ME 451: Cotrol Systems Laboratory Departmet of Mecaical Egieerig Miciga State Uiversity East Lasig, MI 4884-16 ME451 Laboratory Time Respose Modelig ad Experimetal Validatio of a Secod Order Plat: Mass-Sprig Damper System ME451 Laboratory Maual Pages, Last Revised: 9-1-009 Sed commets to: Dr. Clark Radcliffe, Professor Modelig ad Experimetal Validatio of a Secod Order Plat: Mass-Sprig-Damper System page 1
1. Objective Liear time-ivariat dyamical systems are categorized uder first-order systems, secodorder systems, ad iger-order systems. A large umber of secod-order systems are described by teir trasfer fuctio i stadard form". Tis form eables us to ivestigate te respose of a large variety of secod-order systems for ay specific iput. Te qualitative respose depeds primarily o te atural frequecy, ω, ad te dampig ratio, ζ. Bot ω ad ζ are fuctios of system parameters. Te objective of tis experimet is to model a stadard secod-order system ad to ivestigate te effect of system parameters ad feedback o its respose to a step iput. For tis lab, we coose to experimet wit a torsioal mass-sprig-damper system. It will be sow tat ideally tis system acts as a secod order, time-ivariat system. Te trasfer fuctio, specifically te gais, atural frequecy, ad dampig ratio, of te system will be experimetally obtaied. Feedback will be used to vary te effective dampig ratio ad atural frequecy.. Backgroud.1. Secod-order systems Te stadard form of trasfer fuctio of a secod-order system is Y ( s) K! G( s) = = (1) U ( s) s + "! s +! were Y(s) ad U(s) are te Laplace trasforms of te output ad iput fuctios, respectively, ω is te atural frequecy, ad ζ is te dampig ratio. For a uit step iput (U(s) = 1/s), te respose of te system, i Laplace domai, ca be writte as K! Y ( s) = s( s + "! s +! ) Assumig poles of G(s) are complex (ζ < 1) ad te DC gai G(0) = K = 1, te time domai uit step respose ca be writte as 1 % $# t y( t) = 1% e si( "# t +! ), " $ 1 #!, # $ arcta( " /! ) () " Te step respose of a secod-order system wit varyig dampig ratios is sow i Fig. 1. Te plot is ormalized by frequecy ad amplitude. Te respose of te system is oscillatory, but damped. Te amout of dampig is caracterized by te dampig ratio ζ, ad te frequecy of damped oscillatio is βω. Note tat te frequecy of damped oscillatio is differet from te atural frequecy ω. Te settlig time of a secod order system is defied as te time is takes for te system to settle witi % of its steady state value. It is give by te expressio T s = 4*τ, were τ is te time costat, ad is give as τ = 1/ζω. Notice ow te time costat appears i te expoetial term of eq. (). Te peak time, Tp, is te time required for te output to reac its most extreme value, ad is give as # T p = " 1 $! Aoter useful respose caracteristic, Percetage oversoot, is defied as te ratio of te differece y peak1 y( ) to y( ), were y peak1 is te amplitude of te first peak. Ulike te damped Modelig ad Experimetal Validatio of a Secod Order Plat: Mass-Sprig-Damper System page
atural frequecy, settlig time, ad peak time, it is idepedet of ω ; it is oly a fuctio of ζ. Te followig relatio gives percetage oversoot: percetage oversoot = e "#$ / 1" #! 100 Figure1. Normalized Step Respose of a Secod Order System I te evet tat te poles of G(s) are real (ζ 1), te time domai respose ca be critically damped (ζ = 1), or overdamped (ζ > 1). For ζ = 1, te time domai respose of te output takes te form " t /! " t /! y( t) = 1+ k1e + kte (3) were k 1 ad k are costats ad τ is te time costat correspodig to te repeated pole of G(s). For ζ > 1, te time domai respose takes te form " t /! 1 " t /! y( t) = 1+ k1e + ke (4) were τ 1 ad τ are te time costats correspodig to te real distict poles of G(s). Bot Eqs. (3) ad (4) depict cases were te respose is ot oscillatory, ad ece, damped atural frequecy, settlig time, ad peak time are ot relevat. We G(s) as real distict poles (ζ > 1) te time costat of te respose is govered by te pole closer to te origi ad te respose resembles tat of first-order systems... Torsioal mass-sprig-damper system Cosider te torsioal mass-sprig-damper system i Fig.. Te system variables are T exteral torque applied o rotor. Modelig ad Experimetal Validatio of a Secod Order Plat: Mass-Sprig-Damper System page 3
