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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
  • 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


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



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
  • 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 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


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
  • Sine wave
  • Triangle Wave





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
  • 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 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
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
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
Lab 8 Sample Data
  • 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

for given fS and fM?

Maximum frequency that can

be accurately measured using

sampling frequency fS .