## Abstract

We consider the influence of aerodynamic forces on the shape of a whirling filament that is held at both ends, i.e. a jump rope. At high Reynolds numbers, the rope curls out of the plane and towards the axis of rotation—a feature we demonstrate via experiment. We derive a pair of coupled nonlinear differential equations that characterize the steady-state shape of the rope, and the resulting eigenvalue problem is solved numerically. The solution depends on two dimensionless groups: the ratio between the length of the rope and the distance between its ends, and the relative magnitude of the aerodynamic to centrifugal forces. As the latter ratio is progressively increased, the tension in the rope and the out-of-plane deflection increases, until eventually the rope reaches a limiting shape. Finally, we show that the airflow-induced shape change leads to a relative reduction in drag and has implications for successful skipping.

## 1. Introduction

There are numerous examples of slender structures that interact with high-speed fluid flows: wind causes flags to flap, sails to fill and bridges to vibrate (Marchaj 1980; Naudascher & Rockwell 2005; Holmes 2007; Shelley & Zhang 2011). In nature, slender biological structures often exploit their inherent flexibility to survive in wavy or windy environments (Vogel 1994). The stems of crops such as wheat, for example, are bent by the wind, leading to decreased drag and coherent wave-like motion (de Langre 2008). The flexibility of submerged vegetation, such as sea grasses, allows for decreased canopy drag, leading to increased in-canopy velocities and nutrient transport (Ghisalberti & Nepf 2009). Understanding how flexible structures respond to fluid flow is also relevant to possible approaches to energy harvesting (Allen & Smits 2001) and turbulence reduction (Shen *et al.* 2003).

The jump rope is a familiar example of a slender structure that interacts with the fluid through which it moves. Yet, previous studies of the underlying physics have neglected this interaction, in part because an analytical solution exists for the shape or curve of the jump rope in the drag-free limit (Appell 1941; Blackwell & Reis 1974; Nordmark & Essen 1996, 2007; Mohazzabi & Schmidt 1999). Similar to the arched curves of a catenary and a suspension bridge, a jump rope may be described mathematically as a one-dimensional inextensible structure subject to external forces (Yong 2006). In contrast to the former two cases, gravity is not the dominant force responsible for the shape of a jump rope. At sufficiently high rotation speeds, the shape is dominated by centrifugal forces and possibly forces of fluid dynamical origin. In the absence of the latter, the jump rope curve remains planar. This feature of the solution was confirmed experimentally by using a heavy chain as an idealized jump rope so that aerodynamic effects could be neglected (Mohazzabi & Schmidt 1999). Here, we consider the influence of aerodynamic forces on the shape of a jump rope, and show how the airflow-induced shape change leads to both out-of-plane deflection and drag reduction.

A schematic of a jump rope is shown in figure 1. The apparent centrifugal force (per unit length) acting on an element of the jump rope depends on its rotation frequency *ω*, mass per unit length *m* and distance to the axis of rotation *r* according to |**F**_{C}|=*mV*^{2}/*r*, where *V* =*ωr* is the local rope speed. The aerodynamic drag (per unit length) depends on the air density *ρ*, the drag coefficient *C*_{D}, as well as the rope radius *b* and local rope speed according to |**F**_{D}|=*ρV*^{2}*bC*_{D}. The relative importance of the aerodynamic drag to the centrifugal force is thus given by |**F**_{D}|/|**F**_{C}|, or equivalently, by
1.1
where we denote *ϵ* as the dimensionless drag parameter, and we take the characteristic distance to the axis of rotation to be the length of the rope *L*. The drag coefficient *C*_{D} depends on the Reynolds number of a cylindrical element of radius *b* moving in the surrounding fluid. Typically, we expect large Reynolds numbers;^{1} so *C*_{D}=*O*(1). The aerodynamic drag may safely be neglected if *ϵ*≪1. However, we find 0.21≤*ϵ*≤0.61 from measurements of four commercially available jump ropes,^{2} which indicates that drag should have some influence on the shape of the jump rope curve.