ME 451: Cotrol Systems Laboratory θ agular positio of rotor. ω agular velocity of rotor. Te parameters of te system, sow i Fig., iclude J momet of iertia of rotor. b coefficiet of viscous frictio. k sprig costat. Te trasfer fuctio of te mass-sprig-damper system, defied wit exteral torque as iput ad agular positio of te rotor as output, ca be writte as G (s) = # ' (s) (k / J ) 1 1& = = $! T ( s ) Js + bs + k k % s + (b / J ) s + (k / J ) " (5) Figure (b) sows te experimetal setup tat is modeled wit te diagram i Figure. Ecoder #3 Torsioal Sprig (Rod) Ecoder # Servomotor ECP Mass Figure : Diagram of torsioal MSD system Ecoder #1 Figure (b) Experimetal Setup Te trasfer fuctio above bears close resemblace wit te stadard secod-order trasfer fuctio i Eq.(1). Te oly differece is te DC gai of (1/k), wic appears i Eq.(5). Modelig ad Experimetal Validatio of a Secod Order Plat: Mass-Sprig-Damper System page 4
By comparig Eqs.(1) ad (5), te expressios for atural frequecy ad dampig ratio ca be obtaied as k! = ad J b b!" = so tat! = (6) J kj It is clear from Eq.(6) tat ω ad ζ are fuctios of system parameters J, b, ad k, wic are typically fixed. A pysical implemetatio, wit drivig motor, of te torsioal system is available for tis laboratory ad as te block diagram sow i Figure 3. E(s) (volt) Torsioal System Electroics Mecaism T(s) 1 & ( k / J ) # K $!" (N-m) k % s + ( b / J ) s + ( k / J ) ' (s) (rad) Figure 3: Block Diagram of te Ope-Loop Torsioal Mass-Sprig-Damper System Tis system as a trasfer fuctio closely related to te idealized trasfer fuctio (5) wit te additio of a motor to provide iput torque. Tis torsioal system as a motor ardware gai, K (N-m/volt) tat gives te torque to voltage ratio of te motor drive. If we build te system i Fig.3, we will ave a specific secod-order respose of te form. ' ( s) = K E( s) ' ( s) = K T ( s) G( s) = Js K + bs + k K = k & $ % s ( k / J ) #! + ( b / J ) s + ( k / J )" (7) Te atural frequecy ad dampig ratio for tis system are te same as (5) wit a revised DC gai of K k. Figure 4. Block Diagram of te Closed-Loop Torsioal Mass-Sprig-Damper System For studyig te effect of system parameters o te respose, we must be able to cage te apparet J, b, ad k. Cosider te system sow i Figure 4. Tis closed-loop system acieves programmable cages i system dampig ad stiffess. We acieve tis by Modelig ad Experimetal Validatio of a Secod Order Plat: Mass-Sprig-Damper System page 5
programmig te motor to geerate te torques geerated by a additioal sprig ad damper tereby cagig te et stiffess ad dampig of te system. Specifically, te motor is programmed to geerate te torque give by te relatio T = K k K! " K! K!) (8) e ( pf d 1 "! were K is te electromecaical gai of te motor, ke is te electroic gai of te DSP system, ad θ d is te desired output. Note tat K1 θ is te type of torque geerated by a sprig, ad K! is te type of torque geerated by a damper. We tese torques are applied o te rotor, te feedback geerates differet apparet stiffess ad dampig ad te dyamic equatio of te programmed system becomes J! + b! + k! = T! J! + b! + k! + K kek! + K kek1! = K kek pf! d Takig te Laplace Trasform of tis equatio gives us Js!! = K k K! + bs! + k! + K kek s! + K kek1 e pf d Te block diagram of te programmed system is sow i Fig.4. Te trasfer fuctio of tis system, wit T as iput ad θ as output, is ' ( s) G( s) = = ' ( s) d [( K k K ) / J] e [( k + K k K ) / J] e pf 1 & $ % s + [( k + K ] [ ] [ ]! # kek1) / J ( b + K kek ) / J s + ( k + K kek1) / J " wic as te same structure as tat i Eq.(1). Te DC gai of te programmed system is [ K k K ) ( k K k K )] K =, ( e pf + e 1 ad te atural frequecy ad dampig ratio are give by te relatios! = ( k + K kek1) J ad ( ) ( b + K kek )!" = b + K kek J so tat! = (10) ( k + K kek1)j Oe of te objectives of tis experimet is to study te effects of varyig te atural frequecy ω ad te dampig ratio ζ o respose of te torsioal mass. We will vary ω ad ζ by varyig te gais K 1 ad K i software. Te ardware gai K ke ad te rotor iertia J will remai fixed durig experimets. (9) Modelig ad Experimetal Validatio of a Secod Order Plat: Mass-Sprig-Damper System page 6
