On the instability of steady motion

This paper deals with the instability of steady motions of conservative mechanical systems with cyclic coordinates. The following are applied: Kozlov’s generalization of the first Lyapunov’s method, as well as Rout’s method of ignoration of cyclic coordinates. Having obtained through analysis the Maclaurin’s series for the coefficients of the metric tensor, a theorem on instability is formulated which, together with the theorem formulated in Furta (J. Appl. Math. Mech. 50(6):938–944, 1986), contributes to solving the problem of inversion of the Lagrange-Dirichlet theorem for steady motions. The cases in which truncated equations involve the gyroscopic forces are solved, too. The algebraic equations resulting from Kozlov’s generalizations of the first Lyapunov’s method are formulated in a form including one variable less than was the case in existing literature.     

On the stability of equilibria of nonholonomic systems with nonlinear constraints

Lyapunov＇s first method, extended by V. V. Kozlov to nonlinear mechani- cal systems, is applied to the study of the instability of the position of equilibrium of a mechanical system moving in the field of conservative and dissipative forces. The mo- tion of the system is limited by ideal nonlinear nonholonomic constraints. Five cases determined by the relationship between the degree of the first nontrivial polynomials in Maclaurin＇s series for the potential energy and the functions that can be generated from the equations of nonlinear nonholonomic constraints are analyzed. In the three eases, the theorem on the instability of the position of equilibrium of nonholonomic systems with linear homogeneous constraints （V. V. Kozlov （1986）） is generalized to the case of nonlin- ear nonhomogeneous constraints. In the other two cases, new theorems are set extending the result from V. V. Kozlov （1994） to nonholonomic systems with nonlinear constraints.     

Stability for manifolds of equilibrium states of generalized Hamiltonian systems

For a generalized Hamiltonian system, stability for the manifolds of equilibrium states is presented based on Lyapunov’s stability theories. Equilibrium equations, perturbation equations and first approximate equations of the system are given. A theorem for the stability of manifolds of equilibrium states of general autonomous system is used to the generalized Hamiltonian system, and three propositions on the stability of manifolds of equilibrium states of the system are obtained. Two examples are given to illustrate application of the method and results.     

Studying the stability of equilibrium solutions in a planar circular restricted four-body problem

The Newtonian circular restricted four-body problem is considered. We obtain nonlinear algebraic equations determining equilibrium solutions in the rotating frame and find six possible equilibrium configurations of the system. Studying the stability of equilibrium solutions, we prove that the radial equilibrium solutions are unstable, while the bisector equilibrium solutions are stable in Lyapunov’s sense if the mass parameter satisfies the conditions μ ∈ (0, μ₀, where μ₀ is a sufficiently small number, and μ ≠ μ_j, j = 1, 2, 3. We also prove that, for μ = μ₁ and μ = μ₃, the resonance conditions of the third order and the fourth order, respectively, are satisfied and, for these values of μ, the bisector equilibrium solutions are unstable and stable in Lyapunov’s sense, respectively. All symbolic and numerical calculations are done with the Mathematica computer algebra system. Published in Neliniini Kolyvannya, Vol. 10, No. 1, pp. 66–82, January–March, 2007.     

Stability and vibration of a helical rod constrained by a cylinder

The stability and vibration of an elastic rod with a circular cross section under the constraint of a cylinder is discussed. The differential equations of dynamics of the constrained rod are established with Euler’s angles as variables describing the attitude of the cross section. The existence conditions of helical equilibrium under constraint are discussed as a special configuration of the rod. The stability of the helical equilibrium is discussed in the realms of statics and dynamics, respectively. Necessary conditions for the stability of helical rod are derived in space domain and time domain, and the difference and relationship between Lyapunov’s and Euler’s stability concepts are discussed. The free frequency of flexural vibration of the helical rod with cylinder constraint is obtained in analytical form. The project supported by the National Natural Science Foundation of China (10472067). The English text was polished by Yunming Chen.     

Impulsive stabilization of mechanical systems in Takagi–Sugeno models

Takagi–Sugeno fuzzy impulsive systems are analyzed for Lyapunov stability. Lyapunov’s second method is used to establish sufficient stability conditions for such systems. It is shown that these conditions are expressed by a system of matrix inequalities. Impulsive fuzzy control of two coupled pendulums is considered as an example     

Stability properties of equilibrium sets of non-linear mechanical systems with dry friction and impact

In this paper, we will give conditions under which the equilibrium set of multi-degree-of-freedom non-linear mechanical systems with an arbitrary number of frictional unilateral constraints is attractive. The theorems for attractivity are proved by using the framework of measure differential inclusions together with a Lyapunov-type stability analysis and a generalisation of LaSalle’s invariance principle for non-smooth systems. The special structure of mechanical multi-body systems allows for a natural Lyapunov function and an elegant derivation of the proof. Moreover, an instability theorem for assessing the instability of equilibrium sets of non-linear mechanical systems with frictional bilateral constraints is formulated. These results are illustrated by means of examples with both unilateral and bilateral frictional constraints.     

