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Name: Richard H.
Status: Educator
Age: Old
Location: N/A
Country: N/A
Date: January 2004

Physics and/or mechanical engineering question:

Consider the following situation: You have a coil spring that is wound such that when viewed from the end, the wire coils away from you in a clockwise (CW) direction. Now, if as you compress the spring and view the end of the wire (not the coil), which way does the wire twist CW or CCW?

I am not interested in things like Hooke's law and such. What I really want to know is: When a coil spring is stretched or compressed within its design limits, does the wire itself twist? If it does, and I think it does, please point me to a reference that supports the assertion. Thank you for any insights on this matter.

This is not an easy analysis because it depends upon a number of constraints and conditions:

1. Both ends of the coil are "fixed". By definition of the boundary condition the coil does not move.

Not very interesting.
2. There are two "lengths" involved in a coil.

A. The arc length of the coil.
B. The "end to end" length of the coil.

How the spring "stretches" depends upon how the thermal expansion contributes to these two "lengths" and that is going to depend upon the 'pitch' of spring, that is, what is the change "end to end" length change for an incremental change in the angle change. This is probably some sort of tensor because you are looking for the change in motion in say the Z direction as a point on the spring moves in the (X,Y) "plane", but the (X,Y) "plane" is not orthogonal to the Z axis but is tipped along the tangential "plane" of the spring. And this tangential direction is going to change as the temperature is increased.

B1. For very small temperature changes, and for a spring whose arc length >> "end to end" length, the major increase in length is going to be along the axis of the arc. In this case dL/Lo = a*dT that is, it is just a change in the length due to the linear expansion coefficient,
a. And the angular change is given by: dA = dL / R where dA is the change in the angle of the "far" end of the spring relative to the "near" end of the spring and R is the axial radius of the spring, and dL is given by the previous expression for the linear expansion coefficient.

B2. For a spring in which the arc length "end to end" length, the limiting case is just the linear expansion of a thin rod: dL/Lo = a*dT.

B3. The "in between" cases I would think are both mathematically and physically messy. Mathematically for the reasons stated above; physically because the spring has to be uniform. Any impurities that would act as a bi-metallic length of wire puts the problem at a new even higher level of complexity.

I know this is not an "answer" but I do not think there is a simple one. This "back of the envelope" analysis just says the problem is complicated.

Vince Calder

Professor Calder:

Thank you for your detailed analysis of my spring question. I base my assertion that the wire twists on an experiment I did while preparing for a physics class I hosted many years ago.

I took a stiff automotive valve spring (it just happened to be wound CW) and affixed an 18 inch long stiff, slender wire to one of the central coils. I clamped the spring in a vise with the whisker oriented straight up. The ends of the spring were not constrained -- just resting against the smooth steel jaws of the vise. When I gradually compressed the spring, the far end of the whisker swung through an arc that was parallel to the central axis of the coil. This told me that the wire of which the spring was made was twisting along the wire's central axis. Thus, it appears that a coil spring is really just a reconfigured torsion bar.

I thought I would share this experiment information with you because you were kind enough to respond to my question. Thanks again.

Richard Hoffmann

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