Pre-Lab Sample Questios Use te plot below to aswer te followig questios: 1) Wic system as a iger dampig ratio? Aswer: System B ) Wat is te percet oversoot of System A? Aswer: % OS 70% 3) Wat is te peak time of System B? Aswer: T P 0.8 s 4) Wat is te atural frequecy of System A? Aswer: ω 4.19 rad/s 5) Wat is te steady-state gai of System B? Aswer: Steady-state gai = Modelig ad Experimetal Validatio of a Secod Order Plat: Mass-Sprig-Damper System page 7
Read Istructios Carefully!!! 3. Descriptio of Experimetal Setup 3.1. Hardware ad software 1. Electromecaical plat, ECP Model 05 Te electromecaical plat is comprised of te torsio disk, a DC servo motor tat provides a exteral torque to te disk, ad sesors for measuremet of agular positio of te disk.. ECP iput/output electroics uit, Model 05 It cotais te power supply uit for te DC motor ad associated electroic ardware. 3. Digital sigal processor (DSP) Te DSP, istalled i te PC, takes te sesor sigals i digital form, provided by te aalog-todigital covertors (ADC), ad computes te torque to be geerated by te motor. Te digital torque sigal is set to te motor via te digital-to-aalog covertors (DAC). 4. Software ECP executive program i Widows NT eviromet. 3.. Basic setup 1. Place two 500 gm masses o te lowest disk at equal distaces from te ceter. Verify tat te masses are secured to te disk.. Ru te program Ecp3", i te ECP folder. From te File" pull dow meu click o Load Settigs" ad coose me451lab.cfg". Tis file sould be dowloaded from te software sectio of te lab website. Rigt-click o te file ad save it to your desktop. Make sure tat te cotrol loop status is Ope" ad te cotroller status is OK". If ot, click o te Utility pull dow meu ad select Reset Cotroller. Next, click o te Abort Cotrol ico wit te red ad i te lower rigt corer of te scree. I te Setup pull dow widow, select User Uits, te select Radias ad OK. Cautio: I tis work, ad all future work, make sure to stay clear of te mecaism uless oterwise directed. If te system appears to react violetly, you sould immediately click te Abort Cotrol" butto o te ECP program widow. If te problem persists, promptly tur off te ECP iput/output electroic uit by pressig te red butto o te electroics uit. Modelig ad Experimetal Validatio of a Secod Order Plat: Mass-Sprig-Damper System page 8
4. Experimetal Procedures Objective: I tis experimet you will aalyze te ope-loop ad closed-loop respose of te torsioal mass-sprig-damper system to a step iput. You will also vary te system gais ad predict te cage i respose. Part A: Test te ope-loop system wit o servo motor drive a) Tur te ECP electroics uit OFF, b) Displace te disk (rotate) ad observe it s beavior Cautio: Do ot displace te disk more ta 0 because you ca damage te log sleder vertical saft o tis system by displacig it too far Be ice to our saft ad it will be ice to you Questios to aswer i te sort form: A.1. Estimate f (atural frequecy i ertz) ad compute ω (atural frequecy i rad/sec). Estimate a rage of ζ by comparig te system respose to Figure 1. Note: It is importat to compare te atural frequecy of te system to tat of te plot before estimatig ζ. Part B: Test te ope-loop system usig servo motor torque a) Tur te ECP electroics uit ON. Te block diagram sow i Fig. 3 ow models te system. b) I te Setup" pull dow meu, coose Cotrol Algoritm". i) I te ew widow tat opes, select Cotiuous Time" for type, ad State Feedback" for cotrol algoritm. ii) Click o te box Setup Algoritm" ad set all te gais to zero. Click OK. c) Back i te Setup Cotrol Algoritm" widow i) click Implemet Algoritm". Te Cotrol Loop Status" is ow CLOSED" ad all ecoder readigs are zero. Eve toug te Cotrol Loop Status is CLOSED, settig all feedback gais to zero effectively makes it ope-loop. Click OK. d) From te Commad" pull dow meu, select Trajectory". i) I te ew widow tat opes, select Step". ii) Click Setup" to cotiue. Aoter widow will ope. iii) Select te Ope Loop Step. Use te default step size of 0.5 V, te dwell time to 000 msec. ad reps to 1. Tis will take you back to te previous widow. You eed to click OK" agai. iv) Go to "Data", "Setup Data Acquisitio" ad add "Cotrol Effort" to te "selected items" list, click OK. e) From te Commad" pull dow meu, click o Execute". Coose Normal Data Samplig" te click o Ru". f) You sould immediately otice te disk turig. Wait for approximately 15 secods for te disk to retur to its origial cofiguratio ad for data uploadig. After te data as bee successfully uploaded, a ew widow will ope. Click OK" i tis widow to accept te disc displacemet measuremet data. Modelig ad Experimetal Validatio of a Secod Order Plat: Mass-Sprig-Damper System page 9
