Chaos on the conveyor belt

Spring-block type systems are successfully used for modeling various complex systems which exhibit self-organization. Usually, the static friction coefficients exceed the dynamical ones, in systems composed of two or more blocks connected by linear springs the coexistence of these friction types can lead to avalanches, nonlinear dynamics or Self-Organized Criticality (SOC) Bak1987 ; Bak1988 ; Bak1989 ; Carlson1989 .

Spring-block type models have broad interdisciplinary applications (for a recent review see Neda2009 ), and prove to be very useful in describing many complex phenomena in different areas of science. For example, they have been applied successfully to explain elements of the Portevin-Le Chatelier effect Lebyodkin1995 , the fragmentation obtained in drying granular materials in contact with a frictional surface Andersen1994a ; Andersen1994b ; Leung2000 , to understand the formation of self-organized nanostructures produced by capillary effects Jarai-Szabo2005 ; Jarai-Szabo2011 , to model magnetization processes and Barkhausen noise Kovacs2005 , to describe glass fragmentation Horvat2012 , and even for modeling highway traffic Jarai-Szabo2011a ; Jarai-Szabo2012 .

The first members of this model-family were presented in 1967 by R. Burridge and L. Knopoff Burridge1967 to explain the empirical Guttenberg-Richter law of the size distribution of earthquakes Gutenberg1956 . One of their original models (called BK model here) is composed of a chain of many blocks connected with springs and two planes, to model the sliding of tectonic plates (for a recent review see for example Kawamura2012 ). The blocks can slide with friction on the bottom surface and they are all connected by springs to the upper plane which is dragged with a constant velocity. This model presents stick-slip dynamics, and the size distribution of the slipping events exhibits a power-law scaling.

Since the pioneering work of Carlson et al. Carlson1989 ; Carlson1989a , the dynamics of the BK model has been investigated with several type of friction forces. In these studies both chaotic dynamics Xu1994 ; Ryabov1995 ; SousaVieira1999 ; Szkutnik2004 ; Erickson2011 and phase transition-like phenomena SousaVieira1993 have been reported.

The system used here has also been proposed by Burridge and Knopoff Burridge1967 , and later it was referred as ”train model” SousaVieira1992 . As sketched in Fig. 1, a spring-block chain is placed on a platform (conveyor belt) that moves with constant velocity. The first block is connected by a spring to a static external point. As a result, due to the dragging effect of the moving platform, the blocks will undergo complex stick-slip dynamics.

The stick-slip dynamics of one block has been studied by analytical SousaVieira1994 ; Baumberger1995 ; Vasconcelos1996 ; Elmer1997 , numerical Baumberger1995 ; Vasconcelos1996 ; Elmer1997 ; Sakaguchi2003 and experimental methods Johansen1993 ; Baumberger1995 . The results for a chain composed of several blocks are contradictory from the point of view of self-organized criticality SousaVieira1992 ; SousaVieira1996 ; Vasconcelos1996 ; Chianca2009 . Undoubtedly, chaotic dynamics has been found in such systems by many authors SousaVieira1995 ; SousaVieira1999 ; Szkutnik2002 ; Erickson2008 . In these systems nonlinearity is introduced via friction forces, and several models have been studied from velocity-weakening SousaVieira1992 ; SousaVieira1994 ; SousaVieira1995 ; Szkutnik2002 to state-dependent Sakaguchi2003 friction forces. It has been shown that with velocity-weakening friction forces (and a constant static friction), the case of two blocks is the simplest autonomous spring-block system exhibiting chaos SousaVieira1995 . It was also argued, that for systems composed of many blocks, chaotic dynamics and SOC can coexist SousaVieira1996 .

It needs to be mentioned that a somewhat modified version of the train model has also been investigated, where in contrast with the former model, both the first and the last blocks are connected to static points. Additional springs are also introduced. This system also exhibits periodic and chaotic behavior Galvanetto1998 ; Galvanetto2002 .

