Physics 401 Classical Physics Laboratory. Torsional Oscillator. Contents

Transcription

1 University f Illinis at Urbana-Champaign Physics 401 Classical Physics Labratry Department f Physics Trsinal Oscillatr Cntents I. Intrductin II. Experimental Setup Hw t use the drive mtr III. Exercises Determine the trsinal cnstant frm static measurements Determine the trsinal cnstant frm dynamic measurements Demnstrate under and ver viscus damped mtin Demnstrate Culmb damping Demnstrate turbulent damping Demnstrate the phenmenn f beats Measure amplitude and phase f damped, driven scillatr Drive scillatr at sub harmnic f resnant frequency IV. Reprt Appendix I Exact slutin fr Culmb damping Appendix II Buffer bard switch psitins Revised 9/5/007

2 Physics 401 Experiment 6 Page /36 Physics Department, UIUC I. Intrductin The gals f this experiment are threefld: first, t study the transient mtin f a mechanical scillatr; secnd, t study frms f dissipatin besides viscus damping; and third, t study the behavir f a driven scillatr. A damping frce that is prprtinal t the velcity (linear damping) is made by the eddy currents induced by the mtin f the scillatr in a magnetic field. This magnetic damping is analgus t the effect f the resistance in an RLC circuit. Culmb (r dry) frictin is anther frm f damping that can ccur in mechanical systems. Culmb frictin, since it is independent f velcity, will exhibit a different frm f damping. Finally, turbulent dissipatin can ccur in mechanical systems. Turbulent frictin is fund in the mtin f air arund a fast mving car r in the mtin f water arund a bat. Such dissipatin can increase as the square (r larger) pwer f velcity. Turbulent frictin als will exhibit a different frm f damping. The apparatus is illustrated in Fig. 1. Fig. 1 Optical sensrs measure the psitin f the disc and the drive. One sensr is munted n the side f the disc. The secnd sensr is munted n the pian wire at the drive end. Revised 9/5/007

3 Physics 401 Experiment 6 Page 3/36 Physics Department, UIUC II. Experimental Setup This experiment will be carried ut with a 4 inch diameter cpper disc fastened t the center f a pian wire as shwn in Fig. 1. Befre beginning the exercises nte that the large magnet shuld be kept as far away frm the cmputer as pssible. The large field culd affect the cmputer s hard drive. Yur wrist watch and credit cards shuld als be kept far frm the magnet. The psitins f the pt-electrnic sensrs relative t the cde wheels attached t the disc and drive mtr have been adjusted with an scillscpe fr ptimal peratin. The psitining screws have been cvered with red tape. D nt remve the tape and d nt adjust the psitining screws. The cde wheels are very delicate high angular reslutin devices and can be easily damaged. Therefre, whenever yu are mving the trsinal pendulum by hand, r psitining the magnet be extremely careful nt t bump r tuch either f the cde wheels. The 16-bit instantaneus angular psitin data assciated with the trsinal pendulum and mtr cde wheels is stred lcally in memry n a frnt-end electrnics buffer bard, which is read ut via the TORSNHP6 LabWindws/CVI prgram. The frnt-end electrnics buffer bard insures that the cde wheel data is captured at unifrm time intervals. The time interval, r data acquisitin rate, is set in hardware n the buffer bard via a binary cde implemented using fur DIP switches. The default data acquisitin rate is 50 Hz, which crrespnds t a time interval f 0 msec. This sampling rate ptimizes the reslutin n the angular velcities assciated with the trsinal pendulum apparatus. See the appendix fr the psitin f the DIP switches t achieve a given data acquisitin rate, varying frm 5Hz t 00Hz. If the hardware data acquisitin rate is changed frm its default setting, then the TORSIONHP6 prgram must als be infrmed f this change in data acquisitin rate frm the nminal/default setting, via the prgram s Time Base pull-dwn menu. If the hardware setting f the sampling time n the buffer bard vs. what the TORSNHP6 prgram thinks it is, then the prgram will nt analyze the cde-wheel data crrectly. The data acquisitin electrnics must be first initialized using the muse t click n the INIT_DAQ buttn n the TORSNHP6 prgram. A DAQ Initialized LED turns red after initializatin. Next, with bth the trsinal pendulum and mtr mtinless, click n the ZERO_Cunters buttn t zer the cde-wheel up/dwn cunters, thus defining the angular equilibrium/zer psitins fr each {nte: when the mtr is then turned n, unless the mtr is at its true equilibrium psitin, the equilibrium psitins fr bth trsinal pendulum and mtr will be changed pst-fact, nte that yu can als use the SHIFT_DATA buttn t crrect fr a Revised 9/5/007

