This paper is devoted to briefly introducing a class of non-smooth controlled dynamical systems, namely complementarity dynamical systems. Several ways of writing the dynamics of such systems are presented (the hybrid dynamical systems approach and the differential inclusions framework). Some particular features concerning controllability and stabilization are discussed.
{"title":"On the control of non-smooth complementarity dynamical systems","authors":"B. Brogliato","doi":"10.1098/rsta.2001.0856","DOIUrl":"https://doi.org/10.1098/rsta.2001.0856","url":null,"abstract":"This paper is devoted to briefly introducing a class of non-smooth controlled dynamical systems, namely complementarity dynamical systems. Several ways of writing the dynamics of such systems are presented (the hybrid dynamical systems approach and the differential inclusions framework). Some particular features concerning controllability and stabilization are discussed.","PeriodicalId":20023,"journal":{"name":"Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2001-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89783376","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The main features of the non‐smooth contact dynamics (NSCD) method—the dynamical equation, the Signorini relation as a non-smooth modelling of unilateral contact, and the frictional Coulomb's law, treated with fully implicit algorithms—are briefly presented in this paper. By mere changes of variables, it appears that a large class of interface problems, including cohesive interface problems, may be solved using Signorini, Coulomb and standard NSCD algorithms. Emphasis is put on contact between deformable bodies. Examples illustrating numerical simulation are given for fibre-reinforced materials and for buildings made of blocks.
{"title":"Non-smooth contact dynamics approach of cohesive materials","authors":"M. Jean, Vincent Acary, Y. Monerie","doi":"10.1098/rsta.2001.0906","DOIUrl":"https://doi.org/10.1098/rsta.2001.0906","url":null,"abstract":"The main features of the non‐smooth contact dynamics (NSCD) method—the dynamical equation, the Signorini relation as a non-smooth modelling of unilateral contact, and the frictional Coulomb's law, treated with fully implicit algorithms—are briefly presented in this paper. By mere changes of variables, it appears that a large class of interface problems, including cohesive interface problems, may be solved using Signorini, Coulomb and standard NSCD algorithms. Emphasis is put on contact between deformable bodies. Examples illustrating numerical simulation are given for fibre-reinforced materials and for buildings made of blocks.","PeriodicalId":20023,"journal":{"name":"Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2001-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74048768","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The classical theory of rigid-body systems dynamics with perfect bilateral constraints is extended in order to take perfect unilateral constraints into account. A systematic formulation of the dynamics is derived and the most general admissible form of the impact-constitutive equation is obtained. Well-posedness of the evolution problem is proved under the assumption that the data are analytic.
{"title":"Formulation and well-posedness of the dynamics of rigid-body systems with perfect unilateral constraints","authors":"P. Ballard","doi":"10.1098/rsta.2001.0854","DOIUrl":"https://doi.org/10.1098/rsta.2001.0854","url":null,"abstract":"The classical theory of rigid-body systems dynamics with perfect bilateral constraints is extended in order to take perfect unilateral constraints into account. A systematic formulation of the dynamics is derived and the most general admissible form of the impact-constitutive equation is obtained. Well-posedness of the evolution problem is proved under the assumption that the data are analytic.","PeriodicalId":20023,"journal":{"name":"Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2001-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79902332","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A model is presented for the impact of a multiple-angled flexible bar in planar motion. The model consists of a system of nonlinear differential equations that considers the multiple collisions as well as frictional effects at the contacting end, and allows one to predict the rigid and elastic body motion after the impact. The mode functions are selected such that the method can be made computationally as simple as possible, without compromising accuracy. To describe the impact between the elastic bar and the rigid surface, the classical Hertzian contact theory and elasto-plastic indentation theory are used. Particular emphasis is placed on the geometry of the bar because the unit tangent vector and the unit normal vector have first-order discontinuities. Analytical and experimental results are compared to establish the accuracy of the model.