Although the proposed problem has been thoroughly investigated for one, two and many blocks, we believe that a careful experimental and detailed simulation study for an intermediate number of blocks can reveal new and interesting dynamical complexity. Comparisons between analytical, numerical and experimental results exist mainly for the one block system. An exception is the work of Burridge and Knopoff Burridge1967 , where a chain of eight blocks was thoroughly investigated. The main purpose of their work was, however, to investigate the distribution of the potential energy released during avalanches, and they did not considered the problem from a dynamical systems view. Also, a general feature of previous numerical studies is that the model parameters and assumptions are arbitrarily chosen, and usually the only argument for this is to make the calculations or the numerical code simpler.

Here we consider an experimental setup composed of $5$ blocks, and computer simulations of a simple model for systems up to $10$ blocks. In contrast to previous works, the model parameters are realistically estimated from experimental measurements. We also incorporate the surface inhomogeneities of the conveyor belt by using a Coulomb type friction force varying stochastically about a mean. This feature appears to be essential since the temporal intermittency observed in the experiments cannot be recovered with only deterministic friction. A spring force with exponential cutoffs is used, to make the force profile more realistic and to avoid the collisions of the blocks with each other.

To the best of our knowledge this intermediate system size has not yet been investigated experimentally. The choice of intermediate system sizes is motivated by the expectation that this is the range where the theory of collective phenomena and of dynamical systems overlap and thus tools borrowed both from statistical mechanics and chaotic dynamics become both applicable. Despite the relatively small number of elements the system consists of, we surprisingly find a sharp phase transition-like behavior as a function of the conveyor belt velocity. Interestingly, as the size of the system is increased this transition becomes smoother, a phenomenon that can be understood through the intermittency that was revealed in this region. Tuning the level of disorder to a certain value, disorder induced phase transition-like behavior is also observed. We show that by using the Coulomb type friction forces this simple system exhibits a complex, chaotic dynamics accompanied by power-law type avalanche size distribution indicating SOC, noise induced intermittency and also noise induced chaos.

The work is structured as follows. First the used experimental setup is presented and the results are described for a chain of five blocks (Sec. II). In Sec. III the model is detailed. Simulation results (Sec. IV) without and with a disorder in the friction force are discussed. Finally, the influence of the number of blocks (up to $N=10$) on the observed dynamics is investigated (Sec. V) and final conclusions are drawn (Sec. VI).

The experimental setup is sketched in Fig. 1. The chain is built by $N=5$ black wooden blocks of mass $m=115.8$ g, and of dimensions $4$ cm$\times 8$ cm$\times 4$ cm. The blocks are connected by steel springs of rest length $l=7$ cm and spring constant of $k=19.8$ N/m. The chain is placed on the conveyor belt of a treadmill (running machine) with adjustable speed. A digital camera is placed above the chain to record the dynamics of the blocks with $24$ fps (for a snapshot see Fig. 2). In order to make the last block clearly distinguishable from the others, and from the black colored conveyor belt, the top of this block is colored white.

The video recordings are converted by a threshold operation into black and white image sequences as shown in Fig. 3. Then, the position of the last block is detected on these images. The length unit on the image can be determined using the image of a tape measure, which is placed next to the chain. Accordingly, the length $x_{5}(t)$ of the chain as a function of time can be obtained automatically with a simple processing program.

The experiments were carried out with two values of the belt’s speed: $u=0.22$ m/s and $u=0.28$ m/s. The measured average values of friction forces in the experiment are $F_{st_{0}}=1.98$ N for static friction and $F_{k_{0}}=0.89$ N for kinetic friction. Thus, the ratio of the two types of friction forces is $f_{s}=F_{k_{0}}/F_{st_{0}}=0.45.$

