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Homework answers / question archive / Homework 4 Due February 24", 2021 1

Homework 4

Due February 24", 2021

1. The block diagram of a servomechanism is given below. The system parameters are:

K_{s} = 60 volts/radian K_{a} = 20 Volts/volt K_{m} = 30 x 10^{-6} lb-ft/volt

J = 10x 10^{6} slug-ft2 F = 161x10^{6} Ib-ft/radin/sec

a. Find the damping ratio

b. Compute the lack of correspondence between the input and feedback shaft positions for an input velocity of 2 radians/sec at steady state

K_{s } K_{A} K_{M} C_{M}

r→ O→ Synchros → Amplifier → Motor -----→ 1/30 → Load

__ __ 1/20 ¬?

2. The fixed configuration of a 400-cps aircraft feedback control system consists of a synchro error detector, an amplifier, a 400-cps servomotor, and a geared load. The parameters are as follows:

K_{s} — synchro constant = 25 volts/radian. h=a=1/30

K_{M} = motor torque constant = 40 x 10^{-6} Ib-ft/volt

F_{M} = motor viscous friction = 9 x 10^{-6} Ib-ft/ radian/ sec

F_{L} = load viscous friction v= 45 x 10^{-6} Ib-ft/radian/sec

J_{M} = negligible. J_{L} = load inertia = 9x 10^{-4} slug-ft^{2} K_{A} = 3 volts/volt

a). When the system is used as described above, compute the value of the stedy-state velocity error as well as the stead-state actuating signal (€ in radians) for a command velocity of 1 radians/sec.

b). Calculate the damping ratio for the part conditions of part a. What specific information does this quantity convey about the system’s dynamic performance?

c). It is desirable to reduce the magnitude of the steady-state actuating signal to 0.035 radian while maintaining € fixed. Compute the amount of error-rate damping to accomplish this.

3. a). Explain why a feedback control system includes the following two features in its design: negative feedback and sign sensitivity.

b). A servomechanism is represented by the following differential equation:

dc_{M}/dt + 6.4 * dc_{M}/dt_{ }= 160e

Where e = r - 0.4C_{M}

Cu = motor output shaft position

Find the value of the damping ratio. What information does this convey about the transient performance?

4. A servomechanism consists of a motor which produces a torque proportional to the error, an output load having inertia and viscous damping, and a position-sensitive error-measuring means. The system is pictured in Fig. P4-13a. The motor;

_r_+_→ O → Amplifier _________→ Load

-__________________?

(a)

Unit step response t = 0.125sec

(b) FIG. P4-13

Torque T = Ke. The response of the system to a unit-step input is sketched in Fig. P4-13b. Only the time interval t is known accurately.

When a constant velocity input of 1 radian/sec is put into the system, a position lag error of 1/8.333π radians is observed.

When the input shaft is held fixed so that it cannot move and a torque is applied to the output shaft, a steady-state error of 1/π^{2} radians is observed for each ft-lb applied torque. From the above data determine:

(a) The undamped natural frequency w_{π} (Two values are possible.)

(b) The damping ratio ς. (Two values are possible.)

(c) The viscous damping F. (Only one solution is possible.) State the units.

(d) The moment of inertia J. (Two values are possible.) State flip units.

5. Solve only for (a), and (b).

6.14. The block diagram of a servo system is shown in Fig. P6-14.

R(s) Amplifier gain K Motor 1/s^{2} C(s)

As + b

Figure P6-14.

(a) For K = 10, determine the values of a, and b to give an overshoot of 16 per cent and a time constant of 0.1 sec of the system response to a unit step input. Time constant is defined here as the inverse of the damping factor.

(b) If the value of K is decreased slightly, how does it affect the damping ratio of the system?

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