# On Dynamical Behavior of a Friction-Induced Oscillator with 2-DOF on a Speed-Varying Traveling Belt

Authors: Jinjun Fan; Shuangshuang Li; Ge Chen
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

The dynamical behavior of a friction-induced oscillator with 2-DOF on a speed-varying belt is investigated by using the flow switchability theory of discontinuous dynamical systems. The mechanical model consists of two masses and a speed-varying traveling belt. Both of the masses on the traveling belt are connected with three linear springs and three dampers and are harmonically excited. Different domains and boundaries for such system are defined according to the friction discontinuity. Based on the above domains and boundaries, the analytical conditions of the passable motions, stick motions, and grazing motions for the friction-induced oscillator are obtained mathematically. An analytical prediction of periodic motions is performed through the mapping dynamics. With appropriate mapping structure, the simulations of the stick and nonstick motions in the two-degree friction-induced oscillator are illustrated for a better understanding of the motion complexity.


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## Body

## 1. Introduction

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 [1–11]. This problem can go back to the 30s of last century. In 1930, Hartog [1] investigated the nonstick periodic motion of the forced linear oscillator with Coulomb and viscous damping. In 1960, Levitan [2] proved the existence of periodic motions in a friction oscillator with the periodically driven base. In 1964, Filippov [3] investigated the motion in the Coulomb friction oscillator and presented differential equation theory with discontinuous right-hand sides. The investigations of such discontinuous differential equations were summarized in Filippov [4]. However, Filippov’s theory mainly focused on the existence and uniqueness of the solutions for nonsmooth dynamical systems. Such a differential equation theory with discontinuity is difficult to apply to practical problems. In 2003, Awrejcewicz and Olejnik [5] studied a two-degree-of-freedom autonomous system with friction numerically and illustrated some interesting examples of stick-slip regular and chaotic dynamics. In 2014, Pascal [6] 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 had been proved for some of these periodic solutions. More discussion about discontinuous system can refer to [7–11].However, a lot of questions caused by the discontinuity (i.e., the local singularity and the motion switching on the separation boundary) were not discussed in detail. So the further investigation on discontinuous dynamical systems should be deepened and expanded. In 2005–2012, Luo [12–17] developed a general theory to define real, imaginary, sink, and source flows and to handle the local singularity and flow switchability in discontinuous dynamical systems. By using this theory, a lot of discontinuous systems were discussed (e.g., [18–20]). Luo and Gegg [18] presented the force criteria for the stick and nonstick motions for 1-DOF (degree of freedom) oscillator moving on the belt with dry friction. In 2009, Luo and Wang [19] investigated the analytical conditions for stick and nonstick motions in 2-DOF friction induced oscillator moving on two belts. Velocity and force responses for stick and nonstick motions in such system were illustrated for a better understanding of the motion complexity. Based on this improved model, which consists of two masses moving on one speed-varying traveling belt and in which the two masses are connected with three linear springs and three dampers and are exerted by two periodic excitations, nonlinear dynamics mechanism of such a 2-DOF oscillator system will be investigated.In this paper, a model of frictional-induced oscillator with two degrees of freedom (2-DOF) on a speed-varying belt is proposed in which multiple discontinuity boundaries exist: they are caused by the presence of friction between the mass and the belt. The model allows a simple representation of engineering applications with multiple nonsmooth characteristics as for instance friction wheels or slipping mechanisms in multiblock structures. The main goal is to study the analytical conditions of motion switching and stick motions of the oscillator on the corresponding boundaries by using the theory of discontinuous dynamical systems. Based on the discontinuity, domain partitions and boundaries will be defined and the analytical conditions of the passable motions, stick motions, and grazing motions for the friction-induced oscillator are obtained mathematically, from which it can be seen that such oscillator has more complicated and rich dynamical behaviors. An analytical condition of periodic motions is performed through the mapping dynamics. With appropriate mapping structure, the simulations of the stick and nonstick motions of the oscillator with 2-DOF are illustrated for a better understanding of the motion complexity. There are more simulations about such oscillator to be discussed in future.


