Laboratory Exercise 3: Dyamic System Respose Laboratory Hadout AME 50: Fudametals of Measuremets ad Data Aalysis Prepared by: Matthew Beigto Date exercises to be performed: Deliverables: Part I 1) Usig the output from the DO fid τ ) From τ ad C calculate R for the Variable Resistor Part II 3)Complete the RLC respose table. Usig these results ad a program or M-file, costruct two plots o of M(ω) ad other of Φ(ω) versus the ormalized frequecy ratio. These plots must cotai the data alog with the theoretical curves. 4) Does the data support the coclusio the RLC circuit behaves as a secod-order system i both cases? Fially, compare the values of the dampig ratio foud with the value of R/Rc. Do this by comparig the data with the correspodig values determied usig various values of R/Rc i the give equatios. How well do the experimetal ad theoretical values of ζ compare? Part III 5) What is the atural frequecy (ω ) ad dampig ratio( ζ) of the bat? Where you able to chage it? Why or why ot? 6) Plot the experimetal ad theoretical respose of the bat Itroductio: The mai objective of this laboratory exercise is to ivestigate the dyamic respose characteristics of first-order ad secod order measuremet systems. The first week, the dyamic Respose of a first order RC circuit ad secod order RLC circuit will be studied. The secod week you will examie the impulse respose of a alumium bat. Data will be acquired usig a digital oscilloscope (DO). First Order System Respose: For this part of the exercise a DO will be used to determie the time costat of a RC circuit. A fuctio geerator will be used to geerate a square wave of give magitude ad frequecy. This circuit cosists of a resistor (R) ad a capacitor (C) ad is show i Figure 1 below. 1
Variable Resistor: R= 1 Ω to 910 Ω E i C=0.68 µf We kow from Kirchhoff s voltage law Figure 1. First Order Circuit dv RC dt + V = E (t) i (1) Therefore τ=rc, τ beig the time costat ad R beig the value of the variable resistor. Make sure the output cable from the FG is attached to the iput of the RLC box ad i parallel to chael 1 o the DO. The output of the RLC box should be coected to chael of the DO. I this way, both the iput ad output sigals of the RLC box ca be viewed o the DO. Make sure that the toggle switch is set to 1 st Order. Now tur the R kob o the RLC box to that which you chose before ruig the lab. Set the FG ad DO to their iitial settigs, 100 Hz frequecy ad 3.64 V peak to peak amplitude ad o DC offset. O the DO, for both chaels AC with divisioal settigs of 500 mv ad.50 ms. (You ca chage these values i order to view a particular part of the graph better.) Usig the cursors at 63.% of the iput magitude fid the time. This will correspod to the time costat τ ad is illustrated i Figure.
Figure. First Order System Respose Secod Order System: I this part a FG ad DO will be used to determie the respose characteristics (the magitude ratio ad the phase lag as fuctios of the iput frequecy) of a electrical RLC circuit. This circuit cosists of a resistor (R), a iductor (L) ad a capacitor (C) ad has the respose characteristics of a secod-order system. The circuit will be characterized by providig a iput siusoidal wave of kow amplitude ad frequecy from the FG to the RLC circuit ad measurig the circuit s output amplitude ad time delay usig the DO. The electrical diagram of the RLC circuit is show below 3
Variable Resistor: R= 1 Ω to 910 Ω L= 5 mh (R= 9.0 Ω) E i E o Figure 3. RLC Circuit The voltage differeces, V, across each compoet i a AC circuit are V = RI for the resistor, V=L(dI/dt) for the iductor ad V=Q/C for the capacitor, where I = dq/dt. I this circuit, all three compoets are i series. Thus applicatio of Kirchhoff s voltage law for the circuit gives d Q dq Q L R + = E si( t) i ω dt + dt C (5) This secod order-liear differetial equatio ca be solved for Q to yield the steady-state output voltage amplitude E o Q Ei = = (6) C C ( 1/ C Lω ) + ( Rω) From Equatio (3) ad the solutio equatio for Q, the magitude ratio is M ( ω) = E E o i = 1 [ 1 ( ω / ω ) ] + [ ( R / R )( ω / ω )] c (7) Ad the phase lag is φ( ω) = ta 1 ( R / 1 R c )( ω / ω ( ω / ω ) ) (8) 4
