40 Kane: Friction Coefficients for Arborist Ropes gations because findings will help manufacturers design ropes that better withstand the rigors of climbing and opti- mize climber efficiency. In modeling the stopping forces associated with different accelerations of a climber, four simplifying assumptions were made. First, the rope was considered as not having any mass. Although this is clearly not true, a typical 13 mm (0.52 in) climbing line weighs between 0.10 and 0.12 kg/m (0.7 and 0.9 lb/ft), which gives a mass of between 1.5 and 1.9 kg (3.3 and 4.2 lb) when a climber has descended 15 m (49.5 ft). For a climber whose mass is 91 kg (200.2 lb), this is only 4% of the climber’s weight. New, smaller diameter ropes (11 mm [0.44 in]) have a mass of ≈0.9 kg/m (0.6 lb/ft), so a 15 m (49.5 ft) length would have a mass of 1.2 kg (2.6 lb), only 1.5% of the climber’s mass. Second, the rope is assumed not to stretch. All of the ropes do stretch, but the amount is generally less than or equal to 1% at 445 N (100 lb) tension (www.neropes.com, Anonymous 2003; Anonymous 2004), the approximate tension that a 91 kg (200.2 lb) climber would put on each length of rope. Third, acceleration is assumed to be constant as the climber stops, but acceleration varies with time as the climber comes to rest. Fourth, the anchor point in the tree is assumed to be rigid, although there may be some spring effect at the anchor, for example, when a tree limb deflects under load. Modeling the spring effect would be difficult because of the inherent variability of different wood properties, branch sizes, and heights of anchor points. The innate error in our experimental protocol does not seem to justify creating a more precise physical model. CONCLUSION Rope friction is an important consideration in tree care op- erations that affects climber fatigue, rope abrasion, and rig- ging operations. Characteristics of both the rope and surface over which it runs influence friction in complex ways that are not always intuitively clear. The important influence of ma- terial surface roughness of cambium saver rings on friction coefficients suggests that surface roughness may be more important than rope characteristics, especially considering that there were more differences among ropes tested on rings with a rough surface. Future investigations should test fric- tion on branches with varying degrees of bark roughness. Further studies of used ropes are also necessary because rope wear greatly influenced frictional properties. Finally, ropes should be tested at contact stresses approaching those nor- mally encountered in rigging operations because the results from the current study cannot be extrapolated to such situa- tions. APPENDIX A Equation (4) combines equation (2) with the equations of motion for constant acceleration. In the free-body diagram ©2007 International Society of Arboriculture shown in Figure A1, the climber may be considered as a particle. The tension in the rope at the friction hitch (T2) and one-half of the climber’s weight (w) act on the climber. The other half of the climber’s weight (assuming no friction at the cambium saver rings) is supported by the other half of the rope. To determine T2, Newton’s Second Law, F m*a, can be used. In this case, all forces and motion are vertical, so subscripts denoting direction have been omitted. The positive vertical direction is taken to be toward the top of the page. Integrating the acceleration equation with respect to time (t), a a(t) gives the equation for velocity, v a*t + C1 (A1) (A2) where C1 vo, initial velocity, which in this case is −vo because the climber is descending. Integrating the velocity equation with respect to time gives the position equation, y a*t2/2 − vo*t+C2 (A3) where C2yo, the initial position. When the climber stops, v 0, so (A2) can be solved for t, substituting −vo for C1, t vo/a (A4) Using y 0 as the climber’s final position, t (from [A4]) is substituted into (A3), which is solved for a, a vo 2/2yo Next, T2, w, and a are substituted into ∑F m*a: T2 w + m*(vo 2/2yo) (A5) (A6) Figure A1. Free body diagram of a climber (black circle) of weight (w) descending at an initial velocity (vo slowing down and stopping as a result of a force (T2 ), then ) provided by a friction hitch with acceleration (a). The other half of the climber’s weight is accounted for by the second leg of rope that would attach to the saddle.
January 2007
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