141_1_Chapter3_problems_2

141_1_Chapter3_problems_2 - 5 Chapter uynauutnwpmw...

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Unformatted text preview: 5 Chapter; uynauutnwpmw Thickness T Figure 3.51 Continuous rolling mill l : N Gear ratio Vertically adjustable roller Motion of WW’ material . out of rollers Thickness x Fixed roller determine the following: (a) The DC gain; (b) The final value to a step input. tinuous rolling mill depicted in Fig. 3.51. Suppose that the motion of 3.18 Consider the con the adjustable roller has a damping coefficient 19, and that the force exerted by the rolled table roller is proportional to the material’s change in thickness: material on the adjus F s :: C(T ~— x). Suppose further that the DC motor has a torque constant K, and a back emf constant Kg. and that the rack—and-pinion has effective radius of R. (a) What are the inputs to this system? The output? (b) Without neglecting the effects of gravity on the adjustable roller, draw a block dia— gram of the system that explicitly shows the followrng quantities: VS (5), 10 (s), F (s) (the force the motor exerts on the adjustable roller), and X (s). (c) Simplify your block diagram as much as possible while still ide each input separately. nti‘fying output and Probicmsfbr Section 3.2: System Modeling Diagrams 3.19 Consider the block diagram shown in Fig. 3.52. Note that ai and bl- are constantsi Compute the transfer function for this system. This special structure is called the “conth canonical form” and will be discussed further in Chapter 7. % E: i i n igmub all-Wm- Block diagram for Problem 3.19 Figure 3.53 Block diagrams for Problem 3.20 O Y(s) RO (a) i a G7 + + + RC » G1 G3 > G4 ’ G6 + Y G2 1 (.5 .j (b) r G7 > G6 + + R G’ 2 ’G2 ’Ga 2 G4 Gs--><Jr :HY 3.21 Find the transfer functions for the block diagrams in Fig. 3.54. using the ideas of block diagram simplification The s ' ‘ ' . . peeial structure in F1 . 3.54 b ' “ canonlcal form” and will be discussed in Chapter 7. g ( ) 15 called the ObserVer 5 Chapter 5 Uyndmu. [\CDPUIrox. (a) (b) (c) R(s) A(s) > l)(s) ___.__.._..J 3(5) _ Y(S) H (s) L,_,__.._. Figure 3.54 Block diagrams for Problem 3.21 3.22 Use block—diagram algebra t Fig. 3.55. (d) C(s) i‘ 0 determine the transfer function between R(s) an d Y(s) in figure 309;» Block diagram for Problem 3.22 Figure 3.56 Circuit for Problem 3.23 Figure 3.57 Unity feedback system for Problem 3.24 Prablemsfor Section 3.3: Effect ofPole Locations 3.23 For the electric circuit shown in Fig. 3.56, find the following: (a) The time—domain equation relating i(t) and v10); (b) The time—domain equation relating i (t) and v20); (c) Assuming all initial conditions are zero, the transfer function ‘Zf—ESS; and the damping ratio g“ and undamped natural frequency ton of the system; (d) The values of R that will result in v2 (2‘) having an overshoot of no more than 25%, assuming v1 (t) is a unit step, L = 10 mH, and C = 4 ,uF. L R + + v10) 2(1) C v20) 3.24 For the unity feedback system shown in Fig. 3.57, specify the gain K of the proportional controller so that the output 37(1) has an overshoot of no more than 10% in response to a unit step. O Y(s) 3.25 For the unity feedback system shown in Fig. 3.58, specify the gain and pole location of the compensator so that the overall closed—loop response to a unit—step input has an overshoot of no more than 25%, and a 1% settling time of no more than 0.1 sec. Verify your design using MATLAB. Chapter 3 Dynamic Response ire 3.58 ty feedback system Problem 3.25 Figure 3.59 Unity feedback system for Problem 3.28 Figure 3.60 Desired closed—loop pole locations for Problem 3.28 Compensator Plant J" K g g 100 Rmo T 5+4; s+25 Problems for Section 3 3.26 Suppose you desire the Draw the region in the s .4: Time— peak time of a . . , specification tp < tp. 3.27 A certain servomech and no finite zeros. overshoot (M p), and settling time (a) Sketch the region in the s anism s ‘ The time—domain O Yts) plane that corresponds to v ystem has dy nami speci Domain Specification ' ‘ to b iven second—order system g alues of the poles that meet the es dominated b _ fications on the rise time (ts) are given by tr < 0.6 sec, Mp g 17%, ts g 9.2 sec. will meet all three specifications. (b) Indicate on your smallest rise—time and als sign a unity feedback ‘ . 