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Calculus method for finding maximum possible angle of twist

  1. Dec 28, 2011 #1
    1. The problem statement, all variables and given/known data

    I am supposed to design a torsion stepper (using equations τ=Tr/J and θ=TL/JG) . Given a list of materials provided of which the hint is to use one of these materials :

    Polycarbonate : G = 2.3 GPa , τmax = 45 MPa
    Nylon : G = 4.1 GPa , τmax = 60 MPa
    Stainless Steel : G = 77.2 GPa , τmax = 180 MPa

    The torsion shaft can either be hollow or solid, and specifications of length of shaft, perpendicular distance from center of load, inner diameter and outer diameter must be reasonable for actual application. Force applied must be close to value in actual application. All these variables are carefully adjusted so that maximum angle of twist without failing is as high as possible given material limitations.

    2. Relevant equations

    τ=Tr/J and θ=TL/JG , where
    J=π(D4-d4)/32
    T=F.x

    Variables to adjust are
    F - force exerted
    x - perpendicular distance from center of shaft
    D - external diameter of shaft
    d - internal diameter of shaft (optional)

    so that θ angle of twist is suitable for actual application according to material.

    3. The attempt at a solution

    The method taught to solve this problem is to use Excel spreadsheet.
    On one table are the variables to be adjusted F, x, D, d and modulus of material G,

    and another table are the calculations.
    T = F.x
    J=π(D4-d4)/32

    From these values,
    actual angle θ=TL/JG (to find max possible)
    while actual maximum shear stress
    τ=Tr/J must be within material constraint.

    Using Polycarbonate and adjusting parameters slowly
    (and the design must be suitable for human use)
    I get 15.855 degrees
    and 44.99 MPa (within 45 MPa)

    However this method is troublesome and involves manually tweaking many permutations of values in Excel.

    From another subject I learnt that the problem may be approached with calculus and related rates (?) to get a definitive answer... However I don't know how to apply the calculus method especially with so many variables and equations, not sure if i learnt that far... anyone can help?
     
  2. jcsd
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