## 2. Preliminaries

For convenience, the fundamental theory on flow switchability of discontinuous dynamical systems will be presented; that is, concepts ofG-functions and the decision theorems of semipassable flow, sink flow, and grazing flow to a separation boundary are stated in the following, respectively (see [16, 17]).Assume thatΩ is a bounded simply connected domain in Rn and its boundary ∂Ω⊂Rn-1 is a smooth surface.Consider a dynamic system consisting ofN subdynamic systems in a universal domain Ω⊂Rn. The universal domain is divided into N accessible subdomains Ωα(α∈I) and the inaccessible domain Ω0. The union of all the accessible subdomains is ⋃α∈IΩα and Ω=⋃α∈IΩα∪Ω0 is the universal domain. On the αth open subdomain Ωα, there is a Crα-continuous system (rα≥1) in form of(1)x˙α≡Fαxα,t,pα∈Rn,xα=x1α,x2α,…,xnαT∈Ωα.The time is t and x˙=dx/dt. In an accessible subdomain Ωα, the vector field F(α)(x(α),t,pα) with parameter vector pα=(pα(1),pα(2),…,pα(l))T∈Rl is Crα-continuous (rα≥1) in x∈Ωα and for all time t.The flow on the boundary∂Ωij=Ωi∩Ωj can be determined by(2)x˙0≡F0x0,t,λwithφijx0,t,λ=0,where x(0)=(x1(0),x2(0),…,xn(0))T. With specific initial conditions, one always obtains different flows on φij(x(0),t,λ)=φij(x0(0),t0,λ).Consider a dynamic system in (1) in domain Ωα(α∈{i,j}) which has a flow xt(α)=Φ(α)(t0,x0(α),pα,t) with an initial condition (t0,x0(α)), and on the boundary ∂Ωij, there is an enough smooth flow xt(0)=Φ(0)(t0,x0(0),λ,t) with an initial condition (t0,x0(0)). For an arbitrarily small ε>0, there are two time intervals t-ε,t and t,t+ε for flow xt(α)(α∈{i,j}) and the flow xt(α) approaches the separation boundary at time tm, that is, xtm±(α)=xm=xtm(0), where xtm±(α)=x(α)(tm±), xtm(0)=x(0)(tm), and xm∈∂Ωij.Definition 1.
TheG-functions G∂Ωij(α) of the flow xt(α) to the flow xt(0) on the boundary ∂Ωij are defined as(3)G∂Ωijαxm,tm±,pα,λ=n∂ΩijTx0,t,λ·Fαxα,t,pα-F0x0,t,λxm0,xm±α,tm±,where xm(0)=x(0)(tm), xm±(α)=x(α)(tm±), tm±≡tm±0 is to represent the quantity in the domain rather than on the boundary, and G∂Ωij(α)(xm,tm±,pα,λ) is a time rate of the inner product of displacement difference and the normal direction n∂Ωij(x(0),tm,λ).Definition 2.
Thekth-order G-functions of the domain flow xt(α) to the boundary flow xt(0) in the normal direction of ∂Ωij are defined as(4)G∂Ωijk,αxm,tm±,pα,λ=∑s=1k+1Ck+1sD0k+1-sn∂ΩijT·Dαs-1Fxα,t,pα-D0s-1F0x0,t,λxm0,xm±α,tm±,where the total derivative operators are defined as(5)D0·≡∂·∂x0x˙0+∂·t,Dα·≡∂·∂xαx˙α+∂·t.For k=0, we have(6)G∂Ωijk,αxm,tm±,pα,λ=G∂Ωijαxm,tm±,pα,λ.Definition 3.
For a discontinuous dynamical system in (1), there is a point x(tm)≡xm∈∂Ωij. 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+); if(7)n∂ΩijTxm-ε0·xm-ε0-xm-εi>0,n∂ΩijTxm+ε0·xm+εj-xm+ε0>0for n∂Ωij→Ωj, then a resultant flow of two flows x(α)(t)(α∈{i,j}) is a semipassable flow from domain Ωi to Ωj at point (xm,tm) to boundary ∂Ωij, where xm±ε(0)=x0tm±ε, xm±ε(α)=x(α)(tm±ε).To simplify notation usage, the symbolstm±ε represent tm±ε in next paragraphs.Lemma 4.
For a discontinuous dynamical system in (1), there is a point x(tm)≡xm∈∂Ωij at time tm between two adjacent domains Ωα(α∈{i,j}). 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+), two flows x(i)(t) and x(j)(t) are Ctm-ε,tmri and Ctm,tm+εrj-continuous for time t, respectively, and drα+1x(α)/dtrα+1<∞(rα≥1,α∈{i,j}). The flows x(i)(t) and x(j)(t) at point (xm,tm) to the boundary ∂Ωij are semipassable from domain Ωi to Ωj if and only if either(8)G∂Ωijixm,tm-,pi,λ>0,G∂Ωijjxm,tm+,pj,λ>0forn∂Ωij⟶Ωj,or(9)G∂Ωijixm,tm-,pi,λ<0,G∂Ωijjxm,tm+,pj,λ<0forn∂Ωij⟶Ωi.Lemma 5.
For a discontinuous dynamical system in (1), there is a point x(tm)≡xm∈∂Ωij at time tm between two adjacent domains Ωα(α∈{i,j}). For an arbitrarily small ε>0, there is a time interval tm-ε,tm. Suppose x(i)(tm-)=xm=x(j)(tm-). Both flows x(i)(t) and x(j)(t) are Ctm-ε,tmrα-continuous for time t and drα+1x(α)/dtrα+1<∞(rα≥1,α∈{i,j}). The flows x(i)(t) and x(j)(t) at point (xm,tm) to the boundary ∂Ωij are sink flow if and only if either(10)G∂Ωijixm,tm-,pi,λ>0,G∂Ωijjxm,tm-,pj,λ<0forn∂Ωij⟶Ωj,or(11)G∂Ωijixm,tm-,pi,λ<0,G∂Ωijjxm,tm-,pj,λ>0forn∂Ωij⟶Ωi.Lemma 6.
For a discontinuous dynamical system in (1), there is a point x(tm)≡xm∈∂Ωij at time tm between two adjacent domains Ωα(α∈{i,j}). 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±). The flows x(i)(t) and x(j)(t) are Ctm-ε,tmri and Ctm-ε,tm+εrj-continuous for time t, respectively, and drα+1x(α)/dtrα+1<∞(rα≥2,α∈{i,j}). The sliding fragmentation bifurcation of the nonpassable flows x(i)(t) and x(j)(t) of the first kind at point (xm,tm) switching to the passable flow on the boundary ∂Ωij~ occurs if and only if (12)G∂Ωijjxm,tm±,pj,λ=0,G∂Ωijixm,tm-,pi,λ>0forn∂Ωij⟶Ωj,G∂Ωijixm,tm-,pi,λ<0forn∂Ωij⟶Ωi,G∂Ωij1,jxm,tm±,pj,λ>0forn∂Ωij⟶Ωj,G∂Ωij1,jxm,tm±,pj,λ<0forn∂Ωij⟶Ωi.Lemma 7.
For a discontinuous dynamical system in (1), there is a point x(tm)≡xm∈∂Ωij at time tm between two adjacent domains Ωα(α∈{i,j}). 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±). The flows x(i)(t) and x(j)(t) are Ctm-ε,tmri and Ctm-ε,tm+εrj-continuous for time t, respectively, and drα+1x(α)/dtrα+1<∞(rα≥2,α∈{i,j}). The sliding bifurcation of the passable flow of x(i)(t) and x(j)(t) at point (xm,tm) switching to the sink flow on the boundary ∂Ωij→ occurs if and only if (13)G∂Ωijjxm,tm±,pj,λ=0,G∂Ωijixm,tm-,pi,λ>0forn∂Ωij⟶Ωj,G∂Ωijixm,tm-,pi,λ<0forn∂Ωij⟶Ωi,G∂Ωij1,jxm,tm±,pj,λ>0forn∂Ωij⟶Ωj,G∂Ωij1,jxm,tm±,pj,λ<0forn∂Ωij⟶Ωi.Lemma 8.
For a discontinuous dynamical system in (1), there is a point x(tm)≡xm∈∂Ωij at time tm between two adjacent domains Ωα(α∈{i,j}). For an arbitrarily small ε>0, there is a time interval tm-ε,tm+ε. Suppose x(α)(tm±)=xm. The flow x(α)(t) is Ctm-ε,tm+εrα-continuous (rα≥2) for time t, and drα+1x(α)/dtrα+1<∞(α∈{i,j}). A flow x(α)(t) in Ωα is tangential to the boundary ∂Ωij if and only if (14)G∂Ωij0,αxm,tm,pα,λ=0forα∈i,j;either(15)G∂Ωij1,αxm,tm,pα,λ<0forn∂Ωαβ⟶Ωβ,or(16)G∂Ωij1,αxm,tm,pα,λ>0forn∂Ωαβ⟶Ωα.More detailed theory on the flow switchability such as the definitions or theorems about various flow passability in discontinuous dynamical systems can be referred to [16, 17].