g) From te Plottig" pull dow meu, select Setup Plot" ad make sure "Cotrol Effort" ad "Ecoder 1-Positio" are listed i te left axis display, te click o Plot Data to display te system s respose o te scree. ) From te Plottig" pull dow meu, i) select Axis Scalig". I te ew widow tat opes, mark all Plot Attributes" (tis eables orizotal ad vertical grids ad plots all data poits wit visibly large circles makig tem more visible we prited) ad click OK". ii) Select Prit Plot" uder Plottig" pull dow meu to get a ard copy. Questios to aswer i te sort form: B.1. From your values obtaied from your ope-loop plot, compute ω ad dampig ratio ζ. Do tey agree wit questio (1)? Wy or wy ot? B.. From te ope-loop plot compute te ope-loop system s steady state Gai. Part C: Measure Ope-Loop System Parameters Tur te ECP electroics uit OFF, a) Measure te torsioal stiffess k of te torsio system Usig te electroic force idicator, apply a force at rigt agles to a radius of te disc ad record te agular deflectio of te disc. Record te force ad radial distace at wic it was applied. Compute te agular stiffess k of te torsio system i uits of (N-m)/rad. Record your result for k. b) Fid te ardware gai of te servo motor K Usig te DC gai computed i part B, your computed stiffess, ad te Ope-loop model sow i equatio (7), fid te ardware gai of te servo motor K c) Compute b, J of te system usig Ope Loop atural frequecy ad dampig ratio alog wit your computed agular stiffess k. d) Fid te electroic gai k e of te DSP system Wit te ECP electroics box OFF, te DSP system fuctios but te servo motor drive does ot. We will geerate a sigal! d causig a voltage E ad measure te ratio to fid gai k e i) I te Setup" pull dow meu, coose Cotrol Algoritm". I te ew widow tat opes, select Cotiuous Time" for type, ad State Feedback" for cotrol algoritm. Click o te box Setup Algoritm" ad set te gais below. Click implemet te algoritm. =.1; wit K = K = K = K = K = K 0 K pf 0 1 3 4 5 6 = ii) Set te closed-loop iput! d to a 4000 millisecod step of step size 0.1 radias ad execute tis trajectory. Look at te closed loop error o te computer scree (Cotrol Effort) ad ote it s value. Wit te feedback terms all set to zero, te closed-loop error to iput ratio equals te gai product K pf ke (see figure 4). Compute K pf ke iii) From K pf ke, compute te DSP electroic gai k e Modelig ad Experimetal Validatio of a Secod Order Plat: Mass-Sprig-Damper System page 10
Questios to aswer i te sort form: C.1. Apply a kow torque to te ope-loop system, measure te agular deflectio ad compute te system stiffess k i uits of (N-m)/rad. Sow your work. C.. Usig te DC gai computed i part, your computed stiffess, ad te Ope-loop model sow i equatio (7), fid te ardware gai of te servo motor K. Sow your work C.3. Compute b ad J of te system usig ope loop atural frequecy, dampig ratio ad k. C.4 Fid te electroic gai k e of te DSP system. Sow your work. Part D: Test te Closed Loop Respose of te System To complete tis sectio, you will measure te parameters for may of te compoets sow i Figure 4, measure te closed-loop respose ad fially te steady state gai. a) Tur te ECP electroics uit ON. Now te block diagram sow i Figure 4 models te system. b) I te Setup" pull dow meu, coose Cotrol Algoritm". i) I te ew widow tat opes, select Cotiuous Time" for type, ad State Feedback" for cotrol algoritm. ii) Click o te box Setup Algoritm" ad set all te gais to te values below. Click OK. = K =.1; K = 0.00; K = K = K = K 0 K pf 1 0 3 4 5 6 = c) Back i te Setup Cotrol Algoritm" widow i) click Implemet Algoritm". Te Cotrol Loop Status" is