Hagedorn's theorem in some special cases of rheonomic systems

Hagedorn's theorem on instability [Arch. Rational Mech. Anal. 58 (1976) 1], deduced from Jacobi's form of Hamilton's principle, refers to scleronomic mechanical systems. In this paper we shall prove that Hagedorn's methodology can be generalized to a class of rheonomic mechanical systems with differential equations of motion which allow the existence of Painlevé's integral of energy. The application of this methodology to the case of rheonomic systems which allow, together with Painlevé's integral, cyclic integrals, as well as to the mechanical systems having resultant motion, with prescribed transport motion, and, finally, to the systems having Mayer's rheonomic potential, are also considered. Obtained results are illustrated by an example.     

On stability of stationary motion of a nonconservative nonholonomic rheonomic system

One considers, in this paper, the motion of a mechanical system in a nonstationary field of potential and positional forces, subject to the action of rheonomic holonomic and nonholonomic linear homogeneous constraints. Assuming that differential equations of motion of the system considered satisfy the conditions for the existence of Painlevé's integral of energy, formulated in [Painlevé, P., 1897. Leçons sur l'intégration des équations de la Mécanique, Paris] and [Appell, P., 1911. Traité de mécanique rationnelle, T. II, Dynamique des systémes – Mécanique analitique, Gauthier-Villars, Paris] and generalized in [Čović, V., Vesković, M., 2004. On stability of motion of a rheonomic system in the field of potential and positional forces, BAMM-1720/2004, No-2233, 93–100] and [Čović, V., Vesković, M., 2005. Hagedorn's theorem in some special cases of rheonomic systems. Mechanics Research Communications 32 (3), 265–280], the original mechanical system is substituted by an equivalent one whose Lagrangian function, nontransformed with respect to nonholonomic constraints, does not depend on time explicitly. Using the properties of the equivalent system, which, in contrast to the original one, moves in a stationary field of potential forces and in a nonstationary field of gyroscopic forces, the definition of cyclic coordinates is generalized, as well as sufficient conditions for the existence of (cyclic) first integrals, corresponding to coordinates mentioned and linear in velocities are established. Further, the conditions for the existence of steady motion of the system considered are found. In the case of existence of such a motion of the system, the Theorem of Routh's type on stability of that motion, based on the minimum of reduced potential for which it is shown that, in contrast to known cases (see, for example, [Gantmacher, F., 1975. Lectures in Analytical Mechanics. Mir Publisher, Moscow; Neimark, J., Fufaev, N., 1972. Dynamics of Nonholonomic Systems. Amer. Math. Soc., Providence, RI; Pars, L., 1962. An Introduction to Calculus of Variations. Heinemann, London; Karapetyan, A., Rumyantsev, V., 1983. Stability of conservative and dissipative systems. In: Itogi Nauki I Tekhniki: Obschaya Mekh., vol. 6, VINITI, Moscow, pp. 3–128 (in Russian)]), it includes the influence of the positional forces field, is formulated. Thus, the Routh's Theorem on stability of steady motion of a conservative mechanical system is extended to the case of a nonconservative system.     

Shapes of a rotating rheonomic rod

The case of a rotating fluid mass is one of the classical fields of mechanics [1]. In particular, the solution of creep problems for a rotating mass is actual in geophysics in connection with Earth gravity force simulation on rotating samples under laboratory conditions [2]. A special case of a rheonomic rod in a potential field was studied in [3], where it was shown that the main problem about the rod shapes is the problem of determining the relations between the Lagrangian and Euler coordinates in the creep process.In what follows, we show how this problem can be solved for a rotating rod.     

Carnot and Lorenz cycles for dual absorption system

Ideal cycles are a very important tool in performance analysis, comparison and development of actual thermodynamic systems. Carnot cycle has been recognized as one of the bases of thermodynamic science. One of the main draw backs of Carnot cycle has been corrected by Lorenz cycle, namely, the isothermal heat sources have been replaced by polytropic ones.The dual absorption system has all the advantages of the absorption cooling system, however, it avoids the use of the water consuming cooling tower. Absorption system is one of the strong alternatives to CFSs, since it utilizes refrigerants which has no bad effects on the ozone layer. It also, utilizes waste heat and solar energy as main driving power. The characteristics of such a system has been recently studied without introducing the basic ideal cycles. In this work Carnot and Lorenze cycles have been developed for this new absorption system. The graphical representation onT-s, PTX and lnP-1/T diagram have been also reported.Also, the conventional expression used for the coefficient of performance of the ideal absorption cycle has been corrected on a rigorous basis of first and second laws of thermodynamics.

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Carnot- und Lorenz-Kreisprozesse für ein zweistufiges Absorptions-System

Zusammenfassung Idealisierte Kreisprozesse sind ein wertvolles Werkzeug, um reale thermodynamische Systeme hinsichtlich ihrer Wirkungsweise zu analysieren, vergleichende Bewertungen vorzunehmen und Neuentwicklungen voranzutreiben. Der Carnot-Prozeß stellt einen Grundpfeiler der thermodynamischen Wissenschaft dar. Einer seiner Hauptnachteile — die isothermen Wärmereservoire — ist beim Lorenz-Prozeß durch Einführung polytroper Zustandsänderungen beseitigt worden. Das zweistufige Absorptionssystem besitzt alle Vorteile des Absorptions-Kühlsystems, kommt aber ohne wasserverbrauchenden Kühlturm aus. Das Absorptions-Verfahren ist die Alternative zu FCKW-Anlagen, da es Kühlmittel einsetzt, die keine nachteilige Auswirkung auf die Ozonschicht haben. Auch nutzt es Abwärme und Solarenergie als Hauptantriebsenergien. Das charakteristische Verhalten eines solchen Systems wurde kürzlich ohne Zugrundelegung eines idealen Vergleichsprozesses untersucht. In dieser Arbeit werden Carnot- und Lorenz-Prozesse für das neue Absorptions-System entwickelt. Graphische Darstellungen in Form vonT-s Diagrammen,p-T-x Diagrammen undp-1/T Diagrammen werden angegeben. Ferner erfolgt eine Korrektur des üblichen Ausdruckes für den Gütegrad bzw. die Leistungsziffer des idealen Absorptions-Prozesses durch strenge Anwendung des ersten und zweiten Hauptsatzes der Thermodynamik.