4 Physics 401 Experiment 6 Page 4/36 Physics Department, UIUC nn-zer ffset in the trsinal pendulum angular psitin data}. Use the muse t click n the START buttn t initiate data-taking. When the STOP buttn is pressed, the pendulum/mtr cde-wheel data is then read ut frm the frnt-end electrnics buffer bard assciated with the STOP-START time interval. The TORSNHP6 prgram will plt the trsinal pendulum/mtr angular psitin, velcity and acceleratin data n-line via pull-dwn menus and als allws a certain level f data analysis n-line. The raw angular psitin data is simply the cde-wheel sensr reading versus sample number. One revlutin f the disc (π radians) equals a cdewheel sensr reading f 819. The pendulum/mtr angular velcities and acceleratins are calculated using finite time differences f angular psitin data. The TORSNHP6 prgram has a pull-dwn menu ffering a chice f fur algrithms fr calculating angular velcity and acceleratin. The first chice is the simplest using finite differences assciated with nearestneighbr data (n and n-1 elements). The secnd algrithm chice is similar, but uses (n+1, n and n-1 elements). The third algrithm uses a weighted average f 75% frm symmetric nearest neighbrs (n+1, n and n-1 elements) and 5% frm next-t-nearest neighbrs (n+ and n-). The furth algrithm, which is the default/nminal algrithm uses a mre sphisticated apprach knwn as the Verlet integratin methd, which uses leading-rder terms assciated with the Taylr series expansin f the angular psitin finite-difference data. We encurage yu t investigate/cmpare velcity/acceleratin data calculated via each f these fur algrithms. The WRITE_DATA buttn writes ut the trsinal pendulum and mtr data t the disk. The psitin f the disc at the mment when data acquisitin begins is taken t be the zer psitin. This ffset can be crrected fr in Origin when yu analyze yur data. 1. Hw t use the drive mtr: The drive mtr applies trque t the trsinal pendulum thrugh a series f small steps. A pulse applied t the mtr rtates the shaft by a small and well defined angle. By cntrlling the rate f the applied pulses, the angular velcity f the mtr can be accurately cntrlled. This type f mtr is called a stepper mtr. A stepper mtr cntrller receives pulses frm an external pulse generatr and prduces a current pulse that is applied t the stepper mtr. In ur experiment, we will use the Wavetek Functin Generatr t supply the pulses t the setter mtr cntrller. The drive frequency is set by changing the frequency f the Wavetek. T determine the drive frequency, cnnect the CAL OUT cnnectr f the cntrller t the HP Digital Multimeter using the BNC t banana cnnectr. The multimeter shuld be set t FREQ mde. The drive frequency is given by Revised 9/5/007

5 Physics 401 Experiment 6 Page 5/36 Physics Department, UIUC r CAL OUTPUT in Hz drive frequency in Hz =, 1,600 drive angular frequency in rad / s = CAL OUTPUT in Hz 54.6 On the frequency cunter, push the "atten" buttn in, and the "filt" buttn ut. The frequency reading shuld be very stable. When setting the frequency settings f the Wavetek, the stepper mtr cntrller shuld be in the STOP psitin. The functin generatr shuld be set t psitive square wave utput, V ut =.50 V pp, and V ffset = 0 V. Set the frequency t an initial value f 300 Hz. T turn the mtr n, set the cntrller t the RUN psitin. The mechanism that cnnects the drive mtr t the wire is cvered fr safety cncerns. The mtr rtates at a cnstant rate, and the cnnectin mechanism causes the wire t scillate apprximately sinusidally; the linkage between the drive mtr and the wire des nt prduce perfect sinusidal mtin. There are small cmpnents in the trque n the wire at twice and three times the drive frequency. The presence f these cmpnents can be fund by running the drive mtr at ne-half r ne-third f the resnant frequency and bserving that the steady state mtin f the disk is nt purely sinusidal. There may be a backgrund sund, like grinding metal, which is nrmal. It is nt destrying itself. Hwever, if the mtr stalls by being started r run at t high a speed, it Revised 9/5/007

6 Physics 401 Experiment 6 Page 6/36 Physics Department, UIUC shuld be stpped, and its speed reduced befre restarting. In general, yu shuld turn n the mtr at a lw speed and then raise the speed. III. Exercises WARNING: Be very careful when driving the scillatr near resnance. The pian wire can break, which is very dangerus. ALWAYS keep the plastic cvering n when perating the scillatr, regardless f the frequency, fr yur safety. This is a tw-week experiment. It is suggested that yu spend the first week measuring varius surces f damping and the secnd week using the drive mtr; hwever, yu may prceed thrugh the experiment in whatever fashin yu see fit. Immediately fllwing this sectin are sme experiments. Fllwing the experiments is a rather detailed sectin n data analysis, including Furier transfrmatins. Regardless f what experiments yu chse d, it is expected that yu will perfrm this type f data analysis. Exercise 1. Determine the trsinal cnstant frm static measurements. The masses and dimensins f the cpper disc and plastic disc are written n the discs. Calculate the mment f inertia f the disc. Measure the diameter f the pian wire with the precisin micrmeter, using any necessary technique t minimize the uncertainty. Measure and recrd the lengths f wire, L 1 and L (see Fig. in Appendix I) which supprt the disc. T measure the trsinal spring cnstant f the pian wire, yu will use a static displacement f the disc under a knwn trque. The methd is illustrated in Fig. 4. Revised 9/5/007

7 Physics 401 Experiment 6 Page 7/36 Physics Department, UIUC Figure 4 Measurement f Trsinal Spring Cnstant Yu will need t clamp a pulley t the edge f the table and use steps f 0 grams t crank the disc. In rder t stabilize the disc f the trsinal pendulum quickly, psitin the magnet arund the disc t help damp its mtin. Start the TORSNHP6 prgram and add ne mass at a time until yu reach 100g, then remve them ne at a time. The spring cnstant K is determined frm the slpe f the line f angle θ versus mass m gρ θ = m. (1) K In Eq. 1 g is the gravitatinal acceleratin f 9.80 m/sec and ρ is the radius f the disc. Uncertainty in the mass may be taken t be ±0.01 g. Determine the trsinal cnstant, K, and its uncertainty by analyzing these data in Origin. Sample data are shwn in figure 5. Fig. 5 Disk psitin fr varius static trques Exercise. Determine the trsinal cnstant frm dynamic measurements. Carefully measure the natural frequency f the cpper disc. Frm yur data, calculate the natural frequency. Since ω = K I, yu can cmpare this value f K with the value yu fund earlier. Yu can als plt angular acceleratin versus psitin t calculate K. Sample data are shwn in figure 6. Revised 9/5/007

