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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.

This paper deals with the discontinuous dynamic behaviors of a class of three-degree-of-freedom friction and collision system with symmetric bilateral rigid constraints by using the flow switching theory of discontinuous dynamical systems. The model takes into account the inequality of static and dynamic friction forces, which is more common in practice. The external excitation of the object contains a negative feedback, which can change with the variation of the velocity of the object, so that the system can adapt to different external environment. And the connection between the objects adopts nonlinear springs and viscous dampers in order to achieve a better vibration reduction effect. The model can be applied to mechanical equipment such as shock absorbers etc. According to the discontinuity caused by friction and collision, the dynamic domain and boundary of the object’s motion are defined in phase space, which requires the introduction of absolute coordinates and relative coordinates respectively to discuss the motion between objects. The vector field for the respective domain in the system is given, which can control the motion of object in each domain. The corresponding normal vector on the discontinuous boundary is introduced to determine the positive direction of the flow. The sufficient and necessary conditions for the switching motion of such dynamical system are given based on $$\mathrm{G}$$ -function and its higher order. These switching conditions can be used to predict or even control the motion of the object on the separation boundary. In addition, several typical motions, such as sliding, stick, grazing, impact motions and periodic motion, are numerically simulated to better explain the complexity of the switching motion in the discontinuous dynamic system. Finally, the stick bifurcation scenarios for excitation frequency or amplitude are presented. The research results obtained can provide a theoretical basis for better use or control of friction and collision in practical applications.

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.

Starting from the car suspension system, the nonlinear characteristics of a class of two-degree-of-freedom oscillators with strong nonlinearity under a periodic excitation are discussed by the switching theory of flow in discontinuous dynamical systems. Based on the discontinuous forces and different motions of the two masses, the phase plane of each mass is composed of stick domain, nonstick domain (or free domain) and separation boundary in absolute and relative coordinates, respectively. The swithching criteria between the stick and nonstick motions and the conditions of grazing motion in two different regions are developed via the G-functions and switching control laws. The mapping dynamics theory is used to give the four-dimensional transformation set and four-dimensional mapping, and the conditions for periodic motions are explored. In addition, the stick motions, two kinds of grazing motions, periodic motions for this system and a comparison of the velocities, accelerations (or forces responses) of the two masses under the two conditions of control force are simulated numerically. The results show that the stability and comfort of the vehicle can be improved by adjusting the control force, which is generated by the control unit of system or exerted by the external excitation. For further investigating the influence of system parameters on dynamical behaviors, the stick and grazing bifurcation scenarios varying with driving frequency or amplitude are also developed, which can provide useful information for parameter selection of vibration systems with clearance and the optimal design of vehicle suspension systems. This paper also has important reference value for practical applications in other industries or machinery with elastic impacts.

In this paper, the flow switching theory of discontinuous dynamical systems is used as the main tool to study the dynamical behaviors of a two-degree-of-freedom friction oscillator with elastic impacts, where considering that the inequality of static and kinetic friction forces results in the existence of flow barriers at the relative velocity between the mass and conveyor belt tending to zero. The negative feedback is also taken into account, $$\mathrm {i.e.}$$ , when the direction of the relative velocity between the mass and conveyor belt changes, the constant force in the periodic excitation force will also change to adapt to the variation in speed. Due to the discontinuity resulted from the friction and non-smoothness resulted from impact, the motion of the two masses can be divided into the following kinds: non-sliding motion (or free motion), sliding motion, non-sliding–stick motion and sliding–stick motion, and the phase space in system is divided into different domains and boundaries, where the vector field is continuous in each region but discontinuous on the velocity boundary. The corresponding normal vectors and G-functions on the separation boundaries are introduced to acquire the switching conditions for all possible motions such as passable motion, sliding motion, grazing motion and stick motion. The mapping theory provides an effective method for the discussion of periodic motion in discontinuous dynamical systems. The four-dimensional switching sets are defined by the means of the form of direct product of two-dimensional switching sets, and the mapping structures are introduced based on the switching sets to describe periodic motions. Eventually, in order to explain more intuitively the above motions in such a friction–impact system, the velocity, displacement, G-functions time histories and the trajectories in phase plane for the masses are illustrated by simulation numerically.

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.

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.

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.

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.

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.

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.

The deformation behavior and amorphization process of polycrystalline and single-crystalline silicon are investigated using molecular dynamics (MD) simulations and compared with the previous nanoindentation experiments. In order to unravel the grain size effect on the deformation mechanism, the indentation simulations of polycrystalline silicon with the different grain sizes are performed, and all grains follow the inverse Hall-Petch relation. The results reveal that Young’s modulus of polycrystalline silicon is much greater than Young’s modulus obtained in the 〈1 0 0〉 direction single-crystalline silicon due to the random crystallographic orientation and strong anisotropy of polycrystalline silicon at nanoscale. The nucleation of amorphization in both single-crystalline and polycrystalline silicon appears always beneath the indenter due to the maximum shear stress, controlling the plastic deformation during the indentation process. The horizontal expansion of amorphization region is faster and larger than the vertical expansion of amorphization region due to two reasons: the limit of indentation stress along the depth direction is compared with along the width direction; the grain boundary is the most possible path of weak connection, and produces the complex stress distribution for the propagation of amorphization region. The smaller grain size leads to the stronger gradient strain, especially in the region of upper surface.

