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# System Dynamics& Control AE 3515

GPA 3.79

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This 0 page Class Notes was uploaded by Demond Hoppe on Monday November 2, 2015. The Class Notes belongs to AE 3515 at Georgia Institute of Technology - Main Campus taught by Staff in Fall. Since its upload, it has received 6 views. For similar materials see /class/234313/ae-3515-georgia-institute-of-technology-main-campus in Aerospace Engineering at Georgia Institute of Technology - Main Campus.

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Date Created: 11/02/15

Mathematical Modeling of Hydraulic Systems ME3015 System Dynamics 8 Control Hydraulic systems are used in many realworld systems including machine tool applications aircraft control systems automobiles They have high power to weight ratio and fast starting stopping and reversal actions They often operate under a wide pressure range 145psi5000psi Powerweight ratio can be increased by increasing operating pressure Example of a Hydraulic Circuit ME3015 System Dynamics 8 Control Basic Lump Elements of Hydraulic Systems ME3015 System Dynamics amp Control Resistors are used to model pressure drop due to an orifice friction leakage etc Hydraulic resistors are often nonlinear Capacitors are used to model the compressibility ofa hydraulic uid Assuming an incompressible uid uid capacitance may be ignored lnertance is often insigni cant thus ignored Modeling an Orifice ME3015 System Dynamics amp Control M 9 AP Laminar low Ap q ApR R ow resistance Turbulent high Ap Power Steering Model ME3015 System Dynamics amp Control M lsteering t J 9 b Iquot Q f 39 I M I u u v R m l 9mm SUPPly Controlvalve and y h 1quot p werpist t T Ration I T 9 k S b t A miR lquot3939m39 mu i F ngwgw umln 39w quot L t F x M 1quot Am 1 Frame 39 A Hydraulic ServomotorAmplifier ME3015 System Dynamics amp Control Modelinq obiective relate Po 3905 p0 Basic flowpressure relationship output y t0 inPUt X3 T l T 9 X q1Cpsp1 2X 2 C12Cp239po12X 1 C constant 42 q1 Assuming turbulent high T lt 0 pressure differential flow p2 p1 Simple Model M53015 System Dynamics amp Control Assume lncompressible flow lgt No uid leakage q1q2 also implies that pSp1p2popsp1p2 Letting App1p2 then p1 and p2 can be expressed as p1ps APVZ and p2ps AP2 Thus P AP 0 7 XC 5 XfXA 71 p5 pl 2 P Linearizing about 7 0 A5 0 gives Igt ViA q1Cl2Adydt q1K1x K1C A more realistic Model M53015 System Dynamics amp Control We assume incompressible uid q1q2 but allow Fluid leakage Load inertia and damping Q1QLAV qL ow leakage ApR R ow resistance laminar Mechanical subsystem FAAp mdvdtbv Frequency Response FR Principle gt Linear u A sinoot 3112f yss A0 sinogtt cp Mi Ar G Transfer function of the system IGfw P 46103 Bode Diagram Ob39ective Plot of Magnitude db and phase pz 300 vs scale Mdb20ogM decibel Basic Factors siN Ts1 iN Slope 20N dbdec forsi asymptotically as s jw mo Phase 90N dbdec for asymptotically as s jw mo MIGJwl in 0 in semilog 2nd order factor N and Ts1 iN siN and Ts1 iN 2nd order factor 2 Transfer 68 2 function 3 2Q n3 0 032 1 G n 2 Di 2 12CJOJHOJ 1B2 1 M Z f 1i3 2 213 2Q 03 t 1 P an 1B2 B con Asymptotic Behavior 3 M Mdb P M IGUm 2 0 1 0 0 H32 2q5 1 o o P tan 1 2CB2 3 sma 3 m 3 M 2010gMr Resonance Minimizing g31 3222g 32 with respect to 3 Le dgd30 yields 1 294 2010g2 9 Beth 2 0 so707 O large 3392 4010g3 Resonance 1 M 0 180 magnitude 2Q I1 2 0quot Asymptotic Plot Buds Diagram 8 deg Magnitude dB Fha mi i Frequency radsec Bode Diagram for Different Damping Ratios d2 Diagrams Um i i i 4 P 2 i A a r x a 7 a a r g i i g i i 3 J E E i i g i i 2 i i Bode Diagram of Composite Transfer Functions Suppose that GG1G2Gn then Gj lGij llej lGnj M M1M2uMn Mdb M1db M2db quot39 Mndb PZCP1P2H39PH Superposition principle applies Composite Example Asymptotic bode diagram of 500 s 05 35 2 ss 253 2500 Put in timeconstant form GS500 gtltO5 2s1 2500 s2 25s 2500 2500 Break up into basic factors 012s1 1s and 2500s225s2500 20db Magmtude dB Phase deg mm Asymptotic plot Bode Dtagrams HW WU Frequency radsec Factors K01 Vs 231 2500 2 253 2500 d6db Magmtude dB Phase deg TaW Actual Bode Plot Bode Dtagrams mmww t Frequency radsec k Ts1 Frequency Domain Identi cation Bode Diagram of G s 10 0 8 8 m 3 3 E c 5 E g 5 77777777 n g 5 g g H a a D i t t 39 quotquotquot E E g 7 e e e r 7 g g o r r 7 7 r quotiquot U U m e e a n n uni 20 0 10 MA at an we 6ep eseqd gp epmtu zw 66p esqu Another Identification Example 2010gk y u requency radsec ia Transfer Function Estimation Initial slope40 phase180 G1 ls2 Corner frequency at 05 slope changes by 20 G22s1 G3 2nd order wn2 radsec d14 db51l2 thus QOfl Gs4s204s4 l GG1G2G32s1s2s204s4 l Experimental System Identi cation Method Sweeping Sinusoidal f lit fl Magnitude Phase Method ll Random Input rivaling Transfer function is determined by analyzing the spectrum