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Thread: Coil Spring Paradox

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  1. #1
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    Coil Spring Paradox

    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.

    -
    Last edited by megafiddle; 29th September 2018 at 20:22.


  2. #2
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    No paradox whatsoever -- the number of coils DOES matter.

    I got into this in some depth a good many years ago, when I was working with race car suspensions. I used to have some good references bookmarked, but that was several computer crashes back, and I haven't looked at this stuff for years. Here's one link I found in a quick search, but it's not very scientifically or mathematically explaind (even though it has a formula):
    http://www.bluecoilspring.com/rate.htm

    The second of your illustrations is an extreme over-simplification, but lets try to put it into an example that's easier to visualize.

    First -- as you noted, a coil spring is basically a torsion bar that's wrapped around on itself in a helix. So take the spring in your second illustration and uncoil it. It's now a straight torsion bar, of the same length and diameter as the material making up the coil. To make it easier to work with, we'll grind flats on each end so we can put a wrench on it.

    For example number one, clamp one end in a vise, and put a wrench on the free end. Let's say you're able to twist the wrench end of the torsion rod 20 degrees. The end in the vise didn't move at all, so the rotation at the midpoint is 10 degrees, since the stress is uniformly distributed.

    That's pretty straightforward. Now, take the end out of the vise, clamp into the vise an eye or bushing that's just large enough for the torsion rod to go through it, and insert the rod so the exact midpoint is at the bushing. Now we get two guys, each holding a wrench on one end of the rod. Each one turns his wrench clockwise, from his perspective, which means they are working against each other. Since the two guys are exactly equal in strength (because I said so), each guy turns his wrench through an arc of 10 degrees. Now, if each guy went 10 degrees and they went in opposing directions, the total rotation is 20 degrees, but the rotation at the center eye bushing is zero. So there's no movement at the center point, but it's obvious that the rotational stress is present and uniform everywhere in the rod - including at the center.


    Here's another look at the formula: http://drtempleman.com/spring-design...pring-formulas

    https://www.efunda.com/designstandar...signer_eqn.cfm

    By the way -- coil springs are described ("rated') in terms of units of force per unit of length. 1911 recoil springs are discussed in pounds and measured in inches, so let's assume that the rate is expressed in pounds per inch. BUT -- when we say that a Government model takes a 16-pound recoil spring, that is NOT the "rate." That's the load the spring will resist or carry when compressed to its shortest working length. Many people refer to this as the spring's "rate," but that's not a proper use of terminology.

    If I can get it to come in, the Ordnance Department drawing of the M1911 recoil spring may help to relieve the confusion regarding terminology.
    ...
    Annnnnnnnnnd, I can't attach the image. You may be able to find the Odnance drawing of the recoil spring online. We don't need the picture -- it's a coil spring. It's the specs I'm interested in. Here they are:

    [quote]Diameter of wire - - - - - - - - - .043
    Diameter of coil (OD) - - - - - - - - - - -.430 +/- .005
    Free length - - - - - - - - - - - - - - - - 6.55 Ref.
    Active coils - - - - - - - - - - - - - - - - 29 Ref.
    Total coils- - - - - - - - - - - - - - - - - 30 Ref.
    Direction of helix - - - - - - - - - - - - - L H
    Load at compressed length of - - - - - - 3.72 = 8.00 LB +/- 0.50 LB
    Load at compressed length of - - - - - - 1.81 = 13.55 LB +/- 0.60 LB
    Spring rate - - - - - - - - - - - - - - - - -2.88 LB/IN Ref.
    Solid length - - - - - - - - - - - - - - - - 1.375 Max.
    Type of ends - - - - - - - - - - - - - - - Not squared or closed
    Crimp one end coil to .326 - .010 ID

    The specs tell us the spring "rate" is 2.88 pounds per inch. When we talk about a "16-pound" recoil spring, what we're referring to is what the specs call the "Load at compressed length of 1.81 [inches]"

    Has this helped at all, or just added to the confusion?
    Hawkmoon
    On a good day, can hit the broad side of a barn ... from the inside
    Last edited by Hawkmoon; 29th September 2018 at 21:57.


  3. #3
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    Quote Originally Posted by Hawkmoon View Post
    No paradox whatsoever -- the number of coils DOES matter.
    Well yes. That's part of the reason for the paradox.

    Quote Originally Posted by Hawkmoon View Post
    The second of your illustrations is an extreme over-simplification, ...
    Not for the purpose that I included it.

    It's purpose was to show that additional coils would make no difference in the tortional strain at the ends of the spring wire. I could have included any number of coils in the illustration, and the tortional strain at the coil ends would have been exactly the same.

    The whole point there, was that the total torsional strain, end to end, is not affected by the number of coils. The total stress is certainly affected.

    Quote Originally Posted by Hawkmoon View Post
    The second of your illustrations is an extreme over-simplification, but lets try to put it into an example that's easier to visualize.
    First -- as you noted, a coil spring is basically a torsion bar that's wrapped around on itself in a helix. So take the spring in your second illustration and uncoil it. It's now a straight torsion bar, of the same length and diameter as the material making up the coil. To make it easier to work with, we'll grind flats on each end so we can put a wrench on it.

    For example number one, clamp one end in a vise, and put a wrench on the free end. Let's say you're able to twist the wrench end of the torsion rod 20 degrees. The end in the vise didn't move at all, so the rotation at the midpoint is 10 degrees, since the stress is uniformly distributed.

    That's pretty straightforward. Now, take the end out of the vise, clamp into the vise an eye or bushing that's just large enough for the torsion rod to go through it, and insert the rod so the exact midpoint is at the bushing. Now we get two guys, each holding a wrench on one end of the rod. Each one turns his wrench clockwise, from his perspective, which means they are working against each other. Since the two guys are exactly equal in strength (because I said so), each guy turns his wrench through an arc of 10 degrees. Now, if each guy went 10 degrees and they went in opposing directions, the total rotation is 20 degrees, but the rotation at the center eye bushing is zero.
    Very well explained.

