In this note we consider the nonholonomic problem of rolling without slipping and twisting of an ??-dimensional balanced ball over a fixed sphere. This is a ????(??)?Chaplygin system with an invariant measure that reduces to the cotangent bundle ??*?????1. For the rigid body inertia operator r I? = I? + ?I, I = diag(I1,...,In) with a symmetry I1 = I2 = ... =Ir ? Ir+1 = Ir+2 = ... = In, we prove that the reduced system is integrable, general trajectories are quasi-periodic, while for ?? ? 1, ?? ? 1 the Chaplygin reducing multiplier method does not apply.
{"title":"Note on a ball rolling over a sphere: Integrable Chaplygin system with an invariant measure without Chaplygin hamiltonization","authors":"B. Jovanović","doi":"10.2298/TAM190322003J","DOIUrl":"https://doi.org/10.2298/TAM190322003J","url":null,"abstract":"In this note we consider the nonholonomic problem of rolling without slipping and twisting of an ??-dimensional balanced ball over a fixed sphere. This is a ????(??)?Chaplygin system with an invariant measure that reduces to the cotangent bundle ??*?????1. For the rigid body inertia operator r I? = I? + ?I, I = diag(I1,...,In) with a symmetry I1 = I2 = ... =Ir ? Ir+1 = Ir+2 = ... = In, we prove that the reduced system is integrable, general trajectories are quasi-periodic, while for ?? ? 1, ?? ? 1 the Chaplygin reducing multiplier method does not apply.","PeriodicalId":44059,"journal":{"name":"Theoretical and Applied Mechanics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80056525","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 interaction of two parallel shifted cracks in plate bending is considered. The cracks closure has been investigated in the classical two-dimensional statement, using the model of smooth contact along a line. The influence of the relative position of cracks and of the contact of their edges on the forces and moment intensity factors has been studied by the singular integral equations method.
{"title":"Investigation of the interaction of two parallel shifted cracks in plate bending adjusted for their closure","authors":"T. Dalyak","doi":"10.2298/tam190808009d","DOIUrl":"https://doi.org/10.2298/tam190808009d","url":null,"abstract":"The problem of interaction of two parallel shifted cracks in plate bending is considered. The cracks closure has been investigated in the classical two-dimensional statement, using the model of smooth contact along a line. The influence of the relative position of cracks and of the contact of their edges on the forces and moment intensity factors has been studied by the singular integral equations method.","PeriodicalId":44059,"journal":{"name":"Theoretical and Applied Mechanics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78662735","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 the paper we take the first steps in studying the dynamics of systems with one-sided differential constraints defined by inequalities in the phase space. We give a general definition of motion for systems with such constraints. Within the framework of the classical non-holonomic model, and also for systems with servoconstraints (according to B/eghin), we present the conditions under which the system leaves two-sided differential constraints. As an example, we consider the Chaplygin sleigh with a one-sided constraint, which is realized by means of an anisotropic force of viscous friction. Variational principles for the determination of motion of systems with one-sided differential constraints are presented.
{"title":"On the dynamics of systems with one-sided non-integrable constraints","authors":"V. V. Kozlov","doi":"10.2298/TAM190123005K","DOIUrl":"https://doi.org/10.2298/TAM190123005K","url":null,"abstract":"In the paper we take the first steps in studying the dynamics of systems with one-sided differential constraints defined by inequalities in the phase space. We give a general definition of motion for systems with such constraints. Within the framework of the classical non-holonomic model, and also for systems with servoconstraints (according to B/eghin), we present the conditions under which the system leaves two-sided differential constraints. As an example, we consider the Chaplygin sleigh with a one-sided constraint, which is realized by means of an anisotropic force of viscous friction. Variational principles for the determination of motion of systems with one-sided differential constraints are presented.","PeriodicalId":44059,"journal":{"name":"Theoretical and Applied Mechanics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85206659","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 consider a planar motion of a rigid body partially filled with an inviscid liquid and suspended in a uniform horizontal flow. At first, we write the equations of the problem, prove the existence of an equilibrium under a suitable condition and, using a first integral, we give a sufficient condition of stability of this one. Afterwards, we give the equations of the small oscillations of the system about its equilibrium position. Writing these equations in an operatorial form, we prove the existence of a denumerable infinity of complex conjugate pairs of eigenvalues having the infinity as a point of accumulation and obtain the characteristic equation permitting the calculation of the eigenvalues.
