The concept of grooves in PyRolL

The Generalized Groove

All elongation grooves can be traced back to a generalized elongation groove consisting of two straights and four radii. The geometry of this is shown below.

Geometry of Generalized Groove

All geometric key values like cross-sections and perimeters can be calculated on this generalized groove. The generalized groove is implemented in the GrooveBase class, all special groove types are derived from that.

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

\(r\)

Radius

\(\alpha\)

Angle corresponding to a radius

\(\beta\), \(\gamma\)

Angles useful for coordinate calculation

\(b_d\)

Ground width

\(b_d'\)

Even ground width

\(b_\mathrm{ks}\)

Tip width

\(b_\mathrm{kn}\)

Usable width

\(d\)

Depth

\(i\)

Indent

\(s\)

Roll gap

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

1

\(z_2 + r_1 \tan \frac{\alpha_2}{2}\)

\(0\)

2

\(\frac{b_{kn}}{2}\)

\(0\)

3

\(z_1 - r_1 \sin \alpha_1\)

\(r_1 \left( 1 - \cos \alpha_1 \right)\)

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

\(\frac{b_d'}{2}\)

\(d - i\)

8

\(\frac{b_d'}{2}\)

\(y_7 + r_4\)

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

\(z_1\)

\(r_1\)

However, in the current implementation the term “groove” is more narrow. From now on, the term should represent only the shape machined into the roll surface. Therefore, the roll gap \(s\) is no measure of the groove itself but of the RollPass. Also, the tip width \(b_\mathrm{kn}\) is not inherent to the groove, since it depends on the roll gap.

Box-like Grooves

The BoxGroove class

The BoxGroove class represents a rectangular shaped groove as shown in the figure. For wear reasons, the flanks a typically inclined by a small angle.

box groove geometry

Mandatory measures of the box groove are the two radii \(r_1\) and \(r_2\), as well as the depth \(d\). To constrain geometry fully, any two of the following must be given:

  • usable width \(b_\mathrm{kn}\)

  • ground width \(b_d\)

  • flank angle \(\alpha_1\)

So the constructor has the following signature:

BoxGroove(r1, r2, depth, usable_width, ground_width, flank_angle)

The radii are typically small, the depth is \(d\) typically \(\le \frac{b_\mathrm{kn}}{2}\).

\(r_3\) and \(r_4\) are considered to be zero.

\(b_d\) was chosen in favor of the even ground width \(b_d'\), because it does not change when the radii are modified. So the overall geometry remains the same if one modifies only the radii.

The ConstrictedBoxGroove class

The ConstrictedBoxGroove class represents a BoxGroove but with an indent in the ground as shown in the figure.

constricted box groove geometry

Mandatory measures of the box groove are the two radii \(r_1\) and \(r_2\), as well as the depth \(d\) and the indent \(i\). To constrain geometry fully, any two of the following must be given:

  • usable width \(b_\mathrm{kn}\)

  • ground width \(b_d\)

  • flank angle \(\alpha_1\)

So the constructor has the following signature:

ConstrictedBoxGroove(r1, r2, depth, indent, usable_width, ground_width, flank_angle)

The radii are typically small, the depth is \(d\) typically \(\le \frac{b_\mathrm{kn}}{2}\).

\(r_3\) and \(r_4\) are considered to be zero.

Diamond-like grooves

The DiamondGroove class

The DiamondGroove class represents a rhombus shaped groove as shown in the figure.

diamond groove geometry

Mandatory measures of this groove are the two radii \(r_1\) and \(r_2\). To constrain geometry fully, any two of the following must be given:

  • usable width \(b_\mathrm{kn}\)

  • tip depth \(d_\mathrm{t}\)

  • tip angle \(\delta\)

So the constructor has the following signature:

DiamondGroove(r1, r2, usable_width, tip_depth, tip_angle)

The radii are typically small, the depth is \(d_\mathrm{t}\) typically \(< \frac{b_\mathrm{kn}}{2}\) so that the tip angle \(\delta\) is larger than 90°.

\(r_3\) and \(r_4\) are considered to be zero, as well as \(b_d\) and \(b_d'\).

The tip depth \(d_\mathrm{t}\) was chosen in favor of the real depth \(d\), because it does not change, when the radii are modified. So the overall geometry remains the same if one modifies only the radii. The tip depth can be considered as the diagonal of the rhombus with sharp corners.