{"title":"Impact of a planar flexible bar with geometrical discontinuities of the first kind","authors":"D. Marghitu","doi":"10.1098/rsta.2001.0910","DOIUrl":"https://doi.org/10.1098/rsta.2001.0910","url":null,"abstract":"A model is presented for the impact of a multiple-angled flexible bar in planar motion. The model consists of a system of nonlinear differential equations that considers the multiple collisions as well as frictional effects at the contacting end, and allows one to predict the rigid and elastic body motion after the impact. The mode functions are selected such that the method can be made computationally as simple as possible, without compromising accuracy. To describe the impact between the elastic bar and the rigid surface, the classical Hertzian contact theory and elasto-plastic indentation theory are used. Particular emphasis is placed on the geometry of the bar because the unit tangent vector and the unit normal vector have first-order discontinuities. Analytical and experimental results are compared to establish the accuracy of the model.","PeriodicalId":20023,"journal":{"name":"Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2001-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76042535","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The dynamical impact problem in generalized coordinates is approximated by the penalty method, which is often used for numerical approximation. The correct penalty terms are devised to include loss of energy at impact, i.e. an arbitrary restitution coefficient eε [0, 1]. There is a certain freedom in the choice of the penalty term, which permits more convenient practical choices. The convergence of this approximation is proved. The result presented here is much more general than the results already known: beyond generalized coordinates, it includes a smooth time-dependent set of constraints and the possibility of zero restitution coefficients.
{"title":"Penalty method for impact in generalized coordinates","authors":"M. Schatzman","doi":"10.1098/rsta.2001.0859","DOIUrl":"https://doi.org/10.1098/rsta.2001.0859","url":null,"abstract":"The dynamical impact problem in generalized coordinates is approximated by the penalty method, which is often used for numerical approximation. The correct penalty terms are devised to include loss of energy at impact, i.e. an arbitrary restitution coefficient eε [0, 1]. There is a certain freedom in the choice of the penalty term, which permits more convenient practical choices. The convergence of this approximation is proved. The result presented here is much more general than the results already known: beyond generalized coordinates, it includes a smooth time-dependent set of constraints and the possibility of zero restitution coefficients.","PeriodicalId":20023,"journal":{"name":"Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2001-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87048692","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The paper presents the modelling of imperfect joints between plates by accounting for both the clearance and the rotative friction. This model is used to underline some special features of the static behaviour of rolling shutters. The numerical implementation is performed with an hybrid formulation coupled with a Newton-type method. The numerical instabilities may be easily controlled by applying some criteria from analysis of very simple examples.
{"title":"Numerical simulation of the behaviour of a multi-jointed structure","authors":"K. Ach, P. Alart","doi":"10.1098/rsta.2001.0909","DOIUrl":"https://doi.org/10.1098/rsta.2001.0909","url":null,"abstract":"The paper presents the modelling of imperfect joints between plates by accounting for both the clearance and the rotative friction. This model is used to underline some special features of the static behaviour of rolling shutters. The numerical implementation is performed with an hybrid formulation coupled with a Newton-type method. The numerical instabilities may be easily controlled by applying some criteria from analysis of very simple examples.","PeriodicalId":20023,"journal":{"name":"Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2001-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81742824","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, the continuous and numerical formulations of rigid–body dynamics based on measure differential inclusions and time–stepping methods recently developed are described and extended to include a finite number of elastic modes of vibration. The time-stepping methods already incorporate Coulomb friction, and are able to handle situations such as PainlevÉ's famous problem where impulsive forces occur without a collision. The elastic modes of vibration can be incorporated directly into the continuous formulation, but due to the stiffness typical of elastic vibrations, the numerical methods used need to be modified to incorporate them directly. The resulting numerical methods are dissipative in the limit, but only dissipate energy while there is contact.
