{"title":"Stability of a two-layer system of inhomogeneous heavy barotropic fluids","authors":"Sh. A. Mukhamediev, E.I. Ryzhak, S.V. Sinyukhina","doi":"10.1016/j.jappmathmech.2016.07.005","DOIUrl":"10.1016/j.jappmathmech.2016.07.005","url":null,"abstract":"<div><p>Basing on the static energy criterion for a bounded domain<span> of an arbitrary shape and with regard for the boundary conditions at all parts of the boundary, the stability of a two-layer system of inhomogeneous barotropic fluids in the uniform gravity field<span><span> is studied for arbitrary distributions of their densities and elastic properties over depth. Almost coinciding with each other (up to the strictness of one of the two inequalities), equally valid for an arbitrary number of layers, the necessary and sufficient conditions for stability are obtained, that represents a new exhaustive result for the problem considered. Additionally (with </span>compressibility admitted) possible influence of viscosity (which may be anisotropic), and also the case when the layers consist of solid elastic materials, are considered. In the case of instability, the lower estimates for the greatest rate of disturbances growth are obtained.</span></span></p></div>","PeriodicalId":49686,"journal":{"name":"Pmm Journal of Applied Mathematics and Mechanics","volume":"80 3","pages":"Pages 264-270"},"PeriodicalIF":0.0,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.jappmathmech.2016.07.005","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81867531","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}
{"title":"Control of a double-cascade electromechanical system subject to perturbations","authors":"I.M. Anan’evskii, T.A. Ishkhanyan","doi":"10.1016/j.jappmathmech.2017.02.001","DOIUrl":"10.1016/j.jappmathmech.2017.02.001","url":null,"abstract":"<div><p>The control of a precision turntable mounted on an orbital spacecraft and designed to reduce the apparent acceleration of a container with a gravitationally sensitive load fixed on the platform is investigated. The platform is modelled by an electromechanical system consisting of three rigid bodies successively connected via electrical motors. All the bodies are able to rotate about a common axis. It is assumed that in the drive bearings there is friction, the parameters of which are unknown and variable, and the torques created by the friction are comparable in magnitude with the acceleration to be reduced. A feedback control law is proposed, which, within a finite time interval, ensures the specified motion of the container relative to the platform.</p></div>","PeriodicalId":49686,"journal":{"name":"Pmm Journal of Applied Mathematics and Mechanics","volume":"80 5","pages":"Pages 361-368"},"PeriodicalIF":0.0,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.jappmathmech.2017.02.001","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84424711","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the problem of the stability of a Hamiltonian system with one degree of freedom on the boundaries of regions of parametric resonance","authors":"A.P. Markeyev","doi":"10.1016/j.jappmathmech.2016.05.002","DOIUrl":"10.1016/j.jappmathmech.2016.05.002","url":null,"abstract":"<div><p><span><span>A one–degree–of–freedom system that is periodic in time is considered in the vicinity of its equilibrium position<span> in the case of multiple multipliers of the linearized system. It is assumed that the </span></span>monodromy matrix is reduced to diagonal form and, therefore, the equilibrium is stable in a first approximation. An algorithm for constructing a </span>canonical transformation<span> that brings the system into such a form, in which the terms of high (finite) order are eliminated in the expansion of the Hamiltonian into a time series and the second-order terms are totally absent, is described. The stability and instability conditions are found using Lyapunov's second method and KAM (Kolmogorov–Arnold–Moser) theory in one particular case, in which the stability problem is not solvable for the third- and fourth-order forms in the expansion of the original Hamiltonian into a series.</span></p></div>","PeriodicalId":49686,"journal":{"name":"Pmm Journal of Applied Mathematics and Mechanics","volume":"80 1","pages":"Pages 1-6"},"PeriodicalIF":0.0,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.jappmathmech.2016.05.002","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79493420","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}
