The equations of motion

Slacklines are governed by nonlinear dynamics. In this article the equations of motion are presented for a mass attached at the midpoint of a horizontal slackline (Fig. 1). This system corresponds to a two-dimensional nonlinear oscillator with some interesting and peculiar properties. Numerical simulations of trickline bouncing, leash falls and longline surfing are performed (Figs. 2, 3 and 4) reproducing the key characteristics of observed systems. This study shows that geometric nonlinearity is a fundamental aspect characterizing the dynamics of slacklines.

Figure 1:  A horizontal slackline gap (AOB) with a vertical and orthogonally intersecting plane (xOz) at the midpoint. The direction of the restoring force exerted by the slackline is opposite to the displacement vector \(\,\mathbf{r}(x,z)\). This simple model captures all the main aspects of slackline dynamics.

The equations of motion

Many physical systems, such as a simple pendulum or the string of a guitar, have a linear behaviour at the limit of very small displacements around the equilibrium point, thus leading to small-amplitude harmonic oscillations. To leading order, some slackline systems can be approximated similarly, provided that the angle between the webbing and the level of the anchors remains small and the tension of the webbing remains approximately constant. A high-tension longline made with a high-stretch webbing provides a good example. However, at larger angles the upward force exerted on the slackliner is a nonlinear function of the vertical displacement. This nonlinearity has two conceptually separable causes: the change in line angle and the associated change in line tension.

Suppose that we have a piece of slackline webbing of original length \(\,\lambda_o\,\) tensioned to fit a gap of length \(\,\lambda\), so that \(\lambda_o\le\lambda\), and a point mass \(\,m\,\) (the slackliner) attached at the midpoint. To a first approximation, a linear elastic behaviour is assumed for the webbing with an elasticity constant \(\,k\,\) so that the webbing tension is proportional to its elongation (Hooke's law). For any transverse displacement \(\mathbf{r}=\mathbf{i}x + \mathbf{k}z\) (Fig. 1), the restoring force exerted on the mass by the slackline will act opposite to the displacement vector. Then, it is shown that the differential equations of motion can be written in Cartesian coordinates as follows: \begin{align} \label{eqn:duffing_x} m\ddot{x} &\:=\: - (\alpha r + \beta r^3)\sin\phi, \\ \label{eqn:duffing_z} m\ddot{z} &\:=\: mg - (\alpha r + \beta r^3)\cos\phi, \end{align} where \(\,\phi = \cos^{-1}(\mathbf{k}\cdot\mathbf{r}/r)\,\) is the angle between the vertical unit vector (pointing down) and the displacement vector \(\,\mathbf{r}\), and the costants \(\alpha\) and \(\beta\) are: \begin{align} \label{eqn:duffing_alpha} \alpha &\:=\: 4 k\left(1-\frac{\lambda_o}{\lambda}\right) \\ \label{eqn:duffing_beta} \beta &\:=\: \frac{\,8 k\lambda_o\,}{\lambda^3}. \end{align} Ignoring the inertia of the webbing, this restoring force is exerted on the slackliner when he/she is balancing, bouncing, surfing, or taking a leash fall near the midpoint. Due to gravity, the equilibrium point is below the level of the anchors and hence the motion of the mass is described by a non-axisymmetric potential leading to characteristic 2D orbits (Fig. 4). Now, as slackliners do not just stay at the midpoint, a generalization of the equation of motion is needed to account for any possible position between the two anchors. The position of the person along the line can be determined by a factor \(\delta\in(0,1)\) so that the horizontal distance from one and the other anchor is: \(\,\delta\lambda\,\) and \(\,(1-\delta)\lambda\). In this case, assuming that the mass is fixed on the webbing and it moves purely on the transverse plane \((x,z)\), the net force on this plane resulting from the line tension acts opposite to the displacement vector and is equal: \begin{equation} \label{eqn:force_off_center} F \:=\: k\,\lambda_o\,\eta\,\left(\frac{1}{\rho\delta} - \frac{1}{\sqrt{\delta^2+\eta^2}} + \frac{1}{\rho(1-\delta)} - \frac{1}{\sqrt{(1-\delta)^2+\eta^2}}\right), \end{equation} where the ratio \(\,\rho = \lambda_o/\lambda\,\) determines the initial stretch, and the ratio \(\,\eta = r\,/\lambda\,\) expresses the transverse displacement in a non-dimensional form.

Trickline simulations

Figure 2, below, shows results for two cases representing tricklines with different characteristics: a low-stretch line (3% at 10  kN) and a high-stretch line (15% at 10  kN). These were set to have the same static sag (\(z_o=\,\)0.5  m for a mass of \(\,m=\,\)60  kg weighing at the middle of a \(\lambda=\,\)20  m distance). Clearly, to achieve the same sag each line requires a different pretension (Fig. 2.c,d). In both cases, the mass is left to fall on the line from a height of 2  m, simulating a slackliner who bounces very hard. Here a moderate damping has been added in the form of a linear drag. In reality, to maintain a constant bouncing amplitude the slackliner has to provide energy, as a kid does on a traditional swing. On a trickline the energy dissipation takes various forms, such as: visco-elasticity effects in the webbing itself, absorption by the body's inelastic deformation and aerodynamic friction. Yet, accurately parametrizing these effects goes beyond the scope of the present study.

