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Home Mechanical Connection Between Motor and Load Calculation of Accelerating Time  
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Calculation of Accelerating TimeAuthor: E.E. Kimberly It is sometimes necessary to predict the time required to accelerate a load by means of a motor of known speed , torque characteristics, or it may be necessary to produce a given speed in a specified time. The total energy required to accelerate a load to full speed may be allotted to two classifications, namely, the kinetic energy of inertia of the motor and load, and the friction loss or its equivalent in the driven machine. If the power required to drive the machine at any constant speed equal to or less than full speed is not known, it may be determined approximately by analysis of the load or may be determined accurately by test. It is also necessary to know the equivalent moment of inertia of the driven machine as well as that of the driving motor.
Let the torque required to drive a given load at a series of constant speeds be given by the load curve (2) in Fig. 132 (a). For this purpose the relatively simple speedtorque curve of an induction motor (described in Chapter 18) has been used rather than the stepped curve of a directcurrent motor, which would be more difficult to use. Let the speedtorque characteristic of the driving motor also be as shown in Fig. 132 (a) by curve (1). At any speed, the torque of the motor performs two functions, namely, provides the power necessary for steady state at that speed, and provides the power of acceleration. The torque available for acceleration, as obtained by taking the difference between the ordinates to curves (1) and (2) in Fig. 132 (a), is plotted in (b). The acceleration, which is determined by dividing the accelerating torque by the moment of inertia, is plotted in (c). Here, the acceleration a, or is in radians per second per second. In general,
(131)
in which T = torque, in poundfeet; = moment of inertia; a = motor acceleration, in radians per second per second. The curve of Fig. 133 is a reciprocal of the curve of Fig. 132 (c). The total time required to attain any speed from any initial speed is the integral of the function of Fig. 133, with respect to ω, between the initial and desired speeds. The curve ofvs. speed is asymptotic to a vertical line at the steadystate ultimate speed. Theoretically the motor would continue to accelerate indefinitely. However, the time required to attain a speed of, say, 98 per cent of the ultimate speed may be readily calculated and gives results sufficiently accurate for most purposes.
Example 131.  A centrifuge has a steadystate torque requirement as a function of speed, as shown by curve (2) of Fig. 132 (a). The moment of inertia of the motor and all connected parts is 10, Calculate the time required for the motor with the speedtorque characteristic shown by curve (1) in Fig. 132 (a) to accelerate the centrifuge to 98 per cent of its ultimate speed.
Solution.  Fig. 132 (6) is a plot of the difference between curves (1) and (2) of Fig. 132 (a), and represents the accelerating torque. The acceleration is
Fig. 132 (c) is a plot of the acceleration a in radians per second per second against speed. Fig. 133 is a plot of reciprocal values from Fig. 132 (c). Synchronous speed of 1200 rpm is a speed ω of = 125.8 radians per sec. In Fig. 133 the ordinatesare plotted to a scale of 1 in. = 0.05; and the abscissas are plotted to a scale of 1 in. = 400 rpm, or 1 in. =41.9 radians per sec. Hence, 1 sq in. under the curve of Fig. 133 equals 41.9x0.05 = 2.095 sec. There are 5.3 sq in. under that curve to 98% speed. The time required to accelerate to 98% speed is 2.095 X 5.3 = 11 sec.


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