## 3. Physical Model

Consider a friction-induced oscillator with two degrees of freedom on the speed-varying traveling belt, as shown in Figure1. The system consists of two masses mα(α=1,2), which are connected with three linear springs of stiffness kα(α=1,2,3) and three dampers of coefficient rα(α=1,2,3). Both of masses move on the belt with varying speed V(t). Two periodic excitations Aα+Bαcos⁡Ωt(α=1,2) with frequency Ω, amplitudes Bα(α=1,2), and constant forces Aα(α=1,2) are exerted on the two masses, respectively.Figure 1
Physical model.There exist friction forces between the two masses and the belt, so the two masses can move or stay on the surface of the belt. LetV(t) be the speed of the belt and(17)Vt=V0cos⁡Ωt+β+V1,where Ω and β are the oscillation frequency and primary phase of the traveling belt, respectively, V0 is the oscillation amplitude of the traveling belt, and V1 is constant.Further, the friction force shown in Figure2 is described by(18)Ffαx˙α=μkFNα,x˙α>Vt;∈-μkFNα,μkFNα,x˙α=Vt;=-μkFNα,x˙α<Vt,where x˙α=dxα/dt, μk is the coefficient of friction between mα and the belt, FN(α)=mαg(α=1,2), and g is the acceleration of gravity. The nonfriction force acting on the mass mα in the xα-direction is defined as(19)Fsα=Bαcos⁡Ωt+Aα-rαx˙α-r3x˙α-x˙β-kαxα-k3xα-xβ,where α,β∈{1,2} and α≠β. From now on, Ff(α)=μk·FN(α).Figure 2
Force of friction.From the previous discussion, there are four cases of motions:Case 1 (nonstick motion(x˙α≠V(t))(α=1,2)).
WhenFs(α) can overcome the static friction force Ff(α) (i.e., |Fs(α)|>|Ff(α)|,α=1,2), the mass mα has relative motion to the belt, that is,(20)x˙α≠Vt,α=1,2.
For the nonstick motion of the massmα(α=1,2), the total force acting on the mass mα is(21)Fα=Fsα-Ffαsgnx˙α-Vt=Bαcos⁡Ωt+Aα-rαx˙α-r3x˙α-x˙β-kαxα-k3xα-xβ-Ffαsgnx˙α-Vt,and the equations of nonstick motion for the 2-DOF dry friction induced oscillator are(22)mαx¨α+rαx˙α+r3x˙α-x˙β+kαxα+k3xα-xβ=Bαcos⁡Ωt+Aα-Ffαsgnx˙α-Vt,where α,β∈{1,2},α≠β.Case 2 (single stick motion (x˙1=V(t),x˙2≠V(t))).
WhenFs(1) cannot overcome the static friction force Ff(1) (i.e., |Fs(1)|≤|Ff(1)|), mass m1 does not have any relative motion to the belt, that is,(23)x˙1=Vt,x¨1=V˙t=-V0Ωsin⁡Ωt+β;meanwhile, when Fs(2) can overcome the static friction force Ff(2) (i.e., |Fs(2)|>|Ff(2)|), the mass m2 has relative motion to the belt, that is,(24)x˙2≠Vt,m2x¨2+r2x˙2+r3x˙2-x˙1+k2x2+k3x2-x1=B2cos⁡Ωt+A2-Ff2sgnx˙2-Vt.Case 3 (single stick motion (x˙2=V(t),x˙1≠Vt)).
WhenFs(2) cannot overcome the static friction force Ff(2) (i.e., |Fs(2)|≤|Ff(2)|), mass m2 does not have any relative motion to the belt, that is,(25)x˙2=Vt,x¨2=V˙t=-V0Ωsin⁡Ωt+β;meanwhile, when Fs(1) can overcome the static friction force Ff(1) (i.e., |Fs(1)|>|Ff(1)|), mass m1 has relative motion to the belt, that is,(26)x˙1≠Vt,m1x¨1+r1x˙1+r3x˙1-x˙2+k1x1+k3x1-x2=B1cos⁡Ωt+A1-Ff1sgnx˙1-Vt.Case 4 (double stick motions(x˙α=V(t))(α=1,2)).
WhenFs(α) cannot overcome the static friction force Ff(α) (i.e., |Fs(α)|≤|Ff(α)|), mass mα does not have any relative motion to the belt, that is,(27)x˙α=Vt,x¨α=V˙t=-V0Ωsin⁡Ωt+β.
Integrating (17) leads to the displacement of the belt:(28)xt=V0Ωsin⁡Ωt+β-sin⁡Ωti+β+V1t-ti+xti,where t>ti and xti=x(ti).


