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From: Michael Daly <michaeldaly_at_home.com> subject: [Paddlewise] The impact loading of tow lines - corrected version (long) Date: Wed, 20 Oct 1999 23:33:48 -0400 · This archive was generated by hypermail 2.4.0 : Thu Aug 21 2025 - 16:33:04 PDT

Part 1 - Introduction and assumptions

The objective: to design a shock absorbing system for a kayak towing rig.  

The goals are to design a system that will be simple and will limit the loads 
that are experienced by a paddler (for those that tow with lines attached to 
their bodies) or the kayak (for deck mounted rigs).  

In order to achieve this, a simple model is constructed.  A bungie arrangement 
is used to absorb the energy of a moving kayak.  By balancing the energy of 
the moving kayak with the energy absorbing capacity of a bungie, it is hoped 
that we can find a suitable combination of materials and dimensions.

The problem is that if a towed kayak is impeded in its motion, the sudden rise 
in tension in the tow line could injure a paddler if attached to the tower's 
waist.  Alternatively, the force could damage the kayak if the line is fixed 
to it.  In order to analyse this, I have made a few assumptions

1. 	We will look at one worst case scenario - a towed kayak comes to a
 	complete stop while the towing kayak continues.  
2. 	All energy is absorbed by a bungie in the tow line, none by the tow line,
 	kayak etc.
3. 	The bungie is a nice, linear-elastic material. 

The second assumption makes the resulting forces calculated higher that what 
would occur in reality - this is safe.

The third assumption is iffy.  The bungie material I tested has some linear 
behavior, but I know that real rubber is not - I seem to remember that it's bilinear.  
I'm not sure what the bungie's real behavior is over a wide range of loads, nor 
do I know if bungies are typically real rubber or synthetic. 

Part 2 - The Theory:

The kinetic energy in the towing kayak is what must be absorbed and is defined as:

1/2 M V**2    where M = the total mass,  V = the velocity and **2 means squared.

The energy absorbed by an elastic material is defined as:

1/2 k x**2   where k = spring constant, x = change in length of the spring

The force in the elastic material is (Hooke's law)

F = k x    or   x = F/k

Equating the energies:

1/2  M V**2 = 1/2  k x **2  or   M V**2 = k x**2  or  x**2 = V**2 M/k

or  x = V sqrt(M/k) = F/k

Therefore    F = V sqrt(kM)                 

Substituting weight for mass  (M = w/g):

   F = V sqrt( k w/g)  where g = acceleration of gravity.       (1)

Now we just have to figure out what to do about the k and we can figure out 
the forces and such.

The spring constant, k, is the relationship between the force and the change 
in length of a spring for a fixed length of spring.  You can determine this 
experimentally for any material.  Now k has a more formal definition in the 
case of an elastic material that is constant in cross section, like a uniform 
spring (bar, rod or, in this case, bungie). That is :

	k = EA/L     where E = Young's modulus, a basic property of the material,
		   	   A = the cross sectional area of the spring and
			   L = the length of the spring.

Note that this is a nice linear relationship that will allow us to add the k 
values of parallel bungies if they are of the same type and length.

In the case of a measured value of k, say km in my test, and measured L, say Lm, 
the relationship is:

	km = EA/Lm

Multiplying both sides by Lm:

	km Lm = EA  

which is a constant for the object in question - it is in fact the k value for a 
unit length of the spring.  Knowing this, we can determine the k for any length of 
the elastic material by reintroducing the length.  Let's refer to the unit value 
of k as k1 (= EA):

      k = k1/L   for any length L.                   (2)

Now let's consider the force in the spring given by Hooke's Law:

	F = k x 

We can define a value of extension, e, that expresses the change in length x as a 
function of the original length of the spring:

	x = eL

Thus, if we want to have an change of length of 50% of L, e is 0.5.  If we want 75%, 
e is 0.75 and so on.  Now

	F = keL

Substituting from (2) above,

	F = (k1/L) eL 

	F = k1 e                     (3)

The force in the bungie will be equal to its unit k value times the extension as a 
percentage of the original length.

Now substitute from (2)

	k1 e  =  V sqrt( k1 w / L g )

Or	k1 e / V  = sqrt(k1 w / L g )

Or	k1**2 e**2 /V**2  =  k1 w / L g

Or    k1 e**2 / V**2 = w / L g

Or	L =  w V**2 / g k1 e**2                   (4)

Now we can choose a force we want to limit ourselves to, and choose a combination of 
elongation and elastic constant that gives us that force.  Then using (4), we derive 
the length of the bungie that gives us that result.  We can iterate until we get 
a result that we like.

Part 3 - Calculations

I rigged up a simple test arrangement in my workshop in the basement.  I used some 
1/4 inch bungie cord and hung it from the ceiling with a carabiner and attached weights 
to the other end to measure the change in length over an eight inch section in the 
middle of the bungie.  By using several different weights and repeating this for single, 
double and triple strands of bungie and plugging the numbers into a spreadsheet, I 
determined the elastic constants for each.  The results are a little confusing in that 
two strands of bungie is not exactly twice as stiff as one strand.   Due to the degree 
of stretch and such, I was obliged to use a different weight range to calculate the 
numbers for each of the single, double and triple configurations.  I think that the 
results are revealing some of the non-linearity of the material.

