On the mathematical basis of solid friction

Abstract

A piecewise-smooth ordinary differential equation model of a dry-friction oscillator is studied, as a paradigm for the role of nonlinear and hysteretic terms in discontinuities of dynamical systems. The friction discontinuity is a switch in direction of the contact force in the transition between left- and rightward slipping motion. Nonlinear terms introduce dynamics that is novel in the context of piecewise-smooth dynamical systems theory (in particular they are outside the standard Filippov convention), but are shown to account naturally for static friction, and moreover provide a simple route to including hysteresis. The nonlinear terms are understood in terms of dummy dynamics at the discontinuity, given a formal derivation here. The result is a three-parameter model built on the minimal mathematical features necessary to account for the key characteristics of dry friction. The effect of compliance can be distinguished from the contact model, and numerical simulations reveal that all behaviours persist under smoothing and under small random perturbations, but nonlinear effects can be made to disappear abruptly amid sufficient noise.

Appendices

Appendix 1: Switching asymptotics

The main contributions to the integral (3) come from the maxima of the envelope $a(k)$. There may be many such maxima, but near each we obtain expressions of the following form. We expand the phase $\theta (k)$ as a Taylor series, and with a little manipulation we approximate the integral as an envelope which is gaussian $a(k)\approx e^{-k^2/2}$ or gaussian-like (e.g. $a(k)\approx 1/(1+k^2)$), with a linear phase $\theta (k)\approx \rho k$ for constant $\rho$. Taking, for example, $a(k)=e^{-k^2/2}$, we have

$$\begin{aligned}&\mu (v)={\text {Re}}\left[ {\text {Erf}}\left( {\frac{v/\varepsilon -i\rho }{\sqrt{2}}}\right) \right] \nonumber \\&\quad \sim {\text {sign}}(v)-\sqrt{\frac{2}{\pi }}{e^{(\rho ^2-v^2/\varepsilon ^2)/2}}{}{\text {Re}}\left[ \!e^{i\rho v/\varepsilon }\!\sum _{n,m=0}^\infty \frac{(i\rho )^mC_{nm}}{(v/\varepsilon )^{2n+1+m}}\!\right] \nonumber \\&\quad \sim {\text {sign}}(v)\times \left\{ \;1-\sqrt{\frac{2}{\pi }}\frac{e^{(\rho ^2-v^2/\varepsilon ^2)/2}\cos {(\rho v/\varepsilon )}}{|v|/\varepsilon }+\cdots \;\right\} .\nonumber \\ \end{aligned}$$

(37)

The first line denotes the real part of the standard error integral [1] denoted ${\text {Erf}}$, the second (with coefficients $C_{nm}=\frac{(-1)^n(2n-1)!!(2n+1)!}{(2n+1-m)!m!}$) follows from its asymptotic approximation for large argument and $|v|/\varepsilon \gg \rho$, whose leading order terms are shown in (37). The $v/\varepsilon$ limit of the integral introduces a cut-off similar to Gibbs phenomenon in Fourier series or ringing in signal control [27]. Taking $a(k)$ to be any of $e^{-k^2/2},\, 1/(1+k^2)$ or ${\text {sech}}^2(k)$, even when introducing more complex oscillations by replacing $\cos \theta (k)$ with, say, $\cos (\rho _1k)\cos (\rho _2k)$ or $\cos (\rho _1k)\sin (\rho _2k)/k$, all yield the same asymptotic form (2). These all tend towards a curve like Fig. 2(ii) in the limit $\varepsilon \rightarrow 0$, which has peak values $\mu =\pm \mu _s$ which are $\varepsilon$ independent and therefore remain clearly defined in the limit $\varepsilon \rightarrow 0$. In the example (37), the peak has height $\mu _s\sim 1+\sqrt{\frac{2}{\pi }}\frac{4\rho ^3}{\pi ^2}e^{-\pi ^2/8\rho ^2}$ and lies at $v\sim \pm \pi \varepsilon /2\rho$.

