Method Of Integral Manifolds Research Articles

Thermal radiation effect on thermal explosion in gas containing fuel droplets

The effect of thermal radiation on the dynamics of a thermal explosion of a flammable gas mixture with the addition of volatile fuel droplets is studied. This is based on an original physical model of self-ignition. The thermal radiation energy exchange between the evaporating surface of the fuel droplets and burning gas is described using the P-1 model with Marshak boundary conditions. The original system of equations describing the effects of heating, evaporation and the combustion of fuel droplets is simplified to enable their analysis using asymptotic methods. The mathematical formulation is eventually reduced to a singularly perturbed system of ordinary differential equations. This allows us to apply the advanced geometric asymptotic technique (integral manifold method) for the qualitative analysis of the behaviour of the solution. Possible types of dynamic behaviour of the system are classified and parametric regions of their existence are determined analytically. The main attention is concentrated on the situations where delays might occur before the final ignition. Our study is focused on the impact of thermal radiation on the delay time. The dimensionless parameter responsible for the impact of thermal radiation is singled out and analysed. The dependence of the delay characteristics on the physical parameters of the problem under consideration is analysed. An explicit expression for the minimum time delay of the thermal explosion of fuel droplets in the presence of thermal radiation is derived and applied to the thermal explosion of n-decane and tetralin droplets. It is pointed out that the effects of thermal radiation can be significant, especially at high temperatures, and cannot be ignored in the analysis of this phenomenon.

Integral manifolds of singularly perturbed systems with application to rigid-link flexible-joint multibody systems

In this paper, we first review results of integral manifolds of singularly perturbed non-linear differential equations. We then outline the basic elements of the integral manifold method in the context of control system design, namely, the existence of an integral manifold, its attractivity, and stability of the equilibrium while the dynamics are restricted to the manifold. Toward this end, we use the composite Lyapunov method and propose a new exponential stability result which gives, as a by-product, an explicit range of the small parameter for which exponential stability is guaranteed. The results are applied to the control problem of multibody systems with rigid links and flexible joints in which the inverse of joint stiffness plays the role of the small parameter. The proposed controller is a composite control law that consists of a fast component, as well as a slow component that was designed based on the integral manifold approach. We show that the proposed composite controller has the following properties: (i) it enables the exact characterization and computation of an integral manifold, (ii) it makes the manifold exponentially attractive, and (iii) it forces the dynamics of the reduced flexible system on the integral manifold to coincide with the dynamics of the corresponding rigid system (i.e. the one obtained by making stiffness very large) implying that any control law that stabilizes the rigid system would stabilize the dynamics of the flexible system on the manifold. We finally present a detailed stability analysis and give an explicit range of the joint stiffness, in terms of system parameters and controller gains, for which the established exponential stability is guaranteed.

EFFECTS OF STATOR TRANSIENTS ON STEADY-STATE STABILITY REGIONS OF ELECTRIC POWER SYSTEMS

ABSTRACT Using linear integral manifolds, the assumption of neglecting stator transients is reexamined in the context of steady-state stability of electric power systems under varying system parameters and operating conditions. This widely-accepted assumption is shown to give erroneous results in two examples consisting of a synchronous machine or an induction machine connected to an infinite bus by a transmission line with a varying stator resistance. In the first example, the classical quasi-steady-state order reduction yields the same stability regions for two synchronous machines with different inertias. The stability regions, while being adequate for the larger machine, are erroneous for the small machine, for relatively large values of stator resistance. These results are also confirmed by the Routh-Hurwitz criterion of stability. Using a first-order integral manifold correction, almost exact stability regions are reconstructed for both machines. In the second example, the stability regions obtained from the classical quasi-steady-state order reduction are inadequate in generating mode for both small and large induction machines and require a first-order correction. These higher-order corrections using the method of integral manifolds can be applied in conventional small-signal stability programs to obtain better eigenvalue results near stability boundaries without the need for reintroducing the neglected stator transients.

Feedback control of a class of nonlinear singularly perturbed systems with time delay

The feedback control of two classes of nonlinear time-delay singularly perturbed systems with fast actuators is considered. First, by means of the integral manifold method, the asymptotic stability of a class of nonlinear time-delay singularly perturbed systems can be guaranteed and the phase trajectories of the closed-loop systems are steered along the integral manifold to the origin. Second, a continuous state feedback controller is constructed such that the trajectories of a class of uncertain nonlinear time-delay singularly perturbed feedback-controlled systems are uniformly ultimately bounded. We require no a priori information on the uncertain element, except that its restraint set is assumed to be known and compact.

A simple criterion for global stabilizability of a class of nonlinear singularly perturbed systems

In this paper, a simple criterion is proposed for the global stabilizability of a class of nonlinear singularly perturbed systems. Our approach is based on the global nonlinear controller canonical transformation and the integral manifold method. A globally stabilizing feedback control is also proposed such that the pre-specified design manifold becomes an exact slow integral manifold and the trajectories of the closed-loop system, starting from any initial states, are steered along the integral manifold to the origin.

