On the stability of equilibrium of continuous systems

Summary A sufficiency theorem for the stability of a linearly viscoelastic solid subjected to partial follower surface tractions is established. It is shown that an appropriately defined functional metric space must be introduced in order to formulate a well-posed problem. The usual energy method, if applicable, and the Galerkin method, if convergent, yield stability conditions only in a functional space whose metric is defined in an average

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Übersicht Es wird ein hinreichendes Kriterium für die Stabilität eines viskoelastischen festen Körpers abgeleitet, der Kräften unterworfen wird, die teilweise den Verformungen der Oberfläche folgen. Es wird gezeigt, daß ein geeignet definierter metrischer Funktionalraum eingeführt werden muß, um das Problem vernünftig formulieren zu können. Die übliche Energie-Methode, sofern sie anwendbar ist, sowie das GalerkinVerfahren, wenn es konvergiert, ergeben lediglich Stabilitätsbedingungen in einem Funktionalraum, dessen Metrik im Mittel definiert ist.

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This research was supported by the National Aeronautics and Space Administration under Grant NsG 605.     

On the stability of elastic systems under nonconservative follower forces

Summary We consider here the problem of the transition between stability and instability of the equilibrium configuration of a linear elastic system subjected to nonconservative follower(1) forces, when their magnitudes are affected by a common factor which is gradually increased.It is known that critical conditions for the stability of equilibrium may occur either by two natural frequencies of the system coinciding or by one natural frequency becoming zero, and that in the second eventuality they can also be recognised by the ordinary static criteria.We give evidence how, as distinct from the case of the conservative forces, the configuration considered can present itself again as stable for values of the load superior to an eventual critical load after the passing of a subsequent characteristic condition of one of the above-mentioned types and in particular of the second type.We indicate the necessity of the completion from this point of view of the treatment of a classic group of problems set by L. Collatz and A. Pflüger.Finally, we give sufficient conditions for the non-occurrence in problems of this kind of characteristic conditions of the second type.This study was presented at the meeting 19th January, 1964 of the Istituto Veneto di Scienze Lettere ed Arti and included in Vol. CXXII pp. 107–119 of the Acts of the same Institute.     

Influence of mass distribution on the stability of rods in the presence of polygenic forces

Summary In this paper, the influence of a non-uniform mass distribution on the stability behaviour of a rod subjected to follower forces is investigated. It is assumed that the mass contributes to the dead load of the system but does not affect the bending stiffness of the rod. Using Pflüger's rod as an example, it is shown that although there will always be instability by divergence, additional instability by flutter is possible depending on the way the mass is distributed. Specifically, a condition is studied which ensures the absence of flutter instability. Coresponding conclusions are drawn for a rocket type rod system.

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Übersicht In dieser Arbeit wird der Einfluß einer nicht gleichmäßig verteilten Masse auf das Stabilitätsverhalten eines Stabes untersucht, der mitgehenden Kräften unterworfen ist. Es wird angenommen, daß diese Masse wohl zur Last des Stabes, aber nicht zu seiner Biegesteifigkeit beiträgt. Indem der Pflügersche Stab als Beispiel gewählt wird, wird gezeigt, daß obwohl immer Instabilität in der Form von Divergenz auftreten wird, zusätzliche Instabilität in der Form von Flattern möglich ist. Diese Möglichkeit hängt von der Art und Weise ab, in der die Masse verteilt ist. Insbesondere wird eine Bedingung in den Untersuchungen hervorgehoben, die garantiert, daß kein Flattern auftreten wird. Entsprechende Schlüsse werden für ein raketenähnliches Stabsystem gezogen.

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The support of this research by the National Research Council, Canada, under grant No. A 7297 is gratefully acknowledged.     

On dynamics and stability of continuous systems subjected to a distributed moving load

Summary The paper deals with an analysis of different models of continuous systems subjected to a load distributed over a given length and moving at a constant velocity.The general discussion concerns a beam resting on a viscoelastic semi-space. The motion of the body is described by polynomial differential operators. With body forces being disregarded, the motion of the beam lying on a viscoelastic foundation is discussed with either taking into account the effect of shear deflection and the inertia of rotation or with neglecting them. Consideration is also given to the cases of relative motion of two continuous systems and the stability of their interaction. The above cases represent the models of a number of mechanical systems applied in e.g. modern transportation facilities, technology of bonding of layer materials, etc.The analysis is substantially simplified because the equations of motion and the boundary conditions are written in a moving coordinate system related with the load and only stationary solutions are considered. With such an approach, the solutions depend only on the transformed space variable, while the velocity of the load appears as one of the parameters of the system. Interesting conclusions are drawn from the obtained solutions and numerical calculations.

