Nontrivial temporal scaling in a Galilean stick-slip dynamics

E. J. R. Parteli, M. A. F. Gomes, and V. P. Brito

Phys. Rev. E 71, 036137 – Published 24 March 2005

Abstract

We examine the stick-slip fluctuating response of a rough massive nonrotating cylinder moving on a rough inclined groove which is submitted to weak external perturbations and which is maintained well below the angle of repose. The experiments presented here, which are reminiscent of Galileo’s works with rolling objects on inclines, have brought in the last years important insights into the friction between surfaces in relative motion and are of relevance for earthquakes, differing from classical block-spring models by the mechanism of energy input in the system. Robust nontrivial temporal scaling laws appearing in the dynamics of this system are reported, and it is shown that the time-support where dissipation occurs approaches a statistical fractal set with a fixed value of dimension. The distribution of periods of inactivity in the intermittent motion of the cylinder is also studied and found to be closely related to the lacunarity of a random version of the classic triadic Cantor set on the line.

Received 20 August 2004

DOI:https://doi.org/10.1103/PhysRevE.71.036137

Authors & Affiliations

E. J. R. Parteli

Institut für Computerphysik, ICP, Universität Stuttgart, Pfaffenwaldring 27, 70569 Stuttgart, Germany

M. A. F. Gomes

Departamento de Física, Universidade Federal de Pernambuco-50670-901, Recife, PE, Brazil

V. P. Brito

Departamento de Física, Universidade Federal do Piauí-64049-550, Teresina, PI, Brazil

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 71, Iss. 3 — March 2005

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Schematic diagram of the experimental apparatus: the incline A has a length $L_{0} = 1500 mm$ and is rigidly maintained at an inclination $θ < θ_{c}$ ( $= angle$ of repose) with the horizontal. The cross section of A is shown in B; C is a cylinder with length $L$ varying from $L = 5 to 1000 mm$ . Sliding events occur after a variable number of impacts of the hammer D. The time series discussed in the text and illustrated in Fig. 2 are chains of 100 sliding events.Reuse & Permissions
Figure 2
Typical nonhomogeneous intermittent time series of sliding events for massive cylinders of aluminum of length (a) $L = 20 mm$ and (b) $L = 200 mm$ , with $(θ_{c} - θ) ∕ θ_{c} \approx 0.30$ .Reuse & Permissions
Figure 3
Cumulative sum $A_{t} = \sum_{i = 1}^{N} λ (t)$ of the sliding events as a function of time for two different experiments: In the main plot, the corresponding time series was obtained with a cylinder of length $L = 10 mm$ , while in the inset the length of the cylinder used in the experiment was $L = 100 mm$ . The dashed lines in the two plots are guides to the eye meant to show the linear behavior of $A_{t}$ .Reuse & Permissions
Figure 4
Log-log plot of the average number $⟨ N (τ) ⟩$ of one-dimensional boxes of size $τ$ necessary to cover the regions along the time axis where the sliding activity is concentrated. The average ⟨∙⟩ is over the entire experimental set of time series of sliding events (circles). The asterisks denote the same quantity for an ensemble of equal size of synthetic series: random Cantor sets whose number of events is equal to the number (100) of sliding events (see text). The dotted line has as slope the golden mean $(\sqrt{5} - 1) ∕ 2 ≃ 0.618 \dots$ . The inset shows $N (τ)$ for typical experimental time series.Reuse & Permissions
Figure 5
Log-log plot of the average number $⟨ M (τ) ⟩$ of sliding events within an interval of time $τ$ . The average ⟨∙⟩ is over the entire experimental set of time series of sliding events (circles). The line has slope $δ = 0.65 \pm 0.05$ , which is interpreted here as the (mass) dimension of the statistic fractal set defined by the dissipation support of the experimental time series. This value is in agreement, within the fluctuation bars, with the dimension found in Fig. 4 calculated with the box counting for the same values of the time interval $τ$ .Reuse & Permissions
Figure 6
Log-log plot of the lacunarity function $⟨ Λ (τ) ⟩$ for the experimental time series (circles). The dashed line gives the same function for an ensemble of MTCS of order 7 ( $2^{7}$ events, $T = 3^{7}$ ) with 28 events randomly eliminated (see text). The inset shows $Λ (τ)$ for some typical time series. $⟨ Λ (τ) ⟩$ decays effectively as the power law $τ^{- 0.36}$ for $3 \times 10^{- 3} ⩽ τ ⩽ 2 \times 10^{- 1}$ (continuous line).Reuse & Permissions

Physical Review E

covering statistical, nonlinear, biological, and soft matter physics