141_1_Chapter2and3_problems

141_1_Chapter2and3_problems - Chapter 2 Dynamic Models...

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Unformatted text preview: Chapter 2 Dynamic Models Figure 2.39 Mechanical systems A A 11. Why do we approxi a linear model? 12. Give the relationships for (a) heat flow acros (b) heat st 13. Name Problems for Section 2.1: Dyn . . . for differentlal equations ' e ink the system W1 zero initial con 2.1 Write th and (b), state whether youth at all, givon that there are non« for your answer. No friction orage in a substance. and give the equations for mate a physical model of the pl s a substance, and the three relationships governin ant (which is always nonli g fluid flow. PRfingMS Friction (b) armies of Mechanical Systems . n in Fig. 2.39. For (a) i ' l s stems . I the mewamca y ecay so that it has no motion show ll eventually d v ditions for both masses, and give .172 2.2 Write the differential you think the system wil . . tial conditions f0 motion for the double are non—zero im . . f Write the equations 0 2.3 displacement an horizontal. T Assume that the the spring is always and the springs are equation for th attached thre —pendulurn system all en les of the pendulums are sm i6 pendulum rods are taken to be mass e—fourths of the way down. i Fig. 2.40. State whether at it has no motion at all, give ason for your answer. shown in Fig. ough to ensure that near) with _ a T838011 n that there 2.41 . less, of length l. Figure 2.4.0 Mechanical system for Problem 2.2 figure 2.4a Double pendulum 2.4 2.5 2.6 2.7 2.8 No friction No friction Write the equations of motion of a pendulum consisting of a thin, 4 kg stick of length l suspended from a pivot. How long should the rod be in order for the period to be exactly 2 sec? (The inertia I of a thin stick about an end point is %m12. Assume that 9 is small enough that sine % 9.) Why do you think grandfather clocks are typically about 6 ft high? For the car suspension discussed in Example 2.2, plot the position of the car and the wheel after the car hits a “unit bump”(i.e., r is a unit step) using MAT LAB. Assume that m1 = 10 kg, m2 2 350 kg, KW : 500,000 N/m, Ks = 10,000 N/m. Find the value of b that you would prefer if you were a passenger in the car. Write the equations of motion for a body of mass M suspended from a fixed point by a spring with a constant k. Carefully define where the body’s displacement is zero. Automobile manufacturers are contemplating building active suspension systems. The simplest change is to make shock absorbers with a changeable damping, b(u 1). It is also possible to make a device to be placed in parallel with the springs that has the ability to supply an equal force, M27 in opposite directions on the wheel axle and the car body. (a) Modify the equations of motion in Example 2.2 to include such control inputs. (b) Is the resulting system linear? (c) Is it possible to use the forcer L12 to completely replace the springs and shock absorber? Is this a good idea? Modify the equation of motion for the cruise control in Example 21, Eq. (2.4), so that it has a control law; that is, let u = K(vr — v), (2.89) where v, 2 reference speed, (2.90) K = constant. (291) This is a “proportional”control law in which the difference between vr and the actual speed is used as a signal to speed the engine up or slow it down. Revise the equations of motion with vr as the input and v as the output and find the transfer function. Assume that m = lOOO kg and b = 50 N~sec/m, and find the response for a unit step in vr using Figure 2.44 Chapter 2 Dynamic Models Circuit for Problem 2.11 that you think would result in a control or, find a value of K ible to the reference speed MATLAB. Using trial and err system in which the actual speed converges as quickly as poss with no objectionable behavior. 