EE 4314 Lab 2 Handout for Workbench #1 Modeling and Identification of the Double-Mass-Spring-Damper System Fall

Transcription

1 EE 434 Lab Hadou for Workbech # Modelig ad Ideificaio of he Double-Mass-Sprig-Damper Sysem Fall IMPORTANT! This hadou is for hose who are assiged o Workbech #. Please check your lab schedule o see which workbech you are assiged o.. Lab Iformaio a) Dae, Time, Locaio, ad Repor Due Please check your lab schedule o Blackboard for your lab dae, ime, ad workbech. The lab will be held a NH5. Your lab repor is due o Wedesday a :59 pm wo weeks afer your lab sessio. b) Prelimiary Work You are required o complee he Prelimiary Work give i he ex secio of his hadou ad o read he res of he hadou before you come o he lab for he experime. A he begiig of he lab, he TA will check your prelimiary work. Ge a priou of your aswers ad resuls ad show i o he TA whe you go o your lab sessio. c) Lab Repor Afer he lab, you will work o he problems a he ed of he hadou o your ow ad are required o submi a repor o ha. Please check he iformaio abou he assigme i he lab repor secio a he ed. d) Some Noes ad Rules - Do o forge o sig he aedace shee. - Logi o he accou creaed for EE434 ad use he folders provided o save your work. - Do o chage he experime hardware seup uless isruced by he TA or by he lab procedure i he hadou. - Cooperae wih your group maes o icrease your efficiecy i performig he experime. - Brig a USB memory device o ge a copy of he experime daa a he ed of he lab. - Keep clear from he movig pars of he experime seup whe i is ruig. 9/9/ Page

2 . Prelimiary Work This par of he hadou provides he prelimiary iformaio ha you eed o udersad he experime, coduc i, ad ierpre he resuls. Follow he seps below o complee he prelimiary work. Priou your aswers ad resuls ad prese i o he TA whe you go o your lab sessio. The block diagram of he double-mass-sprig-damper (DMSD) sysem ha will be used i his lab is show i Figure. The combiaio of masses, sprigs, ad dampers ca be foud everywhere i idusry ad i our everyday life. Some commo examples are he vehicle suspesio sysems or he ai-vibraio sysems i CD-ROM drives. The followig are he parameers of he sysem: Table Physical parameers of he double-mass-sprig-damper sysem Masses m (.77 kg) m (.59 kg) Cosas for he viscous fricio of he rail ad he damper c (kg/s) c (kg/s) Elasiciy coefficie (siffess) of he sprig k (kg/s ) Moor volage o force geeraio cosa f (C/m) The goal of his lab is o prese a simple mehod for ideificaio of sysems ha ca be spli io a se of secod-order subsysems wih complex-cojugae poles. Sarig from he physical priciples, a se of secod-order rasfer fucios will be creaed for he experimeal sysem. The respose of he sysem o zero ipu ad ozero iiial codiios will be observed ad used o deermie he ukow parameer values c, c, k, ad f. Usig he differeial equaios ad he ideified parameers, a sae-space model for he sysem will be buil. The validaio of he model will be doe by providig he same ipu o he real sysem ad o he model ad by comparig heir oupus. This model will be used i he fifh lab o desig a sae-feedback coroller. Moor Moor force (F) Posiio Ecoder Posiio Ecoder Damper fricio is uable (do o chage) Roaio o raslaio coversio Viscous rail fricio (c ) Mass (m ) x Sprig (k) Mass (m ) Figure Double-mass-sprig-damper sysem seup Combiaio of damper ad rail fricio (c ) The physical sysem show i Figure ca be modeled wih he diagram show i Figure. Noe ha c represes he viscous dampig due o he fricio bewee he rail ad Mass whereas c represes he combiaio of he fricio bewee Mass ad rail ad he fricio due o he damper. I realiy, however, here is also oliear Coulomb fricio ha is egleced i his model. x 9/9/ Page

