Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Compression spring constant k?

  1. Oct 2, 2012 #1

    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:
    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?


    Attached Files:

  2. jcsd
  3. Oct 2, 2012 #2


    User Avatar
    Science Advisor
    Homework Helper

    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
    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


    User Avatar
    Science Advisor
    Homework Helper

    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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook