Periodic,almost periodic and chaotic motions of forced self-sustained oscillators

Steady motions of the Van der Pol oscillator and an oscillator with hysteresis are studied numerically in this paper. Some features of periodic, almost periodic and chaotic motions of forced self-sustained oscillators are investigated. This paper has been presented at the ICTAM XVI Lyngby.     

On the stability of nonlinear vibrations of coupled pendulums

The motion of two identical pendulums connected by a linear elastic spring is studied. The pendulums move in a fixed vertical plane in a homogeneous gravity field. The nonlinear problem of orbital stability of such a periodic motion of the pendulums is considered under the assumption that they vibrate in the same direction with the same amplitude. (This is one of the two possible types of nonlinear normal vibrations.) An analytic investigation is performed in the cases of small vibration amplitude or small rigidity of the spring. In a special case where the spring rigidity and the vibration amplitude are arbitrary, the study is carried out numerically. Arbitrary linear and nonlinear vibrations in the case of small rigidity (the case of sympathetic pendulums) were studied earlier [1, 2].     

Finite motions from periodic frameworks with added symmetry

Recent work from authors across disciplines has made substantial contributions to counting rules (Maxwell type theorems) which predict when an infinite periodic structure would be rigid or flexible while preserving the periodic pattern, as an engineering type framework, or equivalently, as an idealized molecular framework. Other work has shown that for finite frameworks, introducing symmetry modifies the previous general counts, and under some circumstances this symmetrized Maxwell type count can predict added finite flexibility in the structure.In this paper we combine these approaches to present new Maxwell type counts for the columns and rows of a modified orbit matrix for structures that have both a periodic structure and additional symmetry within the periodic cells. In a number of cases, this count for the combined group of symmetry operations demonstrates there is added finite flexibility in what would have been rigid when realized without the symmetry. Given that many crystal structures have these added symmetries, and that their flexibility may be key to their physical and chemical properties, we present a summary of the results as a way to generate further developments of both a practical and theoretic interest.     

On periodic motions in magnetohydrodynamics

Summary A uniqueness theorem and an existence theorem for motions asymptotically stable in the mean and periodically depending on time, are demonstrated in magnetohydrodynamics, on the hypothesis that a motion sufficiently regular and asymptotically stable in the mean exists. In particular, similar theorems for steady motions are deduced. The procedure used is also a simple method for constructing periodic and steady motions.

---

Sommario Si dimostra un teorema di unicità ed un teorema di esistenza per i moti magnetoidrodinamici periodicî nel tempo ed asintoticamente stabili in media, subordinatamente all'ipotesi che esista un moto asintoticamente stabile in media sufficientemente regolare, e si deducono, in particolare, analoghi teoremi per i moti stazionari. Il procedimento adoperato costituisce anche un semplice metodo per la effettiva costruzione dei moti periodici e dei moti stazionari.

---

---

This work was done in the sphere of activity of the C.N.R. groups for mathematical research and, in Italian, has already been presented for publication in Rend. Acc. Sc. Fis. Mat., Napoli.     

Extensional motions of spatially periodic lattices

The behavior of microrheological models for multiphase fluids that have spatially periodic structure depends on certain kinematic properties of the unit cell. Anomalous results associated with identical objects approaching too closely during the flow can be reduced if not eliminated by satisfying lattice compatibility conditions. This is straightforward for simple shearing flow but subtle for extensional flows. Using the connection between lattice compatibility and lattice reproducibility (periodic lattice behavior with the flow) we establish sufficient conditions for compatibility of arbitrary lattices in planar extensional flow. Detailed results for square and hexagonal unit cells include: initial orientations for periodic behavior; strain periods; and minimum lattice spacings D. We identify the orientation of a square unit cell that leads to periodic behavior (with the minimum period) and the largest D of any lattice in planar extensional flow. We show that no lattice exhibits periodic behavior in uniaxial extensional flow (or biaxial extensional flow) even though Adler & Brenner have established the existence of compatibility.     

Stability of periodic motions of quasilinear systems

The stability of linear and quasilinear systems with periodic coefficients is analyzed. The properties of stability are established in terms of matrix-valued Lyapunov functions. An algorithm is developed to set up matrix-valued Lyapunov functions for linear quasiperiodic systems. A numerical example is given to illustrate the application of the algorithm Translated from Prikladnaya Mekhanika, Vol. 44, No. 10, pp. 101–113, October 2008.     

