Flow Switchability Research Articles

Discontinuous dynamics of a class of 3-DOF friction impact oscillatory systems with rigid frame and moving jaws

In this paper, the discontinuous dynamics of a class of 3-DOF (three-degree-of-freedom) friction impact oscillatory systems with rigid frame and moving jaws is investigated by using the high-dimensional flow switchability theory. For the close relationship between any two objects of the 3-DOF system, we directly investigate the dynamics of whole system rather than respective dynamics of each object separately. The flow barriers in such a system are also considered based on the inequalities of coefficients for dynamic friction and static friction. Based on discontinuities/nonsmoothness of the oscillatory system, the six-dimensional phase space of whole system is divided into different domains and boundaries/edges in relative coordinates, and the local singularity and switchability of flows to high-dimensional boundaries and edges are fully investigated. In order to better understand the complexity of dynamics of the 3-DOF system, several typical motions are numerically simulated, such as passable, sliding, grazing, stick, impact and periodic motions. For partial numerical results, a comparison between simulation data obtained in MATLAB and MSC Adams has been performed. The study of such a 3-DOF model enriches the high-dimensional flow switchability theory, and provides a theoretical basis for optimization design and parameter selection of mechanical mechanism with jaws.

Stick Motions and Grazing Flows in a Double Belt Friction Oscillator

This paper is concerned with the dynamics of a double belt friction oscillator which is subjected to periodic excitation, linear spring-loading, damping force and two friction forces using the flow switchability theory of the discontinuous dynamical systems. Different domains and boundaries for such system are defined according to the friction discontinuity, which exhibits multiple discontinuous boundaries in the phase space. Based on the above domains and boundaries, the analytical conditions of the stick motions and grazing motions are obtained mathematically. There are more theories about such friction oscillators to be discussed in future.

Complex switching dynamics and chatter alarm for aerial agents with artificial potential field method

This paper presents a dynamic problem with moving discontinuous boundary for two aerial agents. One of the agents is assigned as the leader, and the other follows such a leader based on artificial potential field. The artificial potential field technique is designed to make the two agents repel each other when the relative distance is less than r1 in order to avoid collision, and such two agents will attract each other when the distance is greater than r1. The interaction between two agents becomes zero when the relative distance is greater than r2(r2>r1) to model the communication disconnected. Since the mathematic function for the artificial potential field is discontinuous, the dynamics of such two aerial agents might be also discontinuous. The main contributions of this paper are: 1) the dynamics of such two aerial agents switching on the discontinuous boundaries are firstly studied based on the complexity theory of flow switchability; 2) the “chatter” phenomenon due to the artificial potential field is explained as the onset of sliding motion on such a moving discontinuous boundary.

Analytical Dynamics of a Discontinuous Dynamical System with a Hyperbolic Boundary

In this paper, analytical dynamics of a discontinuous system with a hyperbolic control boundary is studied. The analytical conditions for flow switchability of the discontinuous system are developed from a flow passability theory at boundaries, which are for a better understanding of dynamics of such a discontinuous dynamical system. With the analytical switchability conditions, the sliding and non-sliding motions are described through generic mappings. Periodic motions in the discontinuous dynamical system with hyperbolic boundary are studied through mapping structures. The bifurcation trees of periodic motions are presented, and the corresponding stability and bifurcation of periodic motions are analyzed. The grazing bifurcation for a flow to the boundary is discussed as periodic motion switching. Numerical simulations are completed for illustration of complex periodic motions and flow switchability at the hyperbolic boundary. This paper is dedicated to Nail Ibragimov as a good friend and colleague for 20 years friendship with Albert Luo.