ow CLOSED" ad all ecoder readigs are zero. Click OK. d) From te Commad" pull dow meu, select Trajectory". i) I te ew widow tat opes, select Step". ii) Click Setup" to cotiue. Aoter widow will ope. iii) Select te Closed Loop Step. Set te step size to 0.1 radias, te dwell time to 1000 msec. ad reps to 1, click OK. Tis will take you back to te previous widow. You eed to click OK" agai. e) From te Commad" pull dow meu, click o Execute". Coose Normal Data Samplig" te click o Ru". f) You sould immediately otice te disk turig. Wait for approximately 15 secods for te disk to retur to its origial cofiguratio ad for data uploadig. After te data as bee successfully uploaded, a ew widow will ope. Click OK" i tis widow to accept te disc displacemet measuremet data. g) From te Plottig" pull dow meu, select Setup Plot". Plot commaded positio ad Ecoder 1 positio o te left vertical axis. Click o Plot Data to display te system s respose o te scree. ) From te Plottig" pull dow meu, Modelig ad Experimetal Validatio of a Secod Order Plat: Mass-Sprig-Damper System page 11
i) select Axis Scalig". I te ew widow tat opes, mark all Plot Attributes" (tis eables orizotal ad vertical grids ad plots all data poits wit visibly large circles makig tem more visible we prited) ad click OK". ii) Select Prit Plot" uder Plottig" pull dow meu to get a ard copy. Questios to aswer i te sort form: D.1. Determie te percetage oversoot (PO), settlig time (Ts), peak time (Tp), atural frequecy (ω) ad dampig ratio (ζ), from te grap you obtaied from te closed-loop experimet. Label your grap ad sow all calculatios. D.. Compute te steady state gai of te closed loop system usig equatio (9). How does tis compare to te value determied from te plot? Part E: Predictig System Respose Kowig k, b, J, K, k e, you will predict ad verify closed loop system respose. a) Usig your kow values above, predict (calculate) te atural frequecy ad dampig ratio of te system. b) Repeat steps D a- usig te gais give below. K pf = K1 = 0.075; K = 0.004; K 3 = K 4 = K 5 = K 6 = 0 Estimate te percetage oversoot (PO), peak time (Tp), atural frequecy (ω) ad dampig ratio (ζ), from te grap you obtaied. Label your grap ad sow all calculatios. How well do tese values compare to your predictios? Questios to aswer i te sort form: E.1. List ad compare your calculated ad estimated values for te atural frequecy ad dampig ratio of te system. Modelig ad Experimetal Validatio of a Secod Order Plat: Mass-Sprig-Damper System page 1
Sort Form Laboratory Report Name: Sectio: Date: A.1. Estimate values for f ad ζ, ad compute ω. B.1. From your values obtaied from your ope-loop plot, compute ω ad dampig ratio ζ. Do tey agree wit questio (1)? Wy or wy ot? B..From te ope-loop plot compute te ope-loop system s steady state Gai (1 rad = 1volt) C.1. Apply a kow torque to te ope-loop system, measure te agular deflectio ad compute te system stiffess k i uits of (N-m)/rad. Sow your work. C.. Usig te DC gai computed i part. ad your computed stiffess, ad te Ope-loop model sow i equatio (7), fid te ardware gai of te servo motor K. Sow your work C.3. Compute b ad J of te system usig ope loop atural frequecy, dampig ratio ad k. Modelig ad Experimetal Validatio of a Secod Order Plat: Mass-Sprig-Damper System page
Sort Form Laboratory Report Name: Sectio: Date: C.4 Fid te electroic gai k e of te DSP system. Sow your work. D.1. Determie te percetage oversoot (PO), settlig time (T s ), peak time (T p ), atural frequecy (ω ) ad dampig ratio (ζ), from te grap you obtaied from te closed-loop experimet. Label your grap ad sow all calculatios. D.. Compute te steady state gai of te closed loop system usig equatio (9). How does tis compare to te value determied from te plot? Modelig ad Experimetal Validatio of a Secod Order Plat: Mass-Sprig-Damper System page
Sort Form Laboratory Report Name: Sectio: Date: E.1. List ad compare your calculated ad estimated values for te atural frequecy ad dampig ratio of te system. Coclusio Summarize te lessos you ave leared from tis laboratory experiece, i a few seteces. Modelig ad Experimetal Validatio of a Secod Order Plat: Mass-Sprig-Damper System page 3