Hollow shaft power transmission

[Ipodbob]

Hello all this is my first post so i hope i can give you the relevant data in the correct format, also thank you for any assistance :)

I am to find the OD of a hollow shaft when the shafts ID is 0.75 of the OD dimension, the shaft transmits 3MW at 200Rpm. The Shear stress max is 55 MN/m² and the modulus of rigidity is 80 GN/m². The OD can be no larger than 270mm / 0.27m.

Power = 2.Pi.N.T/60 n = 200rpm

Torque = (p.60)/(2.Pi.N)

Torque applied = 143.2 KNm

J (second moment of area) = Pi(D⁴-d⁴)/32

I now know i can use Ta/J = τ/r and transpose that,

Ta/J = 2τ/OD

Subbing J so i can then transpose the equation to find the D(OD), by changing out the d(ID) and r for OD parameters.

Ta/[Pi/32(D⁴-(0.75.D)⁴)] = 2τ/D

I need to transpose this equation for D (the outer diameter) but I'm stuck, i hope this makes sense. The problem is i have calculated all the the answer assuming the max size of 0.27 which is strong enough to transmit the torque but i realize now i need to find the outer diameter with the figures i have given above.

Regards

 

[SteamKing]

Well, you know the applied torque Ta and the max. shear stress τ from your calculations and the OP.

you can substitute these values into your equation:

Ta/[Pi/32(D^{4}-(0.75.D)^{4})] = 2τ/D

and solve for D. All it requires is a little algebra.

 

[Ipodbob]

I got this far i am not sure how to move the (0.75)⁴ away so i can leave D⁴ - D⁴ on that side, leaving me with just D on the other side. It seems i need to catch up on some basic algebra.

https://scontent-a-cdg.xx.fbcdn.net/hphotos-prn2/q80/s720x720/1480744_581451648571507_706677602_n.jpg

 

[SteamKing]

Remember, 0.75^4 is a number. You can evaluate numbers. Cross-multiply your expression above to make an algebraic equation in the unknown variable D.