On discontinuous dynamics of a 2-DOF system with bilateral rigid constraints and nonlinear friction

In this paper, the discontinuous dynamic analysis of a two-degree-of-freedom system with bilateral rigid constraints in the coexistence of nonlinear friction and external excitations with negative feedbacks is presented through the theory of flow switchability and flow barrier in discontinuous dynamical systems, where the nonlinear friction force is approximated by a piecewise linear, kinetic friction model with static friction. Considering inequality of coefficients of static and kinetic friction, the flow barriers on velocity boundaries are introduced. Based on friction and collisions, different domains and boundaries are defined in phase plane. The analytical conditions of motion switching at separation boundaries are explored through the force criteria and the flow barrier’s criteria based on the analysis of vector fields in domains. Finally, the numerical simulations of several typical motions are given for a better understanding of switching conditions between different motions. The results obtained in this paper are of great significance to the optimization design of mechanical system with bilateral rigid constraints and negative feedbacks.

Discontinuous dynamical behaviors in a 2-DOF friction collision system with asymmetric damping

Abstract By using the flow switchability theory in discontinuous dynamical systems, this paper deals with the discontinuous dynamical behaviors of a two degrees of freedom system with asymmetric damping, where considering that friction and impact coexist and the static and dynamic friction coefficients are different. Because of the particularity of friction force, the flow barriers on the velocity boundary that affect the leaving flow are considered in this paper. Based on discontinuity that is caused by the sudden change of friction force or the collision between two objects, the phase space of motion for the object is divided into several different domains and boundaries; and with the help of the analysis of vector fields and G-functions on the corresponding discontinuous boundaries or in domains, the analytical conditions for all possible motions are obtained, which is used to determine the switching of motion state in this system. Finally, numerical simulations are presented to better understand the analytical conditions of the stick, grazing, impact, stuck and periodic motions.

Discontinuous dynamic analysis of a class of three degrees of freedom mechanical oscillatory systems with dry friction and one-sided rigid impact

In this paper, the discontinuous dynamical behavior for a three degrees of freedom mechanical oscillatory system with one-sided rigid impact in the presence of dry friction is investigated by using the flow switchability theory in discontinuous dynamical systems. Based on the discontinuity resulted from impact and friction, different domains and boundaries are defined in absolute and relative coordinates, respectively. At the discontinuous boundaries, the necessary and sufficient conditions for motion switchability are developed by means of G-functions. Finally, the passable flows, sliding flows, grazing flows, stick flows, impact motions and periodic motions are illustrated numerically for a better understanding of the motion switching mechanism in such a complex system. The results obtained in this paper reveal the complicated switching mechanism of motion for objects in a discontinuous dynamic system and also provide a reference for the selection of parameters in the system of vibro-impact mechanical devices such as pile drivers and horizontal directional drillings etc.

On discontinuous dynamics of a SDOF nonlinear friction impact oscillator

In this paper, the discontinuous dynamical behaviors of a SDOF nonlinear oscillator in the co-existence of friction and impact are studied via using flow switchability theory of discontinuous dynamical systems, where considering that the static friction coefficient is greater than the kinetic friction coefficient at the object’s velocity tending to zero and the friction coefficient is described by the Stribeck friction model. Because of both friction and collision, the object will have four motion states: free motion, impacting motion, stationary state (or stick motion) and stuck motion. According to discontinuity caused by the abrupt change of friction force and the constraint of the right rigid wall, the phase space is divided into various domains and boundaries. Due to the particularity of friction force, the flow barrier on the segmentation boundary is considered, which may affect the leaving flow at the velocity boundary. The analytical conditions for all possible motions are acquired through G-functions on the corresponding boundaries based on the analysis of vector fields in domains. The mapping structures are introduced and the periodic motions of this system are presented by the basic mappings. In order to understand the motion conversion principium of the SDOF oscillator more intuitively, the time histories of displacement, velocity, G-functions and the corresponding trajectories in phase plane for the object are shown by simulation numerically. Finally, the grazing and stick bifurcation scenarios for excitation frequency and excitation amplitude are presented.

