Question 11.5: Determining the Energy Variation in a Torque-Time Function. ...

1 Calculate the average value of the torque-time function over one cycle, which in this case is 70.2 lb-in. (Note that in some cases the average value may be zero.)

2 Note that the integration on the left side of equation 11.18c is done with respect to the average line of the torque function, not with respect to the θ axis. (From the definition of the average, the sum of positive area above an average line is equal to the sum of negative area below that line.) The integration limits in equation 11.18 are from the shaft angle θ at which the shaft ω is a minimum to the shaft angle θ at which ω is a maximum.

\begin{aligned}&\int_{\theta @ \omega_{\min }}^{\theta @ \omega_{\max }}\left(T_{L}-T_{a v g}\right) d \theta=\int_{\omega_{\min }}^{\omega_{\max }} I \omega d \omega \\&\int_{\theta @ \omega_{\min }}^{\theta @ \omega_{\max }}\left(T_{L}-T_{a v g}\right) d \theta=\frac{1}{2} I\left(\omega_{\max }^{2}-\omega_{\min }^{2}\right)\end{aligned}                         (11.8c)

3 The minimum ω will occur after the maximum positive energy has been delivered from the motor to the load, i.e., at a point (θ) where the summation of positive energy (area) in the torque pulses is at its largest positive value.

4 The maximum ω will occur after the maximum negative energy has been returned to the load, i.e., at a point (θ) where the summation of energy (area) in the torque pulses is at its largest negative value.

5 To find these locations in θ corresponding to the maximum and minimum ω’s and thus find the amount of energy needed to be stored in the flywheel, we need to numerically integrate each pulse of this function from crossover to crossover with the average line. The crossover points in Figure 11-11 have been labeled A, B, C, and D. (Program Linkages does this integration for you numerically, using a trapezoidal rule.)

6 The Linkages program prints the table of areas shown in Figure 11-11. The positive and negative pulses are separately integrated as described above. Reference to the plot of the torque function will indicate whether a positive or negative pulse is the first encountered in a particular case. The first pulse in this example is a positive one.

7 The remaining task is to accumulate these pulse areas beginning at an arbitrary crossover (in this case point A) and proceeding pulse by pulse across the cycle. Table 11-1 shows this process and the result.

8 Note in Table 11-1 that the minimum shaft speed occurs after the largest accumulated positive energy pulse (+200.73 in-lb) has been delivered from the driveshaft to the system. Delivery of energy slows the motor down. Maximum shaft speed occurs after the largest accumulated negative energy pulse (–60.32 in-lb) has been returned from the system by the driveshaft. This return of stored energy will speed up the motor. The total energy variation is the algebraic difference between these two extreme values, which in this example is –261.05 in-lb. This negative energy coming out of the system needs to be absorbed by the flywheel and then returned to the system during each cycle to smooth the variations in shaft speed.