Dynamics of an AMB-rotor with time varying stiffness and mixed excitations

A rotor- active magnetic bearing (AMB) system with a periodically time-varying stiffness subjected to multi- external, -parametric and -tuned excitations is studied and solved. The method of multiple scales is applied to analyze the response of the two modes of the system near the simultaneous sub-harmonic, super-harmonic and combined resonance case. The stability of the steady state solution near this resonance case is determined and studied applying Lyapunov’s first method. Also, the system exhibits many typical nonlinear behaviors including multi-valued solutions, jump phenomenon, softening nonlinearities. The effects of the different parameters on the steady state solutions are investigated and discussed. Simulation results are achieved using MATLAB 7.0 program.     

The theory of stability of an orbiting observatory with gyroscopic stabilization of motion

The method of Lyapunov’s matrix functions is used to establish the stability conditions for a spacecraft. A control system with exesutive devices in the form of three gyroscopic frames orients the spacecraft in inertial space. Translated from Prikladnaya Mekhanika, Vol. 36, No. 5, pp. 131–138, May, 2000.     

Revisiting the theory of stability on time scales of the class of linear systemswith structural perturbations

Linear systems of dynamic equations with periodic coefficients and structural perturbations on time scale are analyzed for Lyapunov stability. Sufficient conditions for the asymptotic stability of the equations are established based on the matrix-value concept of Lyapunov’s direct method for all values of the structural matrix from the structural set. A system of two dynamic equations on time scale is considered as an example of applying the theoretical results obtained     

Dissipation-Induced Instability Phenomena in Infinite-Dimensional Systems

This paper develops a rigorous notion of dissipation-induced instability in infinite dimensions as an extension of the classical concept implicitly introduced by Thomson and Tait for finite degree of freedom mechanical systems over a century ago. Here we restrict ourselves to a particular form of infinite-dimensional systems—partial differential equations—whose inherent function-analytic differences from finite-dimensional systems make uncovering this notion more intricate. In building the concept of dissipation-induced instability in infinite dimensions we found Arnold’s and Yudovich’s nonlinear stability methods, for conservative and dissipative systems respectively, along with some new existence theory for solutions, to be the essential foundation. However, when proving the results for classical solutions, as motivated by their direct physical significance, we had to overcome a number of fundamental difficulties associated with existing stability analysis methods, which has led to new techniques. In particular, in this work we establish the connection of existence and general stability theories in strong and weak topologies and provide new insights into the physics and geometry of the dissipation-induced instability phenomena in infinite-dimensional systems. As a paradigm and the first infinite-dimensional example to be rigorously analyzed, we use a two-layer quasi-geostrophic beta-plane model, which describes the fundamental baroclinic instability in atmospheric and ocean dynamics; early formal linear approximate studies suggested that this system can be destabilized after the introduction of dissipation.     

Vibration reduction of a nonlinear spring pendulum under multi external and parametric excitations via a longitudinal absorber

The vibration of a ship pitch-roll motion described by a non-linear spring pendulum system (two degrees of freedom) subjected to multi external and parametric excitations can be reduced using a longitudinal absorber. The method of multiple scale perturbation technique (MSPT) is applied to analyze the response of this system near the simultaneous primary, sub-harmonic and internal resonance. The steady state solution near this resonance case is determined and studied applying Lyapunov’s first method. The stability of the system is investigated using frequency response equations. Numerical simulations are extensive investigations to illustrate the effects of the absorber and some system parameters at selected values on the vibrating system. The simulation results are achieved using MATLAB 7.0 programs. Results are compared to previously published work.     

Active damping of the vibrations of a plate using sensor data: effect of boundary conditions

A new approach is followed to study the effect of mixed mechanical boundary conditions on the effectiveness of active damping of the forced resonant vibrations of thermoviscoelastic orthotropic plates. The problem is solved by the Bubnov–Galerkin method. Formulas for the voltage that should be applied to the actuator to damp the first vibration mode are derived. It is shown that the mechanical boundary conditions, the dissipative properties of the material, and the dimenstions of the sensors and actuators have a strong effect on the effectiveness of active damping of the vibrations of plates     

Existence of Multi-Pulses of the Regularized Short-Pulse and Ostrovsky Equations

The existence of multi-pulse solutions near orbit-flip bifurcations of a primary single-humped pulse is shown in reversible, conservative, singularly perturbed vector fields. Similar to the non-singular case, the sign of a geometric condition that involves the first integral decides whether multi-pulses exist or not. The proof utilizes a combination of geometric singular perturbation theory and Lyapunov–Schmidt reduction through Lin’s method. The motivation for considering orbit flips in singularly perturbed systems comes from the regularized short-pulse equation and the Ostrovsky equation, which both fit into this framework and are shown here to support multi-pulses.     

