The study of the dynamic contact in ultrasonic motor

In this paper, we consider a mathematical model which describes the dynamic contact between an elastic body and an obstacle. The process is assumed to be dynamic and the contact is modeled with the normal compliance. We present a variational formulation of the problem and prove the existence and the uniqueness of a weak solution. An efficient numerical method is presented to analyze this dynamic contact. This approach exploits the augmented Lagrangian concept and a special time integration algorithm. This is exploited to study the dynamic contact between the rotor and the stator inside an ultrasonic motor SHINSEI USR 60. Numerical results are presented and show the interest of this method to forecast the origin of the principal failure mode of this motor.     

Displacement,velocity, and acceleration dependent damping in a two degree of freedom system

Summary A system consisting of two spinning bars connected by a friction bearing, and undergoing plane motion is considered. Small oscillations superimposed on large displacements are investigated. It is shown that friction results in damping terms proportional to displacements, velocities, and accelerations.

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Résumé On considère un systeme de deux barres tournantes liées par un coussinet à friction. On étudie les oscillations petites superimposées sur les déplacements grands. On demontre que la friction produit des termes amortissants proportionels aux déplacements, aux vitesses et aux accelerations.

     

Control of a system with one degree of freedom under complex restrictions

A very simple dynamical system with one degree of freedom, controlled by a force of bounded magnitude, is considered. It is assumed that the magnitude of the force may increase gradually at a finite rate and that the force is switched off instantaneously. Under these restrictions, which simulate real servo-systems, a control is constructed that steers the system to the origin and has the simplest possible structure.     

Analysis of the stator/rotor-contact in a two stators hybrid transducer type ultrasonic motor

This paper presents the procedure to understand the contact mechanism of the hybrid transducer type piezoelectric ultrasonic motor with two stators by using finite element analysis. First, the vibration mode of a stator will be calculated and analyzed using the commercial finite element program package ANSYS. The computed resonance frequencies of the stator will be presented in the frequency domain, in term of input admittance analysis. After that the distribution of the elliptic path of motion of a particle on the stator surface area will be studied. Finally, the contact condition between the stator and the rotor will be addressed in the computation. The normal displacement, normal velocity and tangential velocity of a point on stator surface area will be calculated. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Non-linear oscillations of a hamiltonian system with one degree of freedom and fourth-order resonance

Non-linear oscillations of a 2π-periodic Hamiltonian system with one degree of freedom are considered . It is assumed that the origin of coordinates is an equilibrium position, the linearized system is assumed to be stable, its characteristic exponents ±iv are pure imaginary, and the value of 4v is close to an integer. When the methods of classical perturbation theory are used, the investigation reduces to an analysis of a model system which can be described by the typical Hamiltonian of problems on the motion of Hamiltonian systems with one degree of freedom in the case of fourth-order resonance. The system is analysed in detail. The results for the model system are applied to the total system using Poincaré's theory of periodic motion and the KAM-theory. The existence, number and stability of 8π-periodic motions of the initial system are investigated. Trajectories of motion which start in a fairly small neighbourhood of the origin of coordinates are bounded. An estimate of the size of that neighbourhood is given. The examples considered are of a point mass above a curve in the shape of an ellipse which collides with the curve, and plane non-linear oscillations of a satellite in an elliptical orbit in the case of fourth-order resonance.     

Geometry of a five link mechanism with two degrees of freedom

The paper is devoted to the solution of straight and inverse geometrical tasks of five link mechanism with two degrees of freedom. The solution of the mentioned problem is very important in order to determine kinematic parameters of actuators. The problem can be divided into two parts. The first part is considered when we are given the coordinates of the output link of the mechanism and the necessity arises of determining the angles of rotation of actuators. On the other hand, it is very important to determine the position of the output link when the angles of rotation of the actuators are known. Here we consider that the mechanism is composed only of five classes of rotating kinematic pairs and the actuators are situated at the junctions of frames and links of the examined mechanism. The solution of the said problem is based on utilization of homogenous coordinates. On the basis of the obtained equations of motion, one can calculate the trajectories of motion of the output link as well angles of rotation of the actuators by taking into consideration preliminary given kinematic parameters of the mechanism. Here we also obtain equations for calculating values of speed and acceleration of the links of the mechanism. The calculations differ from known methods in simplicity and high performance, which would be useful for programming actuators mounted in the joints of the linkage.     

Hamiltonians with two degrees of freedom admitting a singlevalued general solution

Following the basic principles stated by Painleve, we first revisit the process of selecting the admissible time-independent Hamiltonians H=(p12 + p22)/2 + V(q1,q2) whose some integer power qjnj (t) of the general solution is a singlevalued function of the complex time t. In addition to the well known rational potentials V of Henon-Heiles, this selects possible cases with a trigonometric dependence of V on qj. Then, by establishing the relevant confluences, we restrict the question of the explicit integration of the seven (three "cubic" plus four "quartic") rational Henon-Heiles cases to the quartic cases. Finally, we perform the explicit integration of the quartic cases, thus proving that the seven rational cases have a meromorphic general solution explicitly given by a genus two hyperelliptic function.     