In both experiments the system is initialized with blocks sticking to the belt and with undeformed springs. Initially, the conveyor belt is at rest. After it is set in motion, length oscillations of very large amplitude occur on a time-interval up to $100$ s. In order to ensure that these initial transients are not considered and a kind of steady state has set in, a time-interval of $t_{\rm trans}=200$ s is discarded in the data processing. The recorded data for the position of the last block $x_{5}(t)$ as a function of time is then analyzed. For the velocity $u=0.28$ m/s, the recorded data reveal (Fig. 4) two qualitatively different temporal behaviors. In the domain characterized by small amplitudes in $x_{5}(t)$, the dynamics of the system is nearly periodic, while in the domain of large amplitudes the last block exhibits a chaotic-looking behavior. We recall again, that the initial periodic-like behavior in this figure is not a result of the lasting transients, since before this, several chaotic-like regions were already present during the large amplitude bursts following the start of the conveyor belt.

The differences in the dynamics for these domains can be better understood from the Fourier Transforms (FT). We have thus computed separately the FT for the two well distinguishable regions (see Fig. 4). It is clearly observable that for the small amplitude interval the power spectrum $S$ of the FT has peaks at equal distances suggesting periodic dynamics, while in the chaotic region the power spectrum presents a quasi-continuous distribution.

As can be seen in Fig. 5, for $u=0.22$ m/s the dynamics of the system is intermittent. In the first 125 seconds there are large ”avalanches” that result in a simultaneous movement of blocks (they slip and stick to the belt together). In this regime, the length of the chain evolves in time with fluctuations of large amplitudes. Then, without any external influence, the dynamical behavior changes abruptly, and for more than one minute the system remains in a nearly steady state, when all the blocks are slipping continuously relative to the belt. After this time interval, the behavior of the system looks chaotic again, of the type it was at the beginning of the recorded dynamics. The power-spectrum of the FT for the whole plotted interval is similar to the one observed for the chaotic regime for $u=0.28$ m/s.

The model contains the same elements (blocks and springs) as the experimental setup. The main challenge in the modeling effort is however the quantification of the friction and spring forces and the numerical integration of the equations of motion. Let us consider a chain formed by $N$ identical blocks of mass $m$. The blocks are connected by identical springs of linear spring constant $k$ and undeformed length $l$. As Fig. 1 indicates the motion of the blocks of the model is one dimensional and the transverse oscillations present in the experiment are thus neglected.

In the model, dimensionless units are taken. We have chosen these units so that $m=1$, $k=1$, and $l=50$. The value for $l$ was chosen for the sake of an easier graphical visualization (spring length corresponding to $50$ pixels). Being motivated by the experiment, we cannot avoid using a nonzero rest length for the springs. In order to have these dimensionless values to correspond to the experimental situation, the units are chosen as: $[m]=0.1158$ kg for mass, $[k]=19.8$ N/m for spring constant, and $[l]=1.4\cdot 10^{-3}$ m for length. The units of the other quantities follow from dimensional considerations. The time, velocity and force units are thus, $[t]=\sqrt{[m]/[k]}=0.0765$ s, $[u]=[l]/[t]=0.0183$ m/s, and $[F]=[k]\cdot[l]=0.0277$ N, respectively.

The equation of motion for the $i$th block of the chain is written as

where $\Delta l_{-}=x_{i}-x_{i-1}-50$ and $\Delta l_{+}=x_{i+1}-x_{i}-50$, respectively, and $v_{r_{i}}$ is the relative velocity of the block to the conveyor belt. The elastic force, $F_{e}$, and the friction force, $F_{f}$, are defined below.

The elastic force $F_{e}$ of any spring is linear, up to a certain deformation value, $\Delta l_{max}$. For higher deformations, this dependence is assumed to become exponential, with an exponent bigger (in modulus) for negative deformations (see Fig. 6). Accordingly,

where we have chosen $\Delta l_{max}=20$, $b_{1}=0.2$ for $\Delta l<0$ and $b_{2}=0.01$ for $\Delta l\geqslant 0$. With these choices the model is more realistic since the nonlinearity of spring forces is taken into account and collisions between blocks are avoided. At the same time the choice of parameters $b_{1}$ and $b_{2}$ is somewhat ad-hoc and we have not carried out a systematic change of them, other than the fact that the average chain lengths are thus comparable with those seen in the experiment with an error less than 10%.