## 2. Experimental observations

To ascertain how aerodynamic forces affect the shape of a jump rope, we built a mechanical device that gives us precise control of the rotation rate and distance between the hands, i.e. the separation distance, as well as the rope density, diameter and length. The device consists of a 10×33 cm aluminium frame that is rotated using a variable-speed motor and secured to a mounted ball bearing to reduce vibration. A nylon string (chosen for its low density and low extensibility) is affixed to the inside of the frame and its motion is captured using a Nikon D90 Digital SLR Camera with the aid of a Monarch Instrument Digital Stroboscope.

A typical multi-exposure photograph of an experiment with a rotating string is shown in figure 2. While previous studies have predicted that the jump rope would remain entirely in the plane of rotation (Appell 1941; Blackwell & Reis 1974; Nordmark & Essen 1996, 2007; Mohazzabi & Schmidt 1999), we observe that for *ϵ*=0.41 the rope curls out of the plane and in the direction opposite to that of its rotation. Because the points along the string farther from the axis of rotation travel faster, they experience greater aerodynamic drag, which gives rise to the observed out-of-plane deflection. In §§3 and 4, we quantify this statement by extending the previous mathematical model of the jump rope to include fluid dynamical forces, both lift and drag.

## 3. Theoretical formulation

We seek the steady-state shape of an axisymmetric filament that is immersed in air and rotated at frequency *ω*, i.e. a jump rope, as depicted in figure 1. Let the centre of the rope be a curve described parametrically by **x**(*s*,*t*)=(*x*(*s*,*t*),*y*(*s*,*t*),*z*(*s*,*t*)), where *s* is the distance along the centreline, and *t* is time. A balance of contact and body forces along the centreline yields a form of the wave equation:
3.1
where *T* is the tension, and **F**_{C}, **F**_{G}, **F**_{D} and **F**_{L} are the centrifugal, gravitational, drag and lift forces per unit length, respectively (Yong 2006). In writing (3.1), we have assumed that the rope is inextensible, and that it has no resistance to bending, i.e. the rope behaves as if it were a string. The latter assumption may be validated by evaluating the dimensionless number *Λ*=*Eb*^{4}/(*mω*^{2}*L*^{4}), which characterizes the relative importance of bending forces, *B*/*L*^{3}, to centrifugal forces, where *B*=*Eπb*^{4}/4 is the bending stiffness. Bending resistance was negligible in our experiments owing to the fact that *Λ*≈0.08, corresponding to a bending stiffness of roughly 10^{−5} Nm^{2}. We note that in most situations *Λ*≪1 because it involves (*b*/*L*)^{4}.

Next, we move into the rotating frame of reference so that the the term on the left-hand side of (3.1) is zero. In the rotating frame, the apparent centrifugal force per unit length acting on an infinitesimal element of the rope is , where is the distance to the axis of rotation (the *x*-axis), and *V* =*ωr* is the local speed of the element. Because the magnitude of the gravitational force is *mg*, the weight of the rope may be neglected at high rotation speeds, that is, if *V*^{2}/*gr*≈*ω*^{2}*L*/*g*≫1. This condition is satisfied for a jump rope having length *L*=2.5 m provided that the rotation frequency is much greater than one-half revolution per second. For comparison, competitive jump ropers achieve several rotations per second. For the experiment depicted in figure 2, *ω*^{2}*L*/*g*≈30.

Owing to the non-zero thickness of the jump rope and its motion through air, aerodynamic forces necessarily influence the shape. Our approach is to model the air flow over each cross section of the rope as flow past a yawed circular cylinder of infinite length. At high Reynolds numbers, we may write the drag force per unit length as , and the lift force per unit length as , where is the normal component of the velocity. The yaw angle, *β*, is determined by taking the inner product of the incoming airflow vector and the tangent to the rope (*x*_{s},*y*_{s},*z*_{s}), yielding , where *ξ*=(*z*_{s}*y*−*y*_{s}*z*)/*r* and subscripts denote derivatives. At Reynolds number 1000, provided that *β* is in the range of 0–60^{°}, the drag coefficient is roughly constant (*C*_{D}≈1.1) and the time-averaged lift force is negligible (Zhao *et al.* 2009). We thus focus on the effect of drag forces.

After substituting for the body forces **F**_{C} and **F**_{D}, the *x*- *y*- and *z*-components of (3.1) are
3.2a
3.2b
3.2c
which, together with the inextensibility constraint,
3.3
prescribe the shape of the jump rope. Integrating equation (3.2a) gives an expression for the tension, *T*(*s*)=*C*/*x*_{s}, and combining this result with equations (3.2b) and (3.2c) reduces the system to two equations
3.4a
and
3.4b
The tension in the rope is minimum at its midpoint, where *x*_{s} attains its maximum value.

