scholarly journals On the regularization of impact without collision: the Painlevé paradox and compliance

2017 ◽  
Vol 473 (2202) ◽  
pp. 20160773 ◽  
Author(s):  
S. J. Hogan ◽  
K. Uldall Kristiansen
Keyword(s):  
Coulomb Friction ◽  
Exact Formula ◽  
Rigid Bodies ◽  
Painlevé Paradox ◽  
Asymptotic Expressions ◽  
Reaction Forces ◽  
Lift Off ◽  
Stiffness And Damping ◽  
Impact Without Collision ◽  
Painlevé Paradoxes

We consider the problem of a rigid body, subject to a unilateral constraint, in the presence of Coulomb friction. We regularize the problem by assuming compliance (with both stiffness and damping) at the point of contact, for a general class of normal reaction forces. Using a rigorous mathematical approach, we recover impact without collision (IWC) in both the inconsistent and the indeterminate Painlevé paradoxes, in the latter case giving an exact formula for conditions that separate IWC and lift-off. We solve the problem for arbitrary values of the compliance damping and give explicit asymptotic expressions in the limiting cases of small and large damping, all for a large class of rigid bodies.

Identification of Bearing Coefficients of Flexible Rotor-Bearing Systems

2000 ◽  
Author(s):  
Tachung Yang ◽  
Wei-Ching Chaung
Keyword(s):  
Industrial Applications ◽  
Operating Conditions ◽  
Least Square ◽  
Flexible Rotor ◽  
Fem Modeling ◽  
Identification Method ◽  
Reaction Forces ◽  
Manufacturing Assembly ◽  
Stiffness And Damping

The accuracy of stiffness and damping coefficients of bearings is critical for the rotordynamic analysis of rotating machinery. However, the influence of bearings depends on the design, manufacturing, assembly, and operating conditions of the bearings. Uncertainties occur quite often in manufacturing and assembly, which causes the inaccuracy of bearing predictions. An accurate and reliable in-situ identification method for the bearing coefficients is valuable to both analyses and industrial applications. The identification method developed in this research used the receptance matrices of flexible shafts from FEM modeling and the unbalance forces of trial masses to derive the displacements and reaction forces at bearing locations. Eight bearing coefficients are identified through a Total Least Square (TLS) procedure, which can handle noise effectively. A special feature of this method is that it can identify bearing coefficients at a specific operating speed, which make it suitable for the measurement of speed-dependent bearings, like hydrodynamic bearings. Numerical validation of this method is presented. The configurations of unbalance mass arrangements are discussed.

Instant of chair-rise lift-off can be predicted by foot–floor reaction forces

2004 ◽  
Vol 23 (2) ◽  
pp. 121-132 ◽  
Author(s):  
Chris A. McGibbon ◽  
Dov Goldvasser ◽  
David E. Krebs ◽  
Donna Moxley Scarborough
Keyword(s):  
Reaction Forces ◽  
Lift Off ◽  
Chair Rise

scholarly journals High-Speed Rotor Analytical Dynamics on Flexible Foundation Subjected to Internal and External Excitation

2016 ◽  
Vol 46 (4) ◽  
pp. 3-18
Author(s):  
Venelin S. Jivkov ◽  
Evtim V. Zahariev
Keyword(s):  
Gas Turbines ◽  
High Speed ◽  
Asymptotic Methods ◽  
Numerical Procedure ◽  
Rigid Bodies ◽  
Dynamics Simulation ◽  
Geometrical Approach ◽  
Rotating Machine ◽  
Exciting Frequency ◽  
Stiffness And Damping

Abstract The paper presents a geometrical approach to dynamics simulation of a rigid and flexible system, compiled of high speed rotating machine with eccentricity and considerable inertia and mass. The machine is mounted on a vertical flexible pillar with considerable height. The stiffness and damping of the column, as well as, of the rotor bearings and the shaft are taken into account. Non-stationary vibrations and transitional processes are analyzed. The major frequency and modal mode of the flexible column are used for analytical reduction of its mass, stiffness and damping properties. The rotor and the foundation are modelled as rigid bodies, while the flexibility of the bearings is estimated by experiments and the requirements of the manufacturer. The transition effects as a result of limited power are analyzed by asymptotic methods of averaging. Analytical expressions for the amplitudes and unstable vibrations throughout resonance are derived by quasi-static approach increasing and decreasing of the exciting frequency. Analytical functions give the possibility to analyze the influence of the design parameter of many structure applications as wind power generators, gas turbines, turbo-generators, and etc. A numerical procedure is applied to verify the effectiveness and precision of the simulation process. Nonlinear and transitional effects are analyzed and compared to the analytical results. External excitations, as wave propagation and earthquakes, are discussed. Finite elements in relative and absolute coordinates are applied to model the flexible column and the high speed rotating machine. Generalized Newton - Euler dynamics equations are used to derive the precise dynamics equations. Examples of simulation of the system vibrations and nonstationary behaviour are presented.

