Final answer:
To determine the diameter of the shaft, calculate the torque, tension, and maximum shear stress. Use the equation for power transmission through a rotating shaft to find the torque. Determine the tension on the slack and tight sides of the belt and calculate the tension in the belt. Lastly, calculate the diameter of the shaft using the maximum shear stress equation.
Step-by-step explanation:
To determine the diameter of the shaft, we can use the formula for power transmission through a rotating shaft. The power equation is P = (2 * π * N * T) / 60, where P is the power (in watts), N is the rotational speed (in revolutions per minute), and T is the torque (in Newton-meters). Rearranging the equation gives us T = (P * 60) / (2 * π * N). In this case, P = 30,000 watts and N = 160 rpm. We can substitute these values into the equation to find the torque:
T = (30,000 * 60) / (2 * π * 160) = 18,849 Nm
The torque is equal to the product of the force and the radius. In this case, the force is the tension on the slack side of the belt and the radius is half the diameter of the pulley (0.5 m). So we have:
18,849 Nm = Tension * 0.5 m
Solving for the tension, we get:
Tension = 18,849 Nm / 0.5 m = 37,698 N
The tension on the tight side of the belt is 2.5 times that on the slack side, so:
Tight Side Tension = 2.5 * 37,698 N = 94,246.5 N
The tension in the belt is equal to the sum of the tensions on the tight and slack sides, so:
Belt Tension = Tight Side Tension + Tension = 94,246.5 N + 37,698 N = 131,944.5 N
The maximum shear stress is given by the equation τ = T / (π * r^3), where τ is the shear stress, T is the torque, and r is the radius. Rearranging the equation gives us r = (T / (π * τ))^(1/3). In this case, the torque is 18,849 Nm and the maximum shear stress is 56 MPa. Substituting these values into the equation, we find:
r = (18,849 Nm / (π * 56 MPa * 10^6 N/m^2))^(1/3) = 0.0311 m
The diameter of the shaft is twice the radius, so:
Diameter = 2 * 0.0311 m = 0.0622 m = 62.2 mm