stiffness ki(i∈{1,2}), and a switchable damper of coefficient γi(i∈{1,2}) in the domain Di(i∈{1,2}), where D1 is the unbounded outside domain and D2 is the bounded inside domain. The periodic force acting on the mass M in the different domains is (2)Pit=Q0cos⁡Ωt+φ+Bifori∈1,2,where Q0, Ω, φ, and Bi are excitation amplitude, frequency, initial phase, and constant force, respectively. Through prestressing the mass M, the constant-force magnitude of Bi could be adjusted to adapt to different working environments. Furthermore, the mass M slides or rests on the conveyor belt with a constant speed V.Figure 1
(a) Mechanical model; (b) elliptic control law’s boundary.
(a) (b)Since the massM contacts the conveyor belt with friction, the mass M can move along or rest on the conveyor belt. Further, a kinetic friction force is given by (3)Ffx˙=μFN,x˙>V,∈-μFN,μFN,x˙=V,=-μFN,x˙<V,where FN=Mg is the normal force between the mass M and conveyor belt. The frictional force is sketched in Figure 2.Figure 2
Friction force.The equation of motion for such a friction-induced oscillator in the domainDi is (4)x¨+2dix˙+cix=A0cos⁡Ωt+φ+bi-fsgn⁡x˙-Vfori∈1,2,where(5)ci=kiM,di=γi2M,A0=Q0M,bi=BiM,f=μFNM.The nonfrictional force per unit mass in the domainDi is (6)Fnfi=A0cos⁡Ωt+φ+bi-2dix˙-cixfori∈1,2.The motions of the massM in the domain Di are divided into three cases.(i) If|Fnf(i)|≥ff(i) and the displacement and velocity of the mass M do not satisfy the elliptic control law (i.e., x2/a2+x˙2/b2≠1), the corresponding motion is called the nonstick motion or free-flight motion. For this case, the equation of motion is given as follows: (7)x¨+2dix˙+cix=A0cos⁡Ωt+φ+bi-fsgn⁡x˙-Vfori∈1,2,x˙≠V.(ii) If the massM moves together with the conveyor belt (i.e., Fnfi<ff(i)) and the displacement and velocity of the mass M do not satisfy the elliptic control law (i.e., x2/a2+x˙2/b2≠1), the corresponding motion is called the stick motion. For this case, the equation of motion is given by (8)x˙=V,x¨=0.(iii) If the displacement and velocity of the massM satisfy the elliptic control law (i.e., x2/a2+x˙2/b2=1), the corresponding motion is called the sliding motion. For this case, the equation of motion is described by (9)x2a2+x˙2b2=1.


## 3. Domains and Boundaries

Due to the elliptic control law and the friction between the massM and the conveyor belt, the motions of the mass M become discontinuous and more complicated. In order to determine the switching complexity for the motions of the mass M, different domains and boundaries in the absolute coordinates are defined in this section.The origin of the absolute coordinates is set at the equilibrium position of the massM. Based on the discontinuities caused by the switching control law and the friction between the mass M and the conveyor belt, the phase plane for this discontinuous dynamic system is partitioned into four different domains and four boundaries, as shown in Figure 3. In each domain, the motion of the mass M can be described through a continuous dynamical system.Figure 3
Domains and boundaries.Set(10)φx,x˙=x2a2+x˙2b2-1,V<b.The four nonstick domains for the massM can be expressed as (11)Ω1=x,x˙∣φx,x˙=x2a2+x˙2b2-1>0,x˙>V,Ω2=x,x˙∣φx,x˙=x2a2+x˙2b2-1<0,x˙>V,Ω3=x,x˙∣φx,x˙=x2a2+x˙2b2-1<0,x˙<V,Ω4=x,x˙∣φx,x˙=x2a2+x˙2b2-1>0,x˙<V.The corresponding boundaries, including two stick or velocity boundaries and two sliding or control boundaries, can be defined as(12)∂Ω12=∂Ω21=Ω¯1∩Ω¯2=x,x˙∣φ12x,x˙=φ21x,x˙=x2a2+x˙2b2-1=0,x˙-V>0,∂Ω23=∂Ω32=Ω¯2∩Ω¯3=x,x˙∣φ23x,x˙=φ32x,x˙=x˙-V=0,x2a2+x˙2b2-1<0,∂Ω34=∂Ω43=Ω¯3∩Ω¯4=x,x˙∣φ34x,x˙=φ43x,x˙=x2a2+x˙2b2-1=0,x˙-V<0,∂Ω14=∂Ω41=Ω¯1∩Ω¯4=x,x˙∣φ14x,x˙=φ41x,x˙=x˙-V=0,x2a2+x˙2b2-1>0.For the nonstick motions in domains, two vectors are introduced as(13)xλ=xλ,x˙λT=xλ,yλT,Fλ=x˙λ,FλT=yλ,FλT,where λ=1,2,3,4. Using (13), the equations of nonstick motions are rewritten in the vector form of (14)x˙λ=Fλxλ,tforλ∈1,2,3,4,where(15)Fλxλ,t=A0cos⁡Ωt+φ+bλ-fsgn⁡y-V-2dλy-cλx,b1=b4=B1M,b2=b3=B2M,d1=d4=γ12M,d2=d3=γ22M,c1=c4=k1M,c2=c3=k2M.For the stick motions on velocity boundaries, two vectors are introduced as(16)xα1α2=xα1α2,x˙α1α2T=xα1α2,yα1α2T,Fα1α2=x˙α1α2,Fα1α2T=yα1α2,Fα1α2T,where (α1,α2)∈{(2,3),(3,2),(1,4),(4,1)}. The equations of stick motions are rewritten in the vector form of (17)x˙α1α2=Fα1α2xα1α2,tforα1,α2∈2,3,3,2,1,4,4,1,where(18)Fα1α2xα1α2,t=0for  stick  motion,Fα1α2xα1α2,t∈Fα1xα1,t,Fα2xα2,tfor  nonstick  motion.For the sliding motions on elliptic boundaries, two vectors are introduced as(19)xα3α4=xα3α4,x˙α3α4T=xα3α4,yα3α4T,Fα3α4=x˙α3α4,Fα3α4T=yα3α4,Fα3α4T,where (α3,α4)∈{(1,2),(2,1),(4,3),(3,4)}. The equations of sliding motions are rewritten in the vector form of (20)x˙α3α4=Fα3α4xα3α4,tforα3,α4∈1,2,2,1,4,3,3,4,where(21)Fα3α4xα3α4,t=-b2a2xα3α4for  sliding  motion,Fα3α4xα3α4,t∈Fα3xα3,t,Fα4xα4,tfor  non-sliding  motion.


## 4. Analytical Conditions

According to the theory of flow switchability to a specific boundary in discontinuous dynamic systems, the switching conditions of the passable motions, stick or sliding motions, and grazing motions of the massM will be developed in this section.


### 4.1. Basic Theory

For convenience, the concepts of theG-functions and some lemmas are given in the following (see [42]).Consider a dynamic system consisting ofN subdynamic systems in a universal domain ℧∈Rn. The universal domain is divided into N accessible subdomains Ωi and inaccessible subdomains Ω0. And the union of all the accessible subdomains is ⋃i=1NΩi and the universal domain is ℧=⋃i=1NΩi∪Ω0. On the ith open subdomain Ωi, there is a Cri-continuous system (ri≥1) in the form of(22)x˙i≡Fixi,t,Pi∈Rn,xi=x1i,x2i,…,xniT∈Ωi,where the time is t and x˙(i)=dx(i)/dt. In an accessible subdomain Ωi, the vector field F(i)(x(i),t,Pi) with a parameter vector Pi=(pi(1),pi(2),…,pi(l))T∈Rl is Cri-continuous (ri≥1) for all time t, and the flow in (22) xt(i)=Φ(i)(t0,x0(i),Pi,t) with an initial condition (t0,x0(i)) is Cri+1 continuous for time t, and on the boundary of two adjacent domains ∂Ωαβ=Ω¯α∩Ω¯β={x∣φαβ(x,t,λ)=0, φαβ is Cr-continuous (r≥1)}⊂Rn-1, and there is a flow xt(0)=Φ(0)(t0,x0(0),λ,t), which can be determined by (23)x˙0≡F0x0,t,λ∈Rn,x0=x10,x20,…,xn0T∈∂Ωαβwith an initial condition (t0,x0(0)) and a parameter vector λ=(λ1,λ2,…,λk)T∈Rk.Assume that there are two adjacent subdomainsΩα and Ωβ. A flow in subdomain Ωi(i∈{α,β}) is called a local flow if it is governed by the continuous vector field in Ωi(i∈{α,β}) only. A flow is called a passable flow if it switches from a subdomain Ωi(i∈{α,β}) into another one Ωj(j∈{α,β},j≠i) through the boundary ∂Ωij(i,j∈{α,β},i≠j). Obviously, the vector field of a passable flow in subdomain Ωi(i∈{α,β}) will be changed into the one in subdomain Ωj(j∈{α,β},j≠i), accordingly. A flow is called a grazing flow if it is tangential to the boundaries ∂Ωαβ. If a flow cannot pass through but moves along the boundaries ∂Ωαβ, then this flow is called a sliding flow.Definition 1 (see [42]).
The 0th-orderG-functions of the domain flow xt(i)(i∈{α,β}) to the boundary flow xt(0) on the boundary in the normal direction of the boundary ∂Ωαβ are defined as (24)G∂Ωαβixt0,t±,xt±i,Pi,λ≡G∂Ωαβ0,ixt0,t±,xt±i,Pi,λ=Dxt0nt∂ΩαβT·xt±i-xt0+nt∂ΩαβT·x˙t±i-x˙t0.Definition 2 (see [42]).
The1st-order G-functions of the domain flow xt(i)(i∈{α,β}) to the boundary flow xt(0) on the boundary in the normal direction of the boundary ∂Ωαβ are defined as (25)G∂Ωαβ1,ixt0,t±,xt±i,Pi,λ=Dxt02nt∂ΩαβT·xt±i-xt0+2Dxt0nt∂ΩαβT·x˙t±i-x˙t0+n∂ΩαβTt·x¨t±i-x¨t0.In above definitions, the total derivativeDxt0(·)≔∂(·)/∂xt0·x˙t0+∂(·)/∂t, and the normal vector of the boundary surface ∂Ωαβ at point xt(0) is given by (26)nt∂Ωαβxt0,t,λ=▽φαβxt0,t,λ=∂φαβ∂x10,∂φαβ∂x20,…,∂φαβ∂xn0Tt,xt0,and t±=t±0.If the flowxt(i)(i∈{α,β}) contacts with the boundary at time tm, that is, xtm(i)=xm=xtm(0), and the boundary ∂Ωαβ is linear, independent of time t, we can obtain the 0th-order G-functions and the 1st-order G-functions by simple computation (27)G∂Ωαβ0,ixm,tm±,Pi,λ=G∂Ωαβ0,ixtm,tm±,xtm±i,Pi,λ=nt∂ΩαβT·Fixti,t,Pixm0,xm±i,tm±,(28)G∂Ωαβ1,ixm,tm±,Pi,λ=G∂Ωαβ1,ixtm,tm±,xtm±i,Pi,λ=nt∂ΩαβT·DxtiFixti,t,Pixm0,xm±i,tm±.Here tm±=tm±0 is to show the motion in different domains rather than on the boundaries, and tm-, tm+ are the time before approaching and after departing the corresponding boundary, respectively. In this case, the G-functions are simplified as G∂Ωαβ(0,i)(xm,tm±) and G∂Ωαβ(1,i)(xm,tm±).For a discontinuous dynamical system in (22), there is a point x(0)(tm)≡xm∈∂Ωαβ at time tm between two adjacent domains Ωi(i∈{α,β}). For an arbitrarily small ε>0, there are two time intervals [tm-ε,tm) and (tm,tm+ε]. Suppose x(i)(tm-)=xm=x(j)(tm+)(i,j∈{α,β}). Both flows x(i)(t) and x(j)(t) are C[tm-ε,tm)ri- and C(tm,tm+ε]rj-continuous for time t(i,j∈{α,β}), and dri+1x(i)/dtri+1<∞(ri≥1,i∈{α,β}).Based on theG-functions, the decision theorems of semipassable flow, sink flow, tangential flow, sliding bifurcation, and sliding fragmentation bifurcation to the separation boundary are stated in the form of lemma as follows.Lemma 3 (see [42]).
The flowsx(α)(t) and x(β)(t) at point (xm,tm) on the boundary ∂Ωαβ are semipassable from domain Ωα to domain Ωβ if and only if (29)eitherG∂Ωαβαxm,tm-,Pα,λ>0,G∂Ωαββxm,tm+,Pβ,λ>0,forn∂Ωαβ⟶Ωβ,orG∂Ωαβαxm,tm-,Pα,λ<0,G∂Ωαββxm,tm+,Pβ,λ<0,forn∂Ωαβ⟶Ωα.Lemma 4 (see [42]).
The flowsx(α)(t) and x(β)(t) at point (xm,tm) on the boundary ∂Ωαβ are sink flows if and only if (30)eitherG∂Ωαβαxm,tm-,Pα,λ>0,G∂Ωαββxm,tm-,Pβ,λ<0,forn∂Ωαβ⟶Ωβ,orG∂Ωαβαxm,tm-,Pα,λ<0,G∂Ωαββxm,tm-,Pβ,λ>0,forn∂Ωαβ⟶Ωα.Lemma 5 (see [42]).
For an arbitrarily smallε>0, there is a time interval [tm-ε,tm+ε]. Suppose x(i)(tm±)=xm. The flow x(i)(t) is C[tm-ε,tm+ε]ri-continuous for time t, and dri+1x(i)/dtri+1<∞(ri≥2,i∈{α,β}). A flow x(i)(t) in Ωi is tangential to the boundary ∂Ωαβ if and only if (31)G∂Ωαβ0,ixm,tm,Pi,λ=0,eitherG∂Ωαβ1,ixm,tm,Pi,λ<0forn∂Ωαβ⟶Ωj,orG∂Ωαβ1,ixm,tm,Pi,λ>0forn∂Ωαβ⟶Ωi,where i,j∈{α,β},i≠j.Lemma 6 (see [42]).
For an arbitrarily smallε>0, there are two time intervals [tm-ε,tm) and [tm-ε,tm+ε]. Suppose x(α)(tm-)=xm=x(β)(tm±). The flows x(α)(t) and x(β)(t) are C[tm-ε,tm)rα- and C[tm-ε,tm+ε]rβ-continuous for time t and dri+1x(i)/dtri+1<∞(ri≥2andi∈{α,β}). The sliding bifurcation of the passable flows x(α)(t) and x(β)(t) at point (xm,tm) switching to the nonpassable flow of the first kind on the boundary ∂Ω→αβ occurs if and only if (32)G∂Ωαββxm,tm±,Pβ,λ=0,G∂Ωαβαxm,tm-,Pα,λ>0forn∂Ωαβ⟶Ωβ,G∂Ωαβαxm,tm-,Pα,λ<0forn∂Ωαβ⟶Ωα,G∂Ωαβ1,βxm,tm±,Pβ,λ>0forn∂Ωαβ⟶Ωβ,G∂Ωαβ1,βxm,tm±,Pβ,λ<0forn∂Ωαβ⟶Ωα.Lemma 7 (see [42]).
For an arbitrarily smallε>0, there are two time intervals [tm-ε,tm) and [tm-ε,tm+ε]. Suppose x(α)(tm-)=xm=x(β)(tm±). The flows x(α)(t) and x(β)(t) are C[tm-ε,tm)rα- and C[tm-ε,tm+ε]rβ-continuous for time t and dri+1x(i)/dtri+1<∞(ri≥2andi∈{α,β}). The tangential bifurcation of the flows x(α) and x(β) at point (xm,tm) on the boundary ∂Ω~αβ is termed the sliding fragmentation bifurcation if and only if (33)G∂Ωαββxm,tm∓,Pβ,λ=0,G∂Ωαβαxm,tm-,Pα,λ>0forn∂Ωαβ⟶Ωβ,G∂Ωαβαxm,tm-,Pα,λ<0forn∂Ωαβ⟶Ωα,G∂Ωαβ1,βxm,tm∓,Pβ,λ>0forn∂Ωαβ⟶Ωβ,G∂Ωαβ1,βxm,tm∓,Pβ,λ<0forn∂Ωαβ⟶Ωα.More detailed theory on the flow switchability such as high-orderG-functions, the definitions, or the decision theorems about various flow passability in discontinuous dynamical systems can be found in [42].


