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Compression spring constant k?

  1. Oct 2, 2012 #1
    Hi,

    Compression_Springs.JPG
    Does a compression spring’s pitch or rise angle have any relation with its spring constant k?
    I checked various sources and they differ on this. Some sites simply asks you to feed input into a simple formula:
    k=Gd4/3D3na
    in which
    k: spring constant
    G: material’s shear modulus
    d: diameter of the wire
    D: outer diameter of each winding​
    An example of this treatment can be found at http://www.efunda.com/DesignStandards/springs/calc_comp_designer_eqn.cfm.

    On the other hand, some other sites require knowing the pitch between each winding, or equivalently rise angle θ, and result varies with different pitch/θ even all four parameters above remain unchanged. An example is at http://www.planetspring.com/pages/compression-spring-calculator-coil-calculator.php?id=compression.

    I strongly suspect the first type of treatment above is incorrect. Consider extreme case:
    1. θ →0° : This means we are not winding the spring up so all windings remains on the same plane. Of course when approaching this extreme k would approach zero.
    2.. θ →90°: This corresponding the case when we are pulling the string straight without any winding and it points straight upward. Trying to compressing such a “spring” on the two ends is like compressing a stick rod, and we would get extremely large resistance due
    to the rigidity of the material itself. Obviously in this case k → infinity​

    It is then quite clear that θ cannot be overlooked, and the first kind of treatment above is obviously wrong.

    I wonder why so many websites are still providing that answer? Could someone help or give a derivation of the compression spring k formula?


    Bob
     

    Attached Files:

  2. jcsd
  3. Oct 2, 2012 #2

    AlephZero

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    Your first formula includes ##n_a## which is thie number of active coils in the spring. If you change the pitch angle, then for a fixed length of spring you change ##n_a##.

    Actually the first formula tells you something interesting: for a foxed number of coils, the stiffness does NOT depend on the pitch angle. A "long" spring with 10 coils and a big pitch angle has about the same stiffness as a "short" spring with 10 coils and a small pitch angle.

    You can only choose two of the pitch angle, the length of the spring, and the number of coils as independent quantities.
     
  4. Oct 2, 2012 #3
    Aleph,
    This is what I cannot understand: why k(small angle, 10 coils) = k(large angle, 10 coils)? see the below extreme-case reasoning?
     
  5. Oct 3, 2012 #4

    AlephZero

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    Why do you think k approaches 0 when the angle is small? Allowing for the fact that the coils can't intersect each other, when θ → 0 the wire is wound into a spiral. That acts like a curved beam, and its stiffness depends on the length of the wire (i.e. the number of turns).

    For a fixed number of turns (and a fixed length of wire), it doesn't make much difference if the angle is exactly 0 or just close to 0.

    These simple formulas don't really apply when θ → 90, but for a fixed length of spring, as θ → 90 the number of turns ##n_a## → 0 so the first formula does predict the stiffness gets very large.
     
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