A coil spring is basically a torsion rod wound into a coil.
If one end of a straight steel rod is clamped in a vise, and torque is applied to the free end, the rod will function as a basic torsion rod. The applied stress (force) is in the form of torque (twisting force), and the rod's response is in the form of torsional strain (twist). The applied torque could be provided by a lever and might be measured in ft lbs. The resulting strain might be measured in degrees of rotation of the free end. Therefore, the spring rate of a torsion rod might be measured in ft lbs per degree.
The spring rate of a torsion rod is inversely proportion to its active* length. If the length is doubled, the spring rate will be halved. If the length is tripled, the spring rate will be reduced to one third. This characteristic carries over to the coil spring. In the case of the coil spring, it's more convenient to think of the length in terms of number of active* coils rather than length of the spring wire. The relationship is still the same though. The number of active coils is directly related to the active length of wire that makes up those coils. So if the number of coils were doubled, the spring rate would be halved, etc.
One important difference between the torsion rod and coil spring is in the way the stress is applied. For a torsion rod, it's in the form of a torque. For a coil spring, it's in the form of an axial force (the coil's axis, not the wire's axis). Also, the spring rate is therefore expressed differently: ft lbs per degree for the torsion rod, and lbs per axial compression for the coil spring. The response for both is the same though. It results in the spring wire or rod being strained torsionally (twisted).
This torsional strain is most easily seen in this illustration of a single coil of spring wire:
coil_01.jpg
It should be apparent that the ends of the coil wire will rotate as the coil is compressed axially.
In this illustration, another coil has been added:
coil_02.jpg
It should be apparent here that the amount of rotation at the open ends will be the same as that for a single coil, for full compression of the spring. It should also be possible to imagine that we can add as many coils as we like with no change in the amount of rotation at the open ends of the spring wire at full compression. Furthermore, whatever is happening at each individual coil, must be also be happening at every other coil.
And here then is the paradox:
The torsional strain should occur along the entire length of the spring, with each coil undergoing equal strain. And since the coils are effectively in series with each other, and oriented in exactly the same way in relation to each other, there should be an accumulation of torsional strain over the length of the spring. More coils, more total overall rotation.
Yet there is no difference in the total tortional strain over the length of the coil at full compression, regardless of the number of coils. The coils cannot be cancelling each other out; if they were, the number of coils would not matter.
So exactly where is this torsion actually manifesting itself within the coil?
I don't know the answer.
* active refers to that portion of the spring that takes part in the spring's torsion or compression. It would exclude the clamped ends of a torsion rod, or a closed coil at the end of a coil spring.
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