International Journal of Bifurcation and Chaos

In this paper we study the tracking control of Lagrangian systems subject to frictionless unilateral constraints. The stability analysis incorporates the hybrid and nonsmooth dynamical feature of the overall system. The difference between tracking control for unconstrained systems and unilaterally constrained ones, is explained in terms of closed-loop desired trajectories and control signals. This work provides details on the conditions of existence of controllers which guarantee stability. It is shown that the design of a suitable transition phase desired trajectory, is a crucial step. Some simulation results provide information on the robustness aspects. Finally the extension towards the case of multiple impacts, is considered.

Complex vibrations of an Euler–Bernoulli beam with different types of nonlinearities are considered. An arbitrary beam clamping is considered, and deflection constraints (point barriers) are introduced in some beam points along its length. The influence of a constraint, as well as of the amplitude and frequency of excitation on the vibrations is analyzed. Scenarios of transition to chaos owing to the introduced nonlinearities are reported.

In this paper, a special nonsmooth transformation of time is combined with the shooting algorithm for visualization of the manifolds of periodic solutions and their bifurcations. The general class of nonlinear oscillators under smooth, nonsmooth, and impulsive loadings is considered. The corresponding boundary value problems with no singularities are obtained by introducing the periodic piecewise-linear (sawtooth) temporal argument. The Ueda circuit, that is Duffing's oscillator with no linear stiffness, is considered for illustration. It is shown that the temporal mode shape of the input can be responsible for qualitative features of the dynamics, such as transitions between the regular and random motions. The important role of unstable periodic orbits and their links with strange attractors are discussed.

A method for controlling nonlinear dynamics and chaos, previously developed by the authors, is applied to the rigid block on a moving foundation. The method consists in modifying the shape of the excitation in order to eliminate, in an optimal way, the heteroclinic intersections embedded in the system dynamics. Two different cases are examined: (i) generic block under small perturbations and (ii) slender block under generic perturbations, and they are investigated analytically either by a perturbation analysis (former case) or exactly (latter case). Two different strategies are proposed: (i) one-side control, which consists in eliminating the intersections of a single heteroclinic connection, and (ii) global control, which consists in simultaneously eliminating the intersections of both heteroclinic connections. The best excitations permitting the maximum distance between stable and unstable manifolds are determined in both cases. Finally, some numerical investigations aimed at highlighting meaningful aspects of system response under controlled (optimal) and noncontrolled (harmonic) excitations are performed.

The existence of a continuum of many chaotic solutions are shown for certain differential inclusions which are small nonautonomous multivalued perturbations of ordinary differential equations possessing homoclinic solutions to hyperbolic fixed points. Applications are given to dry friction problems.

A self-excited friction pair is modeled by a mechanical system of two degrees-of-freedom with a normal force varying during the block displacement. Two different friction coefficients are used, and a normal force pressing the sliding body to a belt depending on an angle bar motion of the analyzed system is applied. In addition, the numerical analysis is supported by the investigation of a real laboratory object modeling the feedback reinforcement of friction forces acting on the block. Both numerical and experimental results are compared showing good agreement.

The period-1 motion of a piecewise-linear system under a periodic excitation is predicted analytically through the Poincaré mapping and the corresponding mapping sections formed by the switch planes pertaining to the two constraints. The mapping relationship generates a set of nonlinear algebraic equations from which the period-1 motion is determined analytically. The stability and bifurcation of the period-1 motion are determined, and numerical simulations are carried out for confirmation of the analytical prediction of period-1 motion. An unsymmetrical stable period-1 motion is observed. This investigation helps us understand the dynamical behavior of period-1 motion in the piecewise-linear system and more efficiently obtain other periodic motions and chaos through numerical simulations. The similar methodology presented in this paper can be used for other nonsmooth dynamical systems.

Small, discrete, corrective adjustments to foot geometry in a class of bipedal passive walkers are employed to affect the local stability properties of a periodic reference gait. It is demonstrated that successful stabilization can be accomplished for otherwise strongly unstable motions of vertically constrained as well as entirely unconstrained model mechanisms. In particular, recent results by the authors on stabilization of repetitive motion in hybrid dynamical systems are implemented to formulate a rigorous analytical methodology for predicting the stability characteristics of the controlled system. It is demonstrated that these predictions can be explicitly obtained based entirely on knowledge of the local stability properties of the reference motion in the absence of control.