To obtain (5) from (4), we can then define a continuous monotonic approximation of the ${\text {sign}}$ function, a suitable and convenient choice being

$$\begin{aligned} \varLambda _\varepsilon (v)=\frac{v/\varepsilon }{\sqrt{1+(v/\varepsilon )^2}}= {\text {sign}}(v)-\sum _{n=1}^\infty \frac{b_n}{(iv/\varepsilon )^{2n}},\nonumber \\ \end{aligned}$$

(38)

which satisfies (6), with coefficients $b_n=\frac{{\text {sign}}(v)}{(2n)!}\prod _{m=1}^{n}$ $(2m-1)^2$. Using (38) to replace the ${\text {sign}}$ function in (4), we obtain

$$\begin{aligned} \mu (v)\sim & {} \varLambda _\varepsilon (v)+\sum _{n=1}^\infty \left\{ i^{2n}b_n+c_nq(v/\varepsilon )\right\} (\varepsilon /v)^{2n}\nonumber \\:= & {} \varLambda _\varepsilon (v)+\rho L_\varepsilon (v)\varGamma _\varepsilon (v), \end{aligned}$$

(39)

the second line being found by substituting $v/\varepsilon$ with its inverse $v/\varepsilon =\varLambda _\varepsilon /\sqrt{1-\varLambda _\varepsilon ^2}$. Based on the qualitative form of (37), we approximate $L_\varepsilon (v)$ in (39) as $L_\varepsilon (v)=\varLambda _\varepsilon (v)$ for the sake of the friction model in (5). The constant $\rho$ controls the peak height $\mu _s$ of the function $\mu (v)$, the precise relation depending on the functional form of $\mu$. The result (39) is expressed in terms of the function $\varGamma _\varepsilon (v)=1-\varLambda _\varepsilon ^2(v)$ which satisfies (7), and

$$\begin{aligned} L_\varepsilon (v)= & {} \frac{1}{\rho }\sum _{n=1}^\infty \frac{i^{2n}b_n+c_nq\left( {v/\varepsilon }\right) )}{\varLambda _\varepsilon ^{2n}(v)} \left( {1-\varLambda _\varepsilon ^2(v)}\right) ^{n-1}\nonumber \\= & {} \frac{c_1q\left( {v/\varepsilon }\right) -b_1}{\rho \varLambda _\varepsilon ^{2}(v)}\;+\;\mathsf{O}\left( {1-\varLambda _\varepsilon ^2(v)}\right) , \end{aligned}$$

(40)

where $\rho$ serves as a normalization constant at $v=0$. The finiteness of $L_\varepsilon (v)$ at $v=0$ follows from that of $\mu$. In (40) we separate the leading order term in $L_\varepsilon (v)$ from those of order ${1-\varLambda _\varepsilon ^2(v)}$, which vanish as $|v|/\varepsilon \rightarrow \infty$.

The asymptotic form (37) lends some interpretation to the features of the friction curves in Fig. 2. The ${\text {sign}}(v)$ term is the dominant contribution to the integral, arising from wavenumbers associated with lower energies and thus independent of the speed-dependent high-energy cut-off at wavenumber $k=v/\varepsilon$, and this clearly corresponds to the kinetic friction component in Fig. 1(i). At small $v$, the cut-off becomes important, creating decay away from the steady value $|\mu |\approx 1$ towards $\mu \approx 0$, plus an oscillation dominated by wavenumbers near $k_*=v/\varepsilon$ that results in peaks at $\pm \mu _s$; in the Gaussian model, these occur at $v=\pm v_s:=\pi \varepsilon /2\rho$. The peak amplitude $\mu _s$ is $\varepsilon$-independent (since $v_s/\varepsilon$ can everywhere be replaced by $\pi /2\rho$), so the peaks remain at the same height as $\varepsilon \rightarrow 0$ as shown in Fig. 2(ii), reproducing the static friction component in Fig. 1(ii). For $\rho =0$ there are no peaks, while for larger $\rho$ there appear multiple peaks and dips at small but nonzero speeds (not shown); these dips in the friction force may or may not have physical application.

The peak seen in Fig. 2 is just the first of a sequence of oscillations that grow as we take $\rho$ larger (i.e. a faster oscillating integrand), and they need not (though they do in this case) grow so rapidly in amplitude with $\rho$. Such oscillations may be of interest for modelling elsewhere and can be dealt with using similar methods to those we introduce, but we limit this paper to investigating the application of a single peak model to a simple friction problem.

Appendix 2: Perturbation to $\beta >0$ and $\varepsilon >0$

Consider the full discontinuous system (25) for small $\beta >0$, in terms of variables $\left\{ y,v, z\right\}$ and the functions $\varLambda (v),\, \varGamma (v)$, from (6)–(7). On $v=0$ the values $\varLambda (0)$ and $\varGamma (0)$ will be replaced by dummy variables $\lambda$ and $\gamma (\lambda )$ according to (9).

