Optimization and active control of internal gearing power
honing (IGPH) process parameters for excellent and stable gear precision
were carried out using the engagement theory of a conjugate curved face, the
Box–Behnken design of experiments method, and the artificial immune clone
selection algorithm (AICSA). Optimization and active control were carried
out in four stages. In the first stage, the second-order models of tooth
profile deviations were developed considering the nonlinear influence of
IGPH process parameters on tooth profile deviations based on the Box–Behnken
design. In the second stage, a method for solving the multi-objective
optimization of the IGPH process was presented based on building the
synthetic tooth profile deviation model, which considered the different
weighting factors of different tooth profile deviation indexes. In the third
stage, excellent gear precision was obtained by importing the ranges of
synthetic tooth profile deviation and parameters into the AICSA. In the
fourth stage, based on the optimized process parameters, the active control
of IGPH process parameters was realized based on the constant cutting speed
on the fixed position of the gear tooth surface. The total gear profile
error reached a minimum value at the optimal parameters of 1270.4 rpm for spindle speed,
60 mm min-1 for axis
feed velocity, 2.4 µm per oscillation for radial feed velocity, and 2.4 spark-out times. The
gear accuracy test results show that the total gear profile error value from
the above active control method is more stable and lower than that without
active control, indicating that the proposed method is effective.
Introduction
Internal gearing power honing (IGPH) is a gear finishing process via abrasion
that can improve gear tooth surface quality; it is thus widely used in the
field of advanced automobile gearbox systems. The special mechanism of the
IGPH process can form a kind of curved orientation texture and high
compressive residual stress on workpiece gear tooth surfaces (Karpuschewski
et al., 2008). Moreover, these good tooth surface characteristics can reduce the
meshing noise and prolong the service life of gear transmission systems.
Compared with the worm wheel gear grinding process, the IGPH process is
irreplaceable in stepped gear finishing, and its lower cutting velocity can
avoid high thermal load and burns on the tooth surface. However, the gear
precision of the IGPH process ranges in the GB4–GB6 level, while the worm
wheel gear grinding process can reach the GB4–GB5 level, which is the
limitation of the IGPH process. Process parameters have a significant effect
on gear precision, and, thus, in this paper, several experiments on
the optimization of IGPH process parameters for better and more stable gear
precision were carried out.
Recently, many studies have focused on the improvement of gear finishing
with abrasive processes. For instance, Teixeira et al. (2019) reported the
influence of the grains' characteristics and process parameters on the
material removal behavior and gear grinding energy model based on modeling
of the normal force. Through the use of high-order topology expression and
analysis of the numerical loaded tooth contact area, Ding and Tang (2020) proposed
a target grinding flank modeling method to improve the tooth flank
geometric topography and loaded contact performance. Furthermore, Giacomo
et al. (2019) analyzed the relationship between the thermal damage and gear
grinding process parameters on a new dry-grinding machine and found that the
gear module and radial feed rate were the most significant factors affecting the thermal damage. Yoshikoto et al. (2018) developed a high-precision,
high-efficiency internal gear grinding method by setting a large
crossed-axis angle between the grinding wheel spindle and workpiece gear
axis and conducted a series of actual grinding experiments to verify the
new proposed grinding model.
With the wide application of the IGPH process, several studies were carried
out to explore the mechanism and improve the gear tooth surface quality of
the IGPH process. By adding three axes' additional motions, i.e., the honing
wheel spindle axis, swivel axis, and workpiece spindle axis in the internal
gearing power honing machine, Vanquyet and Yuren (2020) built a numerical model of
the closed-loop topology modification for the double-crowned and anti-twist
gear tooth surfaces and verified the validity of the proposed method. A
series of preliminary research works were carried out by our team based on
traditional roughness theory and the internal engagement theory of conjugate
curved surfaces. For example, Han et al. (2017) performed a contrastive analysis
of tooth surface quality between the IGPH process and gear grinding process.
So far, the main point of the IGPH process is ensuring the microscopic
qualities of the tooth surface. In the present work, therefore, to improve
the macroscopic quality of the tooth surface for IGPH gears, three main gear
tooth profile errors and four input IGPH process parameters were chosen as
the factors and indexes of the optimization experiment. The Box–Behnken
design of experiments (DOE) method and artificial immune clone selection
algorithm (AICSA) were the main optimization methods used to find the
predicted model of gear tooth profile deviations and achieve the best gear
precision for the IGPH process. Finally, an active control method was
proposed to deal with the unstable gear precision caused by the variable IGPH
crossed-axis angle parameter.
