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# Linear Systems Theory ME 3253

UCONN

GPA 3.82

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This 27 page Class Notes was uploaded by Mr. Sean MacGyver on Thursday September 17, 2015. The Class Notes belongs to ME 3253 at University of Connecticut taught by Chengyu Cao in Fall. Since its upload, it has received 38 views. For similar materials see /class/205870/me-3253-university-of-connecticut in Mechanical Engineering at University of Connecticut.

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Date Created: 09/17/15

ME 3253 Linear Systems Theory Midterm Review 1 System modeling Mechanical system Electrical systems water level system etc 2 Transient performance of first and second order system curve shape overshot settling time 3 Routh s Stability Criterion 4 Frequency domain analysis Bode plots Mechanical Systems Friction Static friction not moving Sliding friction translational motion sliding friction coefficient smaller than static friction coefficient Rolling friction friction force magnitude unknown Rolling without sliding x R6 Example of Mechanical Svstems 1 Dynamic equation ODE 2 Laplace domain 21 Input output signal 22 Transfer function 3 Laplace transformation of input signal 4 Laplace transformation of output signal 5 Inverse Laplace transformation 9 Output signal Properties 1 Stability real part of components 2 Natural frequency imaginary part of components 3 InitialEnd values Lapace Transform Differentiation Theorem 14 4 sFsf0 udzf 4 s2Fs sf0 f0 alt2 Final Value Theorem if iim f t exists lim ft lim sFs t oo s gt0 Initial Value Theorem f 0 lim SF s Electrical systems Basic Laws continue For single loop circuit loop law is enough For circuits with multiple loops need to use both loop law and current law Dynamic modeling of circuits combination of circuit laws and basic element definitions what does this mean Read Example Figure 610 very very carefully Examples 61 and 62 pp 261263 of Ogata Electrical Systems Complex impedances directly write system equations in Laplace domain E s Z sI s eiR EIR ZR eL ELsI ZLs dt 1 1 1 39dt E I Z e CIl Cs CS Advantage can be used in parallel and series circuit modeling just like resistances Only suitable for Transfer Function Derivation why How to Solve Electrical systems 1 Loop law Voltage equations 2 Node law Current equations 3 Basic element definitions In Laplace domain a set of linear equations Laplace transformation of input signals Solve to get Laplace transformation of output signal InverseLaplace transformation Initial values Basic Elements 1 Resister 2 Capacitor 3 Inductor 4 Source Voltage and Current Sign convention i i i rm 4 Initial Conditions Consider Initial Conditions eiR ifgt EssR di eLE EsLsIs i0 e 2 I W 1s CsEs e0 Current Law Kirchhoff s current law node law the sum of all currents entering and leaving a node is zero sign convention entering leaving Loop Law Basic Laws continue Kirchhoff s voltage law loop law the sum of voltage around any loop is zero Sign convention 1 Choose a loop direction 2 A rise in voltage is positive going through an active element from negative to positive terminal a rise in voltage Example Example problem of Electrical System Review Prob B65 EOM Loop law and node law Rlil R2i1 i2 0 Ii2dtR2i2 il0 Solving for the Inteqral Equation using Laplace Transform Integration theorem page 27 R1I1sR2I1sIzs0 il 0 C R212SI1s0 S S Plug in initial condition i w li2tdtloq0eoc And solve System Modeling First order examples of water level system hydraulic system thermal system Concept of linearization covered in class For a nonlinear system how to perform linearization around equilibrium point to get a LTl system Transient Performance Step response of first and second order systems Steady state value Settling time Time constant Overshot Damping rate of second order system Natural frequency of second order system Routh s Stability Criterion B Given transfer function TOY Ag Stable All roots of As in left half plane How to check stability without solving As0 Routh s Stability Criterion Methodology To Pg 539 In the text book As 1 Write AS 6105quot 6113quot 1 an 2 Let O 27 km J29 52 v a 4 we f gzw w v gz mcx N O y all gt 01 Not Stab39e quot 39 x 13 End a 959 I r 39i Vt Yes SEP3 1 Routh s Stability Criterion Methodology 3 Build pattern S a0 a2 a4 a6 1 S a1 a3 a5 a7 a a a a a a a a 2 Sn b1 2 1 2 0 3 b2 1 4 0 5 a1 a1 S 61 1 b1 b1 0 S g1 Stable First column gt0 Unstable Roots with positive real parts equal To the number of sign changes in first column Frequency analysis Bode Plots Given transfer function T S Outs T sIns Stable system Sinusoidal Input Sinusoidal output in steady state Int sinat Steady state Outt M sinat 5 M Tja Magnitude of T jco angleTja Angle of Tja It is equivalent Tja Mcos 5 j sin 5 Bode plots X axis frequency a in logscale Magnitude 2010g M 2010ngja Phase angleTja Example Bode Plots of First Order System a First order system T S sa a a jaa 6161 101 Ta 2 2 Jaa Jwa Jaa a 0 clecz2w2 a 0 a a 39 a 2 2 39kj 2 2 2 2 J 2 2 aa aa aa aa aa aw Mwcos w J sin w Where a a M a 2 a arcsm W Input Int sinat Frequency response Steady state output response Outt M wsinat Example Bode Plots of First Order System Bode plots Magnitude plot Plot 2010gMa with respect to logw Phase plot Plot a with respect to Ing Magnitude plot when a gt O 2010gMa 2010g L 2010gij 0 la2 Q2 xai2 when a gt oo 2010gMa 2010g L 2010g a laza2 W 2010ga 2010ga Use these two lines to approximate bode plot Example Bode Plots of First Order System Line 1 2010gMa 0 n z 20 40 1 0 1 E 2 3 10 01 1 10 a 100 1000 logm 60 I a 10 Intersection point a a Reason 2010ga 2010ga 0 when a a Example Bode Plots of First Order System Linel 2010gMa 0 2010gMa W fIEXLine 2 2010gMa 2010ga 2010gw n kquot 20 40 1 0 1 i 2 3 M 1 f a lologa Intersection point a a Reason Line 1 is correct when a is small Line 2 is correct when Wis large Red line Approximated bode plot Example Bode Plots of First Order System Line 1 2010gMa 0 2010gMa W fIEXLine 2 2010gMa 2010ga 2010gw n k 20 40 01 60 1 10 6 100 1000 wzlologa Intersection point a a Red line Approximated bode plot Blue line True bode plot Example Bode Plots of First Order System Bode plots Magnitude plot Plot 2010gMw with respect to 10 Phase plot Plot W with respect to 1ng Phase plot when a gt 0 a arcsin l w arcsin 0 Va2 02 Va2 02 7239 a When a gt 00 arCSHl WJ W arcsm 1 3 when a a a arcsin lj arcsin a xaza2 xa2 a2 J 7 arcsm 2 4 Example Bode Plots of First Order System w n A 1 O 2 3 loga quot1 n a lologa 4 2 z i 2 quot pointl a a

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