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Mathematical Problems in Engineering
Volume 2018, Article ID 5301747, 33 pages
https://doi.org/10.1155/2018/5301747
Research Article

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

School of Mathematics and Statistics, Shandong Normal University, Jinan 250014, China

Correspondence should be addressed to Jinjun Fan; moc.621@81jjf

Received 3 December 2017; Revised 24 February 2018; Accepted 15 March 2018; Published 8 May 2018

Academic Editor: Stefano Lenci

Copyright © 2018 Jinjun Fan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 mass through the analysis of the corresponding vector fields and -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.

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 [1632].

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 , 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 [4042] 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 , 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 [5568]. 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 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 mass will be developed through the -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 -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 Figure 1(a). Denote displacement by and time by , and assume that the switching boundary in this discontinuous dynamic system is where , , and , 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 , a switching spring of stiffness , and a switchable damper of coefficient in the domain , where is the unbounded outside domain and is the bounded inside domain. The periodic force acting on the mass in the different domains is where , , , and are excitation amplitude, frequency, initial phase, and constant force, respectively. Through prestressing the mass , the constant-force magnitude of could be adjusted to adapt to different working environments. Furthermore, the mass slides or rests on the conveyor belt with a constant speed .

Figure 1: (a) Mechanical model; (b) elliptic control law’s boundary.

Since the mass contacts the conveyor belt with friction, the mass can move along or rest on the conveyor belt. Further, a kinetic friction force is given by where is the normal force between the mass 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 domain is where

The nonfrictional force per unit mass in the domain is

The motions of the mass in the domain are divided into three cases.

(i) If and the displacement and velocity of the mass do not satisfy the elliptic control law (i.e., ), the corresponding motion is called the nonstick motion or free-flight motion. For this case, the equation of motion is given as follows:

(ii) If the mass moves together with the conveyor belt (i.e., ) and the displacement and velocity of the mass do not satisfy the elliptic control law (i.e., ), the corresponding motion is called the stick motion. For this case, the equation of motion is given by

(iii) If the displacement and velocity of the mass satisfy the elliptic control law (i.e., ), the corresponding motion is called the sliding motion. For this case, the equation of motion is described by

3. Domains and Boundaries

Due to the elliptic control law and the friction between the mass and the conveyor belt, the motions of the mass become discontinuous and more complicated. In order to determine the switching complexity for the motions of the mass , 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 mass . Based on the discontinuities caused by the switching control law and the friction between the mass 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 can be described through a continuous dynamical system.

Figure 3: Domains and boundaries.

Set

The four nonstick domains for the mass can be expressed as

The corresponding boundaries, including two stick or velocity boundaries and two sliding or control boundaries, can be defined as

For the nonstick motions in domains, two vectors are introduced as where Using (13), the equations of nonstick motions are rewritten in the vector form of where

For the stick motions on velocity boundaries, two vectors are introduced as where . The equations of stick motions are rewritten in the vector form of where

For the sliding motions on elliptic boundaries, two vectors are introduced as where . The equations of sliding motions are rewritten in the vector form of where

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 mass will be developed in this section.

4.1. Basic Theory

For convenience, the concepts of the -functions and some lemmas are given in the following (see [42]).

Consider a dynamic system consisting of subdynamic systems in a universal domain . The universal domain is divided into accessible subdomains and inaccessible subdomains . And the union of all the accessible subdomains is and the universal domain is . On the th open subdomain , there is a -continuous system () in the form ofwhere the time is and . In an accessible subdomain , the vector field with a parameter vector is -continuous () for all time , and the flow in (22) with an initial condition is continuous for time , and on the boundary of two adjacent domains , is -continuous , and there is a flow , which can be determined by with an initial condition and a parameter vector .

Assume that there are two adjacent subdomains and A flow in subdomain is called a local flow if it is governed by the continuous vector field in only. A flow is called a passable flow if it switches from a subdomain into another one through the boundary . Obviously, the vector field of a passable flow in subdomain will be changed into the one in subdomain , accordingly. A flow is called a grazing flow if it is tangential to the boundaries . If a flow cannot pass through but moves along the boundaries , then this flow is called a sliding flow.

Definition 1 (see [42]). The 0th-order -functions of the domain flow to the boundary flow on the boundary in the normal direction of the boundary are defined as

Definition 2 (see [42]). The st-order -functions of the domain flow to the boundary flow on the boundary in the normal direction of the boundary are defined as

In above definitions, the total derivative , and the normal vector of the boundary surface at point is given by and .

If the flow contacts with the boundary at time , that is, , and the boundary is linear, independent of time , we can obtain the th-order -functions and the st-order -functions by simple computation Here is to show the motion in different domains rather than on the boundaries, and , are the time before approaching and after departing the corresponding boundary, respectively. In this case, the -functions are simplified as and .

