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J. Lerbet

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Papers by J. Lerbet

Timoshenko beam under Winkler foundation

We are interested in buckling for Timoshenko beam supported along its length by an elastic wall (... more We are interested in buckling for Timoshenko beam supported along its length by an elastic wall (Winkler foundation) and subjected to a longitudinal force. We use analytical methods to determine buckling load and mode shape rather than numerical methods. Haringx and Engesser models are compared. We show that the rigidity of the wall solely gouverns the phenomena and that the two models are equivalent whatever the parameters of the problem.

A New Result in Linear Stability

International audienc

Kinematic Structural Stability

HAL (Le Centre pour la Communication Scientifique Directe), Aug 21, 2016

International audienc

Kinematics of Open Chains, Singularities

Multi-Body Kinematics and Dynamics with Lie Groups, 2018

In the previous chapters all the necessary formula for expanding kinematics or dynamics of any tr... more

Discrete and gradient Elasticity Cosserat Modeling of Granular Chains

International audienceThe present study theoretically investigates the free vibration problem of ... more International audienceThe present study theoretically investigates the free vibration problem of a discrete granular system. This micro structured system consists of uniform grains elastically connected by shear and rotation springs. Such a granular structural system is confined by discrete elastic interactions, to take into account the lateral granular contributions. This discrete repetitive system could be considered as a discrete Cosserat chain or a lattice elastic model with shear interaction. First, from the vibration equation governing the model, the natural frequencies are exactly calculated for the simply supports boundary conditions. Then it is shown that the discrete equations of this granular system, for an infinite number of grains, converge to the differential equations of the Bresse-Timoshenko beam resting on Winkler foundation. A gradient Bresse-Timoshenko model is constructed from continualization of the difference equations of the granular system. The natural freque...

Behaviour of a Kirchhoff rod loaded by a pure moment

Geometric Degree of Non Conservativeness

Geometric Science of Information, 2017

Symplectic structure is powerful especially when it is applied to Hamiltonian systems. We show he... more Symplectic structure is powerful especially when it is applied to Hamiltonian systems. We show here how this symplectic structure may define and evaluate an integer index that measures the defect for the system to be Hamiltonian. This defect is called the Geometric Degree of Non Conservativeness of the system. Darboux theorem on differential forms is the key result. Linear and non linear frameworks are investigated.

Dynamique d'une Chaîne Granulaire en Flexion -Modèles Discrets et Continus

CFM, Aug 26, 2019

La présente étude examine le problème de vibration libre d'un système granulaire discret de taill... more La présente étude examine le problème de vibration libre d'un système granulaire discret de taille finie. Ce système microstructuré est composé de grains uniformes reliés élastiquement par des ressorts de cisaillement et de rotation. Un tel système structurel granulaire est confiné par des interactions élastiques discrètes, pour prendre en compte les contributions granulaires latérales. Ce système peut être perçu comme un modèle de chaîne de Cosserat ou un modèle élastique de réseau avec interaction de cisaillement. Les fréquences propres de ce système discret sont calculées exactement pour la poutre granulaire reposant sur deux appuis, à partir de la résolution d'un problème de valeurs propres linéaire aux différences. Une formulation analytique du calcul des fréquences propres de la chaîne granulaire est donnée pour tous les modes discrets. On montre que les équations discrètes de ce système granulaire convergent, pour un nombre infini de grains, vers les équations différentielles du modèle de poutre de Bresse-Timoshenko reposant sur une fondation élastique de Winkler (ou modèle de Cosserat continu).

Sur les instabilités par divergence et par flottement de systèmes contraints à deux degrés de liberté

International audienceIt is now well ascertained that a variety of instability modes can appear b... more International audienceIt is now well ascertained that a variety of instability modes can appear before the conventional plastic limit condition is met. In this note, both flutter and divergence instability modes are investigated. First, the criterion for detecting their occurrence is established, and the case of kinematically constrained discrete systems is investigated. Based on an illustration example, the competition between the occurrences of each of these instability modes is then analyzed, showing that the prevalence of a given mode is strongly related to both the loading conditions and the stiffness properties of the material system in hand

Stabilité des sytèmes non conservatifs et critère du travail du second ordre

Les systèmes non conservatifs conduisantàconduisantà des matrices de rigidité non symétriques son... more Les systèmes non conservatifs conduisantàconduisantà des matrices de rigidité non symétriques sont connus pour présenter des comportements paradoxaux relativementàrelativementà la viscosité. L'´ etude de la stabilité par divergence sous contraintes cinématiques a récemment débuté pour le cas d' une contrainte et nousétudionsnousétudions ici le cas de deux contraintes pour relier la stabilité par divergence au critère du travail du second ordre. L'exemple d'un système de ZiegleràZieglerà trois degrés de liberté soumisàsoumisà une force suiveuse illustre le résultat g´néralg´néral. Abstract : Non conservative systems leading to a non symmetric stiffness matrix are known to have paradoxical behaviors like destabilization by adding viscosity. The study of divergence instability of non conservative system under kinematic constraints has been reported recently under one kine-matic constraint. We investigate here the case of two constraints, to highlight consequences on div...

