In General, shaft comprised of multiple cross-sectional diameters with different lengths. Some times it is necessary to assume the shaft as a uniform diameter and length for some calculations such as finding the natural frequency. In such cases, we have to replace the different cross-section diameters with the equivalent shaft of uniform diameter. Which will discuss how we can construct such data of torsionally equivalent shaft for calculation purposes.
Torsionally Equivalent Shaft
Consider a shaft of varying cross-sections as shown in below fig. Let this shaft is replaced by an equivalent shaft of uniform diameter d and length l as shown figure.
These two shafts must have the same total angle of a twist when equal opposing torques T are applied at the opposite ends.
- d_{1}, d_{2} and d_{3} = Diameters for the lengths l_{1}, l_{2} and l_{3} respectively,
- θ_{1}, θ_{2}_{ }and θ_{3} = Angles of twist for the lengths l_{1}, l_{2} and l_{3} respectively,
- θ = Angle of twist for the diameter d and length l,
- J_{1}, J_{2} and J_{3} = Polar moment of inertia for the shaft of diameters d_{1}, d_{2 }and d_{3 }respectively.
Since the total angle of the twist of the shaft is equal to the sum of angle of twists of the different lengths.
θ = θ_{1} + θ_{2} + θ_{3}
From the torsion Equation
Where
- τ = Shear stress (MPa)
- r = Radius of the shaft (mm)
- T = Torque (Nm)
- J = Polar moment of inertia
- G = Modulus of rigidity (MPa)
- θ = Angle of twist (rad)
- l = length of the shaft
We can write θ = θ_{1} + θ_{2} + θ_{3} as following
As we want to make the multiple cross-sectional shafts into a uniform diametral shaft, so we have to assume that diameter d of the equivalent shaft should be equal to the one of the diameters of the actual shaft. So we can assume d = d_{1}
Substitute
From this expression, we can evaluate the length of the torsionally equvalent shaft.
Conclusion
We have discussed how the torsionally equivalent shaft diameter and the length can be evaluated from the shaft of varying cross-sectional diameters. If you have further thoughts on this topic, please let us know in the comment section below.
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