Application of a multi-component Eulerian approach to capture interface evolution and vorticity generation in compressible multiphase flow

The hyperbolic Eularian model is used as a mathematical framework for compressible multiphase flows. The formulation was obtained after an averaging process of the single phase Navier-Stokes equations. The closure of multi-component system leads to the volume fraction equation containing a non-conservative term and a pressure equilibrium condition. As a result the model equations cannot be written in a conservative form. To solve the equations a finite volume Godunov type computational approach is developed which uses an approximate Riemann solver combined with a numerical scheme to tackle the non-conservative terms. The approach accounts for pressure non-equilibrium. It enables resolving interfaces separating compressible fluids and captures the baroclinic source of vorticity generation. The computations are performed for various initial conditions and compared with theoretical and experimental data for a shock-bubble interaction problem. The investigated cases include acoustic wave transmission through isolated bubbles of helium and krypton. The numerical results illustrate the characteristic features of the evolving interfaces. The impulsively generated flow perturbations are dominated by the reflection and refraction of the shock and by the vorticity generation within the media. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

时滞Lagrange系统的Lie对称性与守恒量研究

研究了位形间中含单时滞参数的非保守力学系统的Lie对称性和守恒量。首先,利用含时滞的动力学Hamilton原理,建立了含时滞的非保守系统的分段Lagrange运动方程;其次,利用微分方程容许Lie群理论,得到系统的Lie对称确定方程;然后,根据对称性与守恒量之间的关系,通过构造结构方程,得到含时滞的非保守系统的Lie定理;最后,给出了两个具体的算例说明了方法的应用。结果表明:时滞参数的存在使非保守系统的Lagrange方程呈现分段特性,相应的Lie对称性确定方程的个数应是自由度数目的2倍,这对生成元函数提出了更高的限制,同时,守恒量呈现依赖速度项的分段表达。     

The stability of non-conservative systems and an estimate of the domain of attraction

The stability of mechanical systems, on which dissipative, gyroscopic, potential and positional non-conservative forces act, is investigated. The condition for asymptotic stability is obtained using the Lyapunov function and an estimate of the domain of attraction is also found in terms of the system being considered. A precessional system is also examined. It is shown that the condition for the asymptotic stability of a system is the condition of acceptability in the sense of the stability of a precessional system. The results obtained are applied to the problem of the stabilization, using external moments, of the steady motion of a balanced gyroscope in gimbals.     

The inverse problem for Lagrangian systems with certain non-conservative forces

We discuss two generalizations of the inverse problem of the calculus of variations, one in which a given mechanical system can be brought into the form of Lagrangian equations with non-conservative forces of a generalized Rayleigh dissipation type, the other leading to Lagrangian equations with so-called gyroscopic forces. Our approach focusses primarily on obtaining coordinate-free conditions for the existence of a suitable non-singular multiplier matrix, which will lead to an equivalent representation of a given system of second-order equations as one of these Lagrangian systems with non-conservative forces.     

Mechanical systems,equivalent in Lyapunov's sense to systems not containing non-conservative positional forces

Developing results obtained previously (Refs. Koshlyakov VN. Structural transformations of the equations of perturbed motion of a certain class of dynamical systems. Ukr Mat Zh 1997; 49 (4): 535–539; Koshlyakov VN. Structural transformations of dynamical systems with gyroscopic forces. Prikl Mat Mekh 1997; 61 (5): 774–780; Koshlyakov VN, Makarov VL. The theory of gyroscopic systems with non-conservative forces. Prikl Mat Mekh 2001; 65 (4): 698–704; Koshlyakov VN, Makarov VL. The stability of non-conservative systems with degenerate matrices of dissipative forces. Prikl Mat Mekh 2004; 68 (6): 906–913), the general problem of eliminating non-conservative positional structures from the second-order differential equation with constant matrix coefficients, obtained when modelling many mechanical systems, is considered. It is assumed that the matrices of the dissipative and non-conservative positional structures may, in particular, be degenerate. Under fairly general assumptions, theorems containing the necessary and sufficient conditions for a Lyapunov transformation to exist are proved. This converts the initial matrix equation to an equivalent autonomous form (in Lyapunov's sense) with a symmetrical matrix of the positional forces. An illustrative example is considered.     

The application of lyapunov functions to some problems of the acceptability of approximate solutions of differential equations

The acceptability of approximate solutions of differential equations with respect to some of the variables is considered. The notion of acceptability is defined, generalizing a definition used in [1] in a study of the acceptability of precessional equations of gyroscopic systems. Lyapunov functions are introduced accordingly and used to solve the problem of acceptability. As an application, the possibility of reducing the order of the equations of motion for some mechanical systems is discussed.     

The dynamics of systems with large gyroscopic forces and the realization of constraints

Lagrangian systems with a large multiplier N on the gyroscopic terms are considered. Simplified equations of motion of general form with holonomic constraints are obtained in the first approximation with respect to the small parameter ɛ = 1/N. The structure of the solutions of the precessional equations is examined.     

A conservative extension of the method of characteristics for 1-D shallow flows

The method of characteristics (MOC) has been used for a long time in open channels and pipes flows. It is based on non-conservative equations, and hence it cannot be used directly for solving discontinuous shallow flows. In this paper we develop a conservative version of the MOC scheme for 1-D shallow flows by imposing the conservation law at the interpolation step. The conservation property of the scheme ensures the production of an accurate shock modeling and enables the MOC scheme to simulate dam-break type flows. By using a proper interpolation function, the proposed method can also produce quite accurate low-oscillatory results. A number of challenging test cases show considerable improvement compared to the traditional non-conservative MOC scheme in the case of dam-break type and trans-critical flow simulations.     

Stabilization and destabilization in non-conservative gyroscopic systems

Stability of a linear autonomous non-conservative system in presence of potential, gyroscopic, dissipative, and nonconservative positional forces is studied. The cases when the non-conservative system is close to a gyroscopic system or to a circulatory one, are examined. It is known that the marginal stability of gyroscopic and circulatory systems can be destroyed or improved up to asymptotic stability due to action of small non-conservative positional and velocity-dependent forces. The present contribution shows that in both cases the boundary of the asymptotic stability domain of the perturbed system possesses singularities such as “Dihedral angle” and “Whitney umbrella” that govern stabilization and destabilization. Approximations of the stability boundary near the singularities and estimates of the critical gyroscopic and circulatory parameters are found in an analytic form. In case of two degrees of freedom these estimates are obtained in terms of the invariants of matrices of the system. As an example, the asymptotic stability domain of the modified Maxwell-Bloch equations is investigated with an application to the stability problems of gyroscopic systems with stationary and rotating damping, such as the Crandall gyropendulum, tippe top and Jellet's egg. An instability mechanism in a system with two degrees of freedom, originating after discretization of models of a rotating disc in frictional contact and possessing the spectral mesh in the plane ‘frequency’ versus ‘angular velocity’, is described in detail and its role in the disc brake squeal problem is discussed. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Mono-Implicit Runge-Kutta-Nyström Methods with Application to Boundary Value Ordinary Differential Equations

The successful use of mono-implicit Runge—Kutta methods has been demonstrated by several researchers who have employed these methods in software packages for the numerical solution of boundary value ordinary differential equations. However, these methods are only applicable to first order systems of equations while many boundary value systems involve higher order equations. While it is straightforward to convert such systems to first order, several advantages, including substantial gains in efficiency, higher continuity of the approximate solution, and lower storage requirements, are realized when the equations can be treated in their original higher order form. In this paper, we consider generalizations of mono-implicit Runge—Kutta methods, called mono-implicit Runge—Kutta—Nyström methods, suitable for systems of second order ordinary differential equations having the general form, y(t) = f(t,y(t),y(t)), and derive optimal symmetric methods of orders two, four, and six. We also introduce continuous mono-implicit Runge—Kutta—Nyström methods which allow us to provide continuous solution and derivative approximations. Numerical results are included to demonstrate the effectiveness of these methods; savings of 65% are attained in some instances.This revised version was published online in October 2005 with corrections to the Cover Date.     

