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ME 322: Instrumentation Lecture 21 March 9, 2015 Professor Miles Greiner Spectral analysis, Min, max and resolution frequencies, Aliasing,

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Announcements/Reminders 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 3 labs, solo, including performing the measurements, and writing Excel, LabVIEW and PowerPoint.

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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?

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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 car vibration, beam or bell ringing – System response (Vibration Analysis) Resonance Spectral analysis transforms a signal from the time-domain V(t) to the frequency-domain, V RMS (f) – What does this mean?

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Fourier Transform 0 t T 1 V n = 0 n = 1 n = 2 sine cosine

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

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What is the lowest Frequency mode that can be observed during measurement time T 1

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Sampling Rate Theory What discrete sampling rate f S must be used to accurately observe a sinusoidal signal of frequency f M ? Must be greater than f M, but much how larger?

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Lab 8 Aliasing Spreadsheet Example on/Labs/Lab%2008%20Unsteady%20Voltage/Lab8Index.htm on/Labs/Lab%2008%20Unsteady%20Voltage/Lab8Index.htm Measured sine wave, f m = 10 Hz – V(t) = (1volt)sin[2 (10Hz)(t+t shift )] Total sampling time, T 1 = 1 sec How many peaks to you expect to observe in one second? How large does the sampling rate f S need to be to capture this many peaks?

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How to predict indicated (or Alias) Frequency? 3 Frequencies: – f m being measured; f s Sampling frequency; f a indicated frequency f a = f m if f s > 2f m Otherwise using folding chart on page 106 – Let f N = f s /2 be the maximum frequency that can be accurately observed using sampling frequency f s.

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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 f s > 2f m ?

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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 f m that can cause this? ID: what is known? f s and f a 18,000

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

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Fourier Transfer Example Lab 8 site: – ntation/Labs/Lab%2008%20Unsteady%20Voltage/Lab8Index.htm ntation/Labs/Lab%2008%20Unsteady%20Voltage/Lab8Index.htm Dependence of coefficient b (sine transform) on weigh function frequency and phase shift Dependence of V rms on weight function frequency, but not phase shift.

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

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Estimate Maximum Slope PP V PP

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Fig. 3 Sine Wave and Derivative Based on Different Time Steps dV/dt 1 ( t=0.000,020,8 sec) is nosier than dV/dt 10 ( t=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.

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Fig. 4 Sawtooth Wave and Derivative Based on Different Time Steps dV/dt 1 is again nosier than dV/dt 10 dV/dt 1 responds to the step change in slope more accurately than dV/dt 10 The maximum slope from the finite difference method is larger than the ideal value.

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Fig. 5 Measured Spectral Content of 100 Hz Sine Wave for Different Sampling Frequencies The measured peak frequency f P equals the maximum signal frequency f M = 100 Hz when the sampling frequency f S is greater than 2f M f s = 70 and 150 Hz do not give accurate indications of the peak frequency.

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

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Table 3 Signal and Indicated Frequency Data This table shows the dimensional and dimensionless signal frequency f m (measured by scope) and frequency indicated by spectral analysis, f a. For a sampling frequency of f S = 48,000 Hz, the folding frequency is f N = 24,000 Hz.

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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 f a, there are many possible signal frequencies, f m.

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Figure 2 VI Block Diagram

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Figure 1 VI Front Panel

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Lab 8 Sample Data hing/MECH322Instrumentation/Labs/Lab%20 08%20Unsteady%20Voltage/Lab8Index.htm hing/MECH322Instrumentation/Labs/Lab%20 08%20Unsteady%20Voltage/Lab8Index.htm Calculate Derivatives Plot using secondary axes Frequency Domain Plot – Lowest finite frequency f 1 = 1/T 1

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Effect of Random Noise on Differentiation

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Time Dependent Data iT = (∆t s ) i V 00 1∆t s 22∆t s

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