Consider the two-mass system in Figure P2.51. Find the set of differential equations

Very few physical systems are linear within some range of the variables.

The s-plane plot of the poles and zeros graphically portrays the character True or False

The roots of the characteristic equation are the zeros of the closed-loop system.

A linear system satisfies the properties of superposition and homogeneity.

The transfer function is the ratio of the Laplace transform of the output variable to the Laplace transform of the input variable, with all initial conditions equal to zero.

Consider the system in Figure 2.79 where Gc(*) = 10, H(s) = l, and G(s) = s + ** s* + 60.y + 500 If the input R(s) is a unit step input, Td(s) = 0, and N(s) = 0, the final value of the output Y(s) is: a. yss = limy(t) = 100 f-CO b. yss = lira y(t) = 1 t*oo c. yss = lim y(t) = 50 f~0O d. None of the above

Consider the system in Figure 2.79 with Gc(s) = 20, H{s) = 1, and G(s) = s + 4 5 2 - 125 - 65 When all initial conditions are zero, the input R(s) is an impulse, the disturbance Td(s ) ~ 0, and the noise N(s) = 0, the output y(t) is a. y(t) = 10e~5 ' + 10e~3 ' b. y(t) = e'* + 10e~' c. y(t) = 10e~3 ' - 10e_5r d. y(t) = 20e-8' + 5e~15'

Consider a system represented by the block diagram in Figure 2.80. R(s) FIGURE 2.80 Block diagram with an internal loop. The closed-loop transfer function T(s) = Y(s)/R(s) is s 2 s 2 + 55s + 50 10 + 555 + 10 10 b. T(s) = c. T(s) = , v 52 + 505 + 55 d. None of the above

The closed-loop transfer function T(s) = Y(s)/R(s) is: TV ^ 5 0 a. T(s) = b. T(s) = c. T(s) = d. T(s) = s 2 + 10s + 5' s 2 s 2 s 2 s 2 + 5s + 50 20 + 105 + 25 50 + 55 + 56 20 + 105 - 15

The closed-loop unit step response is: a. y{t) =i r + 20 20 _, 5' _ t2e-5t 25 25 b. y{t) = 1 + 20re~5 ' ~5t - 4te~ , s 20 20 5t - 3 M = 25-25 e d. y(t) = 1 - 2e~5 ' - Ate'5 '

The final value of y(t) is: a. y = lim y(t) = 0.8 r-*oo b. yw = lim y(f) = 1.0 f-oo c. y = Kmy(t) = 2.0 /->oo d. yss = lim y(t) = 1.25

Consider the differential equation y + 2y + y = u where y(0) = y(0) = 0 and u(t) is a unit step. The poles of this system are: a. s-i = -1, = - 1 b. 5! = 1/, 52 = -1 ; C S-i = -1 , 52 = 2 d. None of the above

A cart of mass m = 1000 kg is attached to a truck using a spring of stiffness k = 20,000 N/m and a damper of constant 6 = 200 Ns/m, as shown in Figure 2.81. The truck moves at a constant acceleration of a = 0.7 m/s2 . FIGURE 2.81 Truck pulling a cart of mass m. The transfer function between the speed of the truck and the speed of the cart is: 50 a. T(s) = b. T(s) - c. T(s) = 552 + s + 100 20 + s ^ + 105 + 25 100 + 5 5s2 + s + 100 d. None of the above

Consider the closed-loop system in Figure 2.79 with 1000 Gc(s) = 15, H(s) = 1, and G(s) = s 3 + 50s2 + 45005 + 1000' Compute the closed-loop transfer function and the closed-loop zeros and poles. 15000 a. T(s) = - ,5! = -3.70,52 3 = -23.15 61.59/ 5 3 + 5052 + 45005 + 16000 ' ' . ^, N 15000 n,nn b. 7/(5) = , Si = -3.70,5, = -86.29 5052 + 45005 + 16000 1 c. T(5) = -r 5 ,5X = -3.70,52 , = -23.2 63.2/ ' 53 + 5052 + 45005 + 16000 d. 7/(5) = 1_292 s = _3>7 0 s = -23.2, s3 = -63.2

Consider the feedback system in Figure 2.79 with K(s + 0.3) 1 Gc(s) = -, H{s) = 2s, and G(s) = (5 - 2)(52 + 10s + 45)' Assuming R(s) = 0 and N(s) = 0, the closed-loop transfer function from the disturbance 7/rf(5) to the output Y(s) is: a. b. d. Y(s) Td(s) Y(s) Td(s) Y(s) Td(s) Y(s) 1 53 + 8^ + (2 ^ + 25)5 + (0.6K - 90) 100 5 3 + 852 + (2K + 25)5 + (0.6K - 90) 1 852 + (2K + 25)5 + (0.6K - 90) K(s + 0.3) Td(s) s 4 + 853 + {2K + 25)52 + {0.6K - 90)5

A unity, negative feedback system has a nonlinear function y = /(e) = e 2 , as shown in Figure E2.1. For an input r in the range of 0 to 4, calculate and plot the openloop and closed-loop output versus input and show that the feedback system results in a more linear relationship.

A thermistor has a response to temperature represented by R = /^-01- ^ where R0 = 10,000 ft, R = resistance, and T = temperature in degrees Celsius. Find the linear model for the thermistor operating at T = 20C and for a small range of variation of temperature. Answer: AR = -135AF

The force versus displacement for a spring is shown in Figure E2.3 for the spring-mass-damper system of Figure 2.1. Graphically find the spring constant for the equilibrium point of y = 0.5 cm and a range of operation of 1.5 cm.

A laser printer uses a laser beam to print copy rapidly for a computer. The laser is positioned by a control input r(t), so that we have 7(5) - 4(s + 50) s 2 + 30s + 200 R(s). The input r(t) represents the desired position of the laser beam. (a) If r(t) is a unit step input, find the output y(t). (b) What is the final value of y{t)l Answer: (a) y{t) = 1 + 0.6

A noninverting amplifier uses an op-amp as shown in Figure E2.5. Assume an ideal op-amp model and determine v0/vm. Answer: - 1 +

A nonlinear device is represented by the function y = /{X) = e*, where the operating point for the input x is x0 = 1. Determine a linear approximation valid near the operating point. Answer: y = ex

A lamp's intensity stays constant when monitored by an optotransistor-controlled feedback loop. When the voltage drops, the lamp's output also drops, and optotransistor Q\ draws less current. As a result, a power transistor conducts more heavily and charges a capacitor more rapidly [24]. The capacitor voltage controls the lamp voltage directly. A block diagram of the system is shown in Figure E2.7. Find the closed-loop transfer function, I(s)!R(s) where I{s) is the lamp intensity, and R(s) is the command or desired level of light.

A control engineer, N. Minorsky, designed an innovative ship steering system in the 1930s for the U.S. Navy. The system is represented by the block diagram shown in Figure E2.8, where Y(s) is the ship's course, /?(.?) is the desired course, and A(s) is the rudder angle [16]. Find the transfer function Y(s)IR(s).

A four-wheel antilock automobile braking system uses electronic feedback to control automatically the brake force on each wheel [15]. A block diagram model of a brake control system is shown in Figure E2.9, where iy(s) and FR(s) are the braking force of the front and rear wheels, respectively, and R{s) is the desired automobile response on an icy road. Find Ff(s)/R(s).

One of the most potentially beneficial applications of an automotive control system is the active control of the suspension system. One feedback control system uses a shock absorber consisting of a cylinder filled with a compressible fluid that provides both spring and damping forces [17].The cylinder has a plunger activated by a gear motor, a displacement-measuring sensor, and a piston. Spring force is generated by piston displacement, which compresses the fluid. During piston displacement, the pressure unbalance across the piston is used to control damping. The plunger varies the internal volume of the cylinder. This feedback system is shown in Figure E2.10. Develop a linear model for this device using a block diagram model.

A spring exhibits a force-versus-displacement characteristic as shown in Figure E2.ll. For small deviations from the operating point x0, find the spring constant when x0 is (a) -1.4; (b) 0; (c) 3.5.

Off-road vehicles experience many disturbance inputs as they traverse over rough roads. An active suspension system can be controlled by a sensor that looks "ahead" at the road conditions. An example of a simple suspension system that can accommodate the bumps is shown in Figure E2.12. Find the appropriate Plunger Damping orifice Piston rod Piston travel FIGURE E2.10 Shock absorber. FIGURE E2.11 Spring characteristic. gain Kx so that the vehicle does not bounce when the desired deflection is R{s) = 0 and the disturbance is Us).

Consider the feedback system in Figure E2.13. Compute the transfer functions Y(s)/Td(s) and Y(s)/N(s).

Find the transfer function Yj(s) R2(s) for the multivariate system in Figure E2.14.

Obtain the differential equations for the circuit in Figure E2.15 in terms of ^ and i2.

The position control system for a spacecraft platform is governed by the following equations: d 2 p dp f + 2-p + 4p = B dt2 dt Vi=r- p - = 06 * V2 = lV\. The variables involved are as follows: r(t) = desired platform position p{t) = actual platform position V\{l) = amplifier input voltage v2(t) = amplifier output voltage 0(/) = motor shaft position Sketch a signal-flow diagram or a block diagram of the system, identifying the component parts and determine the system transfer function P(s)/R(s).

A spring develops a force /represented by the relation / = kx2 , where x is the displacement of the spring. Determine a linear model for the spring when x0 = j -

The output y and input x of a device are related by y = x + 1.4x3 . (a) Find the values of the output for steady-state operation at the two operating points x0 = 1 and x0 = 2. (b) Obtain a linearized model for both operating points and compare them.

The transfer function of a system is Y(s) _ 15(.f + 1) R(s) ~ s 2 + 9s + 14' Determine y{t) when r(t) is a unit step input. Answer: y(t) = 1.07 + lie - * - 2.57e-7', t s 0

Determine the transfer function VQ(s)/V{s) of the operational amplifier circuit shown in Figure E2.20. Assume an ideal operational amplifier. Determine the transfer function when /?, = R2 = 100 kfl, Cx = 10 jttF, and C2 = 5 fiF.

A high-precision positioning slide is shown in Figure E2.21. Determine the transfer function Xp(s)/Xm(s) when the drive shaft friction is bd = 0.7, the drive shaft spring constant is kd = 2, mc = 1, and the sliding friction is bs = 0.8.

The rotational velocity &> of the satellite shown in Figure E2.22 is adjusted by changing the length of the beam L. The transfer function between

Determine the closed-loop transfer function T(s) = Y(s)/R(s) for the system of Figure E2.23.

The block diagram of a system is shown in Figure E2.24. Determine the transfer function T(s) = Y(s)/R(s).

An amplifier may have a region of deadband as shown in Figure E2.25. Use an approximation that uses a cubic equation y = ax 3 in the approximately linear region. Select a and determine a linear approximation for the amplifier when the operating point is JC = 0.6.

Determine the transfer function X2(s)/F(s) for the system shown in Figure E2.26. Both masses slide on a frictionless surface, and k = 1 N/m.

Find the transfer function Y(s)/Td(s) for the system shown in Figure E2.27. Answer: Y(s) G^s) Td(s) 1 + G,(s)G2(s)H(s)

Determine the transfer function \&(s)/V(s) for the op-amp circuit shown in Figure E2.28 [1]. Let /?j = 167 kfl, R2 = 240 kH, R3 = 1 kH, RA = 100 kH, and C = 1 /iF. Assume an ideal op-amp.

A system is shown in Fig. E2.29(a). (a) Determine G(s) and H(s) of the block diagram shown in Figure E2.29(b) that are equivalent to those of the block diagram of Figure E2.29(a). (b) Determine Y(s)/R(s) for Figure E2.29(b).

A system is shown in Figure E2.30. (a) Find the closed-loop transfer function Y(s)/R(s) 10 when G(s) = -: . s 2 + 2s + 10 (b) Determine Y(

Determine the partial fraction expansion for V(s) and compute the inverse Laplace transform. The transfer function V(s) is given by: V(s) 400 s 2 + Ss + 400

An electric circuit is shown in Figure P2.1. Obtain a set of simultaneous integrodifferential equations representing the network.

A dynamic vibration absorber is shown in Figure P2.2. This system is representative of many situations involving the vibration of machines containing unbalanced components. The parameters M2 and kl2 may be chosen so that the main mass Mi does not vibrate in the steady state when F(t) = a sin(a>0f)- Obtain the differential equations describing the system.

A coupled spring-mass system is shown in Figure P2.3. The masses and springs are assumed to be equal. Obtain the differential equations describing the system.

A nonlinear amplifier can be described by the following characteristic: "o(') = J 4 I 4 m < 0' Tlie amplifier will be operated over a range of 0.5 volts around the operating point for vin. Describe the amplifier by a linear approximation (a) when the operating point is sjj,, = 0 and (b) when the operating point is win = 1 volt. Obtain a sketch of the nonlinear function and the approximation for each case.

Fluid flowing through an orifice can be represented by the nonlinear equation Q = K(P, - A)"2 , where the variables are shown in Figure P2.5 and K is a constant [2]. (a) Determine a linear approximation for the fluid-flow equation, (b) What happens to the approximation obtained in part (a) if the operating point is Pi - P2 = 0?

Using the Laplace transformation, obtain the current I2(s) of Problem P2.1. Assume that all the initial currents are zero, the initial voltage across capacitor C\ is zero, v{t) is zero, and the initial voltage across C2 is 10 volts.

Obtain the transfer function of the differentiating circuit shown in Figure P2.7.

A bridged-T network is often used in AC control systems as a filter network [8]. The circuit of one bridged-T network is shown in Figure P2.8. Show that the transfer function of the network is V&) 1 + IR^Cs + RiRjpV Kn(i') 1 + (2, + R2)Cs + i?,i? 2 cV Sketch the pole-zero diagram when Rx = 0.5, = 1, and C = 0.5.

Determine the transfer function Xi(s)/F(s) for the coupled spring-mass system of Problem P2.3. Sketch the s-plane pole-zero diagram for low damping when M = l,b/k = l,and 4 2- 0.1.

Determine the transfer function Yi{s)jF(s) for the vibration absorber system of Problem P2.2. Determine Problems 143 the necessary parameters M2 and &12 so that the mass Ml does not vibrate in the steady state when F(t) a sin(&)o t).

For electromechanical systems that require large power amplification, rotary amplifiers are often used [8,19]. An amplidyne is a power amplifying rotary amplifier. An amplidyne and a servomotor are shown in Figure P2.ll. Obtain the transfer function 9(s)/Vc(s), and draw the block diagram of the system. Assume vd = k2iq and vq = k{ ic.

