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mibjkk
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Homework Statement
Consider a hollow drive shaft of length L, inner radius b, outer radius a. A constant torque is applied (we can assume one end of the shaft is fixed because we're applying a constant torque T regardless of the angular velocity of the shaft). We know material properties of the shaft, G the shear modulus and E young's modulus, plus any others we might need.
Homework Equations
(i) T/K = G[tex]\Theta[/tex]/L = [tex]\tau[/tex]/r
(ii) K for a hollow cylinder is K = ([tex]\pi[/tex]/2)([tex]a^{4}[/tex] - [tex]b^{4}[/tex])
(iii) Elastic energy stored per unit volume P = (1/2)([tex]\sigma^{2}[/tex]/E) where [tex]\sigma[/tex] is the maximum stress applied to that unit area
The Attempt at a Solution
I'm not sure if these are all the relevant equations, but here is what I've tried.
- At any point distance r from the center of the cylinder, there is a shear stress [tex]\tau[/tex] = rT/K = (rT[tex]\pi[/tex]/2)([tex]a^{4}[/tex] - [tex]b^{4}[/tex]). If I plug this into my original equation for elastic energy per unit volume, then integrate over the volume of the cylinder, I find
E(T) = 1/(2E)[tex]\int\int\int[/tex][([tex]\pi[/tex]rT/2)([tex]a^{4}[/tex] - [tex]b^{4}[/tex])]^2 dr d[tex]\theta[/tex] dL.
I get as a result
E(T) = [tex]((\pi^{3}LT^{2})/(12E)) (a^{4} - b^{4})^{2} (a^{3} - b^{3}))[/tex]
This answer seems fairly reasonable, because it depends on L, but I am a little worried that it would have a factor of a^8, which seems excessively high for an engineering problem. Also, I'm skeptical of being able to substitute shear stress from (i) for tensile stress in (iii). Would I use the shear modulus G instead of young's modulus E? Is this derivation mostly correct, or is there a complete different approach?