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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 instrumentation lecture 22

ME 322: InstrumentationLecture 22

March 12, 2014

Professor Miles Greiner


Announcements reminders
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
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
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 t 1
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
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 8 time varying voltage signals
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
Estimate Maximum Slope during measurement time T

  • Sine wave

  • Triangle Wave

VPP

VPP

P

P


Fig 3 sine wave and derivative based on different time steps
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
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
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
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 f requency data
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.


Figure 6 dimensionless indicated frequency versus signal frequency
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.



Figure 1 FrequencyVI Front Panel


Construct vi
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
Lab 8 Sample Data Frequency

  • 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
Folding Diagram Frequency

for given fS and fM?

Maximum frequency that can

be accurately measured using

sampling frequency fS .


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