ENGR-400 Spring 006 Experiment 4 Experiment 4 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 the material the beam is made f. Finally, we will extend the thery f scillatin t an electrical system, an scillating circuit. Equipment Required: Oscillscpe (HP 5460B Channel 60 MHz Oscillscpe) Functin Generatr (HP 10A 15 MHz Functin/Arbitrary Wavefrm Generatr) Instrumented Beam Parts Kits Helpful links fr this experiment can be fund n the links page fr this curse: http://hibp.ecse.rpi.edu/~cnnr/educatin/eilinks.html#exp4 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 9 vlt battery 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, as pictured in figure A-1, 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 the fllwing figure. 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 acrss a bridge is zer, we say that the bridge is balanced. V1 1k R1 R 1k Vleft Vright R 1k R4 1k 0 Figure A-1 K.A. Cnnr and Susan Bnner 1 Revised: 1/16/006
ENGR-400 Spring 006 Experiment 4 The fllwing equatins apply t the bridge circuit. When dv = 0, the bridge is balanced: V left = R R4 V1 Vright = V R1 + R R+ R4 1 dv = V left V right V left 1K = V1 1K + 1K V right 1K = V1 1K + 1K 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. Lk at the behavir f a balanced bridge circuit. Set up the circuit n the previus page as shwn. Use a 100mV amplitude, 1KHz frequency and n DC ffset. Place vltage markers at Vleft and Vright and 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) Is the difference zer? Lk at the behavir f an unbalanced bridge circuit. Nw, change R4 t be equal t 1.1k hms. 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? Print ut this plt and include it with yur reprt. Analyze this circuit by hand Use vltage dividers 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.] Summary A bridge allws yu t cmpare tw vltages. 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 Ptentimeters and Strain Gauges Backgrund Ptentimeters: A very useful type f vltage divider is the ptentimeter (r pt). The symbl fr the pt is very descriptive f its peratin. K.A. Cnnr and Susan Bnner Revised: 1/16/006
ENGR-400 Spring 006 Experiment 4 Figure B-1 As shwn in figure B-1, a ptentimeter is a resistr that has a tap that can slide up and dwn t prduce tw variable resistrs. Pts are rated just like resistrs. There are 1K pts, 10K pts, etc. The naming system fr pts is based n a number XYZ, where the value is XY 10 Z hms. Fr example, a 10K pt will be marked with 10. The resistance at the tp f the pt and the resistance at the bttm always add up t the rated value f the pt. The wiper (which is usually a screw f sme kind) changes hw much f the ttal is n the tp and hw much is n the bttm. The utput vltage, Vut, can be determined using a simple vltage divider. 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 super glue). When the surface stretches because f an external frce, a tiny difference in the distance between the wires f the strain gauge is created. This results in a small change in wire resistance. The change in resistance f the strain gauge will give a signal prprtinal t the displacement at the end f the beam. Figure B- shws a typical strain gauge. Figure B- The Cantilever Beam: Figure B- shws the cantilever beam we use in class. Figure B- K.A. Cnnr and Susan Bnner Revised: 1/16/006
ENGR-400 Spring 006 Experiment 4 The beam cmes with tw sets f utput wires. One, usually cvered with black insulatin, is cnnected t a cil lcated n the supprt plate near the mveable end f the beam. The ther, usually with twisted wires, is cnnected t a small strain gauge cemented near the fixed end f the beam. We will nly be using the strain gauge in this part. The strain gauge is relatively fragile, s its wires are usually fixed t the supprt plate befre ging t the utside wrld. It is still a gd idea, hwever, t be very careful with such devices. The nes we are using are quite gd (aka nt cheap). Damped Sinusids: The damped sinusid in figure B-4 is gverned by the equatin: v(t)=ce -αt sin(ωt), where α 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 α: v1 = v0e -α (t1 t 0) Experiment Figure B-4 Ptentimeters In this part, we will lk at the behavir f tw pts with different lads n the utput vltage. Remember that the size f the lad resistr can be significant in the perfrmance f a circuit. 5V V1 R1 R R 1MEG R4 00 0 Figure B-5 K.A. Cnnr and Susan Bnner 4 Revised: 1/16/006