On discontinuous dynamics of a class of friction-influenced oscillators with nonlinear damping under bilateral rigid constraints

This article is concerned with the discontinuous dynamical behaviors of a class of single degree of freedom oscillators with linear and nonlinear springs and viscous dampers in the co-existence of bilateral rigid impact and friction through the theory of flow switchability and flow barrier in discontinuous dynamical systems, where considering the inequality of static and kinetic friction coefficients. By the analysis of vector fields and G-functions on the corresponding discontinuous boundaries for such an oscillator, the analytical conditions of all possible motions are established, which are in the form of a set of inequalities or equalities and can be used for the selection of control parameters in this kind of system. Additionally, based on mapping dynamics, the two-dimensional basic mappings are defined, the analytical predictions and stability analysis of different periodic motions are accomplished. Finally, our theoretical results are further demonstrated by numerical simulations by means of dimensional and dimensionless parameters with graphical illustrations of the time histories of displacement, velocity, G-function and the trajectories in phase space for the passable motion, stuck motion, impact motion, grazing motion and periodic motion etc., and stick and grazing bifurcation scenarios varying with excitation frequency and amplitude are also given.

Analysis of dynamical behaviors of a 2-DOF friction-induced oscillator with one-sided impact on a conveyor belt

In this paper, the flow switchability theory of discontinuous dynamical systems is used to illustrate the dynamical behavior of a 2-DOF (two degrees of freedom) friction-induced oscillator with one-sided impact on a conveyor belt. All the possible motion states such as stick and non-stick motions, impact motion, and stuck motion for such an oscillator are introduced. The phase space in system can be divided into different domains and boundaries according to the discontinuity caused by the friction force jumping and the impact between the mass and the rigid obstacle. The vector field in each domain is continuous and different from that in its adjacent domain. The flow barrier on the separation boundary is considered in this paper. Once the boundary flow leaves the boundary, the boundary flow barrier on the velocity boundary may exist and the leaving flow barriers on the velocity boundary may also exist. The G-functions on different separation boundaries are defined to illustrate the flow switching on the corresponding boundaries. The analytical conditions of the passable, stick, grazing, impact, and stuck motions are developed through G-functions and analysis of vector fields. Since the motions of the two masses interact with each other, the four-dimensional switching sets are given by the form of direct product and the four-dimensional mappings are given to describe periodic motions with different mapping structures. The analytical prediction of different periodic motions is performed through the mapping dynamics. For a better understanding of the motion switching mechanism in such a 2-DOF oscillator , the time histories of displacement, velocity, G-function and the trajectories in phase space for the passable motion, stick motion, impact motion, grazing motion, stuck motion, and periodic motion in system are given by simulation numerically. This investigation has important significance in the optimization design of machinery with friction and impact etc.

On discontinuous dynamical behaviors of a 2-DOF impact oscillator with friction and a periodically forced excitation

In this paper, a 2-DOF (two-degree-of-freedom) impact oscillator with friction and a periodically forced excitation is investigated via using the flow switchability theory in discontinuous dynamical systems. Based on the discontinuities caused by friction and impact between the two masses, the phase space is partitioned into different boundaries and domains. Using the G-functions and vector fields, the analytical conditions of grazing motions and passable motions are discussed, and the appearing and vanishing conditions of sliding motions and side-stick motions are also developed. The periodic motions with stick or non-stick are described through the generic mappings. For better understanding of the analytical conditions of periodic motions, grazing motions, stick motions and passable motions, the velocity and displacement time-histories, G-function responses and trajectories are presented. The investigation on such a 2-DOF impact oscillator with friction may be helpful for achieving optimal design of the single row cylindrical roller bearing systems. Besides, it has an important significance to the noise suppression in mechanical systems with clearance.

