Linear Conservative Systems Research Articles

Competitive ternary interactions and relative entropy of solutions

For conservative linear systems (finite-state Markov processes in discrete or continuous time), the relative entropy of two distinct trajectories is a monotonically decreasing function of time. These results naturally raise the question whether distinct trajectories of nonlinear conservative systems also display monotonically decreasing relative entropy. For binary interacting Lotka-Volterra systems with anti-symmetry, the relative entropy oscillates under the motion. The main new result of this paper is that, for ternary interacting Lotka-Volterra systems with anti-symmetry, the relative entropy of two distinct trajectories is a monotonically decreasing function of time near equilibrium. Far from equilibrium, distinct trajectories of ternary Lotka-Volterra systems with anti-symmetry need not have monotonically decreasing relative entropy.

A Sufficient Stability Condition for Linear Conservative Gyroscopic Systems

A sufficient stability condition for linear conservative gyroscopic systems with negative definite stiffness matrices is given. The condition for the stability is stated in terms of the coefficients of system matrices without solving the spectrum of the entire system. An example is given for comparison with existing results.

Optimal positioning of dampers in multi-body systems

An important aspect of the passive and/or active damping of vibrations of multi-body systems is the optimal positioning of the dampers, actuators and sensors. This study is concerned with the problem of finding the optimal positioning of a viscous damper for a linear conservative mechanical system on the basis of an energy criterion. The results obtained are applied to homogeneous oscillating chains.

Stability of Linear Conservative Gyroscopic Systems

Sufficient conditions are obtained for the stability and instability of linear conservative gyroscopic systems. The conditions are nonspectral, involve only the definiteness of certain combinations of the coefficient matrices, and may yield useful design constraints. An example is employed to compare these results with earlier results of the same type.

On the Stability Criteria for Conservative Gyroscopic Systems

Two alternative but equivalent stability criteria for linear conservative gyroscopic systems are presented. The development of the criteria is based on the concept of symmetrizability associated with general matrices, and the results are in terms of the matrices describing the dynamical system. Although the new criteria involves a scalar parameter, the restrictions associated with an earlier result are removed, and the scope of application is broadened. An illustrative example is presented.

Model reduction of linear conservative mechanical systems

An approach for model reduction of linear conservative or weakly damped mechanical systems is proposed. It is based on the balancing of an associated gradient system. It uses the joint knowledge of the system matrix and the input and output matrices of the Hamiltonian system. The key idea is to associate with the Hamiltonian system a gradient (or reciprocal) system.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Stochastically Perturbed Linear Gyroscopic Systems

ABSTRACT Dynamic stability of linear conservative gyroscopic systems under stochastic parametric excitations of small intensity is examined. Conditions for mean square stability of dynamic response are obtained. Results are shown to depend only on those values of the excitation spectral density near twice the natural frequencies and the combination frequencies of the system. These results are applied to the problem of flow induced vibration in a supported pipe conveying fluid with pulsating velocity. The effects of mean flow velocity and virtual mass on the extent of parametric instability regions are then discussed.

INHOMOGENEOUS PLANE WAVES IN A CONSTRAINED ELASTIC BODY

In this paper it is shown how relationships analogous to those obtained earlier for inhomogeneous waves in initially stress-free, unconstrained, elastic bodies and in linear conservative systems apply to inhomogeneous waves in the more general setting in which, besides exhibiting material anisotropy, the transmitting non-heat-conducting elastic body is subject to a primary deformation and a system of internal constraints

Stability analysis of gyroscopic systems by matrix methods

Understanding the behavior of complex gyroscopic systems may be enhanced by formulating a matrix description of the system's equations of motion. In particular, the application of matrix methods to stability analysis often leads to useful constraints on a system's parameter values. In this paper, a new matrix stability condition is derived for linear conservative gyroscopic systems with two degrees of freedom. The work consists first of determining matrix inequalities for the gyroscopic system model. These ineqnalities are then transformed into simple, nonlinear and algebraic relationships among the basic system parameters, thereby providing a quantitative indication of the effects of parameter changes on system stability. This method of stability analysis requires less calculation than the usual procedures of solving the eigenvalue equation or generating the system response. The new stability condition along with standard matrix stability results are applied to a model of a dynamically tuned gyroscope, and the derived regions of stability are discussed. The proposed matrix method may be applicable to the general class of complex gyroscopic systems (flexible structures and rotating devices) which includes the dynamically tuned gyroscope.

The joining of local expansions in the theory of non-linear oscillations

The behaviour of normal modes of oscillation in non-linear conservative systems with a finite number of degrees of freedom, when the amplitude changes from zero to infinity is studied. In the non-linear case, the normal oscillations represent a generalization of the normal oscillations of linear conservative systems (see /1/). It is assumed that the potential of a non-linear system is a polynomial of even degree in all positional variables. One can construct the trajectories of the normal oscillations in configuration space both for sufficiently small amplitude (a quasi-linear expansion), and for sufficiently large amplitude, using the fact that in these cases the system is close to a uniform system (see /2, 3/). The local expansions obtained are joined using rational-fractional Pade representations (see /4/) which enables the behaviour of oscillation modes to be followed when the amplitude changes continuously.

