Niigata University. JAPAN The pair of spiral bevel gears fabricated by Klingelnberg's cyclo-palloid method can be expected to have a good contact pattern because the gear tooth surfaces are fundamentally conjugate. In practice, however, the first gear pair having a good contact pattern is difficult to obtain because of the machine setting errors such as tool setting errors and workpiece setting errors in gear cutting. In this paper, an inspection method for detecting the machine setting errors in a spiral bevel gear cutting in a Klingelnberg cyclo-palloid system is proposed. In this method, the coordinates of many points on the gear tooth surface are measured using a coordinate measuring machine and a geometrical gear tooth surface expressed as the functions of the machine settings is estimated by the method of least squares so that the surface can fit the measured data. The deviations of the estimated machine settings from the design values are considered to be the machine setting errors. The machine setting errors in gear cutting were detected using this method and corrective cutting which compensated for the errors was carried out. Consequently, more accurate gear could be obtained. Key Words: Measurement, Gear, Inspection, Spiral Bevel Gear, Klingernberg Cyclo-Plloid System, Method of Least Squares, Machine Setting Error, Coordinate Measuring Machine 1. Introduction The pair of spiral bevel gears in a Klingelnberg cyclo-palloid system can be expected to have a good contact pattern because the tooth surfaces are fundamentally conjugate. In practice, however, a gear pair with a good contact pattern is difficult to obtain because of machine setting errors such as tool setting and workpiece setting errors in bevel gear cutting. In order to develop a good contact pattern, it is necessary to recut on the basis of contact pattern observation. Such recutting depends heavily on skillful workmanship or on trial-and-error cutting methods. One of the authors has proposed an inspection method for detecting machine setting errors in order to obtain an accurate gear without recutting based on the contact pattern observation. The errors are detected from tooth surface measurement data using a coordinate measuring machine (CMM). In the previous paper, the machine settings in which errors may occur were called error factors and the selection method of the error factors to be detected was presented. In this selection method, an error factor having significant influence on the gear tooth surface must be selected. This method was applied to hypoid gears and the machine setting errors were successfully detected. However, this method was not successful in application to spiral bevel gears in a Klingelnberg cyclo-palloid system. It was clear that the method broke down in the case that the error of the selected error factor was small. We found that the previous result was acceptable when the error of the error factor happened to be large. In this paper, a new method for finding the error factors from the tooth surface measurement data, namely, a selection method of error factors, is proposed, and the inspection method is applied to spiral bevel gears in a Klingelnberg cyclo-palloid system. 2. Basic Concept of Inspection Method In spiral bevel gear cutting, machine setting errors such as tool setting and workpiece setting errors are unavoidable. Each machine setting error influences the gear tooth surface. This study is based on such concepts. The coordinates of many points on the gear tooth surface are measured using a CMM and a theoretical gear tooth surface expressed as a function of the machine settings is estimated by the method of least squares so that the surface fits the measured data. The deviations of the estimated machine settings from the design settings are considered to be the machine setting errors in gear cutting. These errors can be fed back to the gear cutting process, so that more accurate gears are obtained. 