Painleve

Because of the complex nature of the Painleve transcendents and of the resulting difficulties in satisfying the boundary conditions we shall not proceed with the exact analytical solution of b.v.p. (5.3.6) (5.3.8) any further, but rather we turn to an asymptotic and numerical study of this singular perturbation problem. 

A first attack to Painleve s conjecture was done by von Zeipel [26]. von Zeipel considered how the size of an A-body system evolves in time. He chose the moment of inertia I = m,jq, p as the size of the A-body system. He has shown that a necessary condition for a solution having noncolhsional singularity is that the motion of the system becomes unbounded in finite time. 

The Painleve ds2. This involves only one transformation of coordinates the Painleve time r that is given by ... 

A Painleve form exists for this Kerr ds2. Let us define the new variables tn and tpn by... 

Notice that the expression equation (26) has the Painleve form a Minkowski ds2 minus the product of a space-time function by the square of a sum of differentials. Furthermore these three sums of differentials given in equations (19), (25) and (26) have the following common point their ds2 can be written... 

Everybody knows that in several belligerent countries organizations have been established which aim to make scientists help to meet various demands. England has the Advisory Council of Scientific and Industrial Research, America the US National Research Council, in France Min. Painleve has established a similar committee, and we can be sure that in Germany science has been mobilized. In our country, the question has also been raised whether the practitioners of pure and applied science should not be working, more than before, for the benefit of the national welfare and defence. ... 

Analytical studies of Eq. (168) have used different changes of variables to write the electric field in terms of either Painleve transcendents [68] or Jacobian elliptic functions [88]. Alternatively, asymptotic expansions have also been used [68, 87, 89, 90]. The approximate solution methods have neglected different terms of Eq. (168). Thus, while Urtenov and Nikonenko [55, 91, 92] have considered that the space charge density is quasiuniform, dSi/dx 0, Bass [93] assumed that eE RTand I/II 8/Lo to ex-... 

J. S. McCaskill and E. D. Fackerell,/. Chem. Soc. Faraday Trans. 2, 84,161 (1988). Painleve Solution of the Poisson-Boltzmann Equation for a Cylindrical Poly-Electrolyte in Excess Salt Solution. 

Announced in the Portuguese newspaper O Seculo, October 18,1934, as recent news from Paris, October 17— 0 sibio qulmico Aquiles Machado foi eleito presidente do Office International de la Chimie, para a vaga deixada por Painleve. ... 

I.W. Stewart, Painleve analysis for a semi-linear parabolic equation arising in smectic liquid crystals, IMA J. Appl. Math., 61, 47-60 (1998). 

Volume 278 — PAINLEVE TRANSCENDENTS Their Asymptotics and Physical Applications... 

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... 

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. 

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). 

Recent works on the Painleve s classical example - introduced in Sect. 4.3.1 -include [109-111]. Friction impact oscillator which is similar to the system studied in Sect. 4.3.4 but with unilateral contact (creating the possibility of detachment and flight phases) is the subject of many publications see, for example, [53, 112, 113]. 

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. 

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