Spectral properties of linear gyroscopic systems

We study how the natural frequencies of a linear gyroscopic system behave under variations in the kinetic and potential energy matrices as well as in the gyroscopic force matrix.     

Stability of linear time variant dynamic systems with multiple time delays

By means of the frozen time approach and the Kronecker product, two criteria of asymptotic stability are derived for the linear, time variant dynamic systems with either short time delays or with weak feedback involving arbitrary time delays, respectively. It is found that the asymptotic stability of these retarded dynamic systems is governed by the maximal and minimal singular values of the coefficient matrices and their time derivatives. The project supported by the National Natural Science Foundation of China under the Grant of 59625511.     

Physical mechanism of the destabilizing effect of damping in continuous non-conservative dissipative systems

The paper attempts to give a physical explanation of the mechanism behind the so-called destabilizing effect of small internal damping in the dynamic stability of Beck's column. Both internal (material) and external (viscous fluid) damping are considered. An energy equation is derived for the balance between the work done by the non-conservative ‘follower force’ and the energy dissipated by the internal and external damping forces. Evaluated at the critical load, where a flutter instability is initiated, this equation explicitly shows the influence of damping upon flutter frequency, phase angle, and vibration amplitude. The gradient of the phase angle, evaluated at the free end of the column, is found to be the ‘valve’, which controls how much work the follower force can do on the column during each period of oscillation. And a large change in this gradient with increasing—but still small—internal damping is found to be responsible for the destabilizing effect.     

Stability analysis of gyroscopic systems with integral correction

Stability theorems for gyroscopic systems with integral correction under dissipative and nonconservative positional forces are proved. An inertial vertical gyro is used as an example to illustrate the results obtained     

Two degree of freedom gyroscopic systems with linear integrals

In this paper we consider the first integrals, linear in velocities, of conservative gyroscopic systems with two degrees of freedom. A precise criterion which specifies whether a given gyroscopic system possesses a linear integral is derived. When the kinetic energy has the structure of a standard form of the metric of revolution, all the possible systems which admit a linear integral and corresponding integrals are determined explicitly. Two examples are considered to illustrate the usefulness of the derived results.     

Exact solutions for some non-conservative hyperbolic systems

In this paper, we study initial value problem for some non-conservative hyperbolic systems of partial differential equations of first order. The first one is the Riemann problem for a model in elastodynamics and the second one the initial value problem for a system which is a generalization of the Hopf equation. The non-conservative products which appear in the equations do not make sense in the classical theory of distributions and are understood in the sense of Volpert (Math. USSR Sb. 2 (1967) 225). Following Lax (Comm. Pure Appl. Math. 10 (1957) 537) and Dal Maso et al. (J. Math. Pures Appl. 74 (1995) 483), we give an explicit solution for the Riemann problem for the elastodynamics equation. The coupled Hopf equation is studied using a generalization of the method of Hopf (Comm. Pure Appl. Math. 3 (1950) 201).     

Stability analysis for multi-parameter linear periodic systems

Summary This paper is devoted to stability analysis of general linear periodic systems depending on real parameters. The Floquet method and perturbation technique are the basis of the development. We start out with the first and higher-order derivatives of the Floquet matrix with respect to problem parameters. Then the behaviour of simple and multiple multipliers of the system with a change of parameters is studied. Weak and strong interactions of multipliers in the complex plane are treated separately. The presented theory is exemplified and discussed. Received 27 April 1998; accepted for publication 5 November 1998     

Decentralized stabilization of large-scale linear interconnected systems with time-varying delays

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EIGENVALUE PROBLEM OF A LARGE SCALE INDEFINITE GYROSCOPIC DYNAMIC SYSTEM

Gyroscopic dynamic system can be introduced to Hamiltonian system.Based on an adjoint symplectic subspace iteration method of Hamiltonian gyroscopic system, an adjoint symplectic subspace iteration method of indefinite Hamiltonian function gy- roscopic system was proposed to solve the eigenvalue problem of indefinite Hamiltonian function gyroscopic system.The character that the eigenvalues of Hamiltonian gyroscopic system are only pure imaginary or zero was used.The eigenvalues that Hamiltonian function is negative can be separated so that the eigenvalue problem of positive definite Hamiltonian function system was presented,and an adjoint symplectic subspace iteration method of positive definite Hamiltonian function system was used to solve the separated eigenvalue problem.Therefore,the eigenvalue problem of indefinite Hamiltonian function gyroscopic system was solved,and two numerical examples were given to demonstrate that the eigensolutions converge exactly.     

