sensors
Article
A Fast Measuring Method for the Inner Diameter of
Coaxial Holes
Lei Wang, Fangyun Yang, Luhua Fu, Zhong Wang, Tongyu Yang and Changjie Liu *
State Key Laboratory of Precision Measuring Technology and Instruments, Tianjin University, Tianjin 300072,
China; [email protected] (L.W.); [email protected] (F.Y.); [email protected] (L.F.);
[email protected] (Z.W.); [email protected] (T.Y.)
* Correspondence: [email protected]; Tel.: +86-22-8740-1582
Academic Editors: Thierry Bosch, Aleksandar D. Rakić and Santiago Royo
Received: 16 January 2017; Accepted: 17 March 2017; Published: 22 March 2017
Abstract: A new method for fast diameter measurement of coaxial holes is studied. The paper
describes a multi-layer measuring rod that installs a single laser displacement sensor (LDS) on each
layer. This method is easy to implement by rotating the measuring rod, and immune from detecting
the measuring rod’s rotation angles, so all diameters of coaxial holes can be calculated by sensors’
values. While revolving, the changing angles of each sensor’s laser beams are approximately equal
in the rod’s radial direction so that the over-determined nonlinear equations of multi-layer holes
for fitting circles can be established. The mathematical model of the measuring rod is established,
all parameters that affect the accuracy of measurement are analyzed and simulated. In the experiment,
the validity of the method is verified, the inner diameter measuring precision of 28 µm is achieved
by 20 µm linearity LDS. The measuring rod has advantages of convenient operation and easy
manufacture, according to the actual diameters of coaxial holes, and also the varying number of
holes, LDS’s mounting location can be adjusted for different parts. It is convenient for rapid diameter
measurement in industrial use.
Keywords: inner diameter; coaxial holes; measuring rod; laser displacement sensor
1. Introduction
Coaxial holes refer to circular holes widely distributed along the same axis. The most common
of these parts are aircraft wing hinges, internal combustion engine crankshaft holes, etc. [1]. For the
workpiece, the matching accuracy of the holes and shaft is one of the important properties, which is
directly linked to performance and durability, so the measurement of diameters is important for coaxial
holes [2]. In the machining of parts with coaxial holes, the accuracy of the hole’s diameter is sensitive
to the stiffness of the mandrel. Meanwhile, cutting tool elastic deformation appears under the cutting
force. All of this results in a significant impact on machining dimension accuracy of coaxial holes [3].
There are many diameter measuring methods for coaxial holes, such as inside micrometer,
contact probe, and pneumatic gauging, etc. The inside micrometer is the most widely used tool in the
industrial production field, nimble handling but easy to be influenced by individuals [4], the minimum
measuring uncertainty is 5 µm. Common contact probes are inductive displacement transducers [5],
coordinate measuring machines, etc. Contact probes are accurate, being able to achieve the repeatability
at 1 µm, but time consuming [6]. As the confused structure of coaxial hole parts, the contact probes
cannot get all coordinates of measured points in some deep holes. Pneumatic gauging has the
advantages of non-contact and high precision [7], its precision can commonly achieve 0.5 µm accuracy.
Excessive air tightness limits the measuring clearance to less than 100 µm, which results in a small
redundancy space for measuring operations. For different sizes of parts, the modification cost of the
measuring tool is high.
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As a non-contact probe, laser displacement sensors (LDS) are widely used in geometric
measurement [8,9], they can achieve a 0.5 µm uncertainty in a measuring span of 2 mm. It functions
by irradiating a laser beam to the measured surface vertically, and a laser spot is generated, which is
imaged in the linear photoelectric element (PSD, CCD, or CMOS) of the LDS. With the displacement of
the measured surface, the image position will change in the linear photoelectric element [10].
With the decrease in volume and reference distance, LDSs are widely used in the measurement of
small diameters. The usual procedure is installing multiple LDSs in the same cross-section of the hole.
The corresponding point coordinates of the hole can be obtained by only one measurement [11,12].
For a smaller diameter hole, this method would still be limited by the LDS’s volume and reference
distance. For the single LDS diameter measuring method, the rotation angle of the sensor’s axis is
required. In the measuring process, this improves the coaxial requirement between the photoelectric
encoder and the rotation axis [13].
With regard to the diameter measurement of coaxial holes of internal combustion engines,
based on a small coaxial error (0.03 mm) of holes, and ignoring holes’ roundness error (3 µm),
we propose a measuring rod which contains single LDS to measure the diameter for each layer’s
hole [14]. In the process of measurement, as the inclination angle between the measuring rod and
central axis of coaxial holes is small, we can get enough point coordinates of all the holes within
an operation with a number of rotations. For each holes’ cross-section to be measured, the sensors’
rotation angles are approximately equal. The diameters of all the holes are calculated by the least
square fitting method.
For this method, the minimum hole size that can be measured would only be limited by the single
LDS’s volume and reference distance. It significantly improves measuring range for coaxial hole parts,
and expands LDS’s application for diameter measurement in industrial use.
2. Measuring Principle
2.1. Instrument Configuration
In this measuring method, the system is composed of: measuring rod, LDS, vee blocks, baffle,
platform, and the coaxial hole part, as shown in Figure 1. The measuring rod is made of hollow
shaft, which is typically used as a precision guide rail, and has excellent straightness (0.05 mm/M)
and roundness (0.01 mm) [15]. For the measuring rod, according to the number and distribution
of holes in the part being tested, a corresponding number of LDSs are installed in the hollow shaft.
When mounting the LDS in the measuring rod, make sure that the laser beam and its reverse extension
line pass through the hollow shaft’s middle axial line perpendicularly. In the measuring rod’s radial
direction, the angle between the laser beams of each LDS can be any value.
2
3
1
4
5
6
Figure 1. Instrument configuration. (1) Measuring rod; (2) coaxial hole part; (3) LDS; (4) baffle;
(5) vee block; (6) platform.
Before measuring, put the coaxial hole part on the platform, and ensure its centerline is parallel to
the platform. Place the two vee blocks outside the two ends of the coaxial holes, the baffle is installed
on the end of a vee block’s V groove. Get the measuring rod through the coaxial holes, and the two
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ends of the rod arranged on the vee blocks, respectively. Press the measuring rod’s end against the
Sensors 2017, 17, 652
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baffle, which can limit the movement of the rod during rotation in the axial direction. Adjust the
position
that
vee which
blocks,can
so limit
that the
rotaryofaxis
parallel
within
the
part’s
againstof
the
baffle,
the rod’s
movement
the can
rod be
during
rotation
the
axialcenterline
direction. as
much
as
possible.
Adjust the position of that vee blocks, so that the rod’s rotary axis can be parallel with the part’s
During as
themuch
measurement
operation, we can rotate the measuring rod randomly, and read all LDSs’
centerline
as possible.
measuring
values.
All diametersoperation,
of the partwe
can
berotate
calculated
from the measured
values.
During
the measurement
can
the measuring
rod randomly,
and read all
LDSs’ measuring values. All diameters of the part can be calculated from the measured values.
2.2. The Ideal Measurement Model
2.2.InThe
Measurement
Model
theIdeal
ideal
circumstance,
while the measuring rod is revolving to measure diameters, its spinning
axis is stationary
relative
to the part’s
centerline.
We set up
coordinate
system O
In the ideal
circumstance,
while
the measuring
roda global
is revolving
to measure
diameters,
its on
w -xyz based
axis is
stationary
relative
to the2.part’s
centerline.
We setisup
global
system
Owthespinning
coaxial hole
part,
as shown
in Figure
The part’s
centerline
setaas
Ow -Zcoordinate
axis, and the
horizontal
xyz based
coaxial hole
part, asisshown
inwFigure
2. The part’s centerline is set as Ow-Z axis, and
direction
of on
thethe
measuring
platform
set as O
-X axis.
theSet
horizontal
of the
platform
is set as rod’s
Ow-Xrotary
axis. axis. Line FB is paralleled with
F and Bdirection
as the front
andmeasuring
rear end of
the measuring
Set Finand
as the
front and rear
end ofFor
the the
measuring
rod’s hole,
rotarywhen
axis. Line
FB is paralleled
Ow -Z axis
theBideal
measurement
model.
m-th layer’s
the measuring
rod has
with
O
w-Z axis in the ideal measurement model. For the m-th layer’s hole, when the measuring rod
rotated in the n-th time, the LDS’s laser emission point Kmn is relative stationary to the part. The laser
has rotated
in thespot
n-th K
time,’ the
LDS’s laser emission point Kmn is relative stationary to the part. The
beam
gets a laser
mn on the hole wall. LDS measures the length of Kmn Kmn ’, which is the
laser
beam
gets
a
laser
spot
K
mn’ on the hole wall. LDS measures the length of KmnKmn’, which is the
distance between the point on the hole wall and the rotary axis of the measuring rod.
distance between the point on the hole wall and the rotary axis of the measuring rod.
y
z
Kmn'
Kmn
F
Ow
…
K1n
B
x
K1n'
Figure 2. The ideal measurement model.
Figure 2. The ideal measurement model.
The measuring rod is rotated randomly during the measurement. As the two vee blocks and
Thehave
measuring
rotated
randomly
during
twocoordinate
vee blocks
baffle
restrictedrod
theisrod’s
movement
in both
axial the
and measurement.
radial directions.As
In the
global
Owand
baffle
restricted
the rod’s movement
in bothpoint
axialofand
directions.
global
coordinate
xyz, have
we can
get the three-dimensional
coordinate
theradial
laser spot
Kmn’ forIn
each
sensor’
laser
Owbeam,
-xyz, we
cancan
getbe
thewritten
three-dimensional
coordinate point of the laser spot Kmn ’ for each sensor’ laser
which
as:
beam, which can be written as:
 x  l cos(α  φ )
mn cos( αn + m
 x = lmn
ϕm )
n