3. Inspection of Spiral Bevel Gears For measurement, the spiral bevel gear is set arbitrarily on a CMM. The positions of the gear axis and the datum plane must be determined by measurement independent of the tooth surface measurement because the gear is set arbitrarily. We can make the origin ƒg and gear axis zg in the coordinate system ƒg-xgygzg fixed to the gear coincide with the origin ƒm and axis zm in the coordinate system ƒm-xmymzm fixed to the CMM, as shown in Fig.1. However, the angle by which the gear is rotated about its axis is uncertain. Therefore, we must define an unknown angle ƒ³ between axis xg and axis xm. Figure 2 shows that the tooth surface Xm and the spherical probe of radius are in contact with each other at point Qm. The coordinates of the probe center Pm are theoretically expressed as a position vector P(Px,Py,Pz) in ƒm-xmymzm. P = X‚@+ r‚ON‚ ¥¥¥¥¥¥¥(1) On the other hand, the coordinates of the probe center Pm are measured by the CMM. The measured coordinates are expressed as a position vector M(Mx,My,Mz). P and M expressed in the rectangular coordinate system ƒm-xmymzm are transformed into a cylindrical coordinate system ƒm-ƒÁmƒÆmzm the cylindrical axis of which is as shown in Fig.3: P = (P‚’, PƒÆ, P‚š)‚s@@@@@@@@@@ @@@@@@@@@@@@@@@@@ @@ ¥¥¥¥¥¥¥(2) M = (M‚’, MƒÆ, M‚š)‚s Under this transformation, each component (Px,Py,Pz) of P is expressed in the following form : P‚’ = P‚’(ƒË,ƒÕ; C‚P, C‚Q, ¥¥¥ , C‚‹) PƒÆ = PƒÆ(ƒË,ƒÕ;ƒ³, C‚P, C‚Q, ¥¥¥ , C‚‹) (3) P‚š = P‚š(ƒË,ƒÕ; C‚P, C‚Q, ¥¥¥ , C‚‹) where ƒÒ and ƒÕ are the parameters expressing the tooth surface and are variable. On the other hand,C1,C2,¥¥¥¥,Ck are the parameters expressing the machine settings and are invariable because the machine settings do not change during the gear cutting process. When the values of C1,C2,¥¥¥¥,Ck are different from the respective design values, we consider that there are machine setting errors in bevel gear cutting. In this section, we also express the error factors corresponding to these errors as C1,C2,¥¥¥¥,Ck. Let the components Pr and Pz be equal to the components Mr and Mz respectively: M‚’ - P‚’(ƒË,ƒÕ; C‚P, C‚Q, ¥¥¥ , C‚‹) = 0 ¥¥(4) M‚š - P‚š(ƒË,ƒÕ; C‚P, C‚Q, ¥¥¥ , C‚‹) = 0 From Eq.(4), parameters ƒÒ and ƒÕ can be determined, being independent of ƒ³ but being dependent on C1,C2,¥¥¥¥,Ck. The determined parameters ƒÒ and ƒÕ are substituted into PƒÆ. Then, the residual E is defined as the difference between the arguments MƒÆ and PƒÆ:
Fig.1 Relation between coordinate systems ƒm-xmymzm and ƒg-xgygzg
Fig.2 Measurement of tooth surface
Fig.3 Transformation into cylindrical coordinate system
E = MƒÆ - PƒÆ(ƒ³; C‚P, C‚Q, ¥¥¥ , C‚‹) ¥¥¥¥¥¥¥(5)
The coordinates of n points on the gear tooth surface are measured and each
residual Ei(i=1,2,¥¥¥¥,n ) corresponding to the coordinates of the i-th measured
point
Mi is calculated from Eq.(5). We generally cannot obtain the values of ƒ³,C1,C2,¥¥¥¥,Ck
so that all the Ei(i=1,2,¥¥¥¥,n) are equal to zero. Therefore, we estimate the
values of ƒ³,C1,C2,¥¥¥¥,Ck so that the sum of the squares of Ei(i=1,2,¥¥¥¥,n) can be
minimized, namely, we estimate these values by the method of least squares. The sum
of the squares of Ei, denoted by F, is as follows:
n
F = ƒ° E‚‰‚Q@@@@@@@@@@@@@@¥¥¥¥¥¥¥(6)
i=1
However, the extent of the real gear tooth surface is limited compared with the
theoretical gear tooth surface, the gear tooth surface is a generated complex
surface, and the errors of C1,C2,¥¥¥¥,Ck are small, so that the following two cases
may occur. (i) One is that some of the errors in gear cutting have similar
influences on the real tooth surface. In this case, it is difficult to distinguish
the influences of those errors. (ii) The other is that errors having very significant
influences and errors having slight influences on the tooth surface are both
contained in C1,C2,¥¥¥¥,Ck. In this case, the errors with slight influences are
sometimes concealed by the errors with significant influences, so that the errors
with slight influence cannot be distinguished. In either case, it is difficult to
estimate all the values of C1,C2,¥¥¥¥,Ck simultaneously. In the previous paper,
the case (i) was evaluated in terms of the independence among plural error factors,
the case (ii) was evaluated in terms of the influence coefficients, and we selected
the error factor to be detected according to these two evaluation criteria.