The SquareGroove class

The SquareGroove class represents a square shaped groove as shown in the figure.

square groove geometry

Mandatory measures of this groove are the two radii \(r_1\) and \(r_2\). To constrain geometry fully, any two of the following must be given:

  • usable width \(b_\mathrm{kn}\)

  • tip depth \(d_\mathrm{t}\)

  • tip angle \(\delta\)

So the constructor has the following signature:

SquareGroove(r1, r2, usable_width, tip_depth, tip_angle)

The radii are typically small, the depth is \(d_\mathrm{t}\) typically \(\approx \frac{b_\mathrm{kn}}{2}\). The tip angle \(\delta\) is typically a one or two degree larger than 90° for wear reasons.

\(r_3\) and \(r_4\) are considered to be zero, as well as \(b_d\) and \(b_d'\).

The tip depth \(d_\mathrm{t}\) was chosen in favor of the real depth \(d\), because it does not change, when the radii are modified. So the overall geometry remains the same if one modifies only the radii. The tip depth can be considered as the diagonal of the square with sharp corners.

The constructor will raise a warning, if the tip angle significantly deviates from 90°, consider to use a DiamondGroove instead.

Round-like Grooves

The RoundGroove class

The RoundGroove class represents a groove with a circular cross-section as shown in the figure.

round groove geometry

It is defined by two radii \(r_1\) and \(r_2\) and the depth \(d\), so the constructor has the following signature:

RoundGroove(r1, r2, depth)

The geometric constraints are \(r_1 << r_2\) and \(d < r_2\).

\(r_3\) and \(r_4\) are considered to be zero, as well as \(b_d\) and \(b_d'\).

The angles can be calculated as following:

\[ \alpha_1 = \alpha_2 = \arccos \left( 1 - \frac{d}{r_1 + r_2} \right) \]

The usable width is then:

\[ b_\mathrm{kn} = 2 \left( r_1 \sin \alpha_1 + r2 \sin \alpha_2 - r_1 \tan \frac{\alpha_1}{2} \right) \]

The FalseRoundGroove class

The FalseRoundGroove class represents a groove with a roughly circular cross-section, which shows a small straight flank, as shown in the figure.

false round groove geometry

It is defined by two radii \(r_1\) and \(r_2\), the depth \(d\) and the flank angle \(\alpha_1\) , so the constructor has the following signature:

FalseRoundGroove(r1, r2, depth, flank_angle)

The geometric constraints are \(r_1 << r_2\), \(d < r_2\) and \(\alpha_1 < 90°\) .

\(r_3\) and \(r_4\) are considered to be zero, as well as \(b_d\) and \(b_d'\).

The usable width can be calculated as:

\[ b_\mathrm{kn} = 2 \frac{d + \frac{r_2}{\cos \alpha_1} - r_2}{\tan \alpha_1} \]

Oval-like Grooves

The CircularOvalGroove class

The CircularOvalGroove class represents an oval shaped groove consisting of two radii as shown in the figure.

circular oval groove geometry

It is defined by two radii \(r_1\) and \(r_2\) and the depth \(d\), so the constructor has the following signature:

CircularOvalGroove(r1, r2, depth)

The geometric constraints are \(r_1 << r_2\) and \(d << r_2\).

\(r_3\) and \(r_4\) are considered to be zero, as well as \(b_d\) and \(b_d'\).

The topology of this groove is similar to the RoundGroove, with the main difference, that the center of \(r_2\) is not placed in the center of the groove. For this reason \(d\) is typically much smaller than \(`r_2` \).

The FlatOvalGroove class

The FlatOvalGroove class represents an oval shaped groove consisting of two radii and an even ground as shown in the figure.

flat oval groove geometry

Mandatory measures of this groove are the two radii \(r_1\) and \(r_2\), as well as the depth \(d\) and the usable width \(b_ \mathrm{kn}\).

So the constructor has the following signature:

FlatOvalGroove(r1, r2, depth, usable_width)

The depth is \(d\) typically \(\le \frac{b_\mathrm{kn}}{2}\).

\(r_3\) and \(r_4\) are considered to be zero.

