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FIGURE 7.3 (a) Torque-angle or torque-time curve for Example 4. Dashed lines indicate average values, and the numbers in parentheses represent typical values found in Table 7.3. (b) Calculated rotational speed. (c) Calculated rotation angle.
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7.2.5 Speed-Dependent Torques The input and loading torques are in reality a function of angle, time, speed, acceleration, and other factors. In most cases, the assumption that they are functions of angle only is a good one. In some applications, however, different assumptions are necessary. Figure 7.4 shows the torque-speed relationship for an induction motor. The curve can be approximated by a straight line Ts = a + b in the recommended operating
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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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TABLE 7.3 Numerical Integration for Example 4
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7.10 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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range [7.7]. If the torque-angle curve is known for the loading torque Ta, the speed equation is 2 ( ) 2 = 0 2 J
[a ( ) + b Ta( )] d
(7.22)
This equation is implicit in , and the initial value 0 is unknown. Dividing the angle for one cycle into intervals long and letting j denote the speed at the end of j intervals, the trapezoidal rule gives 2j 2 = 0 2 a( 0 + 2 1 + 2 2 + + 2 j 1 + j) 2 J + jb T0 + 2T1 + 2T2 + + 2Tj 1 + Tj 2 (7.23)
where T0 through Tj are the values of Ta at the end of each interval. The unknowns in this equation are j and 0. If a value for 0 is assumed, the quadratic equation gives j = Aj + A2 + Bj j (7.24)
FIGURE 7.4 Torque-speed curve for the induction motor of Example 5.The curve can be approximated by a straight line near the recommended full-torque operating speed of 1000 rpm.
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.
FLYWHEELS 7.12
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where Aj = a 2J 2 a ( + 2 1 + + 2 j 1) + Cj J 2 0 (7.25) (7.26)
Bj = 2 + 0 and Cj = jb
T0 + 2T1 + 2T2 + + 2Tj 1 + Tj 2
(7.27)
Equation (7.24) is solved successively for each j. The calculated value of at the end of the final interval will not in general equal the value that was originally assumed for 0. If the calculation is rerun with this final speed as the assumed value for 0, the numbers will be closer. After a few calculation cycles, the answers will converge. Example 5. An induction motor provides 0.175 lb ft of torque at the recommended full-load operating speed of 1000 rpm. The linear portion of the torquespeed curve also goes through the point of 0.102 lb ft torque at 1025 rpm. Using the torque-angle curve for the load given in Fig. 7.5, find Cs if the load has the same 200-rpm speed as the flywheel and J = 0.085 lb s2 ft referred to the flywheel. The motor has been chosen so that the average loading torque balances the motor torque at the recommended speed; since the speed of the motor is 1000/200 = 5 times the speed of the load, the average load should be approximately T = 5(0.175) = 0.875 lb ft. Referring the motor torques and speeds to the flywheel speed, the two points on the torque-speed curve become 0.175(5) = 0.875 lb ft at 1000/5 = 200 rpm and 0.102(5) = 0.510 lb ft at 1025/5 = 205 rpm. With 200 rpm = 2 (200)/60 = 20.944 rad/s
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