The stability of mechanical systems with positional non-conservative forces

A classical problem is discussed, namely, the influence of the structure of the applied forces on the stability of the equilibrium position of an autonomous mechanical system. Several propositions extending the Thomson-Tate-Chetayev theorems to systems with non-conservative positional forces are proved.     

The stability and stabilization of the motion of non-conservative mechanical systems

Two stability problems are solved. In the first, the stability of mechanical systems, on which dissipative, gyroscopic, potential and positional non-conservative forces (systems of general form) act, is investigated. The stability is considered in the case when the potential energy has a maximum at equilibrium. The condition for asymptotic stability is obtained by constructing Lyapunov's function. In the second problem, the possibility of stabilizing a gyroscopic system with two degrees of freedom up to asymptotic stability using non-linear dissipative and positional non-conservative forces is investigated. Stability of the gyroscopic system is achieved by gyroscopic stabilization. The stability conditions are obtained in terms of the system parameters. Cases when the gyroscopic stabilization is disrupted by these non-linear forces are indicated.     

The exponential stability and stabilization of non-autonomous mechanical systems with non-conservative forces

The exponential stability of the unperturbed motion of a non-autonomous mechanical system with a complete set of forces, that is, dissipative, gyroscopic, potential and non-conservative positional forces, is investigated. The problem of stabilizing a non-autonomous system with specified non-conservative forces is considered with and without the use of potential forces. The problem of stabilizing a non-autonomous system with specified potential forces by the action of the forces of another structure is studied. The domain of stabilizability of the relative equilibrium position of a satellite in a circular orbit is found.     

The Ziegler Spectrum of a Locally Coherent Grothendieck Category

The general theory of locally coherent Grothendieck categoriesis presented. To each locally coherent Grothendieck categoryC a topological space, the Ziegler spectrum of C, is associated.It is proved that the open subsets of the Ziegler spectrum ofC are in bijective correspondence with the Serre subcategoriesof coh C the subcategory of coherent objects of C. This is aNullstellensatz for locally coherent Grothendieck categories.If R is a ring, there is a canonical locally coherent Grothendieckcategory RC (respectively, CR) used for the study of left (respectively,right) R-modules. This category contains the category of R-modulesand its Ziegler spectrum is quasi-compact, a property used toconstruct large (not finitely generated) indecomposable modulesover an artin algebra. Two kinds of examples of locally coherentGrothendieck categories are given: the abstract category theoreticexamples arising from torsion and localization and the examplesthat arise from particular modules over the ring R. The dualitybetween coh-(RC) and coh-CR is shown to give an isomorphismbetween the topologies of the left and right Ziegler spectraof a ring R. The Nullstellensatz is used to give a proof ofthe result of Crawley-Boevey that every character : K₀(coh-C) Z is uniquely expressible as a Z-linear combination of irreduciblecharacters. 1991 Mathematics Subject Classification: 16D90,18E15.     

The control of a wheeled mechanical system

Non-holonomic systems with rolling or wheeled systems are investigated. The investigation is restricted to kinematic models and the dynamics of the drive mechanism of the system are taken into account. A control law is constructed which stabilizes the motion of a wheeled system along a specified trajectory (a plane smooth curve). For the basic variables of the system, the property of stabilizability is substantiated in the large.     

Dynamic analysis of a non-conservative band-wheel system with a moving boundary Part 1: Mathematical modelling

The dynamic band/wheel system with a moving boundary is very different from that with a fixed length, it is a non-conservative mass system and the boundary position is not fixed. In this paper, the moving boundary is one unknown to be determined associated with the string displacement, and the partial differential equation of the transverse vibrations of the band and the transversality condition of the moving boundary are derived by the calculus of variation and Hamilton’s principle. In order to verify the dynamic formulation is correct, we reduce the governing equation and separation point to coincide with the previous studies. It is found that the physical properties of the moving boundary could be obtained from the geometric constraint of the band/wheel system.     

An application of a theorem of Ziegler

The aim of this note is to prove that an artin algebra (resp. a finite dimensional algebra over an algebraically closed field) all of whose algebraically compact (resp. -algebraically compact) indecomposable modules are finitely generated must be of finite-representation type. The result is derived with the aid of a theorem of Ziegler.     

The quadratic integrals of a complex mechanical system in a permanent uniform field

A mechanical system, consisting of an invariable rigid body (a carrier) and a subsystem, the configuration and composition of which may vary with time (the motion of its components with respect to the carrier is specified), is considered. The system moves in a uniform gravitational field around a fixed point of the carrier. The general form of the quadratic integral is obtained when there is no dynamic symmetry, and the necessary and sufficient conditions for it to exist are found. The conditions when the integral can be split into two independent integrals and the equations of motion are reduced to autonomous form, are obtained.     

