ch2 nise - 110 lynholle Hath Symbol“: Math J Chapter 2...

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Unformatted text preview: 110 lynholle Hath Symbol“: Math J Chapter 2 MUdElll'lg Ill the Frequency Domain Problems 1. Derive the Laplace transform for the following time functions: a. ufl) b. m0) c. sin mm“) 11. cos wt 110‘} 2. Using the Laplace transform pairs of Table 2.1 and the Laplace transform theorems of Table 2-2, derive the Laplace transforms for the following time functions. a. 8“” sin (emf!) h. 6“" cos altar!) c. that) 3. Repeat Problem 15 in Chapter 1, using Laplace transforms. Assume that the forcing functions are zero prior to t = 0—. 4. Repeat Problem [6 in Chapter I, using Laplace transforms. Assume that the forcing functions are zero prior to r = 0—- 5. Use MATLAB and the Symbolic Math Toolbox to find the Laplace transform of the iollowrng tlme functions: a. fit) = 5t2cosi3t + 45°) b. lit} = Ste—2’srni4t + 60“) 6. Use MATLAB and the Symbolic Math Toolbox to find the Inverse Laplace transform of the lollowrng frequency functions: [52 + 35 + 7M5 + 2} a. GM = m b Glsl= 53+452+65+5 (s + Site2 + 85 + Falls2 + 55 + 7] 7. A system is described by the following differential equation: d3); day dy d3x dzx dx Er+3d—fl-+SE+}’- EE+4E+6E+SX Find the expression for the transfer function of the system, st) X(s). 8. For each of the following transfer functions, write the corresponding differ- ential equation- .. w - __1 FLS) sz+25+7 {fl = 10 ' F(s) (s+7){s+ 8) XEfi 5+2 '3' H3) - s3+8s2+93+15 Figure 92.2 mm: MITLAI lymholle Hath Problems 111 9. Write the differential equation for the system shown in Figure P2. l. R”) 35 + 2.5""1 + 453 + 52 + 3 6(5) 36+ 79+ 334+ 253+52+3 10. Write the differential equation that is mathematically equivalent to the block diagram shown in Figure P22. Assume that rt!) = 313. fig} 54+ 2.13 + 5:2 + s + 1 CU} 55+ 354+ 253+ 451+ Ss+2 11. A sys1em is described by the following differential equation: d'zx dx F + + 3x - 1 with the initial conditlons x(0} = I, x(0) = —I. Show a block diagram of the system. giving its transfer function and all pertinent inputs and outputs. (Hint: the initial conditions will show up as added inputs to an effective system with zero initial conditions.) 12. Use MATLAB to generate the transfer function M sls + 55llsz + 55 + 301ls + Stills? + 275 + 52) In the following ways: Glsl = a. the raho of factors; I). the ratio of polynomials. 13. Repeat Problem 12 for the following transfer function: 5' + 2551' + 2le2 + 155 + 42 s5 + 135‘ + 953 + 31's2 + 355 + 50 14. Use MATLAB to generate the parhalfraction expansion of the following lunetlon: lofts + lfllls + 60) sls + 40lls + EDJlS2 + 75 + l‘ilIIS-lls2 + 65 + 901 15. Use MATLAB and the Symbolic Math Toolbox to input and form L11 objects In polynomial and factored form for the following frequency tunetlons: 45ls2 + 31's + "Milsf + 2852 + 325 + 16} is + 39lls + fills? + 25 + 100ll53 + 2752 + 185 + 15) 56ls + I‘ll-{53 + 4952 + 625 + 53] {53 + 8152 + 765 + stills? + 385 + 33llsz + 565 + Til Gls) = Flsl = a. Glsl = b. Gls) = 112 Chapter 2 Modeling In the Frequency Bertram 16. Find the transfer function, 6(5) = V43) 14(5). for each network shown in Figure P23 Flgure P23. 1 Q I Q l H + I I I I + -' urn " is") turn 1 F Iain [M (a) IT. Find the transfer functions, 6(3) 2 fits} Vts}. for each of the networks Figure Pu Show" In Figure P14- 1 H 1 o 1 F H I i I Q 19 r -m e _ . + '- (a) m (aural 18. Find the transfer functions, C(s) = vom- mm, for each of the networks O—h— . Figure stgl'fl" shown in Figure P25- Solve the problem using mesh analysis. 19. Repeat Problem 18 using nodal equations 20. 