The SwedishOvalGroove class

The SwedishOvalGroove class represents a hexagonal shaped groove as shown in the figure. The term “hexagonal” is also used for this type of groove, but can be confused with regular hexagon shaped grooves. The current type of groove is used as an oval and therefore the term swedish oval should be used, which is derived from its origin in swedish steel plants.

swedish oval groove geometry

Mandatory measures of this groove are the two radii \(r_1\) and \(r_2\), as well as the depth \(d\). To constrain geometry fully, any two of the following must be given:

  • usable width \(b_\mathrm{kn}\)

  • ground width \(b_d\)

  • flank angle \(\alpha_1\)

So the constructor has the following signature:

SwedishOvalGroove(r1, r2, depth, usable_width, ground_width, flank_angle)

The radii are typically small, the depth is \(d\) typically \(<< \frac{b_\mathrm{kn}}{2}\).

\(r_3\) and \(r_4\) are considered to be zero.

\(b_d\) was chosen in favor of the even ground width \(b_d'\), because it does not change when the radii are modified. So the overall geometry remains the same if one modifies only the radii.

The topology of this groove is similar to the BoxGroove, but typically the flank angles are smaller and the groove is less deep.

The ConstrictedSwedishOvalGroove class

The ConstrictedSwedishOvalGroove class represents a SwedishOvalGroove but with an indent in the ground as shown in the figure.

constricted swedish oval groove geometry

Mandatory measures of this groove are the two radii \(r_1\) and \(r_2\), as well as the depth \(d\) and the indent \(`i` \). To constrain geometry fully, any two of the following must be given:

  • usable width \(b_\mathrm{kn}\)

  • ground width \(b_d\)

  • flank angle \(\alpha_1\)

So the constructor has the following signature:

ConstrictedSwedishOvalGroove(r1, r2, depth, indent, usable_width, ground_width, flank_angle)

The radii are typically small, the depth is \(d\) typically \(<< \frac{b_\mathrm{kn}}{2}\).

\(r_3\) and \(r_4\) are considered to be zero.

The Oval3RadiiGroove class

The Oval3RadiiGroove class represents an oval shaped groove consisting of three radii as shown in the figure.

3 radii oval groove geometry

Mandatory measures of this groove are the three radii \(r_1\), \(r_2\) and \(r_3\), as well as the depth \(d\) and the usable width \(b_\mathrm{kn}\).

So the constructor has the following signature:

Oval3RadiiGroove(r1, r2, r3, depth, usable_width)

The depth is \(d\) typically \(\le \frac{b_\mathrm{kn}}{2}\).

\(r_4\) and \(b_d'\) are considered to be zero.

The Oval3RadiiFlankedGroove class

The Oval3RadiiFlankedGroove class represents an oval shaped groove consisting of three radii and a small straight flank as shown in the figure.

3 radii flanked oval groove geometry

Mandatory measures of this groove are the three radii \(r_1\), \(r_2\) and \(r_3\), as well as the depth \(d\), the usable width \(b_\mathrm{kn}\) and the flank angle \(\alpha_1\).

So the constructor has the following signature:

Oval3RadiiFlankedGroove(r1, r2, r3, depth, usable_width, flank_angle)

The depth is \(d\) typically \(\le \frac{b_\mathrm{kn}}{2}\).

\(r_4\) and \(b_d'\) are considered to be zero.

Reference of Groove Classes

class BoxGroove(r1: float, r2: float, depth: float, ground_width: Optional[float] = None, usable_width: Optional[float] = None, flank_angle: Optional[float] = None)

Represents a box shaped groove.

Exactly two of ground_width, usable_width and flank_angle must be given.

Parameters

  • r1 (float) – radius of the first edge

  • r2 (float) – radius of the second edge

  • depth (float) – depth of the groove

  • ground_width (float) – width of the groove from intersection between two flanks and ground width

  • usable_width (float) – ground width excluding influence of radii

  • flank_angle (float) – angle of the flanks

Raises

ValueError – if not exactly two of ground_width, usable_width and flank_angle are given

property types: 'box'

A tuple of keywords to specify the types of this groove.

class CircularOvalGroove(r1: float, r2: float, depth: float)

Represents an oval shaped groove with one main radius.