Basic gear profile error model
Design of experiments (DOE) is an important branch of mathematical statistics
which has been widely used to find the influence rule between
factors (x0, x1, x2, ..., xn) and
responses (y0, y1, y2, ..., yn) in scientific
research and industry. Response surface methodology is one of the DOE methods,
which was proposed by Box and Wilson in 1951 (Korra et al., 2014).
Generally, the purpose of response surface methodology is to find the
optimal process through building the approximation surface model. If factors
have a linear effect on responses, the approximation surface model between
responses and factors can be expressed by the first-order Taylor series
expansion:
y(x)=β0+β1x1+β2x2+…+βkxk+ε.
If there are some nonlinear factors in the system, the first-order Taylor
series expansion cannot express the system model accurately, and mostly the
system model will be comprised of second-order Taylor series expansion:
y(x)=β0+∑i=1kβixi+∑i=1kβiixi2+∑i<jk∑βijxixj+ε,
where y(x) is the response target function; k is the number of factors;
βi, βii, and βij are the influence
coefficient of each factor; xi and xj are the values of each factor; and
ε is the additive constant.
Diagrams of gear profile errors.
This study aims at achieving minimum gear profile errors. According to the
standards of gear accuracy, a high-precision gear will have a good tooth
profile accuracy, lead accuracy, and rotation accuracy. In the actual
production, the total profile deviation (Fα), the total helix
deviation (Fβ), and the total cumulative pitch deviation (FP)
embody these three kinds of gear accuracies, as shown in Fig. 1. In this
figure, Lβ is the measurement range along tooth longitudinal
direction, b is the tooth width, B is the central position of the range,
Lα is the measurement range of tooth profile, and LAC is the
measuring range of tooth profile. The input parameters of honing process
include the spindle speed SW (rpm), the axis feed velocity fz
(mm min-1), the radial feed velocity fx (µm per oscillation), and the
spark-out times ts. Thus, there are four factors and three
responses in this study, and the basic model of the total profile
deviation (Fα), the total helix deviation (Fβ), and the
total cumulative pitch deviation (FP) were built as in Eq. (3):
y(x)=β0+β1x1+β2x2+β3x3+β4x4+β11x12+β22x22+β33x32+β44x42+β12x1x2+β13x1x3+β14x1x4+β23x2x3+β24x2x4+β34x3x4+ε.
The left hand of y(x) is a polynomial which is the approximate value of
reality, and the coefficients of the model were estimated by the
least-squares method in matrix form. The estimated coefficients of the model
depend on the minimum error, and it can be expressed by Eq. (4):
Qβ0,β1,⋯,β13,β23=∑i=0nyi-β0-β1x1-β2x2-⋯-β13x1x3+β23x2x32→min.
The extreme value of Q(β0,β1,⋯,β13,β23) can be expressed by Eq. (5):
∂Q∂βij=-2∑i=0n(yi-β0-β1x1-β2x2-⋯-β13x1x3+β23x2x3)xixj=0.
Equation (5) can be expressed in matrix form by Eq. (6):
XTXβ=XTy,
and the value of β can be calculated by Eq. (7):
β=(XTX)-1XTy.
After solving the coefficient of the model, it is necessary to verify the
accuracy of the model. The multiple correlation coefficient (R) can be
calculated by Eq. (8–10):
8R=SSR/SSY=(SSY-SSE)/SSY9SSY=∑i=1m(yi-y¯)210SSE=∑i=1m(yi-ỹi)2,
where SSY is the total sum of squares for error, SSE is the total sum
of squares for regression, SSR is the regression sum of the square,
y¯ is the mean of response, ỹi is the value
of the response, and R represents the accuracy of the model; the larger the value, the more
accurate.
To screen out the significant factors, the associated F value or p value to
the factors can solve this problem:
F=R2/m(1-R2)/(n-m-1).