ENGR-400 Spring 006 Experiment 4 Using PSpice, create the circuit shwn in figure B-5. The vltage surce is nw VDC rather than VSIN, since we nly want a DC surce this time. The pts are called POT in the parts list and are lcated in the Breakut Library. Since pts are variable, their values have t be set up befre the analysis can be dne. Unlike ther parts, pts cme with a cmplete set f default attributes. The ttal resistance f the pt will be 1k hm and the tap (als called the wiper) will be set at the half way pint. Cnnecting acrss the entire pt frm tp t bttm will give a resistance f 1k hm. Cnnecting frm the tp t the tap will give a resistance f 500 hms, as will a cnnectin frm the tap t the bttm. Thus, fr the default cnditins, we have, in effect, a 500 hm resistr abve the tap and a 500 hm resistr belw the tap. Place vltage markers at Vut fr bth f the pts shwn and at the surce vltage. Run a transient analysis and yu will see that the vltage at the tap n the left pt is clse t half f the surce vltage, as it shuld be. (Since the surce is DC, all f yur traces shuld be straight lines.) The large 1 M hm resistr des nt lad dwn the pt vltage divider very much. Hwever, the 00 hm lad n the right pt is much smaller than the 500 hm resistance f the bttm half f the pt and thus this vltage divider is laded dwn significantly. The vltage acrss the 00 hm resistr is determined mre by the 00 hms than the 500 hms f the bttm half f the pt. View the PSpice utput file T get additinal infrmatin abut the exact value f yur vltages, yu can lk in the utput file. In the OrCAD/Pspice dem windw (which displays the simulated scpe traces), chse Output File frm the View menu. The display will switch t the text file cntaining the utput. (See the figure B-6.) Figure B-6 Scrll dwn until yu see a heading Initial Transient Slutin. Just belw that, yu shuld see a series f nde names and vltages. Each f the uniquely defined nn-zer ndes shuld have a vltage value. In this case, there are three such ndes, ne fr the vltage at the tp f the vltage surce and ne fr each f the wiper vltages. Once yu have fund these vltages, write them dwn. Lk thrugh the rest f the file and see hw PSpice describes this circuit. K.A. Cnnr and Susan Bnner 5 Revised: 1/16/006
ENGR-400 Spring 006 Experiment 4 T return t the scpe display, use the tab in the lwer left crner f the screen. Draw the equivalent circuit (as just regular resistrs, nt with pts) and then calculate the vltages yu expect t see at the three pints with vltage markers. Check t be sure that yur answer agrees with the PSpice Output File.. DC Sweeps and Parameters in PSpice In this part, we will use PSpice t simulate the turning f the screw n the pt frm ne extreme t the ther. Nw we want t use anther functin f PSpice t see what happens as we vary the psitin f the tap. T d this, we have t set the tap psitin as a variable, rather than having it fixed. Use the fllwing prcedure: Duble click n the left pt and change the value f the SET attribute t setvar. Yu can make the name anything yu want, but this reminds us what we are ding. This makes the value f the left set pint a variable we set when we set up the analysis. Next we have t tell PSpice that we are using sme parameters. 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 value t yur variable setvar using the fllwing prcedure: Duble click n the wrd PARAMETERS: t display the spreadsheet and click n New Clumn. In the Prperty name textbx, enter yur chsen name (here use setvar). Enter 0.5 fr the default value, since that will leave the wiper half way up in its default psitin and click k. While this cell is still selected, click Display. In the Display Frmat windw select Name and Value, 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 B-7 belw.) PARAMETERS: setvar = 0.5 5V V1 R1 R R 1MEG R4 00 0 Figure B-7 Next we will set up the analysis (See figure B-8 n the fllwing page). Create a simulatin. Select DC sweep. Set up the glbal parameter which we have called setvar t vary linearly frm 0 t 1 in steps f 0.01. K.A. Cnnr and Susan Bnner 6 Revised: 1/16/006
ENGR-400 Spring 006 Experiment 4 Figure B-8 Perfrm the simulatin and yu will see the vltages at the taps f the tw pts. Yu shuld print this particular plt and then discuss why it lks the way it des. Include this in yur reprt. Strain Gauges We will nw cmbine ur study f ptentimeters and bridge circuits in a practical hardware applicatin. We will create a bridge circuit and use a strain gauge munted t a beam fr ne f the bridge resistrs. Then we will put a strain n the beam and bserve the results. Befre yu build the circuit, use the multimeter t determine the resistance f yur strain gauge in its rest psitin. D nt make this measurement when the beam is cnnected t the circuit. 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. Deflect the beam dwn until it tuches the supprt plate and measure the resistance with the DMM. 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. Yu shuld nw see why we indicate this resistance with a variable resistr in the figure belw. Build the circuit using the circuit diagram in figure B-9. Use a 1K pt (10), ne 1K resistr, and the strain gauge n the beam. Nte that we have used a 5V DC surce t supply the vltage. The scillatin this time will be created by the beam. K.A. Cnnr and Susan Bnner 7 Revised: 1/16/006
