This article analyses the dynamics of a resonantly excited single–degree–of–freedom linear system coupled to an array of nonlinear autoparametric vibration absorbers (pendulums). Each pendulum is also coupled to the neighbouring pendulums by linear elastic springs. The case of a 1:1:…:2 internal resonance between pendulums and the primary oscillator is studied for stationary (harmonic) and non–stationary (slow frequency sweep) excitations. The method of averaging is used to obtain amplitude equations that determine the first–order approximation to the nonlinear response of the system. The amplitude equations are analysed for their equilibrium as well as non–stationary solutions as a function of the parameters associated with the absorber pendulums. For stationary excitation, most steady–state solutions correspond to modes in which only one pendulum and the primary system execute coupled motions. Conditions for the existence of manifolds of equilibria are revealed when the averaged equations are expressed in modal coordinates. In the non–stationary case with linear frequency sweep through the primary resonance region, delays through pitchforks, smooth but rapid transitions through jumps, and transitions from one stable coupled–mode branch to another are studied using numerical simulations of the amplitude equations. The array of autoparametric pendulums is shown to effectively attenuate the large–amplitude resonant response of structures over a wide band of excitation frequencies.
{"title":"Dynamics of structures with wideband autoparametric vibration absorbers: theory","authors":"A. Vyas, A. Bajaj, A. Raman","doi":"10.1098/rspa.2003.1204","DOIUrl":"https://doi.org/10.1098/rspa.2003.1204","url":null,"abstract":"This article analyses the dynamics of a resonantly excited single–degree–of–freedom linear system coupled to an array of nonlinear autoparametric vibration absorbers (pendulums). Each pendulum is also coupled to the neighbouring pendulums by linear elastic springs. The case of a 1:1:…:2 internal resonance between pendulums and the primary oscillator is studied for stationary (harmonic) and non–stationary (slow frequency sweep) excitations. The method of averaging is used to obtain amplitude equations that determine the first–order approximation to the nonlinear response of the system. The amplitude equations are analysed for their equilibrium as well as non–stationary solutions as a function of the parameters associated with the absorber pendulums. For stationary excitation, most steady–state solutions correspond to modes in which only one pendulum and the primary system execute coupled motions. Conditions for the existence of manifolds of equilibria are revealed when the averaged equations are expressed in modal coordinates. In the non–stationary case with linear frequency sweep through the primary resonance region, delays through pitchforks, smooth but rapid transitions through jumps, and transitions from one stable coupled–mode branch to another are studied using numerical simulations of the amplitude equations. The array of autoparametric pendulums is shown to effectively attenuate the large–amplitude resonant response of structures over a wide band of excitation frequencies.","PeriodicalId":20722,"journal":{"name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2004-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84779357","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 problem of finding a curl–free matrix–valued field E with values in an assigned set of matricesКhas received considerable attention. Given a bounded connected open set Ω, and a compact set Кof m × n matrices, in this paper we establish existence or non–existence results for the following problem: find B ∈ L∞(Ω, М m × n) such that Div B = 0 in Ω in the sense of distributions under the constraint that B(x) ∈ К almost everywhere in Ω. We consider the case of К={A,B}, rank(A−B) = n, and we establish non–existence both for the case of exact solutions described above and for the case of approximate solutions described in §1. We also prove existence of approximate solutions for a suitably chosen triple {A1,A2,A3,} of matrices with rank{Ai −Aj} =n, i≠jandi,j = 1,2,3. We give examples when the differential constraints are of a different type and present some applications to composites.