Such a calculated F value will be compared with the critical value F′. If
F>F′, it means that the associated factor is significant, or
this factor is insignificant. The other way is to look at the p value of the
result, which is calculated by design expert software; if the p value is
smaller than 0.05, it indicates the model has 95 % confidence level, which
means that the model is statistically significant.
Experimental workPilot experiments
The IGPH experiments were conducted on the HMX-400 IGPH machine. Gear
profile errors were measured on the gear measuring center (Klingelnberg P40),
as shown in Fig. 1. Table 1 shows the processing parameter range of the IGPH
machine and the measuring range of the gear measuring center. Table 2 shows
the parameters of the work piece gear and the honing wheel.
HMX-400 IGPH machine and Klingelnberg P40 gear measuring
center.
The main technological parameters of the experiment equipment.
Basic parameters of workpiece gear and honing wheel.
Workpiece gear Honing wheel Material: 20CrMnTi (60HRC)Material: microcrystalline fused aluminaModule: 2.25 mmModule: 2.25Gear number: 73Tooth number: 123Helix angle: 33∘Helix angle: 41.722∘ (not fixed)Pressure angle: 17.5∘Pressure angle: 17.5∘IGPH experiments based on the Box–Behnken DOE method
British biostatistician Ronald Aylmer Fisher first proposed and established the
concept of the DOE (design of experiments) method in the 1920s (Fisher, 1954). The DOE
and ANOVA (analysis of variance) methods have been successfully used in
agricultural and biological tests since then. A good DOE can exclude most of
the interference from non-experimental factors and improve the accuracy of the
predicted model and experiment efficiency. Normally, DOE is composed of
input factors, unavoidable random factors, and subject and output indexes. In
this paper, the input factors are the four input IGPH process parameters;
the random factors are the abrasive wear, the cooling and lubrication
conditions of machine tools, and so on; the subject is the IGPH process; and
the output indexes are the three main gear profile deviations.
Common response surface methodology (RSM) includes the Box–Behnken design,
the central composite design (CCD), the central composite inscribed design (CCI), and the central composite face-centered design (CCF) method. The Box–Behnken design
method is a statistical method which is usually used to model the
relationship between the factors and responses for s nonlinear system.
Compared with the other three methods, the Box–Behnken design method possesses
the advantage of a fewer number of tests, and it can avoid exceeding the
allowable process parameters. Figure 3 shows the comparison of experiment points
between the CCD and Box–Behnken DOE method. It is shown that the star points
would exceed the setting process parameter range, and the Box–Behnken DOE
method can avoid this problem. Thus, the Box–Behnken DOE method is the most
suitable method for this study considering the cost and safety of the
experiment.
Comparison of experiment points between the CCD and Box–Behnken
DOE method.
In this study, the main IGPH process parameters were the spindle speed
SW (rpm), the axis feed velocity fz (mm min-1), the radial feed
velocity fx (µm per oscillation), and the spark-out times ns. The main gear profile deviations were the total profile
deviation (Fα), the total helix deviation(Fβ), and the
total cumulative pitch deviation (Fp). Based on the principle of the Box–Behnken
DOE method and the input IGPH process parameters, the experiment levels were
carried out as shown in Table 3. The common spindle speed ranged from 800 to
1800 (rpm), the axis feed velocity ranged from 60 to 200 (mm min-1), the
radial feed velocity ranged from 2 to 8 (µm per oscillation), and the spark-out times ranged from 1 to 3.
According to the different combinations of IGPH process parameter levels in
Table 3, 29 IGPH processing and gear measuring experiments in total
were carried out using the experimental equipment. Figure 4 shows the 29
workpiece gears from 29 IGPH experiments, and Table 4 shows the
measured gear profile errors for each IGPH experiment.
Workpiece gears of IGPH experiments.
IGPH experiments and measurement results based on
the Box–Behnken DOE method.