This equatio yields positive values of φ(ω). By covetio, because φ(ω) is a phase lag, it is plotted as havig egative values. Further for ω > ω, the phase shift must be refereced correctly. Thus the covetioal plot of φ(ω) ( i o ) versus ω would actually be -φ(ω) for ω ω ad -180 o φ(ω) for ω > ω. Also ote that i Equatios 4 ad 5, the resoat frequecy is give by R c = L / C. Oe other ote, the dampig ratio, ζ, equals R/R c for this circuit. Make sure the output cable from the FG is attached to the iput of the RLC box ad i parallel to chael 1 o the DO. The output of the RLC box should be coected to chael of the DO. I this way, both the iput ad output sigals of the RLC box ca be viewed o the DO. Make sure that the toggle switch is set to d Order. Now tur the R kob o the RLC box to that which you chose before ruig the lab. Set the FG ad DO to their iitial settigs, 100*K (K beig the frequecy multiplier for your group) Hz frequecy ad 3.64 V peak to peak amplitude sie wave ad o DC offset. O the DO, set both chaels AC with divisioal settigs of 500 mv ad.00 ms. (You ca chage these values i order to view a particular part of the graph better.) Data will be aalyzed i the fial form of M(ω) ad φ(ω), each versus the ormalized frequecy ratio, ω/ω. These values will be determied from the raw data. This icludes the iput ad output amplitudes, E i ad E o ad phase lag time, t, which is the time betwee the peak of E i ad the correspodig peak of E o. The phase lag i degrees equals (360 o )( t/t i ) where T i is the iverse of the iput frequecy i HZ ad the mius sig idicates a lag i time. Oce a satisfactory set of sigals has bee captured o the DO display, use the cursors to record the data. Eter all the raw data i the first three colums of Table. The last three colums ca be filled i after the lab. Whe doe with a iput frequecy, set the ext oe o the FG ad repeat the process. Table. RLC Respose Data Table Freq.(Hz) E i (V) E o (V) t (s) ω (rad/s) M(ω ) Φ(ω) ( o ) 100 * K 500 * K 1000 * K 100 * K 1600 * K 000 * K 00 * K 500 * K 650 * K 800 * K 3100 * K 3600 * K 4000 * K 5000 * K 7000 * K 5
10000 * K Whe you are fiished, usig a voltmeter measure the total resistace of the RLC circuit (betwee the ceter pis of the IN ad the OUT coectors). Subtract 9Ω from this value ad record it. This is the value of R for this case, which will be eeded later i the calculatios. Variable Resistor: 7 6 8 9 10 11 5 4 3 1 Max Mi This is a diagram of the variable resistor ad the umber values assiged to each tic mark. Tur the kob to the umber which your group has selected ad ru part II at this resistace. Bat Impulse Repose: I week two you will examie the system respose of bat strikig a ball. This ca be characterized as a impulse respose sice the ball will be statioary. I your text the step respose of a secod order system is reviewed o page 113. The respose to a impulse is the derivative of the step iput respose. Therefore the respose ca be characterized as y δ t) = ω 1 ς e 1 dy s yδ ( t) = (9) dt ς + ς 1 ωt e ς ς 1 ωt ( (10) for cases whe the system is uderdamped (0<ζ<1) this equatio simplifies to the form 6
y δ t ω e ςω ( t) = 1 ς si( ω t) d (11) where the rigig frequecy or damped atural frequecy (ω d ) is defied as ω d = ω 1 ς (1) The impulse respose is plotted i Figure 4 for a rage of dampig ratios ad a atural frequecy of 1.. Figure 4. Secod Order Impulse Respose. The purpose of this lab is determie the atural frequecy ad dampig ratio of a alumium bat. Strai gauges have bee added to the bat to measure ay vibratios as a result of the bat strikig the ball. The orietatio of these strai gauges i relatio to where you strike the bat is importat. A marker will be placed o the bat to idetify the proper face to strike the ball. A schematic of the bat setup is show i Figure 5. 7