The loading-frequency relation of linear conservative systems via a direct energetic (action) method

This paper presents alternative proofs of the well-known strict negative monotonicity and convexity properties of the fundamental frequency-load pair curve of the pure eigenproblem of free vibrations/buckling of linear conservative systems subject to a single loading parameter, by studying the first and second order variations of the system's Hamiltonian action (over the fundamental or highest eigenperiod), as one moves along the fundamental eigenpair curve ( FEC ). Specifically, by setting the action's first and second frequency derivatives equal to zero it is shown that, all along the FEC , both the first and second derivatives of the load with respect to the frequency keep a strict negative sign; these action derivatives are special cases of the general first and second variations of a variable endpoints variational problem. This energetic method opens the way for the development of an asymptotic scheme (by taking third, fourth and higher action derivatives) for the approximate study of the FEC shape in the neighborhood of a given point (reminiscent perhaps of Koiter's asymptotic elastostatic post-buckling theory). At this point, the method does not seem to be restricted by linearity neither in the equations of motion nor in their eigenload(s) dependence.

On the stability of linear conservative gyroscopic systems

A simple criterion concerning the stability of linear gyroscopic conservative systems is developed. First, a sufficient condition for stability is established. The proof is based on a transformation converting the original autonomous system into a nonautonomous system, and applying Liapunov's direct method to the latter. Then a theorem is given which provides a necessary and sufficient condition for a restricted class of systems. Illustrative examples are provided.

Energy flux for trains of inhomogeneous plane waves

The propagation of inhomogeneous plane waves in linear conservative systems is considered. It is assumed that the secular equation governing the propagation of a plane wave of slowness S has the form Q ( S ) = 0 where Q is independent of frequency. Dispersion enters via the boundary condi­tions. By using the point form of the conservation of energy equation results are obtained which relate the mean energy flux vector R ˜ with the mean energy density Ẽ for any number of wave trains. In particular for a single train of inhomogeneous plane waves it is shown that R ˜ . S + = Ẽ . The results are relevant in electromagnetism, elasticity and fluid dynamics.

Energy flux for plane waves in linear conservative systems

AbstractThe point form of the conservation of energy equation is used to give simple and direct proof of results concerning the mean energy flux vector for systems of sinusoidal small amplitude waves in linear conservative systems. No constitutive equation is used explicitly.

Generalization of the rayleigh theorem to gyroscopic systems: PMM vol. 40, n≗ 4, 1976, pp. 606–610

The Rayleigh theorem on the properties of the spectrum of a linear conservative mechanical system is generalized to embrace the gyroscopic systems, i.e. to the case in which the equations of motion contain, in addition to the kinetic and potential energy matrices, an arbitrary skew-symmetric matrix of gyroscopic forces.

The Loading-Frequency Relationship in Multiple Eigenvalue Problems

The free vibrations of a linear conservative system with multiple loading parameters are studied, attention being restricted to pure eigenvalue problems. It is shown that the smallest frequency and external loading parameters of such a system constitute a strictly convex (synclastic) surface which cannot have convexity toward the origin of the “parameter space.” It is further proved that in the case of systems with one degree of freedom only, the surface takes the form of a plane. The practical implications of these results regarding the estimation of the frequencies and/or the stability boundary of the system are discussed. Thus it is observed that, on the basis of the established theorems, lower bounds to the frequencies at any stage of external loading and/or upper bounds to the stability boundary are readily obtainable. A two-degree-of-freedom illustrative example is discussed.

On the stability of a linear nongyroscopic conservative system

Im Anschluss an die Erklarung der Begriffe des Pseudofreiheitsgrades und des quasidynamischen Systems wurden durch Anwendung des Hamiltonschen Prinzips die Bewegungsdifferentialgleichungen solcher Systeme hergeleitet. An einem Beispiel wird gezeigt, dass fur die Stabilitatsuntersuchung bei Systemen von diesem Typus nur die Schwingungsmethode zulassig ist, auch wenn es sich um einen konservativen nichtgyroskopischen Fall handelt.

The response of an oscillatory system to excitation by a transient displacement

The response of a conservative linear system to a transient applied force of some prescribed form is shown, with suitable interpretation, to be the same as that due to excitation by a transient displacement of the same form. Consequently the solution of practical vibration problems involving ‘time-dependent boundary conditions’ (with their attendant mathematical complexity) is greatly simplified and given a readily understood physical significance. In the development of the theory, the concepts of ‘effective modal stiffness’ and ‘effective modal mass’ (which have occasionally been invoked in the past, and in more senses than one) are elucidated.

On energy relations for waves in temporally and spatially dispersive systems

Simple expressions for energy flow and stored energy in a linear conservative system are derived from a special form of the dispersion equation, and some applications are given for temporally and spatially dispersive systems.

Matrix analysis of linear conservative systems

The loop differential equations of any lossless, reciprocal electric circuit are integrated by the use of functions of matrices. It is shown how the transient analysis of such circuits is a generalization of the analysis of the simple single-loop circuit composed of a linear inductance in series with a linear elastance. The scaler trigonometric functions that appear in the analysis of the single-loop circuit are replaced by matric trigonometric functions in the analysis of the general circuit of n loops. The matric functions are then expanded by a form of Sylvester's theorem. The case in which the circuit has multiple eigenvalues is particularly easy to analyze by the use of functions of matrices. The analysis is extended to the forced oscillations of the n-loop linear conservative circuit.