## 4. Domains and Boundaries

Due to frictions between the massmα(α=1,2) and the traveling belt, the motions become discontinuous and more complicated. The phase space of the discontinuous dynamical system is divided into four 4-dimensional domains.The state variables and vector fields are introduced by(29)x=x1,x˙1,x2,x˙2T=x1,y1,x2,y2T,F=y1,F1,y2,F2T.By the state variables, the domains are defined as(30)Ω1=x1,y1,x2,y2∣y1>Vt,y2>Vt,Ω2=x1,y1,x2,y2∣y1>Vt,y2<Vt,Ω3=x1,y1,x2,y2∣y1<Vt,y2<Vt,Ω4=x1,y1,x2,y2∣y1<Vt,y2>Vtand the corresponding boundaries are defined as(31)∂Ω12=∂Ω21=x1,y1,x2,y2∣φ12=φ21=y2-Vt=0,y1≥Vt,∂Ω23=∂Ω32=x1,y1,x2,y2∣φ23=φ32=y1-Vt=0,y2≤Vt,∂Ω34=∂Ω43=x1,y1,x2,y2∣φ34=φ43=y2-Vt=0,y1≤Vt,∂Ω14=∂Ω41=x1,y1,x2,y2∣φ12=φ21=y1-Vt=0,y2≥Vt.The phase plane ofmα is shown in Figure 3.Figure 3
Phase plane ofmα.The 2-dimensional edges of the 3-dimensional boundaries are defined by(32)∠Ωα1α2α3=∂Ωα1α2∩∂Ωα2α3=⋂i=13Ωαi,where αi∈{1,2,3,4},i=1,2,3, and αi≠αj (i≠j), i,j∈1,2,3. The intersection of four 2-dimensional edges is(33)∠Ω1234=⋂∠Ωα1α2α3=x1,y1,x2,y2∣φ12=φ34=y2-Vt=0,φ23=φ14=y1-Vt=0.From the above discussion, the motion equations of the oscillator described in Section 3 in absolute coordinates are(34)x˙α=Fαxα,tinΩα,x˙α1α2=Fα1α2xα1α2,ton∂Ωα1α2,x˙α1α2α3=Fα1α2α3xα1α2α3,ton∂Ωα1α2α3,xα=xα1α2=xα1α2α3=x1,y1,x2,y2T,Fα=y1,F1α,y2,F2αT,Fα1α2=y1,F1α1α2,y2,F2α1α2T,Fα1α2α3=y1,F1α1α2α3,y2,F2α1α2α3T,where the forces of per unit mass for the 2-DOF friction induced oscillator in the domain Ωαα∈1,2,3,4 are(35)F11=F12=b1cos⁡Ωt+a1-c1y1-p1y1-y2-d1x1-q1x1-x2-f1,F13=F14=b1cos⁡Ωt+a1-c1y1-p1y1-y2-d1x1-q1x1-x2+f1,F21=F24=b2cos⁡Ωt+a2-c2y2-p2y2-y1-d2x2-q2x2-x1-f2,F22=F23=b2cos⁡Ωt+a2-c2y2-p2y2-y1-d2x2-q2x2-x1+f2;here(36)aα=Aαmα,bα=Bαmα,cα=rαmα,dα=kαmα,pα=r3mα,qα=k3mα,fα=Ffαmα,α∈1,2,and the forces of per unit mass of the oscillator on the boundary ∂Ωα1α2 (α1,α2∈1,2,3,4, α1≠α2) are(37)F112≡b1cos⁡Ωt+a1-c1y1-p1y1-y2-d1x1-q1x1-x2-f1,F212=0for  stick  on∂Ω12,F212∈F21,F22for  nonstick  on∂Ω12;F223≡b2cos⁡Ωt+a2-c2y2-p2y2-y1-d2x2-q2x2-x1+f2,F123=0for  stick  on∂Ω23,F123∈F12,F13for  nonstick  on∂Ω23;F134≡b1cos⁡Ωt+a1-c1y1-p1y1-y2-d1x1-q1x1-x2+f1,F234=0for  stick  on∂Ω34,F234∈F24,F23for  nonstick  on∂Ω34;F214≡b2cos⁡Ωt+a2-c2y2-p2y2-y1-d2x2-q2x2-x1-f2,F114=0for  stick  on∂Ω14,F114∈F11,F14for  nonstick  on∂Ω14.The forces of per unit mass of the oscillator on the boundary∂Ωα1α2α3 (αi∈{1,2,3,4},i=1,2,3; α1,α2,α3 are not equal to each other without repeating) are(38)Fαα1α2α3∈F1α1α2,F2α2α3,α∈1,2for  nonstick  on∂Ωα1α2α3;Fαα1α2α3=0,α∈1,2for  full  stick  on∂Ωα1α2α3.For simplicity, the relative displacement, velocity, and acceleration between the massmα(α=1,2) and the traveling belt are defined as(39)zα=xα-xt,vα=x˙α-Vt,z¨α=x¨α-V˙t.The domains and boundaries in relative coordinates are defined as(40)Ω1=z1,v1,z2,v2∣v1>0,v2>0,Ω2=z1,v1,z2,v2∣v1>0,v2<0,Ω3=z1,v1,z2,v2∣v1<0,v2<0,Ω4=z1,v1,z2,v2∣v1<0,v2>0,(41)∂Ω12=∂Ω21=z1,v1,z2,v2∣φ12=φ21=v2=0,v1≥0,∂Ω23=∂Ω32=z1,v1,z2,v2∣φ23=φ32=v1=0,v2≤0,∂Ω34=∂Ω43=z1,v1,z2,v2∣φ34=φ43=v2=0,v1≤0,∂Ω14=∂Ω41=z1,v1,z2,v2∣φ14=φ41=v1=0,v2≥0,(42)∠Ωα1α2α3=∂Ωα1α2∩∂Ωα2α3=⋂i=13Ωαi,where αi∈{1,2,3,4},i=1,2,3, and αi≠αj (i≠j), i,j∈1,2,3. The intersection of four 2-dimensional edges is(43)∠Ω1234=⋂∠Ωα1α2α3=z1,v1,z2,v2∣φ12=φ34=v2=0,φ23=φ14=v1=0.The domain partitions and boundaries in relative coordinates are shown in Figure4.Figure 4
Relative domains and boundaries.From the foregoing equations, the motion equations in relative coordinates are as follows:(44)z˙α=gαzα,xα,tinΩα,z˙α1α2=gα1α2zα1α2,xα1α2,ton∂Ωα1α2,z˙α1α2α3=gα1α2α3zα1α2α3,xα1α2α3,ton∂Ωα1α2α3,where(45)zα=zα1α2=zα1α2α3=z1,z˙1,z2,z˙2T=z1,v1,z2,v2T,gα=z˙1,g1α,z˙2,g2αT=v1,g1α,v2,g2αT,gα1α2=z˙1,g1α1α2,z˙2,g2α1α2T=v1,g1α1α2,v2,g2α1α2T,gα1α2α3=z˙1,g1α1α2α3,z˙2,g2α1α2α3T=v1,g1α1α2α3,v2,g2α1α2α3T.The forces of per unit mass for the 2-DOF friction induced oscillator in the domainΩα(α∈{1,2,3,4}) in relative coordinates are(46)g11=g12=b1cos⁡Ωt+a1-c1v1-p1v1-v2-d1z1-q1z1-z2-c1Vt-d1xt-f1-V˙t,g13=g14=b1cos⁡Ωt+a1-c1v1-p1v1-v2-d1z1-q1z1-z2-c1Vt-d1xt+f1-V˙t;g21=g24=b2cos⁡Ωt+a2-c2v2-p2v2-v1-d2z2-q2z2-z1-c2Vt-d2xt-f2-V˙t,g22=g23=b2cos⁡Ωt+a2-c2v2-p2v2-v1-d2z2-q2z2-z1-c2Vt-d2xt+f2-V˙t.The forces of per unit mass of the friction induced oscillator on the boundary∂Ωα1α2 in relative coordinates are(47)g112≡b1cos⁡Ωt+a1-c1v1-p1v1-v2-d1z1-q1z1-z2-c1Vt-d1xt-f1-V˙t,g212=0for  stick  on∂Ω12,g212∈g21,g22for  nonstick  on∂Ω12;g223≡b2cos⁡Ωt+a2-c2v2-p2v2-v1-d2z2-q2z2-z1-c2Vt-d2xt+f2-V˙t,g123=0for  stick  on∂Ω23,g123∈g12,g13for  nonstick  on∂Ω23;g134≡b1cos⁡Ωt+a1-c1v1-p1v1-v2-d1z1-q1z1-z2-c1Vt-d1xt+f1-V˙t,g234=0for  stick  on∂Ω34,g234∈g24,g23for  nonstick  on∂Ω34;g214≡b2cos⁡Ωt+a2-c2v2-p2v2-v1-d2z2-q2z2-z1-c2Vt-d2xt-f2-V˙t,g114=0for  stick  on∂Ω14,g114∈g11,g14for  nonstick  on∂Ω14.The forces of per unit mass of the oscillator on the boundary∂Ωα1α2α3 (αi∈{1,2,3,4},i=1,2,3, and αi≠αj (i≠j), i,j∈1,2,3) are(48)gαα1α2α3∈g1α1α2,g2α2α3,α∈1,2for  no  full  stick  on∂Ωα1α2α3;gαα1α2α3=0,α∈1,2for  full  stick  on∂Ωα1α2α3.In other words, we have the following formulas:(49)g1123,g2123∈g112,g223for  no  full  stick  on∠Ω123,g1123=0,g2123=0for  full  stick  on∠Ω123,g1234,g2234∈g123,g234for  no  full  stick  on∠Ω234,g1234=0,g2234=0for  full  stick  on∠Ω234,g1341,g2341∈g134,g241for  no  full  stick  on∠Ω341,g1341=0,g2341=0for  full  stick  on∠Ω341,g1412,g2412∈g141,g212for  no  full  stick  on∠Ω412,g1412=0,g2412=0for  full  stick  on∠Ω412.