在指定的matricesКhas集合中找到一个无旋流的矩阵值域E的问题受到了相当大的关注。给定一个有界连通开集Ω和一个紧集Кof m × n个矩阵，在约束B(x)∈К几乎处处为Ω的条件下，在分布意义上求B∈L∞(Ω， М m × n)使得Div B = 0在Ω中的存在性或不存在性。我们考虑К={A,B}，秩(A−B) = n的情况，并建立了上述精确解和§1所述近似解的不存在性。对于秩为{Ai−Aj} =n, i≠jandi,j = 1,2,3的矩阵，我们也证明了适当选择的三元组{A1,A2,A3，}的近似解的存在性。我们给出了不同类型的微分约束的例子，并给出了在组合中的一些应用。
{"title":"Rigidity and lack of rigidity for solenoidal matrix fields","authors":"A. Garroni, V. Nesi","doi":"10.1098/rspa.2003.1249","DOIUrl":"https://doi.org/10.1098/rspa.2003.1249","url":null,"abstract":"The problem of finding a curl–free matrix–valued field E with values in an assigned set of matricesКhas received considerable attention. Given a bounded connected open set Ω, and a compact set Кof m × n matrices, in this paper we establish existence or non–existence results for the following problem: find B ∈ L∞(Ω, М m × n) such that Div B = 0 in Ω in the sense of distributions under the constraint that B(x) ∈ К almost everywhere in Ω. We consider the case of К={A,B}, rank(A−B) = n, and we establish non–existence both for the case of exact solutions described above and for the case of approximate solutions described in §1. We also prove existence of approximate solutions for a suitably chosen triple {A1,A2,A3,} of matrices with rank{Ai −Aj} =n, i≠jandi,j = 1,2,3. We give examples when the differential constraints are of a different type and present some applications to composites.","PeriodicalId":20722,"journal":{"name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2004-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74410452","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}
Observational studies of the last 20 years have revealed a spatial distribution of phytoplankton characterized by a wavenumber spectrum whose slope ranges between k−1 and k−3 (with k being the wavenumber), over horizontal scales ranging from 10 to 100 km. This strong spectral variability can be due either to the physics and/or to the biology. We show in this note that the low frequency variability (LFV) linked to the mesoscale dynamics may explain a significant part of the observed variability. Our numerical results also suggest that the difference between the phytoplankton and zooplankton spectral slopes can depend on the dynamical LFV.
{"title":"Does the low frequency variability of mesoscale dynamics explain a part of the phytoplankton and zooplankton spectral variability?","authors":"M. Lévy, P. Klein","doi":"10.1098/rspa.2003.1219","DOIUrl":"https://doi.org/10.1098/rspa.2003.1219","url":null,"abstract":"Observational studies of the last 20 years have revealed a spatial distribution of phytoplankton characterized by a wavenumber spectrum whose slope ranges between k−1 and k−3 (with k being the wavenumber), over horizontal scales ranging from 10 to 100 km. This strong spectral variability can be due either to the physics and/or to the biology. We show in this note that the low frequency variability (LFV) linked to the mesoscale dynamics may explain a significant part of the observed variability. Our numerical results also suggest that the difference between the phytoplankton and zooplankton spectral slopes can depend on the dynamical LFV.","PeriodicalId":20722,"journal":{"name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2004-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73952748","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 method is developed to suppress the intensity of the Rayleigh–Bénard convection by adjusting the heat flux distribution at the bottom boundary under the constraint of constant heat input to the system. The appropriate profile of heat flux, which is changing continually in accordance with the shape of convection cells in the domain, is determined by the model predictive control. The optimal control strategy given by the model predictive control is implemented efficiently by employing the Karhunen–Loéve Galerkin procedure through which the Boussinesq equation is reduced to a low–dimensional dynamic model. This method is found to yield accurate results with a decent requirement of computer time.
{"title":"Stabilization of Rayleigh–Bénard convection by means of mode reduction","authors":"H. Park, M. C. Sung, Jin-Yong Chung","doi":"10.1098/rspa.2003.1175","DOIUrl":"https://doi.org/10.1098/rspa.2003.1175","url":null,"abstract":"A method is developed to suppress the intensity of the Rayleigh–Bénard convection by adjusting the heat flux distribution at the bottom boundary under the constraint of constant heat input to the system. The appropriate profile of heat flux, which is changing continually in accordance with the shape of convection cells in the domain, is determined by the model predictive control. The optimal control strategy given by the model predictive control is implemented efficiently by employing the Karhunen–Loéve Galerkin procedure through which the Boussinesq equation is reduced to a low–dimensional dynamic model. This method is found to yield accurate results with a decent requirement of computer time.","PeriodicalId":20722,"journal":{"name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2004-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87800103","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}
Fermat's principle of stationary travel time serves as a powerful tool for solving direct and inverse problems of wave propagation in media with time–independent parameters. In this paper, Fermat's principle is extended to non–stationary media that support waves with frequency–independent velocity. The approach used to prove the stationarity of travel times with respect to deformation of the actual ray trajectory is based on a comparison of the rays that follow from the variational principle with those from the eikonal equation. Inhomogeneous, moving and anisotropic media are considered. The identities that relate phase and group velocities and their derivatives in general anisotropic, inhomogeneous, non–stationary media are established. It is shown that not only the travel time but also the eikonal is stationary on the actual ray in media with time–dependent parameters if all trial rays are required to arrive at the receiver simultaneously. Some corollaries and applications of Fermat's principle in non–stationary media are discussed briefly.