2.9 In many mechanical positioning syste and another. An example is shown in Fig. 2. Fig. 2.42 depicts such a situation, where a etween one part of the system flexibility of the solar panels. the mass M and another ften modeled by a spring ms there is flexibility b 7 where there is force 14 is applied to n the objects is 0 mass m is connected to it. The coupling betwee constant k with a damping coefficient 19, although the actual situation is usually much more complicated than this. Figure 2.45 Circuit for Problem 2.12 f motion governing this system. tion between the control input u and the output y. (3) Write the equations 0 (b) Find the transfer func “agate 2A2 Schematic of a system with flexibility 2.13 A common CO . nnection for a motor . . haVe the motor c Power amphfier 13 Shown in F’ - . amplifier. Assumlelrltrhialtt tfliilzw the input voltage, and the connectlii'nzijiallltel: Idea 18 to . ense resistor r is a Current reSistor R, l ‘ . S Very small 0 ~ When Rf ”a:: find the transfer function from Vin to Ia A18: 3122:2631: t1th £11176 feedbaCk , w - ' rans er function Problems for Section 2.2: Models ofEiccrric Circuits Figure; 2345 2.10 A first step toward a realistic model of an opdamp is given by the following equations and . OP'amp Circuit for is shown in Fig. 2.43: Pmblem 2.13 V -— 107 [V V ] out —— S + 1 + ~ . Lt. 7: i“ r: 0. Find the transfer function of the simple amplification circuit shown using this model. figure 251.43 2.14 An Op_am CO . _ 1’1 ‘ P nection With feedback to both the negative and the positi t l Ve erminals is Circuit for Problem 2.10 shown in Fi . 2.47 g i If the op—amp has the nonideal transfer function given in Pr b1 2 o em .10, g h a p 7 — r R, 1V6 l 6 III Xllilum llaer p()SSIble {01 the Ofiltllle lfiedback 1:3th 1 Ill tel lllS ()l the negative feedback r3110, N ~— n , f0r the (JICUIt i0 16 all! Stable Rm+Rf 2.15 Write the d ~ ' ynamic e uat‘ , fig 248. q ions and find the transfer functions for the circuits show ' x n in (a) passive lead circuit 4.: ,1 Vin if the Op-aml) (b) active lead circuit hown in Fig. 2.44 results in Vow transfer function of op—amp connection s the transfer function if the op-amp has the nonideal 2.11 Show that the (c) active lag circuit is ideal. Give Problem 2.10. 2.12 Show that, with the non shown in Fig. 2.45 is unstable. ((1) passive notch circuit ideal transfer function of Problem 2.10, the op~amp connection Chapter 2 Dynamic Models sure 2.47 C R -amp circuit for R )blem 2.14 R1 R R v,” ’\/\/\z a w» a 0 V1 V2 V3 Ra R ’\/\/\/ Rb C RC V01” figure 2.48 A?“ . a) Passive lead; (b) + «Ad/V ‘ - - c active a I activelead,( ) _ + Ri R2 y Figure $49 Lag; and (d) passwe u e _ - ' - O Op-amp biquad notch cwcmts o _ (a) R _ f (a) Show that if Ra = R, and R}, 2 RC 2 Rd 2 00, the transfer function from Vin to V0“; can be written as the low—pass filter R - V A R2 1 van _ 0” = _, (2.92) Vin S2 “‘2‘ + 2; m + l a)” can C where R A = —, Rl' _ l (Uri — RC, _ R g ‘" 2R2’ (b) Using the MATLAB command step, compute and plot on the same graph the step responses for the biquad of Fig. 2.49 forA = 1, can 2 l, and g“ = 0.1, 0.5, and 1.0. 2.17 Find the equations and transfer function for the biquad Circuit of Fig. 2.49 if Ra = R, Rd 2 R1, ande = RC = 00. Problems for Section 2.3: Models of Electromechanical Systems 2.18 The torque constant of a motor is the ratio of torque to current and is often given in ounce- inches per ampere. (Ounce—inches have dimension force >< distance, Where an ounce is 1/16 of a pound.) The electric constant of a motor is the ratio of back emf to speed and is often given in volts per 1000 rpm. In consistent units, the two constants are the same for a given motor (d) ause its transfer function lynomials‘ By selecting ow—pass, band—pass, 249 is called a biquad bee «order or quadratic .po he circuit can realize a l ' ' ‘ ~ ‘ Fig. fiexxble Circuit shown in 2.16 The very he ratio of two second can be made to be t different values for Ra. Rb. RC, and Rd. high—pass, or band—reject (a) Show that the units ounce-inches per ampere are proportional to volts per 1000 rpm by reducing both to MKS (SI) units. Chapter 2 Dynamic Models Figure 2.50 Simplified modelfor capacitor microphone 2.19 The electro rpm. What is its torque constant in (b) A certain motor has a back emf of 25 V at 1000 ounce—inches per ampere? (c) What is the torque constant of the motor of part (b) in newto mechanical system shown in Fig. 250 represents a simplified model of a hone. The system consists in part of a parallel plate capacitor connected cuit. Capacitor plate a is rigidly fastened to the microphone frame. d exert a force fs(t) on plate b, which has through the mouthpiece an cted to