3. Write, but do not solve, the mesh and nodal equations for the netWOrk of Figure P16. Problems 1 13 Figure P16 Iymballc "an. I). Use MATLAB, the Symbolic Math Toolboxl and the equations found In {at to solve for the transfer function, Gist = Vols) Vis}. Use both the mesh and nodal equahons and show that either set yields the same transfer tunctlon- 21. Find the transfer function, 0(5) = Vote} Vim, for each of the operational amplifier circuits shown in Figure P17. Figure P2.7ir 100 kfl lUOkQ I}!!! —’\/\/\/—i on IUO kfl l ,uF (b) 1 14 Chapter 2 Modeling In the Frequency Domain 22. Find the transfer function Gm) = V45) 14(3), for each of the operational amplifier circuits shown in Figure P2.8 Figure P23 [00 kg 1 p F 100 kg (5) 23. Find the transfer function, 0(3) = X1 (s‘); Ffs}, for the translational mechani- cal system shown in Figure P29 Figure P23 Problems 115 24. Find the transfer function, Gm = X20”) 'Ffs). for the translational mechani- cal network shown in Figure P2- [0. figure P210 Fnctlonless m 25. Find the transfer function, 6(5) = X20.) Fm. for the translational mechani— """" cal system shown in Figure P1] 1. (Hint: Place a zero mass at 1:20).) Flgure P2.11 ., ‘20] m — +HE 2 me 5 N—sfrn 2 N—sfm 26. For the system of Figure P112 find the transfer function. Gfsl = X; {5] Fiat). Flgure P112 I | 27. Find the transfer function, 0(3) = Xfis) F (s), for the translational mechani- cal system shown in Figure P113. Figure P213 1 N—sfm Fricllonless 28. Find the transfer funetion, 33(3) 3H3), for each of the systems shown in Figure P214. 116 Chapter 2 Modeling In the Frequency Domain Figure P114 Figure P215 M, =4kg 4 N—sfm 3 Nim FI’iL‘llDfllBE-b Frictionlcss / 29. Write, but do not solve, the equations of motion for the translational mechan- ical system shown in Figure P2.15- Problems 11?!' 31]. For each of the rotational mechanical systems shown in Figure P116. write, but do not solve, the equations of motion. 3|“) Tm 5'2") Figure P116 2 N -m-s.l' rad 8 N-mfrad (a) no DI D2 0 W O NH O Kl K2 l I I I K1 (5} . tuml 31. For the rotational mechanical system shown in Figure P117. find the tl'flDSfBF 5"'"" function 6(5) = (92(5)- HS}- Figure F'EJZ'r l N-rn—s had I N-m-s Ira-d 32. For the rotational mechanical system with gears shown in Figure P2- 18, find the transfer function, 6(5) = 03(5),: NS). The gears have inertia and bearing friction as shown. Figure P118 TH] E E | NI Jl-Dl N2 | ' N3 12. D2. Jr3- D: 93“; 118 Chapter 2 Modeling in the Frequency Domain 33. For the rotational syswm shown In Figure P2.19, find the transfer function. 6(5) = 92(3) TL?)- Figure PZJQ Tm N] = 4 3 D] = I N—m-sfrad N2: 12 “93‘” N3=4 a D; = 2 N—m—sirad N4: D3 = 32 N—m—sfrad 34. Find the transfer function, 6(5) = 62(5) “'Tts). for the rotational mechanical system shown in Figure P220. 1000 N -m-s.’rad N3 = 25 ’ zoo leg-ml Figure P220 250 N -mf rad Eililii ' ' . [alllnl 35. Find the transfer function, 015} = 94(5)! HS}, fur the rotational system 5""I" shown in Figure P221. Tm I910] 94!? r] 25 N-m-sfrad 61(1) 6’3th N; = 100 W N3 = 20 I N—mi'rad 36. For the rotational system shown in Figure P222. find the transfer function. (3(3) = 6:15) This). Figure P121 Figure P222 Figure P2 23 Figure P224 Problems 119I l N-m—slrad l N-mi'rad no i 9H“ — 10 N4: 10 3 I 0.04 N—m-sfrad 37. For the rotational system shown in Figure P223, write the equations of motion from which the transfer function, 6(5) = 61(53)- T{s), can be found. Tm Sltn " N W 0' 1 Jrl 38. Given the rorational system shown in Figure P224, find the transfer function, 6(5) = 95(5) 61(3). 3M} 12’ D h D Tm efim - i _ N4 1 0 14.9 “H ‘ D K: 39. In the system shown in Figure P225, the Inertia, J', of radius, r, is constrained to move only about the stationary axis A. A viscous damping force of translational value 1;. exists between the bodies I and M. If an external force, fir), is applied to the mass, find the transfer function. (3(5) = 9159):? (s). 120 Chapter 2 Modeling In the Frequency Domam Figure P225 40. For the combined translational and rotatlonal system shown In Figure P226. find the transfer function, 6(3) = X(s) H5}. Figure F226 1 J' = l kg—rn- Tit] l Nrn-sa'rad ’ l lag-m2 N4 = 60 N2 = N3 = 02— l N—rn-slrad l 41- Given the combined translational and rotational system shown in Figure P227, find the transfer function, G(.