ME 322: Instrumentation Lecture 22

1. ME 322: InstrumentationLecture 22 March 12, 2014 Professor Miles Greiner

2. Announcements/Reminders • HW 8 Due Friday • Josh will hold office hours in PE 215 (and 113) tomorrow after lab, until around 6 PM • This week in lab: • Lab 7 Boiling Water Temperature in Reno • Midterm II, April 2, 2014 (three weeks) • Next week is Spring Break

3. 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 frequencies: , n = 0, 1, 2, … ∞ (integers) (not continuous) • Only admits modes for which an integer number of oscillations span the total sampling time T1. • 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 for a given frequency (from sine and cosine waves) n = 2 n = 1 n = 0 sine cosine

4. Examples (ME 322r Labs) Frequency Domain Time Domain Function Generator 100 Hz sine wave • Wave amplitude does not need to remain constant • Signals may have a wide spectrum of energetic modes Damped Vibrating Cantilever Beam Unsteady Speed Air Downstream from a Cylinder in Cross Flow

5. 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

6. 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.

7. Lab 8: Time Varying Voltage Signals Digital Scope • Produce sine and triangle waves with fm = 100 Hz, VPP = ±1-4 V, T1 = 0.04 sec • 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; and T1 = 1 sec • note: some fS< 2fm • Sample singles between 10,000 Hz < fM < 100,000 Hz using fS = 48,000 Hz • Compare fa 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, 1 sec 4 cycles 192,000 samples

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

9. 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.

10. Fig. 4 Triangle Wave and Derivative Based on Different Time Steps • dV/dtm=1 is again nosier than dV/dtm=10 • dV/dtm=1 responds to the step change in slope more accurately than dV/dtm=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 fP equals the maximum signal frequency fM = 100 Hz when the sampling frequency fSis greater than 2fM • fs = 70 and 150 Hz do not give accurate indications of the peak frequency.

12. 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 frequency is close to fM–fS. • 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 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.

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 fa, there are many possible signal frequencies, fm.

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/MECH322Instrumentation/Labs/Lab%2008%20Unsteady%20Voltage/Lab8Index.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 f1 = 1/T1

20. Folding Diagram for given fS and fM? Maximum frequency that can be accurately measured using sampling frequency fS .