On discontinuous dynamics of a 2-DOF friction-influenced oscillator with multiple elastic constraints

In this paper, the discontinuous dynamical behaviors in a 2-DOF (two-degree-of-freedom) friction-influenced oscillator with multiple elastic constraints are investigated by using flow switchability theory of discontinuous dynamical systems. The mechanical model is composed of two masses driven by periodic forces, which are on the two conveyor belts with constant speeds separately and are connected by two linear springs and two dampers. Both the same springs are fixed on both sides of the first mass. Because of the mutual effect between the two masses and the elastic constraints on both sides of the first mass, the following five cases should be considered: both the masses have nonsliding motion; one of the two masses has sliding motion; both the masses have sliding motion simultaneously; and the first mass with elastic constraints has stick motion. Different domains and boundaries for such system are defined based on the friction discontinuity and multiple elastic constraints, and the state vectors and field vectors of such system are introduced. Based on the above domains and boundaries, the analytical conditions for passable motion, sliding motion, grazing motion and stick motion in the friction-induced system are obtained by means of G-functions on the corresponding boundaries. The mapping structures are presented and the periodic motions of such system are described by the basic mappings. Finally, the time histories of displacement, velocity, G-functions and the corresponding trajectories in phase plane for the two masses are illustrated numerically for a better understanding of the complex dynamical behaviors of such system. This investigation can help us use or control motion of such a friction-influenced oscillator in industry, and has important significance in the optimization design of machinery with friction, impact and multiple constraints.

Analysis of dynamical behaviors of a 2-DOF vibro-impact system with dry friction

In this paper, the discontinuous dynamics in a 2-DOF (two-degree-of-freedom) vibro-impact system with dry friction is investigated by using the flow switchability theory for discontinuous dynamical systems. Multiple domains and discontinuous boundaries are defined according to friction and impact discontinuity. Based on above domains and boundaries, the onset and disappearance conditions of sliding-stick motion are developed and the analytical conditions of side-stick motion and grazing motion are obtained mathematically. The switching sets and mapping structures are adopted to describe the complex motions in such discontinuous system. The numerical simulations are also given to illustrate the analytical results of motion switching on different boundaries. This investigation can help us understand the motion switching mechanism in non-stick, sliding-stick or side-stick motions of the oscillator with friction and impact.

Analysis of discontinuous dynamical behavior of a class of friction oscillators with impact

In this paper, the discontinuous dynamical behaviors in the friction oscillator with impact are investigated by using the theory of flow switchability in discontinuous dynamical systems. On the basis of discontinuity of friction and impact, different domains and boundaries are defined for such system. Based on above domains and boundaries, the analytical conditions of passable motion, stick motion, sliding motion and grazing motion are obtained mathematically. The analytical predictions of the corresponding periodic motion are given by the adequate mapping structures. The passable, stick, sliding, grazing and periodic motions are illustrated through trajectories in phase planes and time histories of displacement, velocity and G-function for a better understanding of motion mechanism of the friction oscillator with impact. In the end, the grazing and sliding bifurcations are determined by the local singularity theory of discontinuous dynamical systems. This model is suitable for nonlinear dynamical description of the vibration in rotor bearing or gear transmission systems with impact and friction.

On discontinuous dynamics of a periodically forced double-belt friction oscillator

In this paper, complex discontinuous dynamics of a periodically forced double-belt friction oscillator are investigated using the theory of flow switchability for discontinuous dynamical systems. Phase plane of such a system is divided into different domains and boundaries due to the discontinuity caused by the friction and the positions of belts. The analytical conditions of the passable, stick and grazing motions on the corresponding discontinuous boundaries are derived for motion switching complexity in detail. Based on the discontinuous boundaries, switching sets and basic mappings are introduced to describe different periodic motions in such a discontinuous dynamical system. The analytical predictions and corresponding local stability analyses for the periodic motions are developed through mapping structures. Stick and non–stick periodic motions with different mapping structures, stick motions and grazing motions on different discontinuous boundaries are presented numerically for a better understanding of physics of the double–belt friction oscillator.

