Investigating Thermal Shock*
Fran Morrissey
DOE ERULF
Kutztown University
Oak Ridge National Laboratory
Oak Ridge, Tennessee 37830
April 29, 1999
CALCULATIONS
Fabry-Perot Interferometric Strain Sensor
Analysis of the signal data is accomplished by several methods.
First each signal waveform must be categorized as either a Multi Fringe or a Partial Fringe. In either case, the signal data is translated into strain data. The ideal signal, for distance changing linearly in time, is represented by a sine function (y = sin(x)), which produces a linear strain (y = mx +b).
Partial Fringes are identified by waveforms which deteriorate into a distorted signal that has cusps or amplitude values that are far below calibration values. Multi Fringes have distinct waveforms that possess amplitude values very close to the calibration values and the signal is sinusoidal in nature. Below is a typical waveform of an interferometric signal as generated by a FOSS I unit under controlled conditions. In particular, this was the result of a fiber probe upon a microcantilever under response to a current induced magnetic field.
The analysis of a partial fringe takes advantage of the small angle approximation for simplifying the calculations. Unfortunately, the ideal location for calculating strain is located in the waveforms curve where the slope is very large and the angle is very small. Regions on the curve where the slope is small, location at maxima or minima, are places where the values of strain can not be accurately calculated with a small angle approximation. Therefore, two sensors at each sensing area limit the chances that both will have data within the undesirable regions.
In the above relationship for the signal, P2P is the Peak to Peak calibration voltage, G.F. is the gauge factor for the sensor, with the phase angle and the DC offset as the value of the voltage need to accurately represent the baseline of the waveform. For the calculations of the strain in a partial fringe configuration , the small angle approximation is used to simplify the calculation of the strain on the target. Below is the strain relationship that was implemented using the small angle approximation to the signal relationship.
The analysis of the Multi Fringe data incorporates the use of modeling to arrive at a representation of the signal in terms of strain. Initially, an ideal strain is assumed that produces a sinusoidal waveform for signal over time. By modifying the strain model, a different signal will result, that will eventually represent the physical data taken during the event. The steps in the analysis include,
- Modifying strain model
- Comparing signal generated to physical data
a) Accurately represents physical data (Strain value is determined)
b) Unacceptable accuracy (Modify strain model and continue with analysis)
To assist in the modeling of the strain data, the values of the strain were mapped in response to the data values between fringes, which varied in each case. As certain values were varied in the strain model an interpolating technique restructured the strain values to produce a smooth (reasonable) curve.
The mapping relationships takes values (epsilon) within the extremes of i and j whereupon it maps them to values (x) within the extremes of a and b. This relationship has the flexibility of mapping to data sets of varying sizes.
The divided difference interpolation technique utilizes Newton's Polynomials (based on a Taylor Series) for n number of points with a sub 0 being the zero order divided difference and a sub n being the nth order divided difference. For an nth degree polynomial, n+1 points are needed.
This formula was used in conjunction with a routine that allowed the user to select turning points and fringe number. The application then calculated the strain based on the human interpretation of the raw strain sensor signal.
Fabry-Perot Interferometric Strain Sensor
Partial Fringe Analysis - Dr. Mike Cates
The partial fringe formulation was headed by Dr. Cates and ultimately my assistance with the construction of the algorithms. The majority of the theory behind the analysis is presented below along with added commentary.
The interferometric signal possesses a sine form which oscillates between Vmax - Vmin that represents the peak to peak voltage calibration. Vref is the voltage at which the value of the sine function equals zero and is the midpoint between Vmax and Vmin. The complexities of partial strain analysis is determination of Vo, which is the zero strain voltage output of the detector that is determined from the base line signal before the system is strained. V(X) is the signal amplitude returned from the Foss I unit and this signal is related to the position of the sensor gap (Fabry-Perot Spacing) which equals X. It is required to know not only the value of Vo, which is the value of V(Xo) (where Xo is the unstrained sensor gap) but also the reference quadrant that the signal started in.
- (V(X) - Vref) / .5 (Vmax - Vmin) corresponds to the value of the sin with the gap X.
- (Vo - Vref) / .5 (Vmax - Vmin) relates the value of the sine function for strain equaling zero.
- X = arcsin [(V(X) - Vref) / .5 (Vmax - Vmin)]
- Xo = arcsin [(Vo - Vref) / .5 (Vmax - Vmin)]
X and Xo express the gap displacement in terms of angles where the signal is comprised for various values of the sine function. The strain (amount of displacement per reference length) is the difference in X and Xo divided by the gauge length, which is the length that the displacement is referenced.
The domain and range of the sine function passes four quadrants before it repeats itself. The values 0 to 1, 1 to 0, 0 to -1, and -1 to 0, correspond to the angles 0 to 90, 90 to 180, 180 to 270, 270 to 360 degrees. Each of these regions of the sine function possess the same shape, being either negatives or mirror images of each other. We define these regions by labeling the first as quadrant 1, the next as quadrant 2, and so forth.
We assume strain goes positive for the first large excursion of the signal. (The experimental data has demonstrated that a small deflection in the opposite direction may occur before a large excursion just after the pulse hits. There may be optical glitches from the beam pulse or other unknown effects that cause this.) If the signal becomes positive in the first large excursion, we know that Vo - Vref is either in quadrant 1 or 4; if the signal is negative, then Vo - Vref is either in quadrant 2 or 3. From this information we can determine the starting quadrant for which zero strain signal is located.
