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

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.


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][2][3][4][5][6][7][8][9][10][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-offreedom 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][8][9][10][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][13][14][15][16][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][19][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 2 Mathematical Problems in Engineering 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 frictioninduced 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.

Preliminaries
For convenience, the fundamental theory on flow switchability of discontinuous dynamical systems will be presented; that is, concepts of -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 that Ω is a bounded simply connected domain in   and its boundary Ω ⊂  −1 is a smooth surface.Consider a dynamic system consisting of  subdynamic systems in a universal domain Ω ⊂   .The universal domain is divided into  accessible subdomains Ω  ( ∈ ) and the inaccessible domain Ω 0 .The union of all the accessible subdomains is ⋃ ∈ Ω  and Ω = ⋃ ∈ Ω  ∪Ω 0 is the universal domain.On the th open subdomain Ω  , there is a   continuous system (  ≥ 1) in form of The time is  and ẋ = x/.In an accessible subdomain Ω  , the vector field F () (x () , , p  ) with parameter vector p  = ( (1)   ,  (2)   , . . .,  ()  ) T ∈   is    -continuous (  ≥ 1) in x ∈ Ω  and for all time .
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.Let () be the speed of the belt and where Ω and  are the oscillation frequency and primary phase of the traveling belt, respectively,  0 is the oscillation amplitude of the traveling belt, and  1 is constant.Further, the friction force shown in Figure 2 is described by where ẋ  =   /,   is the coefficient of friction between   and the belt,  ()  =    ( = 1,2), and  is the acceleration of gravity.The nonfriction force acting on the mass   in the   -direction is defined as where ,  ∈ {1, 2} and  ̸ = .From now on,  ()  =   ⋅  ()  . x Figure 1: Physical model.
From the previous discussion, there are four cases of motions: can overcome the static friction force  ()   (i.e., | ()  | > | ()   |,  = 1, 2), the mass   has relative motion to the belt, that is, For the nonstick motion of the mass   ( = 1, 2), the total force acting on the mass   is and the equations of nonstick motion for the 2-DOF dry friction induced oscillator are where ,  ∈ {1, 2},  ̸ = .

Analytical Conditions
Using the absolute coordinates, it is very difficult to develop the analytical conditions for the complex motions of the oscillator described in Section 3 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 In relative coordinates, the boundary Ω  1  2 is independent on , so n T therefore Equation ( 51) is simplified as represents the time for the motion on the velocity boundary and  ± =   ± 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 Ω  1  2 in the relative coordinates is With ( 41) and (56), we have n Ω 23 = n Ω 14 = (0, 1, 0, 0) T , n Ω 12 = n Ω 34 = (0, 0, 0, 1) T .

Mapping Structures and Periodic Motions
From the boundary Ω  1  2 in (41), the switching sets are where 0 ± = lim →0 + (0 ± ) and the switching set on the edge ∠Ω  1  2  3 is defined by Therefore, eleven basic mappings will be defined as Because the switching set Σ 0 0 is the special case of the switching sets Σ 0 1 and Σ 0 2 , the mappings  3 and  8 can apply to Σ 0 0 , that is, The switching sets and mappings are shown in Figure 5.In all eleven mappings,   are the local mappings when  = 0, 1, 2, 3, 6, 7, 8 and   are the global mappings when  = 4, 5, 9, 10.From the previous defined mappings, for each mapping   ( = 0, 1, . . ., 10), one obtains a set of nonlinear algebraic equations where with the constraints for z  and z +1 from the boundaries.
(110) Such mapping relations provide the nonlinear algebraic equations, that is, where z 1(+5) = z 1() and z 2(+5) = z 2() and  +5 =   +  ( is a period;  = 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.

Numerical Simulations
To illustrate the analytical conditions of stick and nonstick motions, the motions of the 2-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: When  ∈ [1, 1.7450),  1 and  2 move freely in domain Ω 1 , which satisfy  1 >  and  2 > , as shown in Figure 10.In this time interval the time histories of velocities of  1 and  2 are shown in Figures 11(a) and 12(a), respectively.The displacements of  1 and  2 are shown in Figures 11(b) and 12(b), respectively.At the time  1 = 1.7450, the velocity of  2 reached the speed boundary Ω 12 (i.e.,  2 = ).Since the forces  (1)  2− and  (2)  2+ of per unit mass satisfy the conditions of  (1)  2− < V and  (2)  2+ < V (as shown in Figure 12(d)), the analytical condition (59) of the passable motion on the boundary Ω 12 is satisfied in Theorem 9 (b).At such a point the motion enters into the domain Ω 2 relative to  2 < , as shown in Figure 10.Due to the movement in the area Ω 1 at the time  0 = 1 and the movement that reached the boundary Ω 12 at the time  1 = 1.7450, the mapping for this process is  6 .When  ∈ (1.7450, 2.2120),  1 and  2 move freely in F (1)  1 F (1)   1 domain Ω 2 , which satisfy  1 >  and  2 < , as shown in Figure 10.The displacements of  1 and  2 are shown in Figures 11(b) and 12(b), respectively.At the time  2 = 2.2120, the velocity of  1 reached the speed boundary Ω 23 (i.e.,  1 = ).Since the forces  (2)  1− and  (3)  1+ of per unit mass satisfy the conditions of  (2)  1− < V and  (3)  1+ < V (as shown in Figure 11(d)), the analytical condition (62) of the passable motion on the boundary Ω 23 is satisfied in Theorem 9 (e).At such a point the motion enters into the domain Ω 3 relative to  1 < , as shown in Figure 10.Due to the movement on the boundary Ω 12 at the time  1 = 1.7450 and the movement that reached the boundary Ω 23 at the time  2 = 2.2120, the mapping for this process is  5 .When  ∈ (2.2120, 3.3090),  1 and  2 move freely in domain Ω 3 , which satisfy  1 <  and  2 < , as shown in Figure 10.The displacements of  1 and  2 are shown in Figures 11(b) and 12(b), respectively.At the time  3 = 3.3090, the velocity of  2 reached the speed boundary Ω 34 (i.e.,  2 = ).Since the forces  (3)  2− and  (4)  2− of per unit mass satisfy the conditions of  (3) 2− > V and  (4)  2− < V (as shown in Figure 12(d)), the analytical condition (69) of the stick motion on the boundary Ω 34 is satisfied in Theorem 10.At such a point the sliding motion of  2 occurs on the boundary Ω 34 and keeps to  4 = 3.3570.Due to the movement on the boundary Ω 23 at the time  2 = 2.2120 and the movement that reached the boundary Ω 34 at the time  3 = 3.3090, the mapping for this process is  9 .

