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# ME 322: Instrumentation Lecture 22 - PowerPoint PPT Presentation

ME 322: Instrumentation Lecture 22. March 12, 2014 Professor Miles Greiner. 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

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### ME 322: InstrumentationLecture 22

March 12, 2014

Professor Miles Greiner

• 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

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

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

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 during measurement time T

• 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 during measurement time T8: 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

Estimate Maximum Slope during measurement time T

• Sine wave

• Triangle Wave

VPP

VPP

P

P

• 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.

• 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 Different Sampling Frequencies

• 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 Different Sampling FrequenciesFrequency 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.

• 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.

Figure 1 FrequencyVI Front Panel

Construct VI Frequency

• 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 Frequency

• 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 Frequency

for given fS and fM?

Maximum frequency that can

be accurately measured using

sampling frequency fS .