### 4.2. Analytical Conditions

According to the previous definitions and lemmas, the analytical switching conditions on discontinuous boundaries in the friction-induced oscillator with an elliptic control law and the friction force acting on the massM described in Section 3 will be developed in this subsection. From (12), the normal vectors of the absolute boundaries are given as (34)n∂Ωαβ=n∂Ωβα=∇φαβ=∂φαβ∂x,∂φαβ∂x˙T=0,1T,for (α,β)∈{(2,3),(3,2),(1,4),(4,1)}, and (35)n∂Ωαβ=n∂Ωβα=a2b22∇φαβ=a2b22∂φαβ∂x,∂φαβ∂x˙T=b2x,a2yT,for (α,β)∈{(1,2),(2,1),(3,4),(4,3)}, where ∇=(∂/∂x,∂/∂y)T is a Hamilton operator.Theorem 8.
For the friction-induced oscillator with an elliptic control law and the friction force acting on the massM discussed in Section 2, there are the following results.
(i) The necessary and sufficient conditions for passable motion on the boundaryxm≡(xm,ym)∈∂Ω12 at time tm are (36)eitherb2xmym+a2ymF1xm,tm-<0,b2xmym+a2ymF2xm,tm+<0,forΩ1⟶Ω2,(37)orb2xmym+a2ymF2xm,tm->0,b2xmym+a2ymF1xm,tm+>0,forΩ2⟶Ω1.
(ii) The necessary and sufficient conditions for passable motion on the boundaryxm≡(xm,ym)∈∂Ω23 at time tm are (38)eitherF2xm,tm-<0,F3xm,tm+<0,forΩ2⟶Ω3,(39)orF3xm,tm->0,F2xm,tm+>0,forΩ3⟶Ω2.
(iii) The necessary and sufficient conditions for passable motion on the boundaryxm≡(xm,ym)∈∂Ω34 at time tm are(40)eitherb2xmym+a2ymF3xm,tm->0,b2xmym+a2ymF4xm,tm+>0,forΩ3⟶Ω4,(41)orb2xmym+a2ymF4xm,tm-<0,b2xmym+a2ymF3xm,tm+<0,forΩ4⟶Ω3.
(iv) The necessary and sufficient conditions for passable motion on the boundaryxm≡(xm,ym)∈∂Ω14 at time tm are(42)eitherF1xm,tm-<0,F4xm,tm+<0,forΩ1⟶Ω4,(43)orF4xm,tm->0,F1xm,tm+>0,forΩ4→Ω1.Proof.
The massM moves along the conveyor belt with nonzero relative velocity before time tm, and the relative velocity becomes zero at tm; then after tm the relative velocity changes its direction. From the flow switchability theory on the discontinuous dynamical systems, the flow in domain Ωλ(λ∈{α,β}) or Ωγ(γ∈{i,j}) passes through the boundary ∂Ωαβ(αβ∈{12,21,34,43}) or ∂Ωij(ij∈{23,32,14,41}) into domain Ωλ¯(λ¯∈{α,β},λ¯≠λ) or Ωγ¯(γ¯∈{i,j},γ¯≠γ), which can be expressed by Lemma 3. So 0th-order G-functions on such boundaries are needed.
According to (27), the 0th-order G-functions on the boundary ∂Ω12 are(44)G∂Ω120,1xm,tm±=n∂Ω12T·F1xm,tm±,G∂Ω120,2xm,tm±=n∂Ω12T·F2xm,tm±.From (13) and (35), (44) can be computed as(45)G∂Ω120,1xm,tm-=b2xmym+a2ymF1xm,tm-,G∂Ω120,2xm,tm+=b2xmym+a2ymF2xm,tm+.By Lemma 3, the passable motion on the boundary xm∈∂Ω12 at time tm appears if and only if (46)G∂Ω120,1xm,tm-<0,G∂Ω120,2xm,tm+<0,forΩ1⟶Ω2.From (45) and (46), one obtains (47)b2xmym+a2ymF1xm,tm-<0,b2xmym+a2ymF2xm,tm+<0,forΩ1→Ω2.Thus, (36) in (i) holds, and (37) in (i) and (40) and (41) in (iii) can be proved similarly.
From (27), the 0th-order G-functions on the boundary ∂Ω23 are (48)G∂Ω230,2xm,tm±=n∂Ω23T·F2xm,tm±,G∂Ω230,3xm,tm±=n∂Ω23T·F3xm,tm±.From (13) and (34), (48) can be computed as (49)G∂Ω230,2xm,tm-=F2xm,tm-,G∂Ω230,3xm,tm+=F3xm,tm+.By Lemma 3, the passable motion on the boundary xm∈∂Ω23 at time tm appears if and only if (50)G∂Ω230,2xm,tm-<0,G∂Ω230,3xm,tm+<0,forΩ2⟶Ω3.By (49) and (50), one obtains (51)F2xm,tm-<0,F3xm,tm+<0,forΩ2⟶Ω3.Thus, (38) in (ii) holds, and (39) in (ii) and (42) and (43) in (iv) can be proved similarly.Theorem 9.
For the friction-induced oscillator with an elliptic control law and the friction force acting on the massM discussed in Section 2, there are the following results.
(i) The necessary and sufficient conditions for sliding motion on the boundaryxm≡(xm,ym)∈∂Ω12 at time tm are (52)b2xmym+a2ymF1xm,tm-<0,b2xmym+a2ymF2xm,tm->0.
(ii) The necessary and sufficient conditions for stick motion on the boundaryxm≡(xm,ym)∈∂Ω23 at time tm are (53)F2xm,tm-<0,F3xm,tm->0.
(iii) The necessary and sufficient conditions for sliding motion on the boundaryxm≡(xm,ym)∈∂Ω34 at time tm are (54)b2xmym+a2ymF3xm,tm->0,b2xmym+a2ymF4xm,tm-<0.
(iv) The necessary and sufficient conditions for stick motion on the boundaryxm≡(xm,ym)∈∂Ω14 at time tm are (55)F1xm,tm-<0,F4xm,tm->0.Proof.
The displacement and velocity of the massM satisfy the elliptic control law; such motion is called the sliding motion. From the flow switchability theory on the discontinuous dynamical systems, the sliding motion is that the flow in domain Ωλ(λ∈{α,β}) reaches the boundary ∂Ωαβ(αβ∈{12,21,34,43}) and moves along the boundary ∂Ωαβ(αβ∈{12,21,34,43}). Thus it can be predicted by Lemma 4.
From (27), the 0th-order G-functions on the boundary ∂Ω12 are(56)G∂Ω120,1xm,tm±=n∂Ω12T·F1xm,tm±,G∂Ω120,2xm,tm±=n∂Ω12T·F2xm,tm±.From (13) and (35), (56) can be computed as (57)G∂Ω120,1xm,tm-=b2xmym+a2ymF1xm,tm-,G∂Ω120,2xm,tm-=b2xmym+a2ymF2xm,tm-.By Lemma 4, the sliding motion on the boundary xm∈∂Ω12 at time tm appears if and only if (58)G∂Ω120,1xm,tm-<0,G∂Ω120,2xm,tm->0.From (57) and (58), one obtains (59)b2xmym+a2ymF1xm,tm-<0,b2xmym+a2ymF2xm,tm->0.Thus, (i) holds, and (iii) can be proved similarly.
The massM moves together with the conveyor belt and the displacement and velocity of the mass M do not satisfy the elliptic control law; such motion is called the stick motion. From the flow switchability theory on the discontinuous dynamical systems, the stick motion is that the flow in domain Ωγ(γ∈{i,j}) reaches the boundary ∂Ωij(ij∈{23,32,14,41}) and moves along the boundary ∂Ωij(ij∈{23,32,14,41}). Thus it can be predicted by Lemma 4.
By (27), the 0th-order G-functions on the boundary ∂Ω23 are(60)G∂Ω230,2xm,tm±=n∂Ω23T·F2xm,tm±,G∂Ω230,3xm,tm±=n∂Ω23T·F3xm,tm±.From (13) and (34), (60) can be computed as(61)G∂Ω230,2xm,tm-=F2xm,tm-,G∂Ω230,3xm,tm-=F3xm,tm-.By Lemma 4, the stick motion on the boundary xm∈∂Ω23 at time tm appears if and only if (62)G∂Ω230,2xm,tm-<0,G∂Ω230,3xm,tm->0.From (61) and (62), one obtains (63)F2xm,tm-<0,F3xm,tm->0.Thus, (ii) holds, and (iv) can be proved similarly.Theorem 10.
For the friction-induced oscillator with an elliptic control law and the friction force acting on the massM discussed in Section 2, there are the following results.
(i) The necessary and sufficient conditions for grazing motion on the boundaryxm≡(xm,ym)∈∂Ω12 in domain Ω1 or domain Ω2 at time tm are (64)b2xmym+a2ymF1xm,tm±=0,b2xmF1xm,tm±+a2ymDF1xm,tm±>0,inΩ1for∂Ω12,or(65)b2xmym+a2ymF2xm,tm±=0,b2xmF2xm,tm±+a2ymDF2xm,tm±<0,inΩ2for∂Ω12.
(ii) The necessary and sufficient conditions for grazing motion on the boundaryxm≡(xm,ym)∈∂Ω34 in domain Ω3 or domain Ω4 at time tm are (66)b2xmym+a2ymF3xm,tm±=0,b2xmF3xm,tm±+a2ymDF3xm,tm±<0,inΩ3for∂Ω34,or(67)b2xmym+a2ymF4xm,tm±=0,b2xmF4xm,tm±+a2ymDF4xm,tm±>0,inΩ4for∂Ω34.
(iii) The necessary and sufficient conditions for grazing motion on the boundaryxm≡(xm,ym)∈∂Ω14 in domain Ω1 or domain Ω4 at time tm are (68)F1xm,tm±=0,DF1xm,tm±>0,inΩ1for∂Ω14,or(69)F4xm,tm±=0,DF4xm,tm±<0,inΩ4for∂Ω14.
(iv) The necessary and sufficient conditions for grazing motion on the boundaryxm≡(xm,ym)∈∂Ω23 in domain Ω2 or domain Ω3 at time tm are(70)F2xm,tm±=0,DF2xm,tm±>0,inΩ2for∂Ω23,or(71)F3xm,tm±=0,DF3xm,tm±<0,inΩ3for∂Ω23.Proof.
When the massM is moving along the conveyor belt with nonzero relative velocity, the relative velocity equals zero at time tm and then it restores to the original relationship. From the flow switchability theory on the discontinuous dynamical systems, the flow of the motion in domain Ωλ(λ∈{α,β}) or Ωγ(γ∈{i,j}) reaches the boundary ∂Ωαβ(αβ∈{12,21,34,43}) or ∂Ωij(ij∈{23,32,14,41}) and then returns to domain Ωλ(λ∈{α,β}) or Ωγ(γ∈{i,j}). Such motion is called grazing motion on the boundary ∂Ωαβ(αβ∈{12,21,34,43}) or ∂Ωij(ij∈{23,32,14,41}). So the analytical conditions of grazing motion on boundary ∂Ωαβ(αβ∈{12,21,34,43}) or ∂Ωij(ij∈{23,32,14,41}) can be obtained by Lemma 5. Thus the 0th-order G-functions and the 1st-order G-functions on such boundaries are needed.
According to (27) and (28), the 0th-order G-functions and the 1st-order G-functions on the boundary ∂Ω12 in domain Ω1 are (72)G∂Ω120,1xm,tm±=n∂Ω12T·F1xm,tm±,G∂Ω121,1xm,tm±=n∂Ω12T·DF1xm,tm±.From (13) and (35), (72) can be computed as (73)G∂Ω120,1xm,tm±=b2xmym+a2ymF1xm,tm±,G∂Ω121,1xm,tm±=b2xmF1xm,tm±+a2ymDF1xm,tm±.By Lemma 5, the grazing motion on the boundary xm≡(xm,ym)∈∂Ω12 in domain Ω1 appears if and only if (74)G∂Ω120,1xm,tm±=0,G∂Ω121,1xm,tm±>0.From (73) and (74), one obtains (75)b2xmym+a2ymF1xm,tm±=0,b2xmF1xm,tm±+a2ymDF1xm,tm±>0,inΩ1for∂Ω12.Thus (64) in (i) holds, and (65) in (i) and (66) and (67) in (ii) can be proved similarly.
By (27) and (28), the 0th-order G-functions and the 1st-order G-functions on the boundary ∂Ω14 in domain Ω1 are (76)G∂Ω140,1xm,tm±=n∂Ω14T·F1xm,tm±,G∂Ω141,1xm,tm±=n∂Ω14T·DF1xm,tm±.From (13) and (34), (76) can be computed as (77)G∂Ω140,1xm,tm±=F1xm,tm±,G∂Ω141,1xm,tm±=DF1xm,tm±.By Lemma 5, the grazing motion on the boundary xm≡(xm,ym)∈∂Ω14 in domain Ω1 appears if and only if (78)G∂Ω140,1xm,tm±=0,G∂Ω141,1xm,tm±>0.From (77) and (78), one obtains (79)F1xm,tm±=0,DF1xm,tm±>0,inΩ1for∂Ω14.Thus (68) in (iii) holds, and (69) in (iii) and (70) and (71) in (iv) can be proved similarly.Theorem 11.
For the friction-induced oscillator with an elliptic control law and the friction force acting on the massM discussed in Section 2, there are the following results.
(i) The appearance conditions for the sliding motion of the massM at xm≡(xm,ym)∈∂Ω12 are (80)b2xmym+a2ymF1xm,tm-<0,b2xmym+a2ymF2xm,tm±=0,b2xmF2xm,tm±+a2ymDF2xm,tm±<0,forΩ1⟶∂Ω12,or(81)b2xmym+a2ymF1xm,tm±=0,b2xmF1xm,tm±+a2ymDF1xm,tm±>0,b2xmym+a2ymF2xm,tm->0,forΩ2⟶∂Ω12.
(ii) The appearance conditions for the stick motion of the massM at xm≡(xm,ym)∈∂Ω23 are (82)F2xm,tm-<0,F3xm,tm±=0,DF3xm,tm±<0,forΩ2⟶∂Ω23,or(83)F2xm,tm±=0,DF2xm,tm±>0,F3xm,tm->0,forΩ3⟶∂Ω23.
(iii) The appearance conditions for the sliding motion of the massM at xm≡(xm,ym)∈∂Ω34 are (84)b2xmym+a2ymF3xm,tm->0,b2xmym+a2ymF4xm,tm±=0,b2xmF4xm,tm±+a2ymDF4xm,tm±>0,forΩ3⟶∂Ω34,or(85)b2xmym+a2ymF3xm,tm±=0,b2xmF3xm,tm±+a2ymDF3xm,tm±<0,b2xmym+a2ymF4xm,tm-<0,forΩ4⟶∂Ω34.
(iv) The appearance conditions for the stick motion of the massM at xm≡(xm,ym)∈∂Ω14 are (86)F1xm,tm-<0,F4xm,tm±=0,DF4xm,tm±<0,forΩ1⟶∂Ω14,or(87)F1xm,tm±=0,DF1xm,tm±>0,F4xm,tm->0,forΩ4⟶∂Ω14.Proof.
When a flow in a domain reaches the sliding or stick boundary at timetm and then the sliding or stick motion occurs, such time tm may not be the onset time of the sliding or stick motion on this boundary. When conditions (36), (37), (40), (41) or (38), (39), (42), and (43) cannot be satisfied, the passable flow vanishes and the sliding or stick motion will begin at time tm. From the flow switchability theory on the discontinuous dynamical systems, time tm is the switching time from semipassable flow to nonpassable flow. Thus the analytical conditions of the appearance of the sliding or stick motion on the boundary ∂Ωαβ(αβ∈{12,21,34,43}) or ∂Ωij(ij∈{23,32,14,41}) can be obtained by Lemma 6. Therefore, the 0th-order G-functions and the 1st-order G-functions on such boundaries are needed.
From (27) and (28), the 0th-order G-functions and the 1st-order G-functions on the boundary ∂Ω12 in domain Ω1 and domain Ω2 are (88)G∂Ω120,1xm,tm±=n∂Ω12T·F1xm,tm±,G∂Ω121,1xm,tm±=n∂Ω12T·DF1xm,tm±,inΩ1for∂Ω12,G∂Ω120,2xm,tm±=n∂Ω12T·F2xm,tm±,G∂Ω121,2xm,tm±=n∂Ω12T·DF2xm,tm±,inΩ2for∂Ω12.From (13) and (35), (88) can be computed as (89)G∂Ω120,1xm,tm±=b2xmym+a2ymF1xm,tm±,G∂Ω121,1xm,tm±=b2xmF1xm,tm±+a2ymDF1xm,tm±,inΩ1for∂Ω12,G∂Ω120,2xm,tm±=b2xmym+a2ymF2xm,tm±,G∂Ω121,2xm,tm±=b2xmF2xm,tm±+a2ymDF2xm,tm±,inΩ2for∂Ω12.By Lemma 6, the appearance conditions for the sliding motion of the mass M at xm≡(xm,ym)∈∂Ω12 are (90)G∂Ω120,1xm,tm-<0,G∂Ω120,2xm,tm±=0,G∂Ω121,2xm,tm±<0,forΩ1⟶∂Ω12.From (89) and (90), one obtains (91)b2xmym+a2ymF1xm,tm-<0,b2xmym+a2ymF2xm,tm±=0,b2xmF2xm,tm±+a2ymDF2xm,tm±<0,forΩ1⟶∂Ω12.Thus (80) in (i) holds, and (81) in (i) and (84) and (85) in (iii) can be proved similarly.
According to (27) and (28), the 0th-order G-functions and the 1st-order G-functions on the boundary ∂Ω23 in domain Ω2 and domain Ω3 are (92)G∂Ω230,2xm,tm±=n∂Ω23T·F2xm,tm±,G∂Ω231,2xm,tm±=n∂Ω23T·DF2xm,tm±,inΩ2for∂Ω23,G∂Ω230,3xm,tm±=n∂Ω23T·F3xm,tm±,G∂Ω231,3xm,tm±=n∂Ω23T·DF3xm,tm±,inΩ3for∂Ω23.From (13) and (34), (92) can be computed as (93)G∂Ω230,2xm,tm±=F2xm,tm±,G∂Ω231,2xm,tm±=DF2xm,tm±,inΩ2for∂Ω23,G∂Ω230,3xm,tm±=F3xm,tm±,G∂Ω231,3xm,tm±=DF3xm,tm±,inΩ3for∂Ω23.By Lemma 6, the appearance conditions for the stick motion of the mass M at xm≡(xm,ym)∈∂Ω23 are (94)G∂Ω230,2xm,tm-<0,G∂Ω230,3xm,tm±=0,G∂Ω231,3xm,tm±<0,forΩ2⟶∂Ω23.From (93) and (94), one obtains (95)F2xm,tm-<0,F3xm,tm±=0,DF3xm,tm±<0,forΩ2⟶∂Ω23.Thus (82) in (ii) holds, and (83) in (ii) and (86) and (87) in (iv) can be proved similarly.Theorem 12.
For the friction-induced oscillator with an elliptic control law and the friction force acting on the massM discussed in Section 2, there are the following results.
(i) The vanishing conditions for the sliding motion of the massM at xm≡(xm,ym)∈∂Ω12 are (96)b2xmym+a2ymF1xm,tm∓=0,b2xmF1xm,tm∓+a2ymDF1xm,tm∓>0,b2xmym+a2ymF2xm,tm->0,for∂Ω12⟶Ω1,or(97)b2xmym+a2ymF1xm,tm-<0,b2xmym+a2ymF2xm,tm∓=0,b2xmF2xm,tm∓+a2ymDF2xm,tm∓<0,for∂Ω12⟶Ω2.
(ii) The vanishing conditions for the stick motion of the massM at xm≡(xm,ym)∈∂Ω23 are (98)F2xm,tm∓=0,DF2xm,tm∓>0,F3xm,tm->0,for∂Ω23⟶Ω2,or(99)F2xm,tm-<0,F3xm,tm∓=0,DF3xm,tm∓<0,for∂Ω23⟶Ω3.
(iii) The vanishing conditions for the sliding motion of the massM at xm≡(xm,ym)∈∂Ω34 are (100)b2xmym+a2ymF3xm,tm∓=0,b2xmF3xm,tm∓+a2ymDF3xm,tm∓<0,b2xmym+a2ymF4xm,tm-<0,for∂Ω34⟶Ω3,or(101)b2xmym+a2ymF3xm,tm->0,b2xmym+a2ymF4xm,tm∓=0,b2xmF4xm,tm∓+a2ymDF4xm,tm∓>0,for∂Ω34⟶Ω4.
(iv) The vanishing conditions for the stick motion of the massM at xm≡(xm,ym)∈∂Ω14 are (102)F1xm,tm∓=0,DF1xm,tm∓>0,F4xm,tm->0,for∂Ω14⟶Ω1,or(103)F1xm,tm-<0,F4xm,tm∓=0,DF4xm,tm∓<0,for∂Ω14⟶Ω4.Proof.
The massM and the conveyor belt move together for a long time, at time tm, the nonfriction force acting on the mass M is to overcome static friction, and the mass M begins to move with nonzero relative velocity. From the flow switchability theory on the discontinuous dynamical systems, the sliding or stick motion vanishes, which means that the flow on the boundary ∂Ωαβ(αβ∈{12,21,34,43}) or ∂Ωij(ij∈{23,32,14,41}) will go into domain Ωλ(λ∈{α,β}) or Ωγ(γ∈{i,j}), and time tm is the switching time from nonpassable flow of the first kind to semipassable flow. Thus it can be predicted by Lemma 7. So the 0th-order G-functions and the 1st-order G-functions on such boundaries are needed.
According to (27) and (28), the 0th-order G-functions and the 1st-order G-functions on the boundary ∂Ω12 in domain Ω1 and domain Ω2 are (104)G∂Ω120,1xm,tm±=n∂Ω12T·F1xm,tm±,G∂Ω121,1xm,tm±=n∂Ω12T·DF1xm,tm±,inΩ1for∂Ω12,G∂Ω120,2xm,tm±=n∂Ω12T·F2xm,tm±,G∂Ω121,2xm,tm±=n∂Ω12T·DF2xm,tm±,inΩ2for∂Ω12.From (13) and (35), (104) can be computed as (105)G∂Ω120,1xm,tm±=b2xmym+a2ymF1xm,tm±,G∂Ω121,1xm,tm±=b2xmF1xm,tm±+a2ymDF1xm,tm±,inΩ1for∂Ω12,G∂Ω120,2xm,tm±=b2xmym+a2ymF2xm,tm±,G∂Ω121,2xm,tm±=b2xmF2xm,tm±+a2ymDF2xm,tm±,inΩ2for∂Ω12.By Lemma 7, the vanishing conditions for the sliding motion of the mass M at xm≡(xm,ym)∈∂Ω12 are (106)G∂Ω120,1xm,tm∓=0,G∂Ω121,1xm,tm∓>0,G∂Ω120,2xm,tm->0,for∂Ω12⟶Ω1.From (105) and (106), one obtains(107)b2xmym+a2ymF1xm,tm∓=0,b2xmF1xm,tm∓+a2ymDF1xm,tm∓>0,b2xmym+a2ymF2xm,tm->0,for∂Ω12⟶Ω1.Thus (96) in (i) holds, and (97) in (i) and (100) and (101) in (iii) can be proved similarly.
By (27) and (28), the 0th-order G-functions and the 1st-order G-functions on the boundary ∂Ω23 in domain Ω2 and domain Ω3 are (108)G∂Ω230,2xm,tm±=n∂Ω23T·F2xm,tm±,G∂Ω231,2xm,tm±=n∂Ω23T·DF2xm,tm±,inΩ2for∂Ω23,G∂Ω230,3xm,tm±=n∂Ω23T·F3xm,tm±,G∂Ω231,3xm,tm±=n∂Ω23T·DF3xm,tm±,inΩ3for∂Ω23.From (13) and (34), (108) can be computed as(109)G∂Ω230,2xm,tm±=F2xm,tm±,G∂Ω231,2xm,tm±=DF2xm,tm±,inΩ2for∂Ω23,G∂Ω230,3xm,tm±=F3xm,tm±,G∂Ω231,3xm,tm±=DF3xm,tm±,inΩ3for∂Ω23.By Lemma 7, the vanishing conditions for the stick motion of the mass M at xm≡(xm,ym)∈∂Ω23 are (110)G∂Ω230,2xm,tm∓=0,G∂Ω231,2xm,tm∓>0,G∂Ω230,3xm,tm->0,for∂Ω23⟶Ω2.From (109) and (110), one obtains(111)F2xm,tm∓=0,DF2xm,tm∓>0,F3xm,tm->0,for∂Ω23⟶Ω2.Thus (98) in (ii) holds, and (99) in (ii) and (102) and (103) in (iv) can be proved similarly.


