Mathematical Problems in Engineering

Volume 2018, Article ID 5301747, 33 pages

https://doi.org/10.1155/2018/5301747

## 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 [16–32].

On the other hand, the determination of periodic orbits along the switching boundary is very important in the nonsmooth control theory. In 2003, Bernardo et al. [33] showed that sliding bifurcations play an important role in analyzing the dynamics of dry friction oscillator. In , Galvanetto [34] presented several detailed examples to address the fact that the occurrence of sliding bifurcation does not always correspond to the change in the number of periodic orbits at the bifurcation point. In 2005 and 2008, Kowalczyk et al. [35, 36] were concerned with the extension to the case of codimension 2 degenerate sliding bifurcations, numerical continuation, and analytical investigations of sliding bifurcations in Filippov systems. In 2010, Guardia et al. [37] analytically studied the sliding bifurcations of periodic orbits in a dry friction oscillator.

Recently, new development in discontinuous dynamical systems, such as the motion switchability for stick and sliding and grazing motions, has been found. In 2005, Luo [38, 39] developed a theory for nonsmooth dynamical systems on connectable domains and the mapping dynamics of periodic motions for a piecewise linear system under a periodic excitation. In 2005, 2006 and 2008, Luo [40–42] introduced the concepts of imaginary, sink, and source flows in nonsmooth dynamical systems and developed a general theory for the local singularity of a flow to the discontinuous boundary and a theory for flow switchability in discontinuous dynamical systems. In , Luo and Gegg [43] investigated periodic motions in an oscillator moving on a periodically vibrating belt with dry friction. In 2009, Gegg et al. [44] presented the analytical conditions for sliding and passable motions on the periodically time-varying boundary for a friction-induced oscillator through the relative force product. In 2011, Luo [45] systematically presented a new theory for flow barriers in discontinuous dynamical systems, which provides a theoretic base to further develop control theory and stability. As to applications of the aforementioned local singularity theory, Luo and Huang [46] used such theory to determine the flow switchability on the discontinuous boundary for the nonlinear, friction-induced, periodically forced oscillator in 2012. In 2015 and 2017, Zhang and Fu [25, 47, 48] studied the periodic motions, stick motions, grazing flows in an inclined impact oscillator, and the flow switchability of motions in a horizontal impact pair with dry friction. In 2016, Chen and Fan [49] presented the analytical conditions for the motion switchability in a double belt friction oscillator system with a periodic excitation. In 2018, Fan et al. [50] further investigated dynamics of such double belt friction oscillator system with a periodic excitation. In 2017 and 2018, Fan et al. [51, 52] studied a friction-induced oscillator with two degrees of freedom on a speed-varying traveling belt and discontinuous dynamical behaviors in a vibroimpact system with multiple constraints, respectively.

In study of dynamical system, the switching control law also plays an important role. Some researchers have obtained some results by using the theory of discontinuous dynamical systems developed by Luo. In 2009, Luo and Rapp [1] investigated the flow switchability and periodic motions in a periodically forced, discontinuous dynamical system with switching control law of an inclined line. In the next year, Luo and Rapp [2] studied the motions and switchability of an oscillator in a periodically forced, discontinuous system with switching control law of a parabolic boundary. In 2015, Zheng and Fu [53] proposed the switched Van der Pol equation with impulsive effect as switched system and analyzed its features from a discontinuous point of view. In 2017, Fan et al. [54] studied discontinuous dynamics of a friction-induced oscillator with switching control law of a straight line. For the examples of impulsive control and Boolean control as switching control law, see [55–68]. It is worth noting that the switching control law of an elliptic boundary has an important application in practical problem, such as satellite rendezvous and formation flight. The elliptical controller is designed as a reference orbit for formation flight. By selecting the appropriate parameters to design the controller, the orbit of the satellite can be controlled so that it can conduct sliding or periodic motion in the reference orbit. This can effectively improve flight efficiency and save fuel. Compared with the circular reference orbit, the flight mode of elliptical reference orbit has the advantages which can reach the earth layer boundary region of space physics research, travel through specific spaces for longer periods of time, collect more information, and realize multipoint synchronous measurement by three-dimensional configuration and so on. Some scholars do a lot of work. For example, Senqupta [69] investigated elliptic rendezvous in the chaser satellite frame in 2012. In the same year, Chang et al. [70] studied the transfer of satellites between elliptic Keplerian orbits using Lyapunov stability theory. In 2013, Yan et al. [71] investigated pseudospectral feedback control for three-axis magnetic attitude stabilization in elliptic orbits. In 2012, Do [72] presented a design of cooperative controllers that force a group of 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 .