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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 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.

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.

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.

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.