A 7.5-ft-long solid steel shaft (G = 11.2 x 106 psi) is to transmit 15 hp at a speed of 1500 rpm. The shaft is limited to an allowable shearing stress of 4.5 ksi and an allowable angle of twist of 4 degrees.

Determine:

a. The power transmitted by the steel shaft in units of in-lb/sec.
b. The corresponding torque for the steel shaft in units of in-lb.
c. The required diameter of the shaft (Show all units in your calculations).

P = 99000  lb.in / s

T = 630.25 lb-in

dr = 0.9272 in

Explanation:

Given:

- The length of the shaft L= 7.5 ft

- The power transmitted P = 15 hp

- The rotational speed f = 1500 rpm

- Allowable shearing stress of 4.5 ksi

- Allowable angle of twist θ = 4°

Find:

a. The power transmitted by the steel shaft in units of in-lb/sec.

b. The corresponding torque for the steel shaft in units of in-lb.

c. The required diameter of the shaft

Solution:

- The power transmitted can be calculated as:

                         P = 15 hp

                         P = 15*6600 lb.in / s

                        P = 99000  lb.in / s

- The Torque T for the steel shaft can be related to P and w as follows:

                         T = P / 2*pi*f

                         T = 99000 / 2*pi*(1500 / 60 )

                         T = 630.25 lb-in

- Use allowable shearing stress (τ) for design of shaft:

                         τ =  16*T / pi*d^3

Where, d is the diameter of the shaft:

                          d^3 = 16*T / pi*τ

                          d = cbrt ( 16*T / pi*τ )

                          d = cbrt ( 16*630.25 / pi*4.5*10^3 )

                          d = 0.89345 in

- Use allowable angle of twist (θ) for design of shaft:

                           θ = 32*T*L / pi*d^4*G

                           d^4 = 32*T*L / pi*θ*G

                           d = (32*T*L / pi*θ*G)^0.25

                           d = (32*630.25*7.5*12 / pi^2*(4/180)*11.2*10^6)^0.25

                           d = 0.9272 in

- The required diameter should be kept in lieu to both allowable angle of twist and allowable shear stress so dr

                           dr = max ( 0.89345 , 0.9272 )

                          dr = 0.9272 in

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