Painleves Paradox

In Sects. 4.3.1 and 4.3.2, we study the classic Painleve s example and derive the conditions for the occurrence of the paradoxes. In Sect. 4.3.3, the concept of self-locking is introduced which is closely related to the kinematic constraint instability mechanism. In the rigid body systems, this phenomenon is sometimes known as jamming or wedging [97]. As we will see later on, the self-locking is an important aspect of the study of the dynamics of the lead screws. In Sect. 4.3.4, a simple model of a vibratory system is analyzed where the kinematic constraint mechanism leads to instability. In the study of disc brake systems, similar instability mechanism is sometimes referred to as sprag-slip vibration [7]. Some further references are given in Sect. 3.3.5. 

Consider the system shown in Fig. 4.16. A bar of length / is in contact with a rough rigid surface at an angle 6. The equations of motion for this system are written as follows  

When the rod is in contact with the surface we have N 0 and y, = 0. On the other hand, when the rod brakes contact we have N = 0 and 0. This situation can be represented as a linear complementarity problem [52, 96] which is written compactly as 

Differentiating (4.42) twice with respect to time gives 

The solutions to the linear complementarity problem (4.43) can be represented graphically as shown in Fig. 4.17. As it can be seen firom this figure, when A 0 and B 0, the solution is not unique and when A 0 and B 0 no solution exist. 

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Leine RI, Brogliato B, Nijmeijer H (2002) Periodic motion and bifurcations induced by the Painleve paradox. Eur J Mech A Solids 21(5) 869-896... 

The situations where A < 0 are known as Painleve s paradoxes [51]. The paradoxes arise from the violation of the existence and uniqueness conditions of the... 

In Sect. 2.2, we encountered situations where (as a result of frictional constraints), the inertia matrix could become singular or negative definite (i.e., when /I < 0 in (3.19) or when det(M) < 0 in (3.24)). As mentioned above, these situations are known as Painleve s paradoxes. We will study Painleve s paradoxes in detail in Sect. 3.3. 

For the sake of completeness, we consider here the linearized system equation for such cases. Assume that the onset of the Painleve s paradox is at 0 = 0cr (i.e., A(0cr) = 0 or det [M(0cr)] = 0). As the system parameters are varied such that 0 crosses the surface 0 = 0cr an eigenvalue goes to infinity and becomes positive as shown in Fig. 3.3. Beyond the critical value of the parameters, the solution of the linear differential equation (3.8) diverges. 

Finally, in Sect. 4.3, we turn our attention to the kinematic constraint instability. This section begins by the introduction of the Painleve s paradoxes which play an important role in the kinematic constraint instability mechanism. The self-locking property which is another consequence of MctiOTi in the rigid body dynamics is also discussed in this section. This effect has a prominent presence in the study of the lead screws in Chaps. 7 and 8. The concepts presented here form the basis for the study of the kinematic constraint instability in the lead screws in Chap. 8. 

In Sect. 3.2, we have seen that a dynamical system with unilateral or bilateral frictional contact can possess a peculiar characteristic, namely the inertia matrix may be asymmetric and nonpositive definite. Painleve was the first to point out the difficulties that may arise in such cases [53, 95]. As we will see in this section through examples, the presence of a kinematic constraint with friction could lead to situations where the equations of motion of the system do not have a bounded solution (inconsistency) or the solution is not unique (indeterminacy). These situations where the existence and uniqueness properties of the solution of the equations of motion are violated are known as the Painleve s paradoxes. There is a vast literature on the general theory of the rigid body dynamics with frictional constraints... 

Since AT(0eq) >0, instabilities occur if either A/(0eq) <0 (Painleve s Paradox) or C (0eq) < 0. (Negative damping). Starting with the negative damping instability, we can see that if the following conditions are satisfied, the origin of the system (4.63) becomes unstable. 

Fig. 4.22 Unstable steady-sliding equilibrium point, C(0eq) —3 < 0 4.3.4.3 Painleve s Paradoxes...

For the parameter values that satisfy the above inequality and A Vt < 0, Painleve s paradoxes occur and the equation of motion given by (4.59) or the linearized equation given by (4.63) are no longer valid due to violation of existence and uniqueness of the solution. However, as we will see in the numerical examples below, the steady-sliding equilibrium point is indeed unstable for such values of system parameters. 

For these parameter values, we have M(0eq) —6.4 < 0 which indicates the occurrence of the Painleve s paradoxes. Note that, here we have C(0eq) 4.8 > 0. Three curves divide the phase plane into five regions vertical line M+(0) = 0 stick boundary Vt(0,9) = 0 and the curve defined by A(0,0) = 0 where A(0,9) is given by (4.62). 

Of course, (5.40) is equivalent to the system given by (5.30) and (5.31). In this representation, however, the possibility of Painleve s paradox is clearly shown through the appearance of A, given by (5.38), in the equation of motion. 

The third and final instability mechanism in the lead screw drives is the kinematic constraint. In Sect. 4.3, Painleve s paradoxes were introduced and - through simple examples - it was shown that under the conditions of the paradoxes, the rigid body equations of motion of a system with frictional contact do not have a bounded solution or the solution is not unique. We have also discussed the relationship between Painleve s paradoxes and the kinematic constraint instability mechanism. 

Now we consider the possibility of Painleve s paradoxes for the equation of motion of the lead screw drive given by (8.1). Define... 

It will be shown by numerical examples in Sect. 8.5 below that when Fq > 0 and Co < 0 (i.e., negative effective damping), the instability may or may not lead to stick-slip vibrations. In contrast to this case, when Fq < 0 (i.e., Painleve s paradox), the instability is accompanied by stick-slip vibration and impulsive forces. Section 8.6 is dedicated to the study of the kinematic constraint instability and the resulting vibrations. 

From the eigenvalue analysis of Sect. 8.4, it is evident that regardless of the level of linear damping (Co), instability (due to Painleve s paradoxes) occurs whenever... 

Expectedly, the instability conditions given by (8.21) are the same as the necessary conditions for the Painleve s paradoxes discussed in Sect. 8.1. Limiting our study to the case of Q > 0 for simplicity. Fig. 8.7 shows that the phase plane of the system is divided by N = 0 and 6 = 0 lines into four regions. In these regions, the system s equation has either no solution or two solutions when kinematic constraint instability is active (i.e., conditions of (8.21) are satisfied). Based on the discussions in Sect. 8.2, the following conclusions are drawn for the behavior of the lead screw model ... 

Inequality (8.27) is the condition for the occurrence of Painleve s paradoxes or the kinematic constraint instability. 

In this chapter, the role of friction and the kinematic constraint equation (which defines the relative motion of lead screw and nut) in causing friction-induced vibrations in lead screw drives was investigated. Depending on the system parameters (including friction), the kinematic constraint may lead to instability in two distinct ways Negative damping and the occurrence of Painleve s paradoxes. 

Zhao Z, Chen B, Liu C, Jin H (2004) Impact model resolution on painleve s paradox. Acta Mech Sin 20(6) 659-660... 

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