TrailSW (rpm)fz (mm min-1)fx (µm per oscillation)nsFα (µm)Fβ (µm)FP (µm)180060525.673.9840.92180060523.954.2147.93800200526.798.1373.241800200524.858.2378.351300130213.664.8647.461300130815.977.5967.971300130232.543.9737.981300130834.576.8259.99800130516.976.5364.7101800130515.036.9769.311800130536.025.4659.6121800130533.865.7164.113130060222.762.6725.1141300200223.555.8658.115130060824.123.9941.6161300200825.167.6376.317800130224.554.3352.9181800130223.654.9556.119800130827.687.0473.5201800130825.137.1479.121130060513.763.2637.9221300200515.867.2474.723130060532.982.6131.6241300200533.756.5565.4251300130523.714.5754.1261300130523.554.7156.2271300130523.684.7652.9281300130523.843.9854.5291300130523.864.1355.7Mathematical model
After conducting 29 IGPH experiments (Fig. 3), analysis of variance (ANOVA)
was carried out to verify the confidence of the gear profile error models
based on the response surface methodology (RSM). Equations (12)–(14) show the gear
profile error models, and the adjusted R-squared value of the model summary
statistics was below 0.9762, 0.9587, and 0.9902; the closer to 1 the
adjusted R-squared value was, the more accurate the statistical model.
12Fα=(4.181-3.678×10-3×SW+2.287×10-3×fZ-0.102×fX-0.192×nS-6.983×10-8×SW×fZ-4.729×10-5×SW×fX+4.578×10-5×SW×nS+2.258×10-5×fZ×fX-7.492×10-5×fZ×nS+1.148×10-3×fX×nS+1.390×10-6×SW2-2.020×10-6×fZ2+3.281×10-3×fX2+0.036×nS2)213Fβ=11.419-0.012×SW+0.017×fZ-0.140×fX-2.655×nS-9.286×10-7×SW×fZ-8.667×10-5×SW×fX+9.500×10-5×SW×nS+5.357×10-4×fZ×fX-1.429×10-4×fZ×nS+0.01×fX×nS+4.858×10-6×SW2-3.563×10-5×fZ2+0.054×fX2+0.576×nS214Fp=61.297-0.099×SW+0.460×fZ+2.885×fX-3.057×nS-1.357×10-5×SW×fZ-4.000×10-4×SW×fX-5.000×10-5×SW×nS+2.024×10-3×fZ×fX-0.011×fZ×nS+0.125×fX×nS+3.987×10-5×SW2-7.361×10-4×fZ2-0.056×fX2+0.068×nS2
Figure 5a–c show parts of the response surface of the main gear
profile deviation models, while the other two process parameters are set as
the common constant, i.e., the radial feed velocity fx=5 (µm per oscillation) and the spark-out times ns=2. Based on
these response surface results, the roles each IGPH process parameter plays
in influencing the gear profile deviations can be obtained. These three main
gear profile deviations decrease with increasing SW before the peak
value and then increase with increasing SW because the honing wheel
spindle vibration is becoming increasingly more intense with increasing SW.
Once a certain value is exceeded, these three main gear profile deviations
all increase with increasing fz because the cutting amount per
revolution increases. Finally, the comprehensive influence orders were
Fα (fx>SW>fz>ns), Fβ (fz>fx>ns>SW), and Fp (fz>fx>ns>SW).
The influence roles of each IGPH process parameter on the gear
profile deviations.
Optimization
After building the gear profile error models, the optimization studies were
carried out based on the AICSA. First, a total gear profile deviation model
is needed before the optimization process. Because these three profile
deviations have different levels, the results cannot arrive at optimal
values when using a simple addition of three profile deviation functions. In
this work, a total gear profile deviation model Fall was proposed based
on different weighted coefficients of the three gear profile deviation
models according to the same precision as from the Chinese Standard No. GBT
10095.1-2008.
Fall=1SD(Fα)Fα+1SD(Fβ)Fβ+1SD(Fp)Fp=118Fα+118Fβ+150Fp
AICSA is a learning algorithm based on the artificial immunity system which
performs the autoimmune mechanism of antigen and antibody combination.
The AICSA optimization procedure was executed using MATLAB in a Windows
operating system, and the optimization process started with a population
number of 40, a scale of 10 antibody libraries, and an inhibitory factor of 0.5 that evolved up to 500 iterations. The flowchart and calculation processes of the
optimization are shown in Fig. 6. The optimization line stabilized after
75 iterations. The optimal combination of IGPH process parameters
based on the AICSA method is shown in Table 5. The comparison experiment of
IGPH process between the optimization IGPH process parameters and actual
production experience parameters was carried out. Table 6 shows the repeated
IGPH process based on the optimization IGPH process parameters, and the
result shows that the total profile deviation Fα is approximately
flat, and the total helix deviation (Fβ) and the total cumulative pitch
deviation (Fp) reduced 39.5 % and 59.5 % compared with the original
parameters, which achieved the target of improving the gear precision.