{"title":"The contact problem with the bulk application of intermolecular interaction forces: the influence function for an elastic ‘layer–half-space’ system","authors":"I.A. Soldatenkov","doi":"10.1016/j.jappmathmech.2016.09.011","DOIUrl":"10.1016/j.jappmathmech.2016.09.011","url":null,"abstract":"<div><p>The formulation of the contact problem in the presence of bulk forces of intermolecular interaction of contacting bodies is examined. Principal attention is paid to the construction and analysis of the properties of the influence function (Green's function) for an elastic base consisting of a half-space and a layer adhered to it. The behaviour of this function with small and large values of the argument is studied. A numerical analysis of the dependence of the influence function on certain parameters of contact is carried out.</p></div>","PeriodicalId":49686,"journal":{"name":"Pmm Journal of Applied Mathematics and Mechanics","volume":"80 4","pages":"Pages 351-358"},"PeriodicalIF":0.0,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.jappmathmech.2016.09.011","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79557066","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}
{"title":"Analytical and numerical investigation of unsteady flow near a critical point","authors":"A.G. Petrova , V.V. Pukhnachev , O.A. Frolovskaya","doi":"10.1016/j.jappmathmech.2016.07.003","DOIUrl":"10.1016/j.jappmathmech.2016.07.003","url":null,"abstract":"<div><p><span>The problem of unsteady flow<span> of a viscous incompressible fluid<span><span> near a critical point on a plane boundary is investigated. A theorem on the existence and uniqueness of its solution in Hölder classes of functions on an arbitrary time interval with natural restrictions imposed on the initial function is proved. Qualitative properties of the solution are investigated. Results of a </span>numerical analysis demonstrating the possibility of disappearance after a finite time of a </span></span></span>counterflow<span> zone existing at the initial time in the case of a negative pressure gradient at the rigid plane are presented. In the case when the pressure gradient is a periodic function, a periodic mode of motion as well as breakdown of the solution after a finite time is possible.</span></p></div>","PeriodicalId":49686,"journal":{"name":"Pmm Journal of Applied Mathematics and Mechanics","volume":"80 3","pages":"Pages 215-224"},"PeriodicalIF":0.0,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.jappmathmech.2016.07.003","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76060220","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}
{"title":"Solution of axisymmetric hydraulic fracture problem for thinning fluids","authors":"A.M. Linkov","doi":"10.1016/j.jappmathmech.2016.06.009","DOIUrl":"10.1016/j.jappmathmech.2016.06.009","url":null,"abstract":"<div><p><span>A problem of axisymmetric propagation of a penny-shaped crack driven by a thinning fluid is considered. The solution to the accuracy of four significant digits, at least, is obtained on the basis of the modified formulation of hydraulic fracture problem by employing the </span>particle velocity<span><span>, rather than conventionally used flux, that serves the iterations in the opening to be properly organized after reducing the problem to the self-similar form. Numerical results obtained show relatively small dependence of self-similar quantities (fracture radius, propagation speed, opening, particle velocity, pressure, flux) on the behaviour index of a thinning fluid. The results provide benchmarks for the accuracy control of 3D simulators. They are used for assigning an </span>apparent viscosity<span> when simulating the action of a thinning fluid by replacing it with an equivalent Newtonian fluid.</span></span></p></div>","PeriodicalId":49686,"journal":{"name":"Pmm Journal of Applied Mathematics and Mechanics","volume":"80 2","pages":"Pages 149-155"},"PeriodicalIF":0.0,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.jappmathmech.2016.06.009","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72453447","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}