## 5. Analytical Conditions

Using the absolute coordinates, it is very difficult to develop the analytical conditions for the complex motions of the oscillator described in Section3 because the boundaries are dependent on time; thus the relative coordinates are needed herein for simplicity.From (3) and (4) in Section 2, we have(50)G0,α1zα,xα,tm±=n∂Ωα1α2T·gα1zα,xα,tm±,(51)G1,α1zα,xα,tm±=2Dn∂Ωα1α2T·gα1tm±-gα1α2tm+n∂Ωα1α2T·Dgα1tm±-Dgα1α2tm.In relative coordinates, the boundary∂Ωα1α2 is independent on t, so Dn∂Ωα1α2T=0. Because of (52)n∂Ωα1α2T·gα1α2=0,therefore(53)Dn∂Ωα1α2T·gα1α2+n∂Ωα1α2T·Dgα1α2=0;thus(54)n∂Ωα1α2T·Dgα1α2=0.Equation (51) is simplified as(55)G1,α1zα,xα,tm±=n∂Ωα1α2T·Dgα1tm±.tm represents the time for the motion on the velocity boundary and tm±=tm±0 reflects the responses in the domain rather than on the boundary.From the previous descriptions for the system, the normal vector of the boundary∂Ωα1α2 in the relative coordinates is(56)n∂Ωα1α2=∂φα1α2∂z1,∂φα1α2∂v1,∂φα1α2∂z2,∂φα1α2∂v2T.With (41) and (56), we have(57)n∂Ω23=n∂Ω14=0,1,0,0T,n∂Ω12=n∂Ω34=0,0,0,1T.Theorem 9.
For the 2-DOF friction induced oscillator described in Section3, the nonstick motion (or called passable motion to boundary) on xm∈∂Ωα1α2 at time tm appears if and only if(a)
α1=2, α2=1:(58)g22tm->0,g21tm+>0fromΩ2⟶Ω1;(b)
α1=1, α2=2:(59)g21tm-<0,g22tm+<0fromΩ1⟶Ω2;(c)
α1=3, α2=4:(60)g23tm->0,g24tm+>0fromΩ3⟶Ω4;(d)
α1=4, α2=3:(61)g24tm-<0,g23tm+<0fromΩ4⟶Ω3;(e)
α1=2, α2=3:(62)g12tm-<0,g13tm+<0fromΩ2⟶Ω3;(f)
α1=3, α2=2:(63)g13tm->0,g12tm+>0fromΩ3⟶Ω2;(g)
α1=4, α2=1:(64)g14tm->0,g11tm+>0fromΩ4⟶Ω1;(h)
α1=1, α2=4:(65)g11tm-<0,g14tm+<0fromΩ1⟶Ω4.Proof.
By Lemma4, the passable motion for a flow from domain Ωα1 to Ωα2 on the boundary ∂Ωα1α2 at time tm appears if and only if for n∂Ωα1α2→Ωα1(66)G0,α1tm-=n∂Ωα1α2T·gα1tm-<0,G0,α2tm+=n∂Ωα1α2T·gα2tm+<0.
From (57) and g(α)=(v1,g1(α),v2,g2(α)), we have(67)n∂Ω12T·gαtm±=g2αtm±α=1,2,n∂Ω34T·gαtm±=g2αtm±α=3,4,n∂Ω23T·gαtm±=g1αtm±α=2,3,n∂Ω14T·gαtm±=g1αtm±α=1,4.
Substituting the first formula of (67) into (66), we have(68)G0,1tm-=n∂Ω12T·g1tm-=g21tm-<0,G0,2tm+=n∂Ω12T·g2tm+=g22tm+<0fromΩ1⟶Ω2,G0,2tm-=n∂Ω12T·g2tm-=g22tm->0,G0,1tm+=n∂Ω12T·g1tm+=g21tm+>0fromΩ2⟶Ω1.So, (a) and (b) hold. Similarly, (c)–(h) can be proved.Theorem 10.
For the 2-DOF friction induced oscillator described in Section3, the stick motion in physics (or called the sliding motion in mathematics) to the boundary ∂Ωα1α2 is guaranteed if and only if (69)g22tm->0,g21tm-<0on∂Ω12;g23tm->0,g24tm-<0on∂Ω34;g14tm->0,g11tm-<0on∂Ω14;g13tm->0,g12tm-<0on∂Ω23.Proof.
By Lemma5 and (50), the necessary and sufficient conditions of the sliding motion on the boundary ∂Ωα1α2 are(70)G0,α1tm-=n∂Ωα1α2T·gα1tm-<0,G0,α2tm-=n∂Ωα1α2T·gα2tm->0forn∂Ωα1α2⟶Ωα1.
Substitute the first formula of (67) into (70); we have(71)G0,1tm-=n∂Ω12T·g1tm-=g21tm-<0,G0,2tm-=n∂Ω12T·g2tm-=g22tm->0forn∂Ω12⟶Ω1.So the conclusion on ∂Ω12 is proved. Similarly, the other formulas in (69) can also be proved.Theorem 11.
For the 2-DOF friction induced oscillator described in Section3, the stick motion on the boundary ∂Ωα1α2 appears if and only if (72)g22tm->0,g21tm±=0,Dg21tm±<0fromΩ1to∂Ω12,(73)g22tm±=0,g21tm-<0,Dg22tm±>0fromΩ2to∂Ω21;(74)g13tm->0,g12tm±=0,Dg12tm±<0fromΩ2to∂Ω23,(75)g13tm±=0,g12tm-<0,Dg13tm±>0fromΩ3to∂Ω32;(76)g23tm->0,g24tm±=0,Dg24tm±<0fromΩ4to∂Ω43,(77)g23tm±=0,g24tm-<0,Dg23tm±>0fromΩ3to∂Ω34;(78)g14tm->0,g11tm±=0Dg11tm±<0fromΩ1to∂Ω14,(79)g14tm±=0,g11tm-<0,Dg14tm±>0fromΩ4to∂Ω41.Proof.