\) = X0.) Tfs}- Figure P227 Tm Figure P223 9 (it) Figure P229 Problems 121 42. For the motor, load, and torque-speed curve shown in Figure P228, find the transfer function, 6(5) = 6,15), 150(5). NI =50 D] = 2 N-rn-sfrad 61: r- N2 = [50 " a :2 = 13 kg-mz D; = 36 N—m-sftad T(N-m) 100 50 V a) (radial 150 43. The motor whose torque-speed characteristics are shown [11 Figure P229 drives the load shown in the diagram. Some of the gears have inertia. Find the transfer function. Gts} = 62(5). E45]. £11” _ J'l: I kg_m2 N2 = 20 | ‘NJ = 10 _ _ 2 = — 2' :1 i 2 kg m 13 2 k3 '“ 53") D— 32 h—m—sfrad N4=20 If”! 3 J4: 113—1112 T(N—m} 5V RPM 600 If 44. A dc motor develops 50 N-m of torque at a speed of 500 rad s when 10 volts are applied. It stalls out at this voltage with IOU N—rn of torque. If the inertia and damping of the armature are 5 kg—n'i2 and l N-m—s; rad, respectively, 122 Chapter 2 Modeling in theI Frequency Domarn find the transfer function, 6(3) = flash-EGG}, of this motor if it drives an inertia load of 100 kg-m2 through a gear train, as shown in Figure P230. Figure 92.30 9...") + "1 '5111'”! Nl= N4=SU ’ Load w 45. In this chapter We derived the transfer function of a dc motor relating the mm" angular diaplacement output to the armature voltage input. Often we want to control the output torque rather than the displacement. Derive the transfer function of the motor that relates output torque to input armature voltage. 46. Find the transfer function. 6(5) = X(s),-"Ea(s), for the system shown in Figure P231. Figure P231 1 fall} Motor NI = ID ' .r: 1 kg-mz o=1 N—tn—slrad N2 = 20 l— - Radius = 2 In For the motor: 1,, = l ltg-m2 Da = l N—m—slrad Ra=ln fi=lN-sl'm Kb = I V—sl'rad K, = l N-mm 47. Find the series and parallel analogs for the translational mechanical system shown in Figure 2-20 in the text. 48. Find the series and parallel analogs for the rotational mechanical systems shown in Figure P2.ll5(b) in the problems. 49. 50. . :IIIIIIII 51_ o "in Figure P232 52. 53. Problems 123 A system’s output, c, is related to the system's input, r, by the straight-line relationship. c = 51' + '7. Is the system linear? Consider the differential equation (I 2.x d): where f (x) is the input and is a function of the output, I. If f (x) = sin I, linearize the differential equation for small excursions near a. x = 0 b. x = 17 Consider the differential equation :13): dzx dx where f {x} is the input and is a function of the output. )5. lfflx} — e“, linearize the differential equation for 1: near 0. Many systems are piecewise linear. That is. over a large range of variable values, the system can be described linearly. A system with amplifier satura- tion is one such example- Given the differential equation dzx dx E + + 50X assume that f (x) is as shown In Figure P232. Write the differential equation for each of the following ranges of x: a. —00<x<'.—2 h. —2<x<2 c. 2<x<m For the translational mechanical system with a nonlinear spring shown in Figure P233, find the transfer function, 6(5) = X(s) 'F(s), for small 124 Chapter 2 Modehng tn the Frequency Domaln excursrons around f0!) = l. The spring is defined by x3“) = l — 3—15"). where x,(t) is the spring displacement and 1;“) is the spring force. Fiflure P233 Nonlinear —I|- f0] 54. Consider the restaurant plate dispenser shown in Figure P234, which con- sists of a vertical stack of dishes supported by a compressed spring. As each plate is removed, the reduced weight on the dispenser causes the remaining plates to rise. Assume that the mass of the system minus the top plate is M, the viscous friction between the piston and the sides of the cylinder is j; , the spring constant is K, and the weight of a single plate is Wp. Find the transfer function. 113) Fts). where F(s) is the step reduction in force felt when the top plate is removed, and H5) is the vertical displacement of the dispenser in an upward direction. Figure P2 34 Plate dispenser ' Piston Progressive Analysis and Design Problem 55. High-speed rail pantograph. Problem I? in Chapter I discusses active con- trol of a pantograph mechanism for high-speed rail systems. The diagram for the pantograph and catenaty coupling is shown in Figure P2.35(a}. Assume the ...
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ch2 nise - 110 lynholle Hath Symbol“: Math J Chapter 2...

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