When we consider estimation of the value of C1 as an example, the value of ƒ³
(Fig.1) must also be estimated simultaneously because it is an uncertain angle
independent of the machine setting errors. The simultaneous equations (7) are
obtained by the method of least squares:
Ý F
„Ÿ„Ÿ„Ÿ„Ÿ„Ÿ„Ÿ = 0
Ý C‚P
¥¥¥¥¥¥¥(7)
Ý F
„Ÿ„Ÿ„Ÿ„Ÿ„Ÿ„Ÿ = 0
݃³
4. Selection Method of Error Factors
In the previous paper , we selected the error factor having most significant
influence on the tooth surface beforehand and estimated it. In the previous method,
however, the estimated theoretical gear tooth surface does not always fit the
measured data well and accuracy of fit is not satisfactory in the case when the error
of the error factor selected beforehand is small and the errors of other error
factors are large. In this case, if the deviation of the estimated value from the
design value is considered to be the machine setting error and the deviation is fed
back to the gear cutting process, more accurate gears cannot be obtained. Therefore,
the error factors of the occurrence of errors must be found from the measured data.
In this section, a method for finding the error factors of the occurrence of errors,
namely, a selection method of error factors, is proposed for the first time.
We estimate the values of error factor C1 and angle ƒ³ from Eq.(7) by assuming
that the errors of other error factors are zero. Then, we estimate the values of C2
,C3,¥¥¥¥,Ck in the same manner as for C1. After these estimates, we calculate ƒ¢t
according to Eq.(8) for each C2,C3,¥¥¥¥,Ck.ƒ¢t corresponds to standard deviation in
statistics and means the accuracy of fit which expresses whether the estimated tooth
surface fits the measured data or not. Ģt is calculated using the following equation:
F
ƒ¢t = S‚’ ã „Ÿ„Ÿ„Ÿ„Ÿ„Ÿ„Ÿ ¥¥¥¥¥¥¥(8)
n
where Sr denotes the mean radius of the gear. When the mean cone distance of the
gear is denoted by Rm and the pitch cone angle of the gear is denoted by ā,Sr=Rm sinā.
When the estimated tooth surface fits the measured data Mi(i=1,2,¥¥¥¥,n) exactly,ƒ¢t
is zero. In other words, when the value of Ģt is small, we consider that the tooth
surface is estimated well by the measured data Mi. Hence, the error factor
indicating the smallest Ģt is considered first as an error factor to be detected.
However, it is uncertain whether the errors of other error factors are small or
not. Therefore, we estimate the values of other error factors separately and
calculate each Ģt using the estimated value of the first error factor. If there is
another error factor for which Ģt is smaller than the first one, this error factor
is also considered as an error factor to be detected. Otherwise, only the first
error factor is considered as an error factor to be detected.
As mentioned above, the deviations of the estimated values of the selected error
factors from the design values are considered to be the machine setting errors and
corrective cutting which compensates for the errors is carried out.
5. Tooth Surface of Spiral Bevel Gear in Klingelnberg Cyclo-Palloid System
It is necessary to express the equation of the generated gear tooth surface Xm
and that of its unit normal Nm in this method. Therefore, Xm and Nm are expressed
mathematically in this section.
Since the spiral bevel gear in a Klingelnberg cyclo-palloid system is generated
by an imaginary crown gear whose tooth trace is a trochoid curve, the equations of
the generated gear tooth surface and of its unit normal can be expressed by using the
equation of the tooth (tool) surface of the crown gear. The concave tool surface of
the crown gear is considered in this section, but a similar analysis may be applied
to the convex tool surface.
