Questions and Answers of Classical Dynamics Of Particles

a. Design a PI and an I controller with internal feedback for the plant Gp(s) = 1/4s. See Figure PI 1.30. We are given that mmax = 6 and rmax = 3. Set (= 1.b. Evaluate the unit-step response of each
Compare the performance of the critically damped controllers shown in Figure P11.30 with the plant Gp(s) = 1 / I s having the following inputs: a. A unit-ramp disturbance b. A sinusoidal
A certain field-controlled de motor with load has the following parameter values. L = 2 × 10-3 H R = 0.6 Ω KT = 0.04N · m/A C = 0 I = 6 × 10-5 kg · m2 Compute the gains for a state variable
In Figure P11.33 the input u is an acceleration provided by the control system and applied in the horizontal direction to the lower and of the rod. The horizontal displacement of the lower and is y.
Figure P11.34 illustrates an active vibration control scheme for a two-mass system. An electro hydraulic actuator between the two masses provides a force that acts on both and is under feedback
Figure P11.35 a is the circuit diagram of a speed-control system in which the dc motor voltage va is supplied by a generator driven by an engine. This system has been used on locomotives whose diesel
The following equations are the model of the roll dynamics of a missile ([Bryson, 1975]). See Figure P11.36.Figure P11.36WhereÎ´ = aileron deflectionb = aileron effectiveness constantu = command
Many winding applications in the paper, wire, and plastic industries require a control system to maintain proper tension. Figure P11.37 shows such a system winding paper onto a roll. The paper
An electro-hydraulic positioning system is shown in Figure P11.38. Use the following values.Ka = 10 V/A Ki = 10–2 in./V K2
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
Sketch the root locus plot of ms2 + 12s + 10 = 0 for m ≥ 2. What is the smallest possible dominant time constant, and what value of m gives this time constant?
The following table gives the measured open-loop response of a system to a unit-step input. Use the process reaction method to find the controller gains for P. PL and PID control. Time
A liquid in an industrial process must be heated with a heat exchanger through which steam passes. The exit temperature of the liquid is controlled by adjusting the rate of flow of steam through the
Use MATLAB to obtain the root locus plot of 5s2 + cs + 45 = 0 for c ≥ 0.
Use MATLAB to obtain the root locus plot of the system shown in Figure P11.43 in terms of the variable k ≥ 0. Use the values m = 4 and c = 8. What is the smallest possible dominant time constant
Use MATLAB to obtain the root locus plot of the system shown in Figure P11.43 in terms of the variable c ≥ 0. Use the values m = 4 and k = 64. What is the smallest possible dominant time constant
Use MATLAB to obtain the root locus plot of the system shown in Figure P11.45 in terms of the variable k2 ‰¥ 0. Use the values m = 2, c = 8, and k1 = 26. What is the value of k2 required to
Use MATLAB to obtain the root locus plot of the system shown in Figure P11.46 in terms of the variable c2 ≥ 0. Use the values m = 2, C1 = 8, and k = 26. What is the smallest possible dominant time
Use MATLAB to obtain the root locus plot of s3 + 13s2 + 52s + 60 + K = 0 for K ≥ 0. Is it possible for any dominant roots of this equation to have a damping ratio in the range 0.5 ≤ ( ≥ 0.707
(a) Use MATLAB to obtain the root locus plot of 2s3 + 12s2 + 16s + K = 0 for K ≥ 0. (b) Obtain the value of K required to give a dominant root pair having ( = 0.707. (c) For this value of K. obtain
Use MATLAB to obtain 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
In the following equations, identify the root locus plotting parameter K and its range in terms of the parameter p, where p ≥ 0. 6s2 + 8s + 3p = 0 3s2 + (6 + p)s + 5 + 2p = 0 4s3 + 4ps2 + 2s + p = 0
Consider the two-mass model shown in Figure P11.50. Use the following numerical values: m1 = m2 = 1, k1 = 1, k2 = 4, and c2 = 8.a. Use MATLAB to obtain the root locus plot in terms of the parameter
Consider the equations3 + 10s2 + 24s + K = 0a. Use MATLAB to obtain the value of K required to give dominant roots with ( = 0.707. Obtain the three roots corresponding to this value of K.b. Use
Consider the equations3 + 9s2 + (8 + K)s + 2 K = 0a. Use MATLAB to obtain the value of K required to put the dominant root at the breakaway point. Obtain the three roots corresponding to this value
Consider the equation s3 + 10s2 + 24s + K = 0 Use the sgrid function to determine if it is possible to obtain a dominant root having a damping ratio in the range 0.5 ≤ ( ≥ 0.707, and an un-damped
In Example 10.7.4 the steady-state error for a unit-ramp disturbance is 1/KI.For the gains computed in that example, this error is 1 /25. We want to see if we can make this error smaller by
In Example 10.8.3 the steady-state error for a unit-ramp command is -4/KI. For the gains computed in that example, this error is 1 /1000. We want to see if we can make this error smaller by
With the PI gains set to Kp = 6 and K1 = 50 for the plant Gp (s) = 1 / s + 4 The time constant is ( = 0.2 and the damping ratio is ( = 0.707. a. Suppose the actuator saturation limits are ( 5.