## 4.1. Basic Theory

For convenience, the concepts of theG-functions and some lemmas are given in the following (see [42]).Consider a dynamic system consisting ofN subdynamic systems in a universal domain ℧∈Rn. The universal domain is divided into N accessible subdomains Ωi and inaccessible subdomains Ω0. And the union of all the accessible subdomains is ⋃i=1NΩi and the universal domain is ℧=⋃i=1NΩi∪Ω0. On the ith open subdomain Ωi, there is a Cri-continuous system (ri≥1) in the form of(22)x˙i≡Fixi,t,Pi∈Rn,xi=x1i,x2i,…,xniT∈Ωi,where the time is t and x˙(i)=dx(i)/dt. In an accessible subdomain Ωi, the vector field F(i)(x(i),t,Pi) with a parameter vector Pi=(pi(1),pi(2),…,pi(l))T∈Rl is Cri-continuous (ri≥1) for all time t, and the flow in (22) xt(i)=Φ(i)(t0,x0(i),Pi,t) with an initial condition (t0,x0(i)) is Cri+1 continuous for time t, and on the boundary of two adjacent domains ∂Ωαβ=Ω¯α∩Ω¯β={x∣φαβ(x,t,λ)=0, φαβ is Cr-continuous (r≥1)}⊂Rn-1, and there is a flow xt(0)=Φ(0)(t0,x0(0),λ,t), which can be determined by (23)x˙0≡F0x0,t,λ∈Rn,x0=x10,x20,…,xn0T∈∂Ωαβwith an initial condition (t0,x0(0)) and a parameter vector λ=(λ1,λ2,…,λk)T∈Rk.Assume that there are two adjacent subdomainsΩα and Ωβ. A flow in subdomain Ωi(i∈{α,β}) is called a local flow if it is governed by the continuous vector field in Ωi(i∈{α,β}) only. A flow is called a passable flow if it switches from a subdomain Ωi(i∈{α,β}) into another one Ωj(j∈{α,β},j≠i) through the boundary ∂Ωij(i,j∈{α,β},i≠j). Obviously, the vector field of a passable flow in subdomain Ωi(i∈{α,β}) will be changed into the one in subdomain Ωj(j∈{α,β},j≠i), accordingly. A flow is called a grazing flow if it is tangential to the boundaries ∂Ωαβ. If a flow cannot pass through but moves along the boundaries ∂Ωαβ, then this flow is called a sliding flow.Definition 1 (see [42]).
The 0th-orderG-functions of the domain flow xt(i)(i∈{α,β}) to the boundary flow xt(0) on the boundary in the normal direction of the boundary ∂Ωαβ are defined as (24)G∂Ωαβixt0,t±,xt±i,Pi,λ≡G∂Ωαβ0,ixt0,t±,xt±i,Pi,λ=Dxt0nt∂ΩαβT·xt±i-xt0+nt∂ΩαβT·x˙t±i-x˙t0.Definition 2 (see [42]).
The1st-order G-functions of the domain flow xt(i)(i∈{α,β}) to the boundary flow xt(0) on the boundary in the normal direction of the boundary ∂Ωαβ are defined as (25)G∂Ωαβ1,ixt0,t±,xt±i,Pi,λ=Dxt02nt∂ΩαβT·xt±i-xt0+2Dxt0nt∂ΩαβT·x˙t±i-x˙t0+n∂ΩαβTt·x¨t±i-x¨t0.In above definitions, the total derivativeDxt0(·)≔∂(·)/∂xt0·x˙t0+∂(·)/∂t, and the normal vector of the boundary surface ∂Ωαβ at point xt(0) is given by (26)nt∂Ωαβxt0,t,λ=▽φαβxt0,t,λ=∂φαβ∂x10,∂φαβ∂x20,…,∂φαβ∂xn0Tt,xt0,and t±=t±0.If the flowxt(i)(i∈{α,β}) contacts with the boundary at time tm, that is, xtm(i)=xm=xtm(0), and the boundary ∂Ωαβ is linear, independent of time t, we can obtain the 0th-order G-functions and the 1st-order G-functions by simple computation (27)G∂Ωαβ0,ixm,tm±,Pi,λ=G∂Ωαβ0,ixtm,tm±,xtm±i,Pi,λ=nt∂ΩαβT·Fixti,t,Pixm0,xm±i,tm±,(28)G∂Ωαβ1,ixm,tm±,Pi,λ=G∂Ωαβ1,ixtm,tm±,xtm±i,Pi,λ=nt∂ΩαβT·DxtiFixti,t,Pixm0,xm±i,tm±.Here tm±=tm±0 is to show the motion in different domains rather than on the boundaries, and tm-, tm+ are the time before approaching and after departing the corresponding boundary, respectively. In this case, the G-functions are simplified as G∂Ωαβ(0,i)(xm,tm±) and G∂Ωαβ(1,i)(xm,tm±).For a discontinuous dynamical system in (22), there is a point x(0)(tm)≡xm∈∂Ωαβ at time tm between two adjacent domains Ωi(i∈{α,β}). For an arbitrarily small ε>0, there are two time intervals [tm-ε,tm) and (tm,tm+ε]. Suppose x(i)(tm-)=xm=x(j)(tm+)(i,j∈{α,β}). Both flows x(i)(t) and x(j)(t) are C[tm-ε,tm)ri- and C(tm,tm+ε]rj-continuous for time t(i,j∈{α,β}), and dri+1x(i)/dtri+1<∞(ri≥1,i∈{α,β}).Based on theG-functions, the decision theorems of semipassable flow, sink flow, tangential flow, sliding bifurcation, and sliding fragmentation bifurcation to the separation boundary are stated in the form of lemma as follows.Lemma 3 (see [42]).
The flowsx(α)(t) and x(β)(t) at point (xm,tm) on the boundary ∂Ωαβ are semipassable from domain Ωα to domain Ωβ if and only if (29)eitherG∂Ωαβαxm,tm-,Pα,λ>0,G∂Ωαββxm,tm+,Pβ,λ>0,forn∂Ωαβ⟶Ωβ,orG∂Ωαβαxm,tm-,Pα,λ<0,G∂Ωαββxm,tm+,Pβ,λ<0,forn∂Ωαβ⟶Ωα.Lemma 4 (see [42]).
The flowsx(α)(t) and x(β)(t) at point (xm,tm) on the boundary ∂Ωαβ are sink flows if and only if (30)eitherG∂Ωαβαxm,tm-,Pα,λ>0,G∂Ωαββxm,tm-,Pβ,λ<0,forn∂Ωαβ⟶Ωβ,orG∂Ωαβαxm,tm-,Pα,λ<0,G∂Ωαββxm,tm-,Pβ,λ>0,forn∂Ωαβ⟶Ωα.Lemma 5 (see [42]).
For an arbitrarily smallε>0, there is a time interval [tm-ε,tm+ε]. Suppose x(i)(tm±)=xm. The flow x(i)(t) is C[tm-ε,tm+ε]ri-continuous for time t, and dri+1x(i)/dtri+1<∞(ri≥2,i∈{α,β}). A flow x(i)(t) in Ωi is tangential to the boundary ∂Ωαβ if and only if (31)G∂Ωαβ0,ixm,tm,Pi,λ=0,eitherG∂Ωαβ1,ixm,tm,Pi,λ<0forn∂Ωαβ⟶Ωj,orG∂Ωαβ1,ixm,tm,Pi,λ>0forn∂Ωαβ⟶Ωi,where i,j∈{α,β},i≠j.Lemma 6 (see [42]).
For an arbitrarily smallε>0, there are two time intervals [tm-ε,tm) and [tm-ε,tm+ε]. Suppose x(α)(tm-)=xm=x(β)(tm±). The flows x(α)(t) and x(β)(t) are C[tm-ε,tm)rα- and C[tm-ε,tm+ε]rβ-continuous for time t and dri+1x(i)/dtri+1<∞(ri≥2andi∈{α,β}). The sliding bifurcation of the passable flows x(α)(t) and x(β)(t) at point (xm,tm) switching to the nonpassable flow of the first kind on the boundary ∂Ω→αβ occurs if and only if (32)G∂Ωαββxm,tm±,Pβ,λ=0,G∂Ωαβαxm,tm-,Pα,λ>0forn∂Ωαβ⟶Ωβ,G∂Ωαβαxm,tm-,Pα,λ<0forn∂Ωαβ⟶Ωα,G∂Ωαβ1,βxm,tm±,Pβ,λ>0forn∂Ωαβ⟶Ωβ,G∂Ωαβ1,βxm,tm±,Pβ,λ<0forn∂Ωαβ⟶Ωα.Lemma 7 (see [42]).
For an arbitrarily smallε>0, there are two time intervals [tm-ε,tm) and [tm-ε,tm+ε]. Suppose x(α)(tm-)=xm=x(β)(tm±). The flows x(α)(t) and x(β)(t) are C[tm-ε,tm)rα- and C[tm-ε,tm+ε]rβ-continuous for time t and dri+1x(i)/dtri+1<∞(ri≥2andi∈{α,β}). The tangential bifurcation of the flows x(α) and x(β) at point (xm,tm) on the boundary ∂Ω~αβ is termed the sliding fragmentation bifurcation if and only if (33)G∂Ωαββxm,tm∓,Pβ,λ=0,G∂Ωαβαxm,tm-,Pα,λ>0forn∂Ωαβ⟶Ωβ,G∂Ωαβαxm,tm-,Pα,λ<0forn∂Ωαβ⟶Ωα,G∂Ωαβ1,βxm,tm∓,Pβ,λ>0forn∂Ωαβ⟶Ωβ,G∂Ωαβ1,βxm,tm∓,Pβ,λ<0forn∂Ωαβ⟶Ωα.More detailed theory on the flow switchability such as high-orderG-functions, the definitions, or the decision theorems about various flow passability in discontinuous dynamical systems can be found in [42].