{"title":"On the stability of an equilibrium and the small motions of a rigid body containing a liquid, suspended in a uniform flow","authors":"H. Essaouini, P. Capodanno","doi":"10.2298/tam181214007e","DOIUrl":"https://doi.org/10.2298/tam181214007e","url":null,"abstract":"In this paper, we consider a planar motion of a rigid body partially filled with an inviscid liquid and suspended in a uniform horizontal flow. At first, we write the equations of the problem, prove the existence of an equilibrium under a suitable condition and, using a first integral, we give a sufficient condition of stability of this one. Afterwards, we give the equations of the small oscillations of the system about its equilibrium position. Writing these equations in an operatorial form, we prove the existence of a denumerable infinity of complex conjugate pairs of eigenvalues having the infinity as a point of accumulation and obtain the characteristic equation permitting the calculation of the eigenvalues.","PeriodicalId":44059,"journal":{"name":"Theoretical and Applied Mechanics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88961884","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, a circular thick plate made of poroelastic piezoelectric ceramic is studied. The porosities of the plate vary through the thickness and axisymmetric behavior of a piezoelectric disk exhibiting hexagonal material symmetry of class 6 mm. Additionally, external mechanical loads which are in axi-symmetric general form act on the plate. The material properties of the plate vary exponentially as functions of the ?? variable in cylindrical coordinates. Based on an elasticity solution in terms of radial and axial displacements (??, ??), the governing partial differential equations are derived and solved analytically; mechanical stresses and electric displacements are then calculated. Finally an example which illustrates the application of the derived formulas is presented.
{"title":"Axisymmetric elasticity solution for an undrained saturated poro-piezoelastic thick disk","authors":"Ali Abjadi, M. Jabbari, Rezaei Ahmad","doi":"10.2298/tam190911012a","DOIUrl":"https://doi.org/10.2298/tam190911012a","url":null,"abstract":"In this paper, a circular thick plate made of poroelastic piezoelectric ceramic is studied. The porosities of the plate vary through the thickness and axisymmetric behavior of a piezoelectric disk exhibiting hexagonal material symmetry of class 6 mm. Additionally, external mechanical loads which are in axi-symmetric general form act on the plate. The material properties of the plate vary exponentially as functions of the ?? variable in cylindrical coordinates. Based on an elasticity solution in terms of radial and axial displacements (??, ??), the governing partial differential equations are derived and solved analytically; mechanical stresses and electric displacements are then calculated. Finally an example which illustrates the application of the derived formulas is presented.","PeriodicalId":44059,"journal":{"name":"Theoretical and Applied Mechanics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78032371","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 a note at the 1928 International Congress of Mathematicians Cartan outlined how his ?method of equivalence? can provide the invariants of nonholonomic systems on a manifold ?? with kinetic lagrangians [29]. Cartan indicated which changes of the metric outside the constraint distribution ?? ? ???? preserve the nonholonomic connection ?????? = Proj?? ?????, ??,?? ? ??, where ????? is the Levi-Civita connection on ?? and Proj?? is the orthogonal projection over ??. Here we discuss this equivalence problem of nonholonomic connections for Chaplygin systems [30,31,62]. We also discuss an example-a mathematical gem!-found by Oliva and Terra [76]. It implies that there is more freedom (thus more opportunities) using a weaker equivalence, just to preserve the straightest paths: ?????? = 0. However, finding examples that are weakly but not strongly equivalent leads to an over-determined system of equations indicating that such systems should be rare. We show that the two notions coincide in the following cases: i) Rank two distributions. This implies for instance that in Cartan?s example of a sphere rolling on a plane without slipping or twisting, a (2,3,5) distribution, the two notions of equivalence coincide; ii) For a rank 3 or higher distribution, the corank of D in D+[D,D] must be at least 3 in order to find examples where the two notions of equivalence do not coincide. This rules out the possibility of finding examples on (3,5) distributions such as Chaplygin?s marble sphere. Therefore the beautiful (3,6) example by Oliva and Terra is minimal. 1.
