The Generic Elongation Groove

The classic elongation groove shapes can be derived from a generic elongation groove consisting of two straights and four radii as shown below. The actual groove types are special cases of this geometry, where some measures are unequal and some are equal zero.

Geometry of the Generic Elongation Groove.

All geometric key values like cross-sections and perimeters can be calculated on this generic groove. The generic groove is implemented in the GenericElongationGroove class, all classic elongation groove types are derived from that. The constructor of this class expects most of the geometric measures to be given. The calculation of missing measures given others can be a complicated task depending on the segments used, and is therefore implemented in the actual derived groove classes taking advantage of their features and conditions. The GenericElongationGroove class is generally not intended to be used directly by the user, but may be, if one is able to provide all necessary measures, for example from a given drawing.

In the following the measures of the groove are listed, their names are used in source code and throughout the documentation. The radii and angles are numbered from outside to inside.

Symbol	Description
\(r_i\)	Radius
\(\alpha_i\)	Angle corresponding to a radius
\(\alpha\)	Groove flank angle
\(\beta\), \(\gamma\)	Angles useful for coordinate calculation
\(\phi\)	Roll face angle
\(w_\mathrm{g}\)	Ground width
\(w_\mathrm{g}'\)	Even ground width
\(w_\mathrm{t}\)	Tip width
\(w_\mathrm{u}\)	Usable width
\(d\)	Depth
\(i\)	Indent
\(s\)	Roll gap
\(p\)	Roll face padding

For grooves with inclined roll faces (like such for 3 and 4-roll passes), the geometry is slightly different like in the figure below. Note the face line inclined by \(\phi\) and the modified radius \(r_1\). The case shown above is the limit case for \(\phi \rightarrow 0\). If the actual groove has no straight flank line, the resulting geometry of the groove can be heavily influenced by changing the face angle \(\phi\) or the radius \(r_1\).

Geometry of the Generic Elongation Groove for 3-Fold Rolls.

The coordinates of the points 1 to 12 shown in the figure can be calculated as follows, where the angles \(\beta = \alpha_4 - \alpha_3 / 2\) and \(\gamma = \frac{\pi}{2} - \alpha_2 - \alpha_3 + \alpha_4\).

number	z	y
0	\(z_1 + p \cos \phi\)	\(z_1 + p \sin \phi\)
1	\(z_2 + r_1 \tan \frac{\alpha_1}{2} \cos \phi\)	\(r_1 \tan \frac{\alpha_1}{2} \sin \phi\)
2	\(\frac{w_\mathrm{u}}{2}\)	\(0\)
3	\(z_{12} - r_1 \sin \alpha\)	\(y_{12} - r_1 \cos \alpha\)
4	\(z_{11} + r_2 \cos \gamma\)	\(y_{11} + r_2 \sin \gamma\)
5	\(z_{10} + r_3 \sin \left( \frac{\alpha_3}{2} - \beta \right)\)	\(y_{10} + r_3 \cos \frac{\alpha_3}{2}\)
6	\(z_8 + r_4 \sin \alpha_4\)	\(y_8 - r_4 \sin \alpha_4\)
7	\(\frac{w_\mathrm{g}'}{2}\)	\(d - i\)
8	\(\frac{w_\mathrm{g}'}{2}\)	\(y_7 + r_4\)
9	\(0\)	\(y_7\)
10	\(z_6 + r_3 \sin \left( \frac{\alpha_3}{2} + \beta \right)\)	\(y_6 + r_3 \cos \left( \frac{\alpha_3}{2} + \beta \right)\)
11	\(z_{10} + \left( r_3 - r_2 \right) \sin \left( \frac{\alpha_3}{2} - \beta \right)\)	\(y_{10} + \left( r_3 - r_2 \right) \cos \left( \frac{\alpha_3}{2} - \beta \right)\)
12	\(z_1 - r_1 \sin \phi\)	\(y_1 + r_1 \cos \phi\)

The line between points 1 and 0 is the roll face and is in the narrower sense no part of the groove. However, it is included in the grooves contour line for numerical reasons. The extent of this roll face padding is determined by \(p\) and can be given to the constructor by the pad and rel_pad arguments, where pad represents the absolute value and rel_pad the padding relative to the usable width \(w_\mathrm{u}\) of the groove. The default value of rel_pad is given by the pyroll.core.config.GROOVE_PADDING config value. Also, the roll gap \(s\) is no measure of the groove itself but of the roll pass. The tip width \(w_\mathrm{t}\) is consecutively also not inherent to the groove, since it depends on the roll gap.

class GenericElongationGroove(r1: float, r2: float, flank_angle: float | None = None, usable_width: float | None = None, ground_width: float | None = None, depth: float | None = None, r3: float = 0, alpha3: float = 0, r4: float = 0, alpha4: float = 0, indent: float = 0, even_ground_width: float = 0, pad: float = 0, rel_pad: float = 0.2, pad_angle: float = 0, classifiers: Sequence[str] = ())

Represents a groove defined by the generic elongation groove geometry.

Give any three of usable_width, ground_width, flank_angle and depth. All angles are measured in radians. All measures must be non-negative.

Parameters:

• r1 – radius 1 (face/flank)

• r2 – radius 2 (flank/radius 3)

• usable_width – width of flank/face intersections

• ground_width – width of flank/ground-line intersections

• flank_angle – angle of the flanks to the z-axis

• depth – maximum depth

• r3 – radius 3 (radius 2/radius 4)

• r4 – radius 4 (radius 3/ground)

• alpha3 – angle corresponding to r3

• alpha4 – angle corresponding to r4

• indent – indent of the ground

• pad – absolute padding at roll face

• rel_pad – padding at roll face relative to usable_width

• pad_angle – angle of the face padding from horizontal line (commonly 0 for two-roll, π/6 for three-roll and π/4 for four-roll)

• classifiers – sequence of additional type classifiers

local_depth(z) → float | ndarray

Function of the local groove depth in dependence on the z-coordinate.

property classifiers

A tuple of keywords to specify the type classifiers of this groove.

property contour_line: LineString

A line representing the geometry of the groove contour.

property contour_points

An array of the contour line’s points of shape (n, 2).

property cross_section: Polygon

A polygon representing the cross-section of this groove limited by the contour line and y=0.

property depth: float

The maximum depth of the groove.

property usable_width: float

The usable width of the groove, meaning the width of ideal filling.

property width: float

The maximum width of the groove representing the definition region in z-direction.