# Ternary Logic of Motion to Resolve Kinematic Frictional Paradoxes

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## Abstract

**:**

## 1. Introduction

_{xy}= μσ

_{yy}, still remain the most common way to introduce friction into mechanical analysis of both rigid and deformable systems.

## 2. Examples of Frictional Paradoxes

- (a)
- If $P\ge m\ddot{x}$ and $\dot{x}>0$ then $m\ddot{x}=P\frac{1-\mu \mathrm{tan}\phi}{2-\mu \mathrm{tan}\phi}$, which is possible only for $\mu \mathrm{tan}\phi <2$ (otherwise $P<m\ddot{x}$).
- (b)
- If $P\ge m\ddot{x}$ and $\dot{x}<0$ then $m\ddot{x}=P\frac{1+\mu \mathrm{tan}\phi}{2+\mu \mathrm{tan}\phi}$.
- (c)
- If $P<m\ddot{x}$ and $\dot{x}>0$ then $m\ddot{x}=P\frac{1+\mu \mathrm{tan}\phi}{2+\mu \mathrm{tan}\phi}$, which contradicts $P<m\ddot{x}$.
- (d)
- If $P<m\ddot{x}$ and $\dot{x}<0$ then $m\ddot{x}=P\frac{1-\mu \mathrm{tan}\phi}{2-\mu \mathrm{tan}\phi}$, which is possible only for $\mu \mathrm{tan}\phi >2$.

## 3. Ternary Logic of Motion and Rest

## 4. Implications to the Stability

#### 4.1. Resolution of Paradoxes by Introducing Compliance

#### 4.2. Friction-Induced Instabilities and Entropic Stability Criterion

## 5. Discussion

#### 5.1. Note on the History of Mechanics

#### 5.2. Ultraslow Frictional Sliding between Motion and Rest

#### 5.3. Mechanics of Instabilities

## 6. Conclusion

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Setup for the Painlevé paradox: (

**a**) two sliders (one frictional and the other is frictionless) connected by a rigid bar; (

**b**) rigid slender rod falling under gravity.

**Figure 2.**The paradox is resolved if the bar is assumed to be elastically deformable with the compliance k.

**Figure 3.**(

**a**) Three possible dependencies of the coefficient of friction (COF) on sliding velocity (constant COF, decreasing COF, and COF with a minimum value). Decreasing COF can cause the instability; (

**b**) two elastic half-spaces (characterized by the elastic moduli, E

_{1}and E

_{2}, Poison’s ratios ν

_{1}and ν

_{2}and densities ρ

_{1}and ρ

_{2}) slide relative to each other with the velocity V. For slightly dissimilar materials (in terms of their elastic properties) an interface elastic wave can propagate, and the wave becomes unstable when friction is introduced. (

**c**) Schematic of friction induced self-organization of a tribofilm (based on discussion in Reference [20]).

**Figure 4.**The ultraslow sliding friction tribometer MTBM: a general view and the setup (Reproduced from [37], with the permission of AIP Publishing).

**Figure 5.**A typical dependency of the friction force on the frictional path for the ultraslow sliding (37 nm/s, normal load W = 60 N). The friction force drops steadily from the static friction value (F = 12.75 N) to the kinetic friction value (F = 9.1 N) thus demonstrating intermediate values between stick and slip. Based on results discussed in [37].

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Nosonovsky, M.; Breki, A.D.
Ternary Logic of Motion to Resolve Kinematic Frictional Paradoxes. *Entropy* **2019**, *21*, 620.
https://doi.org/10.3390/e21060620

**AMA Style**

Nosonovsky M, Breki AD.
Ternary Logic of Motion to Resolve Kinematic Frictional Paradoxes. *Entropy*. 2019; 21(6):620.
https://doi.org/10.3390/e21060620

**Chicago/Turabian Style**

Nosonovsky, Michael, and Alexander D. Breki.
2019. "Ternary Logic of Motion to Resolve Kinematic Frictional Paradoxes" *Entropy* 21, no. 6: 620.
https://doi.org/10.3390/e21060620