CHAPTER 7 LPV CLOCKING MODELING AND EXPERIMENTAL INTEGRATION
B.2 DATA ANALYSIS-BASIC DEFINITIONS
B.2.4 Time-accurate data: time space, encoder space, and frequency data
As mentioned before, time-averaged data is one of the many statistics that can be generated on the time-resolved data, which is the basic measurement acquired in these experiments. However, time-resolved data can take many forms and it is important to describe the differences here, since later in the dissertation, the relationships between them will become secondary to the data that they are showing. Numerically, this time- resolved data is an array of the form:
F(t) = F(i – 1,i, i+ 1...)
where F is the property and i represents the data sample. For these experiments, these index points (i) represent either monotonically increasing time (for time-space data) or monotonically increasing encoder pulse (for data acquired synchronously with the rotor position, or in encoder-space). For these experiments, all the channels are sampled synchronously with time with the exception of the rotor position, which generates a time pulse (with a resolution of 10 MHz) every time an encoder pulse occurred. Because time is the dependent variable in this case:
Time(t) =Time(i – 1,i,i+ 1...), i =encoder pulse
one can interpolate between time-based and encoder spaced data. This usually is only needed when one is examining the time-resolved data and comparing different sensors and trying to perform an ensemble based average (or an “envelope” which will be discussed later).
Time-resolved data can be shown in many forms. The two main ones are time plots and frequency spectra (or FFT’s). A comparison is shown below in Figure B.1 to Figure B.3.
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68 69 70 71 72 73 74 75 76
120 122 124 126 128
PTDA
Pressure (KPa)
TIME (ms)
Figure B.1 Time-Resolved Data for 1 Revolution, Downstream Total Pressure Rake Figure B.1 shows the time-resolved data for the average of the downstream pressure sensors over one revolution. In this case, the 10 individual measurements (from the two rakes) were averaged together (index point by index point) to generate this pressure history that represents the average time-resolved pressure downstream of the rotor. Just to be clear, although it is unimportant for the issue we are describing, this pressure history was generated in the following manner. There were 10 pressure sensors placed on two rakes (five on each rake) that cover the flow from inner to outer annulus.
If we state these at Pi, i=1-10, then the average pressure downstream of the rotor (Pavg) at any point in time (j) is given by:
Pavg( ) =j nS1 0= 1Pn( )j
10 forall j
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Now back to the main issue. This data was taken over one revolution. As noted earlier there are 72 blades on this machine and 38 vanes. If we look at about 1 ms of data we should see about 8.5 peaks (see Figure B.2).
68 69 70 71 72 73 74 75 76
121 121.2 121.4 121.6 121.8 122
PTDA
Pressure (KPa)
TIME (ms) Between 8 and 9 peaks
Peaks
Figure B.2 Average Downstream Total Pressure over 1 ms Here we see that the individual rotor passing is apparent (even after all the
averaging of the different sensors and after passing through the low vane). There is some variation that occurs from passage to passage which may be due to the averaging
technique, or it may be due for some other reason. While it is easy to see the main frequency here, it is harder to discern any other important frequencies. To do that, the same data that is shown in Figure B.1 can be examined in the frequency domain by using a FFT algorithm on the data to extract the frequency content as shown below.
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0 0.2 0.4 0.6 0.8 1
0 10000 20000 30000 40000 50000
Peak Amplitude (KPa)
Frequency (Hz) Blade Passing
First Harmonic
Second Harmonic
Figure B.3 Power Spectra of Downstream Total Pressure over 1 Revolution From this figure, one can easily see the main frequencies that comprise the data shown in Figure B.1. One can see that the blade passing frequency (or fundamental frequency) is about 1 KPa in value, which is about 1.4% of the average value (1/72). And that while the first and second harmonics are visible, they are about 0.28% of the average value. As will be discussed later, both of these values come close to the calibration accuracy of the pressure sensors.
For those unfamiliar with the concept behind the generation of Figure B.3, a little explanation is in order, although rigorous proof will not be presented here. The idea is that every time-domain signal can be represented by an infinite sum of sinusoidal functions with varying amplitudes and phases. Using a set of algorithms (called Fourier transforms) one can decompose a time base signal into its main frequency components.
One branch of these algorithms is known as Fast Fourier Transforms (FFT’s), because of their numerical efficiencies. It was in fact this innovation that allowed spectral analysis to be done so readily by many numerical routines, and currently these algorithms are
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often imbedded directly into processors and data acquisition boards so that what is stored on the computer after acquisition is not a time based signal, but rather a frequency based one! The raw FFT algorithm produces both the amplitude and the phase. This
information is often combined to produce a power spectrum (or more accurately a single- sided power spectrum), which shows the amplitude at each frequency. There are some filtering and windowing issues that need to be addressed when dealing with discrete data.
This subject will be discussed in more detail later in the dissertation. There are some excellent resources that describe this procedure in more detail than can be provided here.
Adams [55] provides an excellent overview of the relationships between examining data in the frequency domain and time domain. Information regarding actual use of traditional FFT algorithms and other signal processing can be found in [56]. An even more step-by- step tutorial is covered by a signal-processing course published by National Instruments [57]. At the other end of the spectrum, many of the routines used by Matlab or LabView (or any of a host of processing software packages) can be found in [58]. Discussions of how these algorithms were developed and their mathematical basis can be found in many books on digital signal processing (or DSP) such as [59]. The extension of the traditional techniques of DSP into the area of Joint Time-Frequency Analysis has become relatively entrenched in the available software, but is still under-utilized in many areas of research.
While this dissertation does not use wavelet analysis methods, for completeness the reader is referred to some of the many books on wavelets [60].