Stick Motion Research Articles

Numerical analysis of stick–slip induced nonlinear vibration and acoustic responses of composite laminated plates with friction boundaries

This paper is concerned with the development of a numerical model for predicting the nonlinear structural and acoustic responses of a composite laminated plate induced by stick–slip motions. The structural model of the plate is formulated using a higher-order shear deformation zigzag theory for accommodating the analysis of both thin and relatively thick composite plates. A phenomenological macro-slip friction model is adopted to describe the stick and slip motions of the friction boundaries. The nonlinear dynamics model of the composite plate is established based on a finite element method, and the exterior acoustic fluid is modeled using a time-domain boundary element method. The nonlinear vibro-acoustic responses of the plate are calculated by considering the compatibility conditions on the common interface of the plate and the fluid. The role of the frictional forces in the vibration and sound radiation of the plate is explained. It is found that the periodical distortions of the sound pressure waves are mainly due to the stick-to-slip transitions of the friction boundary other than the slip-to-stick transitions. The effects of the preload, friction coefficient and stiffness of the boundary on the nonlinear structural and acoustic responses of composite plates are examined.

Analysis of discontinuous dynamics of a 2-DOF system with constrained spring cushions

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.

Analysis of dynamical behaviors of a 2-DOF friction oscillator with elastic impacts and negative feedbacks

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.

Suppressing the backward motion of a stick–slip piezoelectric actuator by means of the sequential control method (SCM)

For stick–slip piezoelectric actuators working under the friction-inertia principle, the backward motion is nearly unavoidable especially at low driving frequency, which is detrimental to their driving performances. Accordingly, to suppress the backward motion of stick–slip piezoelectric actuators, this paper proposed a novel sequential control method (SCM). To verify the feasibility of SCM, an actuator prototype was designed and fabricated accordingly, and its output performances were tested experimentally. In this prototype, two flexure hinges and piezoelectric stacks were employed for driving and lifting, respectively. Before the generation of backward motion, the whole driving flexure hinge can be lifted up to separate from the mover by the lifting flexure hinge. Comparative experiments of the actuator working under the SCM and traditional stick–slip principle showed that backward motion could be significantly suppressed under the SCM, and the output displacement curve showed good stepping characteristics, proving the effectiveness of the proposed SCM.

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.

Identification of the stick and slip motion between contact surfaces using artificial neural networks

The paper presents work related to nonlinear system parameters identification. The research is focused on systems with hysteretic stiffness characteristics. The identification procedure is developed with use of artificial neural networks. The presented method assumes two separate clusters of neural networks, which are supported by additional signal processing block. Such approach gives an advantage over the conventional identification methods due to its small restrictions. The validation process considers structural responses in time and frequency domains as well as the restoring force plane of the dynamic structure. First, verification of the identification method is performed on the numerical simulation of the system with hysteretic stiffness. Next, the identification of the real dynamic system with contact-related nonlinearity is carried out. The steel samples with contacting surfaces were used in the experiment. Electromagnetic shaker was used to excite the structure and enforce a relative shear motion between surfaces in contact. The system response was recorded using the Polytec laser vibrometer.

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 modern methods for determining the dynamic friction of bulk materials

The coefficient of dynamic friction is investigated in widespread gravitational flows of granular materials of numerous technological processes. The calculation of dynamic friction is necessary in the study of shear flows, the behavior of the sediment layer on an inclined plane, tracking the sticking motion, damping friction in structural dynamics and sliding resistance. The friction coefficients of particles against each other depends on many factors (a set of physical and mechanical properties of particles, chemical and fractional composition, the presence of a lubricant, oxide films, moisture and other conditions, such as volume, angle and height of falling particles), each of which separately difficult to count. The experimental determinations of the value of the coefficient of internal friction during pressing are the most reliable; however, such methods are not applicable in the course of analyzing mixtures in the mining and metallurgical industries due to the fact that bulk materials of this type are characterized by gross porosity and are prone to breaking during similar experiments. To obtain reliable results of mathematical and computer simulations, there is a need to evaluate the mechanics of flow of granular materials or bulk solids. There is a shortage of ways to predict these patterns, that require understanding and conducting extensive basic research. At present, several models are used to describe the connected state of materials (soils, sands, granules and powders): soil mechanics, hydromechanical (continuum mechanics), continuity hypothesis (phenomenological model). All existing methods are based on the average value of the characteristics of a large number of particles and can not be used to describe models with high fractional void volume, and as a result, do not describe each discrete particle in the composition of the flow of bulk materials. This article provides an overview of existing methods for assessing the dynamic friction of solids, their applicability in various situations, as well as the development trends of this scientific field.

On discontinuous dynamics of a 2-DOF oscillator with an one-sided rigid obstacle

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.

Yielding and jerky plasticity of tilt grain boundaries in high-temperature graphene

Graphene is well known for its extraordinary mechanical properties combining brittleness and ductility. While most mechanical studies of graphene focused on the strength and brittle fracture behavior, its ductility, plastic deformation, and the possible brittle-to-ductile transition, which are important for high-temperature mechanical performance and applications, still remain much less understood. Here the mechanical response and deformation dynamics of graphene grain boundaries are investigated through a phase field crystal modeling, showing the pivotal effects of temperature and local dislocation structure. Our results indicate that even at relatively high temperature (around 3350 K), the system is still governed by a brittle fracture and cracking dynamics as found in previous low-temperature experimental and atomistic studies. We also identify another type of failure dynamics with low-angle grain boundary disintegration. When temperature increases a transition to plastic deformation is predicted. The appearance of plastic flow at ultrahigh temperature, particularly the phenomenon of jerky plasticity, is attributed to the stick and climb-glide motion of dislocations around the grain boundary. The corresponding mechanism is intrinsic to two-dimensional systems, and governed by the competition between the driving force of accumulated local stress and the defect pinning effect, without the traditional pathways of dislocation generation needed in three-dimensional materials.