This is all assuming we know the value for Vref. Using our current configuration, in most cases we couldn't determine the true value for Vref since neither the signal data nor the calibration information could give enough information to determine Vmax or Vmin and hence Vref. One reason for this was the nature of the signal which didn't maintain a consistent peak to peak value. This produced another extreme complication that included discontinuities in the strain due to signal not going through a full excursion of the sine function (from peak to peak).
Since the shape of the sine function is the same for any quadrant (differing only by sign or mirror inversion), we can simplify the calculations by analyzing each particular quadrant as presented above. But to relate this relationships to the system used at Los Alamos we need to relate the displacement of the gap in degrees to a physical displacement of the system. From the discussed theory on Fabry-Perot cavities, we know that for one full sine wave cycle relates to the physical displacement of one-half the wavelength of the light used. We can define one full cycle (360 degrees) as a fringe, which is the excursion through four quadrants. For each excursion through one quadrant corresponds to a physical displacement of one-eighth the wavelength of the light used as represented below.
This lends itself to the use of the mod function for calculation of the physical displacement of the system. The only thing one has to consider is when the signal moves beyond quadrant 4. We find that the as the signal continues is simply cycles back through quadrants 1 through 4 with the added condition that we add one-half the wavelength of the light used to the absolute displacement calculated. After the signal transverses the eighth quadrant we must add an additional one-half wavelength of the light used and so forth.
The last complication for determining the displacement is when the signal "turns around", whereupon it retraces its steps back through the quadrant due to the gap getting smaller. The complication arises in one's attempt to identify a "turn around"!
Finally, we can determine strain from the functions presented above with the proviso of adding one-half wavelengths of the light used for further quadrants in the sequence as represented as Fi with i = 1, 2, 3, 4, ... With Fo being the starting quadrant which could be F1 through F4. The given signal is V and the gauge length is GL so that the strain is
- S(V) = [Fi(V) - Fo(V)] / GL
- S(V) = 1000 [Fi(V) - Fo(V)] / GL
Where it is often convenient to express S in terms of microstrains, GL in millimeters, and X in microns.
For data where the signal does not deflect much compared with Vmax and Vmin as compared with peak to peak calibration and where Vo is not too far from Vref, it is reasonable to believe the signal remains monotonic (oscillating between two adjacent quadrants, 1 and 4 or 2 and 3). Therefore, since the signal values for Vmax and Vmin are not known, Vref must be guessed at, where better choices of Vref are expected to result in displacements that sum to approximately zero when the signal has died away. Furthermore, in cases where we can compare data sets with different beam powers (as was the case with the data obtained from Los Alamos), we can expect to produce linear effects on strain; the result is the best choice for Vref should produce the closest scaling of strain to power!
Phosphor Temperature Sensor
The raw sensor data produced from the phosphor was in the form of time amplitude ordered pairs which were retrieved from the oscilloscope. Due to the use of a photomultiplier tub, which outputs negative signals, the waveform generated was inverted from quadrant I to quadrant IV. The signal produced had a initial offset from a zero bas line. By subtracting the offset and multiplying the result by -1, the signal was converted to a meaningful curve that could be analyzed on a logarithmic graph. By finding the maximum intensity of the curve and defining that as the maximum intensity, a plot of ln (intensity/maximum intensity) could be generated. The slope of the curve is equal to the negative of the inverse of the decay time. Using a calibration curve of the phosphor, a relationship for the temperature as a function of decay time can be determined.
The intensity relationship described the electromagnetic emission due to the excitation of the phosphor, which follows an exponential decreasing fluorescence. In the above relationships, I represents the emission intensity, I sub 0 is the maximum intensity, t is the time, and tau is the decay time.
After algebraic manipulation of the intensity relationship the above relationship can be achieved. By comparing this relationship with the equation of a line (y = mx + b) and since the observables are intensity, maximum intensity, and time, the slope can be used to determine the decay time.
After determining the slope from the graphs, the decay time could be calculated which results in ultimately determining the temperature. Prior calibration of Europium doped Lanthanum Oxysulfide Phosphor produced a relationship for decay time and temperature. Due to the amount of distortion in the signal, only part of the data was used in the analysis for determining slope. There is considerable distortion in the beginning of the exponential curve due to the lasing spike produced from the pulsed nitrogen UV laser. The end of the curve shows considerable noise. Therefore, only the middle of the curve was used. The first analysis was done with the median 40% while the second used the median 50% of the curve.
The following information details the calibration data, which was used to determine temperature (x) from the decay time (y). Unfortunately, absolute temperature cannot be easily determined due to a possible shift in the calibration curve for the phosphor used at Los Alamos. But the temperature change can be determined as observed with the 514nm wavelength emission.
High Sensitivity Diaphragm Pressure Sensor
The initial calibration data, which was used to determine optimum positioning of the fiber optic probe located in the cavity, assisted in determining distance away from the diaphragm the probe should be placed. The cavity pressure tests involved applying a range of pressure to the diaphragm and recording the displacement (or deflection) of the diaphragm. These tests were repeated with the fiber optic probe set to different initial distances away from the diaphragm. The data sets produced yielded calibration curves that one could use to interpolate the optimum range of pressures that system could measure with the fiber optic probe set at a particular distance with the specific diaphragm. The fiber optic probe was connected to a Fiberscan 2000, which measured the amount of displacement of the thick diaphragm. The Fiberscan 2000 was ideal for a static situation of sensor calibration but would be impractical to use for transient data acquisition environments where high response times would be required.