We consider a parametrically excited vibro-impact system consisting of two colliding pendula supported on a moving base. Between the impacts the dynamics of the system are described by a nonlinear Mathieu equation, while impacts are represented by an appropriate jump in the state variables, dependent on the coefficient of restitution e. The method of averaging is applied to these equations and a reduced system is developed in which several nontrivial impacting states are identified, including synchronous oscillations, a symmetric impacting state, and a nonsymmetric impacting state. For the symmetric response (including both the synchronous and symmetric impacting states), a two-dimensional map is developed from the averaged equations and for a purely elastic collision (e = 1), a nonlinear Mathieu equation is obtained — the impacts play no role in the amplitude of the response. For inelastic collisions (e ≠ 1) the dissipation induced by the impacts can be combined with viscous damping into an overall dissipation parameter to determine the equilibrium points of the mapping. Finally, for more general nonsymmetric behavior, the existence of the stable nonsymmetric impacting state can be closely tied to the coexistence of the nontrivial state and the origin as stable states in the symmetric response.

We investigate shock isolation designs based on nonlinear energy pumping caused by nonsmooth stiffness elements. In particular, we numerically study the shock isolation properties of a primary linear system of two coupled nonconservative oscillators with weakly coupled attachments possessing clearance nonlinearities. Under shock excitation the nonlinear attachments (termed nonlinear energy sinks — NESs) can be designed to absorb a significant portion of the input energy, thus enhancing the shock isolation performance of the primary system. In contrast to the classical linear vibration absorber whose operation is restricted to narrowband frequency ranges, the NESs are capable of efficiently absorbing energies caused by transient broadband disturbances, a feature that facilitates their implementation in practical applications. Moreover, the nonsmooth nonlinearities considered in this work are easily implementable since they are realized by means of linear stiffness elements.

The paper studies the dynamics of a stop-belt friction oscillator (SBO). The system consists of a visco-elastic oscillator which is dragged by a rough support moving with constant driving velocity and can collide with a rigid obstacle. The peculiarity of this system is that, in the phase space, it exhibits two different discontinuity boundaries. Nonsmooth responses are consequently observed with nonstandard attracting properties. The evolution of steady state limit sets as the position of the obstacle is varied is described. The nonsmoothness sets of the system and, in particular, the presence of discontinuity boundaries caused by friction and impacts, lead to nonstandard bifurcations which are studied here by means of analytical and numerical tools.

The plane dynamics of a rigid block simply supported on a harmonically moving rigid ground is a problem which still needs investigating, although the matter has been the subject of much research since the last century. Unilateral contacts, Coulomb friction and impacts make the system hybrid as it reveals a mixed continuous and discontinuous nature. Thus, stability analysis requires the extension and adaptation of concepts with regard to variational-perturbative procedures. In particular, discontinuous systems exhibit discontinuities or "saltations" in the fundamental solution matrix which must be analyzed carefully. In this paper, the adaptation of numerical methods that permit us to obtain characteristic multipliers and Lyapunov's exponents for the rocking mode of the block will be tackled. Analytical methods are used for the linearized equations of motion; the results are compared with those in the scientific literature.

This paper addresses two main paths of investigations. First, a new numerical method to trace regular and chaotic domains of any nonlinear system governed by ordinary differential equations is proposed. Second, the introduced approach is first testified using the well-known chaotic behavior of a Duffing oscillator and Lorenz system, and is then applied to analysis of discontinuous two-degree-of-freedom self-excited system with friction. Stick-slip and slip-slip chaos is reported, among others.

A single-degree-of-freedom system with one-sided rigid barrier at its equilibrium position is considered. The system is subject to excitation by a sinusoidal-in-time force with random white-noise temporal variations of the excitation frequency. Whilst response analysis had been made previously for the resonant case, i.e. one with expected excitation frequency being close to an even integer multiple of the natural frequency of the system without barrier, the nonresonant case is considered in this paper which is well known to be prone to multiple bifurcations and chaos in the case of perfectly periodic excitation. A simple qualitative explanation of the corresponding phenomenon of frequency multiplication, or "breeding", is provided for this case based on a special piecewise-linear transformation of state variables, which permits to get rid of velocity jump for the transformed velocity. The influence of the imperfect periodicity is then studied by numerical (Monte-Carlo) simulation. It is shown that increasing intensity of the white-noise variations of the excitation frequency, or bandwidth of the excitation power spectral density (p.s.d.), may greatly reduce frequency content of the response with simultaneous increase of the overall response level. The excitation/system bandwidth ratio is identified as a key governing parameter.