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This 18 page Class Notes was uploaded by Jada Daniel on Wednesday September 23, 2015. The Class Notes belongs to MEM255 at Drexel University taught by Staff in Fall. Since its upload, it has received 17 views. For similar materials see /class/212406/mem255-drexel-university in Mechanical Engineering at Drexel University.
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Date Created: 09/23/15
MEM 255 Introduction to Control Systems Review Harry G Kwatny Department of Mechanical Engineering amp Mechanics Drexel University What you should know Computing Laplace transform pairs Computing time responses using partial fraction expansion Solving ode s using Laplace transforms Assembling state variable models Computing transfer functions from state space models Computing state space models from transfer functions T he following are typical exam questions Some with solutions Laplace Transform Pairs Using the Laplace transform de nition derive the transform of the following time functions uI IuI sin quI coszuI Using Laplace transform short table and theorems derive the transform of I3u I e m sin wI u I e m cos an I Example Laplace Transform fft Fs Jowfte 5 dt 51 0 L ulttgte dtl0 0 ramp Jowfte dt I te stdt recall integrtation by parts Judv uv Jvdu pasta J isde 5 Leis S s 0 J e st 0 0 ijoe dt s i s2 Example Laplace Transform sin at I f t e s dt 1 sin mte s dt 1 e dcos at e cos at I cos 0t se dt 1 s no J cosmte lt 0 a a 1 s no 5 1 s X no no X 2J e dsmcot 2 e smmtl sJ smmte dt 0 a 0 a a 0 0 no 1 sz no J sm cote dt 2J sm at e s dt 0 a a 0 0 J sincote39 dt 2 2 0 3 60 Example Laplace Transform 9quot sinwt o from the short table 3 sinwt s2w2 o from Laplace transform Theorems if Fs Sfeimft Fsa Thus f 6 sinwt a sazwz Computing Time Response 115 YS Compute step and Impulse response of drive load load Systems on the system dynamics kinematics 15 5 1 1 115 s 5 s 2 2s 1 L um U I S 51 ls I s Example US Us s1 322 s 8s 64 impulse response US 1 1 31 4 S s2 8s64 341JJ s417j Ys3 7 yt 61841H E cfeim e cle m 6871443 8 2Recle 4 E c 7 mm M S41j S1 7 fang5 17 H 4 1 3 4139J S 407145 741j 417 3 732447145 clei A E 44ij39xgcos4 tijsin4 J t160054J5174xgsin4x51 yt216cos4x5174Jgsin4xgt Example drive load system load dynamics kinematics W m Step response Us1s 55 s2 s 5 5 Y U 7 S s5s2s S s5s2s2 my Cl C2 gg s5 s2 s s2 5 5 1 5 5 cl 2 c2 2 7 s2s 55 725 35 s5s 52 28 c c 1 5 5lt72Sgt 2 32 532Ho 2 31 5 s 5 HO 107ss22 0 100 H yo eetuoygeetuum State Space Models R Determine the state space model for the magnetic levitation system shown The linearized governing equations are d 2h 7 m 2 7 705139 dt dz39 VtL R39 dt 1 equmbnum dh 777777777777777 V d h o 1 o h o J 373 gt iv 0 0 fatm v 0 Vz dz m dt dl R 1 z 0 0 iRL z 1L 7 1 Vt State Space Models The linearized governing equations for the vehicle are my72Cyi7972Cf 7975 0 0 J 2bC39 y7 bg79 72aCf M7975 V0 V0 Put the equations in state space form State Space Models xed plate The governing equations for the capacitive microphone are m56cxkx Lq q0 ft 8A 1 LqRq qqo W 814 Put them in state space form Transfer Function Models II 19 Li Derive the transfer function Qm gt QM for the two tank system shown all1 1 h A dt R 011 2 Qquot dh2 1 hZ A2 dt E011 hi R2 Q0142 My Example Transfer Function assume zero initial conditions 1 1 1 AlsHl 7EH17HZ QM AZsHZ EH17HZ7R ZHZ solve for H2 1 1 A1S jHl HZ Qm R1 R1 1 1 gt R1 Als Azs HZ H27Qm 1 1 1 R1 R R1 7 H1 Azs HZ0 R1 R1 R2 R HZ 1 R1 Alsi Azs i 1 R1 R1 R2 Transfer Function Models I We ring torq e uf I Derife the transfer function u x for the moon lander J u mj T N Transfer Function to State Space 15 Y 322541 S 8S64 Derive state space models For the systems on the right drive load load system dynamics kinematics lS 1 S 2s 1 2 um U I o S 51 Exam le tf2 ss 2s1 2 5 2 5 4 51 1 25mUxgtY55310521654Zs Z 739z 14z 8zuy310216i42 Uv 175 s310sz16s4 Y U S s37sz14s8 S xlz xzxlz x3xzz X3 77x3 714x2 78x1 11 y 77x3 714x2 78x1 110c3 16xZ 4xl x1xz xzx 3 X 7x3 44x2 78x1u y74x12x23x3u Summary Computing Laplace transforms Inverting Laplace transforms via partial fraction expansion Computing system responses Building state variable models from governing equations Building transfer function models from governing equations Converting transfer functions to state variable models
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