The flowchart and the interactions process of AICSA method.
The optimal combination of IGPH process parameters based on
the AICSA method.
Five repeated IGPH experimental results based on the
optimization process parameters.
No.Total profileTotal helixTotal cumulativeTotal geardeviationdeviationpitch deviationprofile deviationFα (µm)Fβ (µm)Fp (µm)Fall (µm)12.32.420.70.67522.32.723.20.74232.12.622.50.71142.42.319.40.64952.62.420.30.684Active control of IGPH process parameters focuses on different crossed-axis
angles
In the IGPH process, the tooth profile precision of the honing wheel would
be reduced with increasing abrasive wear; therefore, a low-precision honing
wheel could increase the rejection rate of workpiece gears. Thus, when a
certain amount of a workpiece gear is honed or the radial force and torque
of a honing wheel are changed abnormally, it is necessary to use a diamond
dressing tool to dress the worn honing wheel during the dressing process.
Changing the helix angle of the honing wheel is usually carried out to
increase the repair number and service life of the honing wheel. The
crossed-axis angle between the honing wheel and workpiece gear also changes
when honing various batches of workpiece gears. However, such a change of
crossed-axis angle could influence the stability of IGPH process quality.
Different IGPH process parameters finally reflect the relative velocity
between the honing wheel and the workpiece gear tooth surface. Thus, the
relationship between the honing relative velocity, crossed-axis angle, and
IGPH spindle speed was built and analyzed in this paper. Machine operators
just need to input a suitable honing speed parameter and the basic
parameters of workpiece gear and honing wheel, and the rotation speed of
the workpiece gear or honing wheel that is needed for the numerical control
program can be obtained by the active control algorithm.
The coordinate systems of the IGPH machine and the relative
velocity on the contact point.
Figure 7a shows the coordinate system of the IGPH process built through
analyzing the structure of the IGPH machine and the motion relationship between the
honing wheel and the workpiece gear, whereby SO (O-x-y-z) and
SP(OP-xP-yP-zP) are the fixed coordinate systems of
the workpiece gear and honing wheel, which can be viewed as the initial position
of the meshing movement between the workpiece gear and the honing wheel;
S1(O1-x1-y1-z1) and
S2(O2-x2-y2-z2) are the following rotation coordinate
systems of the workpiece gear and honing wheel; a is the center distance between the
workpiece gear and honing wheel; φ1 and φ2
are the rotation angle of the workpiece gear and honing wheel; w1 and w2
are the angular velocities of the workpiece gear and honing wheel; and Σ is
the crossed-axis angle between the workpiece axis and honing wheel axis. At a
certain moment, the teeth surface of the workpiece gear and honing wheel come
into a random point M, as shown in Fig. 7b, where
vO1 is the linear velocity vector of point M on
the workpiece gear tooth surface, and vO2 is the
velocity of point M on the honing wheel tooth surface. Thus, the relative
velocity between the honing wheel abrasive and the workpiece gear tooth surface
v12 can be shown as Eq. (16) in the coordinate system
SO(O-x-y-z).
v12=vO1-vO2=wO1×rO1-wO2×rO2,
where wO1 is the angular velocity vector of
the workpiece gear and wO2 is the angular velocity
vector of the honing wheel in the coordinate system SO(O-x-y-z), and
rO1 and rO2 are the position
vector of contact points on the workpiece gear tooth surface and honing wheel
tooth surface in the coordinate system SO(O-x-y-z). The final calculated value
of the relative velocity v12 can be expressed as
Eq. (17).
v12=-ω1y+ω2ycosΣ+zsinΣω1x-ω2x+acosΣ-ω2x+asinΣ
For the convenience of calculation, the relationship between the relative
velocity value v12(M0) of point M0 and the spindle speed value
(SW) of the workpiece gear at a certain crossed-axis angle (Σ) can
be obtained as Eq. (18):
SW=30×v12π×-y+ycosΣ+zsinΣ/i12x-ω2x+acosΣ/i12-x+asinΣ/i12,
where i12 is the transmission ratio between the workpiece gear and honing
wheel.