Discontinuous dynamics of a class of oscillators with strongly nonlinear asymmetric damping under a periodic excitation

Starting from the quarter car suspension system, the discontinuous dynamics of a general class of strongly nonlinear single degree of freedom oscillators are investigated using the flow switchability theory of the discontinuous dynamical systems. The characteristic of this oscillator is that they possess piecewise linear damping properties, which can be expressed in a general asymmetric form. More specifically, the viscous and constant damping properties appearing in the equation of motion depend on the velocity direction. Different domains and boundaries are defined according to the discontinuity. Based on above domains and boundaries, the analytical conditions for motion switchability at the velocity boundary in such oscillators are developed to understand the motion switching mechanism. To describe different motions in domains, the generic mappings and mapping structures are introduced. Based on the appropriate mapping structures, the periodic motions of such discontinuous systems are predicted analytically. Specified periodic and grazing motions for the quarter car model are given through the displacement, velocity and forces responses to illustrate the analytical criteria of complex motions. However, the periodic motions with switching for such nonlinear oscillators cannot be obtained from the traditional analysis, like the perturbation and harmonic balance method. Moreover, the present analysis can be extended to cover wider classes of dynamical systems, like mechanical oscillators with variable stiffness and damping properties.

Discontinuous dynamical behaviors in a vibro-impact system with multiple constraints

In this paper, the discontinuous dynamical behaviors in a two-degree-of-freedom vibro-impact system with multiple constraints are investigated by using the flow switchability theory of discontinuous dynamical systems. Due to the interaction between the two masses in this discontinuous system, the following four cases are taken into account: both the masses are free-flight; one of the two masses is sticking; and both the masses are sticking. Different domains and boundaries are defined in absolute and relative coordinates based on the discontinuity caused by the impact between the masses and the constraints. From the above domains and boundaries, the analytical conditions of switching for stick motions and grazing motions in the vibro-impact system are obtained through the analysis of the corresponding vector fields and G-functions. The switching sets and four-dimensional mappings are introduced to describe different periodic motions and identify the corresponding mapping structures. Periodic motions with different mapping structures in the vibro-impact system are analytically predicted. The time histories of displacement, velocity, G-function and the corresponding trajectories in phase plane for stick, grazing and periodic motions are given to illustrate the dynamics mechanism of complex motions in such a vibro-impact system. A better understanding of the motion switching mechanism in mechanical systems with multiple constraints may be helpful for improving the efficiency of vibro-impact systems.

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.

Flow switchability of motions in a horizontal impact pair with dry friction

Using the flow switchability theory of the discontinuous dynamical systems, the present paper is to develop mechanical complexity in a periodic-excited horizontal impact pair with dry friction. The impact pair studied models the motions of a single bolted connection which is vibrated in the plane perpendicular to the bolt axis. According to motion character, the phase space can be partitioned into several domains and boundaries, in which a continuous dynamical system is defined in each domain, and it possesses dynamical properties different from its adjacent subsystem, the boundaries have different properties and can fall into two kinds – displacement boundaries and velocity boundaries. In this paper, using G–functions defined on separation boundaries to study flow switching on corresponding boundaries, the analytical switching conditions on each boundary are developed: the sufficient and necessary conditions of occurrence and disappearance of sliding-stick motion and side-stick motion are obtained, the sufficient and necessary conditions of grazing motion appearing on velocity boundaries are also obtained, and the analytical conditions of appearance for grazing motion on displacement boundaries are preliminarily discussed. Thus it can be seen that dynamical behaviors of the horizontal impact pair with or without dry friction are essentially different, in particular flow switchability on displacement boundaries depend on whether the conditions of passable flows on velocity boundaries are satisfied. The numerical simulations are given to demonstrate the analytical results of two stick motions and grazing motions in such pair. More details of the motions for the object reaching the intersection point of displacement boundary and velocity boundary need to be considered further in the future.

Stick motions and grazing flows in an inclined impact oscillator

In this paper, the dynamics of an inclined impact oscillator under periodic excitation are investigated using the flow switchability theory of the discontinuous dynamical systems. Different domains and boundaries for such system are defined according to the impact discontinuity. Based on above domains and boundaries, the analytical conditions of the stick motions and grazing motions for the inclined impact oscillator are obtained mathematically, from which it can be seen that such oscillator has more complicated and rich dynamical behaviors. The numerical simulations are given to illustrate the analytical results of complex motions, and several period-1 motions period-2 motion and chaotic motion of the ball in the inclined impact oscillator are also presented. There are more theories about such impact pair to be discussed in future.