{"title":"The effect of surface stresses on the stress–strain state of shells","authors":"N.N. Rogacheva","doi":"10.1016/j.jappmathmech.2016.06.011","DOIUrl":"10.1016/j.jappmathmech.2016.06.011","url":null,"abstract":"<div><p><span>The effect of surface stresses on the stress–strain state (SSS) of an elastic shell is investigated. The surface stresses are represented as a static pre-loading localized in ultra-thin layers of the shell close to its surface. It is assumed that the mechanical properties of the surface layers differ from the properties of the material far from the surface. The three-dimensional elasticity equations are analysed by an </span>asymptotic method<span> using several asymptotic parameters. Equations are obtained by the method of reducing the three-dimensional equations of the theory of elasticity to the two-dimensional equations of shell theory in which surface stresses have to be taken into account for a sufficiently small shell thickness. Asymptotic estimates<span> of the effect of surface stresses on the SSS are obtained as a function of the ratio of the elastic moduli of the shell material and of the layer close to the surface, the ratio of the shell thickness to the thickness of the surface layer and the type of SSS and its variability with respect to the coordinates.</span></span></p></div>","PeriodicalId":49686,"journal":{"name":"Pmm Journal of Applied Mathematics and Mechanics","volume":"80 2","pages":"Pages 173-181"},"PeriodicalIF":0.0,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.jappmathmech.2016.06.011","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85461037","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}
{"title":"The movement of a disc on a rotating horizontal plane with dry friction","authors":"A.V. Karapetyan","doi":"10.1016/j.jappmathmech.2017.02.003","DOIUrl":"10.1016/j.jappmathmech.2017.02.003","url":null,"abstract":"<div><p>In the problem of the movement of a round disc on a horizontal plane uniformly rotating about a vertical with dry friction<span><span>, invariant sets are indicated and their properties determined. The law of motion of the disc is found for the case of a low </span>friction coefficient.</span></p></div>","PeriodicalId":49686,"journal":{"name":"Pmm Journal of Applied Mathematics and Mechanics","volume":"80 5","pages":"Pages 376-380"},"PeriodicalIF":0.0,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.jappmathmech.2017.02.003","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89532149","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}
{"title":"Projection-iterative modification of the method of local variations for problems with a quadratic functional","authors":"E.L. Hart, V.S. Hudramovich","doi":"10.1016/j.jappmathmech.2016.06.005","DOIUrl":"10.1016/j.jappmathmech.2016.06.005","url":null,"abstract":"<div><p>A projection-iterative scheme of realization of the method of local variations for solving variational problems<span> with a quadratic functional is proposed. The convergence of the proposed scheme to the use of functional analysis is investigated. For the case of solving problems of the local stability of a spherical shell, the practical effectiveness of the proposed modification of the method of local variations is shown.</span></p></div>","PeriodicalId":49686,"journal":{"name":"Pmm Journal of Applied Mathematics and Mechanics","volume":"80 2","pages":"Pages 156-163"},"PeriodicalIF":0.0,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.jappmathmech.2016.06.005","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88938285","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}
{"title":"Variational statements of the problem of controlled motions of a system with elastic elements","authors":"G.V. Kostin","doi":"10.1016/j.jappmathmech.2017.02.002","DOIUrl":"10.1016/j.jappmathmech.2017.02.002","url":null,"abstract":"<div><p>Two variational statements of the problem of controlled motions of a linear mechanical system with elastic elements that has a finite number of degrees of freedom are proposed. The first variational statement reduces to nominal minimisation of the non-negative quadratic functional. This functional, the dimension of which is equal to that of the action, comprises an integral residual of constitutive equations of state that define the relations between momenta and velocities of points of the system, and also between elastic forces and relative displacements. The second variational statement, with assignment of certain initial and terminal conditions with respect to time, is related to the Hamilton–Ostrogradskii principle. The variational properties of the two statements of the initial Cauchy problem are shown for the examined type of mechanical system, along with methods for estimating the accuracy of the approximate solutions. An example is given of the numerical calculation of motion of the system and estimates of the accuracy of solution with the chosen control law.</p></div>","PeriodicalId":49686,"journal":{"name":"Pmm Journal of Applied Mathematics and Mechanics","volume":"80 5","pages":"Pages 369-375"},"PeriodicalIF":0.0,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.jappmathmech.2017.02.002","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88552944","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}