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Zur Dynamik und Stabilität von kontinuierlichen Systemen unter verteilten bewegten Lasten

Übersicht Der vorliegende Beitrag behandelt die Analyse von kontinuierlichen Systemen, die verteilten bewegten Lasten ausgesetzt sind. Es wird zunächst ein Balken auf viskoelastischem Halbraum behandelt. Die Bewegung des Körpers wird beschrieben durch Differentialoperatoren in Polynomform, wobei der Einfluß von Schubverformung und Drehträgheit diskutiert wird. Der Fall der Relativbewegung von zwei kontinuierlichen Systemen und ihre Stabilität wird ebenfalls diskutiert. Die untersuchten Fälle stellen Modelle zahlreicher mechanischer Systeme dar, wie zum Beispiel moderne Schnelltransportsysteme, Fügevorrichtungen von Schichtmaterialien, usw.Die Analyse der Probleme wird wesentlich dadurch vereinfacht, daß die Bewegungsgleichungen und Randbedingungen in einem mitbewegten Koordinatensystem formuliert werden und nur stationäre Lösungen betrachtet werden. Auf diese Weise hängen die Lösungen nur von der transformierten Ortsvariablen ab; die Geschwindigkeit der Belastung tritt als ein Parameter des Systems auf.Aufschlußreiche Schlußfolgerungen werden anhand der erhaltenen Lösungen und numerischen Berechnungen dargestellt.

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This paper was presented at the Second Symposium on Inelastic Solids and Structures, Bad Honnef, September 1981     

On the stability of systems with stochastically variable parameters

The paper analyzes questions related to the construction of dynamic stability boundaries of elastic systems subjected to stochastic parametric excitation. It is supposed that the parametric action is a combination of a deterministic static component and a stochastic fluctuating component. The fluctuating component is taken to be a stationary ergodic process. The stability boundaries are built in the region of combination resonance using the stochastic averaging method and a probabilistic approach due to Khasminskii. In this connection, the stochastic averaging method based on the Stratonovich-Khasminskii theory is used. The probabilistic approach consists in using explicit asymptotic expressions for the largest Lyapunov exponent, from which the asymptotic stability boundaries are determined. As an application, the stability of a simply supported thin-walled bar subjected to a stochastically varying longitudinal load is investigated Published in Prikladnaya Mekhanika, Vol. 41, No. 12, pp. 128–138, December 2005.     

Dynamic stability of weak equations of continuous systems

The stability analysis method is developed for distributed dynamic problems with relaxed assumptions imposed on solutions. The problem is motivated by structural vibrations with external time-dependent parametric excitations which are controlled using surface-mounted or -embedded actuators and sensors. The strong form of equations involves irregularities which lead to computational difficulties for estimation and control problems. In order to avoid irregular terms resulting from differentiation of the force and moment terms the dynamics equations are written in a weak form. The weak form of dynamics equations of linear mechanical structures is obtained using variational calculus. The study of stability of stochastic weak system is based on examining properties of Liapunov functional along a weak solution.     

On the use of the Hamiltonian for an evaluation of the stability of elastic systems subjected to follower forces

Summary It is shown that for compressive follower forces q which are smaller in magnitude than the Euler buckling load q _c , the Hamiltonian of certain elastic systems remains either constant or at least bounded. Hence, under such circumstances, the requirement that the Hamiltonian be positive definite can serve as a stability criterion.

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Übersicht Es wird gezeigt, daß das Hamiltonsche Funktional gewisser elastischer Systeme welche mitgehenden Druckkräften ausgesetzt sind, entweder konstant bleibt oder wenigstens beschränkt ist, wenn die Größe q der mitgehenden Kräfte kleiner als die Eulersche Knicklast q _c der Systeme ist. Die Forderung, daß das Hamiltonsche Funktional unter diesen Umständen positiv definit sei, kann dann als Stabilitätskriterium dienen.

