Discontinuous dynamics of a non-linear, self-excited, friction-induced, periodically forced oscillator

The discontinuous dynamics of a non-linear, friction-induced, periodically forced oscillator is studied. The analytical conditions for motion switchability at the velocity boundary in such a nonlinear oscillator are developed to understand the motion switching mechanism. Using such analytical conditions of motion switching, numerical predictions of the switching scenarios varying with excitation frequency and amplitude are carried out, and the parameter maps for specific periodic motions are presented. Chaotic and periodic motions are illustrated through phase planes and switching sections for a better understanding of motion mechanism of the nonlinear friction oscillator. The periodic motions with switching in such a nonlinear, frictional oscillator cannot be obtained from the traditional analysis (e.g., perturbation and harmonic balance method).     

Theoretical model of laminar flow in tubes in rolling motion

The theoretical model of laminar flow in tubes in rolling motion is established. The velocity and temperature correlations are derived, and the frictional resistance coefficient and Nusselt number are also obtained. The oscillation of parameters is induced by the tangential force due to rolling motion. The effect of centrifugal and Coriolis forces on the flow is negligible. The tangential force does not effect on the average parameters. The oscillating amplitude of Nusselt number increase with the Prandtl number increasing. Both the oscillating amplitudes of frictional resistance coefficient and Nusselt number increase with the rolling frequency increasing.     

Non-linear flexings in a periodically forced floating beam

This paper considers periodic flexing in a floating beam, in the presence of a small periodic forcing term. The beam is considered as a vibrating beam with the free-end boundary condition, in the presence of an additional restoring force due to flotation, which becomes zero as soon as the beam lifts out of the water. The equation is therefore non-linear. A theorem is proved which shows that in the presence of small periodic forcing terms, both small- and large-amplitude solutions can exist. Numerical evidence is presented, which shows that the large-amplitude solutions are stable over a wide range of frequency and amplitude, and suggests a cusp-like surface for the multiple solutions.     

Asymptotic Solution of the Autoresonance Problem

We study the problem of forced oscillations near a stable equilibrium of a two-dimensional nonlinear Hamiltonian system of equations. A given exciting force is represented as rapid oscillations with a small amplitude and a slowly varying frequency. We study the conditions under which such a perturbation makes the phase trajectory of the system recede from the original equilibrium point to a distance of the order of unity. To study the problem, we construct asymptotic solutions using a small amplitude parameter. We present the solution for not-too-small values of time outside the original boundary layer.     

Effect of noise on the tuning properties of excitable systems

We analyze the effect of additive dynamical noise on simple phase-locking patterns in the Fitzhugh–Nagumo (FHN) two-dimensional system in the excitable regime. In the absence of noise, the response amplitude for this system displays a classical resonance as a function of driving frequency. This translates into V-shaped tuning curves, which represent the amplitude threshold for one firing per cycle as a function of forcing frequency. We show that noise opens up these tuning curves at mid-to-low frequencies. We explain this numerical result analytically using a heuristic form for the firing rate that incorporates the frequency dependence of the subthreshold voltage response. We also present stochastic phase-locking curves in the noise intensity-forcing period subspace of parameter space. The relevance of our findings for the tuning of electroreceptors of weakly electric fish and their encoding of amplitude modulations of high frequency carriers are briefly discussed. Our study shows that, in certain contexts, it is essential to take into account the frequency sensitivity of neural responses and their modification by sources of noise.     

The solution of three-dimensional friction contact problems

A variational method is developed for solving friction contact problems, in which the friction obeys Coulomb's of friction law in velocities, and numerical solutions of three-dimensional problems of the contact of a sphere, a cylinder of finite length and a cube with an elastic half-space are constructed. It is established that the maximum frictional forces correspond to a boundary point of the regions of adhesion and slippage. When the number of steps,increase this maximum decreases, and the distribution of the frictional forces becomes smoother. Certain undesirable effects that can arise during numerical implementation of the method – numerical artefacts – are described. These effects can occur in the numerical solution of problems with a different physical content, the mathematical structure of which is similar to the structure of the contact problems investigated, as the artefacts are caused by the presence of unilateral constraints and by the dependence on external effects of the region in which unilateral constraints with an equally sign occur. This problem is solved by an appropriate choice of the load-step zero approximations.     

Quasi-periodic solutions and periodic bursters in quasiperiodically driven oscillators

In this paper, we propose a perturbation method to determine an approximation and conditions of existence of quasi-periodic (QP) solutions and bursting dynamics in a quasi-periodically driven system. The QP forcing consists of two periodic excitations, one with a very slow frequency and the other with a frequency of the same order of the proper frequency of the oscillator. A first averaging is done over the fast dynamics, then the quasi-static solutions of the modulation equations of amplitude and phase are determined and their stability analyzed. We show that a necessary condition for the occurrence of periodic bursters is that the slow excitation is parametric.     