8 Physics 401 Experiment 6 Page 8/36 Physics Department, UIUC Fig. 6 Disc psitin versus time fr undamped scillatin Exercise 3. Under- and ver-damped mtin with viscus damping Viscus damping can be achieved by placing the disc between the ples f a magnet. The eddy currents induced in the disc by its mtin are prprtinal the angular velcity f the disc, and these eddy currents will expnentially damp the mtin f the disc. The exact equatins fr viscus damping are discussed in Appendix I. Mve the magnet ples t cmpletely surrund the cpper disc. The mtin f the disc shuld be ver damped, i.e. decay with n scillatin. Recrd this mtin. Next, back the magnet away frm the disc slightly and repeat until yu bserve underdamped mtin. Yu may als try t find the critically damped psitin thugh this may prve difficult. Sample data are shwn in figure 8. Fig. 8 Disc psitin versus time fr viscus under-damped mtin Exercise 4. Demnstrate Culmb (kinetic frictinal) damping Revised 9/5/007

9 Physics 401 Experiment 6 Page 9/36 Physics Department, UIUC The mtin f the disc can be damped by kinetic frictin, als knwn as Culmb damping. Culmb (frictinal) damping is independent f the magnitude but dependent n the directin f the velcity. This type f damping will cause the amplitude t decrease linearly with time. T study Culmb damping remve the magnet and psitin the bristles f the small paint brush clse t the rim f the disc. Sample data are shwn in figure 9. Fig. 9 Disc psitin versus time fr Culmb damped mtin Exercise 5. Demnstrate turbulent damping The undamped scillatr is, f curse, subject t damping mechanism. These damping mechanisms are present with the magnet and with the paint brush, but they are negligible cmpared t the magnetic r the Culmb damping. One mechanism is drag frm the surrunding air. We must increase the drag t demnstrate turbulent damping. The additin f light-weight vanes t the disc will increase the drag frm the air and prduce turbulent damping. Attach either the cardbard r styrfam pieces perpendicular t the disk with tape, then bserve the scillatins. Try t maintain the symmetry f the disk when placing the vanes. Yu will need t take data fr a very lng time t see this effect. This is the end f the experiments fr the first week. We will nw mve n t experiments invlving the drive mtr. Exercise 6. Demnstrate the phenmenn f beats The superpsitin f the undamped transient mtin and the driven mtin can be arranged t demnstrate the phenmenn f beats. Since damping will cause the transient t die away, the magnet shuld be far away frm the disk s that the mtin is undamped. Yu can recall frm Revised 9/5/007

10 Physics 401 Experiment 6 Page 10/36 Physics Department, UIUC the discussin abve that the mtr cntrl bx generates a signal, CAL OUTPUT, which is prprtinal t the drive frequency, CAL OUTPUT in Hz drive frequency in Hz =, 1,600 Yu shuld calculate the CAL OUTPUT in Hz which crrespnds t the natural frequency f the disk. Nte that the frequency cunter displays in khz. Chse a drive frequency just belw resnance, fr example f 0.9 f, t bserve the beats. D nt try t set the drive exactly t resnance. Withut damping, the amplitude f the scillatin can becme dangerusly large, causing the pian wire t break. Sample data are shwn in figure 10. It can be smewhat challenging t prduce a gd beat pattern. Fig. 10 Beat frequency demnstratin (real data) Exercise 7. Measure the amplitude and phase f damped, driven scillatr Yu nw mve n t the majr part f the labratry exercise: the measurement f the amplitude and the phase f damped, driven scillatr as yu vary the driving frequency. Recall the expressins Revised 9/5/007

11 Physics 401 Experiment 6 Page 11/36 Physics Department, UIUC ( ) θ () t = B( ω)cs ωt β( ω) λθω B( ω) = tan βω ( ) = ωγ ( ω ω ) + ω γ ( ω ω ), fr the steady state mtin with amplitude B( ω ) and phase β ( ω ). The natural frequency f the disk is fixed. It is abut 0.5 Hz. The drive amplitude is fixed. It is abut 0.65 rad. Yu d have cntrl ver the damping parameter. If yu wuld like t have a large change in the amplitude f the disk near resnance, yu must chse small damping. Fig. 11 belw shws a damped scillatin. When the magnet is clse t the disk, yu have large damping. When it is far frm the disk, yu have small damping. Yu shuld try t find the magnet psitin which prduces a damping time f abut 5-10 s. Recall that the damping time is the time ver which the amplitude f the scillatin decays by a factr f 1 / e = The damped scillatin in Fig. 11 has a decay time between 7 and 8 secnds. Psitin the magnet a centimeter frm the disk, and measure the decay time f the free scillatin. Observe the free scillatin data and adjust the magnet psitin accrdingly. Recrd the decay time. After yu have fund the crrect magnet psitin, yu must nt mve the magnet fr the entire series f measurements belw. Mving the magnet changes the width f the resnance curve. Fig. 11 Magnetically Damped Oscillatin Revised 9/5/007

