ENGR-00 ELECTRONIC INSTRUMENTATION Experiment 5 Experiment 5 Bridges, Ptentimeters, and Harmnic Oscillatin Purpse: In the fllwing exercises, yu will learn what a bridge is and hw it can be used t measure small changes in resistance. Yu will als learn hw t balance a bridge using a ptentimeter. Then yu will use the bridge t measure small resistances frm a strain gauge munted t an scillating cantilever beam. Yu will use the scillatin frequency and yur knwledge f cantilever beams t determine material cnstants fr the beam (mass and Yung s mdulus) and thrugh tables, the material f which the beam is made. Finally, we will extend the thery f scillatin t an electrical system, an scillating circuit. Backgrund: Befre ding this experiment, students shuld be able t Analyze simple circuits cnsisting f cmbinatins f resistrs, inductrs, capacitrs and p-amps. Measure resistance using a Multimeter and capacitance using a cmmercial impedance bridge. D a transient (time dependent) simulatin f circuits using Capture/PSpice D an AC sweep (frequency dependent) simulatin f circuits using Capture/Pspice, determining bth the magnitude and the phase f input and utput vltages. Determine the general cmplex transfer functin fr circuits. Build simple circuits cnsisting f cmbinatins f resistrs, inductrs, capacitrs, and p-amps n prtbards and measure input and utput vltages vs. time. Perfrm basic mathematical peratins n electrical circuits using p-amps. Review the backgrund fr the previus experiments. Learning Outcmes: Students will be able t Use a cmbinatin f strain gauges and resistrs cnfigured in a bridge circuit t determine the psitin f a cantilever beam. Use a difference p-amp cnfiguratin t amplify the difference signal frm the fixed and variable ndes in a strain gauge Wheatstne bridge. Apply harmnic scillatr analysis t bth a mechanical and an electrical system. Equipment Required: DMM (Keithley 8050A 4-1/ Digital Multimeter the tw Keithley bench meters are NOT the same) Analg Discvery (with Wavefrms Sftware) Oscillscpe (Analg Discvery) Functin Generatr (Analg Discvery) Instrumented Beam, Weights, Clamps, & Scale Parts Kit Helpful links fr this experiment can be fund n the links page fr this curse. Nte: due t the limited number f instrumented beams, students will wrk in teams f 4 n this experiment (as they d fr prjects) and a single reprt will be turned in fr the team. Write dwn the number f the beam and use the same ne thrughut the experiment fr mre cnsistent results. Pre-Lab Required Reading: Befre beginning the lab, at least ne team member must read ver and be generally acquainted with this dcument and the ther required reading materials listed under Experiment 5 n the EILinks page. Hand-Drawn Circuit Diagrams: Befre beginning the lab, hand-drawn circuit diagrams must be prepared fr all circuits either t be analyzed using PSpice r physically built and characterized using yur Analg Discvery bard. K.A. Cnnr, P. Schch - 1 - Rensselaer Plytechnic Institute, Try, New Yrk, USA
ENGR-00 ELECTRONIC INSTRUMENTATION Experiment 5 Part A Bridge Circuits Backgrund Bridges and Vltage Dividers: In Experiment 1, we lked at ne f the simplest useful circuits the vltage divider. In many simple applicatins f electrnics, we have nly a small number f standard vltages in whatever circuits we are building. When we use a 9V battery r V+ n the Analg Discvery as ur surce, we have nly ne vltage level available, unless we use a vltage divider t get smaller vltages. We can als use a divider, t measure resistance, if we have sme device with an unknwn resistance. Fr example, if we cnnect an unknwn resistr in series with a knwn resistr, then the vltage acrss the unknwn resistr can tell us the value f the resistance. An even better measurement can be dne by cmbining tw vltage dividers in a cnfiguratin like the ne shwn in figure A-1, which yu shuld recgnize as being made frm tw dividers. Nte that if R1 = R = R = R4, the vltages at the tw pints marked Vleft and Vright will be equal t half f the surce vltage. Thus, their difference shuld be zer. Whenever the vltage difference dv = Vleft Vright acrss a bridge utput is zer, we say that the bridge is balanced. If ne f the resistrs (e.g. R4) is slightly larger r slightly smaller than the ther three resistrs, the utput vltage dv will nt be zer and becmes, as we shall see, a very sensitive measure f the difference in resistance. V+ R1 50hms R 50hms VOFF = 0 VAMPL = 9 FREQ = 1k V1 Vlef t Vright V- R 50hms R4 50hms 0 Figure A-1. The fllwing equatins apply t the bridge circuit: V left R R4 V1 Vright V R1 R R R4 1 dv V left V right When all fur resistrs are equal t 50Ω, the bridge is balanced: V left 50 V1 50 50 V right 50 V1 50 50 dv V left V right V1 V1 0 Experiment Mdeling a bridge in PSpice In this part, we will set up a bridge in PSpice and lk at the effect f a small change in ne f the resistrs n the difference acrss the bridge. A.1 Lk at the behavir f a balanced bridge circuit. Set up the circuit shwn in Figure A-1. Use a 9V amplitude, 1kHz frequency and n DC ffset. Place vltage markers at Vleft and Vright, r use the vltage difference markers. Then run a transient analysis. (Yu have dne enugh transients nw t be able t find a reasnable run t time and step size. ) Add a Trace f the difference between the tw vltages, (Vleft - Vright). (It will already be there if yu used the vltage difference markers. ) Is the difference zer? K.A. Cnnr, P. Schch - - Rensselaer Plytechnic Institute, Try, New Yrk, USA