Internal Kelvin waves in a two-layer liquid model

The subinertial internal Kelvin wave solutions of a linearized system of the ocean dynamics equations for a semi-infinite two-layer f-plane model basin of constant depth bordering a straight, vertical coast are imposed. A rigid lid surface condition and no-slip wall boundary condition are imposed. Some trapped wave equations are presented and approximate solutions using an asymptotic method are constructed. In the absence of bottom friction, the solution consists of a frictionally modified Kelvin wave and a vertical viscous boundary layer. With a no-slip bottom boundary condition, the solution consists of a modified Kelvin wave, two vertical viscous boundary layers, and a large cross-section scale component. The numerical solutions for Kelvin waves are obtained for model parameters that take account of a joint effect of lateral viscosity, bottom friction, and friction between the layers.     

A class of optimal control problems for piezoelectric frictional contact models

We consider control problems for a mathematical model describing the frictional bilateral contact between a piezoelectric body and a foundation. The material’s behavior is modeled with a linear electro–elastic constitutive law, the process is static and the foundation is assumed to be electrically conductive. Both the friction and the electrical conductivity conditions on the contact surface are described with the Clarke subdifferential boundary conditions. The weak formulation of the problem consists of a system of two hemivariational inequalities. We provide the results on existence and uniqueness of a weak solution to the model and, under additional assumptions, the continuous dependence of a solution on the data. Finally, for a class of optimal control problems and inverse problems, we prove the existence of optimal solutions.     

Dynamics of a two-degrees-of-freedom system with friction and heat generation

A novel thermomechanical model of frictional self-excited stick-slip vibrations is proposed. A mechanical system consisting of two masses which are coupled by an elastic spring and moving vertically between two walls is considered. It is assumed that between masses and walls a Coulomb friction occurs, and stick-slip motion of the system is studied. The applied friction force depends on a relative velocity of the sliding bodies. Stability of stationary solutions is considered. A computation of contact parameters during heating of the bodies is performed. The possibility of existence of frictional auto-vibrations is illustrated and discussed.     

Wavelength selection in convection near a lateral boundary

^** Corresponding author. This paper considers the way in which a lateral boundary restrictsthe wavelength of convective rolls parallel to the boundary.Unlike previous theories of wave number selection, the presentwork allows for perpendicular cross rolls adjacent to the boundary,the existence of which is indicated by stability considerations.The present theory also incorporates forcing at the boundary,equivalent to a wall that is imperfectly insulated, as in manyexperimental investigations of Rayleigh–Bénardconvection. The results are based on weakly non-linear solutionsof amplitude equations derived from the Swift–Hohenbergmodel and are compared with numerical solutions of the fullnon-linear Swift–Hohenberg equation.     

Bifurcation and chaos in a discrete genetic toggle switch system

A discrete genetic toggle switch system obtained by Euler method is first investigated. The conditions of existence for fold bifurcation and flip bifurcation are derived by using center manifold theorem and bifurcation theory. The numerical simulations, including bifurcation diagrams, phase portraits, and computation of Lyapunov exponents, not only show the consistence with the theoretical analysis but also exhibit the rich and complex dynamical behavior. We show the period 3 to 13 windows in different chaotic regions, period-doubling bifurcation or inverse period-doubling bifurcation from period-2 to 12 orbits leading to chaos, different kind of interior crisis and boundary crisis, intermittency behavior, chaotic set, chaotic non-attracting set, coexistence of period points with invariant cycles, and so on. The influence of the amplitude and frequency of excitable forcing on the system are also first considered by using numerical simulation. A different type of quasiperiodic orbits, jumping behaviors of quasiperiodic set from one set to another set, and the processes from quasiperiodic orbits to strange non-chaotic attractor are found.     

Analysis of vibrations in large flexible hybrid systems

The mathematical model of a real flexible elastic system with distributed and discrete parameters is considered. It is a partial differential equation with non-classical boundary conditions. Complexity of the boundary conditions makes it impossible to find exact analytical solutions. To address the problem, we use the asymptotical method of small parameters together with the numerical method of normal fundamental systems of solutions. These methods allow us to investigate vibrations, and a technique for determination of complex eigenvalues of the considered boundary value problem is developed. The conditions, at which vibration processes of different characteristics take place, are defined. The dependence of the vibration frequencies on the physical parameters of the hybrid system is studied. We show that introduction of different feedbacks into the system allows one to control the frequency spectrum, in which excitation of vibrations is possible.     