For the open-loop control system described by the block diagram shown in Figure P2.12, determine the value of K such that y(t) -* 1 as t oo when r(r) is a unit step input. Assume zero initial conditions.

An electromechanical open-loop control system is shown in Figure P2.13. The generator, driven at a constant speed, provides the field voltage for the motor. The motor has an inertia Jm and bearing friction />,. Obtain the transfer function BL{s)fVf{s) and draw a block diagram of the system. The generator voltage can be assumed to be proportional to the field current if .

A rotating load is connected to a field-controlled DC electric motor through a gear system. The motor is assumed to be linear. A test results in the output load reaching a speed of 1 rad/s within 0.5 s when a constant 80 V is applied to the motor terminals. The output steady-state speed is 2.4 rad/s. Determine the transfer function 0{s)/Vf(s) of the motor, in rad/V. The inductance of the field may be assumed to be negligible (see Figure 2.18). Also, note that the application of 80 V to the motor terminals is a step input of 80 V in magnitude.

Consider the spring-mass system depicted in Figure P2.15. Determine a differential equation to describe the motion of the mass m. Obtain the system response x(t) with the initial conditions A(0) = Xg and i(0) = 0.

Obtain a signal-flow graph to represent the following set of algebraic equations where x\ and x2 are to be considered the dependent variables and 6 and 11 are the inputs: h Xi + 1.5 = 6, 2JC, 4A-, 11. Determine the value of each dependent variable by using the gain formula. After solving for JCJ by Mason's signal-flow gain formula, verify the solution by using Cramers rule.

A mechanical system is shown in Figure P2.17, which is subjected to a known displacement x$(t) with respect to the reference, (a) Determine the two independent equations of motion, (b) Obtain the equations of motion in terms of the Laplace transform, assuming that the initial conditions are zero, (c) Sketch a signalflow graph representing the system of equations, (d) Obtain the relationship 7[3(s) between X^(s) and X3(s) by using Mason's signal-flow gain formula. Compare the work necessary to obtain 7^0) by matrix methods to that using Mason's signal-flow gain formula.

An LC ladder network is shown in Figure P2.18. One may write the equations describing the network as follows: h = (Vi ~ Vu)Yh Va = (J, - /0)Z2, I, = K - v2)y3, v2 = / a z 4 . Construct a flow graph from the equations and determine the transfer function K(s)/Vi(.r).

A voltage follower (buffer amplifier) is shown in Figure P2.19. Show that T = vQ/vin = 1. Assume an ideal op-amp.

The source follower amplifier provides lower output impedance and essentially unity gain. The circuit diagram is shown in Figure P2.20(a), and the small-signal model is shown in Figure P2.20(b).This circuit uses an FET and provides a gain of approximately unity. Assume that R2 R] for biasing purposes and that Rg R2. (a) Solve for the amplifier gain, (b) Solve for the gain when gm = 2000 (tl and Rs = 10 kil where Rs = Ry + R2. (c) Sketch a block diagram that represents the circuit equations.

A hydraulic servomechanism with mechanical feedback is shown in Figure P2.21 [18]. The power piston has an area equal to A. When the valve is moved a small amount Az, the oil will flow through to the cylinder at a rate p Az, where p is the port coefficient. The Problems 145 (a) 'in G vgs -> (P*"."* >*a >x i "it Q input oil pressure is assumed to be constant. From the geometry, we find that Az = &-(x - y) - y. h h (a) Determine the closed-loop signal-flow graph or block diagram for this mechanical system, (

Figure P2.22 shows two pendulums suspended from frictionless pivots and connected at their midpoints by a spring [1]. Assume that each pendulum can be represented by a mass Ma t the end of a massless bar of length L. Also assume that the displacement is small and linear approximations can be used for sin 8 and cos 8. The spring located in the middle of the bars is unstretched when fy = 82. The input force is represented by /(r), which influences the left-hand bar only, (a) Obtain the equations of motion, and sketch a block diagram for them, (b) Determine the transfer function T(s) = 8i(s)/F(s). (c) Sketch the location of the poles and zeros of T(s) on the s-plane.

The small-signal circuit equivalent to a commonemitter transistor amplifier is shown in Figure P2.23. The transistor amplifier includes a feedback resistor Rf. Determine the input-output ratio vcJv-m.

A two-transistor series voltage feedback amplifier is shown in Figure P2.24(a). This AC equivalent circuit 146 Chapter 2 Mathematical Models of Systems neglects the bias resistors and the shunt capacitors. A block diagram representing the circuit is shown in Figure P2.24(b).This block diagram neglects the effect of hn., which is usually an accurate approximation, and assumes that R2 + RL R\- (a) Determine the voltage gain vjvin. (b) Determine the current gain ia/lbi(c) Determine the input impedance V\Jib\.