12 Physics 401 Experiment 6 Page 1/36 Physics Department, UIUC Befre beginning the measurement, lk at Fig. 1. Yu can see that yu must chse the drive frequencies carefully t see the resnance peak. Fr an scillatr with small damping, yu culd easily miss the resnance peak if yur frequency steps are t large. Yu shuld plt amplitude and phase versus frequency as yu take the data in rder that yu dn't miss the peak. Remember that after each change f the drive frequency, the transient mtin f the scillatr is excited. Yu must wait at least three decay times fr this transient t decay, i.e. 15 t 30 secnds. Fig. 1 Amplitude and phase f damped, driven scillatr versus frequency (real data) Figure 13 belw shws the mtin f the scillatr and the drive mtr as a functin f time at a drive frequency f Hz as read frm the frequency cunter. The peak-t-peak amplitude is indicated in the figure. Use the cursrs t btain this quantity. When there are tw plts n the screen at nce, the tw cursrs can cause cnfusin. When selecting tw plts, yu will see this ntice n the screen: Fr these plts the user must physically place the YELLOW cursr n the BLUE (Pendulum Data) curve and place the MAGENTA cursr n the MAGENTA (Mtr Data) curve. Only then will X-Y cursr prt data be crrect. Fig. 13 als shws the time difference between the maximum amplitude f the drive mtr and the maximum amplitude f the scillatr. This time difference is dented as Δt in the figure. Yu can cnvert this time difference int a phase. Recall that a time difference f ne perid ( T = 1/ f ) is a phase difference f π. Alternatively, yu can dwnlad the time traces int Origin and fit the data t a sinusidal functin with a phase ffset. This wuld be a mre Revised 9/5/007

13 Physics 401 Experiment 6 Page 13/36 Physics Department, UIUC accurate determinatin f the amplitude, frequency and phase f the drive and the respnse. The phase difference between the drive and respnse can thus be determined. Fig. 13 Oscillatr and drive versus time (real data) Exercise 8. Drive the scillatr at a sub harmnic f the resnant frequency Due t the gemetry f the drive linkage mechanism, the drive mtin is nt purely sinusidal! Empirically, ( ( ) 3 ( )) θdrive( t) = θ csωt+ c cs ωt+ φ + c cs 3ωt+ φ3 where the amplitude f the secnd harmnic is abut 6% and the amplitude f the third harmnic is abut 3%. (The higher rder terms are due t the simple linkage between the stepping mtr and the pian wire. With the mtr stpped, yu may take ff the cver plate and take a lk.) If the drive frequency is set near a sub harmnic f the resnant frequency, the distrtin gets magnified. This effect is further exacerbated by small damping. Set the drive frequency t a Revised 9/5/007

14 Physics 401 Experiment 6 Page 14/36 Physics Department, UIUC sub-harmnic and take sme data. Plt the data and identify the harmnics in the mtin f the disk. Revised 9/5/007

15 Yur reprt shuld address the fllwing pints: 1. Calculate the mment f inertia f the trsinal scillatr. Argue quantitatively as t why the hub and plastic disc may be neglected in this calculatin.. What are the values f the trsinal cnstant frm exercises 1 and? Frm yur measurements f the trsinal cnstant and the radius f the pian wire, calculate the shear mdulus f the material frm which the pian wire was made. Des yur value f the shear mdulus agree with the handbk value? 3. Frm the mment f inertia btained in part 1 and the trsinal cnstant btained in part abve, calculate the natural frequency f the trsinal scillatr. Remember t carry alng errrs fr yur errr analysis. Cmpare the calculated natural frequency t the measured frequency. Include a plt f the measured scillatr mtin. 4. Cmpare the mtin f the disc fr the tw (r three) frms f dissipatin yu demnstrated with the data graphed in an apprpriate way. Nte differences in the mtin in yur discussin. (a) Fr magnetic damping, calculate the attenuatin cnstant, a, the lgarithmic decrement, δ, and the quality factr, Q fr the under damped mtin. (b) Fr Culmb damping, find the frequency f the scillatin and cmpare it t the natural frequency. Determine the Culmb damping cnstant frm the slpe f the maximum amplitude versus time graph. (c) Fr turbulent damping plt the lg decrement vs amplitude. Determine the regin ver which the dependence f lg decrement n amplitude is linear. Over this regin fit the data t a straight line which passes thrugh the rigin. Find the trubulent damping cefficient. These questins pertain t the driven scillatr: 5. Cmpare the measured beat frequency t the expected beat frequency. Include a plt shwing the beat frequency. See Figure 10. Revised 9/5/007

16 Physics 401 Experiment 6 Page 16/36 Physics Department, UIUC 6. Find the decay time f the magnetically damped mtin. Use the semi-lg plt f amplitude versus time. Frm the slpe, determine the decay cnstant, a, the decay time, 1/a, the lgarithmic decrement, δ, and the Q f the scillatr. 7. Plt the amplitude f the scillatr versus frequency. Cmpare the frequency at which the amplitude is a maximum t the natural frequency. Find the frequencies abve and belw the resnant frequency at which the amplitude is 1 f its peak value. The difference f these frequencies is the width f the resnance, γ. Recall the relatin between the decay cnstant, a, and the resnance width fr a magnetically damped scillatr, cmpare a t γ. 8. Plt the phase versus frequency. Find the frequency at which the phase is π/. Cmpare this frequency t expectatins. 9. Plt the mtin f the disc when the drive frequency is a sub harmnic f the natural frequency. Identify the harmnics in the disc mtin. Revised 9/5/007

17 Physics 401 Experiment 6 Page 17/36 Physics Department, UIUC Appendix I II. Thery We will extend the thery presented in the RLC circuit lab t the trsinal scillatr. Fig. Schematic drawing f disc and supprt wires Cnsider a slid unifrm disc f mass M and radius ρ as shwn in Fig. abve. The axis f the disc is alng the hrizntal. The disc is supprted by tw pieces f pian wire f lengths L1 and L. The disc can rtate abut its hrizntal axis. The angular psitin f the disc is dented by the angle θ. The mtin f the disc is btained frm Newtn s Third Law: d θ τ = Iα =, (1) dt where α is the angular acceleratin and in Fig.. I = 1Mρ. Trques are exerted by the wires shwn τ + τ = K θ K θ = Kθ () 1 1 K represents the cmbined trsinal spring cnstant f the tw wires. K1, the trsinal spring cnstant fr the left-hand wire is π 1 = (3) 4 K1 G r L 1 Revised 9/5/007

