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.
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 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 is a bounded simply connected domain in and its boundary is a smooth surface.
Consider a dynamic system consisting of subdynamic systems in a universal domain . The universal domain is divided into accessible subdomains and the inaccessible domain . The union of all the accessible subdomains is and is the universal domain. On the th open subdomain , there is a -continuous system () in form ofThe time is and . In an accessible subdomain , the vector field with parameter vector is -continuous () in and for all time .
The flow on the boundary can be determined bywhere . With specific initial conditions, one always obtains different flows on .
Consider a dynamic system in (1) in domain which has a flow with an initial condition , and on the boundary , there is an enough smooth flow with an initial condition . For an arbitrarily small , there are two time intervals and for flow and the flow approaches the separation boundary at time , that is, , where , , and .
Definition 1. The -functions of the flow to the flow on the boundary are defined aswhere , , is to represent the quantity in the domain rather than on the boundary, and is a time rate of the inner product of displacement difference and the normal direction .
Definition 2. The th-order -functions of the domain flow to the boundary flow in the normal direction of are defined aswhere the total derivative operators are defined asFor , we have
Definition 3. For a discontinuous dynamical system in (1), there is a point . For an arbitrarily small , there are two time intervals and . Suppose ; iffor , then a resultant flow of two flows is a semipassable flow from domain to at point to boundary , where , .
To simplify notation usage, the symbols represent in next paragraphs.
Lemma 4. For a discontinuous dynamical system in (1), there is a point at time between two adjacent domains . For an arbitrarily small , there are two time intervals and . Suppose , two flows and are and -continuous for time , respectively, and . The flows and at point to the boundary are semipassable from domain to if and only if either or
Lemma 5. For a discontinuous dynamical system in (1), there is a point at time between two adjacent domains . For an arbitrarily small , there is a time interval . Suppose . Both flows and are -continuous for time and . The flows and at point to the boundary are sink flow if and only if either or
Lemma 6. For a discontinuous dynamical system in (1), there is a point at time between two adjacent domains . For an arbitrarily small , there are two time intervals and . Suppose . The flows and are and -continuous for time , respectively, and . The sliding fragmentation bifurcation of the nonpassable flows and of the first kind at point switching to the passable flow on the boundary occurs if and only if
Lemma 7. For a discontinuous dynamical system in (1), there is a point at time between two adjacent domains . For an arbitrarily small , there are two time intervals and . Suppose . The flows and are and -continuous for time , respectively, and . The sliding bifurcation of the passable flow of and at point switching to the sink flow on the boundary occurs if and only if
Lemma 8. For a discontinuous dynamical system in (1), there is a point at time between two adjacent domains . For an arbitrarily small , there is a time interval . Suppose . The flow is -continuous for time , and . A flow in is tangential to the boundary if and only if either or
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 Figure 1. The system consists of two masses , which are connected with three linear springs of stiffness and three dampers of coefficient . Both of masses move on the belt with varying speed . Two periodic excitations with frequency , amplitudes , and constant forces are exerted on the two masses, respectively.
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 andwhere and are the oscillation frequency and primary phase of the traveling belt, respectively, is the oscillation amplitude of the traveling belt, and is constant.
Further, the friction force shown in Figure 2 is described bywhere , is the coefficient of friction between and the belt, , and is the acceleration of gravity. The nonfriction force acting on the mass in the -direction is defined aswhere and . From now on, .
From the previous discussion, there are four cases of motions:
Case 1 (nonstick motion ). When can overcome the static friction force (i.e., ), the mass has relative motion to the belt, that is,For the nonstick motion of the mass , the total force acting on the mass isand the equations of nonstick motion for the 2-DOF dry friction induced oscillator arewhere .
Case 2 (single stick motion ()). When cannot overcome the static friction force (i.e., ), mass does not have any relative motion to the belt, that is,meanwhile, when can overcome the static friction force (i.e., ), the mass has relative motion to the belt, that is,
Case 3 (single stick motion ()). When cannot overcome the static friction force (i.e., ), mass does not have any relative motion to the belt, that is,meanwhile, when can overcome the static friction force (i.e., ), mass has relative motion to the belt, that is,
Case 4 (double stick motions ). When cannot overcome the static friction force (i.e., ), mass does not have any relative motion to the belt, that is,Integrating (17) leads to the displacement of the belt:where and .
4. Domains and Boundaries
Due to frictions between the mass 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
By the state variables, the domains are defined asand the corresponding boundaries are defined as
The phase plane of is shown in Figure 3.