Let us consider a two parallel 1/4 inch bungies.  These have a k value of 1.6 pounds 
per inch.  (I regret that, while I am usually more comfortable in metric units, for 
little physics type stuff like this, I am more familiar with Imperial units - this 
reflects the change in units that occurred part way through my education.)  For an 
8 inch length, the unit k1 is:

	k1 = 1.6*8 = 12.8 lb.

Let's do a little check.  Since k1 = EA, and A is pi * r**2, where r is the radius of 
the bungie, we can determine E:

	E = k1/A = 12.8 / ( 3.14 * (1/8)**2 = 261 psi.

According to my solid mechanics textbook (ref 1), soft rubber has a Young's modulus 
between 200 and 800 psi, so my results are within the range that's believable.

Lets assume a paddler weighing 175lb, in a kayak that is 55 lbs carrying 70 lbs of 
gear for a total weight of 300 lb.  We'll also assume a velocity of 3 statute miles 
per hour.  Plugging the results into our equation (with conversions to get 
everything in inches, pounds and seconds)

3 mph =  3 (5280 * 12)/3600 =  52.8 inches per second.

g = 32.2 ft/sec**2 = 32.2 * 12 = 386.4 in/sec**2

	L  = 300 (52.8**2) / (386.4 * k1 e**2) = 2164/(k1 e**2)

Let's assume the maximum force that we want is 50 lb and we will accept a 100% 
elongation, thus:

	F = k1e , or  50 = k1 * 1.0

Thus we need a k1 value of 50.  This is about four times the value I got for a 
double bungie.  This is equivalent to 8 parallel strands of quarter inch bungie.

The length required is 

	L = 2164/(50 * 1**2) = 43.3 inches.

This means that the bungie would start out at 43 inches and would have to stretch 
out to 86 inches before the kayak energy is absorbed.  That seems rather long.

Assuming k1 is 33.3 (equivalent to roughly six parallel quarter inch strands), and 
allowing an elongation of 1.5, the force will still be 50 lb and the length required 
is 29 inches.  The stretched length will be 72 inches (six feet!)

An elongation of 1.5 (or total stretched length of 2.5 time the unstretched) is 
probably at the limit of what I'd consider safe.

Part 4 - Discussion of Results

As can be seen, we need rather long lengths of heavy bungie to absorb the impact 
without unduly loading the paddler or kayak.  As well, the bungies must stretch 
quite a bit.  This can be a bit problematic.  Carting around long, heavy bungie 
in a throw bag can be annoying.

Let us consider quickly the effect of the normal towing loads on the bungie.  
Based on a table by Matt Broze in the August 1998 issue of Sea Kayaker, a fit 
and efficient paddler can work against 3 pounds of drag for a long period of 
time.  Using this as a drag force that the tow line experiences, the change 
in length of the bungie will be (let's use the second example from part 3)

	k1 = 33.3, L = 29, F = 3  therefore:

	x = FL/k1 = 3*29/33.3 = 2.6 inches

Thus we can see that the bungie will likely be stretching about three inches 
or so, or about ten percent, during the towing period.  The period of undamped 
vibration of this bungie will be (ref 2)

	T = 2*pi sqrt( M/k) = 6.28 sqrt ( 300 * 29 / 33.3) = 100 seconds.

This tells me that the system will not be too jerky - that is it won't tend to 
loosen and tighten do to the stretchyness of the bungie.  This is good.

As far as dynamic load versus static load as an analysis technique; this is a 
static analysis.  The bungie will have to stretch a total of 43 inches to absorb 
the energy of the kayak.  At a velocity of 53 inches per second, it would take 
almost a second to reach maximum.  This would not be a very dynamic load and 
I don't think the impact would be significant.

To determine the k1 value of a bungie:

You'll need several items of known weight - a kitchen scale will suffice to 
determine the weight.  Ideally they should be around the same weight.  A good 
measuring tape is also needed.

Hang a length of the bungie from a solid support.  

Attach a small weight to the bottom end to keep it straight and tight.

Mark two positions on the bungie a reasonable distance apart (6-10 inches).  
Use chalk, ink, masking tape etc.

Measure the length between the marks accurately.  Record this as your base length.

Add a weight to the end and measure the new length.

Add another weight, roughly the same as the first, and measure again.

Do this with at least four weights.

Plot the results on a graph (spreadsheets do this easily)

Get the slope of the straight portion of the line - don't be surprised if 
the first section of the graph is curved.

The slope, in pounds per inch, is the k value.  Multiply this by the original 
measured length between the marks,  This is the k1 value.

Use the equations to see what you can do with your bungie.

References:

1) An Introduction to the Mechanics of Solids, Crandall, Dahl & Lardner, 
   Second Edition, 1972, McGraw Hill.

2) Dynamics of Structures, Clough & Penzien, 1975, McGraw Hill
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