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# Calculation of Accelerating Time

Author: 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.

 Fig. 13-2. Acceleration Curves for Motor and Load

Let the torque required to drive a given load at a series of constant speeds be given by the load curve (2) in Fig. 13-2 (a). For this purpose the relatively simple speed-torque curve of an induction motor (described in Chapter 18) has been used rather than the stepped curve of a direct-current motor, which would be more difficult to use. Let the speed-torque characteristic of the driving motor also be as shown in Fig. 13-2 (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. 13-2 (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,

(13-1)

 Fig. 13-3. Reciprocal of Acceleration

in which T = torque, in pound-feet;

= moment of inertia; a = motor acceleration, in radians per second per second.

The curve of Fig. 13-3 is a reciprocal of the curve of Fig. 13-2 (c). The total time required to attain any speed from any initial speed is the integral of the function

of Fig. 13-3, with respect to ω, between the initial and desired speeds. The curve of

vs. speed is asymptotic to a vertical line at the steady-state 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 13-1. - A centrifuge has a steady-state torque requirement as a function of speed, as shown by curve (2) of Fig. 13-2 (a). The moment of inertia of the motor and all connected parts is 10, Calculate the time required for the motor with the speed-torque characteristic shown by curve (1) in Fig. 13-2 (a) to accelerate the centrifuge to 98 per cent of its ultimate speed.

Solution. - Fig. 13-2 (6) is a plot of the difference between curves (1) and (2) of Fig. 13-2 (a), and represents the accelerating torque. The acceleration is

Fig. 13-2 (c) is a plot of the acceleration a in radians per second per second against speed. Fig. 13-3 is a plot of reciprocal values from Fig. 13-2 (c). Synchronous speed of 1200 rpm is a speed ω of

= 125.8 radians per sec. In Fig. 13-3 the ordinates

are 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. 13-3 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.

Last Update: 2010-10-06