{"title":"Cartan meets Chaplygin","authors":"M. K. Ehlers, J. Koiller","doi":"10.2298/TAM190116006E","DOIUrl":"https://doi.org/10.2298/TAM190116006E","url":null,"abstract":"In a note at the 1928 International Congress of Mathematicians Cartan outlined how his ?method of equivalence? can provide the invariants of nonholonomic systems on a manifold ?? with kinetic lagrangians [29]. Cartan indicated which changes of the metric outside the constraint distribution ?? ? ???? preserve the nonholonomic connection ?????? = Proj?? ?????, ??,?? ? ??, where ????? is the Levi-Civita connection on ?? and Proj?? is the orthogonal projection over ??. Here we discuss this equivalence problem of nonholonomic connections for Chaplygin systems [30,31,62]. We also discuss an example-a mathematical gem!-found by Oliva and Terra [76]. It implies that there is more freedom (thus more opportunities) using a weaker equivalence, just to preserve the straightest paths: ?????? = 0. However, finding examples that are weakly but not strongly equivalent leads to an over-determined system of equations indicating that such systems should be rare. We show that the two notions coincide in the following cases: i) Rank two distributions. This implies for instance that in Cartan?s example of a sphere rolling on a plane without slipping or twisting, a (2,3,5) distribution, the two notions of equivalence coincide; ii) For a rank 3 or higher distribution, the corank of D in D+[D,D] must be at least 3 in order to find examples where the two notions of equivalence do not coincide. This rules out the possibility of finding examples on (3,5) distributions such as Chaplygin?s marble sphere. Therefore the beautiful (3,6) example by Oliva and Terra is minimal. 1.","PeriodicalId":44059,"journal":{"name":"Theoretical and Applied Mechanics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85559157","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}
It is known that there is a solution to the Riemann problem for generalized Chaplygin gas model and that it contains the Dirac delta function in some cases. In some cases, usual admissible criteria can not extract a unique weak solution as it was shown in [4]. The aim of this paper is to use a solution to perturbed generalized Chaplygin model by a small constant ?? > 0 and obtain a its unique limit. A weak solution to the unperturbed system that equals that limit is called admissible. The perturbation is made by using the modified model of Chaplygin gas defined in [5].
{"title":"Admissibility of a solution to generalized Chaplygin gas","authors":"M. Nedeljkov","doi":"10.2298/tam190116002n","DOIUrl":"https://doi.org/10.2298/tam190116002n","url":null,"abstract":"It is known that there is a solution to the Riemann problem for generalized Chaplygin gas model and that it contains the Dirac delta function in some cases. In some cases, usual admissible criteria can not extract a unique weak solution as it was shown in [4]. The aim of this paper is to use a solution to perturbed generalized Chaplygin model by a small constant ?? > 0 and obtain a its unique limit. A weak solution to the unperturbed system that equals that limit is called admissible. The perturbation is made by using the modified model of Chaplygin gas defined in [5].","PeriodicalId":44059,"journal":{"name":"Theoretical and Applied Mechanics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89233554","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 the paper we present the new results in the theory of integrable Hamiltonian systems with two degrees of freedom and topological billiards. The results are obtained by the authors, their students, and participants of scientific seminars of the Department of Differential Geometry and Applications, Faculty of Mathematics and Mechanics at Lomonosov Moscow State University.
{"title":"Singularities of integrable Liouville systems, reduction of integrals to lower degree and topological billiards: Recent results","authors":"A. Fomenko, V. V. Vedyushkina","doi":"10.2298/TAM181215001F","DOIUrl":"https://doi.org/10.2298/TAM181215001F","url":null,"abstract":"In the paper we present the new results in the theory of integrable Hamiltonian systems with two degrees of freedom and topological billiards. The results are obtained by the authors, their students, and participants of scientific seminars of the Department of Differential Geometry and Applications, Faculty of Mathematics and Mechanics at Lomonosov Moscow State University.","PeriodicalId":44059,"journal":{"name":"Theoretical and Applied Mechanics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86108330","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}
Z. M. Swalmeh, T. H. Alkasasbeh, A. Hussanan, M. Mamat
The addition of nanoparticles into conventional heat transfer fluids is one of the modern science techniques that offer better heat transfer performance. However, micropolar fluid model is not considered under these nanoparticles effects. Therefore, the main objective of this study is to explore the nanofluids to understand the microstructure and inertial characteristics of nanoparticles. In this paper, heat transfer flow of a micropolar nanofluid mixture containing copper (Cu) and silver (Ag) nanoparticles is investigated over a heated horizontal circular cylinder. The dimensionless governing equations are solved via an implicit finite difference scheme known as Keller-box method. The results of the nanofluid mixture are compared with those with a Newtonian fluid. The effects of different parameters on velocity, angular velocity and temperature are examined graphically for both Cu/Ag-water and Cu/Ag-kerosene oil. Results show that the heat transfer coefficient of the Cu/Ag-kerosene oil nanofluid mixture is larger than that of the Cu/Ag-water nanofluid, when comparison is based on a fixed value of the micro-rotation parameter.