We let , , , and , with , to rewrite (3.4) as
3.5a
and
3.5b
and (3.3) as
3.6
where *ϵ* is defined by (1.1) and where we define the dimensionless tension
3.7
Note that *λ* (or *C*) is unknown *a priori* and corresponds to an eigenvalue.

Five conditions are necessary to specify a solution to (3.5) and (3.6), i.e. to determine the shape functions , and , and the corresponding eigenvalue *λ*. At the origin, , the rope is fixed. Correspondingly, and . In addition, because we have chosen (without loss of generality) to measure the rope’s vertical deflection from the *x*–*y* plane, the slope at . In the absence of bending, the solution is guaranteed to be symmetric. The substitutions and together have no effect on the governing equations (3.5) and (3.6). We thus specify the remaining boundary conditions at the midpoint of the rope, , rather than at its end, , in part because this choice will decrease the time of computation. More important, though, is that we shall observe in §4 that a discontinuity occurs in the slope of the jump rope at its midpoint. Integration from to cannot capture this discontinuity, and thus one must stop the integration at . The origin of the discontinuity comes from the neglect of bending in the derivation of (3.5) because the force balance cannot be satisfied at when *ϵ*>0.

Symmetry is enforced by choosing either or as the boundary condition at . The latter choice, however, is inappropriate. It does not yield a unique solution for the case of *ϵ*=0, nor does it yield a solution for which the minimum tension occurs at the midpoint of the rope. Finally, the midpoint of the rope must be halfway between the two ends of the rope. Correspondingly,
3.8
Note that the shape of the jump rope depends on the aspect ratio *L*/*H* and the drag parameter *ϵ*, and the latter does not depend explicitly on the rotation frequency *ω*, provided we neglect the dependence of *C*_{D} on the Reynolds number and hence *ω*. Consequently, increasing the rotation frequency will increase the tension in the rope *λ*, but not alter its shape. For example, to a first approximation, doubling the rotation frequency will quadruple the tension at each point along the rope, according to (3.7). Finally, we note that for *ϵ*=0, (3.5b) has the trivial solution , i.e. the rope lies entirely in the – plane.

We solve the nonlinear eigenvalue problem for *λ*(*ϵ*,*L*/*H*) given by (3.5) and (3.6) numerically in Matlab by exploiting symmetry and using a shooting method. Specifically, we guess *λ* and and integrate forward in using Matlab’s built-in function, *ode45*, that is based on an explicit Runge–Kutta (4,5) formula. The spatial step we typically use for integration is 10^{−9}. When is reached, corresponding to the midpoint of the rope, we evaluate and . If the sum of these two quantities is less than a specified tolerance, typically 10^{−6}, the shape functions are saved for further analysis. Otherwise, we continue searching using Matlab’s built-in simplex search function, *fminsearch*, for the combination of *λ* and that will satisfy the boundary condition at and the integral constraint. Note that we are only interested in the lowest mode solution, corresponding to the lowest permissible non-zero eigenvalue (*λ*). As an alternative to guessing both *λ* and and then checking (3.8), we could instead choose *λ*, guess and calculate *L*/*H* from the -position of the rope at , thus reducing the dimensionality of the search by one.

## 4. Numerical results and discussion

We first report typical results for given values of *ϵ* and *L*/*H*. In figure 3, we present the shape of five ropes, each having length-to-separation ratio *L*/*H*=4, but different values of *ϵ*. For *ϵ*=0, the curve lies entirely in the – plane. As the drag parameter is progressively increased, the midpoint of the rope deflects further away from the – plane. Plane projections of the curves in figure 3 are shown in figure 4. The discontinuity in the slope at the midpoint of the rope in figure 4*b* arises from the neglect of bending resistance in the derivation of (3.5), which would act to smooth regions of high curvature. Another feature of the solution is that the quantity at the midpoint of the rope for *ϵ*>0.