For a discontinuous dynamical system in (22), there is a point at time between two adjacent domains . For an arbitrarily small , there are two time intervals and . Suppose . Both flows and are - and -continuous for time , and .

Based on the -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 flows and at point on the boundary are semipassable from domain to domain if and only if

Lemma 4 (see [42]). The flows and at point on the boundary are sink flows if and only if

Lemma 5 (see [42]). 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 where

Lemma 6 (see [42]). For an arbitrarily small , there are two time intervals and . Suppose . The flows and are - and -continuous for time and . The sliding bifurcation of the passable flows and at point switching to the nonpassable flow of the first kind on the boundary occurs if and only if

Lemma 7 (see [42]). For an arbitrarily small , there are two time intervals and . Suppose . The flows and are - and -continuous for time and . The tangential bifurcation of the flows and at point on the boundary is termed the sliding fragmentation bifurcation if and only if

More detailed theory on the flow switchability such as high-order -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 mass described in Section 3 will be developed in this subsection. From (12), the normal vectors of the absolute boundaries are given as for , and for , where is a Hamilton operator.

Theorem 8. For the friction-induced oscillator with an elliptic control law and the friction force acting on the mass discussed in Section 2, there are the following results.
(i) The necessary and sufficient conditions for passable motion on the boundary at time are (ii) The necessary and sufficient conditions for passable motion on the boundary at time are (iii) The necessary and sufficient conditions for passable motion on the boundary at time are(iv) The necessary and sufficient conditions for passable motion on the boundary at time are

Proof. The mass moves along the conveyor belt with nonzero relative velocity before time , and the relative velocity becomes zero at ; then after the relative velocity changes its direction. From the flow switchability theory on the discontinuous dynamical systems, the flow in domain or passes through the boundary or into domain or , which can be expressed by Lemma 3. So th-order -functions on such boundaries are needed.
According to (27), the 0th-order -functions on the boundary areFrom (13) and (35), (44) can be computed asBy Lemma 3, the passable motion on the boundary at time appears if and only if From (45) and (46), one obtains Thus, (36) in (i) holds, and (37) in (i) and (40) and (41) in (iii) can be proved similarly.
From (27), the 0th-order -functions on the boundary are From (13) and (34), (48) can be computed as By Lemma 3, the passable motion on the boundary at time appears if and only if By (49) and (50), one obtains 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 mass discussed in Section 2, there are the following results.
(i) The necessary and sufficient conditions for sliding motion on the boundary at time are (ii) The necessary and sufficient conditions for stick motion on the boundary at time are (iii) The necessary and sufficient conditions for sliding motion on the boundary at time are (iv) The necessary and sufficient conditions for stick motion on the boundary at time are

Proof. The displacement and velocity of the mass 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 reaches the boundary and moves along the boundary . Thus it can be predicted by Lemma 4.
From (27), the 0th-order -functions on the boundary areFrom (13) and (35), (56) can be computed as By Lemma 4, the sliding motion on the boundary at time appears if and only if From (57) and (58), one obtains Thus, (i) holds, and (iii) can be proved similarly.
The mass moves together with the conveyor belt and the displacement and velocity of the mass 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 reaches the boundary and moves along the boundary . Thus it can be predicted by Lemma 4.
By (27), the 0th-order -functions on the boundary areFrom (13) and (34), (60) can be computed asBy Lemma 4, the stick motion on the boundary at time appears if and only if From (61) and (62), one obtains 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 mass discussed in Section 2, there are the following results.
(i) The necessary and sufficient conditions for grazing motion on the boundary in domain or domain at time are or(ii) The necessary and sufficient conditions for grazing motion on the boundary in domain or domain at time are or(iii) The necessary and sufficient conditions for grazing motion on the boundary in domain or domain at time are or(iv) The necessary and sufficient conditions for grazing motion on the boundary in domain or domain at time areor

Proof. When the mass is moving along the conveyor belt with nonzero relative velocity, the relative velocity equals zero at time 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 or reaches the boundary or and then returns to domain or . Such motion is called grazing motion on the boundary or . So the analytical conditions of grazing motion on boundary or can be obtained by Lemma 5. Thus the 0th-order -functions and the 1st-order -functions on such boundaries are needed.
According to (27) and (28), the 0th-order -functions and the 1st-order -functions on the boundary in domain are From (13) and (35), (72) can be computed as By Lemma 5, the grazing motion on the boundary in domain appears if and only if From (73) and (74), one obtains 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 -functions and the 1st-order -functions on the boundary in domain are From (13) and (34), (76) can be computed as <