In this paper, the dynamical behaviors of a friction–induced oscillator with switching control law are studied through the flow switching theory of discontinuous dynamical systems. The physical model consists of a mass on the conveyor belt and a spring-damping system with switching control law. Based on the switching control law and the friction between the oscillator and the conveyor belt, multiple domains and discontinuous boundaries are defined. The G–functions are introduced to illustrate the motion switching mechanism and the analytical conditions of the passable motion, stick motion, sliding motion and grazing motion are presented for motion switchability. The switching sets and mapping structures are adopted to describe the complex motions in this discontinuous system. The numerical simulations are also carried out from the analytical conditions and mapping structures in order to better understand the motion switching complexity of this oscillator.

In this paper, by using flow switching theory of discontinuous dynamical systems, the discontinuous dynamic behavior of a 2-DOF (two-degree-of-freedom) friction oscillator with elastic impact and rigid impact on different sides is investigated. For the 2-DOF friction oscillator, when the direction of object’s velocity changes, the negative feedback works and the inequality of maximum static friction force and kinetic friction force leads to the existence of flow barriers. In addition, the elastic impact and rigid impact can change motion states of object, resulting in the discontinuity of this 2-DOF system. Considering the multiple motion states of object and better analyzing the equation of object’s motion, the phase space of each object is divided into different motion domains and boundaries in relative and absolute coordinates by means of the discontinuities resulted from the friction and impact. Moreover, because the elastic impact exists in a time period, there will be a variety of motion behaviors coexisting, such as elastic and rigid impacts occur simultaneously. With that in mind, in absolute coordinate system, the phase space of the system is divided in six cases according to whether there are stick motion and stuck motion. Based on flow switching theory, the switching conditions at discontinuous boundaries for all possible motions are given by using G-functions, and it should be noted that the vanishing conditions of the sliding motion become more complicated due to the existence of flow barriers at speed boundaries. Through defining the switching sets on separation boundaries, the mapping structures are further introduced to demonstrate the global dynamic behavior and periodic motions. For a better understanding of the object’s motion and the switching criteria at separation boundaries, the numerical simulation method is used to demonstrate several typical motions such as passable motion, sliding motion, grazing motion, stick motion (elastic impact), rigid impact and periodic motions etc. The results obtained have reference value for the optimization design and noise control of mechanical systems with gap couplings and negative feedbacks.

We investigate the dynamical characteristics of a two degrees of freedom friction oscillator with constrained spring cushions. Based on the discontinuity resulted from the rough contact surface and the nonsmoothness resulted from fixed spring, different boundaries and domains are given in phase space. The G-functions are introduced to determine whether the passable, grazing, sliding and stick motions occur on separation boundaries. The periodic orbits in such a system are predicted by using mapping dynamical theory after introducing four-dimension switching sets and basic mappings. The passable, grazing, stick, sliding and periodic motions of system are studied numerically by the velocity, displacement and G-functions responses in order to better illustrate the switching conditions of different motions. This investigation can provide effective technical support for the optimization design of system with fixed spring cushions.

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.

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.

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.

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.

A 2-DOF (two-degree-of-freedom) oscillator with an one-sided rigid obstacle in the presence of dry friction and external excitations is presented. Based on friction and the unilateral rigid barrier, the different motions for the masses, i.e., free-flight or nonstick, stick, stuck and impact motions, are discussed, and the different domains are obtained by partitioning the phase spaces. Then the analytical conditions are given for the switching of the above mentioned four motion states. By defining mapping structures and different generic mappings, the periodic motions of this system are developed. Numerical simulations are used to study the grazing, stick, nonstick, stuck and specified periodic motions. The investigation on such system is of great significance to the optimization design of engineering systems with one-sided rigid obstacle.

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 M through the analysis of the corresponding vector fields and G-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.

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.

We consider the existence of positive solutions of nonlinearp-Laplaciandynamic equations with derivative on time scales. Applying the Avery-Peterson fixed point theorem, we obtain at least three positive solutions to the problem. An example is also presented to illustrate the applications of the obtained results.

In this paper. we consider the following Cauchy problem of first order non-linear integro-differential equation of Volterra type (1) x ˙ = H ( t , x , T x ) . x ( t 0 ) = x 0 where (2) ( T x ) ( t ) = ∫ t 0 t K ( t . s ) x ( s ) d s , x ∈ C [ J . Ω ] K ∈ C [ J × J . R ] . | K ( t . s ) | , ≤ k ( k > 0 ) . ( t . s ) ∈ J × J , H ∈ C [ J × Ω × Ω 1 . E ] . J = [ t 0 . t 0 + α ] ( α > ) . Ω = B ( 0. N ) = { x ∈ E : | | x | | N } ( N > 0 ) . Ω 1 = B ( 0. k α N ) . x 0 ∈ Ω . E being real Banach space. The problem (1) is equivalent to the following equation (3) x ( t ) = x 0 + ∫ t 0 t H ( s . x ( s ) , ( T x ) ( s ) ) d s We study the toplogical properties of the problem (3)-continuum. The obtained result is the extention of the results of the ordinary differential equation and the reference [1].