(1) (1)
y y=llmn
sin(ααnn+φmϕ)m )
mn sin(

 z z= H
Hmm

For
the
laserbeam
beamrespectively,
respectively,and
andαnαis
variable
angle
mm isisoriginal
n is
For
them-th
m-thhole,
hole,ϕ
originalangle
angle of
of LDS’s
LDS’s laser
variable
angle
of of
rod’s
rotation
in
radial
direction.
l
is
the
distance
between
laser
emission
point
K
and
the
laser
mn
mn
rod’s rotation in radial direction. lmn is the distance between laser emission point Kmn and the laser
spot
K
’,
which
is
measured
by
LDS.
H
is
the
distance
between
K
and
B.
spot mn
Kmn’, which is measured by LDS. Hmmis the distance between Kmn mn
and B.
Assume
rotaryaxis
axisFB
FBisisperpendicular
perpendicular
to the
cross
section
the hole.
Assumethat
thatthe
the rod’s
rod’s rotary
to the
cross
section
of theofhole.
We canWe
getcan
geta alaser
laserbeam
beam
KK
K’ mn
by rotating
measuring
Several
K’mn
Kmn ’
Kmn
mn
by’ rotating
the the
measuring
beambeam
everyevery
time.time.
Several
laser laser
beamsbeams
KmnKmn
can
mn
can
constitute
a
swept
surface
[16].
This
swept
surface
forms
a
circle
with
the
hole
wall.
In
the
O
constitute a swept surface [16]. This swept surface forms a circle with the hole wall. In the Ow-xy
w -xy
plane,
the
circle
describedbelow:
below:
plane,
the
circlefor
forthe
thecross
crosssection
sectionof
of hole
hole wall
wall is described
2
2
lmn cos
cos(ααnn+ ϕφmm ))
 (yymm +
 llmn
 2 +
sin(ααnn +φϕmm ))2=rmrm 2
( xmxm+lmn
mnsin
2
(2) (2)
theradius
radius of
of the
the hole.
hole. (x
laser
emission
For
them-th
m-thhole,
hole,r rmisisthe
For
the
(xmm, ,yymm) )isisthe
thedifference
differencebetween
between
laser
emission
m
point
K
mn and central coordinate of the hole. Where (xm, ym),  m and rm are unknown coefficients, lmn
point Kmn and central coordinate of the hole. Where (xm , ym ), ϕm and rm are unknown coefficients,
αnαareare
variables,
andand
lmn is
as the
value.value.
lmnand
and
variables,
lmnknown
is known
asmeasured
the measured
n
The
measurement
is
performed
by
rotating
the
measuring
and
discretely,
so so
thethe
The measurement is performed by rotating the measuringrod
rodrandomly
randomly
and
discretely,
calculation
rm can
be transformed
intooptimal
the optimal
solution
of over-determined
nonlinear
calculation
of rofm can
be transformed
into the
solution
of over-determined
nonlinear
equations:
equations:
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∆f =
4 of 12
n
∑
q
2
2
( xm + lmi cos(αi + ϕm )) + (ym + lmi sin(αi + ϕm )) − rm
2
(3)
i =1
We can set ϕm as an arbitrary value, the numerical solution of αn and rm are obtained by the
iterative calculus, and ∆f is the least square of nonlinear equations. The resolution of (xm , ym )
is dependent on ϕm . For numerical solutions of complex over-determined nonlinear equations,
the common calculation methods are neural network, genetic algorithm, and particle swarm
optimization, etc. This paper proposes the global particle swarm optimization algorithm due to
the advantages of generality, global search capability, and high robustness [17]. By using initial random
values to eliminate the relevant amounts, it improves the accuracy of numerical solutions effectively.
The calculation speed is fast, and the algorithm is easy to implement [18].
3. Major Factors Influencing Measuring Uncertainty
From Equation (3), by rotating the measuring rod several times and reading the LDSs’ measured
values, all diameters of a part’s holes can be calculated by the least-square values of ∆f. However,
the ideal experimental conditions are not available in an actual measuring process, there are four factors
that can influence the accuracy of the results: LDS measuring uncertainty, face run-out of the rod,
manufacturing uncertainty, and installation uncertainty of the rod. For a machining workshop, in order
to achieve 30 µm diameter measuring uncertainty—through analysis of the tolerance uncertainty of
diameter—we can effectively reduce the difficulty and cost in the measurement by defining the rod’s
uncertainty factors to a reasonable range.
3.1. LDS Measuring Uncertainty
For the application of LDS, the angle θ sen between LDS’ laser beam and the measured surface’s
normal line should satisfy: θ sen < 5◦ . Accordingly, the distance between the measuring rod’s rotary
axis (FB) and part’s centerline should be less than rm tanθ sen . In the installation of the measuring
rod, it is located in the center of the holes of no more than ±2 mm. For rm < 75 mm, the inclination
angle (θ sen ) caused by the installation of measuring rod is 1.53◦ , which can meet the angle deviation
requirement of LDS [19]. Under the above conditions, the major error of the LDS is its measuring
linear error. In the experiment, the two laser displacement sensors are from SICK Ltd. (Waldkirch,
Germany). Model OD2-P30W04 is used, which has a measuring span of 8 mm, and its uncertainty is
0.02 mm in the full range. For this method, the LDS measurement uncertainty ∆sen is 0.02 mm.
3.2. Face Run-Out of the Measuring Rod
When the rod is rotated, the face run-out error comes principally from the hollow shaft’s roundness
error and vee block’s flatness error [20]. In this measurement system, it is summarized as a random
error. The hollow shafts’ roundness error ∆RD = 10 µm, vee block’s flatness error ∆FL = 2 µm, the face
run-out error of measuring rod can be obtained by:
∆ TR =
q
∆ RD 2 + ∆ FL 2
(4)
Finally, the face run-out error ∆TR = 10.2 µm.
3.3. Manufacturing Uncertainty of the Rod
In the manufacture of the measuring rod, the rotary axis (FB) of the measuring rod is a virtual line,
a line between two ends’ center of the hollow shaft that is substituted as the rotary axis. During the
installation process of LDS, it is difficult to make sure that the laser beam intersects the centerline
perpendicularly. There is a position error between Kmn Kmn ’ and FB, which is composed of a vertical
distance error and a pitching angle error.
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First, we set up a measuring rod coordinate system Os -xyz, the measuring rod’s rear end B is set
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wecoordinate
set up a measuring
Os-xyz,
rod’s
rearbeam
end BKis11set
as originFirst,
of this
system, rod
the coordinate
rotary axissystem
FB is set
as Othe
axis, the first
laser
K11 ’ is
s -Zmeasuring
as
origin
of
this
coordinate
system,
the
rotary
axis
FB
is
set
as
O
s-Z axis, the first laser beam K11K11’ is
set as OsFirst,
-X axis.
shown
in Figure
3.coordinate system Os-xyz, the measuring rod’s rear end B is set
we As
set up
a measuring
rod
set
as Os-X
As shown in
Figurethe
3. rotary axis FB is set as Os-Z axis, the first laser beam K11K11’ is
as origin
ofaxis.
this coordinate
system,
set as Os-X axis. As shown in Figure 3.
Kmn'
y
F
y
Ow
z
z
Ow
Kmn' Kmn
F
y
…
…
Kmn
x
K1n
B
K1n
B
z
K1n'
x
z
y
Os
Os x
x
K1n'
Figure 3. The Global Coordinate System and the Measuring Rod Coordinate System.
Figure 3. The Global Coordinate System and the Measuring Rod Coordinate System.
Figure 3. The Global Coordinate System and the Measuring Rod Coordinate System.
In the Os-xy plane, the laser beam and its reverse extension line cannot intersect the centerline
In the
Osthe
-xy
plane,
the
laserbetween
beam
and
extension
line cannot
intersect
centerline
strictly,
so
vertical
distance
Kmnits
Kits
mnreverse
’reverse
and FBextension
is dm, as shown
in Figure
4. thethe
In the
Os-xy
plane,
the laser
beam and
line cannot
intersect
centerline
strictly,
so the
vertical
distance
between
Kmn’ ’and
andFB
FBisisdmd, mas, as
shown
in Figure
strictly,
so the
vertical
distance
betweenKKmn
mnKmn
shown
in Figure
4. 4.
y
K'mn
y
K'mn
lmn
lmn
αn+φm
os
x
αn+φm
Kmn
dm
os
x
dm
Kmn
Figure 4. The Distance between Laser Beam and Rotary Axis of the Measuring Rod.
Figure
4. The
Distance
andRotary
RotaryAxis
Axisofof
Measuring
Rod.
Figure
4. The
Distancebetween
betweenLaser
Laser Beam
Beam and
thethe
Measuring
Rod.
In the measuring rod coordinate system Os-xyz, the coordinate point of the laser spot Kmn’ is
expressed
In theas:
measuring rod coordinate system Os-xyz, the coordinate point of the laser spot Kmn’ is
In the
measuring
rod coordinate system Os -xyz, the coordinate point of the laser spot Kmn ’ is
expressed as:
 x  l cos(α  φ )  d sin(α  φ )
expressed as:
mn
n
m
m
n
m
 
sin(
φφ
ddmmdm
cos(

lmncos
cos(
sin(
(5)
lmn
(αααnnn+
ϕmmm)))+
sinα(nαn φ+m )ϕm )
 x=yx 
y
=
l
sin
(
α
+
ϕ
)
+
d
cos
(
α
+
ϕ
)
H
l m sin(αnn  φmm)  d m mcos(αn n φm ) m
(5) (5)
 zy mn

 z = H mn
H mbeam is not perpendicular to the rotary axis FB strictly. The
m
 zlaser
For the installation of LDS,
For the
installation
ofmnLDS,
laser
is not perpendicular
the rotary axis FB strictly. The angle
angle
γm the
between
KmnK
’ and
thelaser
Obeam
s-xy plane
shown
in Figureto5.to
For
installation
of
LDS,
beam isisnot
perpendicular
the rotary axis FB strictly. The
γm between
K
K
’
and
the
O
-xy
plane
is
shown
in
Figure
5.
mn
mn
s
angle γm between KmnKmn’ and the Os-xy plane is shown in Figure 5.
y
y
K'mn
K'mn
lmn
lmn
γm
γm K
mn
z
z
dm Kmn
os
Hm
dm
os
Hm
αn+φm
αn+φm
x
x
Figure 5. Angle between the laser beam and rotary axis.
Figure5.5.Angle
Anglebetween
between the
the laser
axis.
Figure
laserbeam
beamand
androtary
rotary
axis.
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So, by adding the angular error γm in Equation (5), the laser spot Kmn ’ is expressed as:
So, by adding the angular error γm in Equation (5), the laser spot Kmn’ is expressed as:


cos
γmγ cos
(αn + ϕm ) + dm sin(αn + ϕm )
 x=
 lmn
cos
 xlmn
m cos(αn  φm )  d m sin(αn  φm )
y=
 lmn cos γm sin(αn + ϕm ) + dm cos(αn + ϕm )

cos γm sin(αn  φm )  d m cos(αn  φm )
 z = yH lmn
m + lmn sin γm
 z  H  l sin γ
m
mn
m

