The notion of discriminantly separable polynomials of degree two in each of three variables has been recently introduced and related to a class of integrable dynamical systems. Explicit integration of such systems can be performed in a way similar to Kowalevski’s original integration of the Kowalevski top. Here we present the role of discriminantly separable polynomials in integration of yet another well known integrable system, the so-called generalized Kowalevski top the motion of a heavy rigid body about a fixed point in a double constant field. We present a novel way to obtain the separation variables for this system, based on the discriminantly separable polynomials.
{"title":"Discriminantly separable polynomials and the generalized Kowalevski top","authors":"V. Dragović, K. Kukić","doi":"10.2298/TAM170926016D","DOIUrl":"https://doi.org/10.2298/TAM170926016D","url":null,"abstract":"The notion of discriminantly separable polynomials of degree two in each of three variables has been recently introduced and related to a class of integrable dynamical systems. Explicit integration of such systems can be performed in a way similar to Kowalevski’s original integration of the Kowalevski top. Here we present the role of discriminantly separable polynomials in integration of yet another well known integrable system, the so-called generalized Kowalevski top the motion of a heavy rigid body about a fixed point in a double constant field. We present a novel way to obtain the separation variables for this system, based on the discriminantly separable polynomials.","PeriodicalId":44059,"journal":{"name":"Theoretical and Applied Mechanics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80166269","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 two-dimensional problem in an infinite microstretch thermoelastic solid with microtemperatures subjected to a mechanical source is studied. The medium is rotating with a uniform angular velocity Ω⃗. The normal mode analysis is used to obtain the exact expressions for the component of normal displacement, microtemperature, normal force stress, microstress tensor, temperature distribution, heat flux moment tensor and tangential couple stress. The effect of microrotation and stretch on the considered variables are illustrated graphically.
{"title":"Plane strain problem in a rotating microstretch thermoelastic solid with microtemperatures","authors":"P. Ailawalia, S. K. Sachdeva, D. Pathania","doi":"10.2298/TAM170102003A","DOIUrl":"https://doi.org/10.2298/TAM170102003A","url":null,"abstract":"A two-dimensional problem in an infinite microstretch thermoelastic solid with microtemperatures subjected to a mechanical source is studied. The medium is rotating with a uniform angular velocity Ω⃗. The normal mode analysis is used to obtain the exact expressions for the component of normal displacement, microtemperature, normal force stress, microstress tensor, temperature distribution, heat flux moment tensor and tangential couple stress. The effect of microrotation and stretch on the considered variables are illustrated graphically.","PeriodicalId":44059,"journal":{"name":"Theoretical and Applied Mechanics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82161101","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 note we consider a method of generating functions for systems with constraints and, as an example, we prove that the billiard mappings for billiards on the Euclidean space, sphere, and the Lobachevsky space are sympletic. Further, by taking a quadratic generating function we get the skew-hodograph mapping introduced by Moser and Veselov, which relates the ellipsoidal billiards in the Euclidean space with the Heisenberg magnetic spin chain model on a sphere. We define analogous mapping for the ellipsoidal billiard on the Lobachevsky space. It relates the billiard with the Heisenberg spin model on the light-like cone in the Lorentz–Poincare–Minkowski space.
{"title":"Billiards on constant curvature spaces and generating functions for systems with constraints","authors":"B. Jovanović","doi":"10.2298/TAM170523005J","DOIUrl":"https://doi.org/10.2298/TAM170523005J","url":null,"abstract":"In this note we consider a method of generating functions for systems with constraints and, as an example, we prove that the billiard mappings for billiards on the Euclidean space, sphere, and the Lobachevsky space are sympletic. Further, by taking a quadratic generating function we get the skew-hodograph mapping introduced by Moser and Veselov, which relates the ellipsoidal billiards in the Euclidean space with the Heisenberg magnetic spin chain model on a sphere. We define analogous mapping for the ellipsoidal billiard on the Lobachevsky space. It relates the billiard with the Heisenberg spin model on the light-like cone in the Lorentz–Poincare–Minkowski space.","PeriodicalId":44059,"journal":{"name":"Theoretical and Applied Mechanics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86752811","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}
Igor Popov, N. Lisitsa, Y. Baloshin, M. Dudin, S. Bober
. Scoliosis, being one of the most widespread spinal diseases among children, has been studied extensively throughout the history of medicine, yet there is no clear understanding of its initiating factors and the mechanogenesis of the monomorphic three-dimensional deformation due to its polyetiological nature. We present a novel mathematical model of the process of emergence of the three-dimensional deformation of the human spine based on variational principles. Typical scoliosis geometry is assumed to be described as minimal curves of a particular energy functional, which are shown to closely resemble actual scoliosis. We investigate the numerical properties of the first stage of scoliosis, which is shown to have the highest influence on the development of the disease.