## 4.2. Analytical Conditions

According to the previous definitions and lemmas, the analytical switching conditions on discontinuous boundaries in the friction-induced oscillator with an elliptic control law and the friction force acting on the massM described in Section 3 will be developed in this subsection. From (12), the normal vectors of the absolute boundaries are given as (34)n∂Ωαβ=n∂Ωβα=∇φαβ=∂φαβ∂x,∂φαβ∂x˙T=0,1T,for (α,β)∈{(2,3),(3,2),(1,4),(4,1)}, and (35)n∂Ωαβ=n∂Ωβα=a2b22∇φαβ=a2b22∂φαβ∂x,∂φαβ∂x˙T=b2x,a2yT,for (α,β)∈{(1,2),(2,1),(3,4),(4,3)}, where ∇=(∂/∂x,∂/∂y)T is a Hamilton operator.Theorem 8.
For the friction-induced oscillator with an elliptic control law and the friction force acting on the massM discussed in Section 2, there are the following results.
(i) The necessary and sufficient conditions for passable motion on the boundaryxm≡(xm,ym)∈∂Ω12 at time tm are (36)eitherb2xmym+a2ymF1xm,tm-<0,b2xmym+a2ymF2xm,tm+<0,forΩ1⟶Ω2,(37)orb2xmym+a2ymF2xm,tm->0,b2xmym+a2ymF1xm,tm+>0,forΩ2⟶Ω1.
(ii) The necessary and sufficient conditions for passable motion on the boundaryxm≡(xm,ym)∈∂Ω23 at time tm are (38)eitherF2xm,tm-<0,F3xm,tm+<0,forΩ2⟶Ω3,(39)orF3xm,tm->0,F2xm,tm+>0,forΩ3⟶Ω2.
(iii) The necessary and sufficient conditions for passable motion on the boundaryxm≡(xm,ym)∈∂Ω34 at time tm are(40)eitherb2xmym+a2ymF3xm,tm->0,b2xmym+a2ymF4xm,tm+>0,forΩ3⟶Ω4,(41)orb2xmym+a2ymF4xm,tm-<0,b2xmym+a2ymF3xm,tm+<0,forΩ4⟶Ω3.
(iv) The necessary and sufficient conditions for passable motion on the boundaryxm≡(xm,ym)∈∂Ω14 at time tm are(42)eitherF1xm,tm-<0,F4xm,tm+<0,forΩ1⟶Ω4,(43)orF4xm,tm->0,F1xm,tm+>0,forΩ4→Ω1.Proof.
The massM moves along the conveyor belt with nonzero relative velocity before time tm, and the relative velocity becomes zero at tm; then after tm the relative velocity changes its direction. From the flow switchability theory on the discontinuous dynamical systems, the flow in domain Ωλ(λ∈{α,β}) or Ωγ(γ∈{i,j}) passes through the boundary ∂Ωαβ(αβ∈{12,21,34,43}) or ∂Ωij(ij∈{23,32,14,41}) into domain Ωλ¯(λ¯∈{α,β},λ¯≠λ) or Ωγ¯(γ¯∈{i,j},γ¯≠γ), which can be expressed by Lemma 3. So 0th-order G-functions on such boundaries are needed.
According to (27), the 0th-order G-functions on the boundary ∂Ω12 are(44)G∂Ω120,1xm,tm±=n∂Ω12T·F1xm,tm±,G∂Ω120,2xm,tm±=n∂Ω12T·F2xm,tm±.From (13) and (35), (44) can be computed as(45)G∂Ω120,1xm,tm-=b2xmym+a2ymF1xm,tm-,G∂Ω120,2xm,tm+=b2xmym+a2ymF2xm,tm+.By Lemma 3, the passable motion on the boundary xm∈∂Ω12 at time tm appears if and only if (46)G∂Ω120,1xm,tm-<0,G∂Ω120,2xm,tm+<0,forΩ1⟶Ω2.From (45) and (46), one obtains (47)b2xmym+a2ymF1xm,tm-<0,b2xmym+a2ymF2xm,tm+<0,forΩ1→Ω2.Thus, (36) in (i) holds, and (37) in (i) and (40) and (41) in (iii) can be proved similarly.
From (27), the 0th-order G-functions on the boundary ∂Ω23 are (48)G∂Ω230,2xm,tm±=n∂Ω23T·F2xm,tm±,G∂Ω230,3xm,tm±=n∂Ω23T·F3xm,tm±.From (13) and (34), (48) can be computed as (49)G∂Ω230,2xm,tm-=F2xm,tm-,G∂Ω230,3xm,tm+=F3xm,tm+.By Lemma 3, the passable motion on the boundary xm∈∂Ω23 at time tm appears if and only if (50)G∂Ω230,2xm,tm-<0,G∂Ω230,3xm,tm+<0,forΩ2⟶Ω3.By (49) and (50), one obtains (51)F2xm,tm-<0,F3xm,tm+<0,forΩ2⟶Ω3.Thus, (38) in (ii) holds, and (39) in (ii) and (42) and (43) in (iv) can be proved similarly.Theorem 9.
For the friction-induced oscillator with an elliptic control law and the friction force acting on the massM discussed in Section 2, there are the following results.
(i) The necessary and sufficient conditions for sliding motion on the boundaryxm≡(xm,ym)∈∂Ω12 at time tm are (52)b2xmym+a2ymF1xm,tm-<0,b2xmym+a2ymF2xm,tm->0.
(ii) The necessary and sufficient conditions for stick motion on the boundaryxm≡(xm,ym)∈∂Ω23 at time tm are (53)F2xm,tm-<0,F3xm,tm->0.
(iii) The necessary and sufficient conditions for sliding motion on the boundaryxm≡(xm,ym)∈∂Ω34 at time tm are (54)b2xmym+a2ymF3xm,tm->0,b2xmym+a2ymF4xm,tm-<0.
(iv) The necessary and sufficient conditions for stick motion on the boundaryxm≡(xm,ym)∈∂Ω14 at time tm are (55)F1xm,tm-<0,F4xm,tm->0.Proof.
The displacement and velocity of the massM satisfy the elliptic control law; such motion is called the sliding motion. From the flow switchability theory on the discontinuous dynamical systems, the sliding motion is that the flow in domain Ωλ(λ∈{α,β}) reaches the boundary ∂Ωαβ(αβ∈{12,21,34,43}) and moves along the boundary ∂Ωαβ(αβ∈{12,21,34,43}). Thus it can be predicted by Lemma 4.
From (27), the 0th-order G-functions on the boundary ∂Ω12 are(56)G∂Ω120,1xm,tm±=n∂Ω12T·F1xm,tm±,G∂Ω120,2xm,tm±=n∂Ω12T·F2xm,tm±.From (13) and (35), (56) can be computed as (57)G∂Ω120,1xm,tm-=b2xmym+a2ymF1xm,tm-,G∂Ω120,2xm,tm-=b2xmym+a2ymF2xm,tm-.By Lemma 4, the sliding motion on the boundary xm∈∂Ω12 at time tm appears if and only if (58)G∂Ω120,1xm,tm-<0,G∂Ω120,2xm,tm->0.From (57) and (58), one obtains (59)b2xmym+a2ymF1xm,tm-<0,b2xmym+a2ymF2xm,tm->0.Thus, (i) holds, and (iii) can be proved similarly.
The massM moves together with the conveyor belt and the displacement and velocity of the mass M do not satisfy the elliptic control law; such motion is called the stick motion. From the flow switchability theory on the discontinuous dynamical systems, the stick motion is that the flow in domain Ωγ(γ∈{i,j}) reaches the boundary ∂Ωij(ij∈{23,32,14,41}) and moves along the boundary ∂Ωij(ij∈{23,32,14,41}). Thus it can be predicted by Lemma 4.
By (27), the 0th-order G-functions on the boundary ∂Ω23 are(60)G∂Ω230,2xm,tm±=n∂Ω23T·F2xm,tm±,G∂Ω230,3xm,tm±=n∂Ω23T·F3xm,tm±.From (13) and (34), (60) can be computed as(61)G∂Ω230,2xm,tm-=F2xm,tm-,G∂Ω230,3xm,tm-=F3xm,tm-.By Lemma 4, the stick motion on the boundary xm∈∂Ω23 at time tm appears if and only if (62)G∂Ω230,2xm,tm-<0,G∂Ω230,3xm,tm->0.From (61) and (62), one obtains (63)F2xm,tm-<0,F3xm,tm->0.Thus, (ii) holds, and (iv) can be proved similarly.Theorem 10.
For the friction-induced oscillator with an elliptic control law and the friction force acting on the massM discussed in Section 2, there are the following results.
(i) The necessary and sufficient conditions for grazing motion on the boundaryxm≡(xm,ym)∈∂Ω12 in domain Ω1 or domain Ω2 at time tm are (64)b2xmym+a2ymF1xm,tm±=0,b2xmF1xm,tm±+a2ymDF1xm,tm±>0,inΩ1for∂Ω12,or(65)b2xmym+a2ymF2xm,tm±=0,b2xmF2xm,tm±+a2ymDF2xm,tm±<0,inΩ2for∂Ω12.
(ii) The necessary and sufficient conditions for grazing motion on the boundaryxm≡(xm,ym)∈∂Ω34 in domain Ω3 or domain Ω4 at time tm are (66)b2xmym+a2ymF3xm,tm±=0,b2xmF3xm,tm±+a2ymDF3xm,tm±<0,inΩ3for∂Ω34,or(67)b2xmym+a2ymF4xm,tm±=0,b2xmF4xm,tm±+a2ymDF4xm,tm±>0,inΩ4for∂Ω34.
(iii) The necessary and sufficient conditions for grazing motion on the boundaryxm≡(xm,ym)∈∂Ω14 in domain Ω1 or domain Ω4 at time tm are (68)F1xm,tm±=0,DF1xm,tm±>0,inΩ1for∂Ω14,or(69)F4xm,tm±=0,DF4xm,tm±<0,inΩ4for∂Ω14.
(iv) The necessary and sufficient conditions for grazing motion on the boundaryxm≡(xm,ym)∈∂Ω23 in domain Ω2 or domain Ω3 at time tm are(70)F2xm,tm±=0,DF2xm,tm±>0,inΩ2for∂Ω23,or(71)F3xm,tm±=0,DF3xm,tm±<0,inΩ3for∂Ω23.Proof.
When the massM is moving along the conveyor belt with nonzero relative velocity, the relative velocity equals zero at time tm and then it restores to the original relationship. From the flow switchability theory on the discontinuous dynamical systems, the flow of the motion in domain Ωλ(λ∈{α,β}) or Ωγ(γ∈{i,j}) reaches the boundary ∂Ωαβ(αβ∈{12,21,34,43}) or ∂Ωij(ij∈{23,32,14,41}) and then returns to domain Ωλ(λ∈{α,β}) or Ωγ(γ∈{i,j}). Such motion is called grazing motion on the boundary ∂Ωαβ(αβ∈{12,21,34,43}) or ∂Ωij(ij∈{23,32,14,41}). So the analytical conditions of grazing motion on boundary ∂Ωαβ(αβ∈{12,21,34,43}) or ∂Ωij(ij∈{23,32,14,41}) can be obtained by Lemma 5. Thus the 0th-order G-functions and the 1st-order G-functions on such boundaries are needed.