Slow motions in problems of the dynamics of a solid with a cavity filled with a viscous liquid

The well-known problem of the motion of a solid with a cavity filled with a viscous liquid is considered. The problem is described by a singularly perturbed system of differential equations, consisting of the equations of motion of the solid and the Navier-Stokes equations. The Reynolds number is the small parameter. It is shown that the method of integral manifolds [1, 2] can be used to investigate this system of equations. It enables the investigation of the complete singularly perturbed system to be replaced by an investigation of a regular abbreviated system of lower dimensions, describing the slow motions of the complete system. The equations of the slow motions are obtained, which turn out to be the equations of motion of the solid with additional terms on the right-hand side which occur as a result of taking into account the effect of the viscous liquid on the motion of the solid.

Duck-trajectories in a thermal explosion problem

The critical values of a parameter in a thermal explosion problem in the autocatalytic case are examined by means of the method of integral manifolds and French-ducks techniques.

Development of the Bogolyubov-Mitropol'skii method of integral manifolds

A survey is presented of some results relating to the development of the Bogolyubov-Mitropol'skii method of integral manifolds.

An asymptotic method for quasilinear vibrational systems with an aperiodic generating solution

Formal methods for constructing asymptotic expansions of quasilinear vibrational systems with an aperiodic generating solution are considered. The proposed asymptotic method assumes the inclusion of damping (antidamping) terms in the generating system which leads to a non-traditional structure of the asymptotic series and of the changes of variables which are used. Although questions concerning the theoretical basis of the method are not considered, certain positive features of the proposed approach compared with the method of averaging are demonstrated by means of a simple example. The connection between the proposed method and the method of integral manifolds is discussed.

Amplitude modulated and chaotic dynamics in resonant motion of strings

The weakly non-linear, resonant motions of a stretched string to a harmonic, planar excitation are investigated. One mode truncation of the in-plane and out-of-plane equation results in a two-degree-to-freedom system which is examined by using the method of averaging and the method of integral manifolds. It is shown that the averaged equations, for small enough damping, possess non-planar constant solutions which become unstable by a Hopf bifurcation. The resulting limit cycle solutions correspond to amplitude modulated whirling or ballooning motions of the string. There are two limit cycle branches for the averaged equations, one arising due to the Hopf bifurcation and the other due to a global saddle-node bifurcation. The two branches merge as the damping is further reduced. Also, with variation in detuning, the isolated branch exhibits period-doubling bifurcations, chaotic attractors and merging of attractors, giving rise to Rössler as well as Lorentz-type solutions. The truncated string equations are also directly integrated. It is shown that there are non-planar periodic responses that bifurcate into amplitude modulated motions on a 2-torus. Changes in detuning result in torus-doubling, merging of tori and then destruction of the torus leading to chaotic amplitude modulations. The results of asymptotic analysis via the method of averaging are found to be in qualitative agreement with the actual response.

Separation of fast and slow motions by method of integral manifolds

Separation of fast and slow motions by method of integral manifolds

On the Method of Averaging, Integral Manifolds and Systems with Symmetry

A unified treatment and generalizations of some of the more important results associated with the method of averaging are given. The generalizations make the results applicable to a larger class of problems. Furthermore, the method of proof is simpler than what is traditional in this field. Examples demonstrating the use of basic results in applications are given.

On a mathematical problem of the theory of elastic stability

There is considered the problem of bifurcation of the self-oscillations from the equilibrium state for nonlinear differential equations of hyperbolic type. A general qualitative analysis is performed and formulas are presented to compute the parameters of periodic solutions. Loss of stability occurs in many nonconservative problems of elastic stability theory (see [1,3], for instance) because the pair of complex-conjugate eigenvalues belonging to the stability spectrum of the linearized problem intersects the imaginary axis under changes in some parameters and goes over into the right half-plane of the complex plane. In this case, the analysis of the appropriate mechanical problem in a nonlinear formulation will permit clarification of the soft or hard nature of loss of stability. One of the best known of such problems is nonlinear “panel flutter” at supersonic speeds, which is simulated by using the law of plane sections [4], The mathematical methods used in this latter problem reduce to the replacement of the nonlinear partial differential equations by their Galerkin approximations with subsequent application of the method of a small parameter of the harmonic balance method. As has been shown in [5], the results obtained by this means are often far from the correct one. A strict method of investigating appropriate nonlinear partial differential equations is proposed below. The mathematical apparatus to be used here is a combination of the method of integral manifolds and a method developed in [6]. The method of integral manifolds is not applicable here in pure form. This is related to the specific hyperbolic-type equations for which the properties of the solutions cannot assure the necessary smoothness of the invariant surface [7]. Let us note that the problem being solved below has actually been solved by V.V. Bolotin (see [1], p. 333).

Nonlinear Behavior of Flutter Unstable Dynamical Systems With Gyroscopic and Circulatory Forces

Postflutter behavior of nonlinear discrete dynamical systems having a combination of gyroscopic and circulatory forces are studied. The study leads to Hopf bifurcations. The method of analysis is based on the method of Hopf and the method of integral manifolds. The results of the analysis are applied to an example and the accuracy of the analysis is checked against numerical solutions of the equations of motion.

Investigation of bounded solutions of linear irregularly perturbed systems by the method of integral manifolds

Investigation of bounded solutions of linear irregularly perturbed systems by the method of integral manifolds

Investigation of bounded solutions of nonlinear irregularly perturbed systems by the integral manifold method

Investigation of bounded solutions of nonlinear irregularly perturbed systems by the integral manifold method