Simulation of Planar Dynamic Mechanical Systems With Changing Topologies: Part I — Characterization and Prediction of the Kinematic Constraint Changes

1987 ◽  
Author(s):  
B. J. Gilmore ◽  
R. J. Cipra
Keyword(s):  
Equations Of Motion ◽  
Mechanical Systems ◽  
Rigid Bodies ◽  
State Variables ◽  
Kinematic Constraint ◽  
Kinematic Constraints ◽  
Force Closure ◽  
Simulation Strategy ◽  
Reaction Forces ◽  
Discontinuous Equations

Abstract Due to changes in the kinematic constraints, many mechanical systems are described by discontinuous equations of motion. This paper addresses those changes in the kinematic constraints which are caused by planar bodies contacting and separating. A strategy to automatically predict and detect the kinematic constraint changes, which are functions of the system dynamics, is presented in Part I. The strategy employs the concepts of point to line contact kinematic constraints, force closure, and ray firing together with the information provided by the rigid bodies’ boundary descriptions, state variables, and reaction forces to characterize the kinematic constraint changes. Since the strategy automatically predicts and detects constraint changes, it is capable of simulating mechanical systems with unpredictable or unforeseen changes in topology. Part II presents the implementation of the characterizations into a simulation strategy and presents examples.

scholarly journals Ternary Logic of Motion to Resolve Kinematic Frictional Paradoxes

Entropy ◽  
2019 ◽  
Vol 21 (6) ◽  
pp. 620 ◽  
Author(s):  
Michael Nosonovsky ◽  
Alexander D. Breki
Keyword(s):  
Dry Friction ◽  
Sliding Friction ◽  
Rigid Bodies ◽  
Newtonian Mechanics ◽  
Stability Criteria ◽  
Ternary Logic ◽  
Elastic Bodies ◽  
Logic Approach ◽  
Frictional System ◽  
Painlevé Paradoxes

Paradoxes of dry friction were discovered by Painlevé in 1895 and caused a controversy on whether the Coulomb–Amontons laws of dry friction are compatible with the Newtonian mechanics of the rigid bodies. Various resolutions of the paradoxes have been suggested including the abandonment of the model of rigid bodies and modifications of the law of friction. For compliant (elastic) bodies, the Painlevé paradoxes may correspond to the friction-induced instabilities. Here we investigate another possibility to resolve the paradoxes: the introduction of the three-value logic. We interpret the three states of a frictional system as either rest-motion-paradox or as rest-stable motion-unstable motion depending on whether a rigid or compliant system is investigated. We further relate the ternary logic approach with the entropic stability criteria for a frictional system and with the study of ultraslow sliding friction (intermediate between the rest and motion or between stick and slip).

Dynamic Modelling and Analysis of Rectangle-Shaped Plastic-Coated Slideways in Machine Tools

2011 ◽  
Vol 50-51 ◽  
pp. 37-41
Author(s):  
Jian Fu Zhang ◽  
Zhi Jun Wu ◽  
Ping Fa Feng ◽  
Ding Wen Yu
Keyword(s):  
Mathematical Model ◽  
Machine Tools ◽  
Vibration Mode ◽  
Dynamic Modelling ◽  
Rigid Bodies ◽  
Theoretical Research ◽  
Damping Characteristics ◽  
Vibration Properties ◽  
Stiffness And Damping ◽  
The Mathematical Model

The plastic-coated slideways have been widely used for form-generating movement in machine tools. Its dynamic behavior plays an important role in the vibration properties of the whole machine. In this work, according to the situation that researches on this subject were rather insufficient, a theoretical research was analyzed concerning the stiffness and damping characteristics of rectangle-shaped plastic-coated slideways. The mathematical model was firstly suggested especially based on the assembly of the saddle and worktable. Both stiffness and damping characteristics on vertical and horizontal directions were theoretically determined. To derive the governing motion equation of the slideway system, the carriage and rail were considered as rigid bodies and connected with a series of spring and damping elements at the joint face. Moreover, through the Lagrange’s approach, the frequencies of the carriage at vertical, pitching, yawing and rolling vibration mode were identified.

scholarly journals Experimental Investigation of the Painlevé Paradox in a Robotic System

2008 ◽  
Vol 75 (4) ◽  
Author(s):  
Zhen Zhao ◽  
Caishan Liu ◽  
Wei Ma ◽  
Bin Chen
Keyword(s):  
Contact Point ◽  
Tangential Velocity ◽  
Rigid Bodies ◽  
Robotic System ◽  
Experimental Results ◽  
Painlevé Paradox ◽  
Dynamical Behaviors ◽  
Tangential Impact ◽  
System Of Rigid Bodies ◽  
Approach Zero