Figure 4 shows the geometrical interrelations of a Klingelnberg
cyclo-palloid system. A rolling circle R rolls on a base circle Q which is on the
pitch surface of the imaginary crown gear. r and q denote the radii of the circles
R and Q respectively. ƒ-xyz is the coordinate system fixed to the crown gear and
axis z is the crown gear axis. Axis y coincides with the pitch surface generator .
P is a point fixed to the circle R. When the rolling circle R rolls on the base
circle Q, the locus on the pitch surface described by the point P is a trochoid
curve.
The coordinate system ƒ-xyz fixed to the circle R is introduced in order to
express the equation of the tool surface of the crown gear. The cutter is attached
to the rolling circle R. ƒPƒc is the machine distance and is denoted by Md. The
radii r and q are represented by the equations:
Z‚— M‚„ sinƒÅ
r = „Ÿ„Ÿ„Ÿ„Ÿ„Ÿ„Ÿ„Ÿ„Ÿ„Ÿ„Ÿ„Ÿ„Ÿ„Ÿ„Ÿ„Ÿ„Ÿ , q = M‚„ - r ¥¥¥¥¥¥¥(9)
Z‚‚ + Z‚— sinƒÅ
where Zb is the number of teeth of the generated gear and Zw is the number of starts
of the cutter head.
The cutter and the coordinate system fixed to the cutter are shown in Fig.5.
Although a cutter with linear cutting edges is considered in this paper, a similar
analysis may be applied to a cutter with circular cutting edges . is the cutter
radius,rc is the pressure angle of the cutter blade, and ƒÁ is the parameter
representing the linear cutting edge. The edge Xc is expressed in ƒc-xcyczc by the
following equation:
0
X‚ƒ(ƒÉ) = m r‚ƒ + ƒÉtanƒÁ n ¥¥¥¥¥¥¥(10)
ă
The surface of the locus described by Xc in ƒ-xyz is expressed as:
X (ƒË,ƒÉ) = C (ƒÆ‚P) X‚ƒ(ƒÉ) + D (ƒË) ¥¥¥(11)
where
cosƒÆ‚O= (M‚„‚Q + R‚‚Q - r‚ƒ‚Q) / 2 M‚„R‚
cosƒ¦‚O= (R‚‚Q + r‚ƒ‚Q- M‚„‚Q) / 2 R‚r‚ƒ
@ƒÆ‚P(ƒË) = M‚„ƒË/ r + ƒ¦‚O ¥¥(12)
- M‚„sin(ƒË-ƒÆ‚O)
@@@@@D (ƒË) = m M‚„cos(ƒË-ƒÆ‚O) n
0
and C is the coordinate transformation matrix with respect to the rotation about axis z:
@@@@@ @cosƒÆ‚P@|sinƒÆ‚P 0
C(ƒÆ‚P) = m sinƒÆ‚P cosƒÆ‚P 0 n ¥¥¥¥¥¥(13)
0 0 1
Fig.4 Tooth trace of imaginary crown gear
Fig.5 Cutter with linear cutting edges
Fig.6 Locus of cutting edge
In Eqs.(11) and (12), ƒÒ is a parameter which represents the swivel angle of the
cutter around axis z (see Fig.6). X expresses the equation of the tooth (tool)
surface of the imaginary crown gear whose tooth trace is a trochoid curve in a
Klingelnberg cyclo-palloid system. The unit normal of X is expressed by N.
The imaginary crown gear is rotated about axis z by angle ƒÕ and generates the
tooth surface of the spiral bevel gear. We call this rotation angle ƒÕ of the crown
gear the generating angle. When the generating angle is ƒÕ, the coordinate system
ƒ-xyz is rotated about axis z by ƒÕ in the coordinate system ƒ-xyz fixed in space.