ENGR-00 ELECTRONIC INSTRUMENTATION Experiment 5 A. Lk at the behavir f an unbalanced bridge circuit. Nw, change R4 t be equal t 60Ω, abut a % change in R. D the analysis again and add the trace f the difference between the tw vltages. What is the amplitude f the difference vltage as a percentage f the surce vltage? Label the plt, cpy it and include it with yur reprt. A. Analyze this circuit by hand Use the vltage divider frmula t find the vltages at the tw pints and their difference. Make sure that yur answer agrees with the PSpice simulatin. Assume that R1 = R = R are knwn resistrs equal t R, and that R4 is unknwn. Derive a frmula fr R4 in terms f R, the surce vltage V1, and the vltage difference between the tw divider vltages (dv=vleft-vright). [Hint: Substitute vltage divider expressins in fr Vleft and Vright and slve fr R4.] A.4 Parameter sweep In PSpice, it is pssible t run simulatins fr several values f a cmpnent. Abve yu ran tw separate simulatins, ne with R4=50Ω and ne with R4=60Ω. Nw yu will d ne run with 11 values fr R4. This is called a parameter sweep. Change the resistance value f R4 frm 60Ω t a variable name set within the curly brackets {Rvar} as shwn in the figure belw. Yu can make the name anything yu want, but use smething that reminds us what yu are ding. The resistance value is nw a variable. Next we have t tell PSpice that we are using a parameter. T d this, we g t the parts list and select PARAM, which we will find in a library called Special. Place this item in an uncluttered spt n yur schematic. The PARAMETERS: part is a list f variables. Yu can nw create and assign a value t yur variable Rvar using the fllwing prcedure: Duble click n the wrd PARAMETERS: t display the spreadsheet Click n New Prperty. In the Prperty name textbx, enter yur chsen name ( Rvar withut the curly brackets). Enter 1k fr the default value. This value is nly used t set a dc perating pint, s the actual value desn t matter. Select the new clumn and then click Display. In the Display Frmat windw select Name and Value and then click OK. Click Apply t update all the changes t the PARAM part. Clse the Parts spreadsheet. When yu have finished, yu will see the parameter and its default value listed under PARAMETERS:. Nw yur circuit shuld lk like the figure belw. When yu have finished, yu will see the parameter and its default value listed under Parameters. (Refer t Figure A- belw.) V+ R1 R 50 00hms 50 00hms VOFF = 0 VAMPL = 9 FREQ = 1k V1 Vlef t R 50 00hms Vright V- R4 {Rv ar} 0 Figure A-. PARAMETERS: Rvar = 1k K.A. Cnnr, P. Schch - - Rensselaer Plytechnic Institute, Try, New Yrk, USA
ENGR-00 ELECTRONIC INSTRUMENTATION Experiment 5 Next we will set up the analysis (See Figure A- n the fllwing page). Create a simulatin. Select Time Dmain (Transient). Check Parametric Sweep. Check the Glbal parameter bx. Enter the variable name in the Parameter name bx. Pick a linear sweep, start value f 40, steps f and end value f 60. Figure A-. Perfrm the simulatin. Yu will have all 11 simulatins listed in the Prbe windw. Use the All ptin. Plt Vleft - Vright. Nte: if yu right click n a trace, click n infrmatin, yu can tell which value f R4 is pltted. Cpy this page and include it in yur reprt. Delete all traces frm the plt windw. Plt just Vright, fr all values f R4. Include this plt in yur reprt Cmment n if it is easier t see the effect f a change in R4 using the bridge (Vleft - Vright) cmpared t just using ne side (Vright.) A.5 Sensitivity calculatin Use the results frm the plt t determine the sensitivity f the bridge circuit t changes in R4. What is the change in (Vleft-Vright) divided by the change in R4? Summary A bridge allws yu t cmpare tw vltages and t detect relatively small changes in a cmpnent value. We will use it in part B t bserve small vltage differences caused by very small changes in the resistance f a strain gauge. Part B Strain Gauges Backgrund Strain Gauges: A strain gauge measures the displacement f a surface when it is subjected t stress. The gauge is munted securely t a surface (usually with smething like super glue). When the surface stretches because f an external frce, the strain gauge als stretches. The cnductrs f the strain gauge are either stretched r cmpressed K.A. Cnnr, P. Schch - 4 - Rensselaer Plytechnic Institute, Try, New Yrk, USA
ENGR-00 ELECTRONIC INSTRUMENTATION Experiment 5 strained. This results in a small change in the strain gauge resistance. In ur case, the change in resistance f the strain gauge will give a signal prprtinal t the displacement at the end f the beam. Figure B-1 shws a typical strain gauge. Natinal Instruments has a very gd discussin f stain gauges and ur applicatin at: http://zne.ni.cm/devzne/cda/tut/p/id/64 G t the link and read up t Figure 8, Full-bridge Circuit. We use Half-Bridge, Cnfiguratin II Bending Strain nly. Figure B-1. The Cantilever Beam: Figure B- shws the cantilever beam we use in class. t l w a Figure B-. Please nte: The beams have tw strain gauges, ne n the tp f the beam and ne n the bttm. This allws fr the use f the Half-Bridge Circuit discussed n the NI website. These beams lk smewhat different than the figure abve but the principles are the same. The beam has tw strain gauges wired tgether, replacing R and R4 in Figure A-1f this dcument. It als has tw matched 50Ω resistrs. These are R1 and R f Figure A-1. The signals frm the stain gauges are small, s a difference amp with a gain f 100 is required. Experiment: Strain Gauges We will nw cmbine ur study f strain gauges, bridge circuits and p-amps in a practical hardware applicatin. We use the tw resistrs and the tw strain gauges munted t a beam t create a bridge circuit t measure the deflectin f the beam. The signals are small s the utput f the bridge will be amplified using a difference amplifier p-amp circuit. B.1 Befre yu build the circuit, use the multimeter t determine the resistance f ne strain gauge in its rest psitin. Unfrtunately it is nt easy t make this measurement when the beam is cnfigured fr the circuit. Rather than discnnect the easily brken strain gauge wires, the best yu can d is measure the strain gauge while K.A. Cnnr, P. Schch - 5 - Rensselaer Plytechnic Institute, Try, New Yrk, USA
ENGR-00 ELECTRONIC INSTRUMENTATION Experiment 5 it is in parallel with series resistrs (the secnd strain gauge and tw 50 resistrs). Yu shuld verify that this is what will actually be measured when cnnecting terminals frm a strain gauge t the Ohm meter. This series/parallel resistr cmbinatin changes by abut 1 frm ne extreme t the ther. Verify the nminal measured resistance value in terms f the series/parallel cmbinatin. Be sure t use the mst accurate DMM (Kiethly 8050A) n the center table fr this measurement. D nt ver extend the beam upwards r dwnwards. Measure the resistance f the strain gauge with the DMM when the beam is at rest. This is the resistance measured between terminals A and B r A and D (with the rest f the bridge cnnected t the circuit). Deflect the beam dwn until yu bserve abut half way t the supprt plate and measure Figure the resistance B- with the DMM. If yu d nt bserve any change, deflect it dwnward until the beam is clse t but nt tuching the supprt plate. Deflect the beam up an equal amunt and measure the resistance with the DMM. Nte that the resistances yu measure are prprtinal t hw much yu mve the beam. Write dwn the maximum and minimum resistance f yur strain gauge circuit. Yu shuld nw see why we indicate this resistance with a variable resistr in the figure belw. Nte that as the strain gauge underges expansin and cmpressin its resistance varies a small amunt. The bridge is designed s that the fur resistances start at a value R. When the beam is flexed, the resistances f the tw strain gauges becme R + ΔR and R ΔR. That is, ne gets larger and ne gets smaller. The parallel resistr cmbinatin measured is given by the fllwing expressin. Thus, the measured change in R is nly half f the resistance change f the strain gauge. Yu shuld als see that the measured resistance is abut ¾ f the strain gauge resistance. R Parallel R RR R R R 4R 4 R 4 R 1 4 R 4 R 1 R D +5V V+ - & GND R1fixed 50hms C Rfixed 50hms R1straingauge Rb1 50hms 100k Ra1 1k U1 + A 1+ 1- Ra Rstraingauge 50hms 1k ua741-7 4 V+ OS OUT OS1 V- 5 6 1 0 Vut + B -5V V- Rb 100k A B C D Figure B-4. Terminal Blck n Beam Base K.A. Cnnr, P. Schch - 6 - Rensselaer Plytechnic Institute, Try, New Yrk, USA
ENGR-00 ELECTRONIC INSTRUMENTATION Experiment 5 Use the ua741 p-amp in the actual circuit. Wire the circuit neatly because it will be used fr the next prject. Nte, befre taking data, identify and label every element in the circuit n yur hand-drawn diagram, especially the resistances in the pre-wired circuit n the beam. Almst all prblems students have with this circuit ccur because they d nt understand what they are building. B. Wire the circuit shwn in Figure B-4. The circuit in the dashed bx is wired n yur prtbard. This shuld be wired neatly. It will be used fr this experiment and fr the next prject. The circuit in the slid bx is prewired n the instrumented beam. Yu need t cnnect 4 wires between the beam and the Prtbard, including V+ (t D) and V- (t B). Dn t enable V+ and V- n the Analg Discvery until yu are certain that the circuit is wired crrectly. Yu must enable these supplies when yu are ready t cllect data. Nte that yu are t measure bth the utput frm the strain gauge bridge (which is als the input t the difference amplifier) and the utput frm the difference amp. Fr the bridge utput, channel 1 is set up t measure in differential mde (neither wire is grunded). B. Repeat the deflectin measurements f part B.1. Remember that the utput f the amplifier (channel ) shuld be 100 times larger than the utput f the bridge (channel 1), s yu shuld nt use the same scales n yur scillscpe. Read the tw utput vltages withut any beam deflectin. Read the tw utput vltages with the beam deflected abut halfway t the plate. Read the tw utput vltages with the beam deflected an equal amunt up. Recrd the three vltage measurements. Hint: Use the measurement functin t read the utput. Keep the signal in the center f the screen. Damped Sinusids: When the beam is deflected and then allwed t scillate, yu shuld measure signals that lk like the damped sinusid in Figure B-5. Damped sinusids are gverned by the equatin: v(t) = Ce -αt sin(ωt), where is called the damping cnstant that determines the rate f decay. T find the damping cnstant, chse tw pints at extreme ends f the sinusid and use the fllwing equatin and slve fr : v1 = v0e -α (t1 t 0) Figure B-5 B.4 When yu have everything hked up, set the beam int free scillatin. Yu shuld bserve decaying sinusidal vltages. The tw signals shuld lk the same, but channel shuld be 100X larger than channel 1. Once yu have bserved tw reasnable decaying sinusids, turn ff channel 1 n yur scpe display. Yu nly need t bserve channel fr the fllwing measurements. K.A. Cnnr, P. Schch - 7 - Rensselaer Plytechnic Institute, Try, New Yrk, USA
ENGR-00 ELECTRONIC INSTRUMENTATION Experiment 5 It will take a little experimenting t get a gd trace n the screen. Start with time base f 10ms/div. Why is this a gd place t start? What is the apprximate frequency f scillatin f yur beam? Pull the beam dwn abut half way and nte if the vltage has drpped r increased. If the vltage has increased, swap the tw wires cnnected t Ra1 and Ra f Figure B-4. That is, swap the tw inputs t the difference amplifier. Pull the beam up abut the same amunt (1cm r ½ ). Nte hw much the vltage changed relative t n deflectin. Call this V trigger. Adjust the trigger level s it is abve zer by abut V trigger r a little less. Nw set the trigger mde t Nrmal. Pull the beam dwn and release. If the Analg Discvery scpe triggered, yu are set. Nw set the time t 100ms/div. Fr lng time traces, 0.1sec/div, yu may need t use Single Sweep. Pull the beam dwn and release. Once yu have btained a clear signal f a decaying sinusid, save the data t a file and plt it with Excel. Yu can als use the measurements r directly click n the trace n the Analg Discvery scpe t measure pints n the trace. Determine the frequency f scillatin and the decay cnstant. Include this plt with yur reprt. Indicate n it hw yu fund the beam frequency and the decay cnstant. Nte that a mre accurate estimatin f the frequency can be fund by averaging ver many cycles. Nte that there is a very gd reference n strain gauges frm the Vishay Measurements Grup, Inc available n the links page. Summary The psitin f a cantilever beam can be measured using strain gauges. The signals frm strain gauges are relatively small s they are typically used in a bridge circuit and with amplificatin. In this experiment, we used the signals t bserve the scillatin f a cantilever beam. Part C Instrumented Beam as a Harmnic Oscillatr Backgrund Simple Pendulums: Befre we address harmnic circuits, we will review sme f the prperties f the simple pendulum. The instrumented beam is a very gd example f a simple pendulum, even thugh it lks mre like a small diving bard. Let us assume that the end f the beam mves in the x-directin. Obviusly, this is a simplificatin, since it really travels alng the arc f a circle. When the beam is statinary, we will assume that it is hrizntal and at x = 0. Again, this is an apprximatin because the beam must bend dwnward slightly due t its wn weight. When the beam is bent, it experiences a restring frce like a spring, F kx, where k is the spring cnstant. Frm Newtn s Law, we can relate this frce t the acceleratin, a, velcity v, and displacement, x, f the beam. F dv d x ma m m kx Using nly the terms related t the displacement, we can derive the harmnic scillatr equatin: x 0. d x k m K.A. Cnnr, P. Schch - 8 - Rensselaer Plytechnic Institute, Try, New Yrk, USA
ENGR-00 ELECTRONIC INSTRUMENTATION Experiment 5 In standard frm, the harmnic scillatr equatin is x 0 where is the frequency f scillatin in rad/sec. Thus, the beam will scillate at d x k. The slutin t this equatin is: x x0 cs( t ) m, where x 0 is the initial deflectin f the beam and 0 is the initial phase. If yu d nt recall that this is the slutin, plug the expressin fr x int the differential equatin and yu will see that it wrks. Fr simplicity, there is n need t include 0. Therefre, we can mdel the scillatin f a beam (with n frictin) using the nw familiar sinusidal equatin: x x 0 cs( t). Using this equatin, we get an scillatin that will g n frever, rather than decaying slwly away, like the actual beam. We can use the cnservatin f energy t verify this result. We knw that the pendulum r any harmnic scillatr wrks by exchanging energy between tw different frms. Nt all frms f energy can be easily cnverted t anther state and then back again, but we knw this is trivial with the kinetic and ptential energy f a mass. 1 The kinetic energy f a mass (the beam) is given by the equatin: KE mv and the ptential energy f a spring 1 system (like the beam) is given by PE kx. Upn initial deflectin, the energy f the beam is all ptential. Since we have assumed n dissipatin (n frictin r ther damping frce), the ttal energy will be cnserved. 1 1 Therefre, the ttal energy, W, will be cnstant and given by, W mv kx at any ne pint in the beam s scillatin. We can start at this expressin f energy cnservatin t determine the equatins f mtin f the beam r any ther simple pendulum. Since the ttal energy is a cnstant, we can take the time derivative f the entire expressin and set it equal t zer: dw 1 m() v dv 1 k() x dx 0 Since dx dv dv v, we can substitute fr dx/ and write mv kxv 0 m kx, which brings us back t Newtn s secnd law: F dv d x ma m m kx cnservatin law, we can use it even t find ut hw things change with time.. Thus, nce we have a Yung s Mdulus: We can use frequency t determine the prperties f an scillating mechanical system. In this part, we will use it t determine Yung s Mdulus fr an scillating beam and use this infrmatin t guess what type f material the beam is made f. We can d this using the relatinship between and the prperties f the scillating system. We already knw the relatinship between frce and the prperties f a spring. We can slve F fr k: F kx and k. Fr the scillating beam, x crrespnds t the displacement at the end f the x beam. This relatinship is defined by the physical prperties f the beam, Fl x EI, where l is the length f the beam, E is Yung s Mdulus, and I is the mment f inertia f the beam. If we slve this relatinship fr F/x, we K.A. Cnnr, P. Schch - 9 - Rensselaer Plytechnic Institute, Try, New Yrk, USA
ENGR-00 ELECTRONIC INSTRUMENTATION Experiment 5 will have a secnd expressin fr ur cnstant, k an bject with a rectangular crss-sectin and substitute it in fr I: F x EI. Nw, we can lk up the mment f inertia fr l wt Ewt Ewt I and k. 1 1l 4l Recalling ur relatinship fr frequency frm befre and slving fr k, we find: k k f and (f ) s k m(f ). Since this is an scillating system, we will m m ignre the negative sign. This gives us ur final result: Ewt 4m l (f ) k m(f ) E 4l wt Therefre, if we can cme up with a reasnable estimate fr the mass f the beam and its resnant frequency, we shuld be able t find Yung s Mdulus and use that t lk up the material frm which the beam is made. The mass f the cantilever beam: Recall that the pendulum r harmnic scillatr equatin hlds fr pint masses lcated at the end f a massless beam. Since the beam has mass, but its center f mass is nt lcated at the end f the beam, this term is multiplied by 0. t give the equivalent mass placed at the end f the beam that prduces the same respnse. The beam may als have a sensr attached t the end which adds extra mass. When we talk abut m in this experiment we are referring t the actual mass f the beam migrated t the end. Thus m is the effective mass f the beam with n lad and m 0. m. Experiment beam Frequency f a laded beam In the last experiment, yu shuld have measured the scillatin frequency f the unladed beam using the strain gauge and the bridge circuit. C.1 Use the circuit frm part B t find the frequency f the beam with varying mass. Set the beam int scillatin. Prduce a plt f the decaying sinusid yu bserve and save it using the Analg Discvery sftware. Cpy it, shw it t a TA r instructr and have them sign yur checklist. Recrd the frequency in the table belw. Mass f bject Mass f clamp m n = Ttal mass at end f beam Frequency (Hz) 0 kg 0 kg m 0 = f = m 1 = f 1 = m = f = m = f = C. Measure the frequency three mre times using additinal masses f yur chice. Chse three bjects frm abut 0 t abut 100 grams. Yu shuld try t get a gd distributin t get discernable data. The actual mass f the beam becmes less imprtant with the heavier masses. S ne mass shuld be heavy (arund 100 grams). Dn t lad the beam with a mass s heavy that it permanently bends the beam. K.A. Cnnr, P. Schch - 10 - Rensselaer Plytechnic Institute, Try, New Yrk, USA
