Sunday, 17 September 2017

POWER TRANSMITTED BY A CIRCULAR HOLLOW SHAFT

POWER TRANSMITTED BY A CIRCULAR HOLLOW SHAFT


Now we are going further to start a new topic i.e. Power transmitted by a circular hollow shaft with the help of this post. We recommend to first review the recent post “Torsion or twisting moment” for having basic information about the concept of torsion or twisting moment.

So, what is torsion or twisting moment?

 
A shaft will said to be in torsion, if it will be subjected with two equal and opposite torques applied at its two ends.

When a shaft will be subjected to torsion or twisting moment, there will be developed shear stress and shear strain in the shaft material.

We will discuss here one case of a hollow circular shaft which will be subjected to torsion and we will secure here the expression for maximum torque transmitted by a hollow circular shaft.
We have following information from above figure
Ro = Outer radius of the hollow circular shaft
Ri = Inner radius of the hollow circular shaft
Do = Outer diameter of the hollow circular shaft
Di = Inner diameter of the hollow circular shaft
dr = Thickness of small elementary circular ring
r = Radius of the small elementary of circular ring
q = Shear stress at a radius r from the centre of the hollow circular shaft
τ = Maximum shear stress at outer surface of shaft
dA = Area of the small elementary of circular ring
dA = 2П x r x dr

Shear stress, at a radius r from the centre, could be determined as mentioned here
q/r = τ / Ro
q = τ x r/ Ro

Turning force due to shear stress at a radius r from the centre could be determined as mentioned here
dF = q x dA
dF = τ x r/ Ro x 2П x r x dr
dF = τ/ Ro x 2П r2dr

 
Twisting moment at the circular elementary ring could be determined as mentioned here
dT = Turning force x r
dT = τ/ Ro x 2П r3dr

Total torque could be easily determined by integrating the above equation between limits Ri and Ro.
Therefore total torque transmitted by a hollow circular shaft will be given by following formula
Let us consider that N is the R.P.M of the shaft and ω is the angular velocity of the shaft. We have already secured here the torque transmitted by circular solid shaft and it is mentioned above.  
Now we will find here the expression for power transmitted by solid circular shaft as mentioned here.
Do you have suggestions? Please write in comment box.

We will now discuss the Torque in terms of polar moment of inertia, in the category of strength of material, in our next post.

Reference:

Strength of material, By R. K. Bansal
Image Courtesy: Google

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