(6)
(6)
In the current mechanical processing conditions, it is easy to meet the requirements: dm < 0.5 mm
In the current mechanical processing conditions, it is easy to meet the requirements: dm < 0.5 mm
and γm < 0.5◦ , so we can obtain the manufacturing error by:
and γm < 0.5°, so we can obtain the manufacturing error by:
q
∆lmn
cos γγm )22+dm
dm2 2−l lmn
(7)
(7)
Δl = (llmn cos
mn
 mn
m

mn
As the
the measuring
measuring rod
rod isis placed
placedin
inthe
themiddle
middleofofcoaxial
coaxialholes,
holes,the
thelaser
laseremission
emissionpoint
point
Kmn
is
As
Kmn
is
closed
to
O
(the
center
of
the
hole
to
be
measured),
then
l
≈
r
,
when
r
<
80
mm,
the
m center of the hole to be measured), then lmn ≈ rm. when
mn rm <m80 mm, the
m manufacturing
closed to Om (the
manufacturing
error ∆lmn < 1.5 µm.
error
Δlmn <1.5 μm.
3.4. Installation Uncertainty of Measuring Rod
3.4. Installation Uncertainty of Measuring Rod
The laser beam Kmn Kmn ’ is revolving around the rotary axis FB while measuring rod is rotating.
The laser beam KmnKmn’ is revolving around the rotary axis FB while measuring rod is rotating.
Spot trajectory {Kmn ’} is formed by laser beams and the wall of the hole, and its shape is affected by
Spot trajectory {Kmn’} is formed by laser beams and the wall of the hole, and its shape is affected by
the installation error of the measuring rod.
the installation error of the measuring rod.
For the position between laser beam Kmn Kmn ’ and rotary axis FB, when Kmn Kmn ’ is perpendicular
For the position between laser beam KmnKmn’ and rotary axis FB, when KmnKmn’ is perpendicular
to FB, the angle γm between Kmn Kmn ’ and the Os -xy plane is equal to zero, so the swept surface formed
to FB, the angle γm
between KmnKmn’ and the Os-xy plane is equal to zero, so the swept surface formed
by laser beams is a circular plane that is perpendicular to FB. When γm 6= 0, and the vertical distance
by laser beams is a circular plane that is perpendicular to FB. When γm ≠ 0, and the vertical distance
d between Kmn Kmn ’ and FB is equal to zero, the swept surface is a cone, and FB is the directrix of the
dmmbetween Kmn
Kmn’ and FB is equal to zero, the swept surface is a cone, and FB is the directrix of the
cone. When γm 6= 0, and dm 6= 0, the swept surface is an irregular conical surface, as shown in Figure 6,
cone. When γm ≠ 0, and dm ≠ 0, the swept surface is an irregular conical surface, as shown in Figure 6,
the generatrix of the conical surface is a curve at the top K , and a straight line near the bottom Kmn ’.
the generatrix of the conical surface is a curve at the top Kmn
mn, and a straight line near the bottom Kmn’.
Kmn'
Kmn
F
B
Figure 6. Spot trajectory formed by laser beams.
Figure 6. Spot trajectory formed by laser beams.
For the position error formed by the installation of the rod relative to the part, when the rotary
For
the
position error
by the
installation
of the
the centerline
rod relative
the part,
when
rotary
axis of
the
measuring
rod isformed
completely
coincident
with
ofto
coaxial
holes,
thethe
irregular
axis
of
the
measuring
rod
is
completely
coincident
with
the
centerline
of
coaxial
holes,
the
irregular
conical surface’s directrix FB and the Ow-Z axis are collinear, so the spot trajectory {Kmn’} formed by
conical
surface’s
directrix
FBideal
and the
Owwith
-Z axis
are collinear,
trajectory
{Kmn
formed
by
laser
beams
is located
in an
circle
radius
rm. As FBsoisthe
notspot
coincident
with
the’} O
w-Z axis,
laser
beams
is
located
in
an
ideal
circle
with
radius
r
.
As
FB
is
not
coincident
with
the
O
-Z
axis,
m as shown in Figure 6.
w
Spot’s trajectory {Kmn’} forms a three-dimensional curve,
Spot’s
trajectory
{K
’}
forms
a
three-dimensional
curve,
as
shown
in
Figure
6.
mn
In the calculation of rm, it is carried out on the assumption that the curve of the spot trajectory
calculation
of ideal
rm , it is
carried
out on
the assumption
thatofthe
curve of the
spot trajectory
{Kmn’}Inisthe
regarded
as an
circle,
which
ignores
the influence
roughness.
However,
in the
{K
’}
is
regarded
as
an
ideal
circle,
which
ignores
the
influence
of
roughness.
However,
in the
mn
installation
and rotation of the measuring rod, it is difficult to ensure that the rotary axis is completely
installationwith
and the
rotation
of theofmeasuring
is difficult
ensure
that the
is
coincident
centerline
the coaxialrod,
holeit part,
so the to
spot
trajectory
{Kmnrotary
’} is a axis
threecompletely
coincident
with
the
centerline
of
the
coaxial
hole
part,
so
the
spot
trajectory
{K
’}
is
mn
dimensional curve. Using a three-dimensional curve to fit the radius of hole, the flatness error and
a
three-dimensional
curve.
Using
a
three-dimensional
curve
to
fit
the
radius
of
hole,
the
flatness
error
roundness error would be introduced [21]. In order to reduce operation difficulty and computation
and roundness
errorawould
introduced
[21]. Inerror,
order we
to reduce
operation
difficulty
complexity
within
certainberadius
calculation
can limit
all error
factorsand
to acomputation
reasonable
complexity
within
a
certain
radius
calculation
error,
we
can
limit
all
error
factors
to
a
reasonable
range
range by simulation.
by simulation.
In measuring rod coordinate system Os-xyz from Equation (6), we can get the point coordinates
in laser beam KmnKmn’:
Sensors 2017, 17, 652
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In measuring rod coordinate system Os -xyz from Equation (6), we can get the point coordinates in
laser beam Kmn Kmn ’:


 xs = dm / sin(αn + ϕm ) + (t − Hm ) cot γm cot(αn + ϕm )
(8)
ys = (t − Hm ) cot γm

 z =t
s
The laser beam Kmn Kmn ’ is revolving around the Os -Z axis, and forms the irregular conical surface.
Set θ as the rotation angle of Kmn Kmn ’, so the parametric equation of this curved surface is set up
as follows:
q



x
=
(dm / sin ϕm + (t − Hm ) cot γm cot ϕm )2 + ((t − Hm ) cot γm )2 cos θ
s



q
(9)
y
=
(dm / sin ϕm + (t − Hm ) cot γm cot ϕm )2 + ((t − Hm ) cot γm )2 sin θ
s





zs = t
In the measuring rod coordinate system Os -xyz, the curved surface equation of the spot trajectory
{Kmn ’} is:
xs 2 + ys 2 = (dm / sin ϕm + (zs − Hm ) cot γm cot ϕm )2 + ((zs − Hm ) cot γm )2
(10)
The spot trajectory {Kmn ’} is formed by the intersection of laser beams and hole wall. In the global
coordinate Ow -xyz, the point Kmn ’ is located on the cylinder surface of the hole:
xw 2 + yw 2 = rm 2
(11)
By Equations (10) and (11), we can get the curve equation of the spot trajectory {Kmn ’}, but it is
necessary to obtain the transition matrix between the measuring rod coordinate system Os -xyz and the
global coordinate system Ow -xyz.
In the space coordinate system conversion [22], the Bursa-Wolf model is widely used in the
form [23]:




xs
xw




(12)
 ys  = λ  yw  R + T
zs
zw
where, R is the rotation matrix from the global coordinate system Ow -xyz to the measuring rod
coordinate system Os -xyz. Set εx , εy , and εz are the three rotation angles around the X-, Y- and Z-axis in
the global coordinate system Ow -xyz. T = [∆x, ∆y, ∆z]T is the transfer matrix from Ow -xyz to Os -xyz.
λ is the scale factor.
In this measurement system, the curved surface is formed by revolving Kmn Kmn ’ around the Os -Z
axis. While calculating the flatness error and roundness error of the spot trajectory {Kmn ’}, the rotation
angle εz can be any value. The baffle limits the movement of the measuring rod in the Os -Z axis, so the
translation parameter ∆z = 0. As the measuring rod is a rigid body, the scale factor λ = 1.
In the transition matrix, the unknowns are ∆x, ∆y, εx , and εy . We only need to calculate the
roundness error and flatness error of the curve {Kmn ’}, so the conversation can be simplified into the
position relationship between the Os -Z axis and the Ow -Z axis, and it is expressed by eccentricity
distance d∆ and deflection angle ω ∆ , as shown in Figure 7.
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yw
yw y s
ys
zs
zs
ωΔ
ωΔ
zw
zw
os
os
ow
ow
dΔ
dΔ
xs
xs
xw
xw
Figure 7. The position relationship between Os -Z and Oww-Z.
-Z.
Figure 7. The position relationship between Os-Z and Ow-Z.
The
The relationship
relationship between
between ddΔ,,Δx,
∆x,Δy,
∆y,ωωΔ, ,εεx, and
, andεyεyare
areasasfollows:
follows:
The relationship between dΔ∆, Δx, Δy, ωΔ,∆εx,xand εy are
as follows:
( d  pΔx2  Δy 2
2
2 2
dd∆Δ = Δ∆x
x2  +
Δy∆y
Δ
ω∆Δ 
=arccos
arccos(cos
cosεεxxcos
cosεεyy )
ω
ωΔ

