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ME 322: Instrumentation Lecture 21

ME 322: Instrumentation Lecture 21. March 10, 2014 Professor Miles Greiner. Announcements/Reminders. HW 7 (L7PP) due now HW 8 Due Friday Then Spring Break! This week in lab: Lab 7 Boiling Water Temperature in Reno

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ME 322: Instrumentation Lecture 21

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  1. ME 322: InstrumentationLecture 21 March 10, 2014 Professor Miles Greiner

  2. Announcements/Reminders • HW 7 (L7PP) due now • HW 8 Due Friday • Then Spring Break! • This week in lab: • Lab 7 Boiling Water Temperature in Reno • Please fully participate in each lab and complete the Lab Preparation Problems • For the final you will repeat one of the last four labs, solo, including performing the measurements, and writing Excel, LabVIEW and PowerPoint.

  3. A/D Converters • Can be used to measure a long series of very rapidly changing voltage • Useful for measuring time-dependent voltage signals and assessment of their dynamic properties • Rates of Change (derivatives) and • Frequency Content (Spectral Analysis) • What can go wrong? • Last time we showed that small random errors (RF noise, IRE) can strongly affect calculation of derivatives • So: Make derivative time-step long enough so that the real signal changes by a larger amount than the random noise. • What is Frequency Content?

  4. Spectral Analysis • Evaluates energy content associated with different frequency components within a signal • Use to evaluate • Tonal Content (music) • You hear notes, not time varying pressure • Dominant or natural frequencies • bell ringing or car shaking • Vibration Analysis • Resonance • Spectral analysis transforms a signal from the time-domain V(t) to the frequency domain, VRMS(f) • What does this mean?

  5. Fourier Transform V 0 t T1 • Any function V(t), over interval 0 < t < T1, may be decomposed into an infinite sum of sine and cosine waves • , • Discrete (not continuous) frequencies: , n = 0, 1, 2, … ∞ (integers) • Only modes for which an integer number of oscillations span the total sampling time T1 are used. • The coefficient’s an and bnquantify the relative importance (energy content) and phase of each mode (wave). • The root-mean-square (RMS) coefficient for each mode quantifies its total energy content (both sine and cosine waves) n = 2 n = 1 n = 0 sine cosine

  6. Examples (ME 322r Labs) Frequency Domain Time Domain Function Generator 100 Hz sine wave • Real signal may have a wide spectrum of energetic modes Damped Vibrating Cantilever Beam Unsteady Speed Air Downstream from a Cylinder in Cross Flow

  7. What is the lowest Frequency mode that can be observed during measurement time T1 • For example, if we measure outdoor temperature for one hour, can we observe variations that require a day to repeat? • The lowest (finite) observable frequency is f1 = 1/T1 • The only other frequencies that can be detected are • What is the frequency resolution? • Smallest change in frequency that can be detected • Increasing the total sampling time T1reduces the lowest detectable frequency and improves frequency resolution,

  8. Sampling Rate Theory • What discrete sampling rate fS must be used to accurately observe a sinusoidal signal of frequency fM? • Must be greater than fM, but much how larger?

  9. Lab 8 Aliasing Spreadsheet Example • http://wolfweb.unr.edu/homepage/greiner/teaching/MECH322Instrumentation/Labs/Lab%2008%20Unsteady%20Voltage/Lab8Index.htm • Measured sine wave, fm = 10 Hz • V(t) = (1volt)sin[2p(10Hz)(t+tshift)] • Total sampling time, T1 = 1 sec • How many peaks to you expect to observe? • How large does the sampling rate fS need to be to capture this mean peaks?

  10. How to predict indicated (or Alias) Frequency? • fa = fmif fs > 2fm • Otherwise using folding chart on page 106 • Let fN = fs/2 be the maximum frequency that can be accurately observed using sampling frequency fs. Maximum frequency that can be accurately measured using sampling frequency fS .

  11. Problem 5.26 (p. 127) • A 1-kHz sine wave signal is sampled at 1.5 kHz. What would be the lowest expected alias frequency? • ID: Is fs > 2fm ?

  12. A more practical example • Using a sampling frequency of 48,000 Hz, a peak in the spectral plot is observed at 18,000 Hz. • What are the lowest 4 values of fm that can cause this? • ID: what is known? fsand fa 18,000

  13. Upper and Lower Frequency Limits • If a signal is sampled at a rate of fS for a total time of T1 what are the highest and lowest frequencies that can be accurately detected? • (f1= 1/T1) < f < (fN = fS/2) • To reduce lowest frequency (and increase frequency resolution), increase total sampling time T1 • To observe higher frequencies, increase the sampling rate fS.

