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.
To find the inside diameter (d) and outside diameter (D) of the hollow shaft, we can follow these steps:
1. Calculate the shaft's angular velocity. Given the shaft speed is 45 RPM, we can convert it to radians per second: 45 * 2π/60 = 4.71 radians per second.
2. Determine the torque transmitted by the shaft. Using the formula torque = power / angular velocity, we can calculate the torque: torque = 1400 kW / 4.71 rad/s = 297240 Nm.
3. Use the torsion equation to relate torque, polar moment of inertia (J), modulus of rigidity (G), angle of twist (θ), and shaft length (L). The torsion equation is T/J = G * θ / L.
4. Rearrange the torsion equation to solve for J: J = T / (G * θ / L). Plugging in the values, we have J = 297240 Nm / (81.75 GPa * 0.01745 / (25 * D)).
5. Simplify the equation for J: J = 297240 * 25D / (81.75 * 10^9 * 0.01745) = 0.005209 D.
6. Use the expression for J, which is J = π/32 * (D^4 - d^4), where d is the inside diameter. Substituting in the value for J we found, 0.005209 D, we have 0.005209 D = π/32 * (D^4 - d^4).
7. Simplify the equation: (0.005209 D * 32) / π = D^4 - d^4.
8. Since we know the relationship between d and D is d = 0.6D, we can substitute d with 0.6D in the equation: (0.005209 D * 32) / π = D^4 - (0.6D)^4.
9. Solve the equation to find D: (0.005209 D * 32) / π = D^4 - 0.1296 D^4.
10. Simplify the equation further: 0.05307 D = 0.8704 D^4.
11. Isolate D by dividing both sides of the equation by 0.8704 D: D^3 = 0.05307 D / 0.8704.
12. Solve for D: D^3 = 0.06097. Taking the cube root of both sides gives D = ∛(0.06097).
13. Calculate D using a calculator or computer: D ≈ 0.3935 m or 39.35 cm.
14. Finally, calculate the inside diameter, d: d = 0.6D ≈ 0.6 * 39.35 cm ≈ 23.61 cm.
Therefore, the inside diameter of the hollow shaft is approximately 23.61 cm, and the outside diameter is approximately 39.35 cm.