Proposition 5

For $\max [\left| \ddot{q}(t)\right| ]\le N\mu _s$, there exists a strip of periodic orbits with fixed

$$\begin{aligned} \left\{ y(t),v(t), z(t)\right\} =\left\{ y_0,0,\gamma (\zeta )\right\} \end{aligned}$$

satisfying $|y_0|\le \left( {N\mu _s-\max [\ddot{q}(t)]}\right) /k$ and with $\zeta$ any solution of

$$\begin{aligned} 0=\ddot{q}-ky_0-N\zeta \left( {1+\rho \gamma (\zeta )}\right) \;:\quad |\gamma (\zeta )|\le 1. \end{aligned}$$

If $\lambda _s>1$ with $\lambda _s$ defined by (16), these periodic orbits are all attracting, if $\lambda _s<1$ those with $|\zeta |<\lambda _s$ are attracting and those with $\lambda _s<|\zeta |\le 1$ are of saddle type.

Proof

A solution of (25) with fixed $\left\{ y,v, z\right\}$ must clearly have $v(t)=0$ and $z(t)=\varGamma (v)$, which implies a sticking trajectory. Therefore, we use the sticking variables (9). By (19), sticking trajectories must satisfy $\ddot{q}(t)-N\mu _s\le ky(t)\le \ddot{q}(t)+N\mu _s$, and for fixed values $y(t)=y_0$ to exist, it is sufficient to have $\max [\ddot{q}(t)]-N\mu _s\le ky_0\le -\max [\ddot{q}(t)]+N\mu _s$, which has solutions for $y_0$ only if $\max [\ddot{q}(t)]\le N\mu _s$. The eigenvalue along the $\zeta$ direction is $\frac{\partial \dot{z}}{\partial z}=-1$ by (25) and therefore attracting, and note that $y$ is stationary in the sticking mode by (21) and (24). Finally, sticking solutions satisfy (24) and therefore lie in the surface $\mathcal {B}( z)$ defined in (22), obeying $\lambda (t)=\zeta$ for some $\zeta$, which implies $0=\ddot{q}-ky_0-N\zeta \left( {1+\rho \gamma (\zeta )}\right)$. Such a fixed point is attracting if $\frac{\partial \lambda '}{\partial \lambda }<0$ and since $\frac{\partial \lambda '}{\partial \lambda }=-N\left( {1+\rho \gamma (\zeta )}\right)$ substituting $\lambda =\lambda _s$ and rearranging gives $|\zeta |<\lambda _s$, which applies if $\lambda _s<1$, and otherwise $|s|<1$ if $\lambda _s\ge 1$, in short $|\zeta |<\max [1,\lambda _s]$. $\square$

As a simple corollary, when $N\mu _s=|\ddot{q}|$ there is only one sticking periodic orbit, given by $\left\{ y(t),v(t),z(t)\right\} \!=\left\{ 0,0,0\right\}$, which passes through at least one point on the sticking boundary, thus undergoing some form of sliding bifurcation. For example, if we take $\ddot{q}(t)=-\sigma \sin (\omega t)$, then the sticking boundary and the periodic orbit contact tangentially at $t=\frac{\pi }{2\omega }$ where the right-slipping speed vanishes, i.e. $\lim _{\delta \rightarrow 0}\left. \dot{v}\right| _{v=+\delta }=0$ with $\delta \ge 0$, and at $t=\frac{3\pi }{2\omega }$ where the right-slipping speed vanishes, i.e. $\lim _{\delta \rightarrow 0}\left. \dot{v}\right| _{v=-\delta }=0$ with $\delta \ge 0$.

When $N\mu _s<|\ddot{q}|$, there no longer exist pure sticking periodic orbits (i.e. purely within $v=0$), but one expects, since the system is dissipative, there to exist at least one attractor. Typical behaviour of piecewise-smooth systems suggests that these will take the form of stick–slip periodic orbits for parameters close to $N\mu _s=|\ddot{q}|$, undergoing brief intervals of sticking between intervals of right- and/or left-slipping motion, though a complete study is beyond our scope here. Stick–slip orbits were found in the case of the undamped $c=0$ and nonhysteretic $\beta =0$ oscillator, and for a sinusoidal forcing $\ddot{q}$, in [28, 37]. Crucially these and other previous works that use Filippov-type solutions assume linear sticking, i.e. $\rho =0$. Some numerical simulations and experiments are given in [15, 55], showing periodic, quasi-periodic, and irregular motions, suggesting that further in-depth analytical studies are an interesting subject for future work; note that the analysis in [55] introduces hysteresis and static friction using multiple switching laws, instead of nonlinear sticking as introduced here, and various multi-period orbits are identified in numerical and experimental data. In simulations similar to those in Sect. 8 but not shown here, periodic left–right slip and periodic stick–slip oscillations are found at a wide range of parameters.