Physics 41. Fall 18 Eugene V. Colla 1/8/18 Physics 41 1
Electrical RLC circuits Torsional Oscillator Damping Data Analysis 1/8/18 Physics 41
V R +V L +V C =V(t) If V(t)= R d d q(t) q(t) dt dt C C L q(t) +R q(t) + =, = V Damping term. Reflects energy dissipation in the resistor V R L V L V(t) scope 1. V C V C V(t).5. -1. -.5..5 1. time 1/8/18 Physics 41 3
1 b < under-damped solution a R R 1, b L L LC U C (V) -1 f 1 1 R LC L -...4.6.8.1 time (ms) 1. b > overdamped solution 1. b = criticallydamped solution U C (V).8.4 U C (V).8.4.....4...4 1/8/18 Physics 41 4
permanent magnet Tension nut Tension bolt Disk comb Motor disk Piano wire Motor comb optical comb for read out Optical sensors 1/8/18 Physics 41 5
M wire L 1 L t R t disk Wires 1 and exert the torques t 1 and t on the disk of mass M t t t K K K K 1 1 1 Gr L 1 4 θ : angular deflection of the disk r : radius of the wires L i : length of the wire I G: shear modulus of the wire 4 1 1 K K1 K G r L1 L A typical shear modulus for steel is 8.3 1 1 N/m K torsional spring constant 1/8/18 Physics 41 6
DAQ rate (Hz) Interface card Program window Program can accept only 1 points. If sampling rate is 5Hz the maximum time of data collection is s! 1/8/18 Physics 41 7
t t 1 4 g step 4 Equation Weight Residual Sum of Squares y = a + b*x No Weighting 8.75999E-4 Pearson's r.99996 Adj. R-Square.99989 Value Standard Error 3 3 B Intercept -.1335.134 Slope 51.315.718 R (rad) (rad) 1 1 1 3 4 5 6 7 slope...4.6.8 m (kg) g=9.81m/s Slope=51.3rad/kg K=.971Nm/rad 1/8/18 Physics 41 8
R R Rope is too short! Avoid the over damping of the pendulum motion and any extra sources of friction. 1/8/18 Physics 41 9
Measuring of the electrostatic forces. t t 1 Charles-Augustin de Coulomb 1736-186 Coulomb's law q1 q 1 F ke ; k e r 4 Coulomb's torsion balance. Courtesy of Wikipedia 1/8/18 Physics 41 1
t1 t Measuring of the gravitational forces. Henry Cavendish (1731 181) F GmM r Gravitational Law Cavendish s result Cavendish torsion balance experiment. Courtesy of Wikipedia Currently accepted value G = 6.74 1 11 m 3 kg 1 s, 6.6748 1 11 m 3 kg 1 s. 1/8/18 Physics 41 11
t t t K K K 1 1 4 4 Gr Gr 1 1 K ; K K K 1 1 L1 L1 L If there is no dissipation: d I K dt If we know I we can calculate K (rad) 1-1 From time trace (t) we can find w it can be done by measuring period but better (and faster!) to perform the nonlinear fitting. 1 3 4 5 6 7 1/8/18 Physics 41 1
1 (rad) -1 1 3 4 5 6 7 rad w w 3.16 ; f.4975 Hz s 1 5 (rad) -1 Count 15 1 5 4 6 8 1 1 -.1 -.5..5.1.15 Regular Residual (pend) 1/8/18 Physics 41 13
From SineDamp fitting f =.497Hz or w f 3.13 rad / s Resonance frequency can be also found by applying FFT on the raw data (rad) 1 1 1.5415-1 1 3 4 5 6 7 1 1.1.1 1 1 1/8/18 Physics 41 14
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Permanent magnet Damping term pendulum The solutions are exactly the same as in case of RLC circuit (three solutions) 1..5 Under damped case (rad). -.5-1. Model Equation Reduced Chi-Sqr SineDamp y=y + A*exp(-x /t)*sin(pi*(x-xc )/w).3473e-5 Adj. R-Square.9996 Value Standard Error pend y.97.11783e-4 pend xc.5818 4.8757E-4 pend w 1.173.55354E-4 pend t 3.1651.751 pend A 1.683.177 1 3 1/8/18 Physics 41 17
1..5 (rad). -.5-1. Type equation here. Model Equation Reduced Chi-Sqr SineDamp y=y + A*exp(-x /t)*sin(pi*(x-xc )/w).3473e-5 Adj. R-Square.9996 Value Standard Error pend y.97.11783e-4 pend xc.5818 4.8757E-4 pend w 1.173.55354E-4 pend t 3.1651.751 pend A 1.683.177 from error propagation analysis 1 3 From SineDamp fitting exponential decay term is exp t t 1/8/18 Physics 41 18
Peaks coordinates are saved in the worksheet and can be used for analysis We can find the amplitudes of the wave using Peak Analyzer.8 ExpDec1.66 Model n (rad).6.4. Equation y = A1*e xp(-x/t1) Value Standard Error Y y.75 3.31661E-4 Y A1 1.834.157 Y t1 3.1358.749 ln(a n+1 /A n ).64.6.6 =.63±.. 1 3.58 1 3 4 5 6 1/8/18 Physics 41 19 peak numbert
1. Fitting to damp exponential decay function. Outcome: resonance frequency and decrement coefficient.. Applying FFT procedure. Result resonance frequency. 3. Using Origin Peak Analyzer we can find amplitudes and positions of the damped sine wave maximum end then plot the envelope. 4. You can directly obtain the envelope of the damped sine wave by using Origin (optional). 1/8/18 Physics 41
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I K t Coulomb t Coulomb C Amplitude decreases by 4C/K per period linearly! ( t) C / K ( (4n 1) C / K)cos( wt) 1 ( n ) T t nt n 1,,... (rad) T/ ( t) C / K ( (4n 3) C / K)cos( wt) 1 ( n 1) T t ( n ) T n 1,,... d/dt (rad/s) - 1 + + - - - -1 1 3 4 5 1/8/18 Physics 41
Amplitude decreases by 4C/K per period linearly! 3 Equation Weight Residual Sum of Squares y = a + b*x No Weighting.463 Pearson's r -.9986 Adj. R-Square.99699 Value Standard Error Y Intercept 3.3967.37 Y Slope -.1177.166 n 1 5 1 15 5 3 1/8/18 Physics 41 3
t Coulomb C K ~ if K C pendulum stops (rad) off.4.. - -. 4 4 44 46 48 (rad) - 4 1/8/18 Physics 41 4
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I K tturb t C sgn( ) turb t n In case of n=1 viscous damping Logarithmic decrement in case of turbulent damping is no more constant and in case n= can be calculated as δ = 8C 3I Θ Expected result decrement decreases with decreasing of the amplitude 1/8/18 Physics 41 6
Analyzing the envelope of the damped oscillating time record we can calculate the log decrement factor 1/8/18 Physics 41 7
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(rad) 4 - Raw data. Our goal: find the positions and amplitudes of the peaks -4-6 4 X i,y i 5 1 15 1 st Technique: using FindPeaks option (rad) - -4 4 6 1/8/18 Physics 41 9
4 (rad) - -4-6 5 1 15 Local Maximum works well for not noisy oscillating dependencies 1/8/18 Physics 41 3
The details related to this project you can find in: \\engr-file-3\phyinst\apl Courses\PHYCS41\Students\6. Torsional oscillator\turbulent damping.opj New plot + labels as a result of finding the peaks Peaks data can be found in a Worksheet and using this data you can plot the dependence of amplitude on time Positive peaks Negative peaks 1/8/18 Physics 41 31
Original Data Envelope 4 (rad) - n (rad) -4-6 5 1 15 5 1 15 1/8/18 Physics 41 3
4 (rad) - -4-6 5 1 15 nd Technique: using Envelope option Origin will create the worksheet with interpolated (defined for the same x s as the raw data) envelope data 1/8/18 Physics 41 33
4 4 (rad) - (rad) -4-6 5 1 15 1 Original data Envelope data 1/8/18 Physics 41 34
All these quasi periodic data can be analyzed using Fast Fourier Transform 4 Our goal: find the resonance frequency of the pendulum (rad) - -4 Origin window -6 5 1 15 1/8/18 Physics 41 35
4 The results of FFT you can find in the same Workbook which contains the raw data (rad) - -4-6 5 1 15 Click on corresponding graph and it will appear in separate window Magnitude vs. frequency plot 1/8/18 Physics 41 36
84 4 (rad) - Magnitude 63 4 1 Linear scale -4-6 5 1 15 1 1 1. 6.8 13.6.4 7. Resonance frequency Frequency.4915 Spectrum better to present in log-log scale 1 1 1 log-log scale.1.1 1/8/18 Physics 41.1.1 1 1 37
Result of measurement Systematic error.35 B.3.5. Correct value Random error P.15.1.5. 4 6 8 1 1 14 X i 1/8/18 Physics 41 38
Systematic error Q F S T 1 t g F 1 3 f c t g 1 1 9d 3 x 3 fc V g t t g g t rise 3 3 9d x 1 1 1 S T V g t t g g t rise 3 1 1 1 dq dq dq dq dq Q F S T S T df ds dt ds dt S T FT S FS T Q S T 1/8/18 Physics 41 39
Systematic error S T Q Q S T S d V 3 x 3 1 1 g d 3 x S d V x g d x 3 1 1 1 1 T t t t t t t t 5 3 g 1 rise g g rise g rise 1/8/18 Physics 41 4
Result of measurement Systematic error.35 B.3.5. Correct value Random error P.15.1.5. 4 6 8 1 1 14 X i Mean of {x i } Standard deviation of of {x i } 1 1 N xi i N 1 N 1 N 1 i x i Standard deviation of mean X N 1 1/8/18 Physics 41 41