A hollow steel shaft has an external diameter of 120 mm and is 10 mm thick. It is coupled to a solid shaft of the same material, 100 mm in diameter. If the length of the hollow shaft is 960 mm, find the length of the solid shaft, so that the total angle of twist is limited to 2.70 under a torque of 120 kN-m, when G = 80 Gpa.
Yes
To find the length of the solid shaft, we need to first calculate the polar moment of inertia for both the hollow and solid shafts. Then, we can use the torsion formula to calculate the length of the solid shaft.
Step 1: Calculate the polar moment of inertia for the hollow shaft.
The polar moment of inertia for a hollow shaft with an external diameter (D) and thickness (t) can be calculated using the formula:
J_hollow = (π/2) * (D^4 - (D - 2t)^4)
Given:
External diameter (D) = 120 mm = 0.12 m
Thickness (t) = 10 mm = 0.01 m
J_hollow = (π/2) * (0.12^4 - (0.12 - 2*0.01)^4)
= (π/2) * (0.12^4 - 0.10^4)
= π/2 * (0.20736 - 0.01)
= π/2 * 0.19736
= 0.31085 m^4
Step 2: Calculate the polar moment of inertia for the solid shaft.
The polar moment of inertia for a solid shaft with diameter (D_solid) can be calculated using the formula:
J_solid = (π/32) * D_solid^4
Given:
D_solid = 100 mm = 0.1 m
J_solid = (π/32) * 0.1^4
= (π/32) * 0.0001
= 0.00000987 m^4
Step 3: Use the torsion formula to find the length of the solid shaft.
The torsion formula relates the angle of twist (ϴ), the torque (T), the length of the shaft (L), and the polar moment of inertia (J):
ϴ = (TL) / (GJ)
Rearranging the formula to solve for the length of the solid shaft:
L = (ϴ * G * J) / T
Given:
Angle of twist (ϴ) = 2.70 degrees = 2.70 * π/180 radians
Torque (T) = 120 kN-m = 120 * 10^3 N-m
G = 80 GPa = 80 * 10^9 Pa
L = (2.70 * π/180 * 80 * 10^9 * 0.00000987) / (120 * 10^3)
= (2.70 * 3.14/180 * 80 * 10^9 * 0.00000987) / (120 * 10^3)
= 2210 m
Therefore, the length of the solid shaft should be approximately 2210 meters.