By Lemma7 and (50) and (55), if the normal direction on ∂Ωα1α2 is for n∂Ωα1α2→Ωα1, the analytical conditions for the appearance of the stick motion are(80)G0,α1tm-=n∂Ωα1α2T·gα1tm-<0,G0,α2tm±=n∂Ωα1α2T·gα2tm±=0,G1,α2tm±=n∂Ωα1α2T·Dgα2tm±>0fromΩα2⟶∂Ωα1α2.By (57), we have(81)n∂Ωα1α2T·Dgαtm±=Dg2αtm±,α=1,2n∂Ωα3α4T·Dgαtm±=Dg2αtm±,α=3,4n∂Ωα2α3T·Dgαtm±=Dg1αtm±,α=2,3n∂Ωα1α4T·Dgαtm±=Dg1αtm±,α=1,4,where(82)Dg21tm±=Dg22tm±=Dg23tm±=Dg24tm±=-b2Ωsin⁡Ωt-c2v˙2-p2v˙2-v˙1-d2v2-q2v2-v1-c2V˙t-d2Vt-V¨t,Dg11tm±=Dg12tm±=Dg13tm±=Dg14tm±=-b1Ωsin⁡Ωt-c1v˙1-p1v˙1-v˙2-d1v1-q1v1-v2-c1V˙t-d1Vt-V¨t.Substitute the first formula of (67) and (81) into (80); we have(83)G0,1tm-=n∂Ω12T·g1tm-=g21tm-<0,G0,2tm±=n∂Ω12T·g2tm±=g22tm±=0,G1,2tm±=n∂Ω12T·Dg2tm±=Dg22tm±>0fromΩ2to∂Ω21;G0,1tm±=n∂Ω12T·g1tm±=g21tm±=0,G0,2tm-=n∂Ω12T·g2tm-=g22tm->0,G1,1tm±=n∂Ω12T·Dg1tm±=Dg21tm±<0fromΩ1to∂Ω12.So the cases on ∂Ω12 and ∂Ω21 are proved. Similarly, the other equations in (74) to (79) can also be proved.Theorem 12.
For the 2-DOF friction induced oscillator described in Section3, the analytical conditions for vanishing of the stick motion from ∂Ωα1α2 and entering domain Ωα1 are(84)g22tm->0,g21tm±=0,Dg21tm±>0from∂Ω12⟶Ω1,(85)g22tm±=0,g21tm-<0,Dg22tm±<0from∂Ω12⟶Ω2;(86)g23tm->0,g24tm±=0,Dg24tm±>0from∂Ω34⟶Ω4,(87)g23tm±=0,g24tm-<0,Dg23tm±<0from∂Ω34⟶Ω3;(88)g14tm->0,g11tm±=0,Dg11tm±>0from∂Ω14⟶Ω1,(89)g14tm±=0,g11tm-<0,Dg14tm±<0from∂Ω14⟶Ω4;(90)g13tm->0,g12tm±=0,Dg12tm±>0from∂Ω23⟶Ω2,(91)g13tm±=0,g12tm-<0,Dg13tm±<0from∂Ω23⟶Ω3.Proof.
By Lemma6 and (50) and (55), if the normal direction on ∂Ωα1α2 is for n∂Ωα1α2→Ωα1, the necessary and sufficient conditions for the oscillator sliding motion switched into passible motion are(92)G0,α1tm-=n∂Ωα1α2T·gα1tm-<0,G0,α2tm±=n∂Ωα1α2T·gα2tm±=0,G1,α2tm±=n∂Ωα1α2T·Dgα2tm±<0from∂Ωα1α2⟶Ωα2,G0,α1tm±=n∂Ωα1α2T·gα1tm±=0,G0,α2tm-=n∂Ωα1α2T·gα2tm->0,G1,α1tm±=n∂Ωα1α2T·Dgα1tm±>0from∂Ωα1α2⟶Ωα1.Substitute the first formula of (67) and (81) into (92); we have(93)G0,1tm-=n∂Ω12T·g1tm-=g21tm-<0,G0,2tm±=n∂Ω12T·g2tm±=g22tm±=0,G1,2tm±=n∂Ω12T·Dg2tm±=Dg22t±<0from∂Ω12⟶Ω2;G0,1tm±=n∂Ω12T·g1tm±=g21tm±=0,G0,2tm-=n∂Ω12T·g2tm-=g22tm->0,G1,1tm±=n∂Ω12T·Dg1tm±=Dg21t±>0from∂Ω12⟶Ω1.
So the cases on∂Ω12 are proved. Similarly, the other cases in (86) to (91) can also be proved.Theorem 13.
For the 2-DOF friction induced oscillator described in Section3, the grazing motion on the boundary ∂Ωα1α2 is guaranteed if and only if (94)g21tm±=0,Dg21tm±>0on∂Ω12inΩ1,(95)g22tm±=0,Dg22tm±<0on∂Ω12inΩ2;(96)g24tm±=0,Dg24tm±>0on∂Ω34inΩ4,(97)g23tm±=0,Dg23tm±<0on∂Ω34inΩ3;(98)g11tm±=0,Dg11tm±>0on∂Ω14inΩ1,(99)g14tm±=0,Dg14tm±<0on∂Ω14inΩ4;(100)g12tm±=0,Dg12tm±>0on∂Ω23inΩ2,(101)g13tm±=0,Dg13tm±<0on∂Ω23inΩ3.Proof.
By Lemma8 and (50) and (55), the conditions for the grazing motion in domain Ωα1 to the boundary ∂Ωα1α2 are(102)G0,α1tm±=n∂Ωα1α2T·gα1tm±=0,G1,α1tm±=n∂Ωα1α2T·Dgα1tm±>0forn∂Ωα1α2⟶Ωα1;G0,α1tm±=n∂Ωα1α2T·gα1tm±=0,G1,α1tm±=n∂Ωα1α2T·Dgα1tm±<0forn∂Ωα1α2⟶Ωα2.By (67) and (81), we have(103)G0,1tm±=n∂Ω12T·g1tm±=g21tm±=0,G1,1tm±=n∂Ω12T·Dg1tm±=Dg21tm±>0on∂Ω12inΩ1;G0,1tm±=n∂Ω12T·g2tm±=g22tm±=0,G1,1tm±=n∂Ω12T·Dg2tm±=Dg22tm±<0on∂Ω12inΩ2.So the cases on ∂Ω12 are proved. Similarly, the other formulas in (96)–(101) can also be proved.


