A solid shaft of 65mm outside diameter and a hollow shaft of 85mm outside diameter are connected by 6 bolts with the mean pitch of thread being 155mm. The shear stress due to the torque on the shafts is equal to the shear stress on the bolts.
a) Find the diameter of the bolts.
b) Find the inside diameter of the hollow shaft.
Where do I start????
Thank you
To solve this problem, you can follow these steps:
Step 1: Understand the problem and its requirements.
The problem asks you to find the diameter of the bolts and the inside diameter of the hollow shaft. To do this, you need to apply the equal shear stress condition between the torque on the shafts and the bolts.
Step 2: Define the variables.
Let's define the variables:
- D1 = outside diameter of the solid shaft (65mm)
- D2 = outside diameter of the hollow shaft (85mm)
- D3 = diameter of the bolts (unknown)
- N = number of bolts (6)
- P = mean pitch of the thread (155mm)
Step 3: Apply the shear stress equation.
The shear stress on the solid shaft can be calculated using the formula:
τ = T / (π/16 * D1^3)
The shear stress on the bolts can be calculated using the formula:
τ = T / (π/4 * D3^2) * N
Step 4: Equate the two shear stress equations.
Since the shear stress on the shafts is equal to the shear stress on the bolts, we can set the two equations equal to each other:
T / (π/16 * D1^3) = T / (π/4 * D3^2) * N
Step 5: Solve for D3.
Simplify the equation by canceling out similar terms and rearranging for D3:
D3^2 = 2 * D1^3 / (N * (D1^2))
D3^2 = 2 * D1 / N
Take the square root of both sides to solve for D3:
D3 = √(2 * D1 / N)
Step 6: Substitute the given values and calculate D3.
Now you can substitute the given value of D1 (65mm) and N (6) into the equations and calculate D3.
D3 = √(2 * 65 / 6)
D3 ≈ 12.8 mm
So, the diameter of the bolts is approximately 12.8 mm.
Step 7: Calculate the inside diameter of the hollow shaft.
To find the inside diameter of the hollow shaft, subtract twice the diameter of the bolts (2 * D3) from the outside diameter of the hollow shaft (D2).
Inside diameter of the hollow shaft = D2 - 2 * D3
Inside diameter of the hollow shaft = 85 - 2 * 12.8
Inside diameter of the hollow shaft ≈ 59.4 mm
So, the inside diameter of the hollow shaft is approximately 59.4 mm.