{"title":"Fermat's principle for non–dispersive waves in non–stationary media","authors":"O. Godin, A. Voronovich","doi":"10.1098/rspa.2003.1231","DOIUrl":"https://doi.org/10.1098/rspa.2003.1231","url":null,"abstract":"Fermat's principle of stationary travel time serves as a powerful tool for solving direct and inverse problems of wave propagation in media with time–independent parameters. In this paper, Fermat's principle is extended to non–stationary media that support waves with frequency–independent velocity. The approach used to prove the stationarity of travel times with respect to deformation of the actual ray trajectory is based on a comparison of the rays that follow from the variational principle with those from the eikonal equation. Inhomogeneous, moving and anisotropic media are considered. The identities that relate phase and group velocities and their derivatives in general anisotropic, inhomogeneous, non–stationary media are established. It is shown that not only the travel time but also the eikonal is stationary on the actual ray in media with time–dependent parameters if all trial rays are required to arrive at the receiver simultaneously. Some corollaries and applications of Fermat's principle in non–stationary media are discussed briefly.","PeriodicalId":20722,"journal":{"name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2004-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91372937","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 collection of resistors with two possible resistivities is considered. This paper investigates the overall or macroscopic behaviour of a square two–dimensional lattice of such resistors when both types coexist in fixed proportions in the lattice. The macroscopic behaviour is that of an anisotropic conductor at the continuum level and the goal of the paper is to describe the set of all possible such conductors. This is thus a problem of bounds, following in the footsteps of an abundant literature on the topic in the continuum case. The originality of the paper is that the investigation focuses on the interplay between homogenization and the passage from a discrete network to a continuum. A set of bounds is proposed and its optimality is shown when the proportion of each resistor on the discrete lattice is½. We conjecture that the derived bounds are optimal for all proportions.
{"title":"Bounds on the effective behaviour of a square conducting lattice","authors":"Andrea Braides, G. Francfort","doi":"10.1098/rspa.2003.1229","DOIUrl":"https://doi.org/10.1098/rspa.2003.1229","url":null,"abstract":"A collection of resistors with two possible resistivities is considered. This paper investigates the overall or macroscopic behaviour of a square two–dimensional lattice of such resistors when both types coexist in fixed proportions in the lattice. The macroscopic behaviour is that of an anisotropic conductor at the continuum level and the goal of the paper is to describe the set of all possible such conductors. This is thus a problem of bounds, following in the footsteps of an abundant literature on the topic in the continuum case. The originality of the paper is that the investigation focuses on the interplay between homogenization and the passage from a discrete network to a continuum. A set of bounds is proposed and its optimality is shown when the proportion of each resistor on the discrete lattice is½. We conjecture that the derived bounds are optimal for all proportions.","PeriodicalId":20722,"journal":{"name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2004-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87387906","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 we develop a theory to describe the Mullins effect in rubber–like solids, based on the notion of limiting chain extensibility associated with the Gent model of rubber elasticity. We relate the theory to the mechanisms of network alteration and to the pseudo–elasticity theory of the Mullins effect. The inherently anisotropic nature of the Mullins effect is also discussed.
{"title":"A theory of stress softening of elastomers based on finite chain extensibility","authors":"C. Horgan, R. Ogden, G. Saccomandi","doi":"10.1098/rspa.2003.1248","DOIUrl":"https://doi.org/10.1098/rspa.2003.1248","url":null,"abstract":"In this paper we develop a theory to describe the Mullins effect in rubber–like solids, based on the notion of limiting chain extensibility associated with the Gent model of rubber elasticity. We relate the theory to the mechanisms of network alteration and to the pseudo–elasticity theory of the Mullins effect. The inherently anisotropic nature of the Mullins effect is also discussed.","PeriodicalId":20722,"journal":{"name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2004-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82849785","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 the Laplacian in curved tubes of arbitrary cross–section rotating together with the Frenet frame along curves in Euclidean spaces of arbitrary dimension, subject to Dirichlet boundary conditions on the cylindrical surface and Neumann conditions at the ends of the tube. We prove that the spectral threshold of the Laplacian is estimated from below by the lowest eigenvalue of the Dirichlet Laplacian in a torus determined by the geometry of the tube.
