P-positive definite matrices and stability of non conservative systems

International audienceThe bifurcation problem of constrained non-conservative systems with non symmetric stiffness matrices is investigated. It leads to study the subset $D_{p,n}$ of $ℳn(ℝ)$ of the so called $p$-positive definite matrices ($1 ≤ p ≤ n$). The main result ($D_{1,n} ⊂ D_{p,n}$) is proved, the reciprocal result is investigated and the consequences on the stability of elastic nonconservative systems are highlighted

Stability and dynamics of non-conservative systems, geometric aspects

Notre travail de doctorat porte sur des questions de stabilité d'une certaine classe de systèmes que nous appelons non conservatifs. Il s'agit de systèmes de corps rigides élastiques soumis à des chargements non conservatifs positionnels. Des formes d'instabilité nouvelles étaient mises en évidence (flottement) et des comportements paradoxaux, Nous nous intéresserons à un type de perturbation consistant en l'ajout de contraintes cinématiques cet ajout de contraintes va dans le sens de la stabilité. Cette problématique peut être un moyen de justifier un critère omniprésent dans ce travail et qui n'est pas à proprement parler un critère de stabilité. Il est appelé critère du travail du second ordre (CTSO). Le CTSO ainsi que la problématique des contraintes additionnelles restent le fil directeur de notre travail. A la vue des résultats obtenus, on peut s'étonner que ces aspects soient si peu connus de la communauté du calcul des structures et l'on a espoir que grâce à ce travail une démarche de réflexion générale sur les aspects non hamiltoniens soit menée. Ce travail, outre ses résultats propres, a également ouvert sans les approfondir des voies originales (matrices p définies positives, degré géométrique de non conservativité,…) et laisse entrevoir des problématiques importantes comme celle des liens entre les instabilités par flottement, le CTSO et l'ajout de contraintes cinématiques, laissant ainsi de nombreux thèmes pour des recherches futures.This paper investigates the linear static stability of constrained nonconservative mechanical systems. More precisely, the systems studied are elastic systems subjected to nonconservative positional forces. It is also well known that such systems may present paradoxical behaviors,. It is, however, less reported that other paradoxical effects may be met for additional constraints. The additional constraint may destabilize the system and preventing the instability by divergence of the constrained system (ie for any kinematic constraint) leads to the second order work criterion (CTSO). The CTSO and the problematic of additional constraints remain the principle of our work. Furthermore, the results obtained (p-positive definite matrices, The geometric degree of nonconservativity,….). THE CTSO AND additional KINEMATIC constraints, THUS LEAVING MANY THEMES FOR FUTURE RESEARCH

Stability of non conservative systems under two kinematic constraints and second order work criterion

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Additional constraints may soften a non-conservative structural system: Buckling and vibration analysis

The effect of additional kinematic constraints on eigenfrequencies of non conservative systems presenting a non symmetric stiffness matrix is investigated with the use of the second order work criterion. It is shown that there are always additional constraints that may soften structural systems, from both buckling and vibration points of view. The steps for building such constraints are given, consequences on stability are discussed and several illustrating examples are presented

Geometric degree of non conservativity

International audienceThis paper deals with non conservative mechanical systems subjected to non conservative positional forces and leading to non symmetric tangential stiffness matrices. The geometric degree of nonconservativity of such systems is then defined as the minimal number $\ell$ of kinematic constraints necessary to convert the initial system into a conservative one. The issue of finding this number and of describing the set of corresponding kinematic constraints is reduced to a linear algebra problem. This index $\ell$ of nonconservativity is the half of the rank of the sew symmetric part $K_a$ of the stiffness matrix $K$ that is always an even number. The set of constraints is extracted from the eigenspaces of the symmetric matrix $K_a^2$. Several examples including the well-known Ziegler column illustrate the results