 arccos  cos ε
x
cos ε y


(13)
(13)
(13)
ωΔ∆,, we
In
distance dd∆Δ and deflection angle ω
In the
the simulation, with difference of eccentricity distance
we can
can get
get
angle ωΔ{K
, we
can get
In the simulation, with
difference of eccentricity distance dΔ and deflection
the
spot trajectory’s
trajectory’s
mn’}
the conversion
conversion matrix
matrix by Equation 13, and the point coordinate of the spot
{Kmn
’} can
can be
be
the
conversion
matrix
by
Equation
13,
and
the
point
coordinate
of
the
spot
trajectory’s
{K
mn’} can be
calculated
calculated in
in the
the global coordinate system Oww-xyz.
-xyz. For
For calculating
calculating the
the flatness
flatness error
error and
and roundness
roundness
calculated
in the trajectory,
global coordinate
system O
w-xyz. For calculating the flatness error and roundness
error
obtained
byby
thethe
spot
trajectory
{Kmn
traj θis
the
error of
of the
thespot
spot trajectory,the
theleast
leastsquare
squareface
facePPtrajtrajis is
obtained
spot
trajectory
{K’}.
is
mnθ’}.
traj
error
ofbetween
the spotPtrajectory,
the
least
squareLface
Pthe
traj is obtained by the spot trajectory {Kmn’}. θtraj is the
angle
traj
and
the
O
w
-xy
plane,
traj
is
crossing
line
between
P
traj
and
the
O
w
-xy
plane,
the angle between Ptraj and the Ow -xy plane, Ltraj is the crossing line between Ptraj and the Ow -xy
angle
betweenWe
Ptraj
and thePO
w-xy plane, Ltraj is the crossing line between Ptraj and the Ow-xy plane,
respectively.
converse
traj toP
the
-xy O
plane
the by
usethe
of use
Rodrigues'
rotation
formula
[24], take
plane, respectively.
We converse
towthe
plane
of Rodrigues'
rotation
formula
[24],
w -xy by
traj O
respectively.
We
converse
P
traj to the Ow-xy plane by the use of Rodrigues' rotation formula [24], take
the
line Lline
traj as
theas
rotation
axis, and
traj isθthe
rotation
angle,
as
shown
in
Figure
8.
takecrossing
the crossing
Ltraj
the rotation
axis,θand
is
the
rotation
angle,
as
shown
in
Figure
8.
traj
the crossing line Ltraj as the rotation axis, and θtraj is the rotation angle, as shown in Figure 8.
zw
zw
Ptraj
Ptraj
{ Kmn'}
{ Kmn'}
yw
yw
ow
ow
xw
xw
{ Kmn'}'
{ Kmn'}'
Ltraj
Ltraj
θtraj
θtraj
Figure 8. Transformation of spatial circle.
Figure
Figure8.8.Transformation
Transformationof
ofspatial
spatialcircle.
circle.
Finally, the 3-D points coordinate of {Kmn’} is transferred to near the Ow-xy plane, and the new
Finally,
thedenoted
3-D points
of {Kmn’} is transferred to near the Ow-xy plane, and the new
3-D points
are
as {Kcoordinate
mn'}'. The flatness error (Δflat) of the laser spot trajectory is the maximum
Finally,
the 3-D points
coordinate
of {Kmn ’} is transferred to near the Ow -xy plane, and the new
3-D
points
are
denoted
as
{K
mn'}'. The flatness error (Δflat) of the laser spot trajectory is the maximum
difference
mn’}’ in the Ow-Z axis. By calculating the least square fitting circle of {Kmn’}’ on the
3-D points of
are{Kdenoted
as {Kmn ’}’. The flatness error (∆flat ) of the laser spot trajectory is the maximum
difference
of {Kmn’}’ in the Ow-Z axis. By calculating the least square fitting circle of {Kmn’}’ on the
O
w-xy plane, the roundness error (Δround) is calculated by the fitting circle and hole’s real radius. The
difference
of {Kmn ’}’ in the Ow -Z axis. By calculating the least square fitting circle of {Kmn ’}’ on the
O
w-xy plane, the roundness error (Δround) is calculated by the fitting circle and hole’s real radius. The
final
radius
error
m of the laser spot trajectory {Kmn’} is given as:
Ow -xy plane,
the Δr
roundness
error (∆round ) is calculated by the fitting circle and hole’s real radius.
final
radius error Δrm of the laser spot trajectory
{Kmn’} is given as:
The final radius error ∆rm of the laser spot trajectory 2{Kmn ’} is2 given as:
(14)
Δr  Δround2  Δ flat2
(14)
Δrmm  qΔround
 Δ flat
∆rm = ∆round 2 + ∆flat 2
(14)
For different holes in the part, the radius
error Δrm is different. Where dΔ and ωΔ are constant,
For different
holesΔr
inmthe
part, the radius
Δrm is different. Where dΔ and ωΔ are constant,
the hole’s
radius error
is proportional
to Herror
m. When analyzing the maximum measuring error of
For different
holes
in
the
part, the radius
error
∆rm is different.
Where
d∆ and
ω ∆ are constant,
the
hole’s
radius
error
Δr
m is
proportional
to
H
m
.
When
maximum
measuring
error of
the radius, the hole near the front end of the rod shouldanalyzing
be chosenthe
to calculate.
the
hole’s
radius
error
∆r
is
proportional
to
H
.
When
analyzing
the
maximum
measuring
error of
m
m
the radius,
the holethe
near
the front
end
the rod should be chosen to calculate.
In calculating
radius
errors
Δrof
m of the curve {Kmn’}, we assume the rod’s manufacturing error
the radius,
the
hole
near
the
front
end
of
the
rod
should
be
chosen
to
calculate.
calculating the radius errors Δrm of the curve {Kmn’}, we assume the rod’s manufacturing error
as: dIn
m = 0.5 mm and γm = 0.5°. The length of the measuring rod is 500 mm. The number of coaxial
as: dm = 0.5 mm and γm = 0.5°. The length of the measuring rod is 500 mm. The number of coaxial
Sensors 2017, 17, 652
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In calculating the radius errors ∆rm of the curve {Kmn ’}, we assume the rod’s manufacturing error
9 of 12
and γm = 0.5◦ . The length of the measuring rod is 500 mm. The number of coaxial
m
holes in the part is two, and all diameters are 150 mm. Based on these, the maximum radius error is
holes
in theunder
part is
two, and
aresimulate
150 mm.results
Based are
on these,
theinmaximum
simulated
different
of all
d∆ diameters
and ω ∆ . The
showed
Figure 9. radius error is
simulated under different of dΔ and ωΔ. The simulate results are showed in Figure 9.
Sensors
17,mm
652
as: d 2017,
= 0.5
Radius error (mm)
0.02
0.015
0.01
0.005
0
2
1.5
1
0.5
Deflection angle (°)
3
0
0
1
4
5
2
Eccentricity distance (mm)
Figure
Figure 9.
9. The
The radius
radius error
error coursed
coursed by
by the
the relative
relative position
position of
of the
the rod.
rod.
Figure
Figure 99 shows
shows the
the final
final radius
radius error
error of
of the
the spot
spot trajectory
trajectory in
in different
different eccentricity
eccentricity distances
distances and
and
deflection
angles.
As
ω
Δ < 1.5° and
d
Δ
<
3
mm,
the
radius
error
is
less
than
10
μm.
While
installing
the
◦
deflection angles. As ω ∆ < 1.5 and d∆ < 3 mm, the radius error is less than 10 µm. While installing
measuring
rod,rod,
for the
between
the rod’s
two ends
and the
centerline,
it is toitbe
the measuring
fordistance
the distance
between
the rod’s
two ends
andpart’s
the part’s
centerline,
isatosmall
be a
range
of
no
more
than
2.5
mm.
Through
this
operation,
the
radius
error
of
the
spot
trajectory
formed
small range of no more than 2.5 mm. Through this operation, the radius error of the spot trajectory
by
the laser
beam
notdoes
exceed
10 μm. 10
If we
achieve
a higher
installation
accuracy,
more
formed
by the
laserdoes
beam
not exceed
µm.can
If we
can achieve
a higher
installation
accuracy,
precision radiuses can be calculated for the coaxial holes.
more precision radiuses can be calculated for the coaxial holes.
3.5.
Diameter Measurement
Measurement Uncertainty
Uncertainty of
of the
the System
System
3.5. Total
Total Diameter
According
According to
to the
the above
above analyses,
analyses, the
the accuracy
accuracy of
of this
