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.     

On some important problems in analytical dynamics of non-holonomic systems

By using deductive method,Chetaev condition is derived in this paper.We pointout that the processes of variation and differentiation are not permutable innon-holonomic dynamics is a misunderstanding.The paper gives two classical relationsof non-holonomic systems and points out integral variational principles can be appliedin non-holonomic systems.     

On the stability of elastic annular rods

The stability of equilibrium of non-linearly elastic rods, whose deformations obey the classical Kirchhoff’s equations, is considered. A variational formulation of the equilibrium problem is given, and the equilibrium equations for infinitesimal deformations superimposed to a finite transformation of a rod are deduced. The stability of annular rings, in which the twisting strain is non-null, is investigated by study of the second variation of the energy functional.     

On the stability of elastic curved bars

A general discussion about the stability of arbitrary elastic curved bars in space under combined actions of bending and twisting is given in this paper. A system of Eqs. (28)–(36) for perturbation functions near some equilibrium state is presented. With appropriate boundary conditions, the nontrivial solution corresponds to the critical state. Five examples are analysed in this paper. Some of them are new results and others are old problems treated using the new method.     

On the field method in non-holonomic mechanics

This paper deals with the generalization of the field method to non-holonomic systems whose motion is subject to either non-linear constraints or those of a higher order, while their motion is modeled by the generalized Lagrange equations of the second kind. Two examples are given to illustrate the theory.The project supported by the Ministry of Science, Technologies and Development, Republic of Serbia (1874)     

On the convection of a cusp in elastic stability

Because of its dominant role in elastic stability, the unstable-symmetric bifurcation, or cusp catastrophe, is worthy of special study. We present here a systematic perturbation analysis of the cusped imperfection-sensitivity in a general bifurcational context, to examine its movement and deformation under the influence of further imperfection parameters. The elimination of passive coordinates allows explicit expressions to be derived for the horizontal and vertical shifts and the tilting of the primary two-thirds power-law cusp. The results throw new light on the earlier experiments of Roorda, which are re-examined in some detail. Contours of constant failure load in a two-dimensional imperfection space finally allow the worst types of compound imperfection to be identified.     

On the elastic stability of beams under unilateral constraints

Summary We analyze the problem of the axially loaded elastic beam on a one-sided elastic foundation. The existence of the critical load is proved and its expression is given.

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Sommario Per il problema della trave elastica su fondazione elastica unilaterale assialmente caricata si prova l'esistenza del carico critico e se ne dà l'espressione.

     

On the stability of an elastic column positioned on an elastic half space

Summary Stability of a heavy elastic column loaded by a concentrated force at the top is analysed. It is assumed that the column is fixed to a rigid circular plate that is positioned on a homogeneous, isotropic, linearly elastic half-space. The constitutive equations for the column are assumed in the form that allows axial compressibility and takes into account the influence of shear stresses. It is shown that eigenvalues of the linearized equations determine the bifurcation points of the full non-linear system of equilibrium equations. Also, the type of bifurcation at the lowest eigenvalue is examined and shown that it could be both super-and sub-critical. The post-critical shape of the column is determined by numerical integration of the equilibrium equations. Received 13 June 1998; accepted for publication 12 November 1998     

The planning of optimal motions of non-holonomic systems

Nonlinear Dynamics - A new method to the planning of optimal motions of the non-holonomic systems is presented. It is based on a non-classical formulation of the Pontryagin Maximum Principle given...     

Variational problem of general displacements in non-holonomic systems

In this paper, the difference of definitional regions of two kinds of q is discussed in analytical dynamics of non-holonomic systems. Its mechanical reason is given. Finally, some difficult problems which have remained unanswered up to now are solved.     

On the foundation of the stability theory of elastic structures

Summary Some difficulties and unsolved problems of the stability theory of continuous systems will be investigated. After having closely examined the meaning and importance of the choices of the distance between two configuration of a continuous system, the problem of the equilibrium stability of elastic structures will be formulated in a proper functional space, where it is possible to demonstrate the validity of the energy criterion as sufficient condition of stability in Lyapunov's sense.Lastly, are pointed out the still open problems on the stability criterion based on the second variation of the potential energy of which a partial proof is given.

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Sommario Vengono esaminate alcune difficoltà e problemi ancora aperti nella teoria della stabilità dei sistemi continui. Dopo aver approfondito il significato e l'importanza della scelta della distanza tra due configurazioni di un sistema continuo, il problema della stabilità dell'equilibrio delle strutture elastiche viene formulato in un opportuno spazio funzionale, nel quale è possibile dimostrare la validità del criterio dell'energia come condizione sufficiente di stabilità nel senso di Lyapunov.Si puntualizzano infine le questioni ancora aperte sul criterio di stabilità basato sulla variazione seconda e del quale si fornisce una parziale giustificazione.

     

On the general theory of stability for elastic bodies

This paper deals with the stability of loaded equilibrium configuration of elastic bodies, within the framework of nonlinear thermoelasticity. Although most of the results are obtained in the context of thermo-mechanical stability, the parallel developments for the purely mechanical stability are also considered. By use of an appropriate thermodynamic restriction, several theorems are proved concerning a sufficient condition for stability of the equilibrium configuration. These theorems hold under a general class of motions and their validity is not limited to small motions and small temperature changes from a loaded equilibrium state. Various generalizations of our main results are discussed and the reasonableness of the proposed stability criterion is demonstrated.     

Reduction of some classical non-holonomic systems with symmetry

Two types of nonholonomic systems with symmetry are treated: (i) the configuration space is a total space of a G-principal bundle and the constraints are given by a connection; (ii) the configuration space is G itself and the constraints are given by left-invariant forms. The proofs are based on the method of quasicoordinates. In passing, a derivation of the Maurer-Cartan equations for Lie groups is obtained. Simple examples are given to illustrate the algorithmical character of the main results.     

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     

Extension of Whittaker equations to non-holonomic mechanical systems

In 1904, using the energy integral Whittaker studied the reduction of a dynamical problem to a problem with fewer degrees of freedom for the holonomic conservative systems and obtained the Whittaker equation^[1].In this article, Whittaker equations are extended to non-holonomic systems and the generalized Whittaker equations are obtained. And then these equations are transformed into Kiel-sen’s form.Finally an example is given.