Consider a unity feedback system with the plant Gp(s) and the controller Gc(s). PID control action is applied to the plant Gp(s) = s + 10 / (s + 1) (s + 2) The PID controller has the transfer
With the PI gains set to Kp = 6 and K1 = 50 for the plant Gp (s) = 1 / s + 4 The time constant is ( = 0.2 and the damping ratio is ( = 0.707. Suppose there is a rate limiter of ± 0.1 between the
A certain dc motor has the following parameter values:L = 2 Ã— 10-3 HR = 0.6 Î©KT = 0.04 N · m/Ac = 0I = 6 Ã— 10-5 kg · m2Figure P11.61Figure P11.61 shows an integral controller using
Consider the liquid-level controller designed in Example 10.10.1, whose Simulink diagram is shown in Figure 10.10.1. Modify the model to include a Rate Limiter block to limit the rate of q1, in front
In parts (a) through (f), sketch the root locus plot for the given characteristic equation for K ≥ 0.s(s + 5) + K .= 0s(s + 7)(s + 9) + K = 0s2 + 3s + 5 + K(s + 3) = 0s(s + 4) + K(s + 5) = 0s(s2 +
PID control action is applied to the plant Gp(s) = s + 10/ (s + 2) (s + 5) The PID controller has the transfer function Gc(s) = Kp (1 + 1/TIs + TDs) Use the values TI = 0.2 and TD = 0.5. Plot the
Consider the following equation where the parameter p is nonnegative. 4s3 + (25 + 5p)s2 + (16 + 30p)s + 40p = 0 Put the equation in standard root locus form and define a suitable root locus parameter
Control of the attitude 0 of a missile by controlling the fin angle cp. as shown in Figure P12.1, involves controlling an inherently unstable plant. Consider the specific plant transfer functionGp(s)
Consider a plant whose open-loop transfer function is G(s) H (s) = 1 / s [(s + 2)2 + 9] The complex poles near the origin give only slightly damped oscillations that are considered undesirable.
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
A certain unity feedback system has the following open-loop system transfer function.G(s) = 5K / s3 + 6s2 + 5sObtain the Bode plots and compute the phase and gain margins fora. K = 2b. K =20c. Use
Figure P12.13 shows a pneumatic positioning system, where the displacement x is controlled by controlling the applied pneumatic pressure p1. Assume thatFigure P12.13The pressure p2 is constant, and
The height h2 in Figure P12.14 can be controlled by adjusting the flow rate q1. Consider the specific plantGp (s) = H2 (s) / Q1 (s) = 25 / 5s2 + 6s + 5With the following series PI compensator is used
Rolling motion of a ship can be reduced by using feedback control to vary the angle of the stabilizer fins, much like ailerons are used to control aircraft roll. Figure PI2.15 is the block diagram of
The following transfer functions are the forward transfer function G(s) and the feedback transfer function H(s) for a system whose closed-loop transfer function is G (s) / 1 + G (s) H (s) For each
Remote control of systems over great distance, such as required with robot space probes, may involve relatively large time delays in sending commands and receiving data from the probe. Consider a
Hot-air heating control systems for large buildings may involve significant dead time if there is a large distance between the furnace and the room beingHeated (Figure P12.19). Proportional control
Figure PI2.2 shows a pneumatic positioning system, where the displacement x is controlled by varying the pneumatic pressure P1. Assume that the pressure P2 is constant, and consider the specific
The block diagram of a position control system is shown in Figure P12.20. Design a compensator for the particular plantGp (s) = 1 / s(s2 + 3s + 2)So that the static velocity error coefficient will be
The speed wi of the load is to be controlled with the torque T acting through a fluid coupling (see Figure P12.5). Design a compensator for the specific plant Gp (s) = Ω2 (s) / T (s) = 4 / s2 +
Design a compensator for the plant Gp (s) = 2 / s2 + 2s So that the static velocity error coefficient will be Cu = 20/sec and the phase margin at least 45o.