11 learn in Chapter 4, the configurati troller.) You are to design the contr wn in Fig. 3.60. . haded regions in Fig. 3.59? (A Simple 3.28 Suppose you are to de Fig. 3.59. (As you wr ro ortional—integral con . p P in the shaded regions she can and g correspond to the s poles lie with (a) What values of estimate from t z a 2: 2. Find values for K an (in) Let Ka . system he w1 6(1) sketch the speci he figure is suffici 0 meet the set —plane where the pole fie locations (de . tling time specification exa order plant depicted in on shown is referred to as a oller so that the closed—loop ent.) thin the shaded regions. controller for a first— d K] so that the poles of e less than t}. y a pair of complex poles (t r), percent s could be placed so that the system noted by x) that will have the ctly. the closed-loop .MW (c) Prove that no matter what the values of Ka and a are, the controller provides enough flexibility to place the poles anywhere in the complex (left-half) plane. 3.29 The open—loop transfer function of a unity feedback system is Gm : Sls + 2)’ The desired system response to a step input is specified as peak time tp = 1 sec and overshoot MP 2 5%. (3) Determine whether both specifications can be met simultaneously by selecting the right value of K. (b) Sketch the associated region in the s-plane where both specifications are met, and indicate what root locations are possible for some likely values of K. (c) Relax the specifications in part (a) by the same factor and pick a suitable value for K, and use MATLAB to verify that the new specifications are satisfied. 3.30 The equations of motion for the DC motor shown in Fig. 2.32 were given in Eqs. (2.52— 2.53) as " KK . K .1QO + (19+ 1 (3)9,” = —tVa. Assume that m = 0.01kg.m2, b = 0.001 N'mtsee, Ke = 0.02 V-sec, Kt = 0.02 N.m/A’ Ra = 10 52. (a) Find the transfer function between the applied voltage va and the motor speed 9m. (b) What is the steady-state speed of the motor after a voltage Va 2: 10 V has been applied? (c) Find the transfer function between the applied voltage Va and the shaft angle 9m. (d) Suppose feedback is added to the system in part (c) so that it becomes a position servo device such that the applied voltage is given by Va : K(9r “' 9m), where K is the feedback gain. Find the transfer function between 9, and 9,”. (e) What is the maximum value of K that can be used if an overshoot Mp < 20% is desired? (D What values of K will provide a rise time of less than 4 sec? (Ignore the Mp constraint.) (g) Use MATLAB to plot the step response of the position servo system for values of the gain K = 0.5, l. and 2. Find the overshoot and rise time for each of the three step responses by examining your plots. Are the plots consistent with your calculations in parts (e) and (f)? 3.31 You wish to control the elevation of the satelliteetracking antenna shown in Figs. 3.61 and 3.62. The antenna and drive parts have a moment of inertia J and a damping B; (c) What is the maximum value of K that can be used if you wish to have an overshoot 1 Chapter 3 Dynamic Response Mp < 10%? .. 1 . . . . we: 3 6 _ (d) What values of K W111 provrde a rise time of less than 80 sec? (Ignore the Mp :elhte—trackmg constraint.) tenna A (e) Use MATLAB to plot the step response of the antenna system for K = 200, 400, Irce: Courtesy Space 1000, and 2000. Find the overshoot and rise time of the four step responses by tems/ LO’ at examining your plots. Do the plots confirm your calculations in parts (c) and (d)? 3.32 Show that the second-order system _ y + 2:010 + wiy = 0, MO) 2 ya. ya» = 0, has the response e—ot YO) = y0— sin(wdt + cos‘1 g). t/ 1 — {2 Prove that, for the underdamped case (g < l), the response oscillations decay at a predictable rate (see Fig. 3.63) called the logarithmic decrement 2 5 lny—0 lne0rd 0rd 7“: g M [1 - C2 ‘ Am An- Ft are 3&2 =1 _.._ 2 ln——, 9 . YI yi Schematic of antenna for Problem 3-31 where 271 27r 17d : -— : is the damped natural period of vibration. The damping coefficient in terms of the logarithmic decrement is then ut mostly from the Figure 3.63 ‘ Definition of logarithmic decrement bearing and aerodynamic friction, b f motion are tent from back emf of the DC drive motor. The equations 0 these arise to some ex 10‘ +30 2 TC, ’ at where TC is the torque from the drive motor. Assume th J = 600.000 kg-m2 B = 20,000 N-msec. and the antenna angle 6. rence command 9r NY d the transfer function between the applied torque Tc a) Fin ‘ f Eb) Suppose the applied torque is computed so that 9 tracks a re e according to the feedback law Tc : Kwr " 9)» e e . W d . h 1‘ K is h t edba k g 1D In 11 t dnbfer functlon bet Ben r (in W 1 e e (2 a F (l t 5 rr W 6 , 0 ...
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