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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

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

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

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

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.

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

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.

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.

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.

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.

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.

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.

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

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

Folding Diagram

for given fS and fM?

Maximum frequency that can

be accurately measured using

sampling frequency fS .

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