Mitsubishi Heavy Industries Technical Review Vol. 57 No. 1 (June 2020)
1Improvement of Transmission Load andNoise Reduction by Optimally Design ofTooth Surfaces Using Principal Component Analysis

KATSUHIKO SHODA*1 KENSUKE NISHIURA*2

With the background of social trends toward energy saving and advances in motor
technology, drivetrains for vehicles such as automobiles have been electrified. Generally speaking,the output torque of a motor used as a prime mover with respect to the torque required for theproduct is small in many cases, so drive systems that combine a motor and a gear unit are often
used to amplify the motor torque. Since these drive systems are required to be small andlightweight and to be very quiet, the gear unit must be improved in terms of the load capability andmust have its vibration and noise reduced. It was difficult for conventional tooth surface
modification design method to simultaneously improve the load capability and reduce vibrationand noise. In this study, however, we made this possible by applying a shape representation method
using principal component analysis to tooth surface modification design. This report describes anoverview of the method and the results of the effectiveness verification.

|1. Introduction
To improve the competitiveness of transmissions, overall optimizations such as noise
reduction, reliability improvement and weight saving are needed. Currently, tooth surface
modification is examined using simple modification parameters such as tooth profile modificationand tooth trace modification(1). It is difficult, however, for this method to generate complex shapes,
so it has been difficult to achieve overall optimization design. Therefore, we applied a shape
representation method using principal component analysis to tooth surface modification design inthis study. As a result, it was made possible to efficiently generate tooth shapes that could not berepresented by conventionally used design parameters and to examine the optimal tooth surface
modification that can achieve both load capability improvement and mating excitation force
reduction. This report describes an overview of the method and the results of the effectivenessverification with analysis.

|2. Purpose of tooth surface modification
The meshing of gears is affected by “deformation due to load,” “gear error” and “stiffness
variation due to meshing” as shown in Figure 1. These cause non-uniform tooth surface contact(tooth contact) and fluctuations in the gear rotational speed. Non-uniform tooth contact increases
stress and reduces reliability, and fluctuations in rotational speed cause vibration and noise. Toothsurface modification is a method of intentionally generating a tooth surface shape that is different
from the theoretical involute tooth surface for the purpose of making the tooth contact uniform andreducing fluctuations in the rotational speed in consideration of “deformation due to load,” “gearerror” and “stiffness variation due to meshing.” Generally, as shown in Figure 2, tooth shapemodification, which is tooth profile modification in the height direction, and tooth trace
modification (crowning), which is tooth modification in the width direction, are applied. It wasdifficult, however, for these to optimally design load capability improvement and mating excitation
force (vibration) reduction.
*1

Research Manager, Machinery Research Department, Research & Innovation Center*2

Machinery Research Department, Research & Innovation CenterMitsubishi Heavy Industries Technical Review Vol. 57 No. 1 (June 2020)
2Figure 1Factors affecting gear meshing

Figure 2 Generally used tooth surface modification method|3. Gear tooth surface design method using principal component
analysis
In this study, we use the parameters obtained by principal component analysis as design
parameters to express the tooth surface shape. The principal component analysis of a shapeprepared in advance makes it possible to obtain new shape definition parameters different from theconventional modification parameters such as tooth profile modification and tooth trace
modification. Figure 3 depicts the flow for examining a tooth surface shape using principal
component analysis. An overview of each process is described below.

Figure 3 Flow for examining tooth surface shape using principal
component analysis(1)

Preparation of tooth surface shape for principal component analysis
The tooth surface shape for principal component analysis is generated using tooth surface
modification parameters such as tooth profile modification, tooth trace modification and biasmodification. At this time, the tooth surface shape is generated using a tooth surface design
guideline used for designing a low-stress and low- mating-excitation-force tooth surface. This isintended to extract the characteristic amount of the tooth surface shape with low stress and lowMitsubishi Heavy Industries Technical Review Vol. 57 No. 1 (June 2020)
3

mating excitation force using principal component analysis.
Principal component analysis
The prepared tooth surface shape data is converted into a matrix, and principal
component analysis is performed on the matrix data. We used the proper orthogonal
decomposition (POD) method(2) for principal component analysis.
(3) Extraction of principal component shape
The principal component shape ( ) is extracted from the results of principal componentanalysis. Figure 4 gives examples of principal component shapes obtained by principal
component analysis. The principal component shapes in Figure 4 are arranged by the amount ofdifference from the theoretical involute tooth surface.
(2)