3 x x F m k m Figure Sysem model digram The dyamical equaios ha model he sysem i Figure are as follows: m x F k x x ) c x, m x k( x x ) c x, F f u () ( a) Usig free body diagrams ad Newo s law of moio, derive he firs wo equaios i (). Derivig he equaios i () meas ha you have o sar wih he fudameal laws of physics ad eveually arrive a he form of equaios give i (). For isace, he acceleraio of Mass is due o he e force acig o i ad he e force ca be foud based o he free body diagram of he mass. The same is rue for Mass as well. b) Apply Laplace rasform o he firs wo differeial equaios i () assumig zero iiial codiios. You ca replace F i he firs equaio wih he expressio i he hird equaio. Obai he s-domai equaios i he followig form X X G G U( s) G X c c 3 X where G, G, ad G 3 represe secod order rasfer fucios. Figure 3 shows block diagram represeaio of he sysem. () U(s) G (s) + X (s) G (s) X (s) + G 3 (s) Figure 3 Separaig he rasfer fucios of he sysem c) Thik abou a way o ideify he parameers, c, c, k, ad f, of he DMSD sysem. The idea i his experime is o aalyze he aural resposes of pars of he sysem represeed by G ad G. Aalyzig oe par of he sysem aloe requires fixig he oher o a zero sae. For isace, he aural respose of he par of he sysem represeed by G ca be observed hrough x () if Mass is fixed o is zero sae (i.e. by o leig Mass move such ha x () = ) ad Mass is le go wih is iiial sae. Similarly, he aural respose of he par of he sysem represeed by G ca be observed hrough x () if Mass is fixed o is zero sae (i.e. by o leig Mass move such ha x () = ) ad Mass is le go wih is iiial sae. 9/9/ Page 3

4 Assumig ha he masses are iiially saioary, he ipu is zero, ad x () is ozero while x () = for all, express X (s) i erms of s, m, c, k, ad x (). Similarly, assumig ha he masses are iiially saioary, he ipu is zero, ad x () is ozero while x () = for all, express X (s) i erms of s, m, c, k, ad x (). d) Rewrie each of he rasfer fucios i he followig sadard form, by makig sure ha s o he deomiaor has a coefficie of : i gi Gi (3) s s i where g i is he DC gai, ω i is he aural frequecy, ad α i is he rae of decay of he rasfer fucio. e) By ideifyig ad comparig he coefficies of he differe powers of s i (3) ad i he G i s you foud, wrie k, c, ad c i erms of m, m, ω, ω, α ad α. For isace, you should be able o fid. f) I MATLAB Simulik, do he simulaio of a secod order sysem ha ca be represeed wih a rasfer fucio as i (3) assumig zero ipu ad ozero iiial codiios. Noe ha alhough he ipu is zero, he oupu of he sysem will o be ideically zero due o he ozero iiial codiios. This is called aural respose. Assume ha he cosas of he rasfer fucio G i are as follows: ω i =, α i =.5, ad g i =. Also, choose a iiial codiio of for he oupu ad zero for everyhig else. Refer o Example 9 i Lab Hadou o see wha kid of a Simulik model you eed o creae. Observe he oupu ad ge a plo of i. Thik abou how you could aalyically solve for his oupu. Would he aalyical soluio be similar o he iverse Laplace rasform of X (s) or X (s) i par (c)? g) Creae he sae space model of he sysem i () by usig he followig defiiio: z x, z z x, z x, z z x z z z z3 z4, z Az Bu 3, i z y z Cz Specify wha A, B, ad C marices are i erms of m, m, c, c, k, ad f. (4) 9/9/ Page 4

5 3. Seig up he Experime (by TA) Simulik Model Commads for he moor Ipu/Oupu Elecroic Ui Readigs of he posiios of he masses Ecoder readigs Amplified drive sigal o he moor No available i our seup Figure 4 Picure of he mass-sprig-damper experime seup This par will have bee doe by he TA before he lab begis. This secio is for your iformaio. The compuer rus a Simulik model i real-ime. The iformaio from he ecoders is se o he compuer ad is processed by he model. The commad o he moor is a volage ha is amplified by he Ipu/Oupu Elecroic Ui ad se o he pla. There are wo icremeal roary shaf ecoders used i he mass versio, Model. Each has a resoluio of 4 pulses per revoluio. The ecoders are all opical ype whose priciple of operaio is depiced i Figure 4. A low power ligh source is used o geerae wo 9 degrees ou of phase siusoidal sigals o he deecors as he movig plae roaes wih respec o he saioary plae. These sigals are he squared up ad amplified i order o geerae quadrae logic level sigals suiable for ipu o he programmable gae array o he real-ime coroller. The gae array uses he A ad B chael phasig o decode direcio ad deecs he risig ad fallig edge of each o geerae 4x resoluio. The Simulik model will receive 6 cous per revoluio from he ecoders. The ecoders are coupled o he masses hrough a mechaism ha rasforms liear moio io roaio. Cosiderig he radius of he ecoder piio o be.3mm, he sofware will hus see 6636 cous per meer whe he masses move. The moor receives a (variable) volage from he I/O Elecroic Ui ad rasforms i io moio. For small speeds, i ca be cosidered wih relaively good approximaio ha he force he moor acs upo he sysem is proporioal wih he ipu volage. You will see i oher labs ha he moor has some olieariies, for example he deadbad, bu his is o impora for he curre lab. The Simulik model for he experime is show i Figure 6. The gree block represes he physical DMSD sysem. I has oe ipu ad oe vecor oupu. The ipu represes he volage ha he moor receives. The oupu is a vecor ha coais he wo ecoder readigs. These readigs represe cous or he umber of pulses sice he simulaio has sared. Whe you coec o he arge, he readigs are rese o zero. The umber of cous is raslaed o he correspodig liear 9/9/ Page 5