  14. How to transform find vs fn? • For • (cosine transform) • (sine transform) • How to evaluate these integrals? • For simple V(t), in closed form • For complex or discretely-sampled signals • Numerically (trapezoid or other methods) • Appendix A, pp 450-2 • LabVIEW Spectral Measurement VI does this • for (f1 = 1/T1) < (fn ) < (fN = fS/2)

  15. Fourier Transfer Example • Lab 8 site: • http://wolfweb.unr.edu/homepage/greiner/teaching/MECH322Instrumentation/Labs/Lab%2008%20Unsteady%20Voltage/Lab8Index.htm • Dependence of coefficient b (sine transform) on weigh function frequency and phase shift • Dependence of Vrms on weight function frequency, but not phase shift.

  16. Lab 8: Time Varying Voltage Signals Digital Scope • Produce sine and triangle waves with fm = 100 Hz, VPP = ±1-4 V • Sample both at fS = 48,000 Hz and numerically differentiate with two different differentiation time steps • Evaluate Spectral Content of sine wave at four different sampling frequencies fS= 5000, 300, 150 and 70 Hz (note: some < 2fm ) • Sample singles between 10,000 Hz < fM < 100,000 Hz using fS = 48,000 Hz (fa compare to folding chart) Function Generator NI myDAQ fM = 100 Hz VPP = ±1 to ± 4 V Sine wave Triangle wave fS = 100 or 48,000 Hz Total Sampling time T1 = 0.04 sec 4 cycles 192,000 samples

  17. Estimate Maximum Slope • Sine wave • Triangle Wave VPP VPP P P

  18. Figure 2 VI Block Diagram

  19. Figure 1 VI Front Panel

  20. Lab 8 Sample Data • http://wolfweb.unr.edu/homepage/greiner/teaching/MECH322Instrumentation/Labs/Lab%2008%20Unsteady%20Voltage/Lab8Index.htm • Calculate Derivatives • Plot using secondary axes • Frequency Domain Plot • Lowest finite frequency f1 = 1/T1

  21. Fig. 3 Sine Wave and Derivative Based on Different Time Steps • dV/dt1 (Dt=0.000,0208 sec) is nosier than dV/dt10 (Dt=0.000,208 sec) • The maximum slope from the finite difference method is slightly larger than the ideal value. This may be because the actual wave was not a pure sinusoidal.

  22. Fig. 4 Sawtooth Wave and Derivative Based on Different Time Steps • dV/dt1 is again nosier than dV/dt10 • dV/dt1 responds to the step change in slope more accurately than dV/dt10 • The maximum slope from the finite difference method is larger than the ideal value.

  23. Fig. 5 Measured Spectral Content of 100 Hz Sine Wave for Different Sampling Frequencies • The measured peak frequency fP equals the maximum signal frequency fM = 100 Hz when the sampling frequency fS is greater than 2fM • fs = 70 and 150 Hz do not give accurate indications of the peak frequency.

  24. Table 2 Peak Frequency versus Sampling Frequency • For fS > 2fM = 200 Hz the measured peak is close to fM. • For fS < 2fM the measured peak is close to the magnitude of fM–fS. • The results are in agreement with sampling theory.

  25. Table 3 Signal and Indicated Frequency Data • This table shows the dimensional and dimensionless signal frequency fm (measured by scope) and frequency indicated by spectral analysis, fa. • For a sampling frequency of fS = 48,000 Hz, the folding frequency is fN = 24,000 Hz.

  26. Figure 6 Dimensionless Indicated Frequency versus Signal Frequency • The characteristics of this plot are similar to those of the textbook folding plot • For each indicated frequency fa, there are many possible signal frequencies, fm.

  27. Effect of Random Noise on Differentiation • Measured voltage has Real and Noise components • VM = VR+VN • For small is large and random • Want • wV decreases as FS gets smaller and N increases • Want to be large enough to avoid random error but small enough to capture real events RF, IRE, other errors, Random, but does not increase with

  28. Lab 8 example Signal Otherwise get false or “alias” frequencies

  29. Time Dependent Data How to find 1st order numerical differentiation (center difference) = differentiation time sampling time m = 1, 2, 3, …

  30. Estimate Maximum Slope Estimate Max Slope

  31. Triangle Demonstrate Lab 8 data processing P

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