1. ME 322: Instrumentation Lecture 22 March 11, 2015 Professor Miles Greiner

2. Announcements/Reminders HW 8 Due Friday How is Lab 7 “Boiling Water Temperature in Reno” going? Midterm II, April 1, 2015 (three weeks) – Next week is Spring Break

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

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

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

6. 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.

7. Lab 8: Time Varying Voltage Signals Produce sine and triangle waves with f m = 100 Hz, V PP = ±1-4 V, T 1 = 0.04 sec – Sample both at f S = 48,000 Hz and numerically differentiate with two differentiation time steps Evaluate Spectral Content of sine wave at four different sampling frequencies f S = 5000, 300, 150 and 70 Hz; and T 1 = 1 sec – note: for some f S < 2 f m Sample singles between 10,000 Hz < f M < 100,000 Hz using f S = 48,000 Hz – Compare f a 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, 1 sec 4 cycles 192,000 samples

8. Estimate Maximum Slope PP V PP

9. Fig. 3 Sine Wave and Derivative Based on Different Time Steps dV/dt 1 (  t=0.000,0208 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 is probably because the actual wave was not a pure sinusoidal.

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

11. 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.

12. 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 frequency is close to f M –f S. The results are in agreement with sampling theory.

13. 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.

14. 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.

15. Figure 2 VI Block Diagram

16. Figure 1 VI Front Panel

17. Construct VI Starting Point VI Spectral Measurement VI – Signal Processing; Waveform Measurement, Result: linear Convert to and from dynamic data – Signal Manipulation Input data type: 1D array of scalars-single channel “Time” of maximum – Mathematics; Probability and Statistics: Statistics

18. Lab 8 Sample Data http://wolfweb.unr.edu/homepage/greiner/teaching/MECH322I nstrumentation/Labs/Lab%2008%20Unsteady%20Voltage/Lab 8Index.htm http://wolfweb.unr.edu/homepage/greiner/teaching/MECH322I nstrumentation/Labs/Lab%2008%20Unsteady%20Voltage/Lab 8Index.htm Calculate Derivatives Plot using secondary axes – Design; Change Chart Type; Combo Scatter with straight line Frequency Domain Plot – The lowest finite frequency and the frequency resolution are both f 1 = 1/T 1

20. Folding Diagram