Exact solutions for the vibration of finite granular beam using discrete and gradient elasticity cosserat models

Journal of Sound and Vibration, 2021

The present study theoretically investigates the free vibration problem of a discrete granular sy... more The present study theoretically investigates the free vibration problem of a discrete granular system. This problem can be considered as a simple model to rigorously study the effects of the microstructure on the dynamic behavior of the equivalent continuum structural model. The model consists of uniform grains confined by discrete elastic interactions, to take into account the lateral granular contributions. This repetitive discrete system can be referred to discrete Cosserat chain or a lattice elastic model with shear interaction. First for the simply supported granular beam resting on Winkler foundations, due to the critical frequencies which concern the nature of the dynamic results, the natural frequencies are exactly calculated, starting from the resolution of the linear difference eigenvalue problem. The natural frequencies of such a granular model are analytically calculated for whatever modes. It is shown that the difference equations governed to the discrete system converge to the differential equations of the Bresse-Timoshenko beam resting on Winkler foundation (also classified as a continuous Cosserat beam model) for an infinite number of grains. A gradient Bresse-Timoshenko model is constructed from continualization of the difference equations. This continuous gradient elasticity Cosserat model is obtained from a polynomial or a rational expansion of the pseudo-differential operators, stemming from the continualization process. Scale effects of the granular chain are captured by the continuous gradient elasticity model. The natural frequencies of the continuous gradient Cosserat models are compared with those of the discrete Cosserat model associated with the granular chain. The results clarify the dependency of the beam dynamic responses to the beam length ratio.

Singular divergence instability thresholds of kinematically constrained circulatory systems

Physics Letters A, 2014

Static instability or divergence threshold of both potential and circulatory systems with kinemat... more Static instability or divergence threshold of both potential and circulatory systems with kinematic constraints depends singularly on the constraints' coefficients. Particularly, the critical buckling load of the kinematically constrained Ziegler's pendulum as a function of two coefficients of the constraint is given by the Plücker conoid of degree n = 2. This simple mechanical model exhibits a structural instability similar to that responsible for the Velikhov-Chandrasekhar paradox in the theory of magnetorotational instability.

Geometric degree of nonconservativity

Mathematics and Mechanics of Complex Systems, 2014

This paper deals with nonconservative mechanical systems subjected to nonconservative positional ... more This paper deals with nonconservative mechanical systems subjected to nonconservative positional forces leading to nonsymmetric tangential stiffness matrices. The geometric degree of nonconservativity of such systems is then defined as the minimal number of kinematic constraints necessary to convert the initial system into a conservative one. Finding this number and describing the set of corresponding kinematic constraints is reduced to a linear algebra problem. This index of nonconservativity is the half of the rank of the skew-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 2 a. Several examples including the well-known Ziegler column illustrate the results.

Coordinate free nonlinear incremental discrete mechanics

ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik

In this paper we extend the Kinematic Structural Stability (KISS) issues obtained in the punctual... more In this paper we extend the Kinematic Structural Stability (KISS) issues obtained in the punctual linear case to a full non linear framework. Kinematic Structural Stability (KISS) refers to the stability of conservative or non-conservative elastic systems under kinematic constraints. We revisite and sometimes introduce some new mechanical and geometrically-related concepts, using the systematic language of vector bundles: transversality-stability, tangent stiffness tensor, loading paths and KISS. We bring an intrinsic geometrical meaning to these concepts and we extend the main results to a general nonlinear framework. The lack of connection on the configuration manifold in order to make derivatives of sections is bypassed thanks to the key rule of the nil section of the cotangent bundle. Two mechanical examples illustrate the interest of the new concepts, namely a discrete nonconservative elastic system under non-conservative loading (Ziegler's column) and, in more details, a four-sphere granular system with nonconservative interactions called diamond pattern.

On the stability of nonconservative continuous systems under kinematic constraints

ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik

In this paper we deal with recent results on divergence kinematic structural stability (ki.s.s.) ... more In this paper we deal with recent results on divergence kinematic structural stability (ki.s.s.) resulting from discrete nonconservative finite systems. We apply them to continuous nonconservative systems which are shown in the well-known Beck column. When the column is constrained by an appropriate additional kinematic constraint, a certain value of the follower force may destabilize the system by divergence. We calculate its minimal value, as well as the optimal constraint. The analysis is carried out in the general framework of inÞnite dimensional Hilbert spaces and non-self-adjoint operators.