ENGR-00 ELECTRONIC INSTRUMENTATION Experiment 5 Measure the masses f the bjects yu have chsen with the scale in the studi. Enter them int the table. Be sure t include the mass f the paper clips used t hld the added mass. The ttal mass at the end f the beam, m n, is the sum f the bject plus the clamp. Place each mass as clse t the end f the beam as pssible using a paper clip. Find the beam frequency fr each mass and recrd it in the table. Try t be as accurate as pssible. Yur values shuld have at least ne decimal pint. Yu shuld check the frequency a cuple f times, since yu shuld ntice that there will be a range f values fr the frequency, primarily because f nise and the smewhat nn-ideal nature f the sinusidal vltage. Analysis f beam data In this part f the experiment, we will analyze the frequency data t determine the mass f the beam, Yung s Mdulus fr the beam, and, finally, the material ut f which the beam is made. C. Find 4 equatins in unknwns. We knw that ur system is gverned by the relatinship between the scillatin frequency and the prperties f a spring. k m m n ( n f ), n=0,1,, We can write ut a k expressin fr each f the fur frequencies yu have measured: k ( m m k ( m m ) (f ) k ( m m k ( m m 0 1 ) (f ) 1 0 ) (f ) ) (f ) C.4 Use these numbers t determine the values f k, and m. Nte that yu are making fur measurements t determine tw cnstants. This means that yu have sme redundancy built in and als that yu will nt btain perfect agreement fr all fur equatins. Nne f yur measurements will be perfect, s it is best t have mre measurements than cnstants t average ut measurement errr. Yu need t find the values f k and m that cme the clsest t satisfying all fur equatins. We are ging t use Excel t plt the frequency f ur system in relatin t the mass added t the end. First we must slve fr f n. Nte that in the equatin belw, m n is the x variable and f n is the y variable. 1 kguess fn m m guess n We need t determine a gd guess fr k (the spring cnstant) and m (the effective mass f the beam) in rder t plt this equatin. Use the data frm nly tw f yur masses and slve tw equatins in tw unknwns t determine a guess fr k and m. (If yu get a negative mass, try using yur smallest and largest mass t d the calculatin.) Keep in mind that these are just guesses, s yu dn t need t get carried away slving all cmbinatins f all the equatins. Yu culd use sme type f statistical analysis instead t get yur guesses fr k and m. Fr example, yu culd determine the standard deviatin f the fur expressins fr k fr a range f realistic values fr the effective beam mass. Nw yu can plt the equatin in Excel. Use k guess and m guess yu just calculated. Chse values fr m n between 0 and 10 grams. Yu are pltting a general functin and matching yur data t it. m n is the dmain f yur functin, yu just need a set f x (m n ) values s yu can calculate y (f n ). Have Excel calculate values fr f n fr each m n and plt the results. Place yur fur data pints n the plt. Hw well d these pints fit the curve yu generated? If yur guesses are exactly perfect, they will lie right n the curve. Since they are nly guesses, there is prbably rm fr imprvement. K.A. Cnnr, P. Schch - 11 - Rensselaer Plytechnic Institute, Try, New Yrk, USA
ENGR-00 ELECTRONIC INSTRUMENTATION Experiment 5 Nw it is time t adjust k guess and m guess t get the curve t match yur data as clsely as pssible. Adjusting ne will mve the plt up and dwn. Adjusting the ther will cause the curvature t change. Play with the numbers until the curve matches yur 4 data pints as clsely as pssible. When yu are ding this, keep in mind that the lcatin where the graph crsses the y axis represents the unladed frequency f the beam. The functin ges up very quickly near zer mass. What is a reasnable estimate fr the unladed beam frequency? Include the final plt with the general curve and the fur data pints marked in yur reprt. Use the values f k guess and m guess that give yu the clsest match in yur final calculatins fr the beam mass and Yung s Mdulus. C.5 Final results. Calculate the mass f the beam using yur best guess fr m, m = 0.m beam. CAREFULLY measure the dimensins f yur beam. A small inaccuracy in yur measurements can lead t a large discrepancy in yur results. Extraplate the frequency fr the beam (with n lad at all n the end) frm yur plt. This is the pint at which mass at the end f the beam is 0 kg. Calculate Yung s Mdulus using yur best estimate fr k and m. Lk up Yung s Mdulus in the table f yur chice and find sme pssible materials fr yur beam. (There may be mre than ne pssibility.) If yu find mre than ne pssible value fr the material, think abut ther prperties f the beam that may narrw the pssibilities. Summary In this part f the experiment, yu used the scillatin frequency and ther physical prperties f a cantilever beam t find infrmatin abut the beam that yu culd nt measure. Yu als learned hw t use curve fitting t find a slutin when there are mre equatins than unknwns. Part D Oscillating Circuits Backgrund Energy strage in inductrs and capacitrs: In passive electrical systems, there are three kinds f circuit elements: resistrs, capacitrs and inductrs. Resistrs turn electrical energy int heat. When a current I flws thrugh a resistr, there will be a vltage drp V acrss the resistr. The pwer dissipated by the resistr is equal t the prduct f I times V. Since resistrs prduce heat, it shuld be n surprise that they play the same rle as frictin in a mechanical system. The ideal pendulum will scillate frever a real pendulum will scillate until all its stred energy is cnverted t heat thrugh frictin. Thus, if we wish t create a circuit analgus t the ideal harmnic scillatr, it can have n resistrs in it. Rather, we will cmbine nly inductrs and capacitrs. A typical inductr cnsists f a cil f wire. If we pass a current thrugh the cil, a magnetic field will be created. Many f us have made simple electrmagnets at sme time in ur lives by wrapping wire arund sme magnetic material like a nail. When a battery is cnnected t the wire, it is pssible t attract small pieces f irn t the nail. The field created by the cil, the magnetic field, can d wrk and thus cntains energy. The energy stred in an inductr is given by the expressin WM = (1/) L I where we have used the subscript M t indicate that the energy is stred in the magnetic field and L is the inductance in units f Henries. Jseph Henry was hnred by using his name fr this unit because f his early wrk in develping practical electrmagnets. He began his wrk here in New Yrk s capital district when he was teaching at the Albany Academy. There is a statue f this great scientist utside the riginal lcatin f the Academy acrss the street frm the Albany City Hall. Henry was a cntemprary f Ams Eatn, the intellectual frce behind the funding f RPI. He eventually left Albany fr Princetn and the Smithsnian. K.A. Cnnr, P. Schch - 1 - Rensselaer Plytechnic Institute, Try, New Yrk, USA
