A motor driving a solid circular steel shaft transmits 30 kW to a gear at B (Fig. 3-34). The allowable shear stress in the steel is 42 MPa.
(a) What is the required diameter d of the shaft if it is operated at 500 rpm?
(b) What is the required diameter d if it is operated at 3000 rpm?

Question

Accepted Answer

(a) Motor operating at 500 rpm. Knowing the horsepower and the speed of rotation, we can find the torque T acting on the shaft by using Eq. (3-43) [latex]H=\frac{2 \pi n T}{60(550)}=\frac{2 \pi n T}{33,000} \quad(n=\mathrm{rpm}, T=\mathrm{b}-\mathrm{ft}, H=\mathrm{hp})[/latex] . Solving that equation for T, we get

[latex]T=\frac{60 P}{2 \pi n}=\frac{60(30 \mathrm{~kW})}{2 \pi(500 \mathrm{~rpm})}=573 \mathrm{~N} \cdot \mathrm{m}[/latex]

This torque is transmitted by the shaft from the motor to the gear.
The maximum shear stress in the shaft can be obtained from the modified torsion formula [Eq. (3-14)]:

[latex] au_{\max }=\frac{16 T}{\pi d^{3}}[/latex]

Solving that equation for the diameter d, and also substituting [latex] au_{ ext {allow }}[/latex] for [latex] au_{ ext {max }}[/latex], we get

[latex]d^{3}=\frac{16 T}{\pi au_{ ext {allow }}}=\frac{16(573 \mathrm{~N} \cdot \mathrm{m})}{\pi(42 \mathrm{MPa})}=69.5 imes 10^{-6} \mathrm{~m}^{3}[/latex]

from which

d = 41.1 mm
The diameter of the shaft must be at least this large if the allowable shear stress is not to be exceeded.

(b) Motor operating at 4000 rpm. Following the same procedure as in part (a), we obtain

[latex]\begin{aligned}T &=\frac{60 \mathrm{P}}{2 \pi n}=\frac{60(30 \mathrm{~kW})}{2 \pi(4000 \mathrm{rpm})}=71.6 \mathrm{~N} \cdot \mathrm{m} \d^{3} &=\frac{16 T}{\pi au_{ ext {allow }}}=\frac{16(71.6 \mathrm{~N} \cdot \mathrm{m})}{\pi(42 \mathrm{MPa})}=8.68 imes 10^{-6} \mathrm{~m}^{3} \d &=20.6 \mathrm{~mm}\end{aligned}[/latex]

which is less than the diameter found in part (a).
This example illustrates that the higher the speed of rotation, the smaller the required size of the shaft (for the same power and the same allowable stress).

Question 3.7: A motor driving a solid circular steel shaft transmits 30 kW...

Related Answered Questions

A circular tube with an outside diameter of 80 mm and an inside diameter of 60 mm is subjected to a torque T = 4.0 kN.m (Fig. 3-30). The tube is made of aluminum alloy 7075-T6. (a) Determine the maximum shear, tensile, and compressive stresses in the tube and show these stresses on sketches of ...

Verified Answer:

A stepped shaft consisting of solid circular segments (D1 = 44 mm and D2 = 53 mm, see Fig. 3-60) has a fillet of radius R = 5 mm. (a) Find the maximum permissible torque Tmax, assuming that the allowable shear stress at the stress concentration is 63 MPa. (b) If the shaft is to be replaced with ...

Verified Answer:

Compare the maximum shear stress in a circular tube (Fig. 3-56) as calculated by the approximate theory for a thin-walled tube with the stress calculated by the exact torsion theory. (Note that the tube has constant thickness t and radius r to the median line of the cross section.) ...

Verified Answer:

Verified Answer:

A shaft of length L = 1.8 m is subjected to torques T = 5 kN m at either end (Fig. 3-48). Segment AB (L1 = 900 mm) is made of brass (Gb = 41 GPa) and has a square cross section (a = 75 mm). Segment BC (L2 = 900 mm) is made of steel (Gs = 74 GPa) and has a circular cross section (d = a = 75 mm). ...

Verified Answer:

A tapered bar AB of solid circular cross section is supported at the right-hand end and loaded by a torque T at the other end (Fig. 3-43). The diameter of the bar varies linearly from dA at the left-hand end to dB at the right-hand end. Determine the angle of rotation ϕA at end A of the bar by ...

Verified Answer:

A prismatic bar AB, fixed at one end and free at the other, is loaded by a distributed torque of constant intensity t per unit distance along the axis of the bar (Fig. 3-42). (a) Derive a formula for the strain energy of the bar. (b) Evaluate the strain energy of a hollow shaft used for drilling ...

Verified Answer:

Verified Answer:

Verified Answer:

Two sections (AB, BC) of steel drill pipe, joined by bolted flange plates at B, are being tested to assess the adequacy of both the pipe and the bolted con-nection (see Fig. 3-19). In the test, the pipe structure is fixed at A and a con-centrated torque 2T0 is applied at x = 2L/5 and uniformly ...

Verified Answer: