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 start by considering the equilibrium of forces on both the solid and hollow shafts.
a) Begin by considering the equilibrium of forces on the solid shaft. The shear stress on the shaft due to torque is equal to the shear stress on the bolts. This means that the total torque applied to the solid shaft is equal to the torque transmitted by the bolts.
You can calculate the torque transmitted by the bolts using the formula:
T = (Shear stress on bolts) * (Area of each bolt) * (Number of bolts)
Next, you need to find the area of each bolt. Since the bolts are assumed to be threaded, you can calculate the area of each bolt using the formula:
Area of each bolt = (π/4) * (diameter of bolts)^2
By rearranging the formulas, you can solve for the diameter of the bolts.
b) Consider the equilibrium of forces on the hollow shaft. Since the shear stress on the hollow shaft is equal to the shear stress on the bolts, you can use the torque transmitted by the bolts and the outside diameter of the hollow shaft to calculate the inside diameter of the hollow shaft.
Using the formula for torque transmitted by the bolts:
T = (Shear stress on bolts) * (Area of each bolt) * (Number of bolts)
By rearranging the formula, you can solve for the inside diameter of the hollow shaft.
Do you want to continue with the calculations?
To find the required diameters, you can use the equations related to shear stress and torque. Here's how you can approach the problem:
a) Finding the diameter of the bolts:
Start by calculating the torque on the shaft using the formula:
Torque = Shear stress × Polar area
Assuming the solid and hollow shafts are made of the same material and have the same shear stress, the torque in both shafts will be equal.
1. Calculate the polar area of the solid shaft:
The polar area, A_solid, can be calculated using the formula:
A_solid = π/32 × (d_solid)^4
Where d_solid is the diameter of the solid shaft (65mm).
2. Calculate the polar area of the bolts:
Since the bolts are assumed to have the same shear stress as the shafts, the torque equation can be written as:
A_bolt × τ = A_solid × τ
Where τ is the shear stress in both the bolts and the shafts, and A_bolt is the polar area of a bolt.
Rearrange the equation to solve for the bolt diameter:
d_bolt = (A_solid / A_bolt)^(1/4)
3. Plug in the values to calculate the bolt diameter:
Substitute the respective values into the formula to obtain the diameter of the bolts.
b) Finding the inside diameter of the hollow shaft:
First, calculate the polar area of the hollow shaft:
A_hollow = π/32 × (d_hollow^4 - d_solid^4)
From the torque equation, the torque on the hollow shaft is equal to the torque on the solid shaft:
A_hollow × τ = A_solid × τ
Then, rearrange the equation to solve for the inside diameter of the hollow shaft:
d_hollow^4 - d_solid^4 = (A_solid / A_hollow)^(1/3)
Solve for d_hollow using the values obtained from previous calculations.
Remember to convert the units to consistent measurements, such as meters or inches, to ensure accuracy in your calculations.