The net external force necessary to keep the rope in motion is equal to the sum of the tension at its two ends. Dimensional analysis shows that we can report this force relative to *mL*^{2}*ω*^{2}, and that it depends on two dimensionless groups: *ϵ* and *L*/*H*. As seen in figure 5, the tension at both the midpoint and the ends of the rope increases monotonically with *ϵ*. The tension at the ends of the rope is roughly independent of *L*/*H*, whereas the tension at the midpoint of the rope depends on *L*/*H* and can be more than an order of magnitude smaller. Our numerical calculation of the tension at steady state indicates that a *fast* rope, i.e. one that is easiest to rotate, is one with small *mL*^{2}*ω*^{2} and even smaller *ρC*_{D}*bL*^{3}*ω*^{2} (so that *ϵ*≪1). This effect may be achieved by (i) reducing the diameter of the rope, (ii) reducing the length of the rope, (iii) reducing the density of the rope, or (iv) reducing the drag coefficient, provided that the above inequalities hold.

### (a) Asymptotic analysis

For *ϵ*≫1, the eigenvalue problem given by (3.5) reduces to
4.1a
and
4.1b
where is the eigenvalue. In this limit, the deflection is produced solely by drag. Thus, as *ϵ* increases, the tension *λ* should become proportional to *ϵ*, as evidenced in figure 5. If the rope is both long and light (*L*/*H*≫1 and *ϵ*≫1), its shape is also described by (4.1) if we rescale *x* by *H* rather than by *L*. In doing so, becomes the eigenvalue, and the length constraint (3.8) reduces to
4.2
Because neither *L*/*H* nor *ϵ* appear explicitly in (4.1) or (4.2), the numerically determined shape of the jump rope in this limit, shown in figure 6, is universal. We find *λ*=0.00941*ϵH*/*L*.

Conversely, for *ϵ*≪1, the shape of the rope does not depend on *ϵ* and thus *λ* reaches a constant value for a given *L*/*H*. In the limit of zero drag (*ϵ*=0), the shape of the curve may be expressed in terms of the Jacobi elliptic function sn (Nordmark & Essen 2007). Owing to the complexity of this solution, we are unable to obtain a regular perturbation expansion of (3.5) and (3.6) at order *ϵ* or higher.

### (b) Drag reduction

Next, we consider the total aerodynamic drag acting on the rope. The total drag is given by , or in dimensionless form,
4.3
Two comments should be made regarding (4.3). First, the value of the integral depends on the shape of the rope, which in turn depends on the drag parameter *ϵ* and the separation ratio *L*/*H*. Second, the integral decreases monotonically with *ϵ* (figure 7, inset), giving rise to a relative drag reduction that is shown in figure 7. This trend indicates that the ability of the rope to deflect out of the – plane gives rise to drag reduction. For long ropes (*L*/*H*≫1), the drag can be reduced by up to 12 per cent, and for short ropes (*L*/*H*≈2) by up to 25 per cent.

In closing, we have determined how the shape of a jump rope is affected by aerodynamic forces, and demonstrated the resulting out-of-plane deflection via experiment. Although the flow-induced deflection of the rope leads to drag reduction, excessive deflection could lead to the rope striking the body of the jumper. More important, though, is the observation that the net external force that is necessary to keep the rope in motion, i.e. the tension at its two ends, still increases with an increase in drag parameter *ϵ*. Therefore, a *fast* rope is one that is light and has a small diameter, a short length and a low drag coefficient.

## Acknowledgements

The authors thank Ian Griffiths and Marcus Hultmark for helpful conversations, and Jiang Li for her assistance with the experiments. We thank an anonymous referee for helpful remarks. J.M.A. gratefully acknowledges the National Science Foundation Mathematical Sciences Post-doctoral Research Fellowship Program.

## Footnotes

↵1 The midpoint of a jump rope having length 2.5 m, diameter 1 cm, revolving twice per second has a Reynolds number

*Re*=*Vb*/*ν*≈10^{4}in air, where*ν*is the kinematic viscosity.↵2 Precision Speed Rope, Cotton Jump Rope and Beaded Jump Rope by Fitness Gear; Aero Speed Jump Rope by Buddy Lee Jump Ropes. The

*ϵ*values were calculated using a drag coefficient of*C*_{D}=1.

- Received June 22, 2011.
- Accepted October 4, 2011.

- This journal is © 2011 The Royal Society