According to (27) and (28), the 0th-order G-functions and the 1st-order G-functions on the boundary ∂Ω12 in domain Ω1 are (72)G∂Ω120,1xm,tm±=n∂Ω12T·F1xm,tm±,G∂Ω121,1xm,tm±=n∂Ω12T·DF1xm,tm±.From (13) and (35), (72) can be computed as (73)G∂Ω120,1xm,tm±=b2xmym+a2ymF1xm,tm±,G∂Ω121,1xm,tm±=b2xmF1xm,tm±+a2ymDF1xm,tm±.By Lemma 5, the grazing motion on the boundary xm≡(xm,ym)∈∂Ω12 in domain Ω1 appears if and only if (74)G∂Ω120,1xm,tm±=0,G∂Ω121,1xm,tm±>0.From (73) and (74), one obtains (75)b2xmym+a2ymF1xm,tm±=0,b2xmF1xm,tm±+a2ymDF1xm,tm±>0,inΩ1for∂Ω12.Thus (64) in (i) holds, and (65) in (i) and (66) and (67) in (ii) can be proved similarly.
By (27) and (28), the 0th-order G-functions and the 1st-order G-functions on the boundary ∂Ω14 in domain Ω1 are (76)G∂Ω140,1xm,tm±=n∂Ω14T·F1xm,tm±,G∂Ω141,1xm,tm±=n∂Ω14T·DF1xm,tm±.From (13) and (34), (76) can be computed as (77)G∂Ω140,1xm,tm±=F1xm,tm±,G∂Ω141,1xm,tm±=DF1xm,tm±.By Lemma 5, the grazing motion on the boundary xm≡(xm,ym)∈∂Ω14 in domain Ω1 appears if and only if (78)G∂Ω140,1xm,tm±=0,G∂Ω141,1xm,tm±>0.From (77) and (78), one obtains (79)F1xm,tm±=0,DF1xm,tm±>0,inΩ1for∂Ω14.Thus (68) in (iii) holds, and (69) in (iii) and (70) and (71) in (iv) can be proved similarly.Theorem 11.
For the friction-induced oscillator with an elliptic control law and the friction force acting on the massM discussed in Section 2, there are the following results.
(i) The appearance conditions for the sliding motion of the massM at xm≡(xm,ym)∈∂Ω12 are (80)b2xmym+a2ymF1xm,tm-<0,b2xmym+a2ymF2xm,tm±=0,b2xmF2xm,tm±+a2ymDF2xm,tm±<0,forΩ1⟶∂Ω12,or(81)b2xmym+a2ymF1xm,tm±=0,b2xmF1xm,tm±+a2ymDF1xm,tm±>0,b2xmym+a2ymF2xm,tm->0,forΩ2⟶∂Ω12.
(ii) The appearance conditions for the stick motion of the massM at xm≡(xm,ym)∈∂Ω23 are (82)F2xm,tm-<0,F3xm,tm±=0,DF3xm,tm±<0,forΩ2⟶∂Ω23,or(83)F2xm,tm±=0,DF2xm,tm±>0,F3xm,tm->0,forΩ3⟶∂Ω23.
(iii) The appearance conditions for the sliding motion of the massM at xm≡(xm,ym)∈∂Ω34 are (84)b2xmym+a2ymF3xm,tm->0,b2xmym+a2ymF4xm,tm±=0,b2xmF4xm,tm±+a2ymDF4xm,tm±>0,forΩ3⟶∂Ω34,or(85)b2xmym+a2ymF3xm,tm±=0,b2xmF3xm,tm±+a2ymDF3xm,tm±<0,b2xmym+a2ymF4xm,tm-<0,forΩ4⟶∂Ω34.
(iv) The appearance conditions for the stick motion of the massM at xm≡(xm,ym)∈∂Ω14 are (86)F1xm,tm-<0,F4xm,tm±=0,DF4xm,tm±<0,forΩ1⟶∂Ω14,or(87)F1xm,tm±=0,DF1xm,tm±>0,F4xm,tm->0,forΩ4⟶∂Ω14.Proof.
When a flow in a domain reaches the sliding or stick boundary at timetm and then the sliding or stick motion occurs, such time tm may not be the onset time of the sliding or stick motion on this boundary. When conditions (36), (37), (40), (41) or (38), (39), (42), and (43) cannot be satisfied, the passable flow vanishes and the sliding or stick motion will begin at time tm. From the flow switchability theory on the discontinuous dynamical systems, time tm is the switching time from semipassable flow to nonpassable flow. Thus the analytical conditions of the appearance of the sliding or stick motion on the boundary ∂Ωαβ(αβ∈{12,21,34,43}) or ∂Ωij(ij∈{23,32,14,41}) can be obtained by Lemma 6. Therefore, the 0th-order G-functions and the 1st-order G-functions on such boundaries are needed.
From (27) and (28), the 0th-order G-functions and the 1st-order G-functions on the boundary ∂Ω12 in domain Ω1 and domain Ω2 are (88)G∂Ω120,1xm,tm±=n∂Ω12T·F1xm,tm±,G∂Ω121,1xm,tm±=n∂Ω12T·DF1xm,tm±,inΩ1for∂Ω12,G∂Ω120,2xm,tm±=n∂Ω12T·F2xm,tm±,G∂Ω121,2xm,tm±=n∂Ω12T·DF2xm,tm±,inΩ2for∂Ω12.From (13) and (35), (88) can be computed as (89)G∂Ω120,1xm,tm±=b2xmym+a2ymF1xm,tm±,G∂Ω121,1xm,tm±=b2xmF1xm,tm±+a2ymDF1xm,tm±,inΩ1for∂Ω12,G∂Ω120,2xm,tm±=b2xmym+a2ymF2xm,tm±,G∂Ω121,2xm,tm±=b2xmF2xm,tm±+a2ymDF2xm,tm±,inΩ2for∂Ω12.By Lemma 6, the appearance conditions for the sliding motion of the mass M at xm≡(xm,ym)∈∂Ω12 are (90)G∂Ω120,1xm,tm-<0,G∂Ω120,2xm,tm±=0,G∂Ω121,2xm,tm±<0,forΩ1⟶∂Ω12.From (89) and (90), one obtains (91)b2xmym+a2ymF1xm,tm-<0,b2xmym+a2ymF2xm,tm±=0,b2xmF2xm,tm±+a2ymDF2xm,tm±<0,forΩ1⟶∂Ω12.Thus (80) in (i) holds, and (81) in (i) and (84) and (85) in (iii) can be proved similarly.
According to (27) and (28), the 0th-order G-functions and the 1st-order G-functions on the boundary ∂Ω23 in domain Ω2 and domain Ω3 are (92)G∂Ω230,2xm,tm±=n∂Ω23T·F2xm,tm±,G∂Ω231,2xm,tm±=n∂Ω23T·DF2xm,tm±,inΩ2for∂Ω23,G∂Ω230,3xm,tm±=n∂Ω23T·F3xm,tm±,G∂Ω231,3xm,tm±=n∂Ω23T·DF3xm,tm±,inΩ3for∂Ω23.From (13) and (34), (92) can be computed as (93)G∂Ω230,2xm,tm±=F2xm,tm±,G∂Ω231,2xm,tm±=DF2xm,tm±,inΩ2for∂Ω23,G∂Ω230,3xm,tm±=F3xm,tm±,G∂Ω231,3xm,tm±=DF3xm,tm±,inΩ3for∂Ω23.By Lemma 6, the appearance conditions for the stick motion of the mass M at xm≡(xm,ym)∈∂Ω23 are (94)G∂Ω230,2xm,tm-<0,G∂Ω230,3xm,tm±=0,G∂Ω231,3xm,tm±<0,forΩ2⟶∂Ω23.From (93) and (94), one obtains (95)F2xm,tm-<0,F3xm,tm±=0,DF3xm,tm±<0,forΩ2⟶∂Ω23.Thus (82) in (ii) holds, and (83) in (ii) and (86) and (87) in (iv) can be proved similarly.Theorem 12.
For the friction-induced oscillator with an elliptic control law and the friction force acting on the massM discussed in Section 2, there are the following results.
(i) The vanishing conditions for the sliding motion of the massM at xm≡(xm,ym)∈∂Ω12 are (96)b2xmym+a2ymF1xm,tm∓=0,b2xmF1xm,tm∓+a2ymDF1xm,tm∓>0,b2xmym+a2ymF2xm,tm->0,for∂Ω12⟶Ω1,or(97)b2xmym+a2ymF1xm,tm-<0,b2xmym+a2ymF2xm,tm∓=0,b2xmF2xm,tm∓+a2ymDF2xm,tm∓<0,for∂Ω12⟶Ω2.
(ii) The vanishing conditions for the stick motion of the massM at xm≡(xm,ym)∈∂Ω23 are (98)F2xm,tm∓=0,DF2xm,tm∓>0,F3xm,tm->0,for∂Ω23⟶Ω2,or(99)F2xm,tm-<0,F3xm,tm∓=0,DF3xm,tm∓<0,for∂Ω23⟶Ω3.
(iii) The vanishing conditions for the sliding motion of the massM at xm≡(xm,ym)∈∂Ω34 are (100)b2xmym+a2ymF3xm,tm∓=0,b2xmF3xm,tm∓+a2ymDF3xm,tm∓<0,b2xmym+a2ymF4xm,tm-<0,for∂Ω34⟶Ω3,or(101)b2xmym+a2ymF3xm,tm->0,b2xmym+a2ymF4xm,tm∓=0,b2xmF4xm,tm∓+a2ymDF4xm,tm∓>0,for∂Ω34⟶Ω4.
(iv) The vanishing conditions for the stick motion of the massM at xm≡(xm,ym)∈∂Ω14 are (102)F1xm,tm∓=0,DF1xm,tm∓>0,F4xm,tm->0,for∂Ω14⟶Ω1,or(103)F1xm,tm-<0,F4xm,tm∓=0,DF4xm,tm∓<0,for∂Ω14⟶Ω4.Proof.
The massM and the conveyor belt move together for a long time, at time tm, the nonfriction force acting on the mass M is to overcome static friction, and the mass M begins to move with nonzero relative velocity. From the flow switchability theory on the discontinuous dynamical systems, the sliding or stick motion vanishes, which means that the flow on the boundary ∂Ωαβ(αβ∈{12,21,34,43}) or ∂Ωij(ij∈{23,32,14,41}) will go into domain Ωλ(λ∈{α,β}) or Ωγ(γ∈{i,j}), and time tm is the switching time from nonpassable flow of the first kind to semipassable flow. Thus it can be predicted by Lemma 7. So the 0th-order G-functions and the 1st-order G-functions on such boundaries are needed.
According to (27) and (28), the 0th-order G-functions and the 1st-order G-functions on the boundary ∂Ω12 in domain Ω1 and domain Ω2 are (104)G∂Ω120,1xm,tm±=n∂Ω12T·F1xm,tm±,G∂Ω121,1xm,tm±=n∂Ω12T·DF1xm,tm±,inΩ1for∂Ω12,G∂Ω120,2xm,tm±=n∂Ω12T·F2xm,tm±,G∂Ω121,2xm,tm±=n∂Ω12T·DF2xm,tm±,inΩ2for∂Ω12.From (13) and (35), (104) can be computed as (105)G∂Ω120,1xm,tm±=b2xmym+a2ymF1xm,tm±,G∂Ω121,1xm,tm±=b2xmF1xm,tm±+a2ymDF1xm,tm±,inΩ1for∂Ω12,G∂Ω120,2xm,tm±=b2xmym+a2ymF2xm,tm±,G∂Ω121,2xm,tm±=b2xmF2xm,tm±+a2ymDF2xm,tm±,inΩ2for∂Ω12.By Lemma 7, the vanishing conditions for the sliding motion of the mass M at xm≡(xm,ym)∈∂Ω12 are (106)G∂Ω120,1xm,tm∓=0,G∂Ω121,1xm,tm∓>0,G∂Ω120,2xm,tm->0,for∂Ω12⟶Ω1.From (105) and (106), one obtains(107)b2xmym+a2ymF1xm,tm∓=0,b2xmF1xm,tm∓+a2ymDF1xm,tm∓>0,b2xmym+a2ymF2xm,tm->0,for∂Ω12⟶Ω1.Thus (96) in (i) holds, and (97) in (i) and (100) and (101) in (iii) can be proved similarly.
By (27) and (28), the 0th-order G-functions and the 1st-order G-functions on the boundary ∂Ω23 in domain Ω2 and domain Ω3 are (108)G∂Ω230,2xm,tm±=n∂Ω23T·F2xm,tm±,G∂Ω231,2xm,tm±=n∂Ω23T·DF2xm,tm±,inΩ2for∂Ω23,G∂Ω230,3xm,tm±=n∂Ω23T·F3xm,tm±,G∂Ω231,3xm,tm±=n∂Ω23T·DF3xm,tm±,inΩ3for∂Ω23.From (13) and (34), (108) can be computed as(109)G∂Ω230,2xm,tm±=F2xm,tm±,G∂Ω231,2xm,tm±=DF2xm,tm±,inΩ2for∂Ω23,G∂Ω230,3xm,tm±=F3xm,tm±,G∂Ω231,3xm,tm±=DF3xm,tm±,inΩ3for∂Ω23.By Lemma 7, the vanishing conditions for the stick motion of the mass M at xm≡(xm,ym)∈∂Ω23 are (110)G∂Ω230,2xm,tm∓=0,G∂Ω231,2xm,tm∓>0,G∂Ω230,3xm,tm->0,for∂Ω23⟶Ω2.From (109) and (110), one obtains(111)F2xm,tm∓=0,DF2xm,tm∓>0,F3xm,tm->0,for∂Ω23⟶Ω2.Thus (98) in (ii) holds, and (99) in (ii) and (102) and (103) in (iv) can be proved similarly.