Figure 4 Examples of principal component shapes obtained by principal
component analysis(4)

Gear analysis using analyzed principal component shape
Gear analysis is performed using the tooth surface shape (∅) as a parameter, and thestress (surface pressure, tooth root stress) and the mating excitation force are evaluated to find atooth surface shape that can both improve load capability and reduce mating excitation force.
In this case, the tooth surface shape (∅) is represented by the linear sum of the product of theprincipal component shape ( ) and the weighting coefficient ( ) as shown in the followingequation.
∅= +∑∙where,: Average shape of tooth surface for which principal component analysis wasperformed: Principal component shape: Weighting coefficient (parameter)|4. Effectiveness verification with analysis
4.1

Comparison between the studied method and the conventional tooth surface
modification design method

We verified the effectiveness of the studied technique by applying it to helical gears, which
are used for industrial products. For the conventional method, we calculated the surface pressure
and the mating excitation force by performing a parameter study on the tooth profile modification
amount, tooth profile modification length, tooth trace modification amount and tooth trace
modification length. For the studied method, we obtained the principal component shape using themethod described in Section 3 and calculated the surface pressure and the mating excitation forceby performing a parameter study on the weighting coefficient . Tooth surface modification wasperformed only on the pinion side.
Figure 5 compares the calculated values of the surface pressure and the mating excitation
force between the studied method and the conventional method. The surface pressure in this figure
is represented by the maximum surface pressure of the tooth surface, and the mating excitation

Mitsubishi Heavy Industries Technical Review Vol. 57 No. 1 (June 2020)
4

force in this figure is represented by the ratio of the primary component of the mating excitation
force to the tooth surface transmitted load. This shows that the studied method attained a tooth
surface modification shape with a lower surface pressure and a lower mating excitation force thanthe conventional method. From these results, the condition of the weighting coefficient αj forreducing the surface pressure and the mating excitation force simultaneously was obtained, and theoptimal tooth surface shape was found.

Figure 5 Analysis results of surface pressure and mating excitation force(comparison between studied method and conventional method)

4.2

Discussion

Figure 6 compares the surface pressure distribution between the tooth surface modification
shape obtained by using principal component analysis and the tooth surface modification shapeobtained by the conventional method (current specification), and Figure 7 compares the mating
excitation force between the two. It is indicated that the tooth surface modification shape obtainedusing principal component analysis had a lower surface pressure than the current specification due
to the increased tooth contact area. In addition, the mating excitation force was reduced due to theoptimization of the meshing length.

Figure 6 Surface pressure distribution withrespect to position on tooth height(comparison between studied methodand conventional method)

4.3

Figure 7 Time series waveform of mating
excitation force (comparison between
studied method and conventionalmethod)

Estimation results of gear weight reduction amountWe estimated the gear weight reduction amount attained by applying the tooth surface
modification shape obtained using principal component analysis. The design was made so that thesurface pressure, PV value and mating excitation force would be equal to or less than the current
specifications to prevent a decrease in strength and performance. Figure 8 presents the estimation
results of the gear weight reduction amount attained by making the module smaller than the current
one and reducing the gear diameter. It was found that the gear weight could be reduced by aboutMitsubishi Heavy Industries Technical Review Vol. 57 No. 1 (June 2020)
521% compared to the current weight by using principal component analysis for the design of tooth
surface modification.

Figure 8

Estimation results of gear weight reduction effect|5. Conclusion
This paper described an overview of a gear tooth surface modification design method using
principal component analysis and the results of effectiveness verification with analysis. The studied
method can attain both higher load capability and lower mating excitation force than theconventional tooth surface modification design method, so it is considered to be effective in designfor the downsizing of gear devices. We are currently verifying the effectiveness of this method byelemental test. After the verification is completed, we will promote its application to our products
and further improve their performance and quality.

References
(1)

(2)

Aizoh Kubo, Kiyohiko Umezawa, et al., On the Power Transmitting Characteristics of Helical Gears withManufacturing and Alignment Errors，The Japan Society of Mechanical Engineers, Transactions of theJSME (in Japanese), Vol.43 No.371 (1977) p. 2771-2783
L Sirovich，Turbulence and the dynamics of coherent structures，Quarterly of applied mathematics, Vol.45
No.3 (1987) 561-571