18 Physics 401 Experiment 6 Page 18/36 Physics Department, UIUC where r is the radius f the wire and G is the shear mdulus f the material frm which the wire is made. (See, fr example, Mechanics by W. Arthur and S. K. Fenster, p. 513.) Using Eq. 3 fr K 1 and K, we btain the ttal spring cnstant, K. π K G r 1 1 ( L L ) 4 = + (4) 1 In the absence f additinal trques the equatin f mtin f the disc is d θ I dt = Kθ. (5) The abve equatin is recgnized as the equatin f mtin f a simple harmnic scillatr. The angular frequency is ω = K I. There will be additinal trques n the disc due t the varius damping frces investigated in the experiments. Viscus damping We expect that resistive damping will have a term prprtinal t dθ dt, the angular velcity. This is called viscus damping, such as is achieved by placing the disk between the ples f a magnet. The trque due t viscus damping is τ = viscus R θ Rθ θ =, (6) where is R is a cnstant with (fr viscus damping) units f N m s. With viscus damping the equatin f mtin is d θ dθ I + R + Kθ = 0. (7) dt dt This equatin is the familiar linear, secnd rder differential equatin. The slutin is f the st frm θ () t = Ae. Slving fr s gives Revised 9/5/007

19 Physics 401 Experiment 6 Page 19/36 Physics Department, UIUC R R K s1, s = a ± b = ± I I I (8) The nature f the slutin depends n the sign. If b > 0, then the slutin is verdamped: θ(t) falls t zer smthly with n scillatins. The slutin is f the frm θ () at bt bt t = e ( A 1 e + B 1 e ). (9) ( Nte that a > b.) The cnstants A 1 and B1 are determined frm the initial cnditins. Fr θ (0) = θ and θ (0) = 0 (initial angular displacement, n initial angular velcity), at a θ a ( a b) t θ( t) θ csh sinh e = bt + bt 1+ e ( a b) t >> 1.(10) b b Fr b = 0, the slutin is critically damped. θ(t) will fall t zer in the minimum time withut scillatin. There are n lnger tw distinct slutins t Eq. 8, and the frm f the slutin is nw ( ) at θ () t = A + B t e (11) Again, the cnstants A and B are determined frm the initial cnditins. Fr θ (0) = θ and θ (0) = 0 (initial angular displacement, n initial angular velcity), ( ) at θ() t = θ 1+ at e. (1) Fr critical damping b = 0 implies that Rcritical 4 = K I and a = K I. If b < 0, b is a cmplex number, and we have an scillatry slutin f the frm at ibt ibt = ( ) θ () t e A e B e (13) Revised 9/5/007

20 Physics 401 Experiment 6 Page 0/36 Physics Department, UIUC Again, the cnstants A3 and B3 are determined frm the initial cnditins. Fr θ (0) = θ and θ (0) = 0 (initial angular displacement, n initial angular velcity), θ() θ at a t = e csbt + sinbt b (14) Culmb damping The mtin f the disc can als be damped by kinetic frictin, als knwn as Culmb damping. Fr Culmb damping as lng as the disc is mving, the trque is a cnstant, independent f the magnitude but dependent n the directin f the velcity. The trque always ppses the mtin f the disc. Culmb damping then is described by the trque, τ Culmb = C θ. (15) θ where C has units f Nm. The differential equatin then becmes I θ θ + C + Kθ = 0. (16) θ There are actually tw differential equatins here. Fr θ > 0 the equatin is I θ + K θ = C, and fr θ < 0 the equatin is I θ + K θ = C. Bth f these equatins are easily slved. They are linear, secnd rder inhmgeneus equatins. If C were zer, the equatins becme the equatin fr a simple harmnic scillatr. We need nly t add the particular slutin t the hmgeneus slutin, and the particular slutin is the cnstant ± CK. Suppse that we have the usual initial cnditins, θ (0) = θ and θ (0) = 0 θ after t = 0 must decrease and, therefre, θ < 0. The slutin fr this case is θ() t = C K + ( θ C K)csωt 0 t π ω. Then initially where ω = K I. This slutin is valid fr half a perid, r, equivalently, until θ ges thrugh zer and changes sign t θ > 0. Then the equatin becmes I θ + Kθ = C, and the slutin is θ( t) = C K + ( θ 3 C K)csωt πω t πω. Revised 9/5/007

21 Physics 401 Experiment 6 Page 1/36 Physics Department, UIUC Nte that after a half f a perid at t π ω = θ( π ω) ( θ CK) =. In ne perid the amplitude f the scillatin has decreased by θ(0) θ( π ω) = C K. After iteratins we btain π 1 π θ( t) =+ C K + ( θ (4n 3) C K)cs ωt ( n 1) t ( n ) n = 1,,... ω ω (17) 1 π π θ() t = C K + ( θ (4n 1) C K)cs ωt ( n ) t n n = 1,,... ω ω In every perid the amplitude decreases by the cnstant 4C K. In cntrast t viscus damping fr which the amplitude decreases expnentially, fr Culmb damping the amplitude decreases linearly. At the end f sme half perid the restring trque will be less than the frictinal trque, i.e. K θ( t = nπ ω) < C fr sme n, and the mtin will stp. The disc des nt return t its equilibrium psitin, θ = 0 Turbulent damping Since the disc mves thrugh air, the mtin f the disc can be damped by the air turbulence. The damping due t turbulence can be apprximated by a pwer f the angular velcity τ turbulent = C θ n θ + 1 (18) where n. The cnstant C nw must have units f N m s n. This gives us a nn-linear, secnd rder differential equatin, 1 n+ I θ θ + C + Kθ = 0, (19) θ which, in general, has n clsed frm slutin. The damping term is mre phenmenlgical than fundamental, and a nn-integer value fr n may better describe the phenmenn. Althugh we cannt in general btain a clsed frm slutin fr Eq. 19, we can btain an apprximate slutin in the situatin that the damping is small s that the mtin is still scillatry, albeit with a decaying amplitude. Rearranging Eq. 19 gives Revised 9/5/007