{"title":"Finite-dimensional contact mechanics","authors":"D. Stewart","doi":"10.1098/rsta.2001.0904","DOIUrl":"https://doi.org/10.1098/rsta.2001.0904","url":null,"abstract":"In this paper, the continuous and numerical formulations of rigid–body dynamics based on measure differential inclusions and time–stepping methods recently developed are described and extended to include a finite number of elastic modes of vibration. The time-stepping methods already incorporate Coulomb friction, and are able to handle situations such as PainlevÉ's famous problem where impulsive forces occur without a collision. The elastic modes of vibration can be incorporated directly into the continuous formulation, but due to the stiffness typical of elastic vibrations, the numerical methods used need to be modified to incorporate them directly. The resulting numerical methods are dissipative in the limit, but only dissipate energy while there is contact.","PeriodicalId":20023,"journal":{"name":"Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2001-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89112011","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the internal constraints on the state quantities and on their velocities. We show that they can be taken into account by the free energy and the pseudopotential of dissipation. Applications to collisions of rigid and deformable solids, to damage, to adhesion, to phase changes with and without discontinuities of temperature, to shape-memory alloys, and finally to collisions of solids and liquids, show how flexible and productive is the method.
{"title":"Internal constraints in mechanics","authors":"M. Frémond","doi":"10.1098/rsta.2001.0853","DOIUrl":"https://doi.org/10.1098/rsta.2001.0853","url":null,"abstract":"We study the internal constraints on the state quantities and on their velocities. We show that they can be taken into account by the free energy and the pseudopotential of dissipation. Applications to collisions of rigid and deformable solids, to damage, to adhesion, to phase changes with and without discontinuities of temperature, to shape-memory alloys, and finally to collisions of solids and liquids, show how flexible and productive is the method.","PeriodicalId":20023,"journal":{"name":"Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2001-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78926621","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper gives a review of mathematical results of existence and uniqueness of solutions to problems of linear elasticity involving friction. Static, steady sliding and quasi-static problems are discussed. Both the case of a continuum body and that of a space-discrete one are treated. The intention is to describe the state of the art for these problems.
{"title":"A review of the theory of static and quasi-static frictional contact problems in elasticity","authors":"L. Andersson, A. Klarbring","doi":"10.1098/rsta.2001.0908","DOIUrl":"https://doi.org/10.1098/rsta.2001.0908","url":null,"abstract":"This paper gives a review of mathematical results of existence and uniqueness of solutions to problems of linear elasticity involving friction. Static, steady sliding and quasi-static problems are discussed. Both the case of a continuum body and that of a space-discrete one are treated. The intention is to describe the state of the art for these problems.","PeriodicalId":20023,"journal":{"name":"Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2001-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73288919","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider vibro–impact problems, i.e. mechanical systems with a finite number of degrees of freedom submitted to perfect unilateral constraints. The dynamics is basically described by a second–order measure differential inclusion for the unknown position completed with a constitutive impact law. Another formulation of the problem as a frictionless sweeping process is possible: the unknown velocity belongs to an appropriate functional space and satisfies a first order measure differential inclusion. The equivalence of these two formulations is studied. They lead to time–discretizations written in terms of positions or in terms of velocities, respectively. We present these different schemes and we compare them on the simple test–problem of a bouncing ball. We recall the convergence results in the single constraint case. Moreover, an example of implementation of the scheme derived from the basic description of the dynamics is presented. Finally, in the multi–constraint case, we point out some theoretical and computational difficulties.
{"title":"Time discretization of vibro‐impact","authors":"L. Paoli","doi":"10.1098/rsta.2001.0858","DOIUrl":"https://doi.org/10.1098/rsta.2001.0858","url":null,"abstract":"We consider vibro–impact problems, i.e. mechanical systems with a finite number of degrees of freedom submitted to perfect unilateral constraints. The dynamics is basically described by a second–order measure differential inclusion for the unknown position completed with a constitutive impact law. Another formulation of the problem as a frictionless sweeping process is possible: the unknown velocity belongs to an appropriate functional space and satisfies a first order measure differential inclusion. The equivalence of these two formulations is studied. They lead to time–discretizations written in terms of positions or in terms of velocities, respectively. We present these different schemes and we compare them on the simple test–problem of a bouncing ball. We recall the convergence results in the single constraint case. Moreover, an example of implementation of the scheme derived from the basic description of the dynamics is presented. Finally, in the multi–constraint case, we point out some theoretical and computational difficulties.","PeriodicalId":20023,"journal":{"name":"Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2001-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86813348","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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