ENGR-00 ELECTRONIC INSTRUMENTATION Experiment 5 A typical capacitr cnsists f tw metal plates f large area separated by sme insulating material such as Tefln r sme ther plastic. When a vltage surce is cnnected t the plates, charge flws frm the surce t the plates with psitive charge depsited n the plate with the highest vltage and negative charge depsited n the plate with the lwest vltage. Since these charges are ppsite in sign and since unlike charges attract ne anther, there is a frce between the tw plates. Again, the existence f this frce tells us that we have t d wrk t charge up the plates and that there is energy stred. In this case, the energy is stred in the electric field created by the charges. The energy stred by a capacitr is given by the expressin WE = (1/) C V where we have used the subscript E t indicate that the energy is stred in the electric field and C is the capacitance in units f Farads. Michael Faraday als wrked n electrmagnets and, being British, gained much mre fame fr his wrk, since America was a scientific backwater at the time. Henry shwed him hw t make better magnets, but Faraday s wrk was much mre far reaching. Henry als shwed Mrse hw t build a telegraph! Aside: It is smewhat interesting t nte that neither Henries nr Farads turns ut t be much f a cmmn practical unit. One Henry is a huge inductr, rarely seen in practice. One Farad is als rare, nw ccasinally seen in highly filtered pwer supplies fr cmputers. We will need t use the prefixes milli-, micr-, nan-, pic-, etc. a lt when dealing with these cmpnents. We als d nt see ne Ohm all that much, but require the ther kind f prefixes (kil-, mega-, etc.). Cnservatin f Energy in a Harmnic Circuit: There are many imprtant lessns we can learn frm the harmnic scillatr, but perhaps ne f the mst useful is the value f cnservatin laws. It is fair t say that the mst pwerful prblem slving technique is t first decide which cnservatin laws hld. Once the cnservatin laws are identified, they can be used t determine a great deal f infrmatin abut any system. Figure D-1. Cnsider the simplest pssible cnfiguratin f a single capacitr and a single inductr cnnected as shwn in Figure D-1. Nte that, since there are nly tw cmpnents, ne can describe this cnnectin as either in parallel r in series. Als assume that the capacitr has been charged up t sme vltage V at time t = 0, at which time it is cnnected t the inductr. The charge will begin t flw creating a current thrugh the inductr. After a shrt time, all the charge that was riginally n the capacitr plates will be gne and the current thrugh the inductr will reach its maximum value. Thus, we began with all the energy stred in the capacitr and nne in the inductr and end with the ppsite cnditin. The current flwing thrugh the inductr will then charge the capacitr back up and the prcess will begin again. The energy is traded back and frth between the tw strage elements. The ttal energy f the system must remain cnstant because there is n dissipative element (n resistr). The ttal energy, W, is a cnstant equal t the energy stred in the capacitr added t the energy stred in the inductr. Thus, 1 1 W CV C LI L. Since this is a cnstant, we can take the time derivative f this expressin and set it equal t zer. dw 1 dvc 1 di L di L dvc C() VC L() I L 0 L I L C VC 0 K.A. Cnnr, P. Schch - 1 - Rensselaer Plytechnic Institute, Try, New Yrk, USA
ENGR-00 ELECTRONIC INSTRUMENTATION Experiment 5 We want t d with this equatin what we did with the energy cnservatin equatin f the beam. This time, thugh, instead f expressing the equatin in terms f the displacement, x, we want t express it in terms f the vltage, V. We can use the general equatins fr the behavir f the capacitr and inductr t make these substitutins: V L di L L I C dv C C Fr the circuit shwn abve, V is the vltage at the tp f the circuit and I is the current flwing arund the circuit. Since this is a series circuit with nly tw elements: V = VC = VL and I = IC = IL. Making these di dv simplificatins, ur equatins becme: V L I C dv I We can further cnclude frm the capacitr equatin that, by slving fr dv/, and, by taking the time C di d V derivative f bth sides, C. Substituting int the cnservatin f energy equatin, we get di dv 0 d V I d V L I C V LC I C V 0 LC V 0 C d V 1 This leads us directly t the harmnic equatin fr scillating circuits: V 0 LC d V If we cmpare this t the general expressin fr harmnic scillatin, 0V 0, we can determine the resnant frequency f the circuit: 1. 0 LC Als recall that this is the equatin fr the resnant frequency f any simple RLC circuit. Damped Oscillatin: The circuit we mdeled abve is unrealistic because it has n resistance at all. This is analgus t a mechanical system with n frictin. T make a mre realistic system, we must add sme resistance, as shwn in Figure D-. Figure D-. The circuit will still scillate, hwever, the scillatin energy will gradually dissipate because f the resistance. The utput signal will be similar t the scillatin behavir f the beam - a damped sinusid. Nte that this circuit has n vltage surce. It needs t have an initial amunt f energy placed int it. This is similar t the initial displacement yu place n the beam t make it scillate. The damped circuit has the fllwing scillatin equatin K.A. Cnnr, P. Schch - 14 - Rensselaer Plytechnic Institute, Try, New Yrk, USA
ENGR-00 ELECTRONIC INSTRUMENTATION Experiment 5 d V dv 0V 0 where is the damping cnstant. It can be shwn that in an ideal damped scillatin circuit, is given by the fllwing equatin belw: R L K.A. Cnnr, P. Schch - 15 - Rensselaer Plytechnic Institute, Try, New Yrk, USA