Gauss色噪声激励下含黏弹力摩擦系统的随机响应分析

研究了Gauss色噪声激励下含黏弹力、弱非线性阻尼的摩擦振子的随机响应.将适用于光滑系统的随机平均法推广到了非光滑摩擦系统,进而得到系统振幅、位移及速度的稳态概率密度函数.同时结合材料的黏弹性,研究了摩擦力和Gauss色噪声对系统响应的影响.研究表明,摩擦力、黏弹力及噪声项的相关参数均可引起随机P-分岔,并且在一定范围内系统响应对摩擦力极为敏感.此外,理论结果与Monte Carlo 模拟结果吻合较好,验证了方法的有效性.     

An electro-viscoelastic frictional contact problem with damage

We consider a quasistatic frictional contact problem between a piezoelectric body and a foundation. The contact is modeled with normal compliance and friction is modeled with a general version of Coulomb's law of dry friction; the process is quasistatic and the material's behavior is described by an electro-viscoelastic constitutive law with damage. We derive a variational formulation for the model which is in the form of a system involving the displacement field, the electric potential field, and the damage field. Then we provide the existence of a unique weak solution to the model. The proof is based on arguments of evolutionary variational inequalities and fixed point.     

Analysis of lumped models with contact and friction

We consider two mathematical models that describe the vibrations of spring-mass-damper systems with contact and friction. In the first model, both the contact and frictional boundary conditions are described with subdifferentials of nonconvex functions. In the second model, the contact is modeled with a Lipschitz continuous function, and the restitution force is described by a differential equation involving a Volterra integral term. The two models lead to second-order differential inclusions with and without an integral term, in which the unknowns are the positions of the masses. For each model, we prove the existence of a solution by using an abstract result for first-order differential inclusions in finite dimensional spaces. For the second model, in addition, we prove the uniqueness of the solution by using a fixed point argument. Finally, we provide examples of systems with contact and friction conditions for which our results are valid.     

Shape optimization for Stokes problem with threshold slip

We study the Stokes problems in a bounded planar domain Ω with a friction type boundary condition that switches between a slip and no-slip stage. Our main goal is to determine under which conditions concerning the smoothness of Ω solutions to the Stokes system with the slip boundary conditions depend continuously on variations of Ω. Having this result at our disposal, we easily prove the existence of a solution to optimal shape design problems for a large class of cost functionals. In order to release the impermeability condition, whose numerical treatment could be troublesome, we use a penalty approach. We introduce a family of shape optimization problems with the penalized state relations. Finally we establish convergence properties between solutions to the original and modified shape optimization problems when the penalty parameter tends to zero.     

Driven response of time delay coupled limit cycle oscillators

We study the periodic forced response of a system of two limit cycle oscillators that interact with each other via a time delayed coupling. Detailed bifurcation diagrams in the parameter space of the forcing amplitude and forcing frequency are obtained for various interesting limits using numerical and analytical means. In particular, the effects of the coupling strength, the natural frequency spread of the two oscillators and the time delay parameter on the size and nature of the entrainment domain are delineated. For an appropriate choice of time delay, synchronization can occur with infinitesimal forcing amplitudes even at off-resonant driving. The system is also found to display a nonlinear response on certain critical contours in the space of the coupling strength and time delay. Numerical simulations with a large number of coupled driven oscillators display similar behavior. Time delay offers a novel tuning knob for controlling the system response over a wide range of frequencies and this may have important practical applications.     

A class of dynamic frictional contact problems governed by a system of hemivariational inequalities in thermoviscoelasticity

In this paper we prove the existence and uniqueness of the weak solution for a dynamic thermoviscoelastic problem which describes frictional contact between a body and a foundation. We employ the nonlinear constitutive viscoelastic law with a long-term memory, which includes the thermal effects and considers the general nonmonotone and multivalued subdifferential boundary conditions for the contact, friction and heat flux. The model consists of the system of the hemivariational inequality of hyperbolic type for the displacement and the parabolic hemivariational inequality for the temperature. The existence of solutions is proved by using recent results from the theory of hemivariational inequalities and a fixed point argument.     

Analysis of lumped models with contact and friction

We consider two mathematical models that describe the vibrations of spring-mass-damper systems with contact and friction. In the first model, both the contact and frictional boundary conditions are described with subdifferentials of nonconvex functions. In the second model, the contact is modeled with a Lipschitz continuous function, and the restitution force is described by a differential equation involving a Volterra integral term. The two models lead to second-order differential inclusions with and without an integral term, in which the unknowns are the positions of the masses. For each model, we prove the existence of a solution by using an abstract result for first-order differential inclusions in finite dimensional spaces. For the second model, in addition, we prove the uniqueness of the solution by using a fixed point argument. Finally, we provide examples of systems with contact and friction conditions for which our results are valid.