6 displaceme of he masses hrough he /6636 meers/cou gai. The vecor sigal is demuliplexed o idividual sigals x ad x ad hese are displayed o he oscilloscopes ad se o he Malab workspace for furher processig if eeded. The Proecio block makes sure ha he moor is o drive wih a high volage ha could damage he pla. Do o chage he parameers of he proecio block! Figure 5 Operaio priciple of opical ecoders Figure 6 Simulik model for he experime The Fial Value of he Sep block specifies he volage applied o he moor. For mos of he experime, he moor will o be drive so he Fial Value ca be se o zero. Click o he Tools meu i he Simulik diagram ad, i he Real-Time Workshop submeu, selec Build Model. You will see he build process will be displayed i Malab commad widow. Oce he build is successfully compleed click he Coec o Targe buo ad afer he coecio is esablished click o he Ru arrow ha appears i he Simulik diagram. This will sar ruig he Simulik code for a umber of secods specified. Noe: For real-ime applicaios, Simulik compiles he model io a fas execuable code ha hadles he daa acquisiio ad processig. Ayime a chage is made i he Simulik file, excep chagig he value of he sep ipu or he Slider Gai, he file mus be rebuil. 9/9/ Page 6

7 x (m) 4. Doig he Experime The objecive of his experime is o ideify he ukow parameers, c, c, k, ad f, of he sysem as iroduced i secio. The, a sae space model of he sysem will be creaed. The ideificaio procedure requires you o aalyze he respose of he G ad G rasfer fucios i () wih ozero iiial codiios ad zero ipu. This meas ha you have o measure he oupu of he rasfer fucio afer you force is ipu o zero ad force a ozero value o a leas oe of he saes. From he diagram i Figure 3, you ca see ha he oupu of he G rasfer fucio is o available for measureme. Sill, i is possible o measure i hrough x if x is forced o be zero. This meas ha if you block Mass ad do o allow i o move, he G 3 rasfer fucio will o have ay effec o x ad you will see he oupu of G o he x measuremes. The exac formula for he respose ca be calculaed usig he Laplace rasforms ad cosiderig ozero iiial codiios as discussed i secio. To experimeally obai he respose, follow he seps below: Ideificaio of he model: a) Le Mass free o move while Mass is blocked by screwig o he frame of he pla. Coec o he arge ad ru he model wih he slider gai se o zero iiially. Slowly icrease he slider gai value ad oe o he firs scope widow ha Mass sars movig oly whe he ipu exceeds a small value such as.5v. You ca call his ipu level as he deadbad volage udb. The apply a es volage u es such as V. You will see a displaceme of Mass ha sabilizes afer some ime o a seady-sae value x ss. As a resul, he DC gai of G ca be calculaed as xss g (5) u es b) While you keep Mass blocked, coec o he arge agai ad ru he model wih he slider gai se o zero. Slowly pull Mass wih your had abou.5 cm. The, suddely release he mass. Afer he mass oscillaios cease, sop he Simulik model. Observe he respose if i looks like he oe i Figure 7. Noe ha α correspods o he same parameer i (3) Iiial agle, x ( ), x ( ) 5, x ( 5 ) e si e ime (sec) Figure 7 Zero ipu, ozero iiial sae respose used for ideificaio c) Choose cosecuive cycles (e.g. = 5) i he higher ampliude rage of he respose such ha hey are o affeced oo much by he oliear fricio effecs. Divide he umber of cycles by 9/9/ Page 7

8 he ime ake o complee hem ( ) beig sure o ake begiig ad ed imes from he same phase of he respecive cycles (i is bes o look a he peaks). You may eed o zoom io he graph or see he daa i a able i he Malab workspace. The followig calculaio gives you he period of oscillaios T i secods ad he associaed frequecies β i radias. T, (6) T d) Measure he iiial cycle ampliude x ( ) ad he las cycle ampliude x ( ) for he cycles measured i he previous sep. Use he relaioships associaed wih he logarihmic decay o obai : x x e e x l x e) Calculae ω wih he followig formula: x l x x l x T (8) f) Usig he formulas you derived i par (e) of he Prelimiary Work, calculae he values for k ad c. Also, fid he value of f. g) Repea seps (b) (f) for Mass. I his case, block Mass by screwig i o he frame of he pla. Calculae he values k ad c agai by referrig back o your Prelimiary Work resuls. Validaio of he model: h) Replace he cosa parameers, m, m, c, c, k, ad f, i your sae space model of par (g) of he Prelimiary Work wih he umerical values you foud hrough he experime. i) Ope he Simulik block diagram amed simulaio.mdl Figure 8. Cofigure he Sae-Space block wih he values of he A, B, ad C marices of your model. Use he wo Sep sources o creae a ipu sigal ha has a pulse widh of secod ad ampliude of V. j) Copy he blocks before he Sae-Space block i Figure 8 o he experime model ad rebuild i. Make sure ha he wo Simulik diagrams are fucioally he same excep you have a simulaio model i Figure 8. Also, free boh masses i he seup. The, ru boh models. If he resposes are similar i meas ha he model ha you jus ideified for he DMSD sysem is fairly good. If he wo resposes are o similar he you have o go back ad fid ou where you migh have made a misake. (7) 9/9/ Page 8

