Fillets are commonly used in mechanical engineering in order to avoid singularities and then stress concentrations.
The goal is here to “put down in writing” methods I’ve been using to analytically calculate fillets; these are one solution among others (and not necessary the best ones or the simplest ones), but I’m good with them and they answer to my needs (if some of you want to share other methods, I’ll be pleased to devote a specific section and the author will be cited).
There are different types of fillets; the simplest of them is the arc of a circle that answers to a practical need, the use of a ball-nose cutter. In the meantime I’ve discovered other methods for curves transition, notably used by topographers in the design of road networks or railway ones for example (it uses clothoids or Cornu’s spirals – the later ones are also called Euler’s spirals especially in the English speaking world).
1- Fillets using an arc of a circle
1-1 The basis
The figure here after presents a fillet in arc of a circle.
It be characterized by 2 straight lines and , by its radius , by its centre and by the points of tangency et we have to calculate.
1-2 The principle
The centre of the arc of a circle is necessarily equidistant from the 2 lines and : it is mandatorily located on the internal bisector line of
Since the arc of a circle is tangent to the lines in point and one, then and are right angles; knowing that is the intersecting point of the 2 lines then and form square triangles where the hypotenuse is none other than : the points , , and are contained in a circle where diameter is and the centre is middle of (see figure bellow).
1-3 Initial check
Previously we verify than the 3 points , , and are neither coincident nor aligned, and that they belong to the same plane (2D).
- coincident points: and (assuming that is the intersecting point of the 2 lines here),
- aligned points: the vector product gives
1-3 Analytical development (general case)
Let us focusing first on the general case, I mean the one where is not the intersecting point of the 2 lines (it’ll be just necessary to simplify after):
(Warning: for the general case, point labeling has been slightly modified compared to the previous schemes)
We need to calculate:
- the linear equation of , and ,
- the intersecting point coordinates,
- the linear equation of the bisector line to / ,
- the coordinates of the point, the centre the arc of a circle/fillet,
- the coordinates of the point, middle of segment,
- the circle equation, its centre and its radius ,
- the intersecting points between and where are the ends of the arc of a circle ().
Let , , and .
Let the point so that ; writing and , then :
Linear equation calculation of line:
Calculation of the internal bisector line equation:
The equation system describing the triangle is then :
The equation system is rewritten:
Where refers to the sign so that
Writing , then if
Ditto for so that
Equation (5) is rewritten as:
To facilitate reading, let and , develop and then
Calculation of the origin of the arc of a circle:
Knowing that the arc of a circle origin is on the bisector line i.e. at a distance equals to of (or ), then:
Then the system becomes:
NB 1: the sign of depends on the one of :
- remember that if , i.e. the points are aligned.
NB2 : calculation of
- multiplying by , it becomes
- knowing that , then the previous equation is rewritten
Arc of a circle ends calculation – linking points with the line and one:
We calculate the coordinates of the tangent points between the arc of a circle and the 2 lines: and .
First point :
By incorporating in the circle equation and by developing it leads to:
Tangency condition, in a single point implies that the equation has a unique root (); the solution is like .
Second point :
We proceed in the same way for :
In a general way, the 2 ends of the arc of a circle have the following form:
We can already notice a specific case i.e. when 2 points are vertical (indeed variables become equal to zero whereas it can be met on the denominator). It’ll be taken into account in a next section.
1-4 Case where , and then are coincident
When the lines and are crossing in , point calculation becomes useless; only the , and () constant values are affected:
The rest remains unchanged.
1-5 Specific case of 2 vertical points
Previous equations remain unchanged up to point calculation (including the later one). Only point calculation requires a specific treatment.
Let us consider the case of the previous figure; constants are expressed as:
The linear equations system becomes:
Basically for we can note that is has the same abscissa than the point (or ), and the ordinate of the point .
point calculation remains unchanged compared to the previous cases.
In a same way for the following example: the calculation of the point remains unchanged and the one of becomes:
1-6 Last specific case : all the lines are horizontal and vertical and form a right angle
This case is the simplest one; it’s not necessary the use the above equations.
Curves were plotted using geogebra; note that:
- the blue points were manually implemented,
- the grey points were calculated by geogebra and they are the result of the intersection of circle (centre and radius