Geometric degree of nonconservativity: Set of solutions for the linear case and extension to the differentiable non-linear case

Applied Mathematical Modelling

This paper deals with nonconservative mechanical systems as those subjected to nonconservative po... more This paper deals with nonconservative mechanical systems as those subjected to nonconservative positional forces and leading to non-symmetric tangential stiffness matrices. In a previous work, the geometric degree of nonconservativity of such systems, defined as the minimal number of kinematic constraints necessary to convert the initial system into a conservative one is found to be, in the linear framework, the half of the rank of the skewsymmetric part of the stiffness matrix. In the present paper, news results are reached. First, a more efficient solution of the initial linear problem is proposed. Second, always in the linear framework, the issue of describing the set of all corresponding kinematic constraints is given and reduced to the one of finding all the Lagrangian planes of a symplectic space. Third, the extension to the local non-linear case is solved. A four degree of freedom system exhibiting a maximal geometric degree of nonconservativity (s = 2) is used to illustrate our results. The issue of the global non-linear problem is not tackled. Throughout the paper, the issue of the effectiviness of the solution is systematically addressed.

Flexible Virtual Structure Consideration in Dynamic Modeling of Mobile Robots Formation

AIP Conference Proceedings, 2009

In cooperative mobile robotics, we look for formation keeping and maintenance of a geometric conf... more In cooperative mobile robotics, we look for formation keeping and maintenance of a geometric configuration during movement. As a solution to these problems, the concept of a virtual structure is considered. Based on this idea, we have developed an efficient flexible virtual structure, describing the dynamic model of n vehicles in formation and where the whole formation is kept dependant. Notes that, for 2D and 3D space navigation, only a rigid virtual structure was proposed in the literature. Further, the problem was limited to a kinematic behavior of the structure. Hence, the flexible virtual structure in dynamic modeling of mobile robots formation presented in this paper, gives more capabilities to the formation to avoid obstacles in hostile environment while keeping formation and avoiding inter-agent collision.

3-DIMENSIONAL Flutter Kinematic Structural Stability

Nonlinear Analysis: Real World Applications, 2016

Having recalled the kinematic structural stability (ki.s.s) issue and its solution for divergence... more Having recalled the kinematic structural stability (ki.s.s) issue and its solution for divergence-type instability, we address the same problem for flutter-type instability for the minimal required configuration of dimensions-meaning 3 degree of freedom systems. We first get a sufficient non optimal condition. In a second time, the complete issue is tackled by two different ways leading to same results. A first way using calculations on Grassmann and Stiefel manifolds that may be generalized for any dimensional configuration. A second way using the specific dimensional configuration is brought back to calculations on the sphere. Differences with divergence ki.s.s are highlighted and examples illustrate the results.

Variational formulation of divergence stability for constrained systems

Applied Mathematical Modelling, 2015

This paper deals with both divergence and second order work criteria and the kinematical structur... more This paper deals with both divergence and second order work criteria and the kinematical structural stability called ki.s.s. In this context, kinematical structural stability means that the criterion remains valid even if the system is subjected to additional kinematic constraints. First some developments about the effect of additional kinematics constraints are presented on divergence instability. Secondly, divergence and second order work criteria are addressed. Using a variational formulation, previous results from a usual algebraic formulation using Schur's complement formula are highlighted and finally translated through the ki.s.s. concept: unconditional ki.s.s. for the second order work and the divergence of conservative systems and conditional ki.s.s. for the divergence of nonconservative elastic systems.

Kinematics of Mechanisms to the Second Order – Application to the Closed Mechanisms

The degree of freedom of a closed mechanism is the dimension of a subset M of R n , M being the i... more The degree of freedom of a closed mechanism is the dimension of a subset M of R n , M being the inverse image of the unity by the closure function f : (q 1 ,. .. , q n) → f (q 1 ,. .. , q n), where q 1 ,. .. , q n are the articular coordinates. We first study the regular points for the mapping f from R n into the Lie group of displacements and, second, study the singularities of the mapping f. The classical theory of mechanisms considers, often implicitly, that f is a subimmersion. Here, the calculations are made in a larger case, up to second order, and the results are then slightly different. The case of such classical mechanisms as Bennett, Bricard, and Goldberg mechanisms, justify the considerations of this more general framework and the example of a Bricard mechanism is chosen as an application of the method.