Parameters

  • r1 (float) – radius of the first edge

  • r2 (float) – radius of the second edge

  • depth (float) – depth of the groove

property types: 'oval', 'circular_oval'

A tuple of keywords to specify the types of this groove.

class ConstrictedBoxGroove(r1: float, r2: float, r4: float, depth: float, indent: float, ground_width: Optional[float] = None, usable_width: Optional[float] = None, flank_angle: Optional[float] = None)

Represents a box shaped groove with an indented ground.

Exactly two of ground_width, usable_width and flank_angle must be given.

Parameters

r1 – radius of the first edge :type r1: float :param r2: radius of the second edge :type r2: float :param depth: depth of the groove :type depth: float :param indent: indentation of the depth of the groove towards the grooves center :type indent: float :param ground_width: width of the groove from intersection between two flanks and ground width :type ground_width: float :param usable_width: ground width excluding influence of radii :type usable_width: float :param flank_angle: angle of the flanks :type flank_angle: float :raises ValueError: if not exactly two of ground_width, usable_width and flank_angle are given

property types: 'box', 'constricted_box'

A tuple of keywords to specify the types of this groove.

class ConstrictedSwedishOvalGroove(r1: float, r2: float, r4: float, depth: float, indent: float, ground_width: Optional[float] = None, usable_width: Optional[float] = None, flank_angle: Optional[float] = None)

Represents a hexagonal shaped groove with an indented ground that is used like an oval groove (swedish oval).

Exactly two of ground_width, usable_width and flank_angle must be given.

Parameters

  • r1 (float) – radius of the first edge

  • r2 (float) – radius of the second edge

  • depth (float) – depth of the groove

  • indent (float) – indentation of the depth of the groove towards the grooves center

  • ground_width (float) – width of the groove from intersection between two flanks and ground width

  • usable_width (float) – ground width excluding influence of radii

  • flank_angle (float) – angle of the flanks

Raises

ValueError – if not exactly two of ground_width, usable_width and flank_angle are given

property types: 'oval', 'swedish_oval', 'constricted_swedish_oval'

A tuple of keywords to specify the types of this groove.

class DiamondGroove(r1: float, r2: float, usable_width: Optional[float] = None, tip_depth: Optional[float] = None, tip_angle: Optional[float] = None)

Represent a diamond shaped groove.

Exactly two of usable_width, tip_depth and tip_angle must be given.

Parameters

  • r1 (float) – radius of the first edge

  • r2 (float) – radius of the second edge

  • usable_width (float) – ground width excluding influence of radii

  • tip_depth (float) – depth of the tip of the groove

  • tip_angle (float) – angle at witch the tip is formed

Raises

ValueError – if not exactly two of usable_width, tip_depth and tip_angle are given

property types: 'diamond'

A tuple of keywords to specify the types of this groove.

class FalseRoundGroove(r1: float, r2: float, depth: float, flank_angle: float)

Represents a round shaped groove with a dedicated flank (false round).

Parameters

  • r1 (float) – radius of the first edge

  • r2 (float) – radius of the second edge

  • depth (float) – depth of the groove

  • flank_angle (float) – angle of the flanks

property types: 'round', 'false_round'

A tuple of keywords to specify the types of this groove.

class FlatGroove(width: float)

Represents a box shaped groove.

Parameters

width (float) – width of the forming zone

property types: 'flat'

A tuple of keywords to specify the types of this groove.

class FlatOvalGroove(r1: float, r2: float, depth: float, usable_width: float)

Represent an oval shaped groove with a flat ground.

Parameters

  • r1 (float) – radius of the first edge

  • r2 (float) – radius of the second edge

  • depth (float) – depth of the groove

  • usable_width (float) – ground width excluding influence of radii

property types: 'oval', 'flat_oval'

A tuple of keywords to specify the types of this groove.

class GenericElongationGroove(usable_width: float = 0, depth: float = 0, r1: float = 0, r2: float = 0, r3: float = 0, r4: float = 0, even_ground_width: float = 0, indent: float = 0, alpha1: float = 0, alpha2: float = 0, alpha3: float = 0, alpha4: float = 0, types: Tuple[str, ...] = ())

Represents a groove defined by the generic elongation groove geometry.

property contour_line: LineString

A line representing the geometry of the groove contour.

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.

local_depth(z) → Union[float, ndarray]

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

property types: Tuple[str, ...]