Acceleration behaviors of Fermi accelerator model excited by Van der Pol oscillator

In this paper, we investigate on dynamics of a generalized Fermi accelerator model with a moving wall described by a nonlinear Van der Pol oscillator. Different acceleration behaviors appear due to the discontinuity brought by impact and the stationary periodic excitation of the moving wall. Utilizing the theory of discontinuous dynamical systems, acceleration mechanism of the particle is studied and the analytical conditions for stick motion and grazing motion are obtained. By the use of generic mappings between different boundaries including stick and non-stick motions, periodic motions of the Fermi accelerator can be constructed, and corresponding local stability and bifurcations can be discussed according to mapping dynamics. Finally, different acceleration behaviors including periodic, chatter and stick motions are presented and illustrated.

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.

Nonlinear flutter of composite laminated panels with local non-smooth friction boundaries

This paper is concerned with the numerical analysis of nonlinear flutter of composite laminated panels with local non-smooth friction interfaces and exposed to supersonic airflow. The Reddy’s third-order shear deformation plate theory with zig-zag effects and the nonlinear von Kármán strains are employed to formulate the structural model of the panel. A series of macro-slip friction models are adopted to impose the friction boundaries, which can describe the non-smooth characteristics of the friction interface. The discretized governing equations of motion of the panel with non-smooth friction boundary are established using the nonlinear finite element method. The effects of the physical parameters of the friction interface including the normal force, the coefficient of friction and the length of the friction interface, on the flutter behaviors of the laminated panel are examined. It is found that the critical flutter dynamic pressure of the panel may increase due to the constraints of the friction boundary. As the dynamic pressure increases, an abrupt variation in the flutter response of the panel may appear due to the coexistence of the stick and slip motions of the friction interface. In addition, the friction boundary has little effects on the location of the maximum deformation of the panel.

Employment of hyper-cycloidal oscillatory motion for finding the coefficient of rolling friction. Part 2: Experimental investigation

The paper presents a method for finding the characteristics of rolling friction between two cylindrical external surfaces. By choosing the motion between two parallel cylinders it results that a point from one of the cylinders describes a hyper-cycloidal curve when rolling over the other cylinder. The study is focused on the friction in ball bearings. Parts of the ball bearings to be studied are directly involved in materializing the hyper-cycloidal contact. Thus, the immobile circles are materialized by the outer surface of the external ring and the mobile circles are materialized by the spherical surfaces of the balls. The motion of a laser stick attached to the pendulum is filmed. Analyzing the movie, the numerical time dependence of angular amplitude decrease is obtained. The concordance between the experimental and theoretical angular amplitude damping directs to the value of coefficient of rolling friction.

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 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.

Diversity and transition characteristics of sticking and non-sticking periodic impact motions of periodically forced impact systems with large dissipation

A forced vibration system with dual bodies and an endstop is considered in this paper. A large energy loss is considered as one of the mass blocks of the system hits the endstop, and the impact associated with the large energy loss is first assumed to be completely plastic. The plastic impact may bring about the occurrence of the sticking phase, which is equivalent to a change in the structure of the forced vibration system at a certain stage after the impact. The incidence relation between dynamical characteristics and model parameters is studied through the multi-target and multi-parameter collaborative simulation analysis for determining the reasonable matching range of parameters. Pattern types, occurrence regions, distribution regularities and bifurcation characteristics of periodic and subharmonic impact vibrations are presented on a series of parameter planes. The key features of Poincare mapping, associated with the plastic impacts, are primarily manifested in piecewise continuity caused by sliding bifurcation and grazing discontinuity induced by grazing bifurcation. Integrative effects of these two nonstandard bifurcations can bring about some abnormal transitions to occur. The large dissipation case associated with small collision recovery coefficient is briefly analyzed, and the induced mechanism of chatter-sticking motion and the incidence relation between the sticking characteristics and the restitution coefficient R are discussed. The nonstandard dynamic characteristics associated with the plastic impacts are further demonstrated by dynamic mechanical behaviors of two practical impact machines applied in engineering.

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.

Global behavior of a vibro-impact system with asymmetric clearances

A simple dynamic model of a vibro-impact system subjected to harmonic excitation with two asymmetric clearances is considered. The Semi-Analytical Method for getting periodic solutions of the vibro-impact system is proposed. Diversity and evolution of the fundamental periodic impact motions are analyzed. The formation mechanism of the complete chatting-impact periodic motion with sticking motion by the influence of gazing bifurcation is analyzed. The transitional law of periodic motions in the periodical inclusions area is presented. The coexistence of periodic motions and the extreme sensitivity of the initial value within the high frequency region are studied. The global distribution of the periodic and chaos motions of the system is obtained by the state-parameter space co-simulation method which very few have considered before. The distribution of the attractor and the corresponding attracting domain corresponding to different periodic motions are also studied.