Dissipation induced instabilities in continuous non-conservative systems

In this contribution we analyze the stabilizing and destabilizing effect of small damping for rather general class of continuous non-conservative systems, described by the non-self-adjoint boundary eigenvalue problems. Explicit asymptotic expressions obtained for the stability domain allow us to predict when a given combination of the damping parameters leads to increase or to decrease in the critical non-conservative load. The results obtained explain why different types of internal and external damping so surprisingly influence on the stability of non-conservative systems. As a mechanical example the stability of a viscoelastic rod with small internal and external damping, loaded by tangential follower force, is studied in detail. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Logarithmic matrix norms in motion stability problems

The problem of the stability of the motions of mechanical systems, described by non-linear non-autonomous systems of ordinary differential equations, is considered. Using the logarithmic matrix norm method, and constructing a reference system, the sufficient conditions for the asymptotic and exponential stability of unperturbed motion and for the stabilization of progammed motions of such systems are obtained. The problem of the asymptotic stability of a non-conservative system with two degrees of freedom is solved, taking for parametric disturbances into account. Examples of the solution of the problem of stabilizing programmed motions – for an inverted double pendulum and for a two-link manipulator on a stationary base – are considered.     

Re-visiting structural optimization of the Ziegler pendulum: singularities and exact optimal solutions

Structural optimization of non-conservative systems with respect to stability criteria is a research area with important applications in fluid-structure interactions, friction-induced instabilities, and civil engineering. In contrast to optimization of conservative systems where rigorously proven optimal solutions in buckling problems have been found, for non-conservative optimization problems only numerically optimized designs were reported. The proof of optimality in the non-conservative optimization problems is a mathematical challenge related to multiple eigenvalues, singularities on the stability domain, and non-convexity of the merit functional. We present a study of the optimal mass distribution in a classical Ziegler's pendulum where local and global extrema can be found explicitly. In particular, for the undamped case, the two maxima of the critical flutter load correspond to a vanishing mass either in a joint or at the free end of the pendulum; in the minimum, the ratio of the masses is equal to the ratio of the stiffness coefficients. The role of the singularities on the stability boundary in the optimization is highlighted and extension to the damped case as well as to the case of higher degrees of freedom is discussed. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Hilbert and Hilbert—Samuel polynomials and partial differential equations

Systems of linear partial differential equations with constant coefficients are considered. The spaces of formal and analytic solutions of such systems are described by algebraic methods. The Hilbert and Hilbert—Samuel polynomials for systems of partial differential equations are defined.     

A new first integral corresponding to Lyapunov's function for a pendulum of variable length

Summary Using Noether's theorem and the generalized Killing equations [1], new first integrals of the differential equation of motion for a class of non-conservative mechanical systems with one degree of freedom, a special case of which is a simple pendulum of variable length, are obtained. These integrals are identified as Lyapunov's functions for non-autonomous systems. The stability conditions are established.

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Zusammenfassung Mit Hilfe des Noetherschen Satzes und den verallgemeinerten Killingschen Gleichungen werden neue erste Integrale der Bewegungsdifferentialgleichungen für eine Klasse von nichtkonservativen mechanischen Systemen mit einem Freiheitsgrad, die das Pendel mit veränderlicher Länge als Sonderfall enthält, hergeleitet. Diese Integrale stellen Ljapunovsche Funktionen dar, mit denen sich die Stabilitätsbedingungen ergeben.

     

Numerical solution of singular Lur'e equations

We consider a generalization of Lur’e equations for differential-algebraic systems. We aim to numerically solve these equations. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The determination of the periodic motions of systems with delay

Continuing the development of results previously obtained for systems with delay described by first-order differential equations with delay [1], a system without delay is constructed which enables the periodic motions of systems with delay to be found.     

The stability of the equilibrium position of scleronomous mechanical systems

The problem of the stability of the equilibrium position of a scleronomic mechanical system is considered. The comparison method enables this problem to be reduced to the problem of the stability of scalar differential equations. The stability conditions are found for certain types of scalar comparison equations (Sections 1–4), and the sufficient conditions for the stability of the equilibrium positions of various scleronomous mechanical systems are determined from these (Sections 5–9).