Lesson 4. Thermomechanical Measurements for Energy Systems (MENR) Measurements for Mechanical Systems and Production (MMER)

Transcription

1 Lesso 4 Thermomehaial Measuremets for Eergy Systems (MENR) Measuremets for Mehaial Systems ad Produtio (MMER) A.Y Zaaria (Rio ) Del Prete

2 RAPIDITY (Dyami Respose) So far the measurad (the physial quatity for whih we wish to measure the itesity) has bee osidered stritly CONSTANT durig the whole measuremet proedure! rom here o it will hae its itesity with time durig the measuremet the measurad beomes a futio of time! i(t) Istrumet u(t) both the iput measurad ad the output measuremet are futios of time i(t) u(t) We may all rapidity of a istrumet the attitude to orretly follow the hages of the measurad durig time Rapidity of mehaial istrumets is always limited by the iertia ad the dampig effets of its movig parts! Rapidity for eletroi istrumets is always limited by the ombiatio of its apaitive ad idutive reataes! A istrumet with isuffiiet rapidity (isuffiiet dyami respose) durig a measuremet will output a measuremet waveform whih will be atteuated ad out of phase (delayed) with respet to the measurad! The output measuremet waveform will be distorted with respet to the iput measurad waveform

3 We have basially two shemes or methods to study the dyami respose of a measuremet istrumet: 1. STEP RESPONSE with whih we a get some importat parameters suh the settlig time ad the time ostat. REQUENCY RESPONSE with whih we a get other importat parameters suh the ut-off frequey ad the dampig fator There is also a third sheme, whih is used less ofte: the ramp respose that leads to the delay time

4 Step Respose At time t the iput sigal hages istataeously its value from i to i 1 The istrumet output will try to follow this hage, form u to u 1, the best he a However, it will tae some time t s = t 1 t to reah the orret fial value withi a ertai error ±ε d! ±ε d is the dyami error, whih has to be established i advae by the operator t s = t 1 t is the settlig time of the istrumet, whih provides a first idea of its dyami respose t SLEW = t sr t is the slew rate, whih is the time the istrumet taes to reah the first overshoot pea ot all the istrumets show a overshoot durig the step respose!

5 requey Respose I the more geeral ase, a iput variable (measurad) hages periodially durig time, the we a refer to simple siusoidal iputs beause every periodi sigal a be deomposed by the ourier Series: f a x f x a osx b sex 1 i( t) I set f iput: with frequey u( t) U se( t ) output: with phase delay U I os t ( b) t tr f ideal frequey respose ad phase respose urves ( a) ostat output/iput ratio for every frequey of the iput! zero phase delay for every frequey of the iput!

6 Real istrumet frequey respose example: Real istrumets always atteuate the output amplitude for higher frequeies (f ꝏ) beause: mehaial istrumet have ier movig parts with iertia whih a ot have ifiite aeleratio eletroi istrumet have ier ompoets whih a ot have idutive reatae X L = jωl = ꝏ apaitive reatae X C = 1/jωC = The extesio of a istrumet frequey respose depeds also from the dyami error the operator aepts: if we aept a dyami error ε d = 3% we have a ut-off frequey f If we aept a dyami error ε d = 5% we have a muh higher ut-off frequey f 1 > f All frequeies betwee f ad f 1 (or f ) form the istrumet badwidth or pass-bad!

7 There are may istrumets where the badwidth does ot start from frequey f = Hz, but starts at a higher value f 1 I these ases, we have two ut-off frequeies ad the pass-bad is betwee these two frequeies: B f ts f ti Most of the times the dyami error (whih a be fairly large ) is expressed i a logarithmi sale, iherited from austis studies ad the logarithmi uit is the deibel : It is world wide aepted that a tolerable dyami error limit is 3dB log 1.77 whih meas the istrumet output A is atteuatig the iput to the 7,7 % of its real amplitude or that it is doig a dyami error of 9,3 %!!

8 We hage ow otatio for the iput ad output variables: y(t) Istrumet x(t) y(t) iput variable (measurad) x(t) output variable (istrumet respose or idiator defletio) We say a istrumet is dyamially liear if we a write its d x dx dyami harateristi equatio as follows : a b x y( t) dt dt The istrumet dyami a be desribed by a ordiary differetial equatio with ostat oeffiiets! The solutio a be writte as: x( t) x ( t) x ( t) tr rg Solutio of the assoiated homogeeous equatio whih represets: x tr (t) the trasiet dyami respose of the istrumet Solutio of a partiular itegral of the equatio whih represets: x rg (t) the dyami steady state of the istrumet

9 To study the TWO differet situatios we will employ the TWO differet experimetal shemes or tools itrodued before: 1. or the dyig out trasiet the STEP RESPONSE x tr (t). or the established steady state the REQUENCY RESPONSE x rg (t)

