# Modeling of Dynamic Systems

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1 Modelng of Dynamc Systems Ref: Control System Engneerng Norman Nse : Chapters & 3 Chapter objectves : Revew the Laplace transform Learn how to fnd a mathematcal model, called a transfer functon Learn how to fnd a mathematcal model, called a state varable representaton How to convert between transfer functon and state space models How to lnearze a non-lnear system Modelng - 1

2 Modelng n Frequency Doman Laplace Transforms Revew 1. Standard notaton n dynamcs and control (shorthand notaton). Converts mathematcs to algebrac operatons 3. Advantageous for block dagram analyss Laplace transforms can be used n process control for: 1. Soluton of dfferental equatons (lnear). Analyss of lnear control systems (frequency response) 3. Predcton of transent response for dfferent nputs Modelng - 3 Modelng - 4

3 Modelng - 5 L(f (t)) = - 0 f -st (t)e dt Examples -st st L(a)= ae dt e bt -bt -st -(b+s)t ( b+ s)t 0 0 df df -st L(f ) L e dt a a a = = s = s s 1 1 L(e )= e e dt = e dt = -e = b+s 0 s+b = = = dt dt 0 sl(f) f(0) Usually defne f(0) = 0 (e.g., the error) Modelng - 6 3

4 Table Laplace Transforms for Varous Tme-Doman Functons f(t) F(s) Modelng - 7 f(t) F(s) Modelng - 8 4

5 f(t) F(s) Modelng - 9 LT Example 1 Solve the ODE, dy 5 4y y( 0) 1 (3-6) dt + = = + = s Frst, take L of both sdes 5( sy ( s) 1) 4Y ( s) Rearrange, Y ( s) Take L -1, y( t) 5s + = s ( 5s + 4) 1 5s + = L s( 5s + 4) (3-34) 0.8 From LT Table, y( t) = + e t (3-37) Modelng

6 Partal fracton expanson 1. Case 1. Roots of the denomnator of F(s) are real and dstnct. Case. Roots of the denomnator of F(s) are real and repeated 3. Case 3. Roots of the denomnator of F(s) are complx or magnary Modelng - 11 LT Example 3 d y d y dy 3 dt dt dt y = 4 y( 0 )= y ( 0 )= y ( 0 )= 0 system at rest (s.s.) To fnd transent response for u(t) = unt step at t > 0 1. Take Laplace Transform (L.T.). Factor, use partal fracton decomposton 3. Take nverse L.T. Step 1 Take L.T. (note zero ntal condtons) s Y(s)+ 6s Y(s)+ 11sY(s) + 6 Y ( s ) = s 3 4 Modelng - 1 6

7 Rearrangng, Y(s)= 3 ( s + 6 s + 11 s + 6) s 4 Step a. Factor denomnator of Y(s) s(s 3 + 6s + 11s+ 6 )=s(s+ 1)(s+ )(s+ 3) Step b. Use partal fracton decomposton 4 α1 α α3 α4 = s(s + 1)(s + )(s + 3 ) s s + 1 s + s + 3 Multply by s, set s = 0 4 α α3 α4 = α1 + s (s+ 1)(s+ )(s+ 3 ) + + s + 1 s + s = α1 = s= 0 s= 0 Modelng - 13 For a, multply by (s+1), set s = -1 (same procedure for a 3, a 4 ) α =, α 3 =, α4 = 3 /3 Step 3. Take nverse of L.T. ( Y(s)= + ) 3 s s + 1 s + s + 3 t t 3t y(t) = e + e e 3 3 t y(t) t = 0 y(0) = 0. (check orgnal ODE) 3 You can use ths method on any order of ODE, lmted only by factorng of denomnator polynomal (characterstc equaton) Practce Matlab.. Modelng

8 Mathematcal model a descrpton of a system n term of equatons Types of model Models of dynamcs system can be of many knds, ncludng the followng: Mental, ntutve or verbal models Graphs and tables Mathematcal models Constructng a model: Mathematcal modelng or frst prncple modelng. Process or system dentfcaton. Modelng - 15 One example of dynamc system automoble Modelng

9 The procedure of dervng models : Understand the physcs and nteracton of the elements Construct a smplfed dagrammatc rep of the system Apply element and nterconnecton laws Draw the FBD Identfy or defne the nputs, outputs and state varables Establsh the system equatons Obtan the desred form of the system model If the model s non-lnear, determne the equlbrum condtons and obtan a lnearzed model. Modelng Mechancal Systems translatonal & rotatonal Translatonal mechancal systems can have only horzontal or vertcal moton a. State varables Varables of trans mechancal systems are : x, dsplacement n meters (m) v, velocty n meters per second (m/s) a, acceleraton n meter per second square (m/s ) f, force n Newton (N) w, energy n Joule (J) p, power n watts (W) all varables are functons of tme Modelng