{"title":"Variational model of scoliosis","authors":"Igor Popov, N. Lisitsa, Y. Baloshin, M. Dudin, S. Bober","doi":"10.2298/TAM170818012P","DOIUrl":"https://doi.org/10.2298/TAM170818012P","url":null,"abstract":". Scoliosis, being one of the most widespread spinal diseases among children, has been studied extensively throughout the history of medicine, yet there is no clear understanding of its initiating factors and the mechanogenesis of the monomorphic three-dimensional deformation due to its polyetiological nature. We present a novel mathematical model of the process of emergence of the three-dimensional deformation of the human spine based on variational principles. Typical scoliosis geometry is assumed to be described as minimal curves of a particular energy functional, which are shown to closely resemble actual scoliosis. We investigate the numerical properties of the first stage of scoliosis, which is shown to have the highest influence on the development of the disease.","PeriodicalId":44059,"journal":{"name":"Theoretical and Applied Mechanics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77511429","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 discuss problems of stability of stationary motions of conservative and dissipative mechanical systems with first integrals. General results are illustrated by the problem of motion of a rotationally symmetric rigid body on a perfectly rough plane. Application of the Routh–Salvadori theorem and its generalizations [1–4] for investigation of stability of stationary motions of mechanical systems with first integrals U0 = c0, U1 = c1, . . . , Uk = ck is reduced to study the type of stationary value of U0 (here U0 can be also a nonincreasing along system trajectories function) for fixed values of U1, . . . , Uk. The effective method of such investigation is proposed in [5]. This method does not take into account equations of motion of the considered system however it is supposed that all first integrals are known explicitly. On the other hand using results by I. M. Mindlin and G. K. Pozharitskii [6] it is possible to distinguish the systems [7] for which the stability analysis does not require the explicit form of all first integrals U1 = c1, . . . , Uk = ck, except U0 = c0. Let equations of motion of a mechanical system have the following form (here T means transposition): (1) d dt (︁∂K ∂?̇? )︁ = ∂K ∂q +G?̇? − ∂W ∂q − Γ ∂W ∂p ,
{"title":"The Routh Theorem for Mechanical Systems with Unknown First Integrals","authors":"A. V. Karapetyan, Alexander S. Kuleshov","doi":"10.2298/TAM170512008K","DOIUrl":"https://doi.org/10.2298/TAM170512008K","url":null,"abstract":"In this paper we discuss problems of stability of stationary motions of conservative and dissipative mechanical systems with first integrals. General results are illustrated by the problem of motion of a rotationally symmetric rigid body on a perfectly rough plane. Application of the Routh–Salvadori theorem and its generalizations [1–4] for investigation of stability of stationary motions of mechanical systems with first integrals U0 = c0, U1 = c1, . . . , Uk = ck is reduced to study the type of stationary value of U0 (here U0 can be also a nonincreasing along system trajectories function) for fixed values of U1, . . . , Uk. The effective method of such investigation is proposed in [5]. This method does not take into account equations of motion of the considered system however it is supposed that all first integrals are known explicitly. On the other hand using results by I. M. Mindlin and G. K. Pozharitskii [6] it is possible to distinguish the systems [7] for which the stability analysis does not require the explicit form of all first integrals U1 = c1, . . . , Uk = ck, except U0 = c0. Let equations of motion of a mechanical system have the following form (here T means transposition): (1) d dt (︁∂K ∂?̇? )︁ = ∂K ∂q +G?̇? − ∂W ∂q − Γ ∂W ∂p ,","PeriodicalId":44059,"journal":{"name":"Theoretical and Applied Mechanics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80940432","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}
Abstract. The problem of the small oscillations of an ideal liquid contained in a vessel in uniform rotation has been studied by Kopachevskii and Krein in the case of an entirely rigid vessel [3]. We propose here, a generalization of this model by considering the case of a vessel closed by an elastic circular plate. In this context, the linearized equations of motion of the system plateliquid are derived. Functional analysis is used to obtain a variational equation of the small amplitude vibrations of the coupled system around its equilibrium position, and then two operatorial equations in a suitable Hilbert space are presented and analyzed. We show that the spectrum of the system is real and consists of a countable set of eigenvalues and an essential continuous spectrum filling an interval. Finally the existence and uniqueness theorem for the solution of the associated evolution problem is proved by means the semigroups theory.