10 Zero-order istrumets Whe the differetial equatio of the dyami harateristi has the simplest form, without the derivative terms: a x b y x b a y The output sigal (or defletio) is always proportioal to the iput sigal! There is o delay betwee output ad iput therefore, zero-order istrumets respod istataeously ad a be osidered ideal istrumet! Example: or the potetiometer we have for the left iruit E = RI while the displaemet y is idiated by the output voltage e(t) whih is: E r( t) y( t)/ S e( t) r( t) I r( t) E E R R Y / S y( t) E Y Y y( t) where R ad r( t) the variable time is oly S S preset as idepedet variable for the positio y = y(t) E Note that e y is also the graduatio urve (stati harateristi Y de E urve) of the potetiometer ad is the stati sesitivity! dy Y

11 irst-order istrumets A first-order istrumet a store eergy i oly oe of its ier elemets! If we displae the idiator S of the simple mehaial system to a positio x, elasti (potetial) eergy is stored i the sprig. If at t = we leave the idiator S free of movig, the elasti eergy stored i the sprig will pull ba the idiator to the rest positio x =. Durig movemet, the damper will oppose the motio. or every time istat t the equatio of fore equilibrium is: x x or x x or every time istat t the veloity is: x x therefore the oeffiiet t is a time! is the time ostat of the first-order istrumet of above!

12 The solutio of the 1 st order differetial equatio x t x x is the well ow dereasig expoetial : ( t) x e the urve of whih a be draw with the iitial oditio: t = x( ) x The taget lie to the urve i the poit t = is: x x ( ) x Observe i the figure the geometrial meaig of the time ostat 1 At the time t = λ we have that x( ) xe. 37 x the idiator S has travelled for 63% of the displaemet ad has still a 37% to go to reah its fial positio x =!! but these are oly itrodutory defiitios ad osideratios

13 I the geeral ase of the step respose for the istrumet of before, a iput variable is preset : a step fore applied to the idiator S at time t = The 1 st order differetial equatio that desribes the idiator defletio therefore is: x x geeral solutio has the form x( t) x tr x rg ad the x rg is a partiular solutio ad desribes the steady state oditio of the istrumet: Note that whe t ꝏ the idiator speed must be zero x therefore the steady state of the idiator S will be desribed by the positio x rg x tr istead, is the solutio of the assoiated homogeeous equatio ad desribes the dyig out trasiet : x tr e t The fial solutio will be the sum of the two otributios: t x( t) x tr xrg e 1 e Observe agai o the figure the meaig of tg / x () At t = λ we have 1 ( ) 1 e. 63 x t

14 Whe we are iterests i studyig the steady state dyami respose of a 1 st order istrumet we have to employ the other tool, the frequey respose. To do so, we have to apply at the istrumet iput a periodi measurad: ( t) set a periodi fore of frequey ω = πf! The differetial equatio whih desribes the idiator S movemets will be: x x set The geeral solutio will always be x( t) x tr x rg but we will be iterested oly i the steady state part, whih we a assume to be: x rg ( t) X se( t ) How does the amplitude of the output idiatio x(t) hage with frequey? Ad what delay will the output x(t) show? To respod to those questios we will employ the phasor otatio: iput: output: ( t) set e ( t) X se t jt jt j X e e x

15 To verify the x rg solutio we have to substitute our assumed steady state solutio i the differetial equatio j jt j jt j X e e Xe e e jt j X e j X e j j j 1 where is the maximum amplitude ad is the time ostat! alulatig the modulus of the ratioalized futio, we will obtai: X 1 X G 1 1 whih is the amplifiatio or gai of the istrumet! artg whih is the phase delay of the istrumet!

16 1 st order istrumet frequey respose 1 st order istrumet phase delay These diagram have o the x axis the o-dimesioal variable ωλ» ad therefore are ormalized diagrams 1 is the harateristi frequey of a 1 st order istrumet for whih we have ω λ = 1 therefore, ω is the ut-off frequey for a - 3db atteuatio : f 1 G 1.77

17 Example: The dyami respose of the merury thermometer or every differetial time iterval dt that preedes the equilibrium oditio we have: Heat amout the eviromet trasfers to the merury: dq A( Ta T) dt Heat the merury reeives from the eviromet: dq mdt Where: m is the total merury mass is the merury speifi heat is the thermal exhage oeffiiet A is the thermal exhage surfae of the thermometer The two heat amouts of above must be the same: m dt AT ( a T) dt m A m dt dt AT a AT is the time ostat for this 1 st order istrumet! m A dt dt T Ta 1 If λ = 1s the ωλ = 1 11 ad quite slow! f 1 f. 16Hz

