Questions and Answers of Classical Dynamics Of Particles

Discuss whether or not the following devices and processes are open-loop or closed-loop. If they are closed-loop, identify the sensing mechanism.a. A traffic light.b. A washing machine.c. A
a. Determine the resistance values to obtain an op-amp PI controller with Kp = 2, TD = 2 s. The circuit should limit frequencies above 5 rad/s. Use a 1-μF capacitor, b. Plot the frequency response
a. Determine the resistance values to obtain an op-amp PID controller with Kp = 10, Kt = 1.4, and KD) = 4. The circuit should limit frequencies above 100 rad/s. Take one capacitance to be 1
Obtain the steady-state response, if any, of the following models for the given input. If it is not possible to determine the response, state the reason. a. Y(s)/F(s) = 6/(7s +
For the following models, the error signal is defined as e(t) = r(t) - c(t).Obtain the steady-state error, if any, for the given input. If it is not possible to determine the response, state the
Given the model3ẍ - (3b + 6) ẋ + (6b + 15)x = 0a. Find the values of the parameter b for which the system is1. Stable.2. Neutrally stable.3. Unstable.b. For the stable case, for what values of b
For the characteristic equation s3+ 9s2 + 26s + K = 0, use the Routh-Hurwitz criterion to compute the range of K values required so that the dominant time constant is no larger than 1/2.
For the following characteristic equations, use the Routh-Hurwitz criterion to determine the range of K values for which the system is stable, where a and b are assumed to be known.a. 2s3 + 2as2 + Ks
The parameter values for a certain armature-controlled motor, load, and tachometer are KT = Kb = 0.2 N.m/A cm = 5 x 10-4 N.m.s/rad cL = 2 x 10-3 Ra = 0.8 Ω La = 4 x 10-3 H lm = 5 x 10-4 It = 10-4 IL
Suppose the plant shown in Figure 10.6.1 has the parameter values I = 2 and c = 3. Find the smallest value of the gain Kp required so that the steady-state offset error will be no greater than 0.2 if
Draw the block diagram of a system using proportional control and feed forward command compensation, for the plant 1/(4s2 + 6s + 3). Determine the transfer function of the compensator. Discuss any
Suppose the plant shown in Figure 10.6.1 has the parameter values I = 2 and c = 3. The command input and the disturbance are unit-ramp functions. Evaluate the response of the proportional controller
For the control system shown in Figure 10.6.2, I = 20, and suppose that only I action is used, so that Kp = 0. The performance specifications require the steady-state errors due to step command and
Suppose that I = c = 4 for the PI controller shown in Figure 10.6.2. The performance specifications require that τ = 0.2. (a) Compute the required gain values for each of the following cases. 1.
For the designs obtained in part (a) of Problem 10.22, use a computer method to plot the actuator torque versus time. Compare the peak torque values for each case.
For the designs found in part (a) of Problem 22, evaluate the steady-state error due to a unit-ramp command and due to a unit-ramp disturbance. In Problem 22 (a) Compute the required gain values for
Consider the PI speed control system shown in Figure 10.6.2, where I = c = 2. The desired time constant is τ = 0.1. (a) Compute the required values of the gains for the following three sets of root
Suppose that I = c = 4 for the I controller with internal feedback shown in Figure 10.6.6. The performance specifications require that τ = 0.2. (a) Compute the required gain values for each of the
For the designs obtained in part (a) of Problem 26, use a computer method to plot the actuator torque versus time. Compare the peak torque values for each case. In problem 26 (a) Compute the required
For the designs found in part (a) of Problem 26, evaluate the steady-state error due to a unit-ramp command and due to a unit-ramp disturbance. In problem 26 (a) Compute the required gain values for
Consider the speed control system using I control with internal feedback shown in Figure 10.6.6, where I = c = 2. The desired time constant is τ = 0.1 a. Compute the required values of the gains for
Investigate the performance of proportional control using feed forward command compensation with a constant gain Kf and disturbance compensation with a constant gain Kd, applied to the plant 10/s.
Modify the diagram shown in Figure to include feed forward command compensation with a constant compensator gain Kf. Determine whether such compensation can eliminate steady-state error for step and
Suppose that I = 10 and c = 5 for the PI controller shown in Figure 10.6.2. The performance specifications require that τ = 2. (a) Compute the required gain values for each of the following
For the designs obtained in part (a) of Problem 31, use a computer method to plot the actuator torque versus time. Compare the peak torque values for each case. In problem 31(a) Compute the required
For the designs found in part (a) of Problem 31, evaluate the steady-state error due to a unit-ramp command and due to a unit-ramp disturbance. In Problem 31(a) (a) Compute the required gain values
Consider the PI speed control system shown in Figure 10.6.2, where I = 5 and c = 4. The desired time constant is τ = 0.5. (a) Compute the required values of the gains for the following three sets of
Suppose that I = 15 and c = 5 for the I controller with internal feedback shown in Figure 10.6.6. The performance specifications require that τ = 0.5. (a) Compute the required gain values for each
For the designs obtained in part (a) of Problem 35, use a computer method to plot the actuator torque versus time. Compare the peak torque values for each case. In Problem 35(a) Compute the required
For the designs found in part (a) of Problem 10.35, evaluate the steady-state error due to a unit-ramp command and due to a unit-ramp disturbance. In Problem 35(a) Compute the required gain values
Consider the speed control system using I control with internal feedback shown in Figure 10.6.6, where I = 15 and c = 5. The desired time constant is τ = 0.5. a. Compute the required values of the
Consider the PD control system shown in Figure 10.7.1. Suppose that I = 20 and C = 10. The specifications require the steady-state error due to a unit-step command to be zero and the steady-state
Derive the output C(s), error E(s), and actuator M(s) equations for the diagram in Figure, and obtain the characteristic polynomial.
Suppose that I = 10 and c = 3 in the PD control system shown in Figure 10.7.1. The performance specifications require that τ = 1 and ζ = 0.707. Compute the required gain values.
Figure 10.7.2 shows a system using proportional control with velocity feedback. Suppose that I = 20 and c = 10. The specifications require the steady-state error due to a unit-step command to be zero
For the system discussed in Problem 40, a. Use a computer method to plot the output θ(t) and the actuator response T(t) for a unit-ramp command input. b. Use a computer method to plot the
Suppose that I = 10 and c = 3 for the PID control system shown in Figure 10.7.3. The performance specifications require that τ = 1 and ζ = 0.707. a. Compute the required gain values. b. Use a
Consider the PD control system shown in Figure 10.7.1. Suppose that I = 20 and c = 10. The specifications require the steady-state error due to a unit-step command to be zero and the steady-state
Modify the PD system diagram shown in Figure 10.7.1 to include feed forward compensation with a compensator gain of Kf. Determine whether such compensation can reduce the steady-state error for step
Consider a plant whose transfer function is l/(20s + 0.2). The performance specifications are 1. The magnitude of the steady-state command error must be no more than 0.01 for a unit-ramp command. 2.
For the system shown in Figure 10.7.1 I = c = 1. Derive the expressions for the steady-state errors due to a unit-ramp command and to a unit-ramp disturbance.
For the PD control system shown in Figure 10.7.1, I = c = 2. Compute the values of the gains Kp and KD to meet all of the following specifications: 1. No steady-state error with a step input 2. A
Consider the PID position control system shown in Figure 10.7.3, where I = 10 and c = 2. The desired time constant is τ = 2. a. Compute the required values of the gains for the following two sets of
For the system shown in Figure, the plant time constant is 5 and the nominal value of the actuator time constant is Ï„a = 0.05. Investigate the effects of neglecting this time constant as the gain
Derive the expression for T(s) in Figure 10.7.6. Using the values given and computed in Example 10.7.5, use MATLAB to plot T(t) for a unit-step command input. Determine the maximum value of T(t).
Integral control of the plant Gp(s) = 3/(5s+ 1) Results in a system that is too oscillatory. Will D action improve this situation?
Modify the system diagram shown in Figure to include feed forward compensation with a compensator gain Kf. Determine whether such compensation can reduce the steady-state error for step and ramp
Consider the PD control system shown in Figure 10.7.1. Suppose that I = 25 and c = 5. The specifications require the steady-state error due to a unit-step command to be zero and the steady-state
Suppose that I = 15 and c = 10 in the PD control system shown in Figure 10.7.1. The performance specifications require that τ = 2 and ζ = 0.707. Compute the required gain values.