## 5. Mapping Structures and Periodic Motions

In order to describe the periodic motions with or without stick or sliding in the friction-induced oscillator with an elliptic control law and the friction acting on the massM discussed in Section 2, the mapping structures are introduced based on the discontinuous boundaries. The switching sets will be defined firstly. From the switching sets, the basic mappings will be defined for this discontinuous dynamical system.Using the discontinuous boundaries∂Ωα1α2(α1α2∈{12,21,23,32,34,43,14,41}) in (12), the switching sets of mass M are defined as (112)Σ12=Σ21=xm,x˙m,tm∣xm2a2+x˙m2b2-1=0,x˙m>V,Σ23=Σ32=xm,x˙m,tm∣xm2a2+x˙m2b2-1<0,x˙m=V,Σ34=Σ43=xm,x˙m,tm∣xm2a2+x˙m2b2-1=0,x˙m<V,Σ14=Σ41=xm,x˙m,tm∣xm2a2+x˙m2b2-1>0,x˙m=V,Σ12+=xm,x˙m,tm∣xm2a2+x˙m2b2-1=0+,x˙m>V,Σ21-=xm,x˙m,tm∣xm2a2+x˙m2b2-1=0-,x˙m>V,Σ23+=xm,x˙m,tm∣xm2a2+x˙m2b2-1<0,x˙m=V+,Σ32-=xm,x˙m,tm∣xm2a2+x˙m2b2-1<0,x˙m=V-,Σ34-=xm,x˙m,tm∣xm2a2+x˙m2b2-1=0-,x˙m<V,Σ43+=xm,x˙m,tm∣xm2a2+x˙m2b2-1=0+,x˙m<V,Σ14+=xm,x˙m,tm∣xm2a2+x˙m2b2-1>0,x˙m=V+,Σ41-=xm,x˙m,tm∣xm2a2+x˙m2b2-1>0,x˙m=V-,where xm and x˙m(m∈N) are switching displacement and velocity on the corresponding discontinuous boundaries for the mass M at time tm, and V±=limδ→0+⁡(V±δ) and 0±=limδ→0+⁡(0±δ).From the above switching sets, the basic mappings can be defined as(113)Pa:Σ12+⟶Σ14+,Pb:Σ12+⟶Σ12+,Pc:Σ12⟶Σ12,Pd:Σ21-⟶Σ21-,Pe:Σ41-⟶Σ43+,Pf:Σ43+⟶Σ41-,Pg:Σ14+⟶Σ12+,Ph:Σ14+⟶Σ14+,Pi:Σ14⟶Σ14,Pj:Σ41-⟶Σ41-,Pk:Σ21-⟶Σ23+,Pl:Σ32-⟶Σ34-,Pu:Σ34-⟶Σ32-,Pn:Σ23+⟶Σ21-,Po:Σ23+⟶Σ23+,Pp:Σ23⟶Σ23,Pq:Σ32-⟶Σ32-,Pr:Σ34-⟶Σ34-,Ps:Σ34⟶Σ34,Pt:Σ43+⟶Σ43+.From the above definitions, the switching sets and basic mappings are sketched in Figure4.Figure 4
The switching sets and basic mappings in phase plane.The corresponding mapping relationships are(114)Pθ:xm,x˙m,tm⟶xm+1,x˙m+1,tm+1forθ∈a,b,c,d,e,f,g,h,i,j,k,l,n,o,p,q,r,s,t,u.In the above basic mappings, Pθ(θ∈{b,c,d,h,i,j,o,p,q,r,s,t}) are the local mappings that map from one switching set to itself. And Pθ(θ∈{a,e,f,g,k,l,u,n}) are the global mappings that map from one switching set to another.Combined with the elliptic control law in (1) and its solution, the governing equations for the sliding mapping Pc are (115)xm+1=asin⁡batm+1-tm+arcsin⁡xma,ym+1=bcos⁡batm+1-tm+arcsin⁡xma,G∂Ω120,ixm+1,tm+1=0,-1iG∂Ω121,ixm+1,tm+1<0,G∂Ω120,1x,t<0,G∂Ω120,2x,t>0,for xm+1=(xm+1,ym+1), xm=(xm,ym), i=1 or 2, x=(x,y), t∈(tm,tm+1), x∈(xm,xm+1), and y∈(ym,ym+1) on ∂Ω12, and the governing equations for the sliding mapping Ps can be defined similarly.In general, the governing equations for basic mappings,(116)Pθθ∈a,b,c,d,e,f,g,h,i,j,k,l,n,o,p,q,r,s,t,u,can be expressed by(117)fθYm,Ym+1=0,with(118)fθ=f1θ,f2θT,Ym=xm,x˙m,tmT,Ym+1=xm+1,x˙m+1,tm+1T,f1θxm,ym,tm,xm+1,ym+1,tm+1=0,f2θxm,ym,tm,xm+1,ym+1,tm+1=0,xm2a2+x˙m2b2=1orx˙m=V,xm+12a2+x˙m+12b2=1orx˙m+1=V.The notation for mapping structure is introduced as(119)Pn1n2⋯nw≡Pn1∘Pn2∘⋯∘Pnw,where ni∈{a,b,c,d,e,f,g,h,i,j,k,l,n,o,p,q,r,s,t,u}, i∈{1,2,3,…,w}. The motion of the N-terms repeating of mapping Pn1n2⋯nw is defined as (120)Pn1n2⋯nwN≡Pn1∘Pn2∘⋯∘Pnw∘⋯∘Pn1∘Pn2∘⋯∘Pnw︸N-terms.Now, consider a periodic orbit formed by generic mappings in phase plane, and a generalized mapping structure is assumed as(121)P=PuKn10eKn9aKn8bKn7cKn6dKn5nKn4qKn3pKn2oKn1⋯uK110eK19aK18bK17cK16dK15nK14qK13pK12oK11︸N-termsfor N∈Z+, Kkτ∈{0,1}, τ∈{1,2,3,…,10}, and k∈{1,2,3,…,n}.In what follows, four simple periodic orbits with or without sliding or stick motions will be presented to explain how to form periodic motions in the discontinuous dynamic system.SupposeN=1, K17=K16=K15=K13=K12=K11=0, and K1(10)=K19=K18=K14=1, and then there is a mapping structure for a passable periodic motion, as shown in Figure 5(a). By using notation in (121), the mapping structure of such a periodic orbit is (122)P=Puean=Pu∘Pe∘Pa∘Pn.The mapping relations are (123)Pn:xm,x˙m,tm⟶xm+1,x˙m+1,tm+1,Pa:xm+1,x˙m+1,tm+1⟶xm+2,x˙m+2,tm+2,Pe:xm+2,x˙m+2,tm+2⟶xm+3,x˙m+3,tm+3,Pu:xm+3,x˙m+3,tm+3⟶xm+4,x˙m+4,tm+4.During N-period time of excitation, the periodicity of mapping P requires (124)xm+4=xm,x˙m+4=x˙m,tm+4=tm+2NπΩ.Therefore, based on the governing equations in (117), the nonlinear algebraic equations for such a mapping structure Puean can be obtained, that is, (124) plus (125)xm+12a2+x˙m+12b2=1,xm+32a2+x˙m+32b2=1,fn=xm,x˙m,tm,xm+1,x˙m+1,tm+1=0,fa=xm+1,x˙m+1,tm+1,xm+2,x˙m+2,tm+2=0,fe=xm+2,x˙m+2,tm+2,xm+3,x˙m+3,tm+3=0,fu=xm+3,x˙m+3,tm+3,xm+4,x˙m+4,tm+4=0.Figure 5
Mapping structures of periodic orbits: (a)P=Puean=Pu∘Pe∘Pa∘Pn, (b) P=Pbcd=Pb∘Pc∘Pd, (c) P=Pqpo=Pq∘Pp∘Po, and (d) P=Pueacnpo=Pu∘Pe∘Pa∘Pc∘Pn∘Pp∘Po.
(a) (b) (c) (d)Consider a mapping structure for a sliding periodic orbit withN=1, K1(10)=K19=K18=K14=K13=K12=K11=0, and K17=K16=K15=1 in (121), as shown in Figure 5(b). Such a periodic orbit can be described as (126)P=Pbcd=Pb∘Pc∘Pd,where(127)Pd:xm,x˙m,tm⟶xm+1,x˙m+1,tm+1,Pc:xm+1,x˙m+1,tm+1⟶xm+2,x˙m+2,tm+2,Pb:xm+2,x˙m+2,tm+2⟶xm+3,x˙m+3,tm+3.During N-period time of excitation, the periodic motion pertaining to such a mapping P requires(128)xm+3=xm,x˙m+3=x˙m,tm+3=tm+2NπΩ.Thus, according to the governing equations in (117), the nonlinear algebraic equations for such a mapping structure Pbcd can be obtained, that is, (128) plus(129)xm2a2+x˙m2b2=1,xm+12a2+x˙m+12b2=1,xm+22a2+x˙m+22b2=1,fd=xm,x˙m,tm,xm+1,x˙m+1,tm+1=0,fc=xm+1,x˙m+1,tm+1,xm+2,x˙m+2,tm+2=0,fb=xm+2,x˙m+2,tm+2,xm+3,x˙m+3,tm+3=0.Consider a mapping structure for a stick periodic orbit withN=1, K1(10)=K19=K18=K17=K16=K15=K14=0, and K13=K12=K11=1 in (121), as shown in Figure 5(c). Such a periodic orbit can be expressed as (130)P=Pqpo=Pq∘Pp∘Po,where(131)Po:xm,x˙m,tm⟶xm+1,x˙m+1,tm+1,Pp:xm+1,x˙m+1,tm+1⟶xm+2,x˙m+2,tm+2,Pq:xm+2,x˙m+2,tm+2⟶xm+3,x˙m+3,tm+3.During N-period time of excitation, the periodic motion pertaining to such a mapping P requires (132)xm+3=xm,x˙m+3=x˙m,tm+3=tm+2NπΩ.Therefore, from the governing equations in (117), one can obtain a set of nonlinear algebraic equations for such a mapping structure Pqpo, that is, (132) plus (133)fo=xm,x˙m,tm,xm+1,x˙m+1,tm+1=0,fp=xm+1,x˙m+1,tm+1,xm+2,x˙m+2,tm+2=0,fq=xm+2,x˙m+2,tm+2,xm+3,x˙m+3,tm+3=0.Consider a mapping structure for stick and sliding periodic orbit withN=1, K1(10)=K19=K18=K16=K14=K12=K11=1, and K17=K15=K13=0 in (121), as shown in Figure 5(d). Such a periodic orbit can be shown as (134)P=Pueacnpo=Pu∘Pe∘Pa∘Pc∘Pn∘Pp∘Po,where(135)Po:xm,x˙m,tm⟶xm+1,x˙m+1,tm+1,Pp:xm+1,x˙m+1,tm+1⟶xm+2,x˙m+2,tm+2,Pn:xm+2,x˙m+2,tm+2⟶xm+3,x˙m+3,tm+3,Pc:xm+3,x˙m+3,tm+3⟶xm+4,x˙m+4,tm+4,Pa:xm+4,x˙m+4,tm+4⟶xm+5,x˙m+5,tm+5,Pe:xm+5,x˙m+5,tm+5⟶xm+6,x˙m+6,tm+6,Pu:xm+6,x˙m+6,tm+6⟶xm+7,x˙m+7,tm+7.During N-period time of excitation, the requirement for the periodic motion of a mapping P is (136)xm+7=xm,x˙m+7=x˙m,tm+7=tm+2NπΩ.So, based on the governing equations in (117), one can show a class of nonlinear algebraic equations for such a mapping structure Pueacnpo, that is, (136) plus (137)xm+32a2+x˙m+32b2=1,xm+42a2+x˙m+42b2=1,xm+62a2+x˙m+62b2=1,fo=xm,x˙m,tm,xm+1,x˙m+1,tm+1=0,fp=xm+1,x˙m+1,tm+1,xm+2,x˙m+2,tm+2=0,fn=xm+2,x˙m+2,tm+2,xm+3,x˙m+3,tm+3=0,fc=xm+3,x˙m+3,tm+3,xm+4,x˙m+4,tm+4=0,fa=xm+4,x˙m+4,tm+4,xm+5,x˙m+5,tm+5=0,fe=xm+5,x˙m+5,tm+5,xm+6,x˙m+6,tm+6=0,fu=xm+6,x˙m+6,tm+6,xm+7,x˙m+7,tm+7=0.


