Comparing two cases
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A rope is applied around a pole with a lot of friction. The very first loop absorbs the tension. The pressure is localized and the rest of the rope is almost free of any mechanical constraint. No tension is propagated to the end of the rope. |
A rope is applied around a pole with zero friction (Of course this is just an assumption). The constraint is transmitted along the rope from one end to the other. The pressure is perfectly distributed. |
Quick links A true but incomplete assertion
<< The perfect rope : the O-link |
The equations
How you can determin the effect of the loops.
To the first order, the normal strength N at a given contact point (i.e. angle coordinate q) between the pole and the rope is proportional to the tension in the rope. The factor a contains curvature information and other physical properties of the rope. (We may also add here some information on external forces applied to the rope towards the pole).
Furthermore, the contribution to the decrease of the tension is the longitudinal friction force applied by the pole to the rope. It is proportional to the normal strength, to the surface in contact and to the friction coefficient f. Notice that dT is a reduction in force and thus is negative.
b is the flattening factor of the rope, and the width of the contact surface is related to the rope diameter by the product w = b.d. The length of the elementary contact surface is related to the radius of the pole r by the equation L = r.dq, where dq is the elementary change in angle around the pole. (we consider that d<<r and neglect the quantity d/2 in the total curvature d/2 + r)
Then we substitute N by its expression to obtain :
We can now calculate the change in strength along the path q1 to q2.
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Lessons learned
In conclusion, we have observed that the tension along the path of the rope drops exponentially. In other words, 95% of the strength is transmitted in the two first loops around the pole…Further rigging is strictly unnecessary. Numerous turns are not the guarantee of a lower pressure. That would only be true for a theoretical situation (no friction), which of course never happens.
How does it apply to rope bondage?
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In order to achieve a uniform distribution of the strength you have first to make your tying perfect. i.e. you need to make your bondage rope look like the O link shown in introduction. It must be closed : Your knots must be perfectly tight. |
Other reasons for doubling the ropes
Maybe you have noticed already that at a given point, the number of rope ends is an even number? This is quite obvious: each rope arriving in a point goes also away from this point. As a result, a knot has an even number of ends whatever the number of ropes that are used...
Read more in the dedicated section >>