For the system discussed in Problem 54, a. Use a computer method to plot the output θ(t) and the actuator response T(t) for a unit-ramp command input. b. Use a computer method to plot the
Suppose that I = 15 and c = 5 for the PID control system shown in Figure 10.7.3. The performance specifications require that τ = 2 and ζ = 0.707. a. Compute the required gain values. b. Use a
Consider the PD control system shown in Figure 10.7.1. Suppose that I = 15 and c = 3. The specifications require the steady-state error due to a unit-step command to be zero and the steady-state
For the PD control system shown in Figure 10.7.1, I = 25 and c = 5. Compute the values of the gains KP and KD to meet all of the following specifications: 1. No steady-state error with a step
We need to stabilize the plant 3/(s2 - 4) with a feedback controller. The closed-loop system should have a damping ratio of ζ = 0.707 and a dominant time constant τ =0.1. a. Use PD control and
In Figure, the block is pulled up the incline by the tension force f in the inextensible cable. The motor torque T is controlled to regulate the speed v of the block to obtain some desired speed vr.
The system shown in Figure represents the problem of stabilizing the attitude of a rocket during takeoff or controlling the balance of a personal transporter. The applied force f represents that from
Figure shows PD control applied to an unstable plant. The gains have been computed so that the damping ratio is Î¶ = 0.707 and the time constant is 2.5 sec, assuming that the transfer functions of
Figure shows PD control applied to an unstable plant. The gains have been computed so that the damping ratio is Î¶ = 0.707 and the time constant is 2.5 sec, assuming that the transfer functions of
Figure shows a proposed scheme for controlling the position of a mechanical system such as a link in a robot arm. It uses two feedback loops-one for position and one for velocity-and a feed forward
Refer to Figure 10.3.9, which shows a speed control system using an armature-controlled dc motor. The motor has the following parameter values. Kb = 0.199 V-sec/rad Ra = 0.43Ω KT = 0.14 lb-ft/A ce =
Using the value of Kp computed in Problem 10.18, obtain a plot of the current versus time for a step-command input of 209.4 rad/s (2000 rpm). In Problem 18 KT = Kb = 0.2 N.m/A cm = 5 x 10-4
Consider Example 10.6.3. Modify the diagram in Figure 10.6.2 to show an actuator transfer function T(s)/M(s) = 1/(0.1s + 1). Use the same gain values computed for the three cases in that example.a.
Consider Example 10.6.3. Use the same gain values computed for the three cases in that example. a. Use MATLAB to plot the command response and the actuator response to the modified unit-step command
Consider Example 10.6.4. Modify the diagram in Figure 10.6.6 to show an actuator transfer function T(s)/M(s) = 1/(0.1s + 1). Use the same gain values computed for the three cases in that example.a.
Figure shows a system for controlling the angular position of a load, such as an antenna. Figure shows the block diagram for PD control of this system using a field-controlled motor. Use the
The diagram in Figure shows a system for controlling the angular position of a load, such as an antenna. There is no disturbance.a. Draw the block diagram of a system using proportional control,
Consider the P, PI, and modified I control systems discussed in Examples 10.6.2, 10.6.3, and 10.6.4. The plant transfer function is 1/(Is + c), where I = 10 and c = 3. Investigate the performance of
A speed control system using an armature-controlled motor with proportional control action was discussed in Section 10.3. Its block diagram is shown in Figure 10.3.8 with a simplified version given
Consider the control system of Problem 71. Use MATLAB to evaluate the following performance measures: energy consumption, maximum current, maximum speed error, rms current, and rms speed error. In
Consider Example 10.6.3. Use the diagram in Figure 10.6.2 to create a Simulink model. Modify the model to use actuator saturation with the limits 0 and 20. Use the same gain values computed in that
Consider Example 10.7.4. Use the diagram in Figure 10.7.3 to create a Simulink model using the same gain values computed in that example. Set the initial position to 3. Plot the command response to a
Consider Example 10.7.4. Use the diagram in Figure 10.7.3 to create a Simulink model. Modify the model to use actuator saturation with the limits 0 and 20. Use the same gain values computed in that
Consider Example 10.7.4. Use the diagram in Figure 10.7.3 to create a Simulink model. Modify the model to use an actuator transfer function Ga(s) = 1 /(0.2s + 1). Use the same gain values computed in
Refer to Figure 10.3.9, which shows a speed control system using an armature-controlled dc motor. The motor has the following parameter values. Create a Simulink model by modifying Figure 10.3.9 to
For the system in Problem 77 part (a), create a Simulink model that has a current limiter of ±10 A. Run the simulation for a step-command input of 104.7 rad/s (1000 rpm). Plot the current and the
In the following controller transfer function, identify the values of Kp, KI, Kp, T1, and TD. Ge(s) = F(s)/E(s) = (15s2 + 6s + 4)/s
Determine the resistance values required to obtain an op-amp PI controller with Kp = 4 and Kt = 0.08. Use a 1-μF capacitor.