ENGR-400 Spring 006 Experiment 4 Figure B-9 Balance the bridge Adjust the pt s that there is n vltage difference between the tw channels f the scpe when the beam is nt deflected. The best way t mnitr the difference between the tw utput pints is t cnnect each pint t ne f the scpe channels and then use the channel math buttn t display the difference between the tw vltages. It is imprtant t balance the bridge because we want t bserve nly the changes in vltage created by the resistance changes f the strain gauge. When yu have everything hked up, set the beam int free scillatin. Yu shuld bserve a decaying sinusidal vltage. Yu will nte that the AUTO SCALE feature will prbably nt wrk at this lw frequency. This is ne f the prblems with mechanical systems. The natural frequencies f mechanical devices are usually lw. Yu will have t learn t set yur scales manually if yu hpe t see anything useful. Once yu have btained a clear signal f a decaying sinusid, stp the scpe scan s that yu can lk at the signal fr sme time. Use the Agilent sftware t btain a picture f the signal. Determine the frequency f scillatin, the decay cnstant, and the range f amplitudes f yur signal. 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 in the links page. T apply any f the materials available there t the experiment we are ding, yu need t knw that the gauge factr f the strain gauge we are using is 15. (Typical fr semicnductr gauges.) Summary A pt is a single cmpnent vltage divider. Pts are ften used in bridges t make them easier t balance. A bridge which is balanced t zer at sme reference lcatin can be used t measure very small changes in resistance. In this experiment, we used ne t bserve the scillatin f a cantilever beam. K.A. Cnnr and Susan Bnner 8 Revised: 1/16/006
ENGR-400 Spring 006 Experiment 4 Part C Instrumented Beam as a Harmnic Oscillatr Backgrund Cil Sensr: The sensr underneath the dughnut at the end f the beam, we call a cil. We call it this because inside is an inductr cil. The metal dughnut hlds a magnet which mves relative t the cil as the beam di scillates. The magnet generates a current in the cil, and thus, a vltage, V = L L L. We can measure this vltage directly with the scpe by cnnecting the leads f the cil t ne f the channels. The signal frm the cil is quite a bit larger than that frm the bridge. Hwever it is nt prprtinal t the psitin f the beam, rather, it is sensitive t the beam velcity. [If the magnet des nt mve, n vltage is generated, and the cil signal will be zer.] 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. dv d x F = ma = m = m = kx d x k Using nly the terms related t the displacement, we can derive the harmnic scillatr equatin: + x = 0 m d x In standard frm, the harmnic scillatr equatin is + ω x = 0 where ω is the frequency f scillatin in k rad/sec. Thus, the beam will scillate at ω =. The slutin t this equatin is: x = x0 cs( ω t + φ), where m x 0 is the initial deflectin f the beam and φ 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 φ 0. Therefre, we can mdel the scillatin f a beam (with n frictin) using the nw familiar sinusidal equatin: x = x 0 cs( ωt). 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. K.A. Cnnr and Susan Bnner 9 Revised: 1/16/006
ENGR-400 Spring 006 Experiment 4 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: 1 dw = 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 dv d x which brings us back t Newtn s secnd law: F = 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 ω 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 Fl beam. This relatinship is defined by the physical prperties f the beam, x =, where l is the length f the EI 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 F EI will have a secnd expressin fr ur cnstant, k = =. Nw, we can lk up the mment f inertia fr x l wt Ewt Ewt an bject with a rectangular crss-sectin and substitute it in fr I.: I = and k = =. 1 1l 4l Recalling ur relatinship fr frequency frm befre and slving fr k, we find: k k ω = πf = and = (πf ) s k = m (πf ). Since this is an scillating system, we will m m ignre the negative sign. This gives us ur final result: Ewt k = 4m l = m(π f ) E = 4l wt (πf ) 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 als has 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. beam K.A. Cnnr and Susan Bnner 10 Revised: 1/16/006