ENGR-00 ELECTRONIC INSTRUMENTATION Experiment 5 Experiment Mdeling a damped scillatr We will nw cnsider an RLC circuit, with all three kinds f passive cmpnents and bserve the damped scillatins. Nte that we are als using a dide, a device that nly allws current t flw in ne directin (t the right in the figure). When a dide is n (frward biased), abut 0.6 0.7V drps acrss it. We will learn mre abut dides in Experiment 8, but, fr nw, yu can cnsider it t be a switch that turns n when a psitive vltage exceeding 0.6 r 0.7V is placed acrss it and is therwise ff. We are using it here s that the cmbinatin f an inductr L and a capacitr C are frced t scillate ver and ver, kind f like ringing a bell. The resistance in the circuit is the DC resistance f the inductr and nt a separate cmpnent. D Vin Vut V1 = -4 D1N4148 V = 1 V TD = 0 TR = 1u TF = 1u PW =.01 PER =.0 R 40 L mh C.1uF 0 Figure D-. Create the scillating circuit in Figure D- abve in PSpice. The vltage surce used is a square wave with a negative ffset s that the psitive vltage is smaller than the negative vltage. Since the perid is 0ms, what is the frequency? Place vltage markers n the circuit between at the pints labeled Vin and Vut. Use PSpice t simulate the transient respnse f this circuit fr a ttal time f ne cmplete perid. Cpy yur results and include them in yur reprt. What features f the vltages reminds yu f the instrumented beam? Use the transient plt t find the scillatin frequency f yur circuit. Hw des it cmpare it t the 1 calculated value f f? LC Use the plt t determine the damping cnstant f the circuit. In a simple RLC circuit, such as this ne, the damping cnstant can als be fund mathematically using the expressin =R/(L). Calculate the damping cnstant and cmpare it t the ne yu fund using the plt. Keep yur simulatin active because yu will be returning t it after the next task. Build the same circuit n yur breadbard and measure the same input and utput vltages. Remember that the resistance and inductr in the circuit are actually nly the inductr in a physical circuit. Yu shuld cnfirm that the DC resistance f the inductr is 40Ω by measuring it with a multi-meter. Set up the functin generatr t prduce a square wave and set the frequency, amplitude and ffset t btain the same values used in the PSpice simulatin. K.A. Cnnr, P. Schch - 16 - Rensselaer Plytechnic Institute, Try, New Yrk, USA
ENGR-00 ELECTRONIC INSTRUMENTATION Experiment 5 Cnnect the wires fr channels 1 and (1+ and +) at the pints labeled Vin and Vut, and grund the ther wires fr each channel (1- and -). Set up the time scale n yur scpe t display ne cmplete perid. Cpy yur results and include them in yur reprt. Answer the same questins as yu did fr yur PSpice results. Describe the similarities and differences between yur simulated and experimental results. Yu shuld have fund excellent agreement between yur simulated and experimental results. If yu did nt, yu shuld identify any prblems with either apprach and repeat the last tw general tasks. The similar experimental and simulated results validate the simulatin and permit us t use features f the simulatin that are nt available in experiments. Fr example, there is n simple way t btain the current thrugh the inductr in the experiment. (Yu may want t think abut why this is the case.) In the simulatin, it is pssible t easily find the current by just adding a current prbe t ne f the ends f the ideal inductr. In the final task f this experiment, yu are t use the simulatin t determine the energy stred in bth the inductr and capacitr and see hw they vary with time. Return t PSpice Delete the vltage prbe n the input and add a current prbe t ne end f the inductr. Yur results plt shuld change withut having t run the simulatin again because PSpice calculates all vltages, currents and pwer each time. Unlike the Analg Discvery scpe, the results display fr PSpice des nt allw fr different vertical scales fr each measurement. Thus, the current signal will be very hard t read because it is s small. T make it easier t see, multiply it by the value f the resistance (40). Yu may als find that it is negative. Change it t a psitive value t be able t better cmpare with the utput vltage. Nw yu have all infrmatin necessary t determine the energy stred in the capacitr and in the inductr. Mdify the tw traces in yur plt s that they display the energy in the inductr 1 LI and the energy in 1 CV. Als, add a third trace that is the sum f the tw energies (e.g. the ttal energy). the capacitr Cpy yur results and include them in yur reprt. There shuld be sme very interesting features in these plts. Be sure that yu anntate them fully s that an bserver can easily identify all features. Describe the key features f yur energy data vs time, including the maximum stred energy in either the inductr r the capacitr. Summary In this part f the experiment, yu have related yur knwledge f scillating mechanical systems t an scillating electrical system and created an scillating circuit. Nte that yu will cntinue yur study f harmnic scillatrs in Prject, where yu will use the strain gauge set up frm this experiment and an accelermeter t determine bth the psitin and acceleratin vs time fr the beam. The beam will be laded a little differently because yu will install the accelermeter near the end. Yu will be using the tw measurements t determine the velcity f the beam. Once yu have bth the beam psitin and velcity, yu will be able t prduce a plt that will lk a gd deal like the energy vs time plt fr the RLC circuit. Checklist and Cnclusins The fllwing shuld be included in yur experimental checklist. Everything shuld be labeled and easy t find. Credit will be deducted fr pr labeling r unclear presentatin. ALL PLOTS SHOULD INDICATE WHICH TRACE CORRESPONDS TO THE SIGNAL AT WHICH POINT AND ALL KEY FEATURES SHOULD BE LABELED. Hand-Drawn Circuit Diagrams fr all circuits that are t be analyzed using PSpice r physically built and characterized using yur Analg Discvery bard. K.A. Cnnr, P. Schch - 17 - Rensselaer Plytechnic Institute, Try, New Yrk, USA