## 6. Simulations

To illustrate the analytical conditions of passable motions, grazing motions, stick motions, sliding motions, and periodic motions, the motions of the friction-induced oscillator with an elliptic control law and the friction force acting on the massM will be demonstrated in the form of phase trajectories, time-displacement curves, time-velocity curves, and the G-functions responses, such as the time histories of 0th-order G-function and 1st-order G-function. The starting points of motions are described by green-solid circular symbols, and the switching points at which the oscillator may change its motion state are depicted by blue-solid circular symbols. Also the displacement curves, velocity curves, G-functions, and the corresponding trajectories in phase space are shown by red curves. Further, the elliptic control law curves and the velocity curves of the traveling belt are depicted by blue and green curves, respectively.Consider a set of system parameters asM = 2 kg, μ=0.5, FN = 20 N, Q0 = 100 N, Ω = 2 rad/s,φ=0, B1 = B2 = 5 N, k1 = 4 N/m, k2 = 3 N/m, γ1 = 3 N·s/m, γ2 = 5 N·s/m, V = 2 m/s,a=8,b=6 to demonstrate a passable motion of the mass M in Figure 6. The initial conditions are selected as x0 = 4.0 m, y0 = 2.0 m/s, and t0 = 0.0743 s. From Figure 6(d), it can be seen that there are F(2)>0 and F(3)>0 at the staring time t0 = 0.0743 s or the starting point (x0,y0) on the velocity boundary ∂Ω23(i.e., y0=V), which satisfies condition (39) in Theorem 8(ii). Thus the flow will move towards domain Ω2, as shown in Figure 6(a). Once the motion of the mass M arrives at the first switching point (x1,y1)=(4.315,5.097) on the elliptic boundary ∂Ω12 with time t1 = 0.1619 s, the corresponding basic mapping of the mass M for this process is Pn, and the velocity of the mass M is greater than the traveling belt’s (i.e., y1>V) in Ω2. At the first switching time t1 = 0.1691 s, it is observed from Figure 6(d) that F(2)>0 and F(1)>0, and thus b2x1y1+a2y1F(2)>0 and b2x1y1+a2y1F(1)>0 by simple computation. The analytical condition (37) of the passable motion on the elliptic boundary ∂Ω12 in Theorem 8(i) is satisfied, so the flow will cross ∂Ω12 from domain Ω2 into domain Ω1, as shown in Figure 6(a). And the displacement and velocity response transitions are placed in Figures 6(b) and 6(c), respectively. In domain Ω1, the flow reaches the second switching point (x2,y2)=(10.93,2) on the velocity boundary ∂Ω14 with time t2 = 0.9375 s, the corresponding basic mapping of the mass M for this process is Pa, and the velocity of the mass M is equal to the traveling belt’s (i.e., y2=V). At the second switching time t2 = 0.9375 s, we have F(1)<0 andF(4)<0 from Figure 6(d), which satisfies condition (42) in Theorem 8(iv), so the motion of the mass M will cross ∂Ω14 from domain Ω1 into domain Ω4. The flow in domain Ω4 intersects the velocity boundary ∂Ω41 at the third switching point (x3,y3)=(-18.69,2.0) with time t3 = 2.6553 s, and the corresponding basic mapping of the mass M for this process is Pj, while the mass M reaches the velocity boundary ∂Ω41(i.e., y3=V). At the third switching time t3 = 2.6553 s, there are F(4)>0 and F(1)>0, from Figure 6(d), which satisfies condition (43) in Theorem 8(iv), so the motion of the mass M will cross ∂Ω41 from domain Ω4 into domain Ω1, as shown in Figure 6(a), and the motion of the mass M in this process is represented by the corresponding basic mapping Ph. The corresponding trajectory in phase space, time histories of displacement, and velocity of the mass M are shown in Figures 6(a)–6(c), respectively.Figure 6
Numerical simulation of a passable motion with a mapping structureP=Phjan: (a) phase trajectory, (b) displacement-time history, (c) velocity-time history, and (d) force F(α)-time history (M=2, μ=0.5, FN=20, Q0=100, Ω=2, φ=0, B1=B2=5, k1=4, k2=3, γ1=3, γ2=5, V=2, a=8, b=6, t0=0.0743, x0=4.0, y0=2.0).
(a) (b) (c) (d)Consider the grazing motion of the elliptic boundary∂Ω21 with parameters as M = 2 kg, μ=0.5, FN = 10 N, Q0 = 100 N, Ω = 5 rad/s, φ=0, B1 = B2 = 0.5 N, k1 = 5 N/m, k2 = 2 N/m, γ1 = 1.0 N·s/m, γ2 = 0.5 N·s/m, V = 2 m/s, a=8, and b=6 and initial conditions as x0 = −2.15 m, y0 = 3.40 m/s, and t0 = 0.1651 s. The trajectory in phase plane, time histories of displacement, velocity, and G-functions are presented in Figure 7. From Figure 7(a), it can be seen that the mass M moves freely in domain Ω2 at the initial point (x0,y0). Once the motion in domain Ω2 arrives at the first switching point (x1,y1)=(-1.415,5.898) on the elliptic boundary ∂Ω21 with t1 = 0.3101 s, which is labeled by a blue circle, the velocity and displacement of the mass M satisfy the conditions of y1>V and x12/a2+y12/b2=1. From Figure 7(d), G∂Ω12(0,2)=G∂Ω21(0,2)=b2x1y1+a2y1F(2)=0 holds, and it is observed from Figure 7(e) that G∂Ω12(1,2)=G∂Ω21(1,2)=b2x1F(2)+a2y1DF(2)<0, which satisfies condition (65) in Theorem 10(i); therefore the flow of the motion of the mass M is to graze the boundary ∂Ω21 and then returns to domain Ω2, as shown in Figure 7(a). In addition, the displacement and velocity response transitions are placed in Figures 7(b) and 7(c), respectively.Figure 7
Numerical simulation of a grazing motion on elliptic switching boundary∂Ω21: (a) phase trajectory, (b) displacement-time history, (c) velocity-time history, (d) 0th-order G-function-time history, and (e) 1st-order G-function-time history (M=2, μ=0.5, FN=10, Q0=100, Ω=5, φ=0, B1=B2=0.5, k1=5, k2=2, γ1=1.0, γ2=0.5, V=2, a=8, b=6, t0=0.1651, x0=-2.15, y0=3.40).
(a) (b) (c) (d) (e)Consider the grazing motion of the velocity boundary∂Ω32 with parameters as M = 2 kg, μ=0.5, FN = 10 N, Q0 = 100 N, Ω = 4 rad/s, φ=0, B1 = B2 = 5 N, k1 = 10 N/m, k2 = 6 N/m, γ1 = 5 N·s/m, γ2 = 4 N·s/m, V = 6.73 m/s, a=14, and b=10 and initial conditions as x0 = 6.30 m, y0 = 5.00 m/s, and t0 = 0.1350 s, labeled by a green circle. The trajectory in phase plane, time histories of displacement, velocity, and G-functions are presented in Figure 8. According to Figure 8(a), it can be seen that the mass M moves freely in domain Ω3 at the starting point (x0,y0). Once the motion in domain Ω3 arrives at the first switching point (x1,y1)=(7.25,6.73) on the velocity boundary ∂Ω32 with t1 = 0.2885 s, which is labeled by a blue circle, the velocity and displacement of the mass M satisfy the conditions of y1=V and x12/a2+y12/b2<1. From Figures 8(d) and 8(e), we have G∂Ω23(0,3)=G∂Ω32(0,3)=F(3)=0 and G∂Ω23(1,3)=G∂Ω32(1,3)=DF(3)<0, satisfying condition (71) in Theorem 10(iv); thus the motion is to graze the ∂Ω32 and then goes back to domain Ω3, as shown in Figure 8(a). Besides, the displacement and velocity response transitions are placed in Figures 8(b) and 8(c), respectively.Figure 8
Numerical simulation of a grazing motion on velocity boundary∂Ω32: (a) phase trajectory, (b) displacement-time history, (c) velocity-time history, (d) 0th-order G-function-time history, and (e) 1st-order G-function-time history (M=2, μ=0.5, FN=10, Q0=100, Ω=4, φ=0, B1=B2=5, k1=10, k2=6, γ1=5, γ2=4, V=6.73, a=14, b=10, t0=0.1350, x0=6.30, y0=5.00).
(a) (b) (c) (d) (e)Using a set of system parameters asM = 2 kg, μ=0.5, FN = 50 N, Q0 = 150 N, Ω = 4.8 rad/s, φ=0, B1 = 3 N, B2 = 2 N, k1 = 4 N/m, k2 = 3 N/m, γ1 = 0.4 N·s/m, γ2 = 0.2 N·s/m, V = 3.3 m/s, a=20, and b=28 and choosing the initial conditions of t0 = 1.9854 s, x0 = −11.9432 m, and y0 = 17.6143 m/s, we obtain a grazing motion on the velocity boundary ∂Ω23 of the mass M, as illustrated in Figure 9. The initial point of the mass M is in the domain Ω2, and then the mass M moves freely from Figure 9(a). And the time histories of displacement and velocity of the mass M are depicted in Figures 9(b) and 9(c), respectively. At time t1 = 2.3045 s, the motion of the mass M reaches the velocity boundary ∂Ω23(i.e., y1=V). For this point, as shown in Figures 9(d) and 9(e), we have G∂Ω23(0,2)=F(2)=0 and G∂Ω23(1,2)=DF(2)>0, which satisfies condition (70) in Theorem 10(iv); therefore the flow of the motion is to graze the boundary ∂Ω23 and then returns to domain Ω2, as shown in Figure 9(a).Figure 9
Numerical simulation of a grazing motion on velocity boundary∂Ω23: (a) phase trajectory, (b) displacement-time history, (c) velocity-time history, (d) 0th-order G-function-time history, (e) 1st-order G-function-time history (M=2, μ=0.5, FN=50, Q0=150, Ω=4.8, φ=0, B1=3, B2=2, k1=4, k2=3, γ1=0.4, γ2=0.2, V=3.3, a=20, b=28, t0=1.9854, x0=-11.9432, y0=17.6143).
(a) (b) (c) (d) (e)Apply a set of system parameters asM = 1 kg, μ=0.5, FN = 10 N, Q0 = 10 N, Ω = 2 rad/s, φ=0, B1 = B2 = 5 N, k1 = 1.0 N/m, k2 = 0.5 N/m, γ1 = 1.0 N·s/m, γ2 = 0.5 N·s/m, V = 2.0 m/s, a=14, and b=10 to show a stick motion of the mass M on the velocity boundary ∂Ω23 in Figure 10. The initial conditions are t0 = 0.12 s, x0 = 8.5 m, and y0 = 2.0 m/s, and the phase trajectory, the time histories of displacement, and velocity are plotted in Figures 10(a)–10(c), respectively. From Figure 10(a), it can be seen that the stick motion appears at the initial point (x0,y0), and the mass M reaches the velocity boundary ∂Ω23 with velocity equaling V and then moves together with the conveyor belt. The sticky portion is shaded by the gray color. At time t1 = 0.7731 s, the blue filled circle is the vanishing point of the stick motion on the boundary ∂Ω23, because the motion of the mass M enters domain Ω3 and is the free-fight motion after such a point. In order to describe the analytical conditions of stick motion on the boundary ∂Ω23 for the mass M, the time histories of G-functions are depicted in Figure 10(d). From Figure 10(d), it can be observed that there are G∂Ω23(0,3)=F(3)>0 and G∂Ω23(0,2)=F(2)<0 at the initial time t0 = 0.12 s on the velocity boundary ∂Ω23. Thus the necessary and sufficient condition (53) of the stick motion on the velocity boundary ∂Ω23 in Theorem 9(ii) is obtained at initial time t0 = 0.12 s. It is worth noting that, within the gray portion, G∂Ω23(0,3)=F(3)>0 and G∂Ω23(0,2)=F(2)<0, so this stick motion continues until t1 = 0.7731 s. But, at time t1 = 0.7731 s, it can be seen that G∂Ω23(0,3)=F(3)=0, G∂Ω23(0,2)=F(2)<0, and G∂Ω23(1,3)=DF(3)<0 from Figures 10(d) and 10(e). Therefore, the necessary and sufficient condition (99) of the stick motion vanishing on the boundary ∂Ω23 in Theorem 12(ii) is satisfied. After this time t1 = 0.7731 s, the motion of the mass M enters the domain Ω3 and is free-flight motion.Figure 10
Numerical simulation of a stick motion on velocity boundary∂Ω23: (a) phase trajectory, (b) displacement-time history, (c) velocity-time history, (d) 0th-order G-function-time history, and (e) 1st-order G-function-time history (M=1, μ=0.5, FN=10, Q0=10, Ω=2, φ=0, B1=B2=5, k1=1.0, k2=0.5, γ1=1.0, γ2=0.5, V=2.0, a=14, b=10, t0=0.12, x0=8.5, y0=2.0).
(a) (b) (c) (d) (e)Use a set of system parameters asM = 1 kg, μ=0.5, FN = 2 N, Q0 = 40 N, Ω = 3.2 rad/s, φ=0, B1 = B2 = 2 N, k1 = 15 N/m, k2 = 25 N/m, γ1 = 2.0 N·s/m, γ2 = 1.0 N·s/m, V = 0.1 m/s, a=3, and b=2 to show a sliding motion of the mass M on the elliptic boundary ∂Ω12 in Figure 11. The initial conditions are t0 = 0.0015 s, x0 = 1.6947 m, and y0 = 1.6503 m/s, and the phase trajectory, the time histories of displacement, and velocity are plotted in Figures 11(a)–11(c), respectively. From Figure 11(a), it can be seen that the sliding motion appears at the initial point (x0,y0) on the elliptic boundary ∂Ω12, and the sliding portion is shaded by the gray color. At time t1 = 0.1888 s, the blue filled circle is the vanishing point of the sliding motion on the elliptic boundary ∂Ω12, because the motion of the mass M enters domain Ω2 and is the free-fight motion after such a point. To describe the analytical conditions of sliding motion on the elliptic boundary ∂Ω12 for the mass M, the time histories of G-functions are depicted in Figure 11(d). From Figure 11(d), it can be observed that there are G∂Ω12(0,1)=b2x0y0+a2y0F(1)<0 and G∂Ω12(0,2)=b2x0y0+a2y0F(2)>0 at the initial time t0 = 0.0015 s on the elliptic boundary ∂Ω12. Thus the necessary and sufficient condition (52) of the sliding motion on the elliptic boundary ∂Ω12 in Theorem 9(ii) is obtained at initial time t0 = 0.0015 s. It is worth noting that, within the gray portion, G∂Ω12(0,1)=b2x0y0+a2y0F(1)<0 and G∂Ω12(0,2)=b2x0y0+a2y0F(2)>0, so this sliding motion continues until t1 = 0.1888 s. But, at time t1 = 0.1888 s, it can be seen that G∂Ω12(0,1)=b2x1y1+a2y1F(1)<0, G∂Ω12(0,2)=b2x1y1+a2y1F(2)=0, and G∂Ω12(1,2)=b2x1F(2)+a2y1DF(2)<0 from Figures 11(d) and 11(e). Therefore, the necessary and sufficient condition (97) of the sliding motion vanishing on the elliptic boundary ∂Ω12 in Theorem 12(ii) is satisfied. After this time t1 = 0.18888 s, the motion of the mass M enters the domain Ω2 and is free-flight motion.Figure 11
Numerical simulation of a sliding motion on elliptic boundary∂Ω12: (a) phase trajectory, (b) displacement-time history, (c) velocity-time history, (d) 0th-order G-function-time history, and (e) 1st-order G-function-time history (M=1, μ=0.5, FN=2, Q0=40, Ω=3.2, φ=0, B1=B2=2, k1=15, k2=25, γ1=2.0, γ2=1.0, V=0.1, a=3, b=2, t0=0.0015, x0=1.6947, y0=1.6503).