On periodic motions of an orbital dumbbell-shaped body with a cabin-elevator

The motion of a dumbbell-shaped body (a pair of massive points connected with each other by a weightless rod along which the elevator, i.e., a third point, is moving according to a given law) in an attractive Newtonian central field is considered. In particular, such a mechanical system can be considered as a simplified model of an orbital cable system equipped with an elevator. The practically most interesting case where the cabin performs periodic ??shuttle??motions is studied. Under the assumption that the elevator mass is small compared with the dumbbell mass, the Poincaré theory is used to determine the conditions for the existence of families of system periodic motions analytically depending on the arising small parameter and passing into some stable radial steady-state motion of the unperturbed problem as the small parameter tends to zero. It is also proved that, for sufficiently small parameter values, each of the radial relative equilibria generates exactly one family of such periodic motions. The stability of the obtained periodic solutions is studied in the linear approximation, and these solutions themselves are calculated up to terms of the firstorder in the small parameter. The contemporary studies of the motion of orbital dumbbell systems apparently originated in Okunev??s papers [1, 2]. These studies were continued in [3], where plane motions of an orbit tether (represented as a dumbbell-shaped satellite) in a circular orbit were considered in the satellite approximation. In [4], in the case of equal masses and in the unbounded statement, the energy-momentum method was used to perform the dynamic reduction of the problem and analyze the stability of relative equilibria. A similar technique was used in [5], where, in contrast to the above-mentioned problems, the massive points were connected by an elastic spring resisting to compression and forming a dumbbell with elastic properties. Under such assumptions, the stability of radial configurations was investigated in that paper. The bifurcations and stability of steady-state configurations of a deformable elastic dumbbell were also studied in [6]. Various obstacles arising in the construction of orbital cable systems, in particular, the strong deformability of known materials, were discussed in [7]. In [8], the problem of orbital motion of a pair of massive points connected by an inextensible weightless cable was considered in the exact statement. In other words, it was assumed that a unilateral constraint is imposed on themassive points. The conditions of stability of vertical positions of the relative equilibria of the cable system, which were obtained in [8], can be used for any ratio of the subsatellite and station masses. In turn, these results agree well with the results obtained earlier in the studies of stability of vertical configurations in the case of equal masses of the system end bodies [3, 4]. One of the basic papers in the dynamics of three-body orbital cable systems is the paper [9]. The steady-state motions and their bifurcations and stability were studied depending on the elevator cabin position in [10].     

On equilibrium configurations and their stability for a system of two coupled pendulums

A mechanical system consisting of two identical mathematical pendulums connected by a linear spring is considered under the assumption that the pendulum suspension points lie on a horizontal straight line and the system is in a homogeneous gravitational field. The equilibrium configurations of this mechanical system and their stability are studied. The results are represented in the form of bifurcation diagrams.     

Nonlinear random coupled motions of structural elements with quadratic nonlinearities

The second-order closure method is used to analyze the nonlinear response of two-degree-of-freedom systems with quadratic nonlinearities. The excitation is assumed to be the sum of a deterministic harmonic component and a random component. The case of primary resonance of the second mode in the presence of a two-to-one internal (autoparametric) resonance is investigated. The method of multiple scales is used to obtain four first-order ordinary-differential equations that describe the modulation of the amplitudes and phases of the two modes. Applying the second-order closure method to the modulation equations, we determine the stationary mean and mean-square responses. For the case of a narrow-band random excitation, the results show that the presence of the nonlinearity causes multi-valued mean-square responses. The multi-valuedness is responsible for a jump phenomenon. Contrary to the results of the linear analysis, the nonlinear analysis reveals that the directly excited second mode takes a small amount of the input energy (saturates) and spills over the rest of the input energy into the first mode, which is indirectly excited through the autoparametric resonance.     

Universal motions for constrained simple materials

The problem of universal motions of a uniform and isotropic simple material subject to a generic isotropic internal constraint is studied. Complete results are achieved for motions with space-dependent strain invariants. As an application, free oscillations of an elastic Bell-constrained spherical shell are investigated.     

On the nonlinear interactions and existence of breathers in a chain of coupled pendulums

The paper considers a chain of linearly coupled pendulums. Continues first order system equations are treated via time and space multiple scale method which lead to nonlinear Schrödinger equation. Further investigations on the nonlinear Schrödinger equation detects systems responses in the form of propagated nonlinear waves as functions of their envelope and phases. This provides information about localization of nonlinear waves and their directions in space and time.     

Isolated large amplitude periodic motions of towed rigid wheels

This study investigates a low degree-of-freedom (DoF) mechanical model of shimmying wheels. The model is studied using bifurcation theory and numerical continuation. Self-excited vibrations, that is, stable and unstable periodic motions of the wheel, are detected with the help of Hopf bifurcation calculations. These oscillations are then followed over a large parameter range for different damping values by means of the software package AUTO97. For certain parameter regions, the branches representing large-amplitude stable and unstable periodic motions become isolated following an isola birth. These regions are extremely dangerous from an engineering point of view if they are not identified and avoided at the design stage. Research Group on Dynamics of Machines and Vehicles, Hungarian Academy of Sciences.     

动力刚化与多体系统刚—柔耦合动力学

首先指出当前柔性多体系统动力学的大量工程研究背景,在回顾柔性多体系统动力学研究进展后指出动力刚化的现象揭示了刚－柔耦合的零次建模方法的局限,认为进一步深入进行柔性多体系统刚－柔耦合动力学的研究是多体系统动力学研究的新阶段,文末提出了刚－柔耦合动力学的研究任务。     

Interaction between periodic disturbances and a turbulent jet

The results of a numerical and experimental investigation of the interaction between harmonic disturbances and a turbulent jet are presented. On the basis of large eddy simulation it is established that the narrow-band noise of a supersonic jet considerably increases, when the forcing amplitude amounts to thousandths and more of the total pressure of the flow within the nozzle. An analysis of the results of a laboratory experiment on the measurement of the longitudinal velocity spectra in the core of a low-velocity jet shows that the acoustic disturbances generated by a fan inside the nozzle lead to the generation of intense tonal hydrodynamic disturbances in the low-velocity jet.     

Bifurcation,periodic and chaotic motions of the modified equal width-Burgers (MEW-Burgers) equation with external periodic perturbation

The modified equal width-Burgers (MEW-Burgers) equation is introduced for the first time. The bifurcation behavior of the MEW-Burgers equation is studied. Considering an external periodic perturbation, the periodic and chaotic motions of the perturbed MEW-Burgers equation are investigated by using phase projection analysis, time series analysis, Poincaré section and bifurcation diagram. The strength (\(f_0\)) of the external periodic perturbation plays a crucial role in the periodic and chaotic motions of the perturbed MEW-Burgers equation.