H. S. Black is noted for developing a negative feedback amplifier in 1927. Often overlooked is the fact that three years earlier he had invented a circuit design technique known as feedforward correction [19], Recent experiments have shown that this technique offers the potential for yielding excellent amplifier stabilization. Black's amplifier is shown in Figure P2.25(a) in the form recorded in 1924. The block diagram is shown in Figure P2.25(b). Determine the transfer function between the output Y(s) and the input R(s) and between the output and the disturbance Td(s). G(s) is used to denote the amplifier represented by fi in Figure P2.25(a).

A robot includes significant flexibility in the arm members with a heavy load in the gripper [6, 20]. A two-mass model of the robot is shown in Figure. P2.26. Find the transfer function Y(s)IF(s).

Magnetic levitation trains provide a high-speed, very low friction alternative to steel wheels on steel rails. The train floats on an air gap as shown in Figure P2.27 [25]. The levitation force FL is controlled by the coil current i in the levitation coils and may be approximated by Pit)' VWWVWA k FIGURE P2.26 The spring-mass-damper model of a robot arm. V where z is the air gap. This force is opposed by the downward force F = mg. Determine the linearized Problems 147 relationship between the air gap z and the controlling current near the equilibrium condition.

A multiple-loop model of an urban ecological system might include the following variables: number of people in the city (P), modernization (M), migration into the city (C), sanitation facilities (S), number of diseases (D), bacteria/area (B), and amount of garbage/area (G), where the symbol for the variable is given in parentheses. The following causal loops are hypothesized: 1. P^G^B^D-^P 2. P-*M^C-^P 3. P-*-MS-*D-*P 4. P^>M-*S^B-*D^>P Sketch a signal-flow graph for these causal relationships, using appropriate gain symbols. Indicate whether you believe each gain transmission is positive or negative. For example, the causal link S to B is negative because improved sanitation facilities lead to reduced bacteria/area. Which of the four loops are positive feedback loops and which are negative feedback loops?

We desire to balance a rolling ball on a tilting beam as shown in Figure P2.29. We will assume the motor Torque motor input current i controls the torque with negligible friction. Assume the beam may be balanced near the horizontal ( = 0); therefore, we have a small deviation of . Find the transfer function X(s)/I(s). and draw a block diagram illustrating the transfer function showing (5), X(s), and T(s).

The measurement or sensor element in a feedback system is important to the accuracy of the system [6]. The dynamic response of the sensor is important. Most sensor elements possess a transfer function T.S + 1 Suppose that a position-sensing photo detector has T = 4,us and 0.999 < k < 1.001. Obtain the step response of the system, and find the k resulting in the fastest responsethat is, the fastest time to reach 98% of the final value.

An interacting control system with two inputs and two outputs is shown in Figure P2.31. Solve for Yt(s)/Ri(s) and Y2(s)/R1(s) when R2 = 0.

A system consists of two electric motors that are coupled by a continuous flexible belt. The belt also passes over a swinging arm that is instrumented to allow measurement of the belt speed and tension. The basic control problem is to regulate the belt speed and tension by varying the motor torques. An example of a practical system similar to that shown occurs in textile fiber manufacturing processes when yarn is wound from one spool to another at high speed. Between the two spools, the yarn is processed in a way that may require the yarn speed and tension to be controlled within defined limits. A model of the system is shown in Figure P2.32. Find J5(s)/i?j{5), Determine a relationship for the system that will make K independent of jRj

Find the transfer function for Y(s)/R(s) for the idlespeed control system for a fuel-injected engine as shown in Figure P2.33.

The suspension system for one wheel of an oldfashioned pickup truck is illustrated in Figure P2.34. The mass of the vehicle is m% and the mass of the wheel is m2-The suspension spring has a spring constant k^ and the tire has a spring constant k2. The damping constant of the shock absorber is b. Obtain the transfer function Y\(s)j'X(s), which represents the vehicle response to bumps in the road.

A feedback control system has the structure shown in Figure P2.35. Determine the closed-loop transfer function Y(s)/R(s) (a) by block diagram manipulation and (b) by using a signal-flow graph and Mason's signal-flow gain formula, (c) Select the gains /C, and K2 FIGURE P2.34 Pickup truck suspension. so that the closed-loop response to a step input is critically damped with two equal roots at s = -10. (d) Plot the critically damped response for a unit step

A system is represented by Figure P2.36. (a) Determine the partial fraction expansion and y{t) for a ramp input, /(f) = t, t > 0. (b) Obtain a plot of y(t) for part (a), and find y(t) for l = 1.0 s. (c) Determine the impulse response of the system v(/) for ( 2 0. (d) Obtain a plot of y(t) for part (c) and find y(i) for ( = 1.0 s.

A two-mass system is shown in Figure P2.37 with an input force u(t). When m | = m2 = land^ i = K2 = 1, find the set of differential equations describing the system.