Mathematical Problems in Engineering
When  ∈ (3.3090, 3.3570),  1 moves freely in domain Ω 3 , satisfying  1 < , as shown in Figure 10.The displacement of  1 is shown in Figure 11(b), correspondingly.However, at such time interval,  2 maintains sliding motion, and the time history of force per unit mass of  2 is shown in Figure 12(e).In this time period, the force product satisfies  (3)  2 ⋅  (4)  2 < 0 relative to V, where  (3)  2 is represented by pink curves and  (4)  2 is represented by light cyan curves.At the time  4 = 3.3570, the forces  (3)  2− and  (4) 2 of per unit mass satisfy the conditions of  (3)  2− > V,  (4) 2 = V and  (4)  2± > V, so the analytical condition (86) of the vanishing of stick motion on the boundary Ω 34 is satisfied in Theorem 12 and the sliding motion of  2 vanishes and the motion of  2 enters the domain Ω 4 .Due to the movement on the boundary Ω 34 from the time  3 = 3.3090 to the time  4 = 3.3570, the mapping for this process is  8 .
When  ∈ (3.3570, 4.5150),  1 and  2 move freely again in domain Ω 4 , which satisfy  1 <  and  2 > , as shown in Figure 10.The displacements of  1 and  2 are shown in Figures 11(b) and 12(b), respectively.At the time  5 = 4.5150, the velocity of  1 reached the speed boundary Ω 14 (i.e.,  1 = ).Since the forces  (4)  1− and  (1)  1+ of per unit mass satisfy the conditions of  (4)  1− > V and  (1)  1+ > V (as shown in Figure 11(d)), the analytical condition (64) of the passable motion on the boundary Ω 41 is satisfied in Theorem 9 (g).At such a point the motion enters into the domain Ω 1 relative to  1 >  as shown in Figure 10.Due to the movement on the boundary Ω 34 at the time  4 = 3.3570 and the movement that reached the boundary Ω 41 at the time  5 = 4.5150, the mapping for this process is  10 .When  ∈ (4.5150, 5.0570),  1 and  2 move freely in domain Ω 1 , which satisfy  1 >  and  2 > , as shown in Figure 10.The displacements of  1 and  2 are shown in Figures 11(b) and 12(b), respectively.At the time  6 = 5.0570, the velocity of  1 reached the speed boundary Ω 41 (i.e.,  1 = ).Since the forces  (1)  1− and  (4)  1+ of per unit mass satisfy the conditions of  (1)  1− < V and  (4)  1+ < V (as shown in Figure 11(d)), the analytical condition (65) of the passable motion on the boundary Ω 41 is satisfied in Theorem 9 (h).At such a point the motion enters into the domain Ω 4 relative to  1 <  as shown in Figure 10.Due to the movement on the boundary Ω 41 at the time  5 = 4.5150 and the movement that reached the boundary Ω 41 at the time  6 = 5.0570, the mapping for this process is  1 .When  ∈ (5.0570, 5.0910),  1 and  2 move freely in domain Ω 4 , which satisfy  1 <  and  2 > , as shown in Figure 10.The displacements of  1 and  2 are shown in Figures 11(b) and 12(b), respectively.At the time  7 = 5.0910, the velocity of  2 reached the speed boundary Ω 34 (i.e.,  2 = ).Since the forces  (4)  2− and  (3)  2+ of per unit mass satisfy the conditions of  (4)  2− < V and  (3)  2+ < V, the analytical condition (61) of the passable motion on the boundary Ω 34 is satisfied in Theorem 9 (d).At such a point the motion enters into the domain Ω 3 relative to  2 <  as shown in Figure 10.Due to the movement on the boundary Ω 41 at the time  6 = 5.0570 and the movement that reached the boundary Ω 34 at the time  7 = 5.0910, the mapping for this process is  10 .
When  > 5.0910, the movement will continue, but, here, the later motion will not be described.In the whole process, the phase trajectories of  1 and  2 are shown in Figures 11(c) and 12(c), respectively.

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.

Figure 6 :Figure 7 :
Figure 6: A mapping structure in absolute space of  1 .

x 2 y 2 Figure 8 :Figure 9 :
Figure 8: A mapping structure in absolute space of  2 .

)
Proof.By Lemma 4, the passable motion for a flow from domain Ω  1 to Ω  2 on the boundary Ω  1  2 at time   appears if and only if for n Ω  1  2 → Ω  1