Dynamics of Mechanical Systems with Coulomb Friction

ISBN-13: 9783642056246 / Angielski / Miękka / 2011 / 272 str.

Le Xuan Anh; Alexander Belyaev

Opis

This book addresses the general theory of motion of mechanical systems with Coulomb friction. In particular, the book focuses on the following specific problems: derivation of the equations of motion, Painleve's paradoxes, tangential impact and dynamic seizure, and frictional self-excited oscillations. In addition to the theoretical results, the book contains a detailed description of experiments that show that, in general, the friction force at the instant of transition to motion is determined by the rate of tangential load and does not depend on the duration of the previous contact. These results are used to develop the theory of frictional self-excited oscillations. A number of industrially relevant mechanisms are considered, including the Painleve-Klein scheme, epicyclic mechanisms, crank mechanisms, gear transmission, the link mechanism of a planing machine, and the slider of metal-cutting machine tools. The book is intended for researchers, engineers and students in mechanical engineering.

Spis treści

1 Development of the theory of motion for systems with Coulomb friction.- 1.1 Coulomb’s law of friction.- 1.2 Main peculiarities of systems with Coulomb friction and the specific problems of the theory of motion.- 1.2.1 The principle peculiarity.- 1.2.2 Non-closed system of equations for the dynamics of systems with friction and the problem of deriving these equations.- 1.2.3 Non-correctness of the equations for systems with friction and the problem of solving Painlevé’s paradoxes.- 1.2.4 The problem of determining the forces of friction acting on particles.- 1.2.5 Retaining the state of rest and transition to motion.- 1.2.6 The problem of determining the property of self-braking.- 1.2.7 Appearance of self-excited oscillations.- 1.3 Various interpretations of Painlevé’s paradoxes.- 1.4 Principles of the general theory of systems with Coulomb friction.- 1.5 Laws of Coulomb friction and the theory of frictional selfexcited oscillations.- 2 Systems with a single degree of freedom and a single frictional pair.- 2.1 Lagrange’s equations with a removed contact constraint.- 2.2 Kinematic expression for slip with rolling.- 2.2.1 Velocity of slip and the velocities of change of the contact place due to the trace of the contact.- 2.2.2 Angular velocity.- 2.3 Equation for the constraint force and Painlevé’s paradoxes.- 2.3.1 Solution for the acceleration and the constraint force.- 2.3.2 Criterion for the paradoxes.- 2.4 Immovable contact and transition to slipping.- 2.5 Self-braking and the angle of stagnation.- 2.5.1 The case of no paradoxes.- 2.5.2 The case of paradoxes (?|L|>1).- 3 Accounting for dry friction in mechanisms. Examples of single-degree-of-freedom systems with a single frictional pair.- 3.1 Two simple examples.- 3.1.1 First example.- 3.1.2 Second example.- 3.2 The Painlevé-Klein extended scheme.- 3.2.1 Differential equations of motion, expression for the reaction force, condition for the paradoxes and the law of motion.- 3.2.2 Immovable contact and transition to slip.- 3.2.3 The stagnation angle and the property of self-braking in the case of no paradoxes.- 3.2.4 Self-braking under the condition of paradoxes.- 3.3 Stacker.- 3.3.1 Pure rolling of the rigid body model.- 3.3.2 Slip of the driving wheel for the rigid body model.- 3.3.3 Speed-up of stacker.- 3.3.4 Pure rolling in the case of tangential compliance.- 3.3.5 Rolling with account of compliance.- 3.3.6 Speed-up with account of compliance.- 3.3.7 Numerical example.- 3.4 Epicyclic mechanism with cylindric teeth of the involute gearing.- 3.4.1 Differential equation of motion, equations for the reaction force and the conditions for paradoxes.- 3.4.2 Relationships between the torques at rest and in the transition to motion.- 3.4.3 Regime of uniform motion.- 3.5 Gear transmission with immovable rotation axes.- 3.5.1 Differential equations of motion and the condition for absence of paradoxes.- 3.5.2 Regime of uniform motion.- 3.5.3 Transition from the state of rest to motion.- 3.6 Crank mechanism.- 3.6.1 Equation of motion and reaction force.- 3.6.2 Condition for complete absence of paradoxes.- 3.6.3 The property of self-braking in the case of no paradoxes.- 3.7 Link mechanism of a planing machine.- 3.7.1 Differential equations of motion and the expression for the reaction force.- 3.7.2 Feasibility of Painlevé’s paradoxes.- 3.7.3 The property of self-braking.- 3.7.4 Numerical example.- 4 Systems with many degrees of freedom and a single frictional pair. Solving Painlevé’s paradoxes.- 4.1 Lagrange’s equations with a removed constraint.- 4.2 Equation for the constraint force, differential equation of motion and the criterion of paradoxes.- 4.2.1 Determination of the constraint force and acceleration.- 4.2.2 Criterion of Painlevé’s paradoxes.- 4.3 Determination of the true motion.- 4.3.1 Limiting process.- 4.3.2 True motions under the paradoxes.- 4.4 True motions in the Painlevé-Klein problem in paradoxical situations.- 4.4.1 Equations for the reaction force.