Dynamics of Mechanical Systems with Coulomb Friction

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.