Figure P12.2 shows a pneumatic positioning system, where the displacement x is controlled by controlling the applied pneumatic pressure p1. Assume that the pressure p2 is constant, and consider the
The block diagram of a position control system is shown in Figure P12.7. Design a compensator for the particular plantGp (s) = 1 / s (s + 5) (s + 1)That will give a static velocity error coefficient
The block diagram of a position control system is shown in Figure P12.7. Design a compensator for the particular plant Gp (s) = 1 / s (s2 + 3s + 2) So that the static velocity error coefficient will
Consider a unity-feedback system having the open-loop transfer functionG(s) = (2n / s(s + 2 ( (n)Derive the following expression for this system's phase margin
Automatic guided vehicles are used in factories and warehouses to transport materials. They require a guide path in the floor and a control system for sensing the guide path and adjusting the
With the PI gains set to Kp = 6 and Kt = 50 for the plant Gp (s) = 1 / s + 4 The time constant is ( = 0.2 and the damping ratio is ( = 0.707. a. Compute the gain and phase margins. b. Suppose there
It is desired to control the angular displacement ( of a space vehicle by controlling the applied torque T supplied by thrusters (Figure PI2.3). The plant model is Gp (s) = ( (s) / T (s) =
12.30 In Example 12.1.4 a lag-lead compensator was designed by canceling the plant pole at s = -3 with a compensator zero. Suppose the plant model is slightly inaccurate and the plant pole is really
When proportional control is applied to the following plant using a gain of Kp = 1, the closed-loop roots arc satisfactory, but the static velocity error coefficient must be increased to Cu =
The speed (1 of the load is to be controlled with the torque Td acting through a fluid coupling (Figure P12.5). Design a compensator for the specific plant Gp(s) = Ω1 (s) / Td (s) = 1 / s2 + s The
The block diagram of a speed control system is shown in Figure P12.6. For a particular system with proportional control, G1(s) = KP, the open-loop transfer function isG(s) = Kp / s (s + 2)With Kp =
The block diagram of a position control system is shown in Figure P12.7. For a particular system with proportional control, G1 (s) = Kp, the open-loop transfer function isG (s) = 2.5Kp / s (s + 2)
It is desired to control the angular displacement ( of a space vehicle by controlling the applied torque T supplied by thrusters (Figure PI2.8a). The plant model isGp (s) = ( (s) / T (s) = 10 /
The plant transfer function for the angular displacement ( of an inertia / subjected to a control torque T is (see Figure PI 2.8a) Gp (s) = ( (s) / T (s) = 1 / Is2 Suppose that I = 5 and that the
The 0.5-kg mass shown in Figure P13.1 is attached to the frame with a spring of stiffness k = 500 N/m. Neglect the spring weight and any damping. The frame oscillates vertically with an amplitude of
Alternating-current motors are often designed to run at a constant speed, typically either 1750 or 3500 rpm. One such motor for a power tool weighs 20 lb and is to be mounted at the end of a steel
When a certain motor is started, it is noticed that its supporting frame begins to resonate when the motor speed passes through 900 rpm. At the operating speed of 1750 rpm the support oscillates with
A 500-lb motor is supported by an elastic pad that deflects 0.25 in. when the motor is placed on it. When the motor operates at 1750 rpm, it oscillates with an amplitude of 0.1 in. Suppose a 1500-lb
A certain pump weighs 50 lb and has a rotating unbalance. The unbalanced weight is 0.05 lb and has an eccentricity of 0.1 in. The pump rotates at 1000 rpm. Its vibration isolator has a stiffness of k
To calculate the effects of rotating unbalance, we need to know the value of the product mu€, where mu is the unbalanced mass and € is the eccentricity. These two quantities are sometimes
A computer disk drive is mounted to the computer's chassis with an isolator consisting of an elastic pad. The disk drive motor weighs 3 kg and runs at 3000 rpm. Calculate the pad stiffness required
Figure P13.16 shows a motor mounted on four springs (the second pair of springs is behind the front pair and is not visible). Each spring has a stiffness k = 2000 N/m. The distance D is 0.2 m. The
A motor mounted on a cantilever beam weighs 20 lb and runs at the constant speed of 3500 rpm. The steel beam is 6 in. long, 4 in. wide, and 3/8 in. thick. The unbalanced part of the motor weighs 1 lb
A motor mounted on a beam vibrates too much when it runs at a speed of 6000 rpm. At that speed the measured force produced on the beam is 60 lb. Design a vibration absorber to attach to the beam.