22 Physics 401 Experiment 6 Page /36 Physics Department, UIUC d d 1 1 E() t = I θ + Kθ = C θ dt dt n+1. (0) The left hand side is recgnized as the time derivative f the ttal energy, the sum f kinetic and ptential energy. The ttal energy decreases at a rate determined by the damping. If we integrate ver ne perid, we btain an expressin fr the energy lss in a perid, T d Δ E = dt E = C dt dt T 0 0 n 1 θ +, (1) which withut lss f generality we assume begins at t = 0. Nw we suppse that ver ne perid the mtin is harmnic s that θ () t = Θ csω t and θ ( t) = ωθ sinωt. Θ is the amplitude at the beginning f the perid, and ω = K I. Our gal is t find the change in the amplitude ver ne perid. We have assumed small damping s we are allwed t assume that the scillatry frequency is that f the undamped mtin. With this assumed frm fr the slutin, the energy lss in ne perid becmes T T /4 n 1 n 1 n 1 ω sin ω + 4 ( ω ) + + sin ω 0 0 Δ E = C dt Θ t = C dt Θ t, () n π / ( Θω) n 1 ( Θ ω + ) ξ ξ + 1 n+ 1 Δ E = 4C d sin = 4C Γn, (3) ω ω 0 where π / 1 sin n + Γ n = dξ ξ is just a definite integral. These integrals can easily be evaluated fr 0 integer n. The first few values are Γ 0 = 1, Γ 1 = π 4, and Γ = 3. Viscus damping has n = 1, and we find the energy lss in a perid t be Δ E = πωθ R. Nte that the damping cnstant, R = ΔE πωθ. We are then lead t define an equivalent damping cefficient fr n 1 as R ( ω Θ ) n+ 1 ΔE 4C 4C n 1 eq = = Γ n = ( ω Θ ) n πωθ π πω Θ Γ, (4) Revised 9/5/007

23 Physics 401 Experiment 6 Page 3/36 Physics Department, UIUC By cnstructin the equivalent (linear) damping trque, R eqθ, has the same energy lss per 1 perid as the (nn-linear) damping trque, C θ n + θ. Assuming an equivalent damping trque, we can define an equivalent attenuatin cefficient, a Req C n 1 ( ) I π I ω = = Θ Γ. (5) eq n With an equivalent attenuatin cefficient we then assume that the amplitude decays in ne perid expnentially s that θ ( T) =Θ exp( aeqt). Then the lg decrement can be fund as Θ n ln ( ) 1 C π δ = aeqt ω n θ ( T) = = Θ Γ πi ω. (6) Fr viscus damping we recver δ = at. Frm Eq. 6 we nte that δ Θ damping, n=, we btain n 1. Fr turbulent 8 C δ = Θ. (7) 3 I This lengthy develpment leads t the cnclusin that fr turbulent damping the lg decrement is linear with the initial amplitude, Θ. A larger initial amplitude has larger velcities and larger damping. Revised 9/5/007

24 Physics 401 Experiment 6 Page 4/36 Physics Department, UIUC Driven Trsinal Oscillatr The equatin f mtin fr the viscus damped, sinusidally driven trsinal scillatr is d θ dθ I + R + Kθ = λkθ cs ωt. (8) dt dt In the abve equatin θ is the angular displacement f the disc, I is the mment f inertia f the disc, R is the viscus damping cefficient, and K is the trsinal spring cnstant f the wire. The right hand side represents the external trque n the disc thrugh the wire frm the sinusidal drive mtr. The amplitude f the drive is θ. The external trque includes the spring cnstant, K, and the factr λ = L ( L + L ) due t the fact that the drive is separated frm the disk by a 1 1 length f wire. The zer f time is chsen s that the trque frm the drive is a maximum at t = 0. Eq. 8 is a linear, secnd-rder nn-hmgeneus differential equatin. The slutin f such an equatin is the sum f the slutin fr n driving trque, the transient slutin, and the steady state slutin due t the presence f the driving trque. The transient slutin is θ ( ω φ) =, (9) at t () t Ae cs 1 t where the decay cnstant and angular frequency are given by R a = and K R ω1 = I I I. (30) The undamped, natural angular frequency is K ω = I, (31) s the damped frequency is equal t R a = I and ω = ω. (3) 1 a Revised 9/5/007

25 Physics 401 Experiment 6 Page 5/36 Physics Department, UIUC The steady state slutin will have the frequency f the driving trque. Since in general ω is nt the same as ω, befre the transient slutin dies ut, the mtin f the disc is the 1 superpsitin f tw scillatins with different frequencies. The superpsitin f tw frequencies prduces the phenmenn f beats. We search fr a steady-state slutin f the frm (where we will take the real part) i t ( t) α e ω α =, (33) where α is a cmplex cnstant. Plugging in, we btain Slving fr α, we btain ( ) ( ) I iω α + R iω α + Kα = λkθ. (34) λkθ λθω α = K Iω + iωr =. (35) ω ω + iωγ ( ) In additin t the undamped natural angular frequency defined in Eq. 31 abve, we als intrduce the quantity, γ = R R a I = I =. (36) The cmplex cefficient, α, a functin f the driving angular frequency, ω, can be written in plar frm as ( ) ( ) α = ω. (37) i B e β ω where B( ω ) and β ( ω ) are given by B ( ω) λθω ω γ = tan β( ω) = ω ω + ω γ ω ω. (38) ( ) We take the real part f α t find the steady state slutin t Eq. 8. Revised 9/5/007

