Dynamics of mechanical systems with two sliding contacts: new facets of Painlevé’s paradox

Abstract

We investigate the dynamics of finite degree-of-freedom, planar mechanical systems with multiple sliding, unilateral frictional point contacts. A complete classification of systems with 2 sliding contacts is given. The contact-mode-based approach of rigid body mechanics is combined with linear stability analysis using a compliant contact model to determine the feasibility and the stability of every possible contact mode in each class. Special forms of non-stationary contact dynamics including “impact without collision” and “reverse chattering” are also investigated. Many types of solution inconsistency and indeterminacy are identified and new phenomena related to Painlevé’s non-existence and non-uniqueness paradoxes are discovered. Among other results, we show that the non-existence paradox is not fully resolvable by considering impulsive contact forces. These findings contribute to a growing body of evidence that rigid body mechanics cannot be developed into a complete and self-consistent theory in the presence of contacts and friction.

Appendix: Eigenvalues of the coefficient matrix in (26)

The eigenvalues and the eigenvectors of the matrix

$$\begin{aligned} \mathbf{M}{\mathop {=}\limits ^{def}} \left[ {{\begin{array}{cc} {{\begin{array}{cc} 0&{} 0 \\ 0&{} 0 \\ \end{array}}}&{} {{\begin{array}{cc} 1&{} 0 \\ 0&{} 1 \\ \end{array} }} \\ {-\mathbf{BK}}&{} {-\mathbf{BQ}} \\ \end{array} }} \right] \end{aligned}$$

(27)

are investigated here. Specifically, we address the following questions:

• 1. do any of the eigenvalues have positive real part?

• 2. are there real, positive eigenvalues such that the two coordinates of the corresponding eigenvector have the same sign?

• 3. is the dominant eigenvalue real, and the corresponding eigenvector in the positive quadrant?

In the special case of Q=0, the block structure of M implies that its eigenvalues are $\pm \uplambda ^{1/2}$ and the eigenvectors are $[\mathbf{x}^\mathrm{T}\uplambda \mathbf{x}^{\mathrm{T}}]^{\mathrm{T}}$ where $\uplambda$ and x are eigenvalues and eigenvectors of the 2 by 2 matrix −BK. Hence, the problem is reduced to the eigenvalue analysis of a 2 by 2 matrix, for which simple, closed-form expressions are available. Below, we provide answers to the three questions without detailed proofs:

• 1. Yes, if and only if −BK has a positive eigenvalue, i.e. if either its determinant is negative or if its trace is positive. This is impossible in classes 12 and 42; true in some regions within classes 13, 14, 23, 31, 41, 46; and always true in the rest of the classes. The eigenvalues with non-positive real parts are always purely imaginary.

• 2. Yes in classes 15, 25, 32-35, 43, 44; yes in some regions within classes 14, 24; no otherwise

• 3. same as for question 2, except for class 25, in which there are two positive real eigenvalues, but only the smaller one has an appropriate eigenvector, implying a negative answer.

In the general case $\mathbf{Q}\ne 0$, the problem is equivalent of a quadratic eigenvalue problem with 2 by 2 matrices. The eigenvalues and eigenvectors can be expressed in closed from; however, they appear to have a much more complicated structure than for Q=0. Instead of analytical calculation, we performed numerical analysis with systematic variations of K and Q. The analysis suggests that the answers outlined above for Q=0 remain true except that the eigenvalues with non-positive real parts are no more purely imaginary, but typically have negative real parts. This is caused by the damping introduced via Q.