the frame by a set of springs and dampers. The capacitance distance x between the r , n—meters per ampere? capacitor microp into an electric Cir Sound waves pass mass M and is conne C is a function of the C05) = 77 where a :: dielectric constant of the material between the plates, A 2 surface area of the plates. The charge q and the voltage 6 across the plates are related by q = C ()06 The electric field in turn produces the following force fe on the movable plate that opposes its motion: 2 fe =3 L ' 28A ribe the operation of this system. (It is acceptable (a) Write differential equations that desc to leave in nonlinear form.) model? f the system? (b) Can one get a linear (c) What is the output 0 MRI" .XM, control is an electric motor driving electromechanical position ises in computer-disk—head 2.20 A very typical problem of a load that has one dominant vibration mode. The problem ar ' , and many other applications. A schematic diagram is control, reel~to- sketched in Fig. 2.5L ical constant Kg, a torque constant Kt, an armature inductance La, has an inertia J1 and a viscous The two inertias are connected by a shaft with a ction B. The load has an inertia 12. an equivalent viscous damping 17. Write the equations of motion. The motor has an electr and a resistance Ra. The rotor fri spring constant k and fignre 2.51 Motor with a flexible load Figure 2&2 (a) Precision table kept level by actuators; (b) side view of one actuator Problems for Section 2.4: Heat and Fluidwflow Model r 2.21 A precisi on tab — ‘ . actuators underliizvelmg scheme shown in Fig. 2.52 relies on thermal ' corners Th comers to level the table by raisin or 1 ~ éXpansmn 0f . 6 parameters are as follows: g 0W€rmg then respective Tact = actuator temperature Tamb = ambient air temperature R = h - ' f eat flow coeffiment between the actuator and the air C = thermal capacity of the actuator, R = resistance of the heater. Assume that (1) the actuator acts - . as a pure electr' ‘ ( actuator 1s r0 0 . 1C TCSIStdnCB, (2 the h - ' to the diffefenge réleipvageto ;he electric power input, and (3) the m)otion é: thg , n act and T due t h ‘ P0 10nal e uations ‘ . amb 0 t ermal ex ans- - . . q relatlng the height of the actuator d versus the apIpliedoilolfmd the dlfferenual age Vi. QC! (a) (b) 2.22 All 611) CO d [[0 6 Sup 6 C d p Ii 1 II T p11 5 01 d1}: at the 53.“ 6 tell! Cldtulel Cat (Km! () ll le 11 Ill 0 h flOOI Oil the lllgoll‘llseolnllldll g Show] HI] 1g. [he floor plan IS ShOWn 11]} 1g. b . t m ) 1116 Cold d1] W p1 duCeS all equal cl Quill ()l heat flow (2 Out () each “)0 . W 116 a bel of dlffel elltlal equdtlons gOVCInng the telllpelatme 111 each IOOID, Whele To 2 temperature outside the building R = . 0 resrstance to heat flow through the outer walls RV = ‘ ' , res1stance to heat flow through the inner walls 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 mm 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 to” and g correspond to the s poles lie with (a) What values of estimate from t (in) Let Ka z: a. 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 em.) 2. Find values for K an 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 G(S) :: 3(3 + ' 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 Jmem + (19+ ’ fie... = _tva. Assume that m = 0.01 kng, b = 0.001 Nmsec, Ke = 0.02 V-sec, Kr = 0.02 N.m/A’ Ra = 10 S). (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 “' am), 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/Loml examining your plots. Do the plots confirm your calculations in parts (c) and (d)? 3.32 Show that the second-order system _ y + 2am + wiy = 0, MO) 2 ya. ya» = 0, has the response e—ot y(t) : yo— 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“: M [1 - t2 ‘ Ayi Ayz- Ft ureBeéfi =ln———-— ; n——, g . 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 edba k g 1D In 11 t dnbfer functlon bet Ben r (in W i e e (2 a F (l t e 1" W 6 , 0 ...
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This note was uploaded on 07/02/2011 for the course EE 141 taught by Professor Balakrishnan during the Spring '07 term at UCLA.

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141_1_Chapter2and3_problems - Chapter 2 Dynamic Models...

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