     

Transverse-longitudinal bending of rheonomic rods

In [1, 2], an energy method for the determination of critical buckling times is developed for rods subjected to compression in the conditions of longitudinal bending. In this case, for given compressive loads, the bending moments in the rod cross-sections depend only on the current deflection of the rod axis. In contrast to longitudinal bending, in the case of transverse-longitudinal bending the bending moment in general depends not only on the deflection but also on the axial coordinate and the reaction forces in the supports. Depending on the rod fixing conditions, the problems of transverse-longitudinal bending can be categorized as statically determinate or statically indeterminate. In the latter case, the derivation of equilibrium conditions for a rod segment is complicated by the indefiniteness of the reactions in the rod buckling process. In the current paper, the energy method developed in [1, 2] is extended to a class of statically indeterminate transverse-longitudinal bending problems. To determine the redundant variables, it is proposed to use the principle of minimum of additional dissipation.     

A METHOD FOR SOLVING THE DYNAMICS OF MULTIBODY SYSTEMS WITH RHEONOMIC AND NONHOLONOMIC CONSTRAINTS

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Fractional dynamical system and its linearization theorem

Nowadays, it is known that the solution to a fractional differential equation can’t generally define a dynamical system in the sense of semigroup property due to the history memory induced by the weakly singular kernel. But we can still establish the similar relationship between a fractional differential equation and the corresponding fractional flow under a reasonable condition. In this paper, we firstly present some results on fractional dynamical system defined by the fractional differential equation with Caputo derivative. Furthermore, the linearization and stability theorems of the nonlinear fractional system are also shown. As a byproduct, we prove Audounet–Matignon–Montseny conjecture. Several illustrative examples are given as well to support the theoretical analysis.     

A theorem for a shear-free sphere in stokes'' flow

A sphere theorem for non-axisymmetric Stokes' flow outside/inside a shear-free sphere is stated and proved. Harper's [2] result for axisymmetric flow is deduced as a special case. Several illusrative examples are given and it is found that the drag for external flow is equal to 4πμ a times the velocity which the singularity induces in undisturbed flow at the centre of the sphere.     

A local existence and uniqueness theorem for a K-BKZ-fluid

We consider the three-dimensional motion of a viscoelastic liquid occupying all of space. The constitutive law is assumed to be of the form suggested by Kaye [13] and Bernstein, Kearsley & Zapas [2]. An existence and uniqueness result for solutions of the initial value problem on sufficiently short time intervals is proved using Kato's theory of quasilinear hyperbolic equations.Sponsored by the United States Army under Contract No. DAAG29-80-C-0041. This material is based upon work supported by the National Science Foundation under Grants Nos. MCS-8210950 and MCS-8215064.     

Existence theorem for a minimum problem with free discontinuity set

We study the variational problem Where is an open set in ⁿ ,n2gL ^q () L (), 1q<+, O<, <+ andH _n–1 is the (n–1)-dimensional Hausdorff Measure.     

关于对动点的动量矩定理------理论力学札记之八

介绍质点系对动点的动量矩定理并讨论它的一些应用.     

Kantorovich theorem for variational inequalities

Kantorovich theorem was extended to variational inequalities by which the convergence of Newton iteration, the existence and uniqueness of the solution of the problem can be tested via computational conditions at the initial point.     

ON A GENERALIZATION OF BERTRAND’S THEOREM

Bertrand’s theorem for the determination of the applied forces to a holonomicsystem from one of its first integrals,is extended to nonholonomic systems.Someinteresting applications of this new result are also given.     

A theorem for a shear stress-free plane boundary in Stokes flow

A theorem for non-axisymmetrical Stokes flow about a shear stress-free plane boundary is established by using a representation for the velocity and pressure fields for the same flow in terms of biharmonic and harmonic functions. A corollary of the theorem is derived which gives the axisymmetrical Stokes flow in terms of the Stokes function about the same boundary. The formulae for drag and torque on the boundary are also given. A few illustrative examples are presented.     

LAGRANGE'S THEOREM FOR A CLASS OF NONHOLONOMICSYSTEMS AND ITS APPLICATION

I.IntroductionSinceE.T.Whittaker.proposedfoestabilit}'problellll'lofnonholononlicsystemsin1904forthefirsttime,thescholarsathomeandabroad11a\'emadealotofresearchesontheequilibriunlstabilityoflinearand11olllinearnonllolollolnicsystems,andhaveobtainedaseriesofimportantresultslZ--7].Hobbled'er,theexpositionandapplicationrelatedtoLagrange'stheorenlinthestabilityanalysisfornonholonomicsystemsisseldonlseenuptonow.Althoughitwasmentionedinreference[3].aspecialdiscussionhasnotbeencarriedoutyet.Asafam…