Therefore, X and N are rewritten as XƒÕ and NƒÕ in ƒ-XYZ :
XƒÕ(ƒÒ,ƒÉ;ƒÕ) = C (ƒÕ) X (ƒÒ,ƒÉ)
¥¥(14)
NƒÕ(ƒÒ,ƒÉ;ƒÕ) = C (ƒÕ) N (ƒÒ,ƒÉ)
When ƒÕ is zero,ƒ-XYZ coincides with ƒ-xyz. Assuming the relative velocity W(XƒÕ)
between the crown gear and the generated gear at the moment when generating angle is ƒÕ
, the generating condition of the gear teeth is as follows:
NƒÕ(ƒÒ,ƒÉ;ƒÕ) ¥ W(ƒÒ,ƒÉ;ƒÕ) = 0 ¥¥¥¥¥¥¥(15)
from which the following equation is obtained:
ƒÉ=ƒÉ(ƒË;ƒÕ) ¥¥¥¥¥¥¥(16)
Substituting Eq.(16) into Eq.(14),XƒÕ{ƒÒ,ƒÉ(ƒÒ;ƒÕ);ƒÕ} which expresses the generating line
of the gear tooth surface and its unit normal NƒÕ{ƒÒ,ƒÉ(ƒÒ;ƒÕ);ƒÕ} in ƒ-XYZ can be obtained.
As shown in Fig.7, XƒÕin ƒ-XYZ is transformed into ƒg-xgygzg and is denoted by Xg.
Since ƒg-xgygzg is rotated by angle -ƒÕ /sinƒÅ,Xg is represented by the equation:
-ƒÕ ƒÎ
X‚‡(ƒË,ƒÕ) = C|‚Pi„Ÿ„Ÿ„ŸjA|‚Pi„Ÿ„Ÿ+ƒÅ jXƒÕ(ƒÒ,ƒÕ)
sinā 2
¥¥¥¥¥¥¥¥¥(17)
where A is the coordinate transformation matrix with respect to the rotation about
axis X:
1 0 0
ƒÎ ƒÎ ƒÎ
Ai\+āj=m 0 cosi\+āj @|sini\+ā n
2 2 2
ƒÎ ƒÎ
0 sini\+āj cosi\+āj
2 2
¥¥¥¥¥¥¥(18)
In Eq.(17), regarding ƒÒ and ƒÕ as parameters, surface Xg expresses the equation of the
generated gear tooth surface. The unit normal Ng of Xg is
-ƒÕ ƒÎ
N‚‡(ƒË,ƒÕ) = C|‚Pi„Ÿ„Ÿ„ŸjA|‚Pi„Ÿ„Ÿ+ƒÅj NƒÕ(ƒÒ,ƒÕ)
sinā 2
¥¥¥¥¥¥¥¥¥(19)
Xg and Ng in ƒg-xgygzg are transformed into ƒm-xmymzm and are denoted by Xm and Nm in
Eq.(1). Equations (17) and (19) include Md,rc,ƒÁ,ƒÅ, etc. which correspond to C1,C2
,¥¥¥¥,Ck in Eq.(3).
Fig.7 Relation between coordinate systems ƒ-XYZ and ƒg-xgygzg
Fig.8 Gear setup on CMM
When these values are different from the design values, there
are machine setting errors in gear cutting.
6. Results of Inspection
An inspection of the spiral bevel gear in a Klingelnberg cyclo-palloid system
was carried out in order to confirm the validity of the proposed method. The
dimensions of the inspected gear are shown in Table 1. The cutter and machine
settings are shown in Table 2. A spherical probe of radius r0=0.997mm was used in
the measurement shown in Fig.8. We measured twenty six points on the tooth surface
in order to avoid the influence of accidental errors.
The error factors in spiral bevel gear cutting may include machine distance Md
with respect to the gear cutting machine, cutter radius rc and pressure angle ƒÁ with
respect to the cutter, and pitch cone angle ā, mounting distance L (see Fig.7), and
offset distance lx along the X axis in ƒ-XYZ, with respect to the workpiece setting.