A winding oscillator consists of two steel spheres on each end of a long slender rod, as shown in Figure P2.38. The rod is hung on a thin wire that can be twisted many revolutions without breaking. The device will be wound up 4000 degrees. How long will it take until the motion decays to a swing of only 10 degrees? Assume that the thin wire has a rotational spring constant of 2 X 10~4Nm/rad and that the K< * l X FIGURE P2.37 Two-mass system. Q-0.5 m4 FIGURE P2.38 Winding oscillator. viscous friction coefficient for the sphere in air is 2 X 10~4 N m s/rad. The sphere has a mass of 1 kg.

For the circuit of Figure P2.39, determine the transform of the output voltage V0(s). Assume that the circuit is in steady state when t < 0. Assume that the switch moves instantaneously from contact 1 to contact 2 at t = 0.

A damping device is used to reduce the undesired vibrations of machines. A viscous fluid, such as a heavy oil, is placed between the wheels, as shown in FIGURE P2.39 Model of an electronic circuit. 1 ~^^77Q 2H /YYY \ 6 V lOe-aV 1 P -If ?2f i < > 0.5 4ft 150 Chapter 2 Mathematical Models of Systems Shaft / % . Outer wheel . Inner wheel ./,,0, Fluid, b FIGURE P2.40 Cutaway view of damping device. Figure P2.40. When vibration becomes excessive, the relative motion of the two wheels creates damping. When the device is rotating without vibration, there is no relative motion and no damping occurs. Find B^s) and 02(s). Assume that the shaft has a spring constant K and that b is the damping constant of the fluid. The load torque is T.

The lateral control of a rocket with a gimbaled engine is shown in Figure P2.41. The lateral deviation from the desired trajectory is h and the forward rocket speed is V. The control torque of the engine is . and the disturbance torque is Ttf. Derive the describing equations of a linear model of the system, and draw the block diagram with the appropriate transfer functions.

In many applications, such as reading product codes in supermarkets and in printing and manufacturing, an optical scanner is utilized to read codes, as shown in Figure P2.42. As the mirror rotates, a friction force is developed that is proportional to its angular speed. The friction constant is equal to 0.06 N s/rad, and the moment of inertia is equal to 0.1 kg m2 . The output variable is the velocity cu(r). (a) Obtain the differential equation for the motor, (b) Find the response of the system when the input motor torque is a unit step and the initial velocity at J = 0 is equal to 0.7.

An ideal set of gears is shown in Table 2.5, item 10. Neglect the inertia and friction of the gears and assume that the work done by one gear is equal to that of the other. Derive the relationships given in item 10 of Table 2.5. Also, determine the relationship between the torques Tm and TL.

An ideal set of gears is connected to a solid cylinder load as shown in Figure P2.44. The inertia of the motor shaft and gear G2 is Jm. Determine (a) the inertia of the load JL and (b) the torque T at the motor shaft. Assume the friction at the load is bL and the friction at the motor shaft is bm. Also assume the density of the load disk is p and the gear ratio is n. Hint: The torque at the motorshaft is given by T = T\ + Tm.

To exploit the strength advantage of robot manipulators and the intellectual advantage of humans, a class of manipulators called extenders has been examined Problems 151 [22]. The extender is defined as an active manipulator worn by a human to augment the human's strength. The human provides an input U(s), as shown in Figure P2.45. The endpoint of the extender is P(s). Determine the output P(s) for both U(s) and F(s) in the form P(s) = T^Uis) + T2(s)F(s).

A load added to a truck results in a force F on the support spring, and the tire flexes as shown in Figure P2.46(a).The model for the tire movement is shown in Figure P2.46(b). Determine the transfer function X,(s)/F(s).

The water level h{t) in a tank is controlled by an open-loop system, as shown in Figure P2.47. A DC motor controlled by an armature current ;' turns a shaft, opening a valve. The inductance of the DC motor is negligible, that is, La = 0. Also, the rotational friction of the motor shaft and valve is negligible, that is, b = 0. The height of the water in the tank is h(t) J [1.60(f) ~ h(t)]dt.

The circuit shown in Figure P2.48 is called a leadlag filter. (a) Find the transfer function V2(s)/\{(s). Assume an ideal op-amp. Determine V2(s)/V^s) when l?j = 100Hl, R2 = 200 kl, Q = 1 /JLF, and C2 = 0.1 fiF. Determine the partial fraction expansion for

A closed-loop control system is shown in Figure P2.49. (a) Determine the transfer function T{s) = Y(s)/R(s). (b) Determine the poles and zeros of T(s). (c) Use a unit step input, .SKY) = 1/s, and obtain the partial fraction expansion for Y(s) and the value of the residues.

A closed-loop control system is shown in Figure P2.50. (a) Determine the transfer function T(s) = Y(s)/R(s). (b) Determine the poles and zeros of T(s). (c) Use a unit step input, R(s) = l/s, and obtain the partial fraction expansion for Y(s) and the value of the residues. (d) Plot y(() and discuss the effect of the real and complex poles of T(s). Do the complex poles or

Consider the two-mass system in Figure P2.51. Find the set of differential equations describing the system.