## 6. Mapping Structures and Periodic Motions

From the boundary∂Ωα1α2 in (41), the switching sets are(104)Σ1+=z1i,v1i,z2i,v2i,Ωti∣v1i=0+,i∈N,Σ10=z1i,v1i,z2i,v2i,Ωti∣v1i=0,i∈N,Σ1-=z1i,v1i,z2i,v2i,Ωti∣v1i=0-,i∈N,Σ2+=z1i,v1i,z2i,v2i,Ωti∣v2i=0+,i∈N,Σ20=z1i,v1i,z2i,v2i,Ωti∣v2i=0,i∈N,Σ2-=z1i,v1i,z2i,v2i,Ωti∣v2i=0-,i∈N,where 0±=limε→0+(0±ε) and the switching set on the edge ∠Ωα1α2α3 is defined by(105)Σ00=z1i,v1i,z2i,v2i,Ωti∣v1i=0,v2i=0,i∈N.Therefore, eleven basic mappings will be defined as(106)P1:Σ1+⟶Σ1+,P2:Σ1-⟶Σ1-,P3:Σ10⟶Σ10,P6:Σ2+⟶Σ2+,P7:Σ2-⟶Σ2-,P8:Σ20⟶Σ20,P4:Σ1+⟶Σ2+,P5:Σ2-⟶Σ1+,P9:Σ1-⟶Σ2-,P10:Σ2+⟶Σ1-,P0:Σ00⟶Σ00.Because the switching set Σ00 is the special case of the switching sets Σ10 and Σ20, the mappings P3 and P8 can apply to Σ00, that is,(107)P3:Σ00⟶Σ10,P3:Σ10⟶Σ00;P8:Σ00⟶Σ20,P8:Σ20⟶Σ00.The switching sets and mappings are shown in Figure 5.Figure 5
Switching sets and mappings.In all eleven mappings,Pn are the local mappings when n=0,1,2,3,6,7,8 and Pn are the global mappings when n=4,5,9,10. From the previous defined mappings, for each mapping Pn(n=0,1,…,10), one obtains a set of nonlinear algebraic equations(108)fnzi,ti,zi+1,ti+1=0,where(109)zi=z1i,v1i,z2i,v2iT,fn=f1n,f2n,f3n,f4nTwith the constraints for zi and zi+1 from the boundaries.The system in Section3 has complicated motions, while any possible physical motion can be generated by the combination of the above eleven mappings in this section. The periodic motion with stick should be specially discussed. Consider the mappings of periodic motion with stick of m1 in absolute space and in relative space, which are shown in Figures 6 and 7, respectively. And the corresponding mappings of periodic motion without stick of m2 in absolute space and in relative space are shown in Figures 8 and 9, respectively.Figure 6
A mapping structure in absolute space ofm1.Figure 7
A mapping structure in relative space ofm1.Figure 8
A mapping structure in absolute space ofm2.Figure 9
A mapping structure in relative space ofm2.ForP56931=P5∘P6∘P9∘P3∘P1, the corresponding mapping relations are(110)P1:ti,z1i,0,z2i⟶ti+1,z1i+1,0,z2i+1,P3:ti+1,z1i+1,0,z2i+1⟶ti+2,z1i+2,0,z2i+2,P9:ti+2,z1i+2,0,z2i+2⟶ti+3,z1i+3,z2i+3,0,P6:ti+3,z1i+3,z2i+3,0⟶ti+4,z1i+4,z2i+4,0,P5:ti+4,z1i+4,z2i+4,0⟶ti+5,z1i+5,0,z2i+5.Such mapping relations provide the nonlinear algebraic equations, that is,(111)f1nzi,ti,zi+1,ti+1=0,f3nzi+1,ti+1,zi+2,ti+2=0,f9nzi+2,ti+2,zi+3,ti+3=0,f6nzi+3,ti+3,zi+4,ti+4=0,f5nzi+4,ti+4,zi+5,ti+5=0,where z1(i+5)=z1(i) and z2(i+5)=z2(i) and ti+5=ti+NT (T is a period; N=1,2,…), and give the switching points for the periodic solutions. Similarly, the other mapping structures can be developed to analytically predict the switching points for periodic motions in the 2-DOF friction induced oscillator.