在传统的传热流体中加入纳米颗粒是提供更好的传热性能的现代科学技术之一。然而，在这些纳米粒子的作用下，微极流体模型没有被考虑。因此，本研究的主要目的是探索纳米流体，了解纳米颗粒的微观结构和惯性特性。本文研究了含铜(Cu)和银(Ag)纳米颗粒的微极性纳米流体在加热的水平圆柱体上的传热流动。无量纲控制方程通过隐式有限差分格式即凯勒盒法求解。将纳米流体混合物的结果与牛顿流体的结果进行了比较。考察了不同参数对Cu/ ag -水和Cu/ ag -煤油速度、角速度和温度的影响。结果表明，在微旋转参数固定的情况下，铜/银-煤油纳米流体的换热系数大于铜/银-水纳米流体的换热系数。
{"title":"Influence of micro-rotation and micro-inertia on nanofluid flow over a heated horizontal circular cylinder with free convection","authors":"Z. M. Swalmeh, T. H. Alkasasbeh, A. Hussanan, M. Mamat","doi":"10.2298/tam181120008s","DOIUrl":"https://doi.org/10.2298/tam181120008s","url":null,"abstract":"The addition of nanoparticles into conventional heat transfer fluids is one of the modern science techniques that offer better heat transfer performance. However, micropolar fluid model is not considered under these nanoparticles effects. Therefore, the main objective of this study is to explore the nanofluids to understand the microstructure and inertial characteristics of nanoparticles. In this paper, heat transfer flow of a micropolar nanofluid mixture containing copper (Cu) and silver (Ag) nanoparticles is investigated over a heated horizontal circular cylinder. The dimensionless governing equations are solved via an implicit finite difference scheme known as Keller-box method. The results of the nanofluid mixture are compared with those with a Newtonian fluid. The effects of different parameters on velocity, angular velocity and temperature are examined graphically for both Cu/Ag-water and Cu/Ag-kerosene oil. Results show that the heat transfer coefficient of the Cu/Ag-kerosene oil nanofluid mixture is larger than that of the Cu/Ag-water nanofluid, when comparison is based on a fixed value of the micro-rotation parameter.","PeriodicalId":44059,"journal":{"name":"Theoretical and Applied Mechanics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75031966","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 use conformal mapping techniques to design two interacting non-elliptical rigid inclusions, each of which is loaded by a couple, which ensure the so-called ?harmonic condition? in which the original mean stress in the matrix remains undisturbed after the introduction of the inclusions. We show that for prescribed Poisson?s ratio and corresponding geometric parameters, several restrictions are necessary on the external loadings to ensure the harmonic condition. It is seen from our analysis that: (i) the interfacial and hoop stresses are uniformly distributed along each of the inclusion-matrix interfaces; (ii) the interfacial normal and hoop stresses along the two interfaces are completely determined by the Poisson?s ratio and the constant mean stress in the matrix whilst the interfacial tangential stress along the two interfaces can be completely determined by the moments of the couples and the areas of the two inclusions; (iii) the existence of the applied couples will influence the non-elliptical shapes of the two rigid harmonic inclusions when the moment to area ratios for the two inclusions differ.
{"title":"Two interacting non-elliptical rigid harmonic inclusions loaded by couples","authors":"Xu Wang, P. Schiavone","doi":"10.2298/tam190228010w","DOIUrl":"https://doi.org/10.2298/tam190228010w","url":null,"abstract":"We use conformal mapping techniques to design two interacting non-elliptical rigid inclusions, each of which is loaded by a couple, which ensure the so-called ?harmonic condition? in which the original mean stress in the matrix remains undisturbed after the introduction of the inclusions. We show that for prescribed Poisson?s ratio and corresponding geometric parameters, several restrictions are necessary on the external loadings to ensure the harmonic condition. It is seen from our analysis that: (i) the interfacial and hoop stresses are uniformly distributed along each of the inclusion-matrix interfaces; (ii) the interfacial normal and hoop stresses along the two interfaces are completely determined by the Poisson?s ratio and the constant mean stress in the matrix whilst the interfacial tangential stress along the two interfaces can be completely determined by the moments of the couples and the areas of the two inclusions; (iii) the existence of the applied couples will influence the non-elliptical shapes of the two rigid harmonic inclusions when the moment to area ratios for the two inclusions differ.","PeriodicalId":44059,"journal":{"name":"Theoretical and Applied Mechanics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76095802","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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