The 2-dimensional edges of the 3-dimensional boundaries are defined bywhere , and (), . The intersection of four 2-dimensional edges isFrom the above discussion, the motion equations of the oscillator described in Section 3 in absolute coordinates arewhere the forces of per unit mass for the 2-DOF friction induced oscillator in the domain arehereand the forces of per unit mass of the oscillator on the boundary (, ) are
The forces of per unit mass of the oscillator on the boundary (; are not equal to each other without repeating) are
For simplicity, the relative displacement, velocity, and acceleration between the mass and the traveling belt are defined as
The domains and boundaries in relative coordinates are defined aswhere , and (), . The intersection of four 2-dimensional edges is
The domain partitions and boundaries in relative coordinates are shown in Figure 4.
From the foregoing equations, the motion equations in relative coordinates are as follows:where
The forces of per unit mass for the 2-DOF friction induced oscillator in the domain in relative coordinates are
The forces of per unit mass of the friction induced oscillator on the boundary in relative coordinates are
The forces of per unit mass of the oscillator on the boundary (, and (), ) areIn other words, we have the following formulas:
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 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 is independent on , so . Because of thereforethusEquation (51) is simplified as represents the time for the motion on the velocity boundary and reflects the responses in the domain rather than on the boundary.
From the previous descriptions for the system, the normal vector of the boundary in the relative coordinates is
Theorem 9. For the 2-DOF friction induced oscillator described in Section 3, the nonstick motion (or called passable motion to boundary) on at time appears if and only if(a), :(b), :(c), :(d), :(e), :(f), :(g), :(h), :
Proof. By Lemma 4, the passable motion for a flow from domain to on the boundary at time appears if and only if for From (57) and , we haveSubstituting the first formula of (67) into (66), we haveSo, (a) and (b) hold. Similarly, (c)–(h) can be proved.
Theorem 10. For the 2-DOF friction induced oscillator described in Section 3, the stick motion in physics (or called the sliding motion in mathematics) to the boundary is guaranteed if and only if
Proof. By Lemma 5 and (50), the necessary and sufficient conditions of the sliding motion on the boundary areSubstitute the first formula of (67) into (70); we haveSo the conclusion on is proved. Similarly, the other formulas in (69) can also be proved.
Theorem 11. For the 2-DOF friction induced oscillator described in Section 3, the stick motion on the boundary appears if and only if
Proof. By Lemma 7 and (50) and (55), if the normal direction on is for , the analytical conditions for the appearance of the stick motion areBy (57), we havewhereSubstitute the first formula of (67) and (81) into (80); we haveSo the cases on and 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 Section 3, the analytical conditions for vanishing of the stick motion from and entering domain are
Proof. By Lemma 6 and (50) and (55), if the normal direction on is for , the necessary and sufficient conditions for the oscillator sliding motion switched into passible motion areSubstitute the first formula of (67) and (81) into (92); we haveSo the cases on 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 Section 3, the grazing motion on the boundary is guaranteed if and only if
Proof. By Lemma 8 and (50) and (55), the conditions for the grazing motion in domain to the boundary areBy (67) and (81), we haveSo the cases on are proved. Similarly, the other formulas in (96)–(101) can also be proved.
6. Mapping Structures and Periodic Motions
From the boundary in (41), the switching sets arewhere and the switching set on the edge is defined byTherefore, eleven basic mappings will be defined asBecause the switching set is the special case of the switching sets and , the mappings and can apply to , that is,The switching sets and mappings are shown in Figure 5.
In all eleven mappings, are the local mappings when and are the global mappings when . From the previous defined mappings, for each mapping , one obtains a set of nonlinear algebraic equationswherewith the constraints for and from the boundaries.
The system in Section 3 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 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 in absolute space and in relative space are shown in Figures 8 and 9, respectively.
For , the corresponding mapping relations areSuch mapping relations provide the nonlinear algebraic equations, that is,where and and ( is a period; ), 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 the -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: = 2 kg, = 0.1 N·s/m, = 1 N/m, = 0.1 N, = −5 N, = 2 kg, = 0.5 N·s/m, = 2 N/m, = −0.5 N, = 5 N, = 0.05 N·s/m, = 0.5 N/m, = 2 rad/s, , = 10 m/s2, , = 5 m/s, = 0.2 m/s.