Figure 2:  Simulations of a low-stretch trickline (left) and a high-stretch trickline (right) following the Eqs. \(\ref{eqn:duffing_x},\ref{eqn:duffing_z}\) for purely vertical displacements (\(x=0\)). Details of the simulated rigs can be found in the full article.

The time intervals in which the line tension remains constant (Fig. 2.c,d) correspond to the free flight of the body above the level of the anchors. It can be seen that the two lines have a very different behaviour in terms of peak tension and total tension change. The tension of the high-stretch line (preferred for trickline, typically made of nylon) barely changes more than 1  kN, whilst the tension of the low-stretch line changes by more than 4  kN in every bounce, peaking above 9.5  kN. This has significant consequences for the dissipation of energy, considering the visco-elasticity of the lines.

The same is true for the force exerted on the body (Fig. 2.e,f), which is stronger for the low-stretch line and applied for a shorter period of time (each peak is shorter and wider for the high-stretch line). These differences are indicative of stronger energy dissipation at the system with the low-stretch line, considering the respective inelastic body deformation. However, in this analysis the applied damping is simply set proportional to the body velocity (similar to an overly strong aerodynamic friction) and cannot reflect correctly the significant differences in the dissipation of energy. This can be seen in Fig. 2.a,b, where the amplitude, or the highest position in each bounce decreases faster for the high-stretch webbing, opposite to what is expected by experience. This discrepancy is due to the unrealistic dissipation used.

Leash fall simulations

The above-presented ordinary differential equations describing the slackline system have been used to simulate also the dynamics of leash falls in two dimensions. This was done by numerical integration starting from initial conditions representing a leash fall occurring slightly to the side of — rather than directly under— the slackline. As mentioned earlier, here the presence of the leash is taken into account by setting the restoring force acting on the mass to zero when the distance between the body and the slackline webbing is smaller than the length of the leash. In effect, the leash here (assumed completely static) simply represents a geometric constrain.

Figure 3:  Numerical simulations of leash falls based on Eqs. \(\ref{eqn:duffing_x},\ref{eqn:duffing_z}\) with (\(\textit{a}\)) zero, (\(\textit{b}\)) weak and (\(\textit{c}\)) moderate damping. For the last case, which seems very realistic compared to an experiment in the laboratory, the vertical displacement of the mass attached to the leash (\(\textit{d}\)), the line tension (\(\textit{e}\)) and the leash tension (\(\textit{f}\)) are also shown as functions of time.

The details of the simulated slackline system are as follows: distance \(\lambda=\,\)22  m, pretension \(T_o=\,\)3.3  kN, falling mass \(m=\,\)60  kg, leash length \(L=\,\)2.0  m, and static sag \(z_o=\,\)1  m. Such a system was also realized in the laboratory and a number of controlled leash falls were performed for comparison. In the absence of energy dissipation (drag coefficient set to zero) the system undergoes wild irregular oscillations (Fig. 3.a). In this case, the trajectory is confined to an area determined by the conservation of energy. Adding some weak damping (Fig. 3.b) leads to a trajectory converging to the equilibrium point determined by the leash length and the sag (\(L+z_o=\,\)3  m). Increasing the damping, a more realistic trajectory emerges (Fig. 3.c) comparable to the observed trajectories recorded in video. In addition, in this case the line tension as a function of time (Fig. 3.e) appears very similar to experimental measurements.

Finally, it is worth noting that the oscillations become more linear as the amplitude decreases (nonlinear terms loose significance compared to the linear term). At the same time, because the maximum angle between the line and the horizontal decreases in every bounce, the leash tension decreases faster than the line tension. The former tends to a value equal to the person's weight \(mg\), while the latter tends to the value corresponding to the static sag \(z_o\).

Surfing simulations

As discussed in the full article, Eqs. \(\ref{eqn:duffing_x},\ref{eqn:duffing_z}\) represent a two-dimensional Helmholtz-Duffing oscillator. In this regard it is important to note that due to gravity the equilibrium point of the mass is below the level of the anchors (standing sag), something that breaks the symmetry of the system in the vertical. As shown below, this has important consequences for the sustained orbits.

The simulations shown below (Fig. 4) are characterized by the following parameters: slackline distance \(\lambda\!=\)100  m, unstretched webbing length \(\lambda_o\!=\)97  m, standing sag \(z_o\!=\)6  m and mass attached at the midpoint \(m\!=\)70  kg. Damping is added only at the last example (Fig. 4.d). These plots show the position of the mass on the vertical plane perpendicular to the line at the midpoint. For reference, the equilibrium position \((0,-z_o)\) is indicated with a diamond marker. The marked points A, B, C, D indicate the initial positions from where the mass is released with zero initial velocity.

