Analysis of DSP, sampling and rotating shafts/machines
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TQM3 - Introduction to Maintainability(DSP)
1. The signal X(t) from an accelerometer, mounted on a pump bearing, contains frequency components up to 500 Hz. The pump shaft rotational speed is 1,179 RPM.
(a) Express the pump shaft rotational frequency in Hz.
(b) At what rate, must X(t) be sampled in order to reconstruct it without error? Choose one answer from below and explain your choice.
A. Less than 500 Hz. B. Greater than 500 Hz. C. Greater than 1,000 Hz.
2. (a) An acoustic signal picked up from a microphone is sampled at 10 kHz. Calculate the maximum information frequency of the digitized signal.
(b) It was later determined that this signal contained an unexpected frequency component at
6 kHz. Calculate the frequency at which aliasing effect is seen in the digitized signal.
(c)State two approaches you may use to avoid the aliasing problem in this case.
3. State four important causes of vibration in rotating machinery.
4. The rotational speed of an electric motor is 3540 RPM. Calculate the frequency of rotation in Hz. Give an example for each of the following:
(a) a harmonic frequency of the rotating speed,
(b) a sub-synchronous frequency of the rotating speed,
(c) a non-synchronous frequency of the rotating speed.
5. Which of the following characteristics may be detected using time waveform analysis of vibration measurements? Explain.
(a) Impacting in the machine components. (b) Signal Modulation. (c) Strong periodic components.
6. (a) The rotational speed of a pump shaft is 1740 RPM. At what frequencies (Hz) in the spectral domain do you expect to see harmonics of the rotating speed due to (i) imbalance, (ii) bent shaft, and (iii) parallel misalignment?
(b) State the directional placement of the accelerometer to measure each of these anomalies.
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Solution Summary
Analysis of DSP, sampling and rotating shafts/machines. This includes problems and solutions on sampling waveforms, aliasing frequency, time waveform analysis, spectral analysis, FFT, bent shafts, mis-aligned shafts, shaft impacts
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1. The signal X(t) from an accelerometer, mounted on a pump bearing, contains frequency components up to 500 Hz. The pump shaft rotational speed is 1,179 RPM.
(a) Express the pump shaft rotational frequency in Hz.
A rotational speed in terms of RPM (denoted by N) may be expressed in terms of radians per minute by multiplication by 2π (since there are 2π radians in one revolution)
Thus
Radians per minute =2πN
As there are 60 seconds in a minute we can write down an equivalent rotational speed in terms of radians per second, ω as
ω=2πN/60=πN/30 rads.s^(-1)
In addition linear frequency, f is expressed in terms of angular frequency by
ω=2πf
⇒f=ω/2π=πN/30.1/2π
f=N/60
Putting in numbers where N=1179 RPM
f=1179/60=19.65Hz
(b) At what rate, must X(t) be sampled in order to reconstruct it without error? Choose one answer from below and explain your choice.
A. Less than 500 Hz. B. Greater than 500 Hz. C. Greater than 1,000 Hz.
In sampling theory the minimum sampling rate of a signal, f_s (for error free reproduction without distortion, anti-aliasing) should be greater or equal to twice the highest frequency in analogue waveform (Nyquist Criteria)
Highest frequency from accelerometer is given as f_h so we can write
Sampling frequency, f_s≥2f_h
f_s≥2×500 Hz
f_s≥1000 Hz (Option C)
2.
(a) An acoustic signal picked up from a microphone is sampled at 10 kHz. Calculate the maximum information frequency of the digitized signal.
Again the Nyquist criteria states that the minimum sampling rate, f_s of a signal for distortion free further reproduction is at least twice the highest frequency, f_h in the information signal or
In the limit
f_s=2f_h
f_h=f_s/2=(10 kHz)/2
Highest frequency in digitized information signal f_h=5kHz
(b) It was later determined that this signal ...
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