{"title":"Small oscillations of an ideal liquid contained in a vessel closed by an elastic circular plate, in uniform rotation","authors":"H. Essaouini, B. El, P. Capodanno","doi":"10.2298/TAM160123002E","DOIUrl":"https://doi.org/10.2298/TAM160123002E","url":null,"abstract":"Abstract. The problem of the small oscillations of an ideal liquid contained in a vessel in uniform rotation has been studied by Kopachevskii and Krein in the case of an entirely rigid vessel [3]. We propose here, a generalization of this model by considering the case of a vessel closed by an elastic circular plate. In this context, the linearized equations of motion of the system plateliquid are derived. Functional analysis is used to obtain a variational equation of the small amplitude vibrations of the coupled system around its equilibrium position, and then two operatorial equations in a suitable Hilbert space are presented and analyzed. We show that the spectrum of the system is real and consists of a countable set of eigenvalues and an essential continuous spectrum filling an interval. Finally the existence and uniqueness theorem for the solution of the associated evolution problem is proved by means the semigroups theory.","PeriodicalId":44059,"journal":{"name":"Theoretical and Applied Mechanics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74553865","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}
L. Kudrjavceva, M. Mićunović, D. Miloradović, A. Obradović
Research of vehicle response to road roughness is particularly important when solving problems related to dynamic vehicle stability. In this paper, unevenness of roads is considered as the source of non-linear vibrations of motor vehicles. The vehicle is represented by an equivalent spatial model with seven degrees of freedom. In addition to solving the response by simulating it within a numerical code, quasi-linearization of nonlinear differential equations of motion is carried out. Solutions of quasi-linear differential equations of forced vibrations are determined using the small parameter method and are indispensable for the study of spatial stability of the vehicle. An optimal stabilization for a simplified two-dimensional model was performed. Spatial stability and internal resonance are considered briefly.
{"title":"Bertolino-Baksa stability at nonlinear vibrations of motor vehicles","authors":"L. Kudrjavceva, M. Mićunović, D. Miloradović, A. Obradović","doi":"10.2298/TAM171128019K","DOIUrl":"https://doi.org/10.2298/TAM171128019K","url":null,"abstract":"Research of vehicle response to road roughness is particularly important when solving problems related to dynamic vehicle stability. In this paper, unevenness of roads is considered as the source of non-linear vibrations of motor vehicles. The vehicle is represented by an equivalent spatial model with seven degrees of freedom. In addition to solving the response by simulating it within a numerical code, quasi-linearization of nonlinear differential equations of motion is carried out. Solutions of quasi-linear differential equations of forced vibrations are determined using the small parameter method and are indispensable for the study of spatial stability of the vehicle. An optimal stabilization for a simplified two-dimensional model was performed. Spatial stability and internal resonance are considered briefly.","PeriodicalId":44059,"journal":{"name":"Theoretical and Applied Mechanics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81239615","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 note is concerned with the problem of determining the completely unstable linear non-conservative undamped (circulatory) dynamical systems. Several conditions that provide the complete instability for such systems are derived using the direct method of Lyapunov and the concept of controllability. The conditions are expressed directly via the matrices describing the dynamical system.
