Existence and Uniqueness Problem

Consider the interacting lead screw and nut pair shown in Fig. 5.3. The equation of motion of this system is given by (5.13)  

Vahid-Araghi and F. Golnaraghi, Friction-Induced Vibration in Lead Screw Drives, 135 DOI 10.1007/978-l-4419-1752-2 8, Springer Science-I-Business Media, LLC 2011 

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

Note that we have 0 and F 0. Assume that the system s parameters are selected such that F+ 0 which requires that 0 and tan X + 7. For to be positive, coefficient of friction must be large enough such that /i tanA. If sgn(0)sgn(7 7 —mrni tan Ar) = 1, then (8.3) has no solution-, setting sgn(AI) = 1 in the RHS of (8.3) results in a negative contact force, and setting sgn(N) = — 1 results in a positive contact force. On the other hand, if sgn(0) sgn(P7— tanAr) = — 1, then (8.3) has two distinct solutions  

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The minimum conditions are recast into different equivalent forms in Section lOJ. Pragmatically and disregarding the theoretical existence and uniqueness problems, the reconciliation problem is reduced to finding some i obeying the conditions (10.3.11). We then expect that this x, making the constraint equation solvable by (10.3.11a), provides also the minimum (10.1.31) thus... 

It has been shown that existence and uniqueness problems arised even for a single-degree-of-freedom system and the conditions under which such problems occur have been determined. A graphical method has been... 

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