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.
I do not get it.
Solid shafts and larger diameter hollow shafts are usally connected throught a flange welded over the smaller diameter of the solid shaft. The flange outside diameter is the same as the hollow shaft. Bolts are then used to secure the two shafts together.
To find the diameter of the bolts, we need to consider the shear stress in the shafts and calculate the shear stress in the bolts.
a) Find the diameter of the bolts:
First, let's calculate the shear stress in the shafts. We know that the shear stress due to torque on the shafts is equal to the shear stress on the bolts.
The formula to calculate shear stress is:
Shear stress = Torque / (Polar moment of inertia * radius)
Where the polar moment of inertia is given by:
Polar moment of inertia = (π/32) * (OD^4 - ID^4)
For the solid shaft:
OD = 65mm (outside diameter)
ID = 65mm (inside diameter)
For the hollow shaft:
OD = 85mm (outside diameter)
ID = ?
Let's assume the inside diameter of the hollow shaft is D (in mm).
The shear stress in the solid shaft is equal to the shear stress in the hollow shaft:
Torque / ((π/32) * (65^4 - 65^4)) = Torque / ((π/32) * (85^4 - D^4))
Cancel out the torque:
1 / ((π/32) * (65^4 - 65^4)) = 1 / ((π/32) * (85^4 - D^4))
Simplify:
32 / (65^4 - 65^4) = 32 / (85^4 - D^4)
Now we can solve for D:
85^4 - D^4 = 65^4 - 65^4
Rearrange the equation:
85^4 - 65^4 = D^4 - 65^4
Simplify:
85^4 - 65^4 = D^4
Now we can calculate D:
D^4 = 85^4 - 65^4
D = (85^4 - 65^4)^(1/4)
Finally, we have the diameter of the bolts.
b) Find the inside diameter of the hollow shaft:
From the previous calculation, we found that the inside diameter of the hollow shaft is D.