The supporting table of a radial saw weighs 160 lb. When the saw operates at 200 rpm it transmits a force of 4 lb to the table. Design a vibration absorber to be attached underneath the table. The
A quarter-car representation of a certain car has a stiffness k = 2000 lb/ft, which is the series combination of the tire stiffness and suspension stiffness, and a damping constant of c = 360
A certain machine of mass 8 kg with supports has an experimentally determined natural frequency of 6 Hz. It will be subjected to a rotating unbalance force with an amplitude of 50 N and a frequency
The operating speed range of a certain motor is from 1500 to 3000 rpm. The motor and its mount vibrate excessively at 2100 rpm. When a vibration absorber weighing 5 lb and tuned to 2100 rpm was
Figure P13.22 shows another type of vibration absorber that uses only mass and damping, and not stiffness, to reduce vibration. The main mass is m1 and the absorber's mass is m2. Suppose the applied
Figure P13.23 shows another type of vibration absorber that uses mass, stiffness, and damping to reduce vibration. The damping can be used to reduce the amplitude of motion near resonance. The main
Find and interpret the mode ratios for the system shown in Figure P13.24. The masses are m1 = 10 kg and m2 = 30 kg. The spring constants are k11 = 104 N/m and k2 = 2Ã—104 N/m.
Find and interpret the mode ratios for the coupled pendulum system shown in Figure P13.25. Use the values m1 = 1, m2 = 4, L1 = 2, L2 =5, and k = 2.
Find and interpret the mode ratios for the torsional system shown in Figure P13.26. Use the values I1 = 1, I2 = 5, k1 = 1, and k2 = 3.
Find and interpret the mode ratios for the system shown in Figure P13.27.
For the roll-pitch vehicle model described in Example 13.4.2, the suspension stiffnesses are to be changed to k1 = 1.95 × 104 N/m and k2 = 2.3 × 104 N/m. Find the natural frequencies the mode
A particular road vehicle weighs 4000 lb. Using the quarter-car model, determine a suitable value for the suspension stiffness, assuming that the tire stiffness is 1300 lb/in.
A certain factory contains a heavy rotating machine that causes the factory floor to vibrate. We want to operate another piece of equipment nearby and we measure the amplitude of the floor's motion
The vehicle model shown in Figure 13.4.2(a) has the following parameter values: weight = 4800 lb, 1G = 1800 slug-ft2, L1 = 3.5 ft, and L2 = 2.5 ft. Design the front and rear suspension stiffnesses to
A 125-kg machine has a passive isolation system for which c = 5000 N-m/s and k = 7 × 106 N/m. The rotating unbalance force has an amplitude of 100 N with a frequency of 2500 rpm. The resonant
A 20-kg machine has a passive isolation system whose damping ratio is 0.28 and whose un-damped natural frequency is 13.2 rad/s. Assuming that the passive system remains in place, calculate the
With the increased availability of powered wheelchairs, improved suspension designs are required for safety and comfort. One chair uses an active suspension like the one shown in Figure P13.33 for
The following model describes a mass supported by a nonlinear spring. The units are SI, so g = 9.81 m/s2. 5 = 5g - (900y + 1700y3) Find the equilibrium position yr, obtain a linearized model using
The following is a model of the velocity of an object subjected to cubic damping. m du / dt = - cu3
Find the equilibria of equation (13.6.10) for m = l, c = 12, k1 = 16, and k3 = -4, and use a numerical method to solve and plot the solution for (0) = 0 and four values of y(0): ± 1, ± 1.9, and ±
Use a numerical method to compute and plot the free response of equation (13.6.10) for m = 1, c = 0, k1 = 2, and k3 = 0.1, for (0) = 0 and two initial conditions: y(0) = 10 and y(0) = 40. Compare
Plot the phase plane plots for the following equations and the initial conditions: x(0) = l, (0) = 0.a. + 0.1 + 2x = 0b. + 2+ 2x = 0c. + 4+ 2x = 0
Plot the phase plane plot for the following equation with the initial conditions: y(0) = l, (0) = 0.+ 2+ 2y + 3y3 =0
An electronics module inside an aircraft must be mounted on an elastic pad to protect it from vibration of the airframe. The largest amplitude vibration produced by the airframe's motion has a
Plot the phase plane plot and identify the limit cycle for Van der Pol's equation (13.6.13) with μ = 5 and the initial conditions: y(0) = 1, (0) = 0.
An electronics module used to control a large crane must be isolated from the crane's motion. The module weighs 2 lb. (a) Design an isolator so that no more than 10% of the crane's motion amplitude

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