ENGR-00 ELECTRONIC INSTRUMENTATION Experiment 5 Part A (1 pints) Include the fllwing: 1. A.1 Nthing t plt. (0 pt). A. a. Plt f Vleft - Vright with R4 mdified. (1 pt) b. Fractin f utput signal cmpared t Vac surce. (1 pt). A. a. Analysis f circuit in part A.. ( pt) b. Derivatin f frmula fr Vut = dv (= Vleft Vright) as a functin f R4, V1,... ( pt) 4. A.4 a. Plt f all 11 Vut traces. ( pt) b. Plt f just Vright fr all 11 cases. (1 pt) c. Cmment n the advantage f using a bridge (Vleft - Vright) vs. using just a divider (Vright). ( pt) 5. A.5 What is the sensitivity f this circuit (change in Vut/change in R)? (1 pt) Part B (16 pints) Include the fllwing: 1. B.1 Min and Max resistance measured. ( pt). B. Nthing.. B. Vltage measurements fr three beam psitins. ( pt) 4. B.4 Analg Discvery plt f beam scillatin with calculatins f beam frequency and damping cnstant n it. (6 pt) Answer the fllwing questins: 1. What is the resnant frequency f the beam? What value did yu find fr the damping cnstant? Write an equatin fr the decaying sinusid utput f the beam in the frm v(t)=ce -αt sin(ωt). (5 pt) Part C (6 pints) Include the fllwing plts/table: 1. Tw Analg Discvery plts f the decaying sinusid btained with different masses added. Fr each write the mass added and the frequency measured. Have a TA r instructr sign and date the plts. (5 pt). A table such as the ne n page 10 f this dcument, listing the mass, ttal mass and frequency. (5pt). Excel plt f frequency vs. lad mass with fur pints marked. (5 pt) Answer the fllwing questins: 1. Explain r shw yur wrk n hw yu did yur analysis t determine a reasnable first guess fr k and m. ( pt). What are the values fr k and m that yu btained by making the plt in Excel? ( pt). Calculate the mass f the beam. ( pt) 4. Calculate Yung s Mdulus fr the beam. Clearly indicate the values yu measured fr the beam s dimensins. ( pt) 5. What d yu cnclude the beam culd be made f? Why? ( pt) Part D (16 pints) Include the fllwing plts: 1. PSpice plt f input and utput frm the scillating circuit. Be sure t label yur plt with the resnant frequency and damping cnstant f the circuit yu analyzed using the PSpice plt. Write an equatin fr the utput in the frm v(t)=ce -αt sin(ωt). (4 pt). Analg Discvery plt f input and utput frm the scillating circuit. Again, be sure t determine the resnant frequency and damping cnstant and include this infrmatin n yur plt. (4 pt) K.A. Cnnr, P. Schch - 18 - Rensselaer Plytechnic Institute, Try, New Yrk, USA
ENGR-00 ELECTRONIC INSTRUMENTATION Experiment 5. PSpice plt f the energy in the inductr, the energy in the capacitr and the ttal energy as functins f time. Be sure t label everything interesting in these plts. (4 pt) Answer the fllwing questins: 1. Cmpare and cntrast bth yur PSpice and Analg Discvery plts with the plts fr the cantilever beam. ( pt). What value did yu calculate fr f using the equatin fr the resnant frequency? Hw clse f an estimate is this t the resnance yu fund in the plt? ( pt). What value did yu calculate fr using the equatin? Hw clse f an estimate is this t the damping cnstant yu fund in the plt? ( pt) Summary (10 pints) 1. Organizatin, cmpleteness, rdering. Is this easy t grade? (8 pt). List member respnsibilities. ( pt) List grup member respnsibilities. Nte that this is a list f respnsibilities, nt a list f what each partner did. It is very imprtant that yu divide the respnsibility fr each aspect f the experiment s that it is clear wh will make sure that it is cmpleted. Respnsibilities include, but are nt limited t, reading the full write up befre the first class; cllecting all infrmatin and writing the reprt; building circuits and cllecting data (i.e. ding the experiment); setting up and running the simulatins; cmparing the thery, experiment and simulatin t develp the practical mdel f whatever system is being addressed, etc. Summary/Overview (0 t -10 pts) There are tw parts t this sectin, bth f which require revisiting everything dne n this experiment and addressing brad issues. Grading fr this sectin wrks a bit differently in that the verall reprt grade will be reduced if the respnses are nt satisfactry. 1. Applicatin: Identify at least ne applicatin f the cntent addressed in this experiment. That is, find an engineered system, device, prcess that is based, at least in part, n what yu have learned. Yu must identify the fundamental system and then describe at least ne practical applicatin.. Engineering Design Prcess: Describe the fundamental math and science (ideal) picture f the system, device, and prcess yu address in part 1 and the key infrmatin yu btained frm experiment and simulatin. Cmpare and cntrast the results frm each f the task areas (math and science, experiment, simulatin) and then generate ne r tw cnclusins fr the practical applicatin. That is, hw des the practical system mdel differ frm the riginal ideal? Be specific and quantitative. Fr example, all systems wrk as specified in a limited perating range. Be sure t define this range. Ttal: 80 pints fr experiment packet 0 t -10 pints fr Summary/Overview 0 pints fr attendance 100 pints Attendance (0 pssible pints) classes (0 pints), 1 class (10 pints), 0 class (0 pints) Minus 5 pints fr each late. N attendance at all = N grade fr this experiment. K.A. Cnnr, P. Schch - 19 - Rensselaer Plytechnic Institute, Try, New Yrk, USA
ENGR-00 ELECTRONIC INSTRUMENTATION Experiment 5 Experiment 5 (5V supplies versin) Sectin: Reprt Grade: Name Name Name Name Checklist w/ Signatures fr Main Cncepts Fr all plts that require a signature belw, yu must explain t the TA r instructr: the purpse f the data (using yur hand-drawn circuit diagram), what infrmatin is cntained in the plt and why yu believe that the plt is crrect. Any member f yur grup can be asked fr the explanatin. PART A: Bridge Circuits A: Behavir f unbalanced bridge circuit 1. Plt f Vleft Vright with R4 mdified. Fractin f utput signal cmpared t Vac surce A: Analyze circuit by hand 1. Analysis f circuit in part A.. Derivatin f frmula fr Vut=dV(=Vleft Vright) A4: Parameter sweep 1. Plt f all 11 Vut traces. Plt f just Vright fr all 11 cases. Cmment n the advantage f using a bridge vs. using just a divider A5: Sensitivity calculatin 1. What is the sensitivity f this circuit? PART B: Strain Gauges B1: Min and Max resistance measured B: Vltage measurements fr three beam psitins B4: Analg Discvery plt f beam scillatin with calculatins Questin 1 K.A. Cnnr, P. Schch - 0 - Rensselaer Plytechnic Institute, Try, New Yrk, USA
ENGR-00 ELECTRONIC INSTRUMENTATION Experiment 5 PART C: Instrumented Beam as a Harmnic Oscillatr 1. Tw Analg Discvery plts f the decaying sinusid with different masses. A table listing the mass, ttal mass and frequency. Excel plt f frequency vs. lad mass with fur pints Questins 1-5 PART D: Oscillating Circuits 1. PSpice plt f utput frm the scillating circuit, with parameters and equatin. Analg Discvery Plt f scillating circuit, with parameters. PSpice Plt f Energy, with explanatin Questins 1- Member Respnsibilities Summary/Overview K.A. Cnnr, P. Schch - 1 - Rensselaer Plytechnic Institute, Try, New Yrk, USA