Relaxation Tribometry: A Generic Method to Identify the Nature of Contact Forces

Abstract

Recent years have witnessed the development of so-called relaxation tribometers, the free oscillation of which is altered by the presence of frictional stresses within the contact. So far, analysis of such oscillations has been restricted to the shape of their decaying envelope, to identify in particular solid or viscous friction components. Here, we present a more general expression of the forces possibly acting within the contact, and retain six possible, physically relevant terms. Two of them, which had never been proposed in the context of relaxation tribometry, only affect the oscillation frequency, not the amplitude of the signal. We demonstrate that each of those six terms has a unique signature in the time-evolution of the oscillation, which allows efficient identification of their respective weights in any experimental signal. We illustrate our methodology on a PDMS sphere/glass plate torsional contact.

Appendix: Two-Times Averaging Method

The right-hand side of Eq. (1) is written $f=\epsilon h(\theta ,\theta ')$ where $\epsilon<<1$ and $\theta '={\dot{\theta }}/\omega _0$ denotes the derivative of $\theta$ with respect to dimensionless time. We seek the solution of the form $\theta =a \cos \varphi$, where $\varphi = \omega _0 t + \phi$ and a(t) and $\phi (t)$ are slowly varying functions. Substituting in Eq. (1) gives:

$$\begin{aligned}&{\ddot{a}} \cos \varphi - 2\dot{a} (\omega _0 + {\dot{\phi }})\sin \varphi - a{\ddot{\phi }} \sin \varphi - 2a{\dot{\phi }} \omega _0\cos \varphi \nonumber \\&\quad - a {\dot{\phi }}^2\cos \varphi = -\epsilon \omega _0^2 h. \end{aligned}$$

(15)

Considering that ${\ddot{a}}$, $\dot{a} {\dot{\phi }}$, ${\ddot{\phi }}$, and ${\dot{\phi }}^2$ are second order terms in $\epsilon$ and can thus be neglected in Eq. (15):

$$\begin{aligned} 2\dot{a} \omega _0 \sin \varphi + 2a {\dot{\phi }} \omega _0 \cos \varphi = \epsilon \omega _0^2 h. \end{aligned}$$

(16)

We must now develop h at order zero in $\epsilon$ since the left-hand side if of order one in $\epsilon$. At order 0, $\theta = a\cos \varphi$ and ${\dot{\theta }}/\omega _0 = -a\sin \varphi$ therefore $h=h(a\cos \varphi ,-a\sin \varphi )$. Then substituting in Eq. (16) and averaging over a time-period (with $\dot{a}$ and ${\dot{\phi }}$ constant) gives the so-called averaged equations:

$$\begin{aligned} \dot{a}= & {} \frac{\epsilon \omega _0}{2\pi } \int _0^{2\pi } h(a\cos \varphi ,-a\sin \varphi )\sin \varphi \,\mathrm {d}\varphi \nonumber \\= & {} \omega _0 \epsilon \langle h(\varphi ) \sin \varphi \rangle \end{aligned}$$

(17)

$$\begin{aligned} a{\dot{\phi }}= & {} \frac{\epsilon \omega _0}{2\pi } \int _0^{2\pi } h(a\cos \varphi ,-a\sin \varphi )\cos \varphi \,\mathrm {d}\varphi \nonumber \\= & {} \omega _0 \epsilon \langle h(\varphi ) \cos \varphi \rangle \end{aligned}$$

(18)

where $\langle \cdot \rangle$ denotes mean value over $2\pi$. These are two first-order ordinary differential equations on a and $\phi$.

For instance, consider the case of a quadratic dissipative force $f=\epsilon {\dot{\theta }}^2\mathrm {sgn}({\dot{\theta }})/\omega _0^2$. Then $h(\theta ,\theta ')= \theta '^2\mathrm {sgn}(\theta ')$ and $h(\varphi ) = -a^2 \cos ^2 \varphi \mathrm {sgn}(\sin \varphi )$. By averaging:

$$\begin{aligned} \langle h \sin \varphi \rangle= & {} -\frac{4a^2}{3\pi }, \end{aligned}$$

(19)

$$\begin{aligned} \langle h \cos \varphi \rangle= & {} 0. \end{aligned}$$

(20)

The two differential equations are therefore $\dot{a} = -\,4a^2 \epsilon \omega _0 / 3\pi$ and ${\dot{\phi }} = 0$. After integration, $a(t) = \left[ a_0^{-1}+4\epsilon \omega _0 t / 3\pi \right] ^{-1}$ (Eq. 7) and $\phi (t) = \phi _0$, $a_0$ and $\phi _0$ being the initial values of a and $\phi$.

A second interesting example is $f=\epsilon \theta ^2\mathrm {sgn}({\dot{\theta }})$ for which $h(\theta ,\theta ')=\theta ^2\mathrm {sgn}(\theta ')$ and $h(\varphi ) = -a^2 \cos ^2 \varphi \mathrm {sgn}(\sin \varphi )$. By averaging:

$$\begin{aligned} \langle h \sin \varphi \rangle= & {} -\frac{2a^2}{3\pi }, \end{aligned}$$

(21)

$$\begin{aligned} \langle h \cos \varphi \rangle= & {} 0, \end{aligned}$$

(22)

which gives the same signature $\dot{a} \propto a^2$ as in Eq. (19).

Similar results for all considered contact forces are summarized in Table 1.