Skip to main content
Log in

Design optimization of external engagement cylindrical gear flowmeter under uncertainty

  • Industrial Application Paper
  • Published:
Structural and Multidisciplinary Optimization Aims and scope Submit manuscript


The purpose of this study is to determine the reasonable structural parameters of the external engagement cylindrical gear flowmeter (EGF) under the interference of random factors, so as to ensure that the EGF system has high structural reliability and robustness of performance index response. In addition to the influence of its own structural parameters (e.g., gear structure parameters, gap height), the random factors (e.g., oil viscosity, pressure, temperature and flow rate) can affect multiple performance indexes (e.g., flow pulsation, internal leakage and total power loss) in different ways, thus reducing the structural robustness of the performance response and increasing the difficulty of optimization design and subsequent accuracy compensation of the EGF. In addition, machining and assembly errors also increase the probability of structural failure of the system. In this paper, based on the Basis-adaptive polynomial chaos expansion (Basis-adaptive PCE) surrogate model technology, an uncertainty optimization design method of EGF is proposed, which improves the robustness of the performance index response while ensuring the structural reliability. Multidisciplinary modeling of the performance index of EGF is performed to derive the performance state function. Based on the design of experiments, the influence trend of random factors on performance index was analyzed. The Basis-adaptive PCE method can significantly improve the efficiency of uncertainty quantification while ensuring high calculation accuracy and can be applied to the subsequent calculation of optimization design. Based on the results of uncertainty analysis, the optimization index of EGF is determined and the uncertainty optimization design is carried out. Compared with the deterministic optimization, the results show that the uncertainty optimization strategy can significantly reduce the probability of undercutting failure of the measuring gear and the fluctuation of the internal leakage response, thus effectively improving the comprehensive performance and can be applied to the optimization design of EGF.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17

Similar content being viewed by others

Data availability

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.


Download references


This work was supported by the “National Natural Science Foundation, China” (Grant No. 52175044); and the “National Key Research and Development Program, China” (Grant No. 2020YFB2007203).

Author information

Authors and Affiliations


Corresponding author

Correspondence to Lintao Wang.

Ethics declarations

Conflict of interest

The authors declare no conflict of interest.

Replication of results

Upon request, the authors will provide the full set of input parameters for each validation problems presented in the paper.

Additional information

Responsible Editor: Zhen Hu

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.








Modules (mm)


Tooth width (mm)


Tooth number (/)

\(\varepsilon_{\alpha }\)

Gear coincidence degree (/)


Graduated circle pressure angle,\(20^{ \circ }\)


Tooth height coefficient,\(h_{a}^{*} = 1\)


Meshing angle (\(^\circ\))


Addendum (mm)


Modification coefficient (/)


Pressure angle of tooth top circle (\(^\circ\))


Round radius of tooth top (mm)


Normal pitch (mm)


Root radius (mm)


Pitch radius (mm)


Center distance modification coefficient (/)


Tooth top linear velocity (m/s)

\(\Delta y\)

Variation coefficient of tooth height (/)


Crest width (mm)




Net displacement (ml/r)


Flow pulsation coefficient (%)


Radial gap (mm)


Axial gap (mm)


Radial gap leakage (L/min)


Axial gap leakage (L/min)

\(\Delta p\)

Pressure loss (MPa)


High Pressure Zone Angle, \(\beta = 45^\circ\)


Hole radius of gear bearing (mm)


Gear shaft radius (mm)


Internal leakage power loss (w)


Crest width (w)


Axial friction loss power (w)


Radial friction loss power (w)


Rotational speed (r/min)


Total internal leakage (L/min)


Viscous friction power loss (w)


Theoretical mean flow rate (L/min)


Internal leakage rate (%)


Oil parameters



Dynamic viscosity (\({\text{Pa}} \cdot {\text{s}}\))


Kinematic viscosity (\({\text{mm}}^{2} {/}s\))

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, L., Zhu, F., Hao, N. et al. Design optimization of external engagement cylindrical gear flowmeter under uncertainty. Struct Multidisc Optim 67, 63 (2024).

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: