Integral variational principles in Poincaré and Chetayev variables

A holonomic mechanical system with k degrees of freedom is considered, its state being characterized by n k defining coordinates, p < k Poincaré parameters [1] and k - p Chetayev parameters [2]. In these variables, generalized Routh equations are introduced and expressions are given for the integral variational principles of Hamilton-Ostrogradskii and Hamilton (the third form), as well as Hölder's principle and the Lagrange and Jacobi versions of the principle of least action.     

Whittaker方程对非完整力学系统的推广

1904年Whittaker利用能量积分将一个完整保守力学系统问题降阶为一个带有较少自由度系统问题.并得到了Whittaker方程[1].本文推导对于非完整力学系统的这类方程.并称之为广义Whittaker方程;然后把这些方程变换为Nielsen形式;最后举例说明新方程的应用.     

On the orbital stability of a periodic solution of the equations of motion of a Kovalevskaya gyroscope

The sufficient conditions for the orbital stability of a periodic solution of the equations of motion of a Kovalevskaya gyroscope in the case of Bobylev-Steklov integrability are obtained.

It is difficult to expect Lyapunov stability for the unsteady motions of a heavy solid having a fixed point since a dependence of the vibrations frequency on the initial conditions is characteristic for the simplest of them, i.e. periodic motions /1/. Moreover, a rougher property of periodic solutions of the Euler-Poisson equations, orbital stability /2/, is not the subject of special investigations in the dynamics of a solid. The algorithm of the present investigation utilizes the treatment ascribed Zhukovskii /3/ of orbital stability as the Lyapunov stability of motion for a special selection of the variable playing the part of time (see /4/ also) and the Chetayev method /5/ of constructing Lyapunov functions from the first integrals of the equations of perturbed motion. This latter circumstance enables the Chetayev method to be put in one series with the methods used in /1, 4, 6–9/, etc.     

Qualitative analysis of systems with an ideal non-conservative constraint

The Hamiltonian form developed in /1/ for the equations of motion of systems with ideal non-conservative constraints enables familiar methods of classical and celestial mechanics to be used to analyse the dynamics of such systems. When this is done certain difficulties arise, due to the fact that the Hamiltonian is not analytic. In this paper one of the possible algorithms applying KAM theory /2/ and Poincaré's theory of periodic motions /3/ to the analysis of systems in which the Hamiltonian is non-analytic in one of the phase variables is described. As an example, some results of /4/ concerning the dynamics of a rigid body colliding with a fixed, absolutely smooth, horizontal plane are refined.     

A note on the disturbed equations of motion in the non-holonomic coordinates

Summary The aim of this article is to establish the disturbed equations of motion in the non-holonomic coordinates. The governing differential equations of non disturbed motion are the Hamel-Boltzman equations. The disturbed equations obtained in this article can be used for consideration in holonomic and non-holonomic dynamics systems. Entrata in Redazione 1'8 maggio 1972.     

Horseshoes in chromosome’s attractors

In this paper, we study dynamics of a class of chromosome’s attractors. We show that these chromosome sequences are chaotic by giving a rigorous verification for existence of horseshoes in these systems. We prove that the Poincaré maps derived from these chromosome’s attractors are semi-conjugate to the 2-shift map, and its entropy is no less than log 2. The chaotic behavior is robust in the following sense: chaos exists when one parameter varies from −5.5148 to −5.4988.     

非完整系统分析动力学中的几个重要问题

本文从变分原理和分析约束的力学性质两个方面入手,首次用演绎法推导出Chetaev条件,并且进行了验证,指出认为对非完整系统分析动力学d-δ交换性不成立的观点实际上是一种误解.在此基础上,首次提出非完整系统分析动力学中的两个经典关系.最后,进一步讨论了积分变分原理应用于非完整系统的问题.     

Reduction of the equations of dynamics of systems with constraints to a given structure

The problem of constructing systems of second-order ordinary differential equations, the solutions of which, with the appropriate initial conditions, satisfy given equations of the constraints, is considered. The conditions for representing the differential equations in the form of Lagrange equations of the second kind are determined. It is shown that, when the equations of the non-holonomic constraints are specified by polynomials of order no higher than two with respect to the generalized velocities, the generalized forces of a system with energy dissipation comprise the sum of the gyroscopic, potential and dissipative forces.     

Horseshoe in a two-scroll control system

In this paper we study a two-scrolls control system and give a rigorous verification for existence of chaos in this system. We show that the dynamics of the Poincaré map derived from the ordinary differential equations of this two-scrolls control system is semiconjugate to the dynamics of 4-shift map.     

Regular self-oscillating and chaotic behaviour of a PID controlled gimbal suspension gyro

The dynamics of a gyro in gimbal with a PID controller to obtain steady state, self-oscillating and chaotic motion is considered in this paper. The mathematical model of the whole system is deduced from the gyroscope nutation theory and from a feedback control system formed by a PID controller with constrained integral action. The paper shows that the gyro and the associated PID feedback control system have multiple equilibrium points, and from the analysis of a Poincaré–Andronov–Hopf bifurcation at the equilibrium points, it is possible to deduce the conditions, which give regular and self-oscillating behaviour. The calculation of the first Lyapunov value is used to predict the motion of the gyro in order to obtain a desired equilibrium point or self-oscillating behaviour. The mechanism of the stability loss of the gyro under small vibrations of the gyro platform and the appearance of chaotic motion is also presented. Numerical simulations are performed to verify the analytical results.     

Generalized solutions of initial-boundary value problems for second-order hyperbolic systems

The method of boundary integral equations is developed as applied to initial-boundary value problems for strictly hyperbolic systems of second-order equations characteristic of anisotropic media dynamics. Based on the theory of distributions (generalized functions), solutions are constructed in the space of generalized functions followed by passing to integral representations and classical solutions. Solutions are considered in the class of singular functions with discontinuous derivatives, which are typical of physical problems describing shock waves. The uniqueness of the solutions to the initial-boundary value problems is proved under certain smoothness conditions imposed on the boundary functions. The Green’s matrix of the system and new fundamental matrices based on it are used to derive integral analogues of the Gauss, Kirchhoff, and Green formulas for solutions and solving singular boundary integral equations.     

Reconnection in a global model of Poincaré map describing dynamics of magnetic field lines in a reversed shear tokamak

Magnetic field lines behaviour in a reversed shear tokamak can be described by a one and a half degree of freedom Hamiltonian system. In order to get insights into its dynamics we study numerically a global model for a Poincaré map associated to such a system. Mainly we investigate the scenario of reconnection of the invariant manifolds of two hyperbolic orbits of the same type n/m and show that it is a generic one. When the two Poincaré–Birkhoff chains involved in this process are aligned in phase, i.e. they are in a nongeneric position, a sequence of two saddle–center bifurcations occur in one of the chains, interfering with the former elliptic orbit of that chain, such that at the reconnection threshold the two chains are in a generic position. Dynamics around the new created configuration at the reconnection appears to vary from a regular motion to a chaotic one.

In connection with the study of this global bifurcation we give the first example of region of instability in the dynamics of a nontwist map.     

Asymptotic Representations for Root Vectors of Nonselfadjoint Operators and Pencils Generated by an Aircraft Wing Model in Subsonic Air Flow

This paper is the second in a series of several works devoted to the asymptotic and spectral analysis of an aircraft wing in a subsonic air flow. This model has been developed in the Flight Systems Research Center of UCLA and is presented in the works by A. V. Balakrishnan. The model is governed by a system of two coupled integrodifferential equations and a two parameter family of boundary conditions modeling the action of the self-straining actuators. The differential parts of the above equations form a coupled linear hyperbolic system; the integral parts are of the convolution type. The system of equations of motion is equivalent to a single operator evolution-convolution equation in the energy space. The Laplace transform of the solution of this equation can be represented in terms of the so-called generalized resolvent operator, which is an operator-valued function of the spectral parameter. This generalized resolvent operator is a finite-meromorphic function on the complex plane having the branch cut along the negative real semi-axis. Its poles are precisely the aeroelastic modes and the residues at these poles are the projectors on the generalized eigenspaces. In the first paper and in the present one, our main object of interest is the dynamics generator of the differential parts of the system. It is a nonselfadjoint operator in the energy space with a purely discrete spectrum. In the first paper, we have shown that the spectrum consists of two branches and have derived their precise spectral asymptotics. In the present paper, we derive the asymptotical approximations for the mode shapes. Based on the asymptotical results of these first two papers, in the next paper, we will discuss the geometric properties of the mode shapes such as minimality, completeness, and the Riesz basis property in the energy space.     

A variational method of deriving the equations of the non-linear mechanics of liquid crystals

The non-linear equations of the dynamics of liquid crystals [1], derived previously by the Poisson brackets method, are derived from the Hamilton-Ostrogradskii variational principle. The variational problem of an unconditional extremum of the action functional in Lagrange variables is investigated. The difference between the volume densities of the kinetic and free energy of the liquid crystal is used as the Lagrangian. It is shown that the variational equations obtained are equivalent to the differential laws of conservation of momentum and the kinetic moment of the liquid crystal in Euler variables, while the Ericksen stress tensor and the molecular field are defined in terms of the derivatives of the free energy.     

The brachistochrone motion of a mechanical system with non-holonomic, non-linear and non-stationary constraints

Further to previous studies /1, 2/ of the brachistochrone motion of non-holonomic mechanical systems with linear homogeneous constraints, consideration is given here to non-holonomic, non-linear and non-stationary mechanical systems. The problem is to formulate the differential equations of the brachistochrone motion of non-holonomic, non-linear and non-stationary mechanical systems and to determine the additional forces which must be introduced in order to implement motion of this type.     

Über die verallgemeinerte Form der Lagrangeschen Gleichungen,welche auch die Behandlung von nicht-holonomen mechanischen Systemen gestattet

Summary A generalized form of the equations of motion of a rheonomic-holonomic mechanical system is proved. As special cases the equations ofNielsen andTzenoff are obtained. By a method of the last author the form ofAppell's equations is derived, in which the kinetic energy, but not the energy of acceleration, appears. The generalized equations can be extended to the case of non-holonomical systems.     

An approach to constructing models of multiphase elastic porous media

A novel approach to describing the behaviour of multiphase elastic porous media is proposed. The average values of the physical quantities needed to describe the motions of porous media are formulated using an integral relation. The validity of this relation is taken as the fundamental hypothesis. The integral definition of the average values enables integral relations to be devised for the average values from the integral laws of conservation of mass, momentum and energy and the increase in entropy. Along with the average values, the integral relations contain new variables that can be identified with generalized thermodynamic forces, which can be used to take into account the phase interaction in a porous medium. The integral relations are used to derive differential equations for the rate of entropy change and Gibbs relations for a porous medium as a basis for obtaining the constitutive relations. Relationships between the thermomechanical parameters of the model are established from the Gibbs relations under additional assumptions. The equation for the rate of entropy change can be used to establish relations between the generalized thermodynamic forces and fluxes. A complete system of differential equations in the defining parameters, which describes the motion of multiphase elastic porous media, is finally obtained.     

非线性非完整空间变质量体的一种运动方程*

引入非线性非完整空间的约束超曲面的基矢量和密歇尔斯基方程点乘,作为非线性非完整系统变质量体的基本动力学方程。它简明、运算简便,而且由它可导出,Nielsou,Appell,Mac-Millan等已有的方程,不必附加关于虚位移的Appell-定义或牛青萍定义。本方程与D'Alembert-Lagrange微分变分原理相容。     

Asymptotic and spectral properties of operator‐valued functions generated by aircraft wing model

The present paper is devoted to the asymptotic and spectral analysis of an aircraft wing model in a subsonic air flow. The model is governed by a system of two coupled integro‐differential equations and a two parameter family of boundary conditions modelling the action of the self‐straining actuators. The differential parts of the above equations form a coupled linear hyperbolic system; the integral parts are of the convolution type. The system of equations of motion is equivalent to a single operator evolution–convolution equation in the energy space. The Laplace transform of the solution of this equation can be represented in terms of the so‐called generalized resolvent operator, which is an operator‐valued function of the spectral parameter. More precisely, the generalized resolvent is a finite‐meromorphic function on the complex plane having a branch‐cut along the negative real semi‐axis. Its poles are precisely the aeroelastic modes and the residues at these poles are the projectors on the generalized eigenspaces. The dynamics generator of the differential part of the system has been systematically studied in a series of works by the second author. This generator is a non‐selfadjoint operator in the energy space with a purely discrete spectrum. In the aforementioned series of papers, it has been shown that the set of aeroelastic modes is asymptotically close to the spectrum of the dynamics generator, that this spectrum consists of two branches, and a precise spectral asymptotics with respect to the eigenvalue number has been derived. The asymptotical approximations for the mode shapes have also been obtained. It has also been proven that the set of the generalized eigenvectors of the dynamics generator forms a Riesz basis in the energy space. In the present paper, we consider the entire integro‐differential system which governs the model. Namely, we investigate the properties of the integral convolution‐type part of the original system. We show, in particular, that the set of poles of the adjoint generalized resolvent is asymptotically close to the discrete spectrum of the operator that is adjoint to the dynamics generator corresponding to the differential part. The results of this paper will be important for the reconstruction of the solution of the original initial boundary‐value problem from its Laplace transform and for the analysis of the flutter phenomenon in the forthcoming work. Copyright © 2004 John Wiley & Sons, Ltd.