ADDITIVE MANUFACTURING PROCESSES QUALITY MANAGEMENT
3 EVALUATION OF AN OPTIMIZED AREA DENSITY GAUGE
3.1 Calibration and data evaluation
The first rethinking of area density gauges was done by [14]. He enhanced the measuring accuracy carrying out the device calibration by the fiber mat instead of the ready pressed panel, common so far. According [13]
calibration and data evaluation mode has to meet the requirements given by material and radiation properties.
Considering these experiences and common approaches, cf. [21], for radiometric area density determination, an optimized X-ray measuring device was developed prior to the measurements of the following section.
This multiple energy system allows the choice of X-ray energy range and spectrum (automatically) according to area density range. The respective calibration data and data evaluation algorithms are stored in the gauge controller. The calibration data were acquired by stacks of fixed thin furnishes mats with predefined area densities and raw densities.
For the investigations below, the following setup was applied:
X-ray tube (up to 65 kV, 40 W) with tungsten target,
radiation pre-filters on demand,
one of the most common forming belts,
Scintillator with photomultiplier as detector.
3.2 Measuring signal considerations
To determine the achievable accuracy and measurable resolution, i. e. the distinguishable area density, the confidence limits of the measurement have to be computed. Therefore, the spread of the measuring signals (count rates, i. e. counts per period cpp) was evaluated. The measurements were done over the whole range of the multiple energy system with an integration time of 3000 ms. Data from <25 kV tube power level is exemplary presented.
With this energy preset and the described setup, a narrow X-ray spectrum with peak energy of 13 keV (measured) is obtained. Thus, area density measurements in the range ρA = 3…5 kg/m² are feasible according equation (4).
After system warm-up, energy adjustment and an appropriate waiting time, measurements of the initial I0 and transmitted I intensities were carried out. A specimen, cut out of a homogeneous lab panel (no raw density profile, fibers, UF resin), with ρA = 4.5 kg/m² and ρ = 400 kg/m³ was used for defined attenuation of the radiation. The count rates were read from the gauge controller in sequence (n = 30 each). Figure 3 shows the measured count rates and their expected Gaussian distribution of transmitted and initial signal. The summarized data are given in Table 3.
A χ²-test under expectation of Gaussian distribution with the same parameters as the measured data showed that the condition is fulfilled in case of I. For I0 the hypothesis has to be rejected. But the result is quite close to the critical value and with more measurement iterations the expectation might be proved. Thus, a Gaussian distribution with the observed parameters in Table 3 is assumed for all measuring signals.
Figure 3: Histogram and expected Gaussian distribution of the count rates of transmitted and initial intensities (n = 30) on <25 kV level.
Table 3: Summarized measuring data of the count rates of transmitted and
A Gaussian shaped distribution of the measured count rates within a certain range was observed, whereas the occurrence of limit values is highly unlikely. Thus, a worthy statistical approach, generally used for high sophisticated tolerance analysis, is applied as follows.
Basically, it is just a matter of mathematically precise superimposing the statistical distributions of the count rates of I0 and I to compute the resulting spread of the measuring result and its statistical distribution. These are typical considerations of statistical tolerance analysis and the subsequent convolution of distribution functions. This method investigates according [23]
interacting single tolerances regarding the resulting function of the considered assembly and can be applied to other measured variables, cf.
[24]. The approach is adapted by [25] to take swelling and shrinkage explicitly into account of wood related tolerance calculations, as well.
Therefore, not maximum and minimum values are added (no worst-case limit consideration) but the assumed distributions of observed data are taken into account. The comprehensive computation is carried out computer aided with input data as histogram or probability density function and the numerical solution of the convolution integral. Furthermore, a simplified calculation assuming Gaussian distributed results exists by means of the Gaussian failure propagation law
Hereafter, this approach is adapted to combine the distributions of measured count rates and to compute the subsequent spread as range of area density.
However, it is also common within metrology (cf. [26]) to evaluate measuring failures occurring from several influence factors by failure propagation calculations.
The functional context f, i. e. the measuring signal, for the following statistical considerations is defined as the logarithmic ratio of the intensities
in terms of equation (3) and furthermore evaluated according the very same by means of known µ/ρ. Hence, the partial derivations are
0 spread as range of area density considering a probability of 99.73 % (±3s of the estimated Gaussian result distribution) can be computed as ρA = (4.5 ± 0.0094) kg/m². More general, the fluctuations of the count rates on the observed energy level lead to variations of the measuring results in the
±3s range equals ±9.4 g/m² or ±0.208 %.