An armature-controlled DC motor is driving a load. The input voltage is 5 V. The speed at ( = 2 seconds is 30 rad/s, and the steady speed is 70 rad/s when t*oo. Determine the transfer function

A system has a block diagram as shown in Figure AP2.2. Determine the transfer function T(s) = g(f) It is desired to decouple Y(s) from R\(s) by obtaining T(s) = 0. Select C5 ( J) in terms of the other Gj(s) to achieve decoupling.

Consider the feedback control system in Figure AP2.3. Define the tracking error as E(t) = R(s) - Y(s). (a) Determine a suitable H(s) such that the tracking error is zero for any input R(s) in the absence of a disturbance input (that is, when Tlt(s) = 0). (b) Using H{s) determined in part (a), determine the response Y(s) for a disturbance T,j(s) when the input R(s) = 0. (c) Is it possible to obtain Y(s) = 0 for an arbitrary disturbance T^(s) when G,i(s) > 0? Explain your answer.

Consider a thermal heating system given by g(') _ 1 q(s) C,s + (QS + l/R.Y where the output 3"(.?) is the temperature difference due to the thermal process, the input q(s) is the rate of A'(.v) Gc(s) I GdU) L -o - H(s) G(0 *- n-o FIGURE AP2.3 Feedback system with a disturbance input. heat flow of the heating element. The system parameters are C ?, S, and Rr The thermal heating system is illustrated in Table 2.5. (a) Determine the response of the system to a unit step q(s) = 1/s. (b) As t*oo. what value does the step response determined in part (a) approach? This is known as the steady-state response, (c) Describe how you would select the system parameters C Q, 5, and R, to increase the speed of response of the system to a step input.

For the three-cart system illustrated in Figure AP2.5, obtain the equations of motion.The system has three inputs j, 1. and u3 and three outputs JC-,, JC2- and v3. Obtain three second-order ordinary differential equations with constant coefficients. If possible, write the equations of motion in matrix form.

Consider the hanging crane structure in Figure AP2.6. Write the equations of motion describing the motion of the cart and the payload. The mass of the cart is M, the mass of the payload is m, the massless rigid connector has length L, and the friction is modeled as Ft , = b'x where x is the distance traveled by the cart.

Consider the unity feedback system described in the block diagram in Figure AP2.7. Compute analytically the response of the system to an impulse disturbance. 154 Chapter 2 Mathematical Models of Systems FIGURE AP2.6 (a) Hanging crane supporting the Space Shuttle Atlantis (Image Credit: NASA/Jack Pfaller) and (b) schematic representation of the hanging crane structure. (a) (b) FIGURE AP2.7 Unity feedback control system with controller Gc(s) - K. , EM Controller K + + ( Td(s) o Plant 1 s + 20 + Y(s) Determine a relationship between the gain K and the minimum time it takes the impulse disturbance response of the system to reach y(r) < 0.1. Assume that K > 0. For what value of K does the disturbance response first reach at y{t) = 0.1 at r = 0.05?

Consider the cable reel control system given in Figure AP2.8. Find the value of A and K such that the percent overshoot is P.O. 10% and a desired velocity of 50 m/s in the steady state is achieved. Compute the closed-loop response v(f) analytically and confirm that the steady-state response and P. O. meet the specifications.

Consider the inverting operational amplifier in Figure AP2.9. Find the transfer function VJ,s){Vls), Show that the transfer function can be expressed as G(s) V,(s) = K, + + K&, where the gains KP, Kh and KD are functions of Cj, C2, JRI, and R2, This circuit is a proportional-integral-derivative (PID) controller (more on PID controllers in Chapter 7).

We want to accurately position a table for a machine as shown in Figure CDP2.1. A traction-drive motor with a capstan roller possesses several desirable characteristics compared to the more popular ball screw. The traction drive exhibits low friction and no backlash. However, it is susceptible to disturbances. Develop a model of the traction drive shown in Figure CDP2.1(a) for the parameters given in Table CDP2.1. The drive uses a DC armature-controlled motor with a capstan roller attached to the shaft.The drive bar moves the linear slide-table. The slide uses an air bearing, so its friction is negligible. We are considering the open-loop model, Figure CDP2.1(b), and its transfer function in this problem. Feedback will be introduced later.

A control system is shown in Figure DP2.1. The transfer functions G2(s) and H2(s) are fixed. Determine the transfer functions G{(s) and //](.?) so that Table CDP2.1 Typical Parameters for the Armature-Controlled DC Motor and the Capstan and Slide Ms M,, 'm r bm Kn K Rm L'm Mass of slide Mass of drive bar Inertia of roller, shaft, motor and tachometer Roller radius Motor damping Torque constant Back emf constant Motor resistance Motor inductance 5.693 kg 6.96 kg 10.91 lfT3 kg m2 31.75-10- 0.268 N ms/rad 0.8379 N m/amp 0.838 Vs/rad 1.36 Q, 3.6 mH the closed-loop transfer function Y(s)!R($) is exactly equal to 1.

