On the oil-whipping of a rotor-bearing system by a continuum model

The nonlinear dynamic behavior of a rotor-bearing system is analyzed based on a continuum model. The finite element method is adopted in the analysis. Emphasis is placed on the so-called “oil-whip phenomena” which might lead to the failure of the rotor system. The dynamic response of the system in unbalanced conditions is approached by a direct integration method. It is found that a typical “oil-whip phenomenon” is successfully simulated, and the effect of the refinement of the finite element mesh is also checked. Furthermore, the bifurcation behavior of the oil-whip phenomenon that is of much concern in recent nonlinear dynamics research is analyzed. The rotor-bearing system is also examined by a simple discrete model. Significant differences are found between these two models. It is suggested that a careful examination should be made in modeling the nonlinear dynamic behavior of a rotor system.     

Rolling of a ball without spinning on a plane: the absence of an invariant measure in a system with a complete set of integrals

In the paper we consider a system of a ball that rolls without slipping on a plane. The ball is assumed to be inhomogeneous and its center of mass does not necessarily coincide with its geometric center. We have proved that the governing equations can be recast into a system of six ODEs that admits four integrals of motion. Thus, the phase space of the system is foliated by invariant 2-tori; moreover, this foliation is equivalent to the Liouville foliation encountered in the case of Euler of the rigid body dynamics. However, the system cannot be solved in terms of quadratures because there is no invariant measure which we proved by finding limit cycles.     

Optimal Motion of a Two-Body System in a Resistive Medium

Locomotion of a mechanical system consisting of two rigid bodies, a main body and a tail, is considered. The system moves in a resistive fluid and is controlled by angular oscillations of the tail relative to the main body. The resistance force acting upon each body is assumed to be a quadratic function of its velocity. Under certain assumptions, a nonlinear equation is derived that describes the progressive motion of the system as a whole.     

Evolution of a regular precession of a solid body carrying a visco-elastic disk

The motion of inertia is studied of a system consisting of an axisymmetric solid body with fixed point and a homogeneous visco-elastic disk lying in the equatorial plane of the ellipsoid of inertia of the solid body (the center of disk coincides with the fixed point). In the case of a solid disk immobilized relative to the solid body the system accomplishes a regular precession (the case of Euler motion of a symmetric solid body with a fixed point /1/). The deformation of the disk is taking place in the plane of the disk, and is accompanied by energy dissipation is the cause of the regular precession finishing by steady rotation about the vector of the moment of momentum of the system /2/.     

Vibrations of a beam-string complex system under a moving force

In this paper, the dynamic response of an Euler-Bernoulli beam and string system traversed by a constant moving force is considered. The force is moving with a constant velocity on the top beam. The complex system is finite, simply supported, parallel one upon the other and continuously coupled by a linear Winkler elastic element. The classical solution of the response of a beam-string system subjected to a force moving with a constant velocity has a form of an infinite series. The main goal of this paper is to show that in the considered case the aperiodic part of the solution can be presented in a closed, analytical form instead of an infinite series. The presented method of finding the solution in a closed, analytical form is based on the observation that the solution of the system of partial differential equations in the form of an infinite series is also a solution of an appropriate system of ordinary differential equations. The dynamic influence lines of complex systems may be used for the analysis the complex models of moving load. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Mathematical study of a tissue-implant connection in a ventral hernia repair in a context of the system's parameters

Ventral hernia repair is nowadays a well recognized matter in surgery and advanced techniques of operating are developed. However, mechanical properties of the system after an operation are not known so recurrences of the illness happen. The paper describes a simple mathematical model of a human fascia-synthetic implant system. The synthetic mesh is modeled by a cable structure and its elastic supports simulate the human fascia. The junction force in the fascia-implant system is sought. A relation of the force change to variations of some parameters of the system is also studied by applying a sensitivity analysis. The authors conclude that a crucial role for the junction force and a patient's comfort plays the implant's elasticity modulus and initial tension of the mesh is the least important factor. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Null controllability of a system of viscoelasticity with a moving control