ENGR-400 Spring 006 Experiment 4 Experiment 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. Since the cil and the strain gauge are subject t identical scillatins, we can use either t measure the frequency f the beam. Fr this experiment, we will use the cil.. Use the cil t find the frequency f the beam Hk the cil leads directly int ne f the scpe channels. Set the beam int scillatin. Prduce a plt f the decaying sinusid yu bserve using the Agilent-Intuilink sftware. Recrd the frequency in the first rw f table belw The mass at the end f the unladed beam is the mass f the dughnut plus the magnet. The dughnut s mass is 1 grams. It is glued t the beam, s yu will have t take ur wrd n that mass. The mass f the magnet is abut 4 grams. These masses vary a bit. Yu can measure the mass f the magnet directly by remving it frm the dughnut, hwever yu MUST replace it at the same height as it was befre yu tk it ut. If yu want yu can use ur estimatin f 4 grams and then, m 0 = 4g + 1g = 7g = 0.07 kg. Mass f bject Mass f clamp Mass f sensr 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 = Measure the frequency three mre times using additinal masses f yur chice. Chse three bjects frm abut 100 t abut 500 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 500 grams). Dn t lad the beam with a mass s heavy that it permanently bends the beam. Measure the masses f the bjects yu have chsen with the scale in the studi. Enter them int the table. Measure the mass f a c-clamp and enter that in the table. The ttal mass at the end f the beam, m n, is the sum f the bject plus the clamp plus the sensr. Place each mass as clse t the end f the beam as pssible using a c-clamp ideally they shuld be lcated at the very end f the beam. Orient the c-clamp and mass s that their center f mass is near the end f the beam, as shwn in figure C-1. Fr example, attach the c-clamp frm the side rather than frm the end f the beam.. Magnet C-Clamp Figure C-1 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. K.A. Cnnr and Susan Bnner 11 Revised: 1/16/006
ENGR-400 Spring 006 Experiment 4 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. 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 = ( πf n) n=0,1,, We can write ut a k expressin fr each f the fur frequencies yu have measured: k = ( m + m f 0 ) (π 0) ( 1) ( π 1) ( ) ( π ) ( ) ( π ) k = m+ m f k = m+ m f k = m+ m f 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 sme mre measurements than cnstants t determine 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. f n 1 = π m k guess guess + m 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 600 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. 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 K.A. Cnnr and Susan Bnner 1 Revised: 1/16/006
ENGR-400 Spring 006 Experiment 4 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. Final results Calculate the mass f the beam using yur best guess fr m. m = 0. mbeam 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 W M = (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. 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 K.A. Cnnr and Susan Bnner 1 Revised: 1/16/006
ENGR-400 Spring 006 Experiment 4 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 W E = (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. There is an excellent applet n the links page which illustrates this principle. 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 and Susan Bnner 14 Revised: 1/16/006
ENGR-400 Spring 006 Experiment 4 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 di = L L L C = I dv 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 = V C = V L and I = I C = I L. 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 takingthe time C di d V derivative f bth sides, = C. Substituting int the cnservatin f energy equatin, we get C di L dv 0 d V I d V I + C V = LC I + C V = 0 LC + V C = 0 This leads us directly t the harmnic equatin fr scillating circuits: + V = 0 If we cmpare this t the general expressin fr harmnic scillatin, 0 we can determine the resnant frequency f the circuit: d V 1 LC d V + ω 0V =, ω 0 = 1. 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- K.A. Cnnr and Susan Bnner 15 Revised: 1/16/006