18 Seod-order istrumets A seod-order istrumet a store eergy i two of its ier elemets! the elasti sprig stores potetial eergy m the movig mass stores ieti eergy The differetial equatio that desribes the free movemet of the idiator m d x m dt dx dt x If we osider the very partiular ase for whih = we get the importat limit situatio desribed by : where ad ω is the atural frequey of the istrumet! m d x dt x m A system lie this has NO dampig ad, if pulled from its equilibrium positio x =, it will osillate idefiitely with a frequey ω ad will NEVER stop! Of ourse, it s a IDEAL system

19 I the real ase of the step respose for the istrumet of before, a iput variable is preset : a step fore applied to the idiator m at time t = The d order differetial equatio that desribes the idiator defletio therefore is: m x x x the geeral solutio has agai the form x( t) x tr x rg ad x rg is a partiular solutio ad desribes the steady state oditio of the istrumet: Note that whe t ꝏ the idiator speed ad aeleratio must be zero x x steady state of the idiator S will be desribed by the positio x rg therefore the x tr istead, is the solutio of the assoiated homogeeous equatio ad desribes the dyig out trasiet : x () t Ae A e tr t 1 t 1 1, with ad the disrimiat! m 4m m 4m m Depedig o the sig of the disrimiat Δ >, Δ =, Δ < we will have differet behaviors of the istrumet. Therefore, the d order istrumet will exhibit more tha oe type of step respose! The urves of the step respose will deped o a parameter obtaied from the disrimiat Δ : 4m m 4m m the dampig fator r where r m

20 or ξ > 1 (whih is the same of > r ) the istrumet is over-damped ad it will reah the fial idiatio with a slow expoetial path! or ξ = 1 (whih is = r ) the istrumet has a ritial dampig ad reahes the fial idiatio with the fastest expoetial path! or ξ < 1 (whih is < r ) the istrumet is uder-damped, it reats quily to the step iput but exhibits several damped over-osillatios before reahig the fial idiatio! Note that for the ase ξ < 1 the frequey of the over-osillatios seems to deped from the value of the dampig fator ξ Istrumet desiger geerally prefer to hoose a dampig fator ξ =.7.8 so the idiator m will have oly oe over-osillatio ad reahes the fial idiatio / from the upper side

21 To obtai the values of ω ad ξ oe should ow the values of the oeffiiets m,, ad of the d order istrumet whih is ofte almost impossible! I ase the istrumet has ξ < 1 (whih is the most ommo) oe a employ a experimetal proedure that leads to the logarithmi deay δ : Displae the istrumet idiator from the zero positio, leave it free to move ba ad measure the dereasig osillatios l x x m m1 1 l x x m m If the deay is small, it a also be measured after waves, as showed i the equatio above. The logarithmi deay δ is depedet from the dampig fator ξ : l 1 1 Ad the frequey ω = π/t of the dereasig osillatio is smaller tha the atural frequey ω depedig also from the dampig fator δ : 1 A A

22 Whe we are iterests i studyig the steady state dyami respose of a d order istrumet we have to employ the other tool, the frequey respose. To do so, we have to apply at the istrumet iput a periodi measurad: ( t) set a periodi fore of frequey ω = πf! The differetial equatio whih desribes the idiator m movemets will be: m x x x set The geeral solutio will always be x( t) x tr x rg but we will be iterested oly i the steady state part, whih we a assume to be: ( t) X se( t ) x rg How does the amplitude X of the output idiatio x(t) hage with frequey? Ad what phase delay ϕ will the output x(t) show? Same as doe for the 1 st order istrumets we substitute the iput jt j ( t) X e e i the differetial equatio: x ( t) e jt ad the istrumet output with m m( jt j jt j jt j ) X e e ( j) X e e Xe e X e j ( m j ad ) j m m r j j j j X e j e jt m j m j 1

23 j j j e X 1 1 whih is the omplex frequey respose for the d order istrumet Calulatig the modulus ad the phase lag of this futio we obtai 4 1 X the amplitude of the frequey respose of the istrumet! Normalized diagram of the amplifiatio or gai of the istrumet: These urves are also depedet from the dampig fator ξ or the amplifiatio is: wih for «small values» of ξ a be «dagerous», these are the resoae oditios X G / 1 1 G

24 artg 1 the phase delay of the istrumet output! The urves deped o the dampig fator ξ, however they all pass through the poit : 1 whih maes it useful to experimetally determie the atural frequey ω without owig the dampig fator ξ If you osider x rg ( t) the steady state output for the step respose, the stati sesitivity will result: S du di dx d 1 1 m whih maes the dyami respose of a istrumet always iversely proportioal to its stati sesitivity!!

25 Example: the galvaometer Stati harateristi (graduatio urve) il B ilb motor torque: resistig torque: C m C r b C m C r ilb b lbb i S d di lbb lbb J the sesitivity is ostat ad iversely proportioal to ω Dyami harateristi (dyami respose) The output idiatio is a rotatio θ therefore : J ad C( t) C for t > (step respose) d d J C( t) dt dt C( t) Cset (frequey respose)