Alex Thorn, Delphi Diesel Systems
alexthornpictonhall@gmail.com
The Application of Planar Geometry to Measure Armature Angle
Figure 1
Introduction
For my YinI placement, I am working in the Development
department at Delphi Diesel Systems. Delphi designs,
develops and manufactures diesel fuel injection
equipment typically for truck engines. In electronically
controlled injectors, a solenoid is placed above an
armature, and when a current flows through the
solenoid, the armature is magnetically attracted. The
movement of the armature and pin is triggers the
injector to inject fuel.
One property of the armature that hugely affects its
performance is how parallel it is to the solenoid. Consequently, it is essential to check that the
armatures are being assembled to the correct specification and are as parallel as possible. It is not
possible to measure the angle between the armature and the solenoid directly in an assembled
injector. Instead, we measure the angle between the armature and the guide when the injector is
disassembled.
Exaggerated Diagrams:
In figure 2.1, armature and pin are
perfectly square to the guide as desired.
In figure 2.2, armature and pin are not
square to the guide and are offset by θo.
This is not desirable.
θ
Figure 2.1
Figure 2.2
Existing Measuring Technique:
Before my project, Delphi measured the angle using a 3D optical scanning
microscope. This method was very time consuming, and the number of machines
was limited. Delphi therefore wanted to make this process quicker for production.
New Measuring Technique:
In the new measuring technique, six LVDTs (Linear variable differential
transformers- precise distance measuring probes) are placed on to the
component as seen in Figure 3. Three LVDTs will form an equilateral triangle and
measure the position of the armature, and three will measure the position of
the guide. I used these six measurements to mathematically calculate the angle
between the armature plane and the guide plane.
Figure 4: Orthographic projection
exaggerating the offset angle.
Solution:
Figure 3
To find the angle between the armature
plane and the Guide plane, I decided to:
1. Find two vectors on each of the planes.
2. Use the two vectors to find a perpendicular vector for each
plane.
3. Find the angle between the perpendicular vectors of both
planes.
j
k
i
Alex Thorn, Delphi Diesel Systems
alexthornpictonhall@gmail.com
A
1. Finding two Vectors on the Guide and the Armature
Plane
21.74mm
E
D
I labelled each probe A to F, and let the distance that
probes measure be their lowercase letter. So probe A will
measure a mm. The probes for each plane had been placed
60o apart, hence forming an equilateral triangle. The probes
measuring the guide were placed 30.14mm apart, and the
probes measuring the armature were placed 21.74mm
apart. I set the origin to a position in the centre of the
armature and then used trigonometry in an Excel spread
sheet to calculate the position vectors of the probes.
. Origin
(O)
C
B
F
30.14mm
Position Vectors of Probes
i/mm
j/mm
k/mm
OA 0
17.40
a
OB 15.07
-8.70
b
OC -15.07 -8.70
c
OD -10.87 6.28
d
OE 10.87
6.28
e
OF 0
-12.55 f
I then found two vectors lying
on each plane: AB, AC, DE &DF
using the simple equation:
AB=OB-OA
Again, I used Excel to speed up
the process.
Two Vectors Lying on Each Plane
i/mm
j/mm
k/mm
AB
15.07
-26.10
b-a
AC
-15.07
-26.10
c-a
DE
21.74
0.00
e-d
DF
10.87
-18.83
f-d
2. Find a perpendicular vector for each plane
Next, I used these vectors to calculate a perpendicular vector for each plane using the cross product.
Let m be a vector perpendicular to the guide plane.
m= AB x AC=
i
15.07
-15.07
j
-26.10
-26.10
k
b-a
c-a
=
26.10(b-c)
15.07(2a-b-c)
-786.71
To find the cross product
quicker, I used 3x3 matrix
determinant calculation as
indicated by enclosing the
matrix in straight brackets.
Let n be a vector perpendicular to the armature plane.
n=DE x DF =
i
21.74
10.87
j
0.00
-18.83
k
e-d
f-d
18.83(e-d)
= 10.87(d+e-2f)
-409.31
3. Finding the angle between these perpendicular vectors
Finally I found the angle (θ) between the two planes is using the dot product:
| || |
It is at this point, when I substitute in the data that has been measured using the LVDTs:
Probe
a
b
c
d
e
f
Distance/mm
0.0016
0.0023
0.0012
0.0739
0.0729
0.0616
Therefore:
n=
-0.0188
0.2565
-409.3075
m=
√
0.0287
-0.0045
-786.7145
√
(
)
My angle calculation method was later implemented in production. Although the angles measured may
seem insignificant, they do drastically affect injector performance and it is imperative that the quality of
the armature is frequently checked.