this measurement
measurement method
method depends
depends on
on several
several
factors.
rod,
the
radius
error
Δr∆r
m of
factors. By
Byevaluating
evaluatingthe
theerror
errorcaused
causedby
byinstallation
installationofofthe
themeasuring
measuring
rod,
the
radius
error
m
laser
spot
trajectory
has
been
controlled
in
a
small
range
on
the
diameters
measurement
result.
Thus
of laser spot trajectory has been controlled in a small range on the diameters measurement result.
Δ
sen, which
is caused
by the measurement
error of error
LDS, is
main
factor
that
influence
the diameter
Thus
∆sen , which
is caused
by the measurement
of the
LDS,
is the
main
factor
that influence
the
measurement
accuracy.accuracy.
While the
coaxial
holes holes
are considered
ideal
circles,
diameter measurement
While
the coaxial
are considered
ideal
circles,the
thetolerance
tolerance of
of
roundness
should
be
taken
as
the
source
of
measuring
uncertainty,
and
we
set
is
as
Δ
Hole
.
The
total
roundness should be taken as the source of measuring uncertainty, and we set is as ∆Hole . The total
diameter
error is
is approximately
approximately calculated
calculated by:
by:
diameter measurement
measurement error
Δ
∆sumsum≈
 Δ sen22  ΔTR22  Δlmn 22  Δrm 2 2 Δ Hole2
q
∆sen + ∆ TR + ∆lmn + ∆rm + ∆ Hole 2
(15)
(15)
by using
From the Equation (15), as the installation of LDSs fulfills: dm < 0.5 mm and γm < 0.5°,
◦ , by using
From
the
Equation
(15),
as
the
installation
of
LDSs
fulfills:
d
<
0.5
mm
and
γ
<
0.5
m
m
LDSs with measurement linearity of 20 μm, so the radius error Δrm caused by the installation position
LDSs
measurement
linearity
µm, so
radius
errorthe
∆rroundness
installation
m caused by
of
the with
measuring
rod is limited
in of
the20range
ofthe
10 μm.
While
ofthe
holes
ΔHole is 3 position
μm, the
of
the
measuring
rod
is
limited
in
the
range
of
10
µm.
While
the
roundness
of
holes
∆
3 µm,
Hole
measurement error Δsum is less than 24.8 μm. For a general machining workshop, it can is
achieve
the measurement
error error
∆sum is
than 24.8
For a general machining workshop, it can achieve
diameter
measurement
ofless
no more
thanµm.
30 μm.
diameter measurement error of no more than 30 µm.
4. Experiments and Discussion
4. Experiments and Discussion
To verify the measuring method for the diameters of coaxial holes, in this paper, two 150 mm
To verify the measuring method for the diameters of coaxial holes, in this paper, two 150 mm ring
ring gauges are chosen as the coaxial hole part, and they are clamped on the platform. The length of
gauges are chosen as the coaxial hole part, and they are clamped on the platform. The length of the
the hollow shaft for the measuring rod is 500 mm. For mounting LDS on the hollow shaft, two square
hollow shaft for the measuring rod is 500 mm. For mounting LDS on the hollow shaft, two square
holes were machined on the shaft by a CNC, it can satisfy the precision requirement of dm and γm in
holes were machined on the shaft by a CNC, it can satisfy the precision requirement of dm and γm in
section 3.3. Fixtures are mounted on the hollow shaft to fix the LDSs, they can also be used to change
the position of the LDS in the radial direction of the hollow shaft, which can extend the measurement
range of the measuring rod for different size coaxial holes.
On the platform, a rectangular groove with 90 mm in width and 5 mm in depth was machined
by an NC milling machine. The widths of vee blocks and clamps of the ring gauge are both 90 mm,
Sensors 2017, 17, 652
10 of 12
Section 3.3. Fixtures are mounted on the hollow shaft to fix the LDSs, they can also be used to change
the position of the LDS in the radial direction of the hollow shaft, which can extend the measurement
range of the measuring rod for different size coaxial holes.
On the platform, a rectangular groove with 90 mm in width and 5 mm in depth was machined
Sensors
652
10 of
12
by an2017,
NC 17,
milling
machine. The widths of vee blocks and clamps of the ring gauge are both 90
mm,
and they were embedded in the rectangular groove, and the edge of the rectangular groove was the
and
they were
in the rectangular
groove,
and
the edge
of the rectangular
groove
benchmark
forembedded
the installation.
Two vee blocks
were
formed
by longitudinal
cutting
of anwas
old the
vee
benchmark
for
the
installation.
Two
vee
blocks
were
formed
by
longitudinal
cutting
of
an
old
vee
block, which ensured that they had the same groove depth, so the altitude difference between the
block,
ensured
that they
had
the
grooveofdepth,
so not
the more
altitude
difference
between
the
middlewhich
axis of
the measuring
rod
and
thesame
centerline
part was
than
1 mm. With
regard
to
middle
axis
of
the
measuring
rod
and
the
centerline
of
part
was
not
more
than
1
mm.
With
regard
to
locating the coaxial hole part on the measurement platform, it is necessary to make the baseline of part
locating
the with
coaxial
part on groove
the measurement
platform,
is necessary
to make the
baseline
of
to coincide
thehole
rectangular
of the platform.
The it
baseline
is the reference
datum
line for
part
to
coincide
with
the
rectangular
groove
of
the
platform.
The
baseline
is
the
reference
datum
line
auxiliary machining the coaxial holes on the outer surface of the part. With the help of vee blocks on
for
machining
coaxial
holes
the outer surface
of the
part.
With the help
vee blocks
theauxiliary
rectangular
groove, the
coaxial
holes
are on
approximately
parallel
to the
measuring
rod’sofrotation
axis.
on
the
rectangular
groove,
coaxial
holes
are
approximately
parallel
to
the
measuring
rod’s
rotation
Through the high-precision rectangular groove, the deviation and inclination of the measuring rod
axis.
Through
the high-precision
rectangular
the deviation and inclination of the measuring
achieved
the accuracy
requirement
in Sectiongroove,
3.4.
rod achieved
accuracy requirement
Sectionin
3.4.
The finalthe
experimental
equipment in
is shown
Figure 10.
The final experimental equipment is shown in Figure 10.
Ring gauge
LDS
Fixture
Baffle
Hollow shaft
Cable
Rectangular groove
Vee block
Clamp
Platform
Data acquisition board
Figure 10. The diameter measurement system for coaxial holes.
Figure 10. The diameter measurement system for coaxial holes.
In the experiment, the measuring rod’s rotation count is n, and the number of coaxial holes is m,
the experiment,
the measuring
rod’s rotation
count
is being
n, and mn.
the number
coaxial holes
whichIndetermines
the number
of equations
in Equation
(3),
In radialofdirection
of theis
m, which determines
equations
Equation
(3),tobeing
mn. Inholes,
radialwhile
direction
the
measuring
rod,  m is the
the number
angle ofofLDS’s
laser in
beam
relative
the coaxial
it isof
only
measuring rod, ϕm is the angle of LDS’s laser beam relative to the coaxial holes, while it is only
correlated with (xm, ym) and is independent of the radius result. In order to simplify the calibration
correlated with (xm , ym ) and is independent of the radius result. In order to simplify the calibration
process, we set  m = 0, which also reduces the computational complexity of iterative operations. The
process, we set ϕm = 0, which also reduces the computational complexity of iterative operations.
final
over-determined
nonlinear
equations
are obtained
by: by:
The final
over-determined
nonlinear
equations
are obtained
n


