Machine tools are often automatically controlled as shown in Figure P9.10. These

Chapter 0, Problem P9.10

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Machine tools are often automatically controlled as shown in Figure P9.10. These automatic systems are often called numerical machine controls [9]. On each axis, the desired position of the machine tool is compared with the actual position and is used to actuate a solenoid coil and the shaft of a hydraulic actuator. The transfer function of the actuator (see Table 2.7) is Ga{s) = K Y(S) 5(VV + 1)" where K = 1 and ra = 0.4 s. The output voltage of the difference amplifier is ,,(5) = Kx{X{s) - Xd(s)l where xlt(t) is the desired position input. The force on the shaft is proportional to the current /, so that F = K2i(t). where K2 = 3.0. The spring constant K, is equal to 1.5,7? = 0.1, and L = 0.2. (a) Determine the gain K] that results in a system with a phase margin of 30. (b) For the gain Kx of part (a), determine M[m, con and the closed-loop system bandwidth, (c) Estimate the percent overshoot of the transient response to a step input X( t(s) = l/.y, and the settling time (to within 2% of the final value).