Chaos in vibroimpact systems with one degree of freedom in a neighborhood of chatter generation: II

We study the rectilinear motion of a mass point with impacts against a stopper. The motion between the impacts is described by a second-order ordinary differential equation with a parameter. The impact recovery coefficient also depends on the parameter of the vibroimpact system. We describe a bifurcation that leads to the generation of a Smale horseshoe in a parametric neighborhood of the chatter phenomenon. We prove the existence of an invariant set described by symbolic dynamics.     

Chaos in vibroimpact systems with one degree of freedom in a neighborhood of chatter generation: I

We study the rectilinear motion of a mass point with impacts against a delimiter. The motion between the impacts is described by a second-order ordinary differential equation with a parameter. The impact recovery coefficient also depends on the parameter of the vibroimpact system. We describe a bifurcation that leads to the generation of a Smale horseshoe in a parametric neighborhood of the chatter phenomenon. We prove the existence of an invariant set described by symbolic dynamics.     

Numerical modelling and analysis of high frequency dynamic contact in the hybrid transducer type ultrasonic motor

The finite element model for describing the intermittent contact dynamics between the stator and the rotor of a hybrid transducer type piezoelectric ultrasonic motor is presented. The contact simulations are performed using an augmented Lagrangian method to determine the normal contact force, the interface condition between the stator and the rotor as dependency of the other factors such as pre-load force on the stator, the thickness of friction layer, the electrical driving voltage and the operational frequency of the stator. Effects of friction layer, both for elastic and viscoelastic frictional materials, are numerically investigated. The numerical results provide some aspects to make the impact that may occur between the contact interface pairs effectively softer. Furthermore, the noncontact phase between the stator and the rotor of this kind of piezoelectric ultrasonic motor will also be optimized. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On stability in Hamiltonian systems with two degrees of freedom

We consider the stability of the equilibrium position at the origin of coordinates of a Hamiltonian system with two degrees of freedom whose unperturbed part describes oscillators with restoring force of odd order greater than 1. It is proved that if the exponents of the restoring force of the oscillators are not equal, then the equilibrium position is Lyapunov stable. If the exponents are equal, then the equilibrium position is conditionally stable for trajectories not belonging to some level surface of the Hamiltonian. The reduction of the system to this surface shows that the equilibrium position is stable in the case of general position.     

On ideals of the group algebra of a free group of degree of freedom two over the field of complex numbers

In this paper, we prove the existence of an element of the group algebra A=F of a free groupF with two generatorsx andy over the field of complex numbersC such that, for any complexa andb for which ¦a¦=¦b¦=1, we haveA _a,b ()A=0, where _a,b ( is an automorphism ofA that mapsx,y intoax, by, respectively. Thus, we give a negative answer to question 12.46 of P. A. Linnel from Kourovka Notebook.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No.4, pp. 571–572, April, 1995.     

An upper bound for the restrained domination number of a graph with minimum degree at least two in terms of order and minimum degree

Let G=(V,E) be a graph. A set S⊆V is a restrained dominating set if every vertex in V−S is adjacent to a vertex in S and to a vertex in V−S. The restrained domination number of G, denoted γ_r(G), is the smallest cardinality of a restrained dominating set of G. We will show that if G is a connected graph of order n and minimum degree δ and not isomorphic to one of nine exceptional graphs, then .     

The bringing of two rigid bodies into collision-free contact by bounded controls in a finite time

The control problem of moving a rigid body onto the surface of another rigid body in such a way that no collision occurs in investigated. For this, phase constraints must be satisfied in the sense that the surface of one body must not overlap the other, and here an important condition must be met: contact of the bodies is ensured by bounded controls and in a finite time. No impact of the rigid bodies occurs at the instant of contact because a strict rule is observed: if any point of one body lies on the surface of the other, then the velocity of the point is directed along this surface (or away from it). To observe this rule, special controls are used. These controls realize assigned accurate values of the coordinates and velocities of the mechanical system in a finite time.     

Differential invariants of a class of Lagrangian systems with two degrees of freedom

We consider systems of Euler–Lagrange equations with two degrees of freedom and with Lagrangian being quadratic in velocities. For this class of equations the generic case of the equivalence problem is solved with respect to point transformations. Using Lie?s infinitesimal method we construct a basis of differential invariants and invariant differentiation operators for such systems. We describe certain types of Lagrangian systems in terms of their invariants. The results are illustrated by several examples.     

Thermomechanical behaviour of a damageable beam in contact with two stops

A model for the thermomechanical behaviour of a beam which allows for the general evolution of material damage is presented and investigated. One end of the beam is fixed while the other is constrained to move between two stops. The contact of the free tip with the stops is modelled by the normal compliance condition. The thermal interaction between the stops and the free tip is described by a heat exchange condition where the heat transfer coefficient is a general function of the gaps between the tip and the stops. The effects on the mechanical properties of the material due to crack expansion are described by a damage field, which measures the decrease in the load-bearing capacity of the material. The damage evolves as a constrained diffusion process in which the microcracks that develop may grow or disappear. The mathematical model consists of a coupled system of energy--elasticity equations together with a nonlinear parabolic inclusion for the damage field. The existence of a local solution is established using truncation, penalization, and a priori estimates.