{"title":"A lower bound to the spectral threshold in curved tubes","authors":"P. Exner, P. Freitas, D. Krejčiřík","doi":"10.1098/rspa.2004.1356","DOIUrl":"https://doi.org/10.1098/rspa.2004.1356","url":null,"abstract":"We consider the Laplacian in curved tubes of arbitrary cross–section rotating together with the Frenet frame along curves in Euclidean spaces of arbitrary dimension, subject to Dirichlet boundary conditions on the cylindrical surface and Neumann conditions at the ends of the tube. We prove that the spectral threshold of the Laplacian is estimated from below by the lowest eigenvalue of the Dirichlet Laplacian in a torus determined by the geometry of the tube.","PeriodicalId":20722,"journal":{"name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2004-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76388651","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 investigate the stability and geometrically nonlinear dynamics of slender rods made of a linear isotropic poroelastic material. Dimensional reduction leads to the evolution equation for the shape of the poroelastica where, in addition to the usual terms for the bending of an elastic rod, we find a term that arises from fluid–solid interaction. Using the poroelastica equation as a starting point, we consider the load–controlled and displacement–controlled planar buckling of a slender rod, as well as the closely related instabilities of a rod subjected to twisting moments and compression when embedded in an elastic medium. This work has applications to the active and passive mechanics of thin filaments and sheets made from gels, plant organs such as stems, roots and leaves, sponges, cartilage layers and bones.
{"title":"Dynamics of poroelastic filaments","authors":"J. Skotheim, Lakshminarayanan Mahadevan","doi":"10.1098/rspa.2003.1270","DOIUrl":"https://doi.org/10.1098/rspa.2003.1270","url":null,"abstract":"We investigate the stability and geometrically nonlinear dynamics of slender rods made of a linear isotropic poroelastic material. Dimensional reduction leads to the evolution equation for the shape of the poroelastica where, in addition to the usual terms for the bending of an elastic rod, we find a term that arises from fluid–solid interaction. Using the poroelastica equation as a starting point, we consider the load–controlled and displacement–controlled planar buckling of a slender rod, as well as the closely related instabilities of a rod subjected to twisting moments and compression when embedded in an elastic medium. This work has applications to the active and passive mechanics of thin filaments and sheets made from gels, plant organs such as stems, roots and leaves, sponges, cartilage layers and bones.","PeriodicalId":20722,"journal":{"name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2004-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82795486","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 treats a simplified model of combustion that takes into account only the exchange of momentum between the fuel particles and ambient gas. Even in the absence of particles there are no rigorous mathematical results available in three dimensions, so our attention is restricted to the two-dimensional case. Treating the particles as points leads to mathematical problems that are circumvented by an appropriate regularization. The latter sections of the paper study the behaviour of the model as this regularization parameter tends to zero under some appropriate scalings, showing that in the limit the standard Navier-Stokes equations are obtained, and that for numerical (Galerkin) approximations the paths of the particles will approach fluid streamlines.
{"title":"A coupled particle-continuum model: well-posedness and the limit of zero radius","authors":"James C. Robinson","doi":"10.1098/rspa.2003.1207","DOIUrl":"https://doi.org/10.1098/rspa.2003.1207","url":null,"abstract":"This paper treats a simplified model of combustion that takes into account only the exchange of momentum between the fuel particles and ambient gas. Even in the absence of particles there are no rigorous mathematical results available in three dimensions, so our attention is restricted to the two-dimensional case. Treating the particles as points leads to mathematical problems that are circumvented by an appropriate regularization. The latter sections of the paper study the behaviour of the model as this regularization parameter tends to zero under some appropriate scalings, showing that in the limit the standard Navier-Stokes equations are obtained, and that for numerical (Galerkin) approximations the paths of the particles will approach fluid streamlines.","PeriodicalId":20722,"journal":{"name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2004-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85841101","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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