Wheat Stoe Bridge/Amplifier Digital Oscilloscope Strai Gauges Figure 5. Schematic of baseball bat setup The bat will be properly coected whe you arrive for the lab. O the DO, set the iput chael to AC. Press auto set o the DO. A steady lie with some oise that reads the output from the amplifier should appear. Adjust the voltage scale o the chael 1 to be 50 mv/div ad adjust the time scale to be 10ms per divisio. Now set up the DO trigger correctly i respose to a step-iput forcig: Press the trigger meu butto. Select the slope meu ad choose, source = ch1 (the trigger is lookig for a sigal from chael 1), mode = Normal, the trigger is waitig for a sigle evet to occur), couplig = DC. Move the trigger level kob so that the trigger level is at 0mv. This esures that the DO will ot trigger util the sigal slope falls to this level. Above the graph readout o the right had side there will be some text. You must wait for the text to read Trig? before proceedig. Now hit the ball with the bat. Oce the sigal is acquired hit the start/stop butto ad save the file to a disk *. Oce you have doe this repeat this same procedure with your lab parter. 8
Determiig Dampig Ratio ad Natural Frequecy I Equatio 11 there is a expoetial term e -ωζt which dictates the decay of the sigal as a fuctio of the dampig ratio ad atural frequecy. It is possible to solve for either the dampig ratio or atural frequecy from the time trace if oe quatity is kow. I Figure 5 the system respose is plotted versus time for a dampig ratio of 0.1 ad atural frequecy of 1.. The first two peaks are idetified as y 1 ad y. Figure 5. System Impulse Respose for ζ= 0.1 & ω =1. We could the take the ratio of y /y 1 which would result i, y y ςω + ςω t 1 = e t1 (13) Therefore if the atural frequecy is kow we ca solve for the dampig ratio, l y y ς = ω t 1 (14) 9
WE ca fid the atural frequecy of the sigal by usig pwelch or FFTs i Matlab ad plottig the power spectre of the sigal. The frequecy with the largest magitude correspods to the atural frequecy of the system. Remember that ω=πf! Whe plottig the theoretical ad experimetal resposes of the bat it might be ecessary to shift ad scale the magitude of the theoretical respose. Error Aalysis Due to the iaccuracy of some of the equipmet used as well as the refiemet of your curve fit there is error associated with both the measuremets ad results i this lab. It would be helpful to try ad quatify these sice it will affect your results. Refereces: [1] Du, P. F., Measuremet ad Data Aalysis for Egieerig ad Sciece, First Editio, McGraw-Hill, New York, NY, 005 * Save the froze trace by pressig the SAVE/RECALL butto at the top of the cotrol pael. Make sure that CH1 is selected ad that the Save Waveform CH1 is selected by pressig the correspodig butto o the bottom of the scree. Press the butto o the right side of the scree correspodig to To Ref 1 (the date ad time should chage). - Tur CH1 off by holdig dow the CH1 butto ad pressig off. - Tur REF1 trace o by pressig the white REF butto ad select Ref 1 by pressig the correspodig butto at the bottom of the scree. - Isert your floppy disk ito the oscilloscope. - Press the SAVE/RECALL butto agai, but this time select the butto correspodig to To File from the buttos to the right of the scree.make sure that Spreadsheet File Format is selected i the meu o the right side of the scree, the use the kob you used to chage the cursor positio to highlight the TEK?????.CVS file. Select the Save Ref1 To Selected File optio from the meu o the right side of the scree. (It will take a couple miutes to save the data to the disk ad the????? will be replaced by the ext sequetially available umber startig at 00000. I would suggest writig dow the file ame it is saved as for future referece). 10