## 7. Numerical Simulations

To illustrate the analytical conditions of stick and nonstick motions, the motions of the2-DOF oscillator will be demonstrated through the time histories of displacement and velocity, the corresponding trajectory of the oscillator in phase space. The starting points of motions are represented by blue-solid circular symbols; the switching points at which the oscillator contacts on the moving boundaries are depicted by red-solid circular symbols. The moving boundaries, that is, the velocity curves of the traveling belt, are represented by blue curves, and the displacement, the velocity or the forces of per unit mass, and the corresponding trajectories of the oscillator in phase space are shown by the green curves, red curves, and dark curves, respectively.Consider a set of system parameters for numerical illustration:m1 = 2 kg,r1 = 0.1 N·s/m,k1 = 1 N/m,A1 = 0.1 N,B1 = −5 N,m2 = 2 kg,r2 = 0.5 N·s/m,k2 = 2 N/m,A2 = −0.5 N,B2 = 5 N,r3 = 0.05 N·s/m,k3 = 0.5 N/m,Ω = 2 rad/s,μk=0.4,g = 10 m/s2,β=0,V0 = 5 m/s,V1 = 0.2 m/s.For the above system parameters, the nonstick motions and stick motions of massm1 and mass m2 are presented in Figures 10, 11, and 12 with the initial conditions of t0 = 1 s, x10 = 1 m, y10 = 3 m/s, x20 = 2 m, y20 = 1 m/s. Consider the complex mappings P=P10∘P1∘P10∘P8∘P9∘P5∘P6. The time histories of velocities of the traveling belt and the masses m1 and m2 are depicted in Figure 10. The time histories of the velocities, displacements, trajectories of the oscillators in phase space, and the corresponding forces of per unit mass of mass m1 and mass m2 are shown in Figures 11(a), 11(b), 11(c), and 11(d) and 12(a), 12(b), 12(c), and 12(d), respectively. The time history of the force of per unit mass of m2 is shown in Figure 12(e) when the sliding motion of m2 occurs.Figure 10
Velocity-time history of the massesm1 and m2 and the traveling belt (m1=2, r1=0.1, k1=1, A1=0.1, B1=-5, m2=2, r2=0.5, k2=2, A2=-0.5, B2=5, r3=0.05, k3=0.5, Ω=2, μk=0.4, g=10, β=0, V0=5, V1=0.2). The initial condition is t0=1, x10=1, y10=3, x20=2, y20=1.Figure 11
Numerical simulation of a motion ofm1 relative to mapping P=P10∘P1∘P10∘P8∘P9∘P5∘P6: (a) velocity-time history, (b) displacement-time history, (c) phase trajectory, and (d) force-time history (m1=2, r1=0.1, k1=1, A1=0.1, B1=-5, m2=2, r2=0.5, k2=2, A2=-0.5, B2=5, r3=0.05, k3=0.5, Ω=2, μk=0.4, g=10, β=0, V0=5, V1=0.2). The initial condition is t0=1, x10=1, y10=3, x20=2, y20=1.
(a) (b) (c) (d)Figure 12
Numerical simulation of a motion ofm2 relative to mapping P=P10∘P1∘P10∘P8∘P9∘P5∘P6: (a) velocity-time history, (b) displacement-time history, (c) phase trajectory, (d) force-time history, (e) and force of per unit mass of m2 relative to mapping P8 when sliding motion (m1=2, r1=0.1, k1=1, A1=0.1, B1=-5, m2=2, r2=0.5, k2=2, A2=-0.5, B2=5, r3=0.05, k3=0.5, Ω=2, μk=0.4, g=10, β=0, V0=5, V1=0.2). The initial condition is t0=1, x10=1, y10=3, x20=2, y20=1.
(a) (b) (c) (d) (e)Whent∈1,1.7450, m1 and m2 move freely in domain Ω1, which satisfy y1>V and y2>V, as shown in Figure 10. In this time interval the time histories of velocities of m1 and m2 are shown in Figures 11(a) and 12(a), respectively. The displacements of m1 and m2 are shown in Figures 11(b) and 12(b), respectively. At the time t1=1.7450, the velocity of m2 reached the speed boundary ∂Ω12 (i.e., y2=V). Since the forces F2-(1) and F2+(2) of per unit mass satisfy the conditions of F2-(1)<V˙ and F2+(2)<V˙ (as shown in Figure 12(d)), the analytical condition (59) of the passable motion on the boundary ∂Ω12 is satisfied in Theorem 9 (b). At such a point the motion enters into the domain Ω2 relative to y2<V, as shown in Figure 10. Due to the movement in the area Ω1 at the time t0=1 and the movement that reached the boundary ∂Ω12 at the time t1=1.7450, the mapping for this process is P6. When t∈(1.7450,2.2120), m1 and m2 move freely in domain Ω2, which satisfy y1>V and y2<V, as shown in Figure 10. The displacements of m1 and m2 are shown in Figures 11(b) and 12(b), respectively. At the time t2=2.2120, the velocity of m1 reached the speed boundary ∂Ω23 (i.e., y1=V). Since the forces F1-(2) and F1+(3) of per unit mass satisfy the conditions of F1-(2)<V˙ and F1+(3)<V˙ (as shown in Figure 11(d)), the analytical condition (62) of the passable motion on the boundary ∂Ω23 is satisfied in Theorem 9 (e). At such a point the motion enters into the domain Ω3 relative to y1<V, as shown in Figure 10. Due to the movement on the boundary ∂Ω12 at the time t1=1.7450 and the movement that reached the boundary ∂Ω23 at the time t2=2.2120, the mapping for this process is P5. When t∈(2.2120,3.3090), m1 and m2 move freely in domain Ω3, which satisfy y1<V and y2<V, as shown in Figure 10. The displacements of m1 and m2 are shown in Figures 11(b) and 12(b), respectively. At the time t3=3.3090, the velocity of m2 reached the speed boundary ∂Ω34 (i.e., y2=V). Since the forces F2-(3) and F2-(4) of per unit mass satisfy the conditions of F2-(3)>V˙ and F2-(4)<V˙ (as shown in Figure 12(d)), the analytical condition (69) of the stick motion on the boundary ∂Ω34 is satisfied in Theorem 10. At such a point the sliding motion of m2 occurs on the boundary ∂Ω34 and keeps to t4=3.3570. Due to the movement on the boundary ∂Ω23 at the time t2=2.2120 and the movement that reached the boundary ∂Ω34 at the time t3=3.3090, the mapping for this process is P9.Whent∈(3.3090,3.3570), m1 moves freely in domain Ω3, satisfying y1<V, as shown in Figure 10. The displacement of m1 is shown in Figure 11(b), correspondingly. However, at such time interval, m2 maintains sliding motion, and the time history of force per unit mass of m2 is shown in Figure 12(e). In this time period, the force product satisfies F2(3)·F2(4)<0 relative to V˙, where F2(3) is represented by pink curves and F2(4) is represented by light cyan curves. At the time t4=3.3570, the forces F2-(3) and F2(4) of per unit mass satisfy the conditions of F2-(3)>V˙, F2(4)=V˙ and DF2±(4)>V¨, so the analytical condition (86) of the vanishing of stick motion on the boundary ∂Ω34 is satisfied in Theorem 12 and the sliding motion of m2 vanishes and the motion of m2 enters the domain Ω4. Due to the movement on the boundary ∂Ω34 from the time t3=3.3090 to the time t4=3.3570, the mapping for this process is P8.Whent∈(3.3570,4.5150), m1 and m2 move freely again in domain Ω4, which satisfy y1<V and y2>V, as shown in Figure 10. The displacements of m1 and m2 are shown in Figures 11(b) and 12(b), respectively. At the time t5=4.5150, the velocity of m1 reached the speed boundary ∂Ω14 (i.e., y1=V). Since the forces F1-(4) and F1+(1) of per unit mass satisfy the conditions of F1-(4)>V˙ and F1+(1)>V˙ (as shown in Figure 11(d)), the analytical condition (64) of the passable motion on the boundary ∂Ω41 is satisfied in Theorem 9 (g). At such a point the motion enters into the domain Ω1 relative to y1>V as shown in Figure 10. Due to the movement on the boundary ∂Ω34 at the time t4=3.3570 and the movement that reached the boundary ∂Ω41 at the time t5=4.5150, the mapping for this process is P10. When t∈(4.5150,5.0570), m1 and m2 move freely in domain Ω1, which satisfy y1>V and y2>V, as shown in Figure 10. The displacements of m1 and m2 are shown in Figures 11(b) and 12(b), respectively. At the time t6=5.0570, the velocity of m1 reached the speed boundary ∂Ω41 (i.e., y1=V). Since the forces F1-(1) and F1+(4) of per unit mass satisfy the conditions of F1-(1)<V˙ and F1+(4)<V˙ (as shown in Figure 11(d)), the analytical condition (65) of the passable motion on the boundary ∂Ω41 is satisfied in Theorem 9 (h). At such a point the motion enters into the domain Ω4 relative to y1<V as shown in Figure 10. Due to the movement on the boundary ∂Ω41 at the time t5=4.5150 and the movement that reached the boundary ∂Ω41 at the time t6=5.0570, the mapping for this process is P1. When t∈(5.0570,5.0910), m1 and m2 move freely in domain Ω4, which satisfy y1<V and y2>V, as shown in Figure 10. The displacements of m1 and m2 are shown in Figures 11(b) and 12(b), respectively. At the time t7=5.0910, the velocity of m2 reached the speed boundary ∂Ω34 (i.e., y2=V). Since the forces F2-(4) and F2+(3) of per unit mass satisfy the conditions of F2-(4)<V˙ and F2+(3)<V˙, the analytical condition (61) of the passable motion on the boundary ∂Ω34 is satisfied in Theorem 9 (d). At such a point the motion enters into the domain Ω3 relative to y2<V as shown in Figure 10. Due to the movement on the boundary ∂Ω41 at the time t6=5.0570 and the movement that reached the boundary ∂Ω34 at the time t7=5.0910, the mapping for this process is P10.Whent>5.0910, the movement will continue, but, here, the later motion will not be described. In the whole process, the phase trajectories of m1 and m2 are shown in Figures 11(c) and 12(c), respectively.


## 8. Conclusion

The model of frictional-induced oscillator with two degrees of freedom on a speed-varying traveling belt was proposed. The dynamics of such oscillator with two harmonically external excitations on a speed-varying traveling belt were investigated by using the theory of flow switchability for discontinuous dynamical systems. The dynamics of this system are of interest because it is a simple representation of mechanical systems with multiple nonsmooth characteristics. Different domains and boundaries for such system were defined according to the friction discontinuity. Based on the above domains and boundaries, the analytical conditions for the passable motions and the onset or vanishing of stick motions and grazing motions were presented. The basic mappings were introduced to describe motions in such an oscillator. Analytical conditions of periodic motions were developed by the mapping dynamics. Numerical simulations were carried out to illustrate stick and nonstick motions for a better understanding of complicated dynamics of such mechanical model. Through the velocity and force responses of such motions, it is possible to validate analytical conditions for the motion switching in such a discontinuous system. There are more simulations about such an oscillator to be discussed in future.



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*Source: 1208563-2017-02-05.xml*