Stability and Boundedness Results for Solutions of Certain Third Order Nonlinear Vector Differential Equations

In this paper, we investigate the asymptotic stability of the zero solution and boundedness of all solutions of a certain third order nonlinear ordinary vector differential equation. The results are proved using Lyapunov’s second (or direct method). Our results include and improve some well known results existing in the literature.     

Dissipation-induced instabilities and symmetry

The paradox of destabilization of a conservative or non-conservative system by small dissipation,or Ziegler’s paradox(1952),has stimulated a growing interest in the sensitivity of reversible and Hamiltonian systems with respect to dissipative perturbations.Since the last decade it has been widely accepted that dissipation-induced instabilities are closely related to singularities arising on the stability boundary,associated with Whitney’s umbrella.The first explanation of Ziegler’s paradox was given(much earlier)by Oene Bottema in 1956.The aspects of the mechanics and geometry of dissipation-induced instabilities with an application to rotor dynamics are discussed.     

Experimentally Evaluating Equilibrium Stress in Uniaxial Tests

Many models use the equilibrium stress, also sometimes known as the back stress, in characterizing the response of both polymeric and non-polymeric materials. We study the characteristics of the equilibrium and show that the tangent modulus and local Poisson’s ratio at equilibrium both are rate independent for common modeling assumptions. This fact is used to propose a method based on uniaxial tension or compression to measure the equilibrium stress, and the associated point’s tangent modulus and local Poisson’s ratio. The method is based on cyclic loading and identification of similar states with vastly different loading rates. The method is used to characterize the equilibrium stress in glassy polycarbonate, and the results are studied in regard to the possible error for such a measurement. The method is faster than most other proposed methods for calculating the equilibrium stress, and provides additional measurements of parameters at equilibrium that are normally not obtained.     

Convective instability of a plane horizontal layer of weakly conducting fluid in alternating and modulated electric fields

The electrothermoconvective instability of a plane horizontal layer of weakly conducting fluid in a modulated vertical electric field is investigated. The analysis is based on the electrohydrodynamic approximation. The stability threshold in the linear approximation is found using Floquet’s theory. The effect of periodic modulation on the fluid behavior is studied in both the presence and the absence of the constant component of the electric field. It is shown that modulation can stabilize the unstable ground state or destabilize fluid equilibrium, depending on the amplitude and frequency. In addition to a synchronous or subharmonic response to an external forcing, the instability may be associated with two-frequency (quasiperiodic) perturbations. The cases of weightlessness and a transversely stratified fluid in a static gravity field are considered. Madrid, Perm’. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 31–38, May–June, 2000. The investigations whose results are presented in this paper were supported by the Russian Foundation for Basic Research (project No. 98-01-00507).     

The minimum plane path for movable end-points and the nonsimultaneous variations

The paper deals with the problem of existence of the minimum path for movable end-points in the one-of-degree-of-freedom mechanical system. The criteria for obtaining of extremum path for movable end-points is extended with new criteria for minimum. The nonsimultaneous variational calculus is applied. It is assumed that the actual path belongs to sub-set C ² of admissible curves. The series expansion up to the second order small values is applied and the first and the second variation of functional are calculated. It is proved that the necessary and sufficient conditions for the minimum path are that the first order variation is zero and the second order variation is positive. The second conditions are based on the arbitrary solution of Riccati’s differential equation and also the known Legender’s and Jacobi criteria for minimum for the case of fixed end-points. Two examples are solved: the problem of the minimal length of a curve joining two fixed boundary curves and problem of motion of a particle between variable boundaries for which the Hamilton action integral is minimal.     

Active damping of vibrations of plates subject to unknown pressure

An approach to the active damping of the forced resonant vibrations of orthotropic thermoviscoelastic plates with distributed sensors and actuators is proposed. The mechanical load is assumed unknown and is determined from the sensors’ indications. The problem of active damping of an isotropic thermoviscoelastic rectangular plate with hinged edges is solved as an example. A formula for the voltage to be applied to the actuator to damp the forced vibrations in the first mode is derived. The effect of the dimensions of the sensor and actuator and the dissipative properties of the materials on the effectiveness of active damping is studied