This paper aims at experimentally investigating the dynamical behaviors when a system of rigid bodies undergoes so-called paradoxical situations. An experimental setup corresponding to the analytical model presented in our prior work Liu et al. [2007, “The Bouncing Motion Appearing in a Robotic System With Unilateral Constraint,” Nonlinear Dyn., 49(1–2), 217–232] is developed, in which a two-link robotic system comes into contact with a moving rail. The experimental results show that a tangential impact exists at the contact point and takes a peculiar property that well coincides with the maximum dissipation principle stated in the work of Moreau [1988, “Unilateral Contact and Dry Friction in Finite Freedom Dynamics,” Nonsmooth Mechanics and Applications, Springer-Verlag, Vienna, pp. 1–82] the relative tangential velocity of the contact point must immediately approach zero once a Painlevé paradox occurs. After the tangential impact, a bouncing motion may be excited and is influenced by the speed of the moving rail. We adopt the tangential impact rule presented by Liu et al. to determine the postimpact velocities of the system, and use an event-driven algorithm to perform numerical simulations. The qualitative comparisons between the numerical and experimental results are carried out and show good agreements. This study not only presents an experimental support for the shock assumption related to the problem of the Painlevé paradox, but can also find its applications in better understanding the instability phenomena appearing in robotic systems.

Numerical Location of Painlevé Paradox-Associated Jam and Lift-Off in a Double-Pendulum Mechanism

2017 ◽  
Vol 12 (6) ◽  
Author(s):  
Shane J. Burns ◽  
Petri T. Piiroinen
Keyword(s):  
Numerical Simulation ◽  
Mechanical System ◽  
Coefficient Of Friction ◽  
Double Pendulum ◽  
Painlevé Paradox ◽  
Forward Solution ◽  
Pendulum System ◽  
Graphical Technique ◽  
Lift Off

In this article, we will introduce the phenomenon known as the Painlevé paradox and further discuss the associated coupled phenomena, jam and lift-off. We analyze under what conditions the Painlevé paradox can occur for a general two-body collision using a framework that can be easily used with a variety of impact laws, however, in order to visualize jam and lift-off in a numerical simulation, we choose to use a recently developed energetic impact law as it is capable of achieving a unique forward solution in time. Further, we will use this framework to derive the criteria under which the Painlevé paradox can occur in a forced double-pendulum mechanical system. First, using a graphical technique, we will show that it is possible to achieve the Painlevé paradox for relatively low coefficient of friction values, and second we will use the energetic impact law to numerically show the occurrence of the Painlevé paradox in the double-pendulum system.

A Set-Valued Force Law for Spatial Coulomb-Contensou Friction

2003 ◽  
Author(s):  
Remco I. Leine ◽  
Christoph Glocker
Keyword(s):  
Coulomb Friction ◽  
Augmented Lagrangian ◽  
Rigid Bodies ◽  
Friction Torque ◽  
Sliding Velocity ◽  
Time Stepping ◽  
Extended Contact ◽  
Tippe Top ◽  
Augmented Lagrangian Approach ◽  
Pseudo Potential

The aim of this paper is to develop a contact law for combined spatial Coulomb friction and normal friction torque (drilling friction) as a function of sliding velocity and spin. We will call this extended contact law the Coulomb-Contensou friction law and derive it from a non-smooth velocity pseudo potential. A Runge-Kutta time-stepping method is briefly presented for the numerical simulation of rigid bodies with Coulomb-Contensou friction. The algebraic inclusion describing the contact problem is solved with an Augmented Lagrangian approach. The theory and numerical methods are applied to the Tippe-Top, which illustrates the importance of Coulomb-Contensou friction for the dynamics of systems with friction.

A Risk Model with Delayed Claims

2013 ◽  
Vol 50 (03) ◽  
pp. 686-702 ◽  
Author(s):  
Angelos Dassios ◽  
Hongbiao Zhao
Keyword(s):  
Asymptotic Formula ◽  
Poisson Process ◽  
Ruin Probability ◽  
Risk Model ◽  
Poisson Model ◽  
Exact Formula ◽  
Poisson Models ◽  
Asymptotic Expressions ◽  
Special Case

In this paper we introduce a simple risk model with delayed claims, an extension of the classical Poisson model. The claims are assumed to arrive according to a Poisson process and claims follow a light-tailed distribution, and each loss payment of the claims will be settled with a random period of delay. We obtain asymptotic expressions for the ruin probability by exploiting a connection to Poisson models that are not time homogeneous. A finer asymptotic formula is obtained for the special case of exponentially delayed claims and an exact formula is obtained when the claims are also exponentially distributed.

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