The television beam circuit of a television is represented by the model in Figure DP2.2. Select the unknown conductance G so that the voltage v is 24 V. Each conductance is given in Siemens (S).

An input r(t) = t, t a 0, is applied to a black box with a transfer function G(s). The resulting output response, when the initial conditions are zero, is y(0 = e -'-^- 2 ' - ^ + |/,/^0 . Determine G(s) for this system.

An operational amplifier circuit that can serve as a filter circuit is shown in Figure DP2.4. Determine the transfer function of the circuit, assuming an ideal op-amp. Find vt)(t) when the input is Uj(f) = At, t >0 .

Consider the clock shown in Figure DP2.5. The pendulum rod of length L supports a pendulum disk. Assume that the pendulum rod is a massless rigid thin rod and the pendulum disc has mass m. Design the length of the pendulum, L, so that the period of motion is 2 seconds. Note that with a period of 2 seconds each "tick" and each "tock" of the clock represents 1 second, as desired. Assume small angles, "< -o + FIGURE DP2.4 Operational amplifier circuit. analysis so that sin

Consider the two polynomials /?(X = .2 s 1 + 7s + 10 and q(s) = s + 2. Compute the following (a) p(s)q(s) (b) poles and zeros of G(s) = Pis) (C) /7(-1)

Consider the feedback system depicted in Figure CP2.2. (a) Compute the closed-loop transfer function using the series and feedback functions. (b) Obtain the closed-loop system unit step response with the step function, and verify that final value of the output is 2/5.

Consider the differential equation y + 4y + 3y = u, where y(0) = y(0) = 0 and u(t) is a unit step. Determine the solution y(t) analytically and verify by coplotting the analytic solution and the step response obtained with the step function.

Consider the mechanical system depicted in Figure CP2.4.The input is given by/(i). and the output is y(t). Determine the transfer function from f(t) to y(t) and, using an m-file, plot the system response to a Forcing function Spring <* constant .I' Mass in Friction constant b J, Mass displacement y(?) FIGURE CP2.4 system. A mechanical spring-mass-damper unit step input. Let in = 10, k = 1, and b = 0.5. Show that the peak amplitude of the output is about 1.8.

A satellite single-axis attitude control system can be represented by the block diagram in Figure CP2.5. The variables k, a, and b are controller parameters, and J is the spacecraft moment of inertia. Suppose the nominal moment of inertia is J = 10.8E8 (slug ft2 ), and the controller parameters are k = 10.8E8, a = 1, and 6 = 8. (a) Develop an m-file script to compute the closedloop transfer function T(s) = 0(s)/0,i(s). (b) Compute and plot the step response to a 10 step input. (c) The exact moment of inertia is generally unknown and may change slowly with time. Compare the step response performance of the spacecraft when / is reduced by 20% and 50%. Use the controller parameters k = 10.8E8, a = 1, and b = 8 and a 10 step input. Discuss your results.

Consider the block diagram in Figure CP2.6. (a) Use an m-file to reduce the block diagram in Figure CP2.6, and compute the closed-loop transfer function.

For the simple pendulum shown in Figure CP2.7, the nonlinear equation of motion is given by 0(0 + sin 6 0, where L = 0.5 m, m = 1 kg, and g = 9.8 m/s~. When the nonlinear equation is linearized about the equilibrium point 6 = 0, we obtain the linear time-invariant model, 0 + j6 0. Create an m-file to plot both the nonlinear and the linear response of the simple pendulum when the initial angle of the pendulum is 0(0) = 30 and explain any differences.

A system has a transfer function X(s) (20/z)(s + z) B{s) ~ s 2 + 3s + 20" Plot the response of the system when R(s) is a unit step for the parameter z = 5,10, and 15.

Consider the feedback control system in Figure CP2.9, where G(s) = ^ ^ and H(s) s + 2 s+ 1 (a) Using an m-file, determine the closed-loop transfer function. (b) Obtain the pole-zero map using the pzmap function. Where are the closed-loop system poles and zeros? (c) Are there any pole-zero cancellations? If so, use the minreal function to cancel common poles and zeros in the closed-loop transfer function. (d) Why is it important to cancel common poles and zeros in the transfer function?

Consider the block diagram in Figure CP2.10. Create an m-file to complete the following tasks: (a) Compute the step response of the closed-loop system (that is, R(s) = Vs and 7",,(i) = 0) and plot the steady-state value of the output Y(s) as a function of the controller gain 0 < K s 10. (b) Compute the disturbance step response of the closed-loop system (that is, R(s) = 0 and Terms and Concepts 159 Td(s) lis) and co-plot the steady-state value of the output Y (s) as a function of the controller gain 0 < K < 10 on the same plot as in (a) above. (c) Determine the value of K such that the steadystate value of the output is equal for both the input response and the disturbance response.