In this paper, we consider the wave equation with both viscous Kelvin–Voigt and frictional damping as a model of viscoelasticity in which we incorporate an internal control with a moving support. We prove the null controllability when the control region, driven by the flow of an ODE, covers all the domain. The proof is based upon the interpretation of the system as, roughly, the coupling of a heat equation with an ordinary differential equation (ODE). The presence of the ODE for which there is no propagation along the space variable makes the controllability of the system impossible when the control is confined into a subset in space that does not move. The null controllability of the system with a moving control is established in using the observability of the adjoint system and some Carleman estimates for a coupled system of a parabolic equation and an ODE with the same singular weight, adapted to the geometry of the moving support of the control. This extends to the multi-dimensional case the results by P. Martin et al. in the one-dimensional case, employing 1-d Fourier analysis techniques.     

The rolling motion of a truncated ball without slipping and spinning on a plane

This paper is concerned with the dynamics of a top in the form of a truncated ball as it moves without slipping and spinning on a horizontal plane about a vertical. Such a system is described by differential equations with a discontinuous right-hand side. Equations describing the system dynamics are obtained and a reduction to quadratures is performed. A bifurcation analysis of the system is made and all possible types of the top’s motion depending on the system parameters and initial conditions are defined. The system dynamics in absolute space is examined. It is shown that, except for some special cases, the trajectories of motion are bounded.     

Input-output behaviour of a stochastic Galerkin method for a low pass filter

We apply the stochastic Galerkin method to a linear dynamical system, which includes random variables to quantify uncertainties in physical parameters. The input-output behaviour of the stochastic Galerkin system is described by a transfer function in the frequency domain. The importance of each output component can be estimated by Hardy norms. We investigate a Hardy norm in the case of a linear dynamical system modelling the electric circuit of a low pass filter. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimization of the motion in a resistant medium of a body with a movable internal mass

The motion, in a resistant medium, of a system consisting of a rigid body and movable internal mass is considered. The external medium acts on the body by a force that piecewise linearly depends on its speed. The class of periodic motions of the internal mass for which the speed of this mass relative to the body is piecewise constant is studied. It is shown that, under certain conditions, the forward movement of the whole system in the medium is possible. The average speed of this movement over a period is determined. Optimal parameters of the motion of the internal mass for which the average speed of the system movement is maximal are found.     

The quasistatic motions of a three-body system on a plane

A controlled three-body system on a horizontal plane with dry friction is considered. The interaction forces between each pair of bodies are controls that are not subject to prior constraints but must be chosen in such a way that the motions of the system generated by them are quasistatic, that is, the total force acting on each of the bodies must be close to zero. All motions in which one body moves and the other two are fixed are found in the class of quasistatic motions. The problem of the optimal displacement of a moving body between two specified positions on a plane such that the absolute magnitude of the work of the friction forces along the trajectory is a minimum is solved. The quasistatic controllability of a three-body system is demonstrated and algorithms for bringing it into a specified position are discussed. The system considered simulates a mobile robot consisting of three bodies between which control forces act that can be realized by linear motors. The sizes of the bodies are assumed to be negligibly small compared with the distances between them so that the bodies are treated as particles.     

Fast computation of a rational point of a variety over a finite field

We exhibit a probabilistic algorithm which computes a rational point of an absolutely irreducible variety over a finite field defined by a reduced regular sequence. Its time-space complexity is roughly quadratic in the logarithm of the cardinality of the field and a geometric invariant of the input system. This invariant, called the degree, is bounded by the Bézout number of the system. Our algorithm works for fields of any characteristic, but requires the cardinality of the field to be greater than a quantity which is roughly the fourth power of the degree of the input variety.

     

Stability of a thermoelastic laminated system subject to a neutral delay

In this work, we consider a system of two identical beams of uniform thickness modeled as a Timoshenko system. The slip between the beams is taken into account, and the system is coupled with a heat equation. Moreover, the slip equation is subject to a distributed delay of neutral type. Delays are known to be of a destructive nature in general. Therefore, here, the delay will compete with a frictional damping and the dissipation produced by the heat equation. We provide sufficient conditions ensuring exponential and polynomial stability of the structure.     

