*2.6. Roughness Model*

A surface roughness model is proposed to explain its causes and origins, based on the model suggested by Rubio-Mateos et al. [14].

In the experimental setup employed, the roughness in the samples can have three origins. Firstly, the theoretical average roughness (*Rh*) generated by the tool geometry and the feed per tooth. Secondly, the roughness generated as a consequence of the forced vibration and the axial displacement of the sample (*Rf*). The axial displacement can happen due to the deflection of the sample or to the relative displacement between sample and fixture. And thirdly, the roughness caused by chatter vibration (*Rc*). So, the global average roughness (*Ra*) can be defined as:

$$R\_{\mathfrak{a}} = R\_{\mathfrak{h}} + R\_{\mathfrak{f}} + R\_{\mathfrak{c}} \tag{2}$$

The floor theoretical average roughness is defined as:

$$R\_{\rm li} = \frac{f\_z^2}{32r} \tag{3}$$

If chatter-free conditions are guaranteed, roughness caused by chatter is zero, so the difference between measured average roughness and theoretical average roughness will be caused by the axial displacement of the sample. Also, this axial displacement can be considered constant for a given setup and cutting conditions, so the chatter roughness will be the roughness surplus generated at chatter conditions. Cross effects, as well as the axial displacement and the vibration of the tool, are neglected.