For the above system parameters, the nonstick motions and stick motions of mass and mass are presented in Figures 10, 11, and 12 with the initial conditions of = 1 s, = 1 m, = 3 m/s, = 2 m, = 1 m/s. Consider the complex mappings . The time histories of velocities of the traveling belt and the masses and 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 and mass 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 is shown in Figure 12(e) when the sliding motion of occurs.
(a)
(b)
(c)
(d)
(a)
(b)
(c)
(d)
(e)
When , and move freely in domain , which satisfy and , as shown in Figure 10. In this time interval the time histories of velocities of and are shown in Figures 11(a) and 12(a), respectively. The displacements of and are shown in Figures 11(b) and 12(b), respectively. At the time , the velocity of reached the speed boundary (i.e., ). Since the forces and of per unit mass satisfy the conditions of and (as shown in Figure 12(d)), the analytical condition (59) of the passable motion on the boundary is satisfied in Theorem 9 (b). At such a point the motion enters into the domain relative to , as shown in Figure 10. Due to the movement in the area at the time and the movement that reached the boundary at the time , the mapping for this process is . When , and move freely in domain , which satisfy and , as shown in Figure 10. The displacements of and are shown in Figures 11(b) and 12(b), respectively. At the time , the velocity of reached the speed boundary (i.e., ). Since the forces and of per unit mass satisfy the conditions of and (as shown in Figure 11(d)), the analytical condition (62) of the passable motion on the boundary is satisfied in Theorem 9 (e). At such a point the motion enters into the domain relative to , as shown in Figure 10. Due to the movement on the boundary at the time and the movement that reached the boundary at the time , the mapping for this process is . When , and move freely in domain , which satisfy and , as shown in Figure 10. The displacements of and are shown in Figures 11(b) and 12(b), respectively. At the time , the velocity of reached the speed boundary (i.e., ). Since the forces and of per unit mass satisfy the conditions of and (as shown in Figure 12(d)), the analytical condition (69) of the stick motion on the boundary is satisfied in Theorem 10. At such a point the sliding motion of occurs on the boundary and keeps to . Due to the movement on the boundary at the time and the movement that reached the boundary at the time , the mapping for this process is .
When , moves freely in domain , satisfying , as shown in Figure 10. The displacement of is shown in Figure 11(b), correspondingly. However, at such time interval, maintains sliding motion, and the time history of force per unit mass of is shown in Figure 12(e). In this time period, the force product satisfies relative to , where is represented by pink curves and is represented by light cyan curves. At the time , the forces and of per unit mass satisfy the conditions of , and , so the analytical condition (86) of the vanishing of stick motion on the boundary is satisfied in Theorem 12 and the sliding motion of vanishes and the motion of enters the domain . Due to the movement on the boundary from the time to the time , the mapping for this process is .
When , and move freely again in domain , which satisfy and , as shown in Figure 10. The displacements of and are shown in Figures 11(b) and 12(b), respectively. At the time , the velocity of reached the speed boundary (i.e., ). Since the forces and of per unit mass satisfy the conditions of and (as shown in Figure 11(d)), the analytical condition (64) of the passable motion on the boundary is satisfied in Theorem 9 (g). At such a point the motion enters into the domain relative to as shown in Figure 10. Due to the movement on the boundary at the time and the movement that reached the boundary at the time , the mapping for this process is . When , and move freely in domain , which satisfy and , as shown in Figure 10. The displacements of and are shown in Figures 11(b) and 12(b), respectively. At the time , the velocity of reached the speed boundary (i.e., ). Since the forces and of per unit mass satisfy the conditions of and (as shown in Figure 11(d)), the analytical condition (65) of the passable motion on the boundary is satisfied in Theorem 9 (h). At such a point the motion enters into the domain relative to as shown in Figure 10. Due to the movement on the boundary at the time and the movement that reached the boundary at the time , the mapping for this process is . When , and move freely in domain , which satisfy and , as shown in Figure 10. The displacements of and are shown in Figures 11(b) and 12(b), respectively. At the time , the velocity of reached the speed boundary (i.e., ). Since the forces and of per unit mass satisfy the conditions of and , the analytical condition (61) of the passable motion on the boundary is satisfied in Theorem 9 (d). At such a point the motion enters into the domain relative to as shown in Figure 10. Due to the movement on the boundary at the time and the movement that reached the boundary at the time , the mapping for this process is .
When , the movement will continue, but, here, the later motion will not be described. In the whole process, the phase trajectories of and 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.
Competing Interests
The authors declare that they have no competing interests.
Acknowledgments
This research was supported by the National Natural Science Foundations of China (no. 11471196, no. 11571208).