According to a review of experimental results presented in Ref. Johansen1993 , ”the classical friction law, where the friction force is proportional to the load, will only exist in an average sense” and it is important to take into account that in many cases the friction is determined by surface asperities, which means that ”deterministic friction-velocity relations at best only exist in an average sense”. In Johansen1993 , the authors also argue that in the case of stick-slip dynamics on a plane surface, the friction forces may have normal distributions.

In the present paper Coulomb’s law of friction Bowden1954 is used with a noisy extension reflecting the surface irregularities. Both the static and kinetic friction forces are independent of velocity modulus. A block remains in a stick state until the resultant external force $F_{ex}$ exceeds the value of the static friction force, $F_{st}$. For higher external force values the block starts to slide in the presence of the kinetic friction force $F_{k}$. We assume that, the ratio of the static and kinetic friction forces $F_{k}/F_{st}=f_{s}$ is constant. The friction force $F_{f}$ acting on the block depends both on $\operatorname{sgn}{(v_{r})}$, where $\operatorname{sgn}{}$ is the signum function and $v_{r}$ is the block’s speed relative to the conveyor belt, and on the value of the resultant external forces $F_{ex}$ acting on it. In our 1D setup, the friction force orientation is given only by its sign:

where $v_{r}=v-u$ and $v$ is the velocity of the block relative to the laboratory frame.

Since the surface of the conveyor belt is not perfectly smooth, the already mentioned argument of Ref. Johansen1993 is implemented by using randomly distributed friction forces. This is incorporated into the model by randomly generating a new static friction force value for every different position of the block relative to the belt. These random friction force values are generated according to a normal distribution with a fixed mean $F_{st_{0}}$ and standard deviation: $\sigma$. This standard deviation (together with the integration time-step, which is however the same for all simulations), will quantify the amount of disorder introduced in the model. Although the origin of the disorder is the spatial inhomogeneity of the belt’s surface, from the point of view of the dynamics the fluctuations of the friction force appear as temporal noise. We shall thus call $\sigma$ as the noise strength.

The kinetic friction force value is always automatically updated together with the static friction force value, using their fixed ratio $f_{s}$. As a result, both the kinetic and static friction forces will fluctuate in time during the sliding of the particular block. In Fig. 7 we illustrate the variation of the friction force as a function of the resultant external force $F_{ex}$ for one stick-slip event. Such type of friction forces also proved to be useful for modeling highway traffic Neda2009 ; Jarai-Szabo2011a .

In order to use the same friction force value as in the experiments, in the dimensionless units the average value of the static friction force is $\langle F_{st}\rangle=F_{{st}_{0}}=71.4$. The ratio of the two types of friction forces is also chosen in agreement with the experiments as $f_{s}=0.45$.

Computer simulations of the model start from initial conditions similar to the experimental ones. The blocks are placed on the belt with undeformed springs between them. At this initial setup the blocks are stuck to the belt which means that their initial velocity in the laboratory reference frame is equal to the conveyor belt’s constant velocity $u$. Computer graphics is also used to visualize in real-time the dynamics of the blocks.

The numerical methods used to solve Newton equations (1) are presented in the Appendix. The time-step of integration is fixed to $dt=0.001$.

In order to characterize statistically the dynamics of the chain, a parameter measuring the fluctuation of the chain length in dynamical equilibrium is introduced. The chain’s length is determined by the position of the last block in the row. The disorder parameter $r$ is defined as the standard deviation of the coordinate ${x_{N}}$ compared to the time averaged length $\langle x_{N}\rangle$ of the chain:

This quantity takes on large values for large fluctuations which explains the term disorder parameter. Among other relevant dynamical measures, this disorder parameter is investigated as a function of the belt velocity $u$ (Section IV.1.), the noise level $\sigma$ (Section IV.2.), and the number of blocks $N$ (Section V.).

