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.
To find the diameter of the bolts, we need to first calculate the shear stress on both the solid shaft and the hollow shaft, and then set them equal to each other.
Let's start by calculating the shear stress on the solid shaft.
The torque on a shaft can be calculated using the formula:
T = (pi/16) * τ_max * d^3
where T is the torque, τ_max is the shear stress, and d is the diameter of the shaft.
Given that the outside diameter of the solid shaft is 65mm, we can calculate the torque on the solid shaft.
T_solid = (pi/16) * τ_max * (65^3)
Next, let's calculate the shear stress on the bolts.
The shear stress on a bolt can be calculated using the formula:
τ_bolt = (2 * T_bolt) / (π * d_bolt * l_bolt)
where τ_bolt is the shear stress on the bolt, T_bolt is the torque transmitted by the bolt, d_bolt is the diameter of the bolt, and l_bolt is the lever arm of the bolt.
Since we have 6 bolts, the total torque transmitted by the bolts is:
T_bolts = 6 * T_bolt
Setting the shear stress on the solid shaft equal to the shear stress on the bolts, we have:
τ_max = (2 * T_bolts) / (π * d_bolt * l_bolt)
Now we can solve for the diameter of the bolts (d_bolt).
To find the inside diameter of the hollow shaft, we need to subtract the thickness of the hollow shaft from its outside diameter.
Let's denote the inside diameter of the hollow shaft as d_hollow and the thickness of the hollow shaft as t. We can write the equation:
d_hollow = 85mm - 2t
To find the value of t, we need to recognize that the total diameter of the hollow shaft is equal to the outside diameter of the solid shaft, plus twice the thickness of the hollow shaft.
Therefore, we can write another equation:
65mm = d_hollow + 2t
Solving these two equations simultaneously will allow us to find the inside diameter of the hollow shaft (d_hollow).
Let's proceed with solving part a) to find the diameter of the bolts.
1. Calculate the torque on the solid shaft using the formula T_solid = (pi/16) * τ_max * (65^3).
2. Substitute the values into the equation for shear stress on the bolts: τ_max = (2 * T_bolts) / (π * d_bolt * l_bolt).
3. Rearrange the equation to solve for d_bolt.
4. Calculate the inside diameter of the hollow shaft by subtracting the thickness of the hollow shaft (t) from the outside diameter of the hollow shaft (85mm).
5. Set up the equation 65mm = d_hollow + 2t to find the value of t.
6. Substitute the value of t into the equation for d_hollow.
7. Solve the equations simultaneously to find the values of d_bolt and d_hollow.