10 b. The element laws nclude n trans systems are mass, frcton and stffness. They relate the external force to the acceleraton, velocty and dsplacement assocated wth the element. Mass Frcton when two bodes slde over each other there s a frctonal force f between them that s a functon of the relatve velocty between the sldng surface Modelng - 19 Modelng

11 Stffness when a mechancal element s subjected to a force f and goes through a change n length x, t can be characterzed by a stffness element. Modelng - 1 c. Interconnecton Laws D Alembert s law s developed from Newton s law for translatonal system. For a constant mass : dv ( fext ) = M dt where the summaton over the ndex ncludes all the external forces (f ext ) actng on the body. dv ( fext ) M = 0 dt The sum of the forces s zero provded -Mdv/dt s thought of as an addtonal forces ths fcttous force s called nertal force or D Alembert force. f = 0 D Alembert s law Modelng - 11

12 The law of reacton force accompanyng any force of one element on another, there s a reacton force on the frst element of equal magntude and opposte drecton (Newton s thrd law : reacton forces) Modelng - 3 The law for dsplacements f the ends of two elements are connected, those ends are forced to move wth the same dsplacement and velocty. Modelng - 4 1

13 d. Obtanng the system model FBD Example 1 Modelng - 5 Example Modelng

14 Example 3 Example 4 Modelng - 7 Seres and parallel combnaton k k 1 k eq = k1 + k = b b b 1 eq b + 1 b k eq = k 1 + k b eq = b 1 + b Modelng

15 Rotatonal mechancal systems are modeled usng the same technques as those for translatonal mechancal systems a. State varables Varables of trans mechancal systems are : θ, angular dsplacement n radans (rad) ω, velocty n radans per second (rad/s) α, angular acceleraton n radans per second squared (rad/s ) τ, torque n Newton-meters (N.m) all varables are functons of tme Modelng - 9 b. The element laws nclude n rotatonal systems are moment of nerta, frcton, stffness, levers and gears. Moment of Inerta J n klogram-meters (kg.m ) J = r dm The net torque appled about the fxed axs of rotaton s gven by d τ ( Jω) dt = where Jω s the angular momentum of a body. Parallel axs theorem states J = J 0 + ma where r s the dstance from the axs of reference and dm s the mass of the small element. where a s the dstance between the parallel axes and J 0 s the moment of nerta about the prncpal axs Modelng

16 Frcton A rotatonal frcton element s one for whch there s an algebrac relatonshp between the torque and the relatve angular velocty between two surfaces. Rotatonal vscous frcton arses when two rotatng bodes are separated by a flm of ol. ( ) τ = b ω = b ω ω 1 Rotatonal devces characterzed by vscous frcton Modelng - 31 Stffness s usually assocated wth a torsonal sprng, such as the mansprng of a clock, or wth a relatvely thn shaft. For a lnear torsonal sprng or flexble shaft, τ = k θ where k s the stffness constant wth unts of newtonmeters (N.m) and θ=θ -θ 1 (a) Rot stffness el wth one end fxed. (b) Rot stffness el wth θ=θ -θ 1 Potental energy s stored n a twsted element and for a lnear sprng or shaft s gven by 1 W = kθ Modelng

17 The Lever an deal lever s assumed to be a rgd bar pvoted at a pont and havng no mass, no frcton, no momentum and no stored energy. Let θ be the angular dsplacement of the lever from the horzontal poston. The lever For small dsplacements d d d x = v = v1 f = f1 d d x1 d1 1 1 Modelng - 33 Gears an deal gear s assumed to have no moment of nerta, no frcton, no stored energy, and a perfect meshng of the teeth. N s a gear rato r n N = r n = where r and n denote the radus and number of teeth 1 1 A par of gears θ 1 and θ are the angular dsplacements for the gears Modelng

18 θ r r 1θ 1 = rθ ; 1 = N ; θ r = 1 ω1 ω r = N r = 1 θ 1 and θ are the angular dsplacements for the gears Ideal gears (a) ref poston, (b) after rotaton τ ω + τ ω 1 1 = 0 FBD for a par of deal gears Modelng - 35 c. Interconnecton Laws D Alembert s law for a body wth constant moment of nerta rotatng about a fxed axs, ( τ ext ) J & ω = 0 where the summaton over ncludes all the torques actng on the body, and the term Jω can be consdered an nertal torque. τ = 0 D Alembert s law the torque Jω s drected opposte to the postve sense of θ, ω and α Modelng

19 The law of reacton force for bodes that are rotatng about the same axs, any torque exerted by one element on another s accompaned by a reacton torque of equal magntude and opposte drecton on the frst element. The law for angular dsplacements Rot sys to llustrate the law of ang dsp The reference marks on the rms are at the top of the two dsks when no torque s appled. The net angular dsplacement for the shaft K wth respect to ts unstressed condton s θ -θ 1. ( θ ) = 0 around any closed path Modelng - 37 d. Obtanng the system model FBD (a) Rotatonal system, (b) & (c) Its correspondng FBD Modelng

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