Budynas-Nisbett:Shigley's lll.Design of Mechanical 7.Shafts and Shaft ©The McGraw-Hfl 365 Mechanical Engineering Elements Components Companies,2008 Design,Eighth Edition 362 Mechanical Engineering Design EXAMPLE 7-2 This example problem is part of a larger case study.See Chap.18 for the full context. A double reduction gearbox design has developed to the point that the gen- eral layout and axial dimensions of the countershaft carrying two spur gears has been proposed,as shown in Fig.7-10.The gears and bearings are located and supported by shoulders,and held in place by retaining rings.The gears transmit torque through keys.Gears have been specified as shown,allowing the tangential and radial forces transmitted through the gears to the shaft to be determined as follows W2s 540lbf W54=-2431bf W5=-1971bf w54=-8851bf where the superscripts t and r represent tangential and radial directions respectively:and,the subscripts 23 and 54 represent the forces exerted by gears 2 and 5 (not shown)on gears 3 and 4,respectively. Proceed with the next phase of the design,in which a suitable material is selected,and appropriate diameters for each section of the shaft are estimated,based on providing sufficient fatigue and static stress capacity for infinite life of the shaft,with minimum safety factors of 1.5. Bearing A Giear 3 d=12 Gear 4 d=2.67 O CA D E F G H 1 KLM B N Figure 7-10 Shoft layout for Example 7-2.Dimensions in inches
Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 7. Shafts and Shaft Components © The McGraw−Hill 365 Companies, 2008 362 Mechanical Engineering Design EXAMPLE 7–2 This example problem is part of a larger case study. See Chap. 18 for the full context. A double reduction gearbox design has developed to the point that the general layout and axial dimensions of the countershaft carrying two spur gears has been proposed, as shown in Fig. 7–10. The gears and bearings are located and supported by shoulders, and held in place by retaining rings. The gears transmit torque through keys. Gears have been specified as shown, allowing the tangential and radial forces transmitted through the gears to the shaft to be determined as follows. Wt 23 = 540 lbf Wt 54 = −2431 lbf Wr 23 = −197 lbf Wr 54 = −885 lbf where the superscripts t and r represent tangential and radial directions, respectively; and, the subscripts 23 and 54 represent the forces exerted by gears 2 and 5 (not shown) on gears 3 and 4, respectively. Proceed with the next phase of the design, in which a suitable material is selected, and appropriate diameters for each section of the shaft are estimated, based on providing sufficient fatigue and static stress capacity for infinite life of the shaft, with minimum safety factors of 1.5. Datum 0.25 0.75 1.25 1.75 2.0 2.75 3.50 7.50 8.50 9.50 9.75 10.25 10.75 11.25 11.50 C A D EF D1 D2 D3 D4 D5 D6 D7 Gear 3 d3 12 Bearing A Bearing B Gear 4 d4 2.67 G H I J KL M B N Figure 7–10 Shaft layout for Example 7–2. Dimensions in inches.
6 Budynas-Nisbett:Shigley's lll.Design of Mechanical 7.Shafts and Shaft ©The McGraw-Hill Mechanical Engineering Elements Components Companies,2008 Design,Eighth Edition Shafts and Shaft Components 363 Solution Perform free body diagram W2 analysis to get reaction forces at the bearings. RA:=115.0lbf R4y=356.7I6f W RB2=1776.01bf G Rg=725.31bf From Mx,find the torque in the shaft between the gears, 3240 T=W5.(d43/2)=54012/2)= 32401bf.in Generate shear-moment 655 diagrams for two planes 115 x-Plane -1776 3341 20 357 160 -725 14721 r-y Planc 62 1713 907 3651 Combine orthogonal planes as 4316 vectors to get total moments. 208 e.g.at,√39962+16322= 4316bf.in Start with Point I,where the bending moment is high,there is a stress con centration at the shoulder,and the torque is present. At /Ma=3651 Ibf-in,T =3240 Ibf-in,Mm =Ta=0
Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 7. Shafts and Shaft Components 366 © The McGraw−Hill Companies, 2008 Shafts and Shaft Components 363 Start with Point I, where the bending moment is high, there is a stress concentration at the shoulder, and the torque is present. At I, Ma = 3651 lbf-in, Tm = 3240 lbf-in, Mm = Ta = 0 A B G I JK RBy RBz RAz RAy W54 t W23 t W23 r W54 r 655 3996 3341 230 2220 x-z Plane 1776 115 V M 160 1632 1472 713 907 725 x-y Plane 357 V M 4316 3651 749 2398 MTOT 3240 T y z x Solution Perform free body diagram analysis to get reaction forces at the bearings. RAz = 115.0 lbf RAy = 356.7 lbf RBz = 1776.0 lbf RBy = 725.3 lbf From Mx , find the torque in the shaft between the gears, T = Wt 23(d3/2) = 540 (12/2) = 3240 lbf · in Generate shear-moment diagrams for two planes. Combine orthogonal planes as vectors to get total moments, e.g. at J, √39962 + 16322 = 4316 lbf · in.