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The energy-storage capacity of a flywheel is determined from its polar moment of inertia J and its maximum safe running speed. The necessary inertia depends on the cyclic torque variation and the allowable speed variation or, in the case of energystorage flywheels, the maximum energy requirements. The safe running speed depends on the geometry and material properties of the flywheel.
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7.1 FLYWHEEL USAGE
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Flywheels store energy. Indeed, flywheels are used as energy reservoirs, and this use will be discussed in Sec. 7.4. Their principal use in machine design, however, is to smooth the variations in shaft speed that are caused by loads or power sources that vary in a cyclic fashion. By using its stored kinetic energy 0.5J 2 to absorb the variations in torque during a machine cycle, a flywheel smooths the fluctuating speed of a machine and reduces undesirable transient loads. The effect of a flywheel is therefore fundamentally different from that of a regulator: A flywheel limits the speed variation over one cycle and has minimal effect on the average speed; a regulator uses negative feedback to maintain a selected average speed with only secondary effects on the speed during a cycle. The flywheel has other features which have to be considered in design. Its size, speed, and windage effect can all be used to advantage in providing a secondary function as part of a clutch, gear, belt pulley, cooling fan, pump, gyroscope, or torsional damper.
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7.2 SIZING THE FLYWHEEL
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7.2.1 Coefficient of Speed Variation A certain amount of fluctuation in shaft speed will not cause harmful torques or reduce the usefulness of a machine. The coefficient of speed fluctuation Cs is defined as Cs = max min avg (7.1)
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where = rotational speed at the flywheel and avg = average of max and min. Ranges for Cs for several categories of speed variation are given in Table 7.1. Assume that the system is stiff (the speeds of all shafts are proportional), that the external torque input or load is constant, and that avg is close to the constant speed at which the energy from the average torque balances the external energy (this is usually a good assumption for values of Cs up to about 0.2). The energy equation U = 0.5J( 2 2 ) and the definition of Cs combine to give the equation for the max min required mass moment of inertia [7.11]: J= U 2 Cs avg (7.2)
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This inertia includes the flywheel inertia and the inertia of all rotating parts, referred to the flywheel speed by multiplying by the square of the ratio of the shaft speeds.
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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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FLYWHEELS 7.4
MACHINE ELEMENTS THAT ABSORB AND STORE ENERGY
TABLE 7.1 Suggested Values for the Coefficient of Speed Fluctuation Cs
Example 1. During each punching cycle, the cranking shaft for a punching operation does 270 J of work while rotating 30 degrees, as shown in Fig. 7.1. No work is done during the remaining 330 degrees. What size flywheel is necessary if the speed at the location of the flywheel is 20 rad/s and the inertia of the other rotating parts referred to the flywheel is 0.51 N s2 m The average work required is 270/(2 ) = 43.0 J/rad. The motor will supply this constant torque throughout the cycle. Referring to Fig. 7.1, the flywheel will give up some of its stored energy during the 30 degrees of actual punching.This is the shaded area above the average-torque line; the motor will supply the additional 43 J/rad. During the remaining 330 degrees, the motor will resupply the flywheel, as shown by the shaded area below the average-torque line. The flywheel speed reaches its maximum and minimum where the loading torque crosses the average-torque line.
FIGURE 7.1 Torque-angle curve for punching operation in Example 1. The dashed line indicates the average torque of 43 J/rad, and the shaded areas are each equal to the maximum energy variation of 247 J.
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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