The dashed contours represent lines of equal potential with the outermost contour corresponding to zero potential. Considering the elastic energy associated to the restoring force \(\,\alpha r + \beta r^3\,\) together with the gravitational energy, the respective total potential is: \begin{equation} \label{eqn:duffing_potential} P(r,\phi) \:=\: \frac{\,\,\alpha\,r^2\,}{2} \:+\: \frac{\,\,\beta\,r^4\,}{4} \:-\: mg\;\!r\cos\phi, \end{equation} where \(\,r\) and \(\phi\) are defined as in Fig. 1. At the equilibrium point this potential is \(P_o=-mgz_o\). For zero damping the energy is conserved, and thus the mass crosses every given iso-potential line always with the same speed. If the mass is initially at rest along an iso-potential line, it will always approach this iso-potential line perpendicularly and it will reach it at zero speed. This can be seen in Fig. 4.b\(--\)c,\, at the points B' and E'.

Figure 4:  Simulated surfing and two-dimensional bounces on a longline. The grey patches in (\(\textit{a}\)) show the areas in which the respective trajectories (starting from points B, C, D with zero initial velocity) are confined to. The trajectories starting from B and E are shown for four periods in (\(\textit{b}\)) and (\(\textit{c}\)). The red curve corresponds to a symmetric surfing motion starting from A, in (\(\textit{d}\;\!\)) with weak damping included.

The superimposed grey patches in Fig. 4.a show the areas in which the respective trajectories are confined to. Each patch has four angles touching the same iso-potential line, and it is only at these points where the mass returns at zero speed; at any other permitted point inside this iso-potential line the mass has non-zero kinetic energy. Fig. 4.b provides a view of the trajectories initiated at points A and B. In the former case (starting from A) the mass stays on a single curve (red line) oscillating back and forth, which is the equivalent of a symmetric surfing motion. Instead, in the latter case, starting from B, the trajectory covers a wide range (light grey patch) and reaches B' after four periods.

Figure 4.c is a close-up in the vertical direction showing a narrower trajectory starting from E. Finally, Fig. 4.d shows the resulting surfing motion starting from A when weak damping in added. As expected, this converges with time towards the equilibrium position. In reality, to maintain a constant surfing amplitude the slackliner has to provide energy with a periodic forcing applied at each extreme end of the trajectory. This is done by extending the knees, which corresponds to forcing each acceleration phase to start from the same iso-potential line.

As an anonymous reviewer pointed out, the trajectories shown in Fig. 4 resemble Lissajous curves. In fact, these trajectories are the result of the planar non-axisymmetric potential of the slackline system and the respective initial conditions (zero angular momentum). In galactic dynamics such trajectories are referred to as box orbits and are characterized by the fact that in the course of time they pass arbitrarily close to every point in the defining box. In contrast, for planar non-axisymmetric potentials there exist also loop orbits. In the latter, conservation of energy and anglular momentum makes it impossible to cross through an area at the center of the orbit.

Exploring the trajectories for initial conditions of non-zero angular momentum, the author found that loop orbits are also possible for slackline systems (not shown). The latter seemed to involve vertical accelerations that exceed the gravitational one, and therefore, while trying to "ride the wave" the slackliner would loose contact with the line. Yet, loop surfing is possible with short and high-tension lines provided that the trajectory of the center of mass of the slackliner is shallower in respect to the trajectory of the point of contact with the line, something that can be achieved by contracting and extending the legs in synchrony with the orbital motion. A slackliner performing such a peculiar surfing motion has been evidenced by the author.

Discussion

The simple model derived in this study (Helmholtz-Duffing oscillator in one and two dimensions) represents a highly accurate approximation of the dynamical equations describing slackline systems. It helps with understanding the fundamental aspects of the associated dynamics, including nonlinearity and the dependence on parameters such as the elasticity constant, the initial stretch and the slackline distance. It also provides insight into a number of empirically known facts related to slackline bouncing, surfing, leash falls and more.

Regarding tricklines, the nonlinearity of the oscillations depends on the polynomial coefficients (\(\alpha\) and \(\beta\)) and the amplitude of the oscillation. For a given line sag and bouncing energy (maximum height reached above the slackline), the nonlinearity increases with the webbing stiffness (\(k\). High-stretch webbings facilitate a more linear behaviour (larger \(\alpha\) and smaller \(\beta\) that translates to smaller body forces acting for longer times, as opposed to higher forces acting for shorter times for low-stretch webbings. This difference has a direct impact on not only comfort but also bouncing efficiency (referred to as rebounce).

Viscoelasticity effects and hysteresis along the elongation cycles were completely ignored in this analysis. Carefully designed laboratory experiments measuring these effects in controlled leash falls have been conducted (check here). Results from these experiments will be published in a separate article. It should be noted here that the strain rates encountered in slackline (i.e., the rate of change of the relative elongation of the webbing) are typically 100 times lower than those encountered by dynamic ropes in rock climbing. Therefore, energy dissipation is generally weaker in slacklines.

The physics of this rapidly developing sport include numerous different and completely unexplored aspects. The present study only scratched the surface of the subject. There is substantial potential and a need for experimental and theoretical studies, as well as for research on particular applications and safety issues. The International Slackline Association (ISA) is currently investigating initiatives for providing norms and standards and promoting understanding and safety.

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© Panos Athanasiadis, Bologna, Italy, 2018.