The problem of the reinforcement of a plate with a cutout by a two-dimensional patch

The problem of the reinforcement of an elastic plate with a cutout by means of a two-dimensional patch, which completely covers the cutout and is rigidly fixed to the plate along its boundary, is considered. The cutout and the patch can be of arbitrary shape. The problem is reduced to a system of three singular integral equations using of special integral representations which describe the stress state in the plate and in the patch. The unique solvability of the system is proved. Examples are presented.     

Approximate Controllability of a Semilinear System Involving a Fully Nonlinear Gradient Term

This paper concerns a control system governed by a semilinear degenerate equation involving a fully nonlinear gradient term. The equation may be weakly degenerate and strongly degenerate on a portion of the lateral boundary, and the gradient term can be controlled by the diffusion term. The linearized system is shown to be approximately controllable by constructing a control by means of its conjugate problem. By doing a series of precise compactness estimates, we prove that the semilinear system is approximately controllable.     

一类串联可修复系统的稳态解

本文讨论了一类具有两不同部件串联的可修复系统。利用系统算子生成的Banach空间中的正压缩C_o半群的性质,证明了系统的非负稳态解恰是系统算子的0本征值所对应的规范化后的本征向量；同时通过对系统算子谱点分布情况的分析,证明了系统算子的谱点均位于复平面左半平面且在虚轴上除0点外无其它谱,作为线性算子半群稳定性的—个直接结果,得出了该串联可修复系统的渐近稳定性。     

Rolling of a homogeneous ball over a dynamically asymmetric sphere

We consider a novel mechanical system consisting of two spherical bodies rolling over each other, which is a natural extension of the famous Chaplygin problem of rolling motion of a ball on a plane. In contrast to the previously explored non-holonomic systems, this one has a higher dimension and is considerably more complicated. One remarkable property of our system is the existence of “clandestine” linear in momenta first integrals. For a more trivial integrable system, their counterparts were discovered by Chaplygin. We have also found a few cases of integrability.     

Output feedback vibration control of a string driven by a nonlinear actuator

In this paper, we design an observer-based output feedback controller to exponentially stabilize a system of nonlinear ordinary differential equation-wave partial differential equation-ordinary differential equation. An observer is designed to estimate the full states of the system using available boundary values of the partial differential equation. The output feedback controller is built via the combination of the ordinary differential equation backstepping which is applied to deal with the nonlinear ordinary differential equation, and the partial differential equation backstepping which is used for the wave partial differential equation-ordinary differential equation. The controller can be applied into vibration suppression of a string-payload system driven by an actuator with nonlinear characteristics. The global exponential stability of all states in the closed-loop system is proved by Lyapunov analysis. The numerical simulation illustrates the states of the actuator, string, payload and the observer errors are fast convergent to zero under the proposed output feedback controller.     

On the nature of dynamic chaos in a neighborhood of a separatrix of a conservative system

In the present paper, we give a new treatment of the mechanism of generation of chaotic dynamics in a perturbed conservative system in a neighborhood of the separatrix contour of a hyperbolic singular point of the unperturbed system. We theoretically prove and justify by three numerical examples of classical Hamiltonian systems with one and a half degrees of freedom and by an example of a simply conservative three-dimensional system that the complication of the dynamics in a conservative system as the perturbation increases is caused by a nonlocal effect of multiplication of hyperbolic and elliptic cycles (and the tori surrounding them), which has nothing in common with the mechanism of separatrix splitting in classical Hamiltonian mechanics.     

Chaotic response of a quarter car model forced by a road profile with a stochastic component

The Melnikov criterion is used to examine a global homoclinic bifurcation and transition to chaos in the case of a quarter car model excited kinematically by a road surface profile consisting of harmonic and noisy components. By analyzing the potential an analytic expression is found for the homoclinic orbit. The road profile excitation including harmonic and random characteristics as well as the damping are treated as perturbations of a Hamiltonian system. The critical Melnikov amplitude of the road surface profile is found, above which the system can vibrate chaotically. This transition is analyzed for different levels of noise and illustrated by numerical simulations.