- 4.4.2 True motions for the paradoxes.- 4.5 Elliptic pendulum.- 4.6 The Zhukovsky-Froude pendulum.- 4.6.1 Equation for the reaction force and condition for the non-existence of the solution.- 4.6.2 The equilibrium position and free oscillations.- 4.6.3 Regime of joint rotation of the journal and the pin.- 4.7 A condition of instability for the stationary regime of metal cutting.- 4.7.1 Derivation of the equations of motion.- 4.7.2 Solving the equations.- 4.7.3 Relationship between instability of cutting and Painlevé’s paradox.- 4.7.4 Boring with an axial feed.- 5 Systems with several frictional pairs. Painlevé’s law of friction. Equations for the perturbed motion taking account of contact compliance.- 5.1 Equations for systems with Coulomb friction.- 5.1.1 System with removed constraints.- 5.1.2 Solving the main system.- 5.1.3 The case of n = 1, m = 1.- 5.2 Mathematical description of the Painlevé law of friction.- 5.2.1 Accelerations due to two systems of external forces.- 5.2.2 Improved Painlevé’s equations.- 5.2.3 Improved Painlevé’s theorem.- 5.3 Forces of friction in the Painlevé-Klein problem.- 5.4 The contact compliance and equations of perturbed trajectories.- 5.4.1 Lagrange’s equations for systems with elastic contact joints.- 5.4.2 Equations for perturbed reaction forces.- 5.5 Painlevé’s scheme with two frictional pairs.- 5.5.1 Lagrange’s equations, reaction forces and the equations of motion with eliminated reaction forces.- 5.5.2 Feasibility of Painlevé’s paradoxes.- 5.5.3 Expressions for the frictional force in terms of the friction coefficients.- 5.5.4 Painlevé’s scheme for compliant contacts.- 5.6 Sliders of metal-cutting machine tools.- 5.6.1 Derivation of equations of motion and expressions for the reaction forces.- 5.6.2 Signs of the reaction forces and feasibility of paradoxes.- 5.6.3 Forces of friction.- 5.7 Concluding remarks about Painlevé’s paradoxes.- 5.7.1 On equations of systems with Coulomb friction.- 5.7.2 On conditions of the paradoxes.- 5.7.3 On the reasons for the paradoxes.- 5.7.4 On the laws of motion in the paradoxical situations.- 5.7.5 On the initial motion of an immovable contact.- 5.7.6 On self-braking.- 5.7.7 On the mathematical description of Painlevé’s law.- 5.7.8 On examples.- 6 Experimental investigations into the force of friction under self-excited oscillations.- 6.1 Experimental setups.- 6.1.1 The first setup.- 6.1.2 The second setup.- 6.1.3 The third setup.- 6.2 Determining the forces by means of an oscillogram.- 6.3 Change in the force of friction under break-down of the maximum friction in the case of a change in the velocity of motion.- 6.4 Dependence of the friction force on the rate of tangential loading.- 6.5 Plausibility of the dependence F+(f).- 6.5.1 Control tests.- 6.5.2 Estimating the numerical characteristics.- 6.5.3 Statistical properties of the dependences.- 6.5.4 Test data of other authors.- 6.6 Characteristic of the force of sliding friction.- 7 Force and small displacement in the contact.- 7.1 Components of the small displacement.- 7.1.1 Definition of break-down and initial break-down.- 7.1.2 Reversible and irreversible components.- 7.1.3 Influence of the intermediate stop and reverse on the irreversible displacement.- 7.1.4 Dependence of the total small displacement on the rate of tangential loading.- 7.1.5 Small displacement of parts of the contact.- 7.1.6 Comparing the values of small displacement with existing data.- 7.2 Remarks on friction between steel and polyamide.- 7.2.1 On critical values of the force of friction.- 7.2.2 Time lag of small displacement.- 7.2.3 Immovable and viscous components of the force of friction.- 7.3 Conclusions.- 8 Frictional self-excited oscillations.- 8.1 Self-excited oscillations due to hard excitation.- 8.1.1 The case of no structural damping.- 8.1.2 Including damping.- 8.2 Self-excited oscillations under both hard and soft excitations.- 8.2.1 Equations of motion.- 8.2.2 Critical velocities.- 8.2.3 Amplitude of auto-oscillation.- 8.2.4 Period of auto-oscillation.- 8.2.5 Self-excitation of systems.- 8.3 Accuracy of the displacement.- References.

Blurb

This book addresses the general theory of motion of mechanical systems with Coulomb friction. In particular, the book focuses on the following specific problems: i) derivation of the equations of motion, ii) Painleve's paradoxes, iii) tangential impact and dynamic seizure, and iiii) frictional self-excited oscillations.

In addition to theoretical results, the book contains a detailed description of experiments that have been performed. These show that, in general, the friction force at the instant of transition to motion is determined by the rate of tangential load and does not depend on the duration of the previous contact. These results are used to develop the theory of frictional self-excited oscillations. A number of industrially relevant mechanisms are considered, including the Painleve-Klein scheme, epicyclic mechanisms, crank mechanisms, gear transmission, the link mechanism of a planing machine, and the slider of metal-cutting machine tools.

The book is intended for researchers, engineers and students in mechanical engineering.