# Efficiency of Machine Sanding of Wood
## 4 Discussion
The spread of sanding efficiency values for softwood is much greater than in hardwood species. This is due to the difference in tribological properties of softwood and hardwood. During sanding, what is important is not only the density but also the specific physical and mechanical properties of individual wood species and their morphologies themselves (hardwood ring-porous, hardwood scattered porous, with resin content or without, chemical composition, etc.). The resin content of the softwood has a great influence on the tribological properties of wood and it may be a cause of quickly blunting of the sanding tool [27]. The mean values of the measurements were statistically analyzed. Confidence intervals were calculated using a t-distribution table (\(\alpha=0.90\)). It was found that the measurement uncertainty of the means was always greater for softwood and its maximum value was independent of the sandpaper gradation. This uncertainty for hardwood was, respectively: 12.9% (P60), 13.2% (P120), 12.5% (P180). For softwood, it was: 19.1% (P60), 21.6% (P120) and 29.1% (P180). The main hypothesis of our study is that hardwood differs from softwood in terms of the efficiency of the sanding process. For this purpose, changes in average weight loss were compared in the function of sanding time (Figures 9-11). The equations for these abrasive belts were found. Then, to adjust and compare the course of the functions, the set theory approach was used to narrow down the functions. In the first stage, for the functions to be considered equal, they must satisfy the first condition of the equality of the functions, which says that the functions \(f_{1}\left(x\right)\) and \(f_{2}\left(x\right)\) are equal to each other if, and only if, they have the same domains and for each point of the common domain, they assume these are the same values \(f_{1}=f_{2}\leftrightarrow D_{f1}=D_{f2}\) and for each \(x\in D_{f1}=D_{f2}\), and \(f_{1}\,=f_{2}\)[24]. For all three grit sizes of sanding belts P60, P120, and P180, and for both types of wood (hardwood and softwood), the domains of functions being a square function with the general formula were calculated as \(y=ax^{2}+bx+c\) using the procedure: (1) calculation of the root of a function \(\Delta\) (\(\Delta=b^{2}-4 ac\)); (2) determination of parameters \(p\,\mathrm{i}\,q\) (\(p=\frac{-\sqrt{\Delta}}{2a}\), \(q=\frac{-\Delta}{4a}\)). In the case of sanding belt P60, significant differences were found between the examined functions because their domains assumed values \(D_{f1}(-5,\,+\infty)\) and \(D_{f2}(-1,\,+\infty)\), thus \(D_{f1}\
eq D_{f2}\), so the functions are not equal. Then the second condition for the definition of function equality was checked, indicating that for each \(x\in D:\,\,f_{1}(x)=f_{2}(x)\)[28]. These functions do not satisfy the equality condition and are therefore different, which  
Figure 11: Comparison of average results for hardwoods (beech, oak, ash, hornbeam, alder) and softwoods (pine, spruce, larch) species; P180 sandpaper. indicates that hardwood differs from softwood in terms of the efficiency of the sanding process for the grit size P60. For the sanding belt P120, it was observed that the first condition of the equality of the function was not satisfied (\(D_{f1}(-1,\ +\infty)\) and \(D_{f2}(-0,1,\ +\infty)\)). However, the second condition assuming that for every \(x\in D:\ f_{1}(x)=f_{2}(x)\) was satisfied. Thus, a restriction was applied for functions on the selected set of points belonging to the set A (\(-89.4\); \(-100\)) where for every \(x\in A\), the functions are equal : \(\ f_{1}(x)=f_{2}(x)\)[29]. Based on the calculations related only to this set of arguments, it was found that both functions are similar to each other in the indicated range, and their domains are the same for this set of arguments. A similar situation was observed for the sanding belt P180. Using the second condition of function equality, set A was determined (\(-81.7\); \(-100\)) for which these functions are equal.