Stabilization and destabilization in non-conservative gyroscopic systems

Stability of a linear autonomous non-conservative system in presence of potential, gyroscopic, dissipative, and nonconservative positional forces is studied. The cases when the non-conservative system is close to a gyroscopic system or to a circulatory one, are examined. It is known that the marginal stability of gyroscopic and circulatory systems can be destroyed or improved up to asymptotic stability due to action of small non-conservative positional and velocity-dependent forces. The present contribution shows that in both cases the boundary of the asymptotic stability domain of the perturbed system possesses singularities such as “Dihedral angle” and “Whitney umbrella” that govern stabilization and destabilization. Approximations of the stability boundary near the singularities and estimates of the critical gyroscopic and circulatory parameters are found in an analytic form. In case of two degrees of freedom these estimates are obtained in terms of the invariants of matrices of the system. As an example, the asymptotic stability domain of the modified Maxwell-Bloch equations is investigated with an application to the stability problems of gyroscopic systems with stationary and rotating damping, such as the Crandall gyropendulum, tippe top and Jellet's egg. An instability mechanism in a system with two degrees of freedom, originating after discretization of models of a rotating disc in frictional contact and possessing the spectral mesh in the plane ‘frequency’ versus ‘angular velocity’, is described in detail and its role in the disc brake squeal problem is discussed. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The effect of variability in a closed-loop conveyor system

A production system consisting of a work station, a loading and unloading stations linked by a closed-loop material-handling system is considered. The material-handling system consists of two continuous line conveyors. Each conveyor is assumed to have a specified constant velocity, length and capacity. The work station is assumed to have a single machine, an unloading station and no local storage. In this production system the work pieces, which at their instant of arrival find the work station busy, are blocked. Those work pieces bypass the work station and are transported by the conveyor to the loading station to merge with the incoming work pieces to be transported to the work station again. The above production system is modeled by a G/G/1/0 queueing loss system with retrials, stationary counting arrival process, generally distributed service times, a single server and no waiting room. The flow of work pieces inside the system is modeled by a point process and is approximated by a renewal process. To analyze the asymptotic performance of the above system, a recursive procedure is developed. Furthermore, an expression for the asymptotic distribution of the number of work pieces along each conveyor is derived and is used to control the congestion along the material handling system. Finally numerical results are provided and compared against those from a simulation study.     

A limit cycle in a non-conservative sytem in 1:2 resonance

Systems with two degrees of freedom under the action of non-conservative forces and small linear viscous friction forces with complete dissipation are considered. A limit cycle that exists under specific conditions in the vicinity of an isolated equilibrium of the system is constructed using asymptotic methods in the case of 1:2 resonance. A criterion for asymptotic stability of this cycle is obtained to within equality-type relations. An estimate of the region of attraction of the limit cycle in a truncated system is given. Oscillations of a two-link rod system on a plane in 1:2 resonance are investigated. ©2011     

The inverse problem for Lagrangian systems with certain non-conservative forces

We discuss two generalizations of the inverse problem of the calculus of variations, one in which a given mechanical system can be brought into the form of Lagrangian equations with non-conservative forces of a generalized Rayleigh dissipation type, the other leading to Lagrangian equations with so-called gyroscopic forces. Our approach focusses primarily on obtaining coordinate-free conditions for the existence of a suitable non-singular multiplier matrix, which will lead to an equivalent representation of a given system of second-order equations as one of these Lagrangian systems with non-conservative forces.     

Dissipation induced instabilities in continuous non-conservative systems

In this contribution we analyze the stabilizing and destabilizing effect of small damping for rather general class of continuous non-conservative systems, described by the non-self-adjoint boundary eigenvalue problems. Explicit asymptotic expressions obtained for the stability domain allow us to predict when a given combination of the damping parameters leads to increase or to decrease in the critical non-conservative load. The results obtained explain why different types of internal and external damping so surprisingly influence on the stability of non-conservative systems. As a mechanical example the stability of a viscoelastic rod with small internal and external damping, loaded by tangential follower force, is studied in detail. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The linear mechanical model for hand-arm system

The paper presents a study regarding the linear mechanical models corresponding to hand-arm system. These models was simplified unused the joint in wrist, elbow and shoulder. Also, for these mechanical models effectuated the equations system that would integrate, it obtained the solutions (displacements and velocities) finally. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Drift and diffusion for a mechanical system

Summary We consider a mechanical system in the plane, consisting of a vertical rod of length , with its center moving on the horizontal axis, subject to elastic collisions with the particles of a free gas, and to a constant forcef. Assuming a suitable initial measure we show that the evolution of the system as seen from the rod is described by an exponentially ergodic irreducible Harris chain, implying convergence to a stationary invariant measure ast. We deduce that in the proper scaling the motion of the rod is described as a drift plus a diffusion. We prove in conclusion that the diffusion is nondegenerate and that the drift is nonzero iff0 and has the same sign off.Partially supported by C.N.R. (G.N.F.M.) and M.U.R.S.T. research funds.     

The analysis of the degenerate case of oscillations of a mechanical system

In this paper, the nonlinear model of the mechanism with two degrees of freedom will be studied. An approximate analytical solution of the differential equation of motion in the series showed the presence of features in the aspiration of the mass of one of the bodies to zero. It also gives an algorithm for finding the points of degeneracy of communication between small perturbations of the function of the problem and the derivatives of these functions at a time. Copyright © 2016 John Wiley & Sons, Ltd.     

The thermodynamic irreversibility of a mechanical system possessing the property of recurrence

The transition from one thermodynamic state to another by a mechanical system possessing the property of recurrence is investigated. It is established that the concept of a state in phenomenological thermodynamics corresponds to the concept of an attainability set in mechanics. It is shown that the presence of states that are reversible within the framework of classical mechanics, does not contradict the irreversibility of thermodynamic states.