Find the Inside diameter(d) and outside diameter(D) of a hollow shaft given:
Power transmitted by shaft=1400kW
Shaft speed=45 RPM
Angle of twist on shaft=1 degree
Shaft lenght(L)=25D
Assume d=0.6D
Modulus of Rigidity(G)=81.75GPa
My answer:
Shaft ang. vel.=45*2pi/60s
A.V.=4.71rads/s
Torque transmitted by shaft=Power/A.V.
Torque=1400000W/4.71
T=297240Nm
Then using the Torsion equation:
T=297240
J=pi/32*(D^4-d^4)
G=81.75X10^9
Angle of twist=1*pi/360=0.01745
L=25D
Using the general torsion equation:
T/J=G*Twist ang./L
287240Nm/J=(81.75X10^9)*0.01745/25D
J=297240X25D/(81.75X10^9)*0.01745
J=0.005209D
Since J=pi/32*(D^4-d^4)
Then:
0.005209D=pi/32*(D^4-0.6D^4)
(0.005209D*32)/pi=1D^4-0.1296D^4
0.05307D=0.8704D^4
D^3=0.05307D/0.8704D
D^3=0.06097
D=39.35cm(ans.)
d=39.35*0.6=23.61cm(ans.)
Hope this is correct!
Thank you for checking it out.
The angular velocity is correct.
The torque (T) is correct
The angle of twist in radians is pi/180, not pi/360, but the 0.01745 rad is correct.
The annulus polar moment of inertia (J) formula is correct.
The stress/strain formula is correct.
D has been calculated properly and cnverted to centimeters. I did not check all of the numbers, but all steps seem to have been followed correctly.
Thank you so much
To find the inside diameter (d) and outside diameter (D) of the hollow shaft, you can follow these steps:
Step 1: Calculate the angular velocity of the shaft.
The given shaft speed is 45 RPM (revolutions per minute). To convert RPM to rad/s, you multiply by (2π/60).
Angular Velocity (ω) = 45 * (2π/60) = 4.71 rad/s
Step 2: Calculate the torque transmitted by the shaft.
The power transmitted by the shaft is given as 1400 kW (kilowatts). To calculate torque, use the equation:
Torque (T) = Power / Angular Velocity
= 1400000 W / 4.71 rad/s = 297240 Nm
Step 3: Calculate the polar moment of inertia (J) of the hollow shaft.
The equation for the polar moment of inertia is: J = (π/32) * (D^4 - d^4)
Here, d is given as 0.6D.
Substituting d = 0.6D into the equation, we get:
J = (π/32) * (D^4 - (0.6D)^4)
Step 4: Calculate the angle of twist (θ) on the shaft.
The given angle of twist is 1 degree. Convert this to radians by multiplying by π/180:
θ = 1 * π/180 = 0.01745 rad
Step 5: Calculate the length of the shaft (L).
The given length is 25D, where D is the outside diameter. So, L = 25D.
Step 6: Use the general torsion equation to solve for J.
The general torsion equation is: T / J = G * θ / L
Here, G is the modulus of rigidity given as 81.75 GPa (GigaPascals).
Substituting the values and solving for J, we get:
J = (T * L) / (G * θ) = (297240 Nm * 25D) / (81.75 GPa * 0.01745)
Step 7: Solve for D by equating J calculated in Step 3 to the J calculated in Step 6.
Set the two equations equal to each other:
(π/32) * (D^4 - (0.6D)^4) = ((297240 Nm * 25D) / (81.75 GPa * 0.01745))
Now, solve the equation (0.005209D * 32) / π = (D^4 - 0.1296D^4)
Simplify the equation and solve for D using numerical methods or algebraic techniques.
Once you solve for D, you can calculate d by substituting the value of D into the equation d = 0.6D.
The final values for D and d are: D = 39.35 cm and d = 23.61 cm (rounded to two decimal places).