In Sect. 4, the optimized spindle speed (SW=1270 rpm) was obtained at
the crossed-axis angle (Σ=8.722∘), and the relative
velocity value at point M0 can be calculated (v12op=2650 mm s-1) at
this fixed crossed-axis angle. Based on the relationship between the
relative velocity value v12(M0) of point M0 and the spindle
speed value (SW) of the workpiece gear at a certain crossed-axis angle
(Σ), the adaptable active control of the spindle speed value (SW)
can be realized at different crossed-axis angles. Figure 8 shows the active
control matched the curve of spindle speed value (SW) with the change of
the crossed-axis angle (Σ) from 0 to 20∘. It can
be seen that the spindle speed (SW) of the workpiece gear is 8328.8077 rpm which can be translated to the spindle speed of the honing wheel,
SH=4943.1135 rpm. But the maximum honing wheel spindle speed is
1500 rpm, which cannot support the needed spindle speed; thus, when
choosing the initial design parameters of the honing wheel, low-speed crossed-axis angle zones need to be considered as an important effect factor.
Rotation speed of the workpiece gear versus the crossed-axis
angle for stable relative velocity.
To verify the effect of this active control method, IGPH experiments were
carried out. When the crossed-axis angle arrived at 9.745,
11.745, and 13.745∘, if using the method of keeping the
stable honing velocity, the workpiece spindle speed should be suitable for
the crossed-axis angle; the adaptive honing wheel spindle speed was 669.1476 rpm, 537.2859 rpm, and 443.5825 rpm. Table 7 shows the results of the gear
profile errors based on the active control method.
The gear profile error results of IGPH experiments based on
the active control method.
The stable spindle speed of 1450 rpm was used when the crossed-axis angle
was changing. The results of the gear profile errors are shown in Table 8.
It can be seen that the total gear profile error value from the active
control method is more stable and lower than that without the active control
method. In addition, a lower spindle speed is needed for the IGPH process when the crossed-axis angle is used, which has the advantage of reducing the
power consumption and production cost.
Conclusion
In order to improve the machining precision and stability for IGPH gear,
this paper proposed a theoretical and experimental strategy for IGPH process
parameter optimization. Simultaneously, the research results of this paper
have an important theoretical basis and engineering application value to
guide the development of the gear finishing process. The following conclusions
are drawn from the experimental results:
The influences of the input IGPH parameters on the gear profile
deviation were obtained based on the Box–Behnken DOE method through 29 IGPH
experiments. The comprehensive influence orders were Fα(fx>SW>fz>ns),
Fβ(fz>fx>ns>SW), and Fp (fz>fx>ns>SW), which could provide a theoretical basis for the
improvement of gear honing quality.
A novel total gear profile deviation modeling method was presented
considering different weight coefficients on each IGPH process parameter,
which could give a new method for multi-objective optimization for the gear
honing process.
An active control method of IGPH parameters was proposed for stable
relative velocity to solve the problem of the crossed-axis angle changing
throughout the total life cycle, which could give the suitable IGPH process
parameters and honing wheel basic parameters for different situations.
Data availability
All the data used in this paper can be obtained from the corresponding author upon request.
Author contributions
BY and LX carried out the research on the algorithm and experiment of the paper, and
JH and XT provided the experiment conditions.
Competing interests
The contact author has declared that neither they nor their co-authors have any competing interests.
Disclaimer
Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Acknowledgements
The authors would like to thank the doctoral initiation fund
project of Anhui Jianzhu University (grant no. 2018QD42), the National Natural Science Foundation of China (grant nos. 51575154 and 51875161), the National Science and Technology Major Project (grant no. 2013ZX04002051), and the Fundamental Research Funds
for the Central Universities (grant no. CHD300102252507) for supporting this research financially.
Financial support
This research has been supported by the Anhui Jianzhu University (grant no. 2018QD42), the National Natural Science Foundation of China (grant no. 51575154), and the National Major Science and Technology Projects of China (grant no. 2013ZX04002051).
Review statement
This paper was edited by Jeong Hoon Ko and reviewed by Zwolak Prof. Jan and one anonymous referee.
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