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Digital twin system of thermal error control for a large-size gear profile grinder enabled by gated recurrent unit

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Abstract

Large-size gear profile grinder is extremely important for high-accuracy machining of large-scale and high-performance gears. A lot of heat is generated during the service process, leading to thermal errors of large-size gear profile grinder. In recent years, the data-based modeling and prediction methods have been widely used in manufacturing systems to control thermal errors. However, a limited bandwidth and latency characteristic of industrial Internet has brought serious challenges to the real-time and efficient processing of a large-volume data. To solve the above problems, a digital twin system of thermal error control is proposed for large-size gear profile grinders. The theoretical-based modeling method is used to prove the memory behavior, and then the feasibility of applying gated recurrent unit (GRU) to thermal error control is demonstrated due to its strong long-term memorizing capability. Then the hyper-parameters of GRU model is optimized by an improved bat algorithm (IBA), and the self-learning IBA-GRU error model is proposed. Finally, a digital twin system of thermal error control is established based on a new haze-cloud computing architecture to improve the executing efficiency, and the self-learning IBA-GRU error control model is embedded into the digital twin system. With the implementation of the digital twin system, the reduced percentages for the maximum tooth profile tilt deviations of \(f_{H\alpha l}\) and \(f_{H\alpha r}\) for the left and right tooth flanks are 58.43% and 64.16%, respectively, and the reduced percentages for the maximum total tooth profile deviations of \(F_{\alpha l}\) and \(F_{\alpha r}\) for the left and right tooth flanks are 28.78% and 34.53%, respectively. The saved ratio of the transferred data is up to 76.30% in 5 months. The executing efficiency of the digital twin system with IBA-GRU as control model is higher than that with other models and architectures.

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Abbreviations

c :

Specific heat capacity

ρ :

Density

\(Q_{{\text{v}}}\) :

Internal heat source

\(T\) :

Temperature

\(t\) :

Time

\(x\), \(y\), \(z\) :

Coordinates

\(M_{0}\) :

Lubricating friction torque

\(M_{1}\) :

Load friction torque

\(v\) :

Kinematic viscosity of lubricant

\(d_{m}\) :

Pitch diameter of bearings

\(P_{1}\) :

Comprehensive load

\(f_{1}\) :

Coefficient related to bearing type and lubricating method

\(A_{c}\) :

Actual contact surface

\(A\) :

Nominal contact surface

\(A_{v}\) :

Area of the non-contact area

\(k_{1}\),\(k_{2}\), and \(k_{f}\) :

Thermal conductivity

\(L_{g}\) :

Thickness of uncontacted space

\(p\) :

Contact pressure between the contact surfaces

\(G\), \(Lu\), and \(D\) :

Fractal parameters of joint surface

\(H\) :

Hardness of softer material in the two contact parts

\(E\) :

Equivalent elastic modulus of the two contact materials

\(k_{x}\) and \(k_{y}\) :

Ratio coefficients

\(A_{i}\) :

Loudness

\(v_{i}^{t}\) and \(x_{i}^{t}\) :

The velocity and position at the time of \(t\)

\(f_{i}\) :

Pulse frequency of the bat \(i\)

\(\overline{A}^{t}\) :

Average loudness at time of \(t\) for bat \(i\)

\(r_{t}\) :

Reset gate

\(z_{t}\) :

Update gate

\(h_{t - 1}\) :

The previous hidden state

\(v_{k}\) :

Velocity of the bat at the time of \(k\)

\(b\) :

Constant pulse frequency

\(\varepsilon_{x}\), \(\varepsilon_{y}\), and \(\varepsilon_{z}\) :

Deformation components

\(\sigma_{x}\), \(\sigma_{y}\), and \(\sigma_{z}\) :

Stress components

\(E\) :

Elastic modulus

\(\mu\) :

Poisson’s ratio

\(M\) :

Total friction torque

\(n\) :

Rotational speed

\(h\) :

Convective coefficient

\(\lambda\) :

Thermal conductivity of air

\(\Pr\) :

Prandtl Number

\(d\) :

Equivalent diameter of spindle;

\(v\) :

Kinematic viscosity of air

\(x^{*}\) :

Current global optimal position

\(f_{\max }\) and \(f_{\min }\) :

Upper and lower limits of the pulse frequency

\(\alpha\) :

Attenuation coefficient of pulse loudness

\(\gamma\) :

Increase coefficient of pulse frequency

J :

Minimization function of the structural risk

\(\omega\) :

Weight vector

\(\xi_{i}\) :

Error variable

\(\gamma\) :

Adjustable parameters

\(x_{i}\) :

Input variable

\(y_{i}\) :

Objective value

\(\varphi \left( \cdot \right)\) :

Mapping function

b :

Deviation

\(h_{t}\) :

New memory content

\(\alpha_{i}\), \(\beta_{i}\), and \(\eta_{i}\) :

Lagrangian coefficients

\(b_{1}\), \(b_{2}\), and \(b_{3}\) :

Deviations

\(K(x,x_{i} )\) :

Kernel function

\(x_{t}\) :

Current input

\(x_{k}\) :

Position of the bat at the time of \(k\)

\(p\) :

Current global optimal position

\(a\), \(c\) and \(d\) :

Weight coefficients

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Acknowledgements

This research was supported by the National Natural Science Foundation of China (51905057), the Natural Science Foundation Project of Chongqing, Chongqing Science and Technology Commission (cstc2019jcyj-msxmX0050), the Fundamental Research Funds for the Central Universities (2020CDJQY-A036), the Venture & Innovation Support Program for Chongqing Overseas Returnees (cx2019054), and State Key Laboratory for Manufacturing Systems Engineering (sklms2020016).

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Authors and Affiliations

  1. College of Mechanical and Vehicle Engineering, Chongqing University, Chongqing, 400044, China

    Jialan Liu, Hongquan Gui & Chi Ma

  2. State Key Laboratory of Mechanical Transmission, Chongqing University, Chongqing, 400044, China

    Jialan Liu, Hongquan Gui & Chi Ma

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  1. Jialan Liu
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Correspondence to Chi Ma.

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We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work; there is no professional or other personal interest of any nature or kind in any product or company that could be construed as influencing the position presented in, or the review of, the manuscript. Jialan Liu and Hongquan Gui contributed equally to this manuscript.

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Liu, J., Gui, H. & Ma, C. Digital twin system of thermal error control for a large-size gear profile grinder enabled by gated