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Dedicated to Prof. Dr. S. Falk in honour of this 60th birthday

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The support of this research by the Natural Sciences and Engineering Research Council Canada under Grant No. A7297 is gratefully acknowledged     

On the destabilizing effect of non-linear damping in non-conservative systems with follower forces

In this paper, a mechanical model with two degrees of freedom previously studied by Ziegler [1] is examined, but Ziegler's linear damping is replaced by a non-linear term, as is usual for structural damping. In the critical case, the stability problem must then be solved by the second method of Liapunov. The undamped system is also examined, but in this case the stability question in the critical case cannot be decided by the terms of lowest order. Non-linear damping gives rise to a behaviour similar to that due to linear damping; however, the stability of the zero position depends only on the ratio of the two damping coefficients.B paбoтe пccлeдyeтcя мeчaничecкця мoдeль co двyмя cтeпeнями cвoбoды paccмaт pпвaeмaя yзe Циглepoм (1), oдпaкo здecь лпнeйпoe дeмпфпpoвaнпe зaмeпeпo пeшнeйпым, кoтopoe oбычнo в cлyчaяч cтpyктypнoгo дeмпфиpoвaнпя.B кpнтпчecкoм cлyчae, пpoблeмa ycтoйчивocти. дoлзпa быть зaтeм peшeпa втopым мeтoдoм ляпyнoвa. Paccмaтpивaeтcя тoзe пeзaтyчayщaя cпcтeмa, oдпaкo в этoм cлyчac вoпpoc ycтoйчпвocти в кpптичecкoм cлyчae пe мoзeт быть peшeн c пoмoщьy члeпoв мaлeйшeгo пopядкa. пeлипeйпoe дeмпфиpoвaнпe выэывaeт пoвeдeпиe пoчoзee пзвecнoмy c линeйнoгo дeмПфиpoвaпия, oднaкo ycтoйчивocть нyлeвoгo пoлoзeпия зaвиcит тoгдa тoлькo oт cooтнoпeния двyч кoзффициeптoв дeмПфиpoвaния.     

On the stability boundary of hamiltonian systems

The criterion for the points in the parameter space being on the stability boundary of linear Hamiltonian system depending on arbitrary numbers of parameters was given, through the sensitivity analysis of eigenvalues and eigenvectors. The results show that multiple eigenvalues with Jordan chain take a very important role in the stability of Hamiltonian systems. Foundation item: the National Natural Science Foundation of China (10072012); the National Natural Science Foundation of Russia Biography: QI Zhao-hui (1964-)     

ON THE STABILITY BOUNDARY OF HAMILTONIAN SYSTEMS

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On the stability of stochastic difference systems

In this paper the stability of linear stochastic difference equations and a class of weakly non-linear stochastic difference equations is considered. For the linear systems explicit criteria are derived for the stability of the moments of any order. We also show how the moments of a linear stochastic difference system can be computed when a certain Lie-algebraic condition is satisfied.     

On the stability of equilibria of nonholonomic systems with nonlinear constraints

Lyapunov＇s first method, extended by V. V. Kozlov to nonlinear mechani- cal systems, is applied to the study of the instability of the position of equilibrium of a mechanical system moving in the field of conservative and dissipative forces. The mo- tion of the system is limited by ideal nonlinear nonholonomic constraints. Five cases determined by the relationship between the degree of the first nontrivial polynomials in Maclaurin＇s series for the potential energy and the functions that can be generated from the equations of nonlinear nonholonomic constraints are analyzed. In the three eases, the theorem on the instability of the position of equilibrium of nonholonomic systems with linear homogeneous constraints （V. V. Kozlov （1986）） is generalized to the case of nonlin- ear nonhomogeneous constraints. In the other two cases, new theorems are set extending the result from V. V. Kozlov （1994） to nonholonomic systems with nonlinear constraints.     

On approximation of systems with delay and their stability

We study the problem of the approximation of differential equations with delay by a system of ordinary differential equations. We analyze the qualitative behavior of solutions of the original system and the approximating system and construct an algorithm for the investigation of the stability of solutions of systems with delay.Translated from Neliniini Kolyvannya, Vol. 7, No. 2, pp. 208–216, April–June, 2004.