{"title":"A note on the complete instability of linear non-conservative undamped systems","authors":"R. Bulatović","doi":"10.2298/TAM170620010B","DOIUrl":"https://doi.org/10.2298/TAM170620010B","url":null,"abstract":"The note is concerned with the problem of determining the completely unstable linear non-conservative undamped (circulatory) dynamical systems. Several conditions that provide the complete instability for such systems are derived using the direct method of Lyapunov and the concept of controllability. The conditions are expressed directly via the matrices describing the dynamical system.","PeriodicalId":44059,"journal":{"name":"Theoretical and Applied Mechanics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75348378","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}
M. Dehsara, H. Fu, S. Mesarovic, D. P. Sekulic, M. Krivilyov
Phase field (diffuse interface) models accommodate diffusive triple line motion with variable contact angle, thus allowing for the no-slip boundary condition without the stress singularities. We consider two commonly used classes of phase field models: the compositionally compressible (CC) model with compressibility limited to the fluid mix within the diffuse interface, and the incompressible (IC) model. First, we show that the CC model applied to fluids with dissimilar mass densities exhibits the computational instability leading to the breakup of the triple line. We provide a qualitative physical explanation of this instability and argue that the compositional compressibility within the diffuse interface is inconsistent with the global incompressible flow. Second, we derive the IC model as a systematic approximation to the CC model, based on a suitable choice of continuum velocity field. Third, we benchmark the IC model against sharp interface theory and experimental kinetics. The triple line kinetics is well represented by the triple line mobility parameter. Finally, we investigate the effects of the bulk phase field diffusional mobility parameter on the kinetics of the wetting process and find that within a wide range of magnitudes the bulk mobility does not affect the flow.
{"title":"In)compressibility and parameter identification in phase field models for capillary flows","authors":"M. Dehsara, H. Fu, S. Mesarovic, D. P. Sekulic, M. Krivilyov","doi":"10.2298/TAM170803009D","DOIUrl":"https://doi.org/10.2298/TAM170803009D","url":null,"abstract":"Phase field (diffuse interface) models accommodate diffusive triple line motion with variable contact angle, thus allowing for the no-slip boundary condition without the stress singularities. We consider two commonly used classes of phase field models: the compositionally compressible (CC) model with compressibility limited to the fluid mix within the diffuse interface, and the incompressible (IC) model. First, we show that the CC model applied to fluids with dissimilar mass densities exhibits the computational instability leading to the breakup of the triple line. We provide a qualitative physical explanation of this instability and argue that the compositional compressibility within the diffuse interface is inconsistent with the global incompressible flow. Second, we derive the IC model as a systematic approximation to the CC model, based on a suitable choice of continuum velocity field. Third, we benchmark the IC model against sharp interface theory and experimental kinetics. The triple line kinetics is well represented by the triple line mobility parameter. Finally, we investigate the effects of the bulk phase field diffusional mobility parameter on the kinetics of the wetting process and find that within a wide range of magnitudes the bulk mobility does not affect the flow.","PeriodicalId":44059,"journal":{"name":"Theoretical and Applied Mechanics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85894312","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}
Buoyancy driven, adiabatic and compressible flow in relatively high solar chimneys is treated in the paper analytically by using one-dimensional model of flow. General equations written suitably in a non-dimensional form are used for a qualitative discussion pertaining to the mutual effects of gravity, viscosity and varying cross section of the chimney. It is shown that in case of low Mach number flow these equations possess exact solutions obtainable by ordinary mathematical methods for any given chimney shape. Also shown, and demonstrated on an example, is the procedure of evaluation of the chimney shape that satisfies a condition imposed beforehand upon the flow. For better insight into the role of various parameters the solutions are presented in the form of power series expansions.
{"title":"Compressible flow through solar chimneys with variable cross section - an exact solution","authors":"D. Djordjevic, S. A. Ćoćić","doi":"10.2298/TAM170815014D","DOIUrl":"https://doi.org/10.2298/TAM170815014D","url":null,"abstract":"Buoyancy driven, adiabatic and compressible flow in relatively high solar chimneys is treated in the paper analytically by using one-dimensional model of flow. General equations written suitably in a non-dimensional form are used for a qualitative discussion pertaining to the mutual effects of gravity, viscosity and varying cross section of the chimney. It is shown that in case of low Mach number flow these equations possess exact solutions obtainable by ordinary mathematical methods for any given chimney shape. Also shown, and demonstrated on an example, is the procedure of evaluation of the chimney shape that satisfies a condition imposed beforehand upon the flow. For better insight into the role of various parameters the solutions are presented in the form of power series expansions.","PeriodicalId":44059,"journal":{"name":"Theoretical and Applied Mechanics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74980961","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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