We can now briefly discuss the influence of the system size $N$ on the observed phase transition-like phenomena. We know from statistical physics that the observed phase transitions become sharper as the system size is increased. Although the investigated system is a non-equilibrium dynamical system and not a system in thermal equilibrium, one might expect that the phase transition-like behavior sharpens for larger system sizes. In our system, however, the opposite happens. As shown in Fig. 19 for increasing system sizes, the sharp phase transition-like behavior is transformed into a smoother and smoother one, the transitional region broadens with $N$.

This observation can be explained through a quantitative change in the systems dynamics in the presence of noise. As we have seen before, the dynamics changes from chaotic to periodic or quasi-periodic via an intermittent behavior. For $N=10$, intermittency occurs even without noise. This is nicely illustrated by the $x_{N}(t)$ graphs for different $u$ values in Fig. 20. The intermittency found in the transition interval $27<u<40$ explains why the sharp phase transition-like behavior disappears for larger systems. As the disorder parameter measures the fluctuations of the chain length, in the intermittent region smaller disorder parameters will be observed. Fig. 20 indicates that larger and larger periodic or quasi-periodic time intervals appear at increased chain velocities. Therefore, the sharp transition becomes smoother. For increasing system sizes, the previously observed noise induced phase transition-like behavior becomes also less evident, as illustrated by Fig. 21, for driving velocities above the transition region.

Bifurcation diagrams generated for different system sizes are shown in Fig. 22. These results augment those obtained from disorder parameter. They indicate that for $N<5$, the system has only periodic or quasi-periodic attractors. The behavior of the disorder parameter does not suggest any phase transition-like behavior here. Dominant chaotic regimes appear for $N\geqslant 5$. As the size of the system is increased, the number of periodic windows decreases, and the transition region is enlarged. Also, for larger systems the effect of noise is less pronounced.

The dynamics of a simple spring-block chain placed on a running conveyor belt was investigated both by simple experiments and through computer simulations. Despite its simplicity, the dynamics of the system proved to be quite complex exhibiting- chaotic, periodic or quasi-periodic behavior as a function of the conveyor belt’s velocity and the amount of noise in the friction forces. The experiments and computer simulations indicate that the transition from chaotic to periodic or quasi-periodic dynamics is typically realized through an intermittent chaotic state.

In the chaotic regime, the avalanche-size distribution function shows a scale-free nature, indicating also the presence of SOC, along with a $1/f$ type of stochasticity in the chain length. Another aspect of the systems complexity is the diversity of the possible dynamical states reflected by the convoluted graph of the bifurcation diagram, and a phase transition-like behavior in the disorder parameter. The presence of noise adds to this picture fascinating phenomena, like noise induced chaos, noise induced intermittency and noise induced phase transition.

Interestingly, the computer simulations suggest that the maximal complexity of the dynamical states is achieved for a relatively small number of blocks in the vicinity of $N=5$ for $\sigma<2$. For $N<5$ there is practically no chaos in the system. With $N=5$ the observed transition is the sharpest one, and for $N>5$ it becomes smoother in a somewhat counter intuitive way. It is remarkable that this collective behavior finds its explanation in terms of dynamical system properties.

Finally we would like to draw the attention to the fact that in several recent experimental studies Mudrock2011 ; Roth2012 performed on metallic alloys, it has been observed that just before the onset of the plastic instability the local strain rate as a function of the corresponding force response shows a multi periodic behavior. This is somehow similar with the graphs shown in Fig. 11. This observation brings us back to the possibility of modeling the Portevin- Le Chatelier effect with a spring-block system Lebyodkin1995 . Indeed, if the plastic deformation is macroscopically uniform in the absence of plastic instability, one can assume that it is governed by only a small number of collective degrees of freedom, similarly with the dynamics of the spring-block chain considered here. In such view the studied model and the obtained results can gain interest in the field of material science as well.

*