26 Physics 401 Experiment 6 Page 6/36 Physics Department, UIUC θs () t = B( ω) cs( ωt β( ω) ). (39) The full slutin fr the damped, driven trsinal scillatr is the sum f the transient and steady state slutins, Eq. 9 and Eq. 39. ( ) at () t = () t + () t = Ae cs( t ) + B( ) cs t ( ) θ θ θ ω φ ω ω β ω t s 1. (40) The cnstants A and φ are determined by matching the slutin t the initial cnditins. We have assumed that the scillatr is underdamped s that the scillatry slutin, Eq. 9, describes the transient. We culd, in principle, als use the critically damped and verdamped slutins here, but they die ut in ne perid r less and the steady state becmes the nly slutin. B. Beats Tw scillatrs f equal amplitude and f similar, but nt identical, frequencies generate waves. The resultant wave at a given pint in space is then the superpsitin f tw frequencies. Then with a judicius chice f phase f the tw independent scillatrs, we btain ω ω ω+ ω A sin ( ωt) Asin ( ω t) Asin t cs t. (41) The difference in the frequencies is small fr ω ω, s ω +ω ω. The perceptin is f a sund with a single frequency but with an unsteady amplitude. It is pssible t demnstrate the same phenmenn with the trsinal scillatr with a judicius chice f frequencies and initial cnditins. Suppse we hld the disc at its equilibrium psitin with the drive mtr running. Since the disc is at its equilibrium psitin and is at rest, bth θ = 0 and θ = 0. If we release the disc when the drive is exerting its maximum trque, we impart an initial angular acceleratin. Cnsider the superpsitin f the transient and steady state mtin, Eq. 40, fr an scillatr with small damping and fr times fr which the transient has nt yet decayed away. Then the Revised 9/5/007

27 Physics 401 Experiment 6 Page 7/36 Physics Department, UIUC expnential functin in Eq. 16, exp( at) exp( γ t ) ω fr 1 =, is clse t ne. We can als substitute ω. With these cnditins the mtin f the disc, Eq. 40, becmes, using as initial θ = 0and θ ( 0) = 0. cnditins ( 0) () t = Bcs( ) cs( t) B sin ( ) sin ( t) + Bcs( t ) θ β ω ω β ω ω ω β ( t) = B cs( ) sin ( t) B sin ( ) cs( t) B sin ( t ) θ ω β ω ω β ω ω ω β. (4) If we are clse t resnance we can set ω ω = 1 and β = π, we btain π θ() t Bsin ( ω t) + Bcs ωt = Bsin t + B t ( ω ) sin ( ω ) ω ω ω + ω = Bsin t cs t (43) Recall that the amplitude B depends n ω and ω, s that λθω ω ω ω + ω θ () t sin t cs t ( ω ω ) (44) Fr ω ω we see in additin t the sinusidal scillatin at ω, a mdulatin f the amplitude at the beat (angular) frequency ω ω. Nte that in many texts the beat (angular) frequency is defined as ω ω. It is als pssible t get cnvincing beats by just turning n the drive mtr with the disc at rest. Revised 9/5/007

28 Physics 401 Experiment 6 Page 8/36 Physics Department, UIUC C. Resnance If we assume that the transient slutin has died away, the mtin f the disc is given by λθω θ () t = cs( ωt β( ω) ) t >> 1 γ. (45) ω ω + ω γ ( ) In the high frequency limit ω >> and ω >> 1 γ Eq. 45 gives ω λθω θ () t = cs ( ωt π). (46) ω The amplitude decreases as 1 ω as shwn in the figure abve. In the lw frequency limit, ω 0, the amplitude appraches λθ. Fr ω clse t ω and with sufficiently small damping that the term, γ ω, in the denminatr can be neglected, we find the amplitude n either side f 1 ω ω. Very clse t resnance, ω ω, the γ ω term in the ω falls ff as ( ) denminatr cannt be neglected, and its magnitude determines the line-width f the resnance, i.e. the range f frequencies fr which the amplitude is large. The width f the resnance depends strngly n the damping cnstant, as shwn in the figure belw. Fig. 1 Amplitude and phase f damped, trsinal scillatr versus frequency (calculatin) Revised 9/5/007

29 Physics 401 Experiment 6 Page 9/36 Physics Department, UIUC We might nte that the maximum amplitude f scillatin des nt fall precisely at resnance, which we have defined as ω = ω. This is a very small pint, but perhaps wrth nting. The maximum amplitude is given by the (angular) frequency at which db dω = 0. With B( ω ) frm Eq. 38 after a little algebra we find that Data Analysis ω = ω γ. (47) max A. Quality factr and lg decrement There are many equivalent ways t discuss the mtin f the scillatr. Recall that the frequency f scillatin fr under damped scillatry mtin is 1 1 = = f ω 1 K R π π I I. (48) The natural (undamped) frequency is f ω 1 K = =. π π I If the mtin is scillatry and the damping is nt large, there are tw auxiliary parameters used t measure the rate at which the scillatins f a system are damped ut. One parameter is the lgarithmic decrement, δ, and the ther is the quality factr, Q. The lgarithmic decrement, δ, is defined by atmax θ ( tmax ) e ln at ( max + T1 ) θ ( tmax T1 ) e δ = ln = = at1. (49) + where t max is the time when θ is at its maximum value and T 1 =perid. Nte that fr viscus damping, since a and T 1 are cnstants, δ is the same frm initial t late times in the mtin. The Q f the circuit, r quality factr, is defined as ttal stred energy Q = π. decrease in energy per perid Revised 9/5/007