Sketch the root locus plot of 3s2 + 12s + k = 0 for k ≥ 0. What is the smallest possible dominant time constant, and what value of k gives this time constant?
In the following equation, K ≥ 0. s2(s + 9) + K(s+ 1) =0 Obtain the root locus plot. Obtain the value of K at the breakaway point, and obtain the third root for this value of K. What is the
Consider the following equation where the parameter K is nonnegative. (2s + 5)(2s2 + 14s + 49) + Ks (2s +1)(2s + 3) = 0 Determine the poles and zeros, and sketch the root locus plot. Use the plot to
In the following equations, identify the root locus plotting parameter K and its range in terms of the parameter p, where p > 0.a. 9s3 + 6s2-5ps + 2 = 0b.
In parts (a) through (f), Obtain the root locus plot for K ≤ 0 for the given characteristic equation.s(s + 5) + K = 0s2 + 3s + 3 + K(s + 3) = Qs(s2 + 3.v + 3) + K = 0s(s + 5)(s + 7) + K =0s(s + 3)
The plant transfer function for a particular process is G p (s) = 8 – s / s2 + 2s + 3. We wish to investigate the use of proportional control action with this plant.a. Obtain the root locus and
The plant transfer function for a particular process isGp (s) = 26 + s – 2s2 / s(s + 2) (s + 3)We wish to investigate the use of proportional control action with this plant.a. Obtain the root
Control of the attitude q of a missile by controlling the tin angle f, as shown in Figure P11.16, involves controlling an inherently unstable plant. Consider
The use of a motor to control the rotational displacement of an inertia I is shown in Figure P11.17. The open-loop transfer function of the plant for a specific application isGp (s) = 6 / s (2s + 2)
Proportional control action applied to the heat flow rate qi can be used to control the temperature of the oven shown in Figure P11.18. Consider the specific plantGp (s) = TI(s) / Qi (s) = s + 10 /
Proportional control action applied to the flow rate qmi can be used to control the liquid height, as shown in Figure P11.19. Consider the specific plantGp (s) = H2(s) / Qmi (s) = 5 (s + 4) / (s + 3)
Sketch the root locus plot of 3s2 + cs + 12 = 0 for c ≥ 0. What is the smallest possible dominant time constant, and what value of c gives this time constant? What is the value of ωn if ζ < 1?
Proportional control action applied to the flow rate qmi can be used to control the liquid height of the system shown in Figure P11.20. Consider the specific plant Gp (s) = H2 (s) / Qmi (s) = 1 / s2
Design a PID controller applied to the motor torque T to control the robot arm angle ( Shon in Figure P11.21 Consider the specific plant Gp (s) = ( (s) / T (s) = 4 / 3s2 + 3 The dominant closed-loop
a) The equations of motion of the inverted pendulum model were derived in Example 3.5.6 in Chapter 3. Linearize these equations about ( = 0, assuming that ( is very small.b) Obtain the linearized
Use of a motor to control the position of a certain load having inertia, damping, and elasticity gives the following plant transfer function. See Figure P11.23.Gp (s) = ( (s) / V (s) = 0.5 / (s2 + s
Figure P11.24 shows an electro hydraulic position control system whose plant transfer function for a specific system is Gp (s) = Y (s) / F (s) = 5 / 2s3 + 10s2 + 2s + 4 a. Use the ultimate cycle
A certain plant has the transfer function Gp (s) = 4p / (s2 + 4(s + 4) (s + p) Where the nominal values of ( and p are ( = 0.5 and p = 1. a. Use Ziegler-Nichols tuning to compute the PID gains.
The plant transfer function of the system in Figure P11.26 for a specific case isGp (s) = 8 / (2s + 2) (s + 2) (4s + 12)a. Use the ultimate cycle method to compute the PID gains.b. Plot the unit-step
Consider the PI-control system shown in Figure P11.27 where I = 5 and c = 0. It is desired to obtain a closed-loop system having ( = 1 and ( = 0.1. Let mmax = 20 and rmax = 2. Obtain Kp and KI.
Consider the PI-control system shown in Figure P11.27 where I = 10 and c = 20. It is desired to obtain a closed-loop system having ( = 1 and ( = 0.1.a. Obtain the required values of KP and KI
Consider the PI-control system shown in Figure P11.27 where I=7 and c = 5. It is desired to obtain a closed-loop system having z = 1 and t = 0.2. Let mmax
Sketch the root locus of the armature-controlled dc motor model in terms of the damping constant c, and evaluate the effect on the motor time constant. The characteristic equation is LaIs2 + (RaI +

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