P‐positive definite matrices and stability of nonconservative systems

Abstract

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 M n(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

On the stability of nonconservative continuous systems under kinematic constraints

“…As long as P <P * , no additional kinematic constraint may destabilize the system. The first proof of this results has been established from a trick (see [7,21]) using the determinant of the augmented system…”

“…Many research papers have recently described the instability issue of non conservative elastic systems with a new point of view (see for example [7,[21][22][23][24][25][26][27][28][29]). They mainly build a bridge between the so-called second order work criterion and the usual Lyapounov stability criterion.…”

Geometric degree of nonconservativity

“…There are 2 m different independent systems A of constraints converting the nonconservative free into a conservative cons (A). This result may be considered as a sort of dual to the result about the destabilizing effect of adding kinematic constraints in nonconservative systems (see again [Challamel et al 2009;Nicot et al 2011;Lerbet et al 2012]): by adding a suitable constraint in a suitable eigenspace of K s , one can destabilize a stable nonconservative system. Here, by choosing appropriate constraints in suitable eigenspaces of K 2 a , one can convert a nonconservative system into a conservative one.…”

“…The genesis of the used approach lies in several papers [Challamel et al 2009;Nicot et al 2011;Lerbet et al 2012] that investigated the deep rule of the second-order work criterion proposed in [Hill 1958] for solids in the framework of nonassociated plasticity, and independently also proposed for instabilities for systems subjected to nonpotential forces in [Absi and Lerbet 2004]. This criterion performs especially well for nonconservative systems because, contrary to the divergence criterion, it remains "stable" under the action of additional kinematics constraints: if this criterion holds for a free system and for a value p of the load parameter, it still holds for the same value p and for a system subjected to any family of additional kinematic constraints.…”

“…Thus, extending the above-mentioned works concerning the effects of additional constraints on such n-DOF (degree of freedom) nonconservative mechanical systems [Challamel et al 2010;Lerbet et al 2012;, we focus on families of kinematic constraints that could convert free into a conservative system. More precisely, we address both the problems of the existence of a minimal family (according to the number of constraints) of such constraints and that of building the set of such families.…”