Examining the Relationship between Forces During Stereolithography 3 D Printing and Geometric Parameters of the Model

In the case of stereolithography 3D printing technology, detaching formed model from the tank with photopolymer is a lengthy process. Forces, which appear during removing of solid photopolymer layerformed in stereolithography 3D DLP printer, can destroy the built model. In this article the detachment force is measured, obtained results arestatistically analyzed and relation between detach force, area of produced layer and thickness of the layer are verified. Linear dependence between detach force and built area is determined. On the other hand, relation between detach force and thickness of the layer is not confirmed.


Introduction
One of the main areas in 3D printing research nowadays is increasing the production speed of a model.In case of stereolithography, one of the most time-consuming processes is detaching of solid layer from the bottom of the tank.Some methods allow preventingstrongadhesion to the tank [1].But mostly in classical stereolithography, its necessary remove the last layer of building material from the tank.This process is likely to causedamage to the whole model.Continuous measurement of this force brings the possibility of acceleration of detaching process.This force can be kept below the maximal level of breaking point to prevent model from damage.

Stereolithography
Stereolithography (SLA, SL) is a technology based on curing of photopolymer resin layer by layer using optical radiation in ultraviolet wavelength range.It was patented in 1986 by Charles (Chuck) W. Hull [2].In our case UV power source is Digital Light Processing (DLP) projector.DLP 3D printing technology descends from the name of technology that was patented by Texas Instrument Inc. in 1986 by Dr. Larry Hornbeck [3].The model of experimental printer is shown on Figure 1.

Description of the experiment
Experiment was made onthe DLP 3D printer, which is shown on Figure 2.This device was designed for testing of new detachment methods.Data projector is placed under the tank with photopolymer and illuminate bottom side of the tank.This solution allows economizing of building material, but on the other hand makes detaching process more complicated.A force sensor strain gagewas used.It was placed between platform brackets and building platform.Assembled experimental platform is shown on Figure 3.

Measurement results
Data for statistical treatment were obtained from three models with different layer thickness.Results of measurements are shown in the Table 1-3.Also the primary statistical processing is represented.After the first analysis of obtained data we noticed, that first three values of take apart force in all models are significantly higher than the rest of values.These deviations came from the adhesion of platform to the bottom of the tank.These values can distort results of statistical treatment, that's why we discarded them from samples before processing.
For further analysis, we have to know, if data in measured samples are normally distributed.For this purpose, arithmetical means and medians in each sample are calculated (see Table 1-3).The relative differences between these values (see last rows in the Table 1-3) are calculated for each measured sample using this formula [4]: where ‫ݕ‬ ത is arithmetical mean, ‫ݕ‬ ො is median.

ICMES 2015
For normally distributed data next equation effects: For making an assumption, that data are normally distributed, relative difference between median and mean value has to be less than 10 %.As we can see, all values of relative differences satisfy this condition.Thus, we accept normal distribution for measured data.

Statistical processing
The goal of mentioned measurement was to check, if there is any linear dependence between obtained values: • Detach force vs building area (Table 4), • Detach force vs layer thickness (Table 5-6).
For this purpose, correlation coefficients between mentioned sets of data are calculated.Since normality of measured samples was approved, we can use Pearson's correlation coefficient for testing of dependency between them.Sample Pearson's correlation coefficient can be found as follows [5]: where ܵ ‫ݔ‬ is sample standard deviation of x, ܵ ‫ݕ‬ is sample standard deviation of y, ‫ܥ‬ ‫ݕ,ݔ‬ is covariance of variables x and y. (5) For making a conclusion about linear dependence we should verify significance of correlation coefficientfor each couple of obtained samples.Test statistics looks as follows: This value is compared to quantile of Student's tdistribution with degrees of freedom (n-2).If following condition is fulfilled, then correlation coefficient is significant and linear dependence between investigated values can be stated: where ߙ = 0.05 (significance level).
Since condition (8) is fulfilled for all samples in Table 5, linear dependence between detach force and built area is confirmed.Thus, linear regression model can be applied on obtained data.
Using the least-squares method, parameters of approximating line can be found as follows [6]:

Conclusion
The first task of this research was to check linear dependence between detach force and built area (Table 4).This dependence is confirmed and appropriate linear regression models are derived.
The second task was to check linear dependence between detach force and layer thickness (Table 5-6).Condition is fulfilled just for the last sample, linear dependence is not approved for rest of the samples.But since most of values of Pearson's correlation coefficient are very close to unity, we can make a conclusion, that measured samples do not contain enough measurements for approving the linear dependence.

Figure 1 .
Figure 1.The functional diagram of DLP 3D printer.

Figure 2 .
Figure 2. Experimental 3D DLP printer.Platform can move in Z axe and has a minimal movement about 10 µm.It allows making thin layers from solidify resin.A force sensor strain gagewas used.It was placed between platform brackets and building platform.Assembled experimental platform is shown on Figure3.

Figure 3 .
Figure 3.Building platform with strain gage sensor.The forces are measured for the pyramidal CAD model, which is shown on Figure4.It has different area of the stairs which are equal to 2500 mm 2 , 1875 mm 2 , 1250 mm 2 and 625 mm 2 .

Figure 4 .
Figure 4.The printed model.This model was sliced to layers with different thickness 25, 50 and 100 µm.Detaching forcesare measured ten times in the vertical directionfor each built area.
which fit obtained data are shown on the Figure 5.

Table 1 .
Measurement of detach force with constant layer thickness of 25 µm, [N].

Table 2 .
Measurement of detach force with constant layer thickness of 50 µm, [N].

Table 3 .
Measurement of detach force with constant layer thickness of 100µm, [N].