9 Figure 8 Simulaio model of he DMSD sysem i Simulik 9/9/ Page 9

10 5. Lab Repor Submi your idividual repor for he followig problems via EE434 Blackboard (hps://elear.ua.edu) i oe of he followig file formas:.doc,.docx, or.pdf. Your repor should iclude if ay your formulaios, lisigs of MATLAB code, picures of Simulik block diagrams, ad resulig graphs for each problem. Make sure ha you show all your work ad provide all he iformaio eeded o make i a sadaloe ad self-sufficie submissio. Have a appropriae repor forma for your submissio. This lab hadou is a good example as o how you should forma your repor. Make sure ha you iclude he followig iformaio i your repor: Repor ile, your ame, ID umber, lab secio, submissio due dae ad ime. Aswers o he problems wih your o Mahemaical derivaios ad formulaios. I is highly recommeded ha you use he Equaio Edior of Microsof Word, MahType, or a similar edior o wrie your equaios. You ca also sca hadwrie equaios ad merge hem io your repor. o Picures of Simulik block diagrams ad propery widows of impora blocks i he simulaio. o Lisigs of MATLAB codes wih ilie commes ad explaaio i ex. o Picures of daa plos wih appropriae axis labelig, iles, ad clearly visible axis values. Scree priig is o well acceped. Commes if ay o le us kow how we ca make your learig experiece beer i his lab. Noe ha you will eed o use he resuls preseed i his lab repor for he experime i Lab 4. Below is he assigme policy: This assigme is due o Wedesday a :59 pm wo weeks afer your lab sessio. You mus upload a sigle file i.doc,.docx, or.pdf forma via he Lab Repor Submissio lik a EE434 Blackboard (hps://elear.ua.edu). Lae repors will ge % deduced score from he ormal score for each lae day (-4 hr) sarig righ afer he due dae ad ime. For example, a paper ha is worh 8 pois ad is days lae (4hr 48hr) will ge 8 8 (/) = 48 pois. A paper ha is lae for 5 or more days will ge score. You will have wo chaces of aemp o submi your repor via Blackboard ad oly he las submissio will be cosidered. Gradig is ou of pois ad ha icludes % ( pois oal) for he forma. A ice forma refers o a clear, cocise, ad well orgaized preseaio of your work. 9/9/ Page

11 Quesio (35 ps): Provide your aswers o he prelimiary work. The followig rubric will apply for pars (a) o (g) of he prelimiary work. (a) 5 ps (b) 5 ps (c) 5 ps (d) 5 ps (e) 5 ps (f) 5 ps (g) 5 ps Quesio (55 ps): Show your resuls, calculaios, ad plos ha you obaied i pars (a) (j) of he experimeal procedure. Wha are he parameers k, c, c, ad f ha you foud from he experime? Show your calculaio seps ad measuremes o he plos. Give he sae space model ha you foud wih he real values plugged i. Fid he poles of your model. For he validaio par of he model, compare your simulaio ad experimeal resuls o he same plo. Discuss he differeces bewee he wo. Quesio 3 ( ps): a) (5 ps) Assumig ha he masses are iiially saioary, he ipu is zero, ad x () =.5m while x () = for all ad usig he expressio you foud for X (s) i erms of s, m, c, k, ad x () i par (c) of he Prelimiary Work, fid he ime domai expressio for x (). Plug i he values of he sysem parameers ad plo x () i MATLAB. Comme o wheher i looks similar o your experimeal observaio of x () i par (b) of he experime. b) (5 ps) Similarly, assumig ha he masses are iiially saioary, he ipu is zero, ad x () =.5m while x () = for all ad usig he expressio you foud for X (s) i erms of s, m, c, k, ad x () i par (c) of he Prelimiary Work, fid he ime domai expressio for x (). Plug i he values of he sysem parameers ad plo x () i MATLAB. Comme o wheher i looks similar o your experimeal observaio of x () i par (b) of he experime. 9/9/ Page