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(b) Divide the shaft into six segments, each 8 in long, and from the equation for the deflection at x of a uniformly loaded, simply supported beam, develop an expression for the deflection at x. y= wx (2 x2 x 3 24EI
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1.11x [2(48)x2 x3 483] 24(30)(106 )(3.19)
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= 0.483(10 9 )(x)(96x2 x 3 483 ) Prepare a table for x, y, and y2 at the six stations.
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xi 4 12 20 28 36 44 yi 0.000 210 8 0.000 570 9 0.000 774 7 0.000 774 7 0.000 570 9 0.000 210 8 0.003 112 8 y2 i 4.44(10 8) 32.6(10 8) 60.0(10 8) 60.0(10 8) 32.6(10 8) 4.44(10 8) 194(10 8)
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From Eq. (17.37), = g yi = y2 i 386(0.003 112 8) = 787 rad/s = 7515 r/min 194(10 8 )
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which agrees with the result of part (a), but is slightly higher, as expected, since the static deflection shape was used. Since most shafts are of variable diameter, Eq. (17.37) will be more useful for estimating the first critical speed, treating simultaneously the contributions of concentrated masses (gears, pulleys, sprockets, cams, etc.) and the distributed shaft mass as well. Example 6. Assume that the shaft of Example 2 has been established with its final geometry as shown in Fig. 17.7. The shaft is decomposed into 2-in segments. The weight of each segment is applied as a concentrated force wi at the segment centroid. Additionally, the left-side gear weighs 30 lbf and the right-side gear weighs 40 lbf. Estimate the first critical speed of the assembly. Solution. Bearing in mind the tabular deflection method of Sec. 17.2, 12 stations are established. Also, bending moment diagrams will be superposed. For the distributed shaft mass load, the shaft weight is estimated as W = 24.52 lbf, and it follows that bearing reactions are R1 = 11.75 lbf and R2 = 12.77 lbf. Because each reaction is opposed by a bearing seat weight of 1.772 lbf, the net reactions are R1 = 11.75 1.722 = 9.98 lbf and R2 = 12.77 1.772 = 11.0 lbf. The bending moments Mi due to shaft segment weights are shown in column 3 of Table 17.5. For the gears, R1 = 31.25 lbf and R2 = 38.75 lbf, and the resulting bending moments are shown in column 4. The superposition of the moment diagrams for these two sources of bending occurs in column 5. Column 6 displays the shaft segment weights at the station of application. Column 7 shows the concentrated gear weights and their station of application. Column 8 is the superposition of columns 6 and 7. Column 9 is obtained by using the tabular method of Sec. 17.2 and imposing the bending moment diagram of column 5. Columns 10 and 11 are extensions of columns 8 and 9. The sums of columns 10 and 11 are used in Eq. (17.37): = 386(2.348)(10 3) = 3622 rad/s = 34 588 r/min 6.91(10 8 )
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Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
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FIGURE 17.7 The final geometry of the shaft of Ex. 17.2. For critical speed estimation, weights of shaft segments and affixed gears generate separate and combined bending moments. The static deflection under such loading found by the tabulation method provides the deflections used in Rayleigh s critical speed equation. See Table 17.5 and Ex. 17.6.
The methods of Secs. 17.2 and 17.3 and this section can be programmed for the digital computer for rapid and convenient use.
17.9 HOLLOW SHAFTS
Advantages accruing to hollow shafting include weight reduction with minor increase in stress (for the same outside diameter), ability to circulate fluids for lubrication or cooling, and the use of thick-walled tubing as shaft stock. However, unbalance must be checked and corrected, and thick-walled tubing may not have enough material in its wall to accommodate the desired external geometry. For a shaft section with outside diameter d, inside diameter di, and K = di/d, for torsional and bending loading, d(1 K4)1/3 may be substituted for diameter d in equations such as (17.25), (17.26), (17.29), and (17.30). Equations (17.25) and (17.29) can no longer be solved explicitly for diameter d unless K is known. In cases where it is not known, iterative procedures must be used.
Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
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