The asymptotic stability of the linear discrete large-scale systems

In this paper,we directly use the linear norm Liapunov function to investigate the stability of the linear discrete large-scale systems and obtain some criteria for the asymptotic stability of such a system.     

On stability boundary of linear multi-parameter Hamiltonian systems

In this paper an approximate equation is derived to describe smooth parts of the stability boundary for linear Hamiltonian systems, depending on arbitrary number of parameters. With this equation, we can obtain parameters corresponding to the stability boundary, as well as to the stability and instability domains, provided that one point on the stability boundary is known. Then differential equations describing the evolution of eigenvalues and eigenvectors along a curve on the stability boundary surface are derived. These equations also allow us to obtain curves belonging to the stability boundary. Applications to linear gyroscopic systems are considered and studied with examples. The project supported by the National Science Foundations of Russia and China (10072012)     

Characterization of bifurcating non-linear normal modes in piecewise linear mechanical systems

The non-linear modal properties of a vibrating 2-DOF system with non-smooth (piecewise linear) characteristics are investigated; this oscillator can suitably model beams with a breathing crack or systems colliding with an elastic obstacle. The system having two discontinuity boundaries is non-linearizable and exhibits the peculiar feature of a number of non-linear normal modes (NNMs) that are greater than the degrees of freedom. Since the non-linearities are concentrated at the origin, its non-linear frequencies are independent of the energy level and uniquely depend on the damage parameter. An analysis of the NNMs has been performed for a wide range of damage parameter by employing numerical procedures and Poincaré maps. The influence of damage on the non-linear frequencies has been investigated and bifurcations characterized by the onset of superabundant modes in internal resonance, with a significantly different shape than that of modes on fundamental branch, have been revealed.     

陀螺效应对转子横向振动的影响分析

举例说明了在动力转子系统中陀螺效应对实际模型的影响。着重分析了转子陀螺效应对进动角速度、振型以及临界角速度的影响。并应用状态空间法求解陀螺系统的本征值问题。数值结果表明，在一些工程问题中，陀螺力对于转子系统振动特性的影响是不能忽略的。     

Normal modes for piecewise linear vibratory systems

A method to construct the normal modes for a class of piecewise linear vibratory systems is developed in this study. The approach utilizes the concepts of Poincaré maps and invariant manifolds from the theory of dynamical systems. In contrast to conventional methods for smooth systems, which expand normal modes in a series form around an equilibrium point of interest, the present method expands the normal modes in a series form of polar coordinates in a neighborhood of an invariant disk of the system. It is found that the normal modes, modal dynamics and frequency-amplitude dependence relationship are all of piecewise type. A two degree of freedom example is used to demonstrate the method.     

Asynchronous relaxed iterative methods for solving linear systems of equations

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STABILITY ANALYSIS OF LINEAR AND NONLINEAR PERIODIC CONVECTION IN THERMOHALINE DOUBLE－DIFFUSIVE SYSTEMS

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Analysis and control of a class of uncertain linear periodic discrete-time systems

Feedback control problems for linear periodic systems （LPSs） with interval- type parameter uncertainties are studied in the discrete-time domain. First, the stability analysis and stabilization problems are addressed. Conditions based on the linear matrices inequality （LMI） for the asymptotical stability and state feedback stabilization, respec-tively, are given. Problems of L2-gain analysis and control synthesis are studied. For the L2-gain analysis problem, we obtain an LMI-based condition such that the autonomous uncertain LPS is asymptotically stable and has an L2-gain smaller than a positive scalar γ. For the control synthesis problem, we derive an LMI-based condition to build a state feedback controller ensuring that the closed-loop system is asymptotically stable and has an L2-gain smaller than the positive scalar γ. All the conditions are necessary and sufficient.     

On the decomposition of forces

We show that any continuously differentiable force is decomposed into the sum of a Rayleigh force and a gyroscopic force. We also extend this result to piecewise continuously differentiable forces. Our result improves the result on the decomposition of forces in a book by David Merkin and further extends it to piecewise continuously differentiable forces.