Stability of the periodic motions of the vibro-impact systems

In many other papers a new direct method is developed for the study of vibro-impact system stability, a case in which the stability conditions are directly and rapidly obtained. This new method is stated and applied for systems with one or several degrees of freedom whose motion is analysed by Lagrange's equations. Thus, the equations in variations determined from the differential equations are considered, which by applying the boundary conditions lead by integration to the linearized equation in perturbations necessary to write the stability conditions. As example, is analysed in detail a particular case with one degree of freedom realised by the rod-crank mechanism. Also, a general system with two degrees of freedom is completely studied.     

Existence and stability of the grazing periodic trajectory in a two-degree-of-freedom vibro-impact system

This paper investigates the existence and stability of the grazing periodic trajectory in a two-degree-of-freedom vibro-impact system. The criterion for existence of grazing period-n motion is presented. A local analysis based on the discontinuity-mapping approach is employed to derive a normal form Poincaré mapping near the grazing trajectory. Based on the above approach, a condition of stability can be formulated, such that a grazing trajectory will be discontinuous if the condition is unfulfilled. The predicted grazing bifurcations are in agreement with the numerical results. In particular, comparison of the grazing bifurcation diagrams of the normal form Poincaré mapping and the simulation diagrams of the original differential equation illustrates the validity of the discontinuity-mapping approach.     

Chaotic dynamics of the vibro-impact system under bounded noise perturbation

In this paper, chaotic dynamics of the vibro-impact system under bounded noise excitation is investigated by an extended Melnikov method. Firstly, the Melnikov method in the deterministic vibro-impact system is extended to the stochastic case. Then, a typical stochastic Duffing vibro-impact system is given to application. The analytic conditions for occurrence of chaos are derived by using the random Melnikov process in the mean-square-value sense. In addition, the numerical simulations confirm the validity of analytic results. Also, the influences of interesting system parameters on the chaotic dynamics are discussed.     

On the sum of distances along a circle

The arc distance between two points on a circle is their geodesic distance along the circle. We study the sum of the arc distances determined by n points on a circle, which is a useful measure of the evenness of scales and rhythms in music theory. We characterize the configurations with the maximum sum of arc distances by a balanced condition: for each line that goes through the circle center and touches no point, the numbers of points on each side of the line differ by at most one. When the points are restricted to lattice positions on a circle, we show that Toussaint's snap heuristic finds an optimal configuration. We derive closed-form formulas for the maximum sum of arc distances when the points are either allowed to move continuously on the circle or restricted to lattice positions. We also present a linear-time algorithm for computing the sum of arc distances when the points are presorted by the polar coordinates.     

Long-time behavior of almost periodically forced parabolic equations on the circle

We study the long-time behavior of the skew-product semiflow generated by scalar reaction-diffusion equation on the circle with almost periodic forcing:

$u_t=u_{xx}+f(t,u,u_x),t>0,x\in S^1=\mathbb{R}/2\pi \mathbb{Z},$

where $f(t,u,u_x)$ is uniformly almost-periodic in t. Almost periodic environmental forcing exhibits the external effects which are roughly but not exactly periodic.Contrary to the time-periodic cases (for which any ω-limit set Ω can be embedded into a periodically forced circle flow), we show that, for almost-periodic forcing, the problem that whether Ω can be embedded into an almost-periodically forced circle flow is strongly related to the dimension of the center space $V^c(\mathrm{\Omega })$ associated with Ω. On the one hand, if $\dim V^c(\mathrm{\Omega })\le 1$ then Ω is either spatially-inhomogeneous or spatially-homogeneous; and moreover, any spatially-inhomogeneous Ω can be embedded into an almost periodically-forced circle flow. On the other hand, when $\dim V^c(\mathrm{\Omega })>1$, it is shown that the above embedding property cannot hold anymore. These reveal that for such system there are essential differences between time periodic forcing and non-periodic forcing.     

Modelling of the forced motions of an elastic beam using the method of integrodifferential relations

A variational approach to the numerical modelling of forced lateral motions of an Euler–Bernoulli elastic beam is developed for a number of linear boundary conditions using the method of integrodifferential relations. A class of linear boundary actions is considered. A family of quadratic functionals, connecting the displacement field of points of the beam with the bending-moment functions in the cross section and the momentum density is proposed. Variational formulations of the original initial-boundary value problem on the motion of the beam are given and the necessary conditions for the functionals introduced to be stationary are analysed. The integral and local quality characteristics of the admissible approximate solutions are determined. The relation between the variational problems, formulated for the beam model, with the classical Hamilton–Ostrogradskii variational principles is demonstrated. An algorithm for constructing approximate systems of ordinary differential equations is developed, the solution of which yields stationary (minimum) values of the functionals introduced on a specified set of displacement fields, moments and momenta. Examples of calculations of the displacements for an elastic beam and an analysis of the quality of the numerical solutions obtained are presented.     