ENGR-400 Spring 006 Experiment 4 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 d V dv + α + ω0v where α is the damping cnstant. It can be shwn that in an ideal damped scillatin circuit, α is given by the fllwing equatin belw: R α = L Experiment Mdeling a damped scillatr We will nw cnsider an RLC circuit, with all three kinds f passive cmpnents and bserve the damped scillatins. = 0 Figure D- Create the scillating circuit in figure D- abve in PSpice In rder t place sme initial energy int the circuit, we use tw switches. Befre time t=0, switch U is clsed and switch U1 is pen. This places an initial charge n the capacitr. Nw the capacitr has enugh energy stred in it t start the scillatin. At time t=0, we discnnect the vltage surce frm the circuit and let it scillate n its wn. (Just like yu release the beam and watch it vibrate until it stps.) The switches are in the EVAL library (Sw_tClse will clse at a specified time perid after t=0 and Sw_tOpen will pen at a specified time perid after t=0). Place vltage markers n the circuit between R1 and L1 and between L1 and C1. Use PSpice t simulate the transient respnse f this circuit fr a ttal time f 1ms. Print ut 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 calculated value f f = 1/[π (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 α=r/(l). Calculate the damping cnstant and cmpare it t the ne yu fund using the plt. K.A. Cnnr and Susan Bnner 16 Revised: 1/16/006
ENGR-400 Spring 006 Experiment 4 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. Reprt and Cnclusins The fllwing shuld be included in yur reprt. Everything shuld be labeled and easy t find. Partial credit will be deducted fr pr labeling r unclear presentatin. Part A (10 pints) Include the fllwing plts: 1. PSpice vltage divider plt with R4 = 1.1K hms ( pt) Answer the fllwing questins: 1. What is the value f the difference vltage as a fractin f the input vltage fr the case where ne f the 1k resistrs is replaced with a 1.1k resistr? Shw that yu get the same answer when yu analyze the circuit by hand, applying the apprpriate frmulas t d the analysis. (4 pt). Assuming 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). (4 pt) Part B (18 pints) Include the fllwing plts: 1. PSpice setvar sweep plt ( pt). Agilent plt f beam scillatin with calculatins f beam frequency and damping cnstant n it. ( 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). What is the resistance f yur strain gauge when the beam is in its undeflected psitin? when it is deflected fully dwnward? when it is deflected upward? ( pt). Why must resistance measurements be made while cmpnents are nt cnnected t the circuit? ( pt) 4. Draw the equivalent circuit f the unbalanced bridge circuit. (Use tw resistrs fr each pt.). Calculate the vltages yu expect t see at the tw wipers. (Assume the pts have a set value f 0.5) Check t be sure that yur answer agrees with the PSpice Output File.(4 pt) Part C (0 pints) Include fllwing plts: 1. Agilent plt f the decaying sinusid btained with the cil fr the beam with nly the sensr n the end. ( pt). Excel plt f frequency vs. lad mass with fur pints marked (6 pt) Answer fllwing questins: 1. Include a cpy f the table f yur mass and frequency data. (6 pt). Explain 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? D yur results seem plausible? Why? (4 pt) 4. Calculate the mass f the beam. ( pt) 5. Estimate the frequency f the beam with a 0 kg lad (with n sensr n the end). Explain hw yu did this. ( pt) K.A. Cnnr and Susan Bnner 17 Revised: 1/16/006
ENGR-400 Spring 006 Experiment 4 6. Calculate Yung s Mdulus fr the beam. Clearly indicate the values yu measured fr the beam s dimensins. ( pt) 7. What d yu cnclude the beam culd be made f? Why? ( pt) Part D (10 pints) Include fllwing plts: 1. PSpice plt f utput frm scillating circuit. ( pt) Answer fllwing questins: 1. Determine the resnant frequency and damping cnstant f the circuit yu analyzed using the PSpice utput plt 1. Write an equatin fr the utput in the frm v(t)=ce -αt sin(ωt). (4 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 α using the equatin? Hw clse f an estimate is this t the damping cnstant yu fund in the plt? ( pt) Summary (1 pints) 1. Summarize key pints (8 pts). Discuss mistakes and prblems ( pts). List member respnsibilities ( pts) Ttal: 80 pints fr write up + 0 fr attendance = 100 pints Attendance: classes (0 pints) classes (10 pints) 1 class (0 pints) ut f 0 pssible pints Minus 5 pints fr each late. N attendance at all = N grade fr experiment. K.A. Cnnr and Susan Bnner 18 Revised: 1/16/006