30 Physics 401 Experiment 6 Page 30/36 Physics Department, UIUC The Q can be related t the lgarithmic decrement. (Recall fr the RLC circuit ω L ω I Q R R 1 1 =.) = ω = ω = π ω = π 1 = π δ (50) Q RI a a π at 1 Fr small R, the lgarithmic decrement is small and therefre Q is large. at The amplitude f the scillatin decays expnentially as e. Frm the definitins f δ and Q, we als find that the amplitude decays expnentially as exp( π tqt 1 ). Thus the amplitude decays by a factr f 1 e in a time t( 1 e) = QT1 π. Q π is the number f cycles f damped mtin needed t reduce the amplitude t 1 e f its riginal value. This definitin is equivalent t the definitin f Q in terms f energy lss abve. There is a third definitin f Q which is apprpriate fr frequencies near resnance, as in part C abve. We have that Q = ω = ω RI = π ω π π a a π = = at δ (51) Fr small damping ω1 ω and using γ = a, we btain an alternative interpretatin f Q. Q ω. (5) γ Fr large damping the resnance line is far frm symmetric, and Eq. 5 is nt accurate. B. Furier analysis Since the mtin f the trsinal scillatr is peridic in time i.e. harmnic, Furier analysis (FFT) can be a very useful tl fr investigating the varius frequency cmpnents f the scillatr. The Furier transfrm decmpses a functin in the time dmain int its frequency cmpnents in the frequency dmain. By carrying ut a Furier transfrm n yur Revised 9/5/007

31 Physics 401 Experiment 6 Page 31/36 Physics Department, UIUC data, yu will be able t determine which frequencies are present in yur system. Fr example, when investigating the phenmenn f beats in the trsinal pendulum data, tw distinct frequencies shuld be bservable, ne that is assciated with the shrter-perid scillatin and ne assciated with the lnger-perid scillatin. In rder t carry ut high-quality Furier transfrms f the trsinal pendulum data, yu shuld take stable, steady-state data fr quite a lng time, therwise frequency artifacts can be intrduced that aren't really there. The Furier transfrm is defined in the fllwing way: + ( ) ( ) i ω ω t X = x t e dt 1 + x() t = X ( ) e ω π i ω t d where x() t is the riginal functin in the time dmain and ( ) in the frequency dmain. ω X ω is the crrespnding functin In Origin, Furier transfrms are implemented by ging t the Analysis menu and clicking n FFT. There are a range f different parameters yu can chse, thugh the defaults wrk quite well. Belw is a sample set f data and its Furier transfrm. The large DC cmpnent cmes frm the fact that the last pint in the data set is nt equal t the first pint in the data set. As ne can see there are tw peaks: ne crrespnding t the higher frequency/shrter perid scillatin, the ther crrespnding t the lwer frequency/lnger-perid scillatin. It is smetimes als pssible t see higher frequency cmpnents e.g. a ω,4 ω,... These are caused by the departure f the mechanical linkage f the drive mtr frm being a perfect sinusid. Furier analysis can be a very useful tl fr understanding yur data and shuld be used in yur reprt. Revised 9/5/007

32 Physics 401 Experiment 6 Page 3/36 Physics Department, UIUC C. Fitting functins Origin, in additin t ding Furier transfrms, can als fit a functin t yur data. The data t fit is the data frm magnetic damping, thugh in principle any data can be fit. The thery suggests that the best type f fit t d wuld be a sinusidal wave whse amplitude is damped expnentially. Befre fitting yur data, it is imprtant that yu make sure the data scillates abut zer. In rigin, this type f fit can be fund in the Analysis menu under Nn-linear curve fit. This will bring up a dialg bx. Click the Functin menu and select Select. Under Categries select Wavefunctin and under Functins select Sinedamp. By selecting the secnd buttn in the tp rw, yu can see hw the functin is defined (see figure belw fr an example). Revised 9/5/007

33 Physics 401 Experiment 6 Page 33/36 Physics Department, UIUC There are fur parameters that the functin will fit: xc, w, t0, and A. xc is the y-value f yur functin at x=0. A is the initial amplitude f the functin (the verall highest r lwest pint). P1 is defined t be π. w is the angular frequency (s w determines the number f secnds in ne perid). t0 determines hw lng it takes fr the amplitude t decay t 1/e f its riginal value. T fit yur functin, click n the green light buttn. Revised 9/5/007

34 Physics 401 Experiment 6 Page 34/36 Physics Department, UIUC A dialg bx will pp up asking if yu wish t fit the current data set. Click yes. Yu need t input yur guesses fr the fur fitting parameters depending n what yur data lks like. Yu may get an errr suggesting yu set xc as fixed. G ahead and d this, since the value f xc shuldn't change. T fit the data, click n 10 Iter. This will d 10 iteratins f fitting and then shw yu the final results. Cntinue fitting until the errr n each parameter des nt change r, alternately, until Chi-sqr is n lnger reduced. Chi-squared is a general measure f hw gd the fit is. The lwer the chi-squared, the better the fit, in general. In the case f this fitting, it is nt such a useful measure f gdness f the fit. A better measure is hw small the errr n each parameter is and the value f R, which yu will see nce the fit is ver. R shuld be as clse t 1 as pssible. Revised 9/5/007

35 Physics 401 Experiment 6 Page 35/36 Physics Department, UIUC In the graph abve is an example f an rigin fit t magnetic damping data. Yu can see that R = This means that the fit is rather gd, even thugh chi-squared is s large. Frm this fit, yu can get a gd feel fr the time cnstant and the frequency f the sine wave. Revised 9/5/007

36 Physics 401 Experiment 6 Page 36/36 Physics Department, UIUC Appendix II Buffer Bard Switch Psitins Revised 9/5/007