(a) (b) (c) (d) (e)Consider a periodic motion with a mapping structure ofPoq=Po∘Pq for the mass M. The system parameters as M = 1 kg, μ=0.5, FN = 6 N, Q0 = 60 N, Ω = 4.5 rad/s, φ=0, B1 = B2 = 10 N, k1 = 10 N/m, k2 = 24.5 N/m, γ1 = 1.0 N·s/m, γ2 = 1.0 N·s/m, V = 2.0 m/s, a=15, and b=60 and the initial conditions t0 = 0.1102 s, x0 = 12.5061 m, and y0 = 2.0 m/s are given to demonstrate a periodic motion in Figure 12. The phase trajectory, displacement-time history, and velocity-time history of the mass M are presented in Figures 12(a)–12(c), respectively. And the corresponding motions are labeled by mappings Po and Pq. Meanwhile, the periodicity can also be observed in displacement-time history and velocity-time history in Figures 12(b) and 12(c), and the interval between the two adjacent dashed vertical lines is one period. To understand the analytical conditions, the time histories of the G-functions are presented in Figure 12(d). From Figure 12(d), it can be seen that there are G∂Ω23(0,2)=F(2)<0 and G∂Ω23(0,3)=F(3)<0 at the initial time t0 = 0.1102 s on the velocity boundary ∂Ω23, which satisfies the condition (38) in Theorem 8(ii), so the flow of the periodic motion will pass through the velocity boundary ∂Ω23 and enter into domain Ω3 from domain Ω2. At the first switching time t1 = 0.8231 s marked by the blue filled circle, there are G∂Ω23(0,2)=F(2)>0 and G∂Ω23(0,3)=F(3)>0 from Figure 12(d); that is, the condition (39) in Theorem 8(ii) is satisfied. Thus the flow of the periodic motion will cross the velocity boundary ∂Ω23 and turn into domain Ω2 from domain Ω3. Similarly, for the second switching time t2 = 1.5065 s, we have G∂Ω23(0,2)=F(2)<0 and G∂Ω23(0,3)=F(3)<0, which satisfies condition (38) in Theorem 8(ii); therefore the flow of the periodic motion is to pass through the velocity boundary ∂Ω23 and enters into domain Ω3 from domain Ω2, as shown in Figure 12(a). In the meantime, the motion of the mass M in domain Ω2 goes back to the starting point. Therefore, the periodic motion with a mapping structure Poq=Po∘Pq under one periodic is formed.Figure 12
Numerical simulation of a periodic motion with a mapping structure ofPoq=Po∘Pq: (a) phase trajectory, (b) displacement-time history, (c) velocity-time history, and (d) 0th-order G-function-time history (M=1, μ=0.5, FN=6, Q0=60, Ω=4.5, φ=0, B1=B2=10, k1=10, k2=24.5, γ1=1.0, γ2=1.0, V=2.0, a=15, b=60, t0=0.1102, x0=12.5061, y0=2.0).
(a) (b) (c) (d)Consider another periodic motion with a mapping structure ofPhj=Ph∘Pj. The system parameters M = 1 kg, μ=0.5, FN = 6 N, Q0 = 60 N, Ω = 4.5 rad/s, φ=0, B1 = B2 = 10 N, k1 = 24.5 N/m, k2 = 10 N/m, γ1 = 1.0 N·s/m, γ2 = 1.0 N·s/m, V = 2.0 m/s, a=9, and b=40 and the starting points x0 = 12.5061 m and y0 = 2.0 m/s with t0 = 0.1102 s are chosen to illustrate a periodic motion of mass M. The phase trajectory, displacement-time history, and velocity-time history of the mass M are presented in Figures 13(a)–13(c), respectively, and the corresponding motions are labeled by mappings Ph and Pj. From the phase trajectory in Figure 13(a), it can be seen that the motion of the mass M is a periodic motion. In addition, the periodicity can also be observed in displacement-time history and velocity-time history in Figures 13(b) and 13(c). For a better understanding of the analytical conditions, the time histories of the G-functions are presented in Figure 13(d). At the initial point marked by the green filled circle, it can be observed from Figure 13(d) that G∂Ω14(0,1)=F(1)<0 and G∂Ω14(0,4)=F(4)<0, which satisfies condition (42) in Theorem 8(i); thus the flow of the periodic motion is to cross the velocity boundary ∂Ω14 and enters into domain Ω4 from domain Ω1. From Figure 13(d), it can be seen that there are G∂Ω14(0,1)=F(1)>0 and G∂Ω14(0,4)=F(4)>0 at the first switching point marked by the blue filled circle on the velocity boundary ∂Ω41(i.e., y1=V), which is satisfies condition (43) in Theorem 8(iv). So the flow of the periodic motion is to pass through the velocity boundary ∂Ω41 and enters into domain Ω1 from domain Ω4. Similarly, for the second switching time t2 = 1.5065 s, we have G∂Ω14(0,1)=F(1)<0 and G∂Ω14(0,4)=F(4)<0; therefore the periodic motion flow is to pass through the velocity boundary ∂Ω14 and enters into domain Ω4 from domain Ω1, as shown in Figure 13(a). At the same time, the motion of the mass M in domain Ω1 goes back to the starting point. Therefore, the periodic motion with a mapping structure Phj=Ph∘Pj under one period is formed.Figure 13
Numerical simulation of a periodic motion with a mapping structure ofPhj=Ph∘Pj: (a) phase trajectory, (b) displacement-time history, (c) velocity-time history, and (d) 0th-order G-function-time history (M=1, μ=0.5, FN=6, Q0=60, Ω=4.5, φ=0, B1=B2=10, k1=24.5, k2=10, γ1=1.0, γ2=1.0, V=2.0, a=9, b=40, t0=0.1102, x0=12.5061, y0=2.0).
(a) (b) (c) (d)Consider another periodic motion with a mapping structure ofPnutlkb with parameters as M = 1 kg, μ=0.5, FN = 1.4 N, Q0 = 60 N, Ω = 4.2 rad/s, φ=0, B1 = B2 = 1 N, k1 = 90 N/m, k2 = 50 N/m, γ1 = 2.0 N·s/m, γ2 = 1.0 N·s/m, V = 1.0 m/s, a=3,b=2. The initial conditions are adopted on the elliptic boundary ∂Ω12 as x0 = −1.2269 m, y0 = 1.8250 m/s with t0 = 0.8895 s. The trajectory in phase plane, time histories of displacement, velocity, and forces are presented in Figure 14. And the starting points of this motion are labeled by the green filled circles in Figures 14(a)–14(c). It is observed from Figure 14(d) that there are F(1)>0 and F(2)>0; thus b2x0y0+a2y0F(1)>0 and b2x0y0+a2y0F(2)>0 hold by simple calculation, which satisfies condition (37) in Theorem 8(i). Therefore the mass M will move towards domain Ω1, as shown in Figure 14(a). At this time, the corresponding basic mapping of the mass M is Pb. In addition, the red solid curves and magenta dotted curves denote the real and imaginary normal vector fields in Figure 14(d), respectively. For the first switching time t1 = 1.2491 s, it is observed from Figure 14(d) that F(1)<0 and F(2)<0, and thus b2x1y1+a2y1F(1)<0 and b2x1y1+a2y1F(2)<0 through simple computation; that is, condition (36) in Theorem 8(i) is satisfied. Thus the flow of the periodic motion will pass through the elliptic boundary ∂Ω12 and enter into domain Ω2 from domain Ω1. At this time, the corresponding basic mapping of the mass M is Pk. And, at the second switching time t2 = 1.5991 s, from Figure 14(d), we have F(2)<0 and F(3)<0; in other words, condition (38) in Theorem 8(ii) is satisfied. Hence the flow of the periodic motion will cross the velocity boundary ∂Ω23 and enter into domain Ω3 from domain Ω2; the corresponding basic mapping of the mass M for this process is Pl, as shown in Figure 14(a). The motion of the mass M in domain Ω3 reaches the third switching point (x3,y3)=(1.3340,-1.8010) on the elliptic boundary ∂Ω34 with time t3 = 1.7275 s, and it is observed that F(3)<0 and F(4)<0 from Figure 14(d) and b2x3y3+a2y3F(3)>0 and b2x3y3+a2y3F(4)>0 by simple computation, which satisfies condition (40) in Theorem 8(iii). Thus the flow of the periodic motion will pass through the elliptic boundary ∂Ω34 and turn into domain Ω4 from domain Ω3; the corresponding basic mapping of the mass M for this process is Pt, as shown in Figure 14(a). Once the motion in domain Ω4 arrives at the fourth switching point (x4,y4)=(-1.2770,-1.9270) on the elliptic boundary ∂Ω43 with time t4 = 2.0792 s, we have F(3)>0 and F(4)>0 from Figure 14(d) and b2x4y4+a2y4F(3)<0 and b2x4y4+a2y4F(4)<0 by simple computation; therefore condition (41) in Theorem 8(iii) is satisfied. So the flow of the periodic motion will pass through the elliptic boundary ∂Ω43 and enter into domain Ω3 from domain Ω4, and the corresponding basic mapping of the mass M for this process is Pu, as shown in Figure 14(a). The flow in domain Ω3 intersects the velocity boundary ∂Ω32 at the fifth switching point (x5,y5)=(-1.3378,1.0) with time t5 = 2.2985 s; inequalities F(3)>0 and F(2)>0 follow from Figure 14(d), which satisfies condition (39) in Theorem 8(ii), so the motion of the mass M will cross ∂Ω32 from domain Ω3 into domain Ω2, as shown in Figure 14(a). At this time, the corresponding basic mapping of the mass M is Pn. And the flow of motion of the mass M in domain Ω2 goes back to the starting point (x0,y0) at time t6 = 2.3855 s, as shown in Figure 14(a). Therefore, the periodic motion with a mapping structure Pnutlkb under one period is formed.Figure 14
Numerical simulation of a periodic motion with a mapping structure ofPnutlkb: (a) phase trajectory, (b) displacement-time history, (c) velocity-time history, and (d) force F(α)-time history (M=1, μ=0.5, FN=1.4, Q0=60, Ω=4.2, φ=0, B1=B2=1, k1=90, k2=50, γ1=2.0, γ2=1.0, V=1.0, a=3, b=2, t0=0.8895, x0=-1.2269, y0=1.8250).
(a) (b) (c) (d)Consider another periodic motion with a mapping structure ofPacgfse with parameters as M = 1 kg, μ=0.5, FN = 6 N, Q0 = 60 N, Ω = 4.5 rad/s, φ=0, B1 = B2 = 10 N, k1 = 24.5 N/m, k2 = 10 N/m, γ1 = 1.0 N·s/m, γ2 = 1.0 N·s/m, V = 2.0 m/s, a=10, and b=60. The initial conditions are adopted on the velocity boundary ∂Ω14 as x0 = 13.0205 m, y0 = 2.0 m/s with t0 = 0.1262 s. The trajectory in phase plane, time histories of displacement, velocity, and forces are presented in Figure 15. And the starting points of this motion are labeled by the green filled circles in Figures 15(a)–15(c). It is observed from Figure 15(d) that there are F(1)<0 and F(4)<0, which satisfies condition (42) in Theorem 8(iv). Therefore the mass M will move towards domain Ω4, as shown in Figure 14(a). At this time, the corresponding basic mapping of the mass M is Pe. In addition, the red solid curves and magenta dotted curves denote the real and imaginary normal vector fields in Figure 15(d), respectively. Once the motion of the mass M arrives at the first switching point (x1,y1)=(4.0871,-54.7601) on the elliptic boundary ∂Ω43 with time t1 = 0.4155 s, it is observed from Figure 15(d) that F(4)>0 and F(3)>0, and thus b2x1y1+a2y1F(4)<0 and b2x1y1+a2y1F(3)<0 through simple computation; that is, condition (41) in Theorem 8(iii) is satisfied. Thus the flow of the periodic motion will pass through the elliptic boundary ∂Ω43 and enter into domain Ω3 from domain Ω4. At this time, the corresponding basic mapping of the mass M is Ps. And at the second switching point (x2,y2)=(-4.5141,-53.6501) on the elliptic boundary ∂Ω34 with time t2 = 0.5722 s, we have F(3)<0 and F(4)<0 from Figure 15(d) and b2x2y2+a2y2F(3)>0 and b2x2y2+a2y2F(4)>0 by simple computation, which satisfies condition (40) in Theorem 8(iii). Hence the flow of the periodic motion will cross the boundary ∂Ω34 and enter into domain Ω4 from domain Ω3, and the corresponding basic mapping of the mass M for this time is Pf, as shown in Figure 15(a). The motion of the mass M in domain Ω4 reaches the third switching point (x3,y3)=(-12.3805,2.0) on the velocity boundary ∂Ω41 with time t3 = 0.8386 s; it is observed from Figure 15(d) that F(4)>0 and F(1)>0, which satisfies condition (43) in Theorem 8(iv). Thus the flow of the periodic motion will pass through the boundary ∂Ω41 and turn into domain Ω1 from domain Ω4, and the corresponding basic mapping of the mass M for this time is Pg, as shown in Figure 15(a). Once the motion in domain Ω1 arrives at the fourth switching point (x4,y4)=(-4.5511,53.5601) on the elliptic boundary ∂Ω12 with time t4 = 1.0899 s, from Figure 15(d), we have F(2)<0 and F(1)<0; therefore b2x4y4+a2y4F(2)<0 and b2x4y4+a2y4F(1)<0 hold by simple computation, which satisfies condition (36) in Theorem 8(i). So the flow of the periodic motion will pass through the boundary ∂Ω12 and enter into domain Ω2 from domain Ω1, and the corresponding basic mapping of the mass M for this time is Pc, as shown in Figure 15(a). The flow in domain Ω2 intersects the elliptic boundary ∂Ω21 at the fifth switching point (x5,y5)=(3.7611,55.5801) with time t5 = 1.2405 s; inequalities F(1)>0 and F(2)>0 follow from Figure 15(d) and b2x5y5+a2y5F(1)>0 and b2x5y5+a2y5F(2)>0 hold by simple computation, which satisfies condition (37) in Theorem 8(i). So the motion of the mass M will cross ∂Ω21 from domain Ω2 into domain Ω1, as shown in Figure 15(a). At this time, the corresponding basic mapping of the mass M is Pa. And the flow of motion of the mass M in domain Ω1 goes back to the starting point (x0,y0) on the velocity boundary ∂Ω14 at time t6 = 1.5225 s, as shown in Figure 15(a). Therefore, the periodic motion with a mapping structure Pacgfse under one period is formed.Figure 15
Numerical simulation of a periodic motion with a mapping structure ofPacgfse: (a) phase trajectory, (b) displacement-time history, (c) velocity-time history, and (d) force F(α)-time history (M=1, μ=0.5, FN=6, Q0=60, Ω=4.5, φ=0, B1=B2=10, k1=24.5, k2=10, γ1=1.0, γ2=1.0, V=2.0, a=10, b=60, t0=0.1262, x0=13.0205, y0=2.0).
(a) (b) (c) (d)


## 7. Conclusions

In this paper, the discontinuous dynamical behaviors of a friction-induced oscillator with elliptic control law and the friction force acting on the massM were investigated through the theory of flow switchability for discontinuous dynamical systems. Different domains and boundaries for such a system were defined according to the discontinuities caused by the friction and the elliptic control law. Based on the above domains and boundaries, the analytical conditions for the passable motions, sliding or stick motions, grazing motions, and the onset or vanishing of sliding or stick motions were presented through the G-functions of the vector fields. The basic mappings and switching sets were introduced to describe motions in such an oscillator. Analytical conditions of periodic motions were developed by the mapping dynamics, and numerical simulations for illustration of analytical conditions and periodic motions were carried out for a better understanding of complicated dynamics of such mechanical model. There are more simulations about such an oscillator to be discussed in future.



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