A worst-case computation considering the ±3s range of the measured intensities (Table 3) leads to a respective range of the measuring result equals ±13.3 g/m² or ±0.295 %. Thus, in comparison to the statistical results a reduction of the range of 29 % by the applied statistical tolerance analysis approach can be obtained.
3.4 Practice oriented implications
The above computed limits represent the uncertainty of the area density measurement on <25 kV energy level with the mentioned setup. Figure 4 shows a plot of the assumed Gaussian result distribution of the area density and makes the difference between statistical and worst-case results (ranges) more obvious. Hence, the occurrence of limit values is highly unlikely and a statistical consideration is the subsequent method. Furthermore, if the certainty is reduced to, e. g, a still acceptable value of 95.44 % (±2s), the range of the measurement uncertainty can be decreased to ±6.3 g/m² or
±0.139 %, which is less than half of the worst-case limits.
Figure 4: Plot of the assumed Gaussian result distribution of the area density, with mean and ranges (limit deviations), determined under
statistical (99.73 % probability) and worst case considerations.
The resolution of the measurement result could theoretically be increased to infinity by extending the number of digits of the displayed area density, because the value is computed out of the measuring signal via the stored calibration function. Nevertheless, the reliable distinguishability between area density values lies above the determined measurement uncertainty.
Thus, a feasible resolution is subsequently a multiple of the theoretically determined limit deviation. Below this value, a displayed change can only be due to chance.
The real measuring sensitivity should rather be determined via experiment, which is nontrivial in practical realization and includes undefined uncertainty due to the inhomogeneity of the measured furnish material. Hence, the sensitivity and the subsequent resolution of an X-ray area density measuring device have to be derived from the explained theoretical basis. However, the achievable measuring accuracy, which depends on appropriate device setup and energy selection, is rather of practice oriented interest. Furthermore, the above exemplary determined values can be enhanced by increasing integration time of the measurement, i. e. traversing frequency of the device.
On the other hand, the measuring velocity can be accelerated accepting a slightly lower accuracy.
4 CONCLUSION
4.1 Needs
The forming machines and their preceding supplying devices are responsible for accurate metering and uniform spread of furnish on the forming belt. Further importance of mat forming is pointed out by [27].
Requirements on mat forming are given by [28], according to which a capable process should lead to a homogeneous material distribution with CV < 4 %, i. e. for a 4.5 kg/m² mat a range of ±0.36 kg/m² or ±8 % (±2s, 95.44 % certainty) and ±0.54 kg/m² or ±12 % (±3s, 99.73 % certainty). A CV > 6…7 % represents a forming process out of control. Similar values are given by [29]. At this time (1995), ±4 % deviations in transverse area density profile caused by mat forming were guaranteed. Additionally, deviations along production feed direction occur. He [29] further points out, that mat forming differences lead to effective loss in material. The context has not changed, but reliable values for achievable accuracy are lacking currently.
Only unofficial information from machinery and equipment constructors for mat forming accuracy is known in the range from
thickness
between 3.8 %...5.3 % could be achieved (but unknown certainty range).Requirements on process control metrology are already given by [30], where area density measuring devices for particle board production should be contactless and continuous with an accuracy of ±1 % (unknown certainty as well). Furthermore, there are still conventional electromechanical mat or panel weighing systems in use, occasionally of old manufacturing date.
These devices apply huge scale platforms directly within the forming machines and behind the pre-press (both furnish mat) as well as behind the hot press (finished panel). Their considerable disadvantages are:
high tare weight compared to weighed portion (especially thin panels),
high feed rates cause increasing vibrations,
only mean area density of the whole panel (no distribution).
The accuracy of such electromechanical scales is reported by [27] to be better than ±2.27 kg for the scale itself resulting in mat weights control of about ±6.82 kg. E. g, a 4.5 kg/m² panel with typical full size dimensions of 5.6 × 2.07 m² has a weight of 52.2 kg. Thus, the relative scale accuracy is in the range of ±13 % for such a light panel. In comparison, [27] quantifies also the accuracy and resolution of nuclear gauges at that time with ±54 g/m² (mean 2.44 kg/m²), i. e. ±2.2 %, which seems doubtful from the present point of knowledge.
As contributed, optimized measuring devices are able to provide process control with data of high accuracy and reliability, because their measuring uncertainty is more than a number of magnitudes lower than the achievable