The values of error factors and ƒ³ were estimated from the measured data
separately in order to find the error factors of the occurrence of errors and each
accuracy of fit Ģt was calculated. These results are shown in Table 3. The
deviations of the estimated values of error factors from the design values are shown
as errors. Since the error factor whose Ģtis the smallest is the mounting distance L
, the theoretical gear tooth surface corresponding to the mounting distance L+ĢL
=101.00-0.25=100.75mm can be considered to fit the measured data best. In other
words, the gear tooth surface is actually cut with the mounting distance L in which
the error ĢL=-0.25mm occurs although the surface should be cut with L equal to
101.00mm.
Table 1 Dimensions of spiral bevel gear Table 2 Cutter and machine settings
Table 3 Results of inspection Table 4 Results of inspection for ĢL=-0.250mm
Table 5 Results of inspection after corrective cutting
However, it is uncertain whether the errors of other error factors are
small or not. Therefore, we estimated the values of error factors Md,rc,ƒÁ,ƒÅ,lx
and each value of ƒ³ separately, and calculated each value of ƒ¢t for L of 100.75mm in
order to determine the values of other error factors. These results are shown in
Table 4 in the same manner as those in Table 3. No errors occur in Md,rc,ƒÁ,ƒÅ and lx
because each accuracy of fit ƒ¢t is nearly equal to 3.7ƒÊm which is ƒ¢t of L in Table 3.
In other cases, if one of the Ģt shown in Table 4 is smaller than the smallest
value 3.7ƒÊm of ƒ¢t in Table 3, we consider that the error of the corresponding error
factor will occur. From Table 4, we can consider that error occurs only in L.
Therefore, corrective cutting increasing L by 0.250mm was carried out. Then, the
values of L and ƒ³ were estimated, and the accuracy of fit ƒ¢t was calculated. The
results after the corrective cutting are shown in Table 5. The value of Ģt with
respect to L is very small. Furthermore, the gear tooth surface is cut accurately
because the estimated error of L is 0.004mm which is very small. These results show
the efficacy of the corrective cutting.
As mentioned above, the validity of the proposed method was confirmed.
7. Conclusion
Although the pair of spiral bevel gears in a Klingelnberg cyclo-palloid system
can be expected to have a good contact pattern, the first accurate gear pair is
difficult to obtain because of machine setting errors such as tool setting and
workpiece setting errors in gear cutting. Therefore, gear recutting based on
observation of the contact pattern, which depends heavily on skillful workmanship or
on trial-and-error cutting methods is necessary in order to develop a good contact
pattern.
In this paper, an inspection method for detecting the machine setting errors in
spiral bevel gear cutting in a Klingelnberg cyclo-palloid system was proposed in
order to obtain an accurate gear without recutting based on contact pattern
observation. In this method, a theoretical gear tooth surface expressed as a
function of the machine settings is estimated from the tooth surface measurement data
using a coordinate measuring machine and the machine setting errors are detected.
In this study, we evaluated many error factors with respect to the machine
settings from the measured data on the gear tooth surface. As a result, we could
detect the machine setting errors. Furthermore, corrective cutting which compensated
for the errors was carried out, so that more accurate gears could be obtained.
We acknowledge the manager Koreaki Ichino of Nagaoka Gear Works Co., Ltd. in
Japan for his support in gear inspection and gear cutting.
References
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(3) Tamura, H., A Study on the Method of Inspecting a Spiral Bevel Gear, Trans. Jpn.
Soc. Mech. Eng., (in Japanese), Vol.52, No.478, C(1986), p.1798-1804.
(4) Oya, M., Yoshimura, K. and Tamura, H., A New Method of Inspecting an Hourglass Worm,
Trans. Jpn. Soc. Mech. Eng., (in Japanese), Vol.58, No.547, C(1992), p.864-869.
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