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Optimization and prediction for gear microgeometry modification considering fabrication errors

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Abstract

Transmission error (TE) in multistage gear transmission is the main excitation source of gear mesh noise. Gear microgeometry modification can effectively reduce TE, while the determination of the modification amount is a challenge. The stochastic fabrication errors of modification dimensions lead to uncertain results. To anticipate and evaluate this uncertainty during the design stage, a convolution optimization method is proposed that takes both the nominal modification values and the probable deviations of fabrication errors into consideration. To best simulate the distribution of deviations of fabricated dimensions, a multi-dimensional skew-normal Gaussian kernel is studied. By calculating the convolution of the Gaussian kernel and TE levels of candidate modification range, one obtains a prediction that is closer to the reality than simply treating the nominal values as design variables. To verify the effectiveness of this method, a case study of vehicle’s transmission is conducted. Monte Carlo stochastic simulation and experimental bench test are performed. Compared with the single-point nominal value optimization, the average level of peak–peak TE and vibration acceleration is reduced. The quality of products is increased for large-scale production. It provides new ideas for both gear microgeometry modification and other industries that need to estimate the uncertain influence of fabrication errors and predict the massive manufacturing results.

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Abbreviations

\(\delta_{1} ,\delta_{2}\) :

Rolling angle for driving and driven gear

\(N_{1} ,N_{2}\) :

Number of teeth for driving and driven gear

\(F_{h\alpha }\) :

Gear profile slope modification

\(F_{h\beta }\) :

Gear lead slope modification

\(C_{\alpha }\) :

Gear profile crowning modification

\(C_{\beta }\) :

Gear lead crowning modification

\(k_{1} ,k_{2} ,k_{3} ,k_{4}\) :

Value of \(F_{h\alpha } , \;F_{h\beta } ,\; C_{\alpha } \;{\text{and}}\;C_{\beta }\)

\({\text{PPTE}}\) :

PPTE database tensor

\(p\) :

Probability density function

\({\varvec{p}}\) :

Tolerance tensor

\(\rho\) :

Dependency factor

\({\varvec{\rho}}\) :

Tensor of \(\rho\)

\(X\) :

Standardized gear modification variable

\({\varvec{X}}\) :

Vector of \(X\)

\({\upvarepsilon }\) :

Skewness

\({\upsigma }\) :

Standard deviation

\(m\) :

Dimension of Gaussian kernel

\(N\) :

Number of load cases

\({\text{TH}}_{{{\text{PPTE}}}}\) :

Threshold of qualified PPTE level

\({\varvec{F}}_{{\varvec{k}}}\) :

Qualification tensor

\({\text{QR}}_{{\varvec{k}}}\) :

Qualification rate for step \({\varvec{k}}\)

\(\overline{{{\text{QR}}_{{\varvec{k}}} }}\) :

Load-case averaged qualification rate

\(\overline{{{\mathbf{QR}}_{{\varvec{k}}} }}\) :

Tensor of \(\overline{{{\text{QR}}_{{\varvec{k}}} }}\)

\(C_{{\text{g}}}\) :

Usage frequency coefficient

\(T\) :

Single-side tolerance range

TE:

Transmission error

STE:

Static transmission error

DTE:

Dynamic transmission error

PPTE:

Peak–peak static transmission error

NVH:

Noise, vibration, and harshness

ESN:

Epsilon-skew-normal distribution

CPK:

Process capability index

DCT:

Dual clutch transmission

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Funding

No funding was received to assist with the preparation of this manuscript.

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Authors and Affiliations

  1. Geely Royal Engine Components Co., Ltd., Ningbo, 315336, China

    Lingyu Zhang & **aohua Cai

  2. InfiMotion Propulsion Technology Co., Ltd., Wuxi, 214000, China

    Guomin Peng, Junfeng Hu, Haisheng Yu & Tianbao Tang

Authors
  1. Lingyu Zhang
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  2. **aohua Cai
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  3. Guomin Peng
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  4. Junfeng Hu
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  5. Haisheng Yu
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  6. Tianbao Tang
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Contributions

LZ performed methodology, software, formal analysis, writing—original draft, and writing—review & editing. XC done conceptualization, formal analysis, and writing—review & editing. GP contributed to data curation and validation. JH and HY did investigation and project administration. TT was involved in data curation and investigation.

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Correspondence to Tianbao Tang.

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Technical Editor: Andre T. Beck.

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Zhang, L., Cai, X., Peng, G. et al. Optimization and prediction for gear microgeometry modification considering fabrication errors. J Braz. Soc. Mech. Sci. Eng. 46, 48 (2024). https://doi.org/10.1007/s40430-023-04609-9

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