2
2
q ( x  l cos α )2  ( y  l sin α )2  r
Δfn 
m
mi
i
m
mi
i
m
∆ f = ∑ i 1 ( xm + lmi cos αi )2 + (ym + lmi sin αi )2 − rm
i =1
(16)
(16)
For the m-th hole, (xm, ym) is the coordinate difference between the laser emission point Kmn and
For the m-th
hole, (x
) isLDS’s
the coordinate
difference
between
thevalue,
laser the
emission
point result
Kmn and
the centerline
of coaxial
holes.
original angle
 m is
a default
calculation
of
m , ymAs
the
centerline
of coaxial
As LDS’s
original
angle in
ϕmthe
is aover-determined
default value, the
calculation
of
(x
m, y
m) is not credible.
Forholes.
Equation
(16), the
unknowns
equations
are:result
coaxial
(x
,
y
)
is
not
credible.
For
Equation
(16),
the
unknowns
in
the
over-determined
equations
are:
coaxial
m
m
holes’ radius rm, coordinate difference (xm, ym), and the rotation angle of rod αn.
holes’
radius
rm , coordinate
difference
(xm , ym
), and
the+rotation
rodnumber
αn.
The
number
of unknowns
in Equation
(15)
is 3m
n, onlyangle
whenofthe
of equations is
of unknowns inequations
Equationcan
(15)converge.
is 3m + n,For
only
the number
of equations
mn ≥ The
3m +number
n, the over-determined
thewhen
two holes
in the experiment,
theis
mn
≥
3m
+
n,
the
over-determined
equations
can
converge.
For
the
two
holes
in
the
experiment,
time of the rod’s rotation should be 6. While rotating the measuring rod manually, in order to reduce
theoperational
time of the errors
rod’s rotation
should
be 6. While
rotating
measuring
manually,
in order
to
the
and improve
calculating
precision,
thethe
rod’s
rotation rod
count
is much more
than
the last
operational
and improve
precision, the rod’s
rotation
countthe
is much
more
6,reduce
and the
result iserrors
the average
of thecalculating
multiple measurements.
Figure
11 shows
results
of
different rotation counts in each measurement.
Sensors 2017, 17, 652
11 of 12
than 6, and the last result is the average of the multiple measurements. Figure 11 shows the results of
different
Sensors
2017,rotation
17, 652 counts in each measurement.
11 of 12
Maximum radius error (μm)
55
50
45
40
35
30
25
20
4
6
8
10
12
14
16 18 20
Rotary times
22
24
26
28
30
Figure
Figure11.
11.The
Themeasurement
measurementresults
resultsfor
fordifferent
differentrotation
rotation times
times of
of the
the measuring
measuring rod.
rod.
The comparison between the measurement result and rotation counts of the measuring rod is
The comparison between the measurement result and rotation counts of the measuring rod is
shown in Figure 11. It can be seen that: as the rotation counts of the measuring rod exceeded 18, the
shown in Figure 11. It can be seen that: as the rotation counts of the measuring rod exceeded 18,
measurement accuracy stopped around 28 μm.
the measurement accuracy stopped around 28 µm.
5.
5. Conclusions
Conclusions
For
coaxialholes
holeswith
with
roundness
error—such
as the crankshaft
of an
internal
For coaxial
lowlow
roundness
error—such
as the crankshaft
hole of anhole
internal
combustion
combustion
engine—this
paper
proposes
a
simple
inner
diameter
measurement
method
for
engine—this paper proposes a simple inner diameter measurement method for coaxialcoaxial
holes.
holes.
A multi-layer
diameter
measurement
is designed,
has a single
onlayer.
each layer.
A multi-layer
diameter
measurement
rod is rod
designed,
whichwhich
has a single
sensorsensor
on each
In the
In
the measurement
process,
we adjusted
the machining
datum
line of hole
coaxial
hole
part,itso
it is
measurement
process,
we adjusted
the machining
datum line
of coaxial
part,
so that
is that
collinear
collinear
axis of measuring
By revolving
the measuring
rod and
immune
detecting
with the with
axis the
of measuring
rod. Byrod.
revolving
the measuring
rod and
immune
fromfrom
detecting
the
the
measuring
rod’s
rotation
angle,
all
diameters
of
coaxial
holes
can
be
calculated
by
sensors'
measuring rod’s rotation angle, all diameters of coaxial holes can be calculated by sensors'values.
values.
For
For the
the measurement
measurement process,
process, the
the influence
influence of
of the
the installation
installation posture
posture of
of the
the measuring
measuring rod
rod to
to the
the
measurement
results
is
analyzed
by
numerical
analysis,
and
the
tolerance
range
of
measuring
rod
measurement results is analyzed by numerical analysis, and the tolerance range of measuring rod
installation
installation error
error is
is obtained
obtained by
by simulation.
simulation. Two
Two 150
150 mm
mm ring
ring gauges
gauges are
are used
usedto
toverify
verifythe
themeasuring
measuring
method
in
the
experiment,
by
the
comparison
between
the
measurement
results
and
indicating
method in the experiment, by the comparison between the measurement results and indicatingvalue
value
of
measurement
precision
of of
thisthis
method
hashas
achieved
30 μm
by
ofthe
thering
ringgauge,
gauge,ititisisproven
proventhat
thatthe
the
measurement
precision
method
achieved
30 µm
the
use
of
the
20
μm
linearity
LDS.
For
coaxial
holes
with
different
sizes
and
number
of
holes,
this
by the use of the 20 µm linearity LDS. For coaxial holes with different sizes and number of holes,
method
is simple
to implement
the the
diameter
measurement.
The
retrofit
ofofthe
this method
is simple
to implement
diameter
measurement.
The
retrofit
themeasuring
measuringrod
rod is
is
inexpensive
and
simple,
which
can
be
easily
applied
in
industrial
use
for
rapid
measurement.
inexpensive and simple, which can be easily applied in industrial use for rapid measurement.
Acknowledgments: This work was supported in part by the National Science and Technology Major Project of
Acknowledgments:
Thisin
work
supported
part
by theInstrument
National Science
and Technology
Major Project
Project
China
(2016ZX04003001),
part was
by the
National in
Key
Scientific
and Equipment
Development
of China (2016ZX04003001), in part by the National Key Scientific Instrument and Equipment Development
(2013YQ170539),
and in part
Ship Research
Project (1st
phase
Low-speed
Marine Engine
Project (2013YQ170539),
andby
in the
partHigh-tech
by the High-tech
Ship Research
Project
(1stofphase
of Low-speed
Marine
Engineering).
Engine Engineering).
Author
L.W.,F.Y.
F.Y.
and
conceived
and designed
the experiments;
L.W.,
F.Y. performed
and T.Y.
Author Contributions: L.W.,
and
L.F.L.F.
conceived
and designed
the experiments;
L.W., F.Y.
and T.Y.
the experiments;
L.W. analyzed
data; Z.W.
contributed
materials; L.W.
wrote L.W.
the paper.
performed
the experiments;
L.W.the
analyzed
the and
data;C.L.
Z.W.
and C.L. contributed
materials;
wrote the paper.
Conflicts of
of Interest:
Interest: The
The authors
authors declare
declare no
no conflict
conflict of
of interest.
interest.
Conflicts
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