A tuple of keywords to specify the types of this groove.

property usable_width: float

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

class GrooveBase

Abstract base class for all grooves.

abstract property contour_line: LineString

A line representing the geometry of the groove contour.

abstract property cross_section: Polygon

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

abstract property depth: float

The maximum depth of the groove.

abstract local_depth(z: Union[float, ndarray]) → Union[float, ndarray]

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

abstract property types: Tuple[str, ...]

A tuple of keywords to specify the types of this groove.

abstract property usable_width: float

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

class Oval3RadiiFlankedGroove(r1: float, r2: float, r3: float, depth: float, usable_width: float, flank_angle: float)

Represents an oval shaped groove with 3 main radii and a dedicated flank.

Exactly two of ground_width, usable_width and flank_angle must be given.

Parameters

  • r1 (float) – radius of the first edge

  • r2 (float) – radius of the second edge

  • depth (float) – depth of the groove

  • usable_width (float) – ground width excluding influence of radii

  • flank_angle (float) – angle of the flanks

Raises

ValueError – if not exactly two of ground_width, usable_width and flank_angle are given

property types: 'oval', 'oval_3_radii', 'oval_3_radii_flanked'

A tuple of keywords to specify the types of this groove.

class Oval3RadiiGroove(r1: float, r2: float, r3: float, depth: float, usable_width: float)

Represents an oval shaped groove with 3 main radii.

Parameters

  • r1 (float) – radius of the first edge

  • r2 (float) – radius of the second edge

  • r2 – radius of the third edge

  • depth (float) – depth of the groove

  • usable_width (float) – ground width excluding influence of radii

property types: 'oval', 'oval_3_radii'

A tuple of keywords to specify the types of this groove.

class RoundGroove(r1: float, r2: float, depth: float)

Represents a round shaped groove.

Parameters

  • r1 (float) – radius of the first edge

  • r2 (float) – radius of the second edge

  • depth (float) – depth of the groove

property types: 'round'

A tuple of keywords to specify the types of this groove.

class SplineGroove(contour_points: Union[Iterable[Tuple[float, float]], ndarray], types: Iterable[str], usable_width: Optional[float] = None)

Represents a groove defined by a linear spline contour.

Parameters

  • contour_points – an iterable of contour points to be used for the spline

  • types – an interable of string keys used as type classifiers

  • usable_width – the usable width to assume for this instance, if None, the maximum width will be used

property contour_line: LineString

A line representing the geometry of the groove contour.

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.

local_depth(z) → Union[float, ndarray]

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

property types: Tuple[str, ...]

A tuple of keywords to specify the types of this groove.

property usable_width: float

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

class SquareGroove(r1: float, r2: float, usable_width: Optional[float] = None, tip_depth: Optional[float] = None, tip_angle: Optional[float] = None)

Represents a square shaped groove (diamond with tip angle near 90°).

Exactly two of usable_width, tip_depth and tip_angle must be given.

Parameters

  • r1 (float) – radius of the first edge

  • r2 (float) – radius of the second edge

  • usable_width (float) – ground width excluding influence of radii

  • tip_depth (float) – depth of the tip of the groove

  • tip_angle (float) – angle at witch the tip is formed

Raises

  • ValueError – if not exactly two of usable_width, tip_depth and tip_angle are given

  • ValueError – if tip angle is <85° or >95° (no matter if given or calculated internally)

property types: 'diamond', 'square'

A tuple of keywords to specify the types of this groove.

class SwedishOvalGroove(r1: float, r2: float, depth: float, ground_width: Optional[float] = None, usable_width: Optional[float] = None, flank_angle: Optional[float] = None)

Represents a hexagonal shaped groove that is used like an oval groove (swedish oval).

Exactly two of ground_width, usable_width and flank_angle must be given.

Parameters

  • r1 (float) – radius of the first edge

  • r2 (float) – radius of the second edge

  • depth (float) – depth of the groove

  • ground_width (float) – width of the groove from intersection between two flanks and ground width

  • usable_width (float) – ground width excluding influence of radii

  • flank_angle (float) – angle of the flanks

Raises

ValueError – if not exactly two of ground_width, usable_width and flank_angle are given

property types: 'oval', 'swedish_oval'

A tuple of keywords to specify the types of this groove.