    My second illustration attempted to show that, showing no net strain at the spring midpoint.

    Quote Originally Posted by Hawkmoon View Post
    So there's no movement at the center point, but it's obvious that the rotational stress is present and uniform everywhere in the rod - including at the center.
    I suspect that the resolution of the paradox lies somewhere in there.

    But I believe there is still something remaining to the paradox:

    You cannot have strain without stress. And you cannot have stress without strain. What I was trying to show in the illustrations, is that the strain must be the same in each individual coil at full compression, since nothing is really different between the coils of a 10 coil spring, and the coils of a 20 coil spring, for example. Each fully compressed coil is physically identical.

    Obviously, stress must be uniformally present along the full active length of the spring wire. But where is the strain?

    In the case of a torsion rod, if there were a certain amount of strain per unit length, there would be a total strain at the end of the rod consisting of the sum of the unit strains.

    So if each individual coil in a spring had a certain strain at full compression. shouldn't the total strain, as evident at the ends of the spring wire also be a sum of the individual coil strains?

    It's not though. The strain at the ends of a coil spring is the same regrdless of number of coils. Hence the paradox.

    -

  4. #4
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    How are you defining (or envisioning) "strain"? It seems to me that you are equating strain with movement. I don't think that's a good way to view it. In your second illustration (and in my second torsion rod example) there is no movement at the center of the spring or rod, yet there is torsional stress at the points of zero movement.

    Then take a handgun. The aft end of the spring is fixed, and as the slide retracts all the other coils move toward the rear. As the spring compresses, the space between coils becomes gradually less, so each coil moves less in linear distance (compression) than the coil ahead of it, but in rotation each coil moves the same amount. And since a coil spring is a helical torsion rod, it's the rotation that matters, not the linear compression travel.
    Hawkmoon
    On a good day, can hit the broad side of a barn ... from the inside

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    Quote Originally Posted by Hawkmoon View Post
    How are you defining (or envisioning) "strain"? It seems to me that you are equating strain with movement. I don't think that's a good way to view it.
    I believe "strain" involves movement by definition. Young's modulus for example is stress/strain, where strain is proportional deformation. Or more specifically, change in length divided by original length. If the movement, and therefore the strain, were ever zero, the modulus would be infinite. The material would be unyieldingly rigid. The modulus in the case of coil springs is torsional rather than elastic, but it still involves deformation. It is just twisting rather than stretching.

    I think I understand what you're saying though. It's clear that the strain must be of such a form that it doesn't accumulate. I'm just unable to imagine it.

    In a torsion rod, the strain does accumulate. Make the rod long enough, and a given torque can produce a full revolution at the end. What changes when it's coiled up? Where does all that twisting go?

    Quote Originally Posted by Hawkmoon View Post
    In your second illustration (and in my second torsion rod example) there is no movement at the center of the spring or rod, yet there is torsional stress at the points of zero movement.
    That's true, but isn't that only true over zero distance? In other words, at a single point?

    _

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    You know, you two gave me a headache!!!
    John Caradimas SV1CEC
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    Quote Originally Posted by John View Post
    You know, you two gave me a headache!!!
    Me too, John, I didn't read it all but, did anyone mention the diameter of wire used makes a difference, also?
    Steve

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    Quote Originally Posted by S.B. View Post
    Me too, John, I didn't read it all but, did anyone mention the diameter of wire used makes a difference, also?
    Steve
    That's in the links I provided, that include the formula for spring rate.
    Hawkmoon
    On a good day, can hit the broad side of a barn ... from the inside

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    Okay a couple things to grasp, the 1.81" of compression is part of this recoil springs description, and not the full recoil spring weight, at the full recoil of 1.625" of the GM. It is at this full recoil space of compression that the typical #16 recoil spring provides #16 of tension. The number of coils only matters as to the springs active length, which is the difference between free length and compressed length. The tension will be the same. In other words if you place two of these #16 springs end to end and compress both of them together to 3.25" the tension will still be #16, just as if you cut one in half and compressed it to .8125" , it would provide the same #16. So more coils provide a greater range of motion, but do not affect the total tension at full compression.

    CAW
    “If it ain't broke, don't fix it' is the slogan of the complacent, the arrogant or the scared. It's an excuse for inaction, a call to non-arms.” Colin Powell

  10. #10
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    Quote Originally Posted by Hawkmoon View Post
    Then take a handgun. The aft end of the spring is fixed, and as the slide retracts all the other coils move toward the rear. As the spring compresses, the space between coils becomes gradually less, so each coil moves less in linear distance (compression) than the coil ahead of it, but in rotation each coil moves the same amount. And since a coil spring is a helical torsion rod, it's the rotation that matters, not the linear compression travel.
    This sounds as if you believe that the coils in front are weaker and are compressing before the coils to the rear begin to add their tension. The spring hasn't any knowledge as to which end is moving, front to rear, or rear to front, but does respond to being compressed, all coils compress the same at the same time as they are the same. This being true for a standard wound spring where all coils are wound with the same distance between coils.

    Your description sounds close to a variable rate spring, where say half the coils are wound half as close to each other as the other half. Then the close coils will move more in relation to the coils spaced further apart. This spring the allows for the slide to more easily begin to retract, but will still have the higher weight of the coils of full space when fully compressed.

    CAW
    “If it ain't broke, don't fix it' is the slogan of the complacent, the arrogant or the scared. It's an excuse for inaction, a call to non-arms.” Colin Powell
    Last edited by CAWalter; 7th October 2018 at 10:44.


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