Fractional dynamics in the trajectory control of redundant manipulators

Under the pseudoinverse control, robots with kinematical redundancy exhibit an undesirable chaotic joint motion which leads to an erratic behavior. This paper studies the complexity of fractional dynamics of the chaotic response. Fourier and wavelet analysis provides a deeper insight, helpful to know better the lack of repeatability problem of redundant manipulators. This perspective for the study of the chaotic phenomena will permit the development of superior trajectory control algorithms.     

Asymptotic properties of the motions of a heavy gyroscope with internal dissipation

The limiting motions of a heavy gyroscope, simulated by a system of rigid bodies, are considered when there is internal friction. The whole set of limiting motions is determined and the nature of their stability is studied in detail for cases when the carried body of the gyroscope has a) three degrees and b) one degree of freedom with respect to the supporting body. The results of an analysis of case a are extended to the motion of a gyroscope with a fluid filling. For case b, the values of the parameters are determined for which the gyroscope, apart from steady rotations, has unsteady limiting motions that are integrable motions in the special Bobylev-Steklov case.     

Exact solutions of the problem of the vibro-impact oscillations of a discrete system with two degrees of freedom

Non-smooth time transformations are used to investigate strongly non-linear periodic free oscillations of a vibro-impact system with two degrees of freedom. Allowance for the boundary conditions at collision times enables the singularities induced by these transformations to be eliminated. The smoothed equations of motion turn out to be linear. Investigation of the periodic solutions reveals vibro-impact states with one- an two-sided collisions, including localized states (only one of the masses experiences collisions with stopping devices), and their bifurcation structure.     

Offset approximation based on reparameterizing the path of a moving point along the base circle

This paper presents a novel algorithm for planar curve offsetting. The basic idea is to regard the locus relative to initial base circle, which is formed by moving the unit normal vectors of the base curve, as a unit circular arc first, then accurately to represent it as a rational curve, and finally to reparameterize it in a particular way to approximate the offset. Examples illustrated that the algorithm yields fewer curve segments and control points as well as C^1 continuity, and so has much significance in terms of saving computing time, reducing the data storage and smoothing curves entirely.     

Smoothness for the collision local time of two multidimensional bifractional Brownian motions

Let ${B^{{H_i},{K_i}}} = \{ B_t^{{H_i},{K_i}},t \ge 0\} ,{\rm{ }}i = 1,2$ be two independent, d-dimensional bifractional Brownian motions with respective indices H _i ∈ (0, 1) and K _i ∈ (0, 1]. Assume d ? 2. One of the main motivations of this paper is to investigate smoothness of the collision local time $${l_T} = \int_0^T {\delta (B_s^{{H_1},{K_1}} - B_s^{{H_2},{K_2}}} )ds,{\rm{ }}T > 0,$$ , where δ denotes the Dirac delta function. By an elementary method we show that l _T is smooth in the sense of Meyer-Watanabe if and only if min{H ₁ K ₁,H ₂ K ₂} < 1/(d + 2).     

Phase space methods and the Hamilton-Jacobi form of dynamics

A general analysis of the Hamilton-Jacobi form of dynamics motivated by phase space methods and classical transformation theory is presented. The connection between constants of motion, symmetries, and the Hamilton-Jacobi equation is described.     

Analysis of the set of trajectories of nonlinear dynamics: Stability and boundedness of motions

For a set of differential equations with the Hukuhara derivative, we obtain sufficient conditions for various types of boundedness of trajectories and stability of the set of stationary solutions. To this end, we use scalar and vector Lyapunov functions constructed on the basis of an auxiliary matrix-valued function.     

On the dynamics of a periodic Colpitts oscillator forced by periodic and chaotic signals

In this paper, we have examined effects of forcing a periodic Colpitts oscillator with periodic and chaotic signals for different values of coupling factors. The forcing signal is generated in a master bias-tuned Colpitts oscillator having identical structure as that of the slave periodic oscillator. Numerically solving the system equations, it is observed that the slave oscillator goes to chaotic state through a period-doubling route for increasing strengths of the forcing periodic signal. For forcing with chaotic signal, the transition to chaos is observed but the route to chaos is not clearly detectable due to random variations of the forcing signal strength. The chaos produced in the slave Colpitts oscillator for a chaotic forcing is found to be in a phase-synchronized state with the forced chaos for some values of the coupling factor. We also perform a hardware experiment in the radio frequency range with prototype Colpitts oscillator circuits and the experimental observations are in agreement with the numerical simulation results.     

On the determinacy of motions for the displacement problem in the dynamics of elastic bodies with viscosity

Summary For a class of materials in which the stress depends upon the deformation and its rate, we show-without the aid of linearizing assumptions-that the solution to the displacement initial boundary-value problem is unique.

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Zusammenfassung Für eine Klasse von Materialien, deren Spannung von ihrer Verformung und der Geschwindigkeit derselben abhängt, zeigen wir-ohne die Hilfe von Linearisierungsannahmen- dass die Lösung des Anfangs- und Randwertproblems für die Verschiebungen eindeutig ist.