of the input and output Random Input Method The transfer function is estimated by findingthe ratio of Yjw Fourier transform of y and Ujn Gum Z FFT f9 tp9t jog FFTof Input Suitable filtering is often used to reduce noise and other uncertainties Random Input Response Matlab Commands to get Bode plot i gt 96 Create Random Input 0 UWWWWWW input gt 96 Collect system response y to input 0 gt zdetrendYU gt Gspaz gt GssettGTs 9sspecify sampling time Ts Output gt bodeplot Gs 2 a Time sec Experimental Bode Plot AMPUTUDE PLOT mput 1 outputf 1072 H m a III II 10 uu39 H H a w I a 1076 LL V g AMA H i H W 33951 LLle 3 0 we 10 10 10 T PHASE RWGMFWeEQApmw m n w m I m m ml 2 m H 39 I m n o 10 frequency radsec Photo Receptor Drive Test Fixture Experimental Bode Plot System Models Magmtude ea Frequency Hz Frequency Hz RootLocus Revisited Consider a unity feedback loop with openloop transfer function Gs KBltsAs H The rootlocus plot shows the location of all possible CLP s and is as the loopgain Kvaries from 0 to infinity satisfy Examgle 33 2 RootLocus Plot lrriag Axis r 70 5 Real Axis Constitutive Relationships for Basic Mechanical Elements ME3015 System Dynamics amp Control Element Translational Rotational Mass F lgt X T A Inertia M J 6 FmVmgt lt TJcbJ Spring LEW Lsz T W F kX1 X2 T Ke192 Damper X1 X1 T 91 9 T FLELFW F b lt1 lt2 T B 162 Modeling Procedure ME3015 System Dynamics amp Control 1 Draw a FreeBodyDiagram of each element showing all the forcesmoments acting on it 2 Write down the constitutive relationship between the exerted or transmitted force or torque and the corresponding motion variables for each element including power transforming elements 3 Eliminate auxiliary variables to obtain a system of differential equations for the system with the same number of equations and unknowns 4 If desired reduce the system of diff eqs in 3 to a single differential eq relating the output to the input of the system MassDamper Example ME3015 System Dynamics amp Control Derive a mathematical model relating the angular velocity of the rotor to the input torque of the rotor shown FreeBodvDiaqram Other Examples ME3015 System Dynamics amp Control I MassSpringDamper System I 2 Degree of Freedom Systems onslational o Rotational I Gear Train Example School of Aerospace Engineering Georgia Institute of Technology AE3515 Modeling of Mechanical Systems Mathematical Modeling of Mechanical Systems Objective Use applicable physical laws to derive a set of differential equations describing the behavior of a mechanical system Basic Elements Mass or Inertia Spring Damper Modeling Procedure Draw the freebodydiagram of each element 2 Write down the constitutive relationship between the exerted or transmitted force or torque and the corresponding motion variables for each element including power transforming elements 3 Eliminate auxiliary variables to obtain a system of differential equations for the system with the same number of equations and unknowns 4 If desired reduce the system of diff eqs in 3 to a single differential eq relating the output to the input of the system Lagrangian Mechanics Motivation Hamilton s Principle of Least Action Lagrange s Equation of Motion Examples Motivation Lagrangian formulation provides an alternative method to Newtonian formulation with the following advantages It is energy based Requires no freebody diagram It can be formulated based on kinematics analysis Lagrangian formulation leads to equations of motion without any need for freebody diagram Generalized Coordinates Degrees of Freedom The minimum number of coordinates needed to uniquely describe the motion of mechanical system Generalized coordinates denoted by q i1 n are a set of n independent coordinates that uniquely determine the configuration of an ndegree of freedom given mechanical system Hamilton s Principle 1834 Ofall the possible paths along which a dynamical system may move from one point to another within a speci ed time interval consistent with any constraints the actual path followed is that which makes the integral ofthe difference between the kinetic and potential energies stationary J fLqlttqlttgttdt where LLagrangian Tqqe Uq and q is the vector of generalized position Variational Principal According to Hamilton s principle J is stationary t t 2 2 0L 0L 51 8Lqqtdt Sq 5q jdt 0 Where 6 denotes functional variation By integration by parts r 12 r 2 0L 0L 2 d 0L 8th 8q 7 8th aq aq h dt Enforcing zero variations at the boundaries 6qt1 6qt20 2 5JI i 75tho VESq dt aq 0q 1 Lagrange s Equation of Motion No external nonconservative forces 7 i O i 12 n dt aqi aqi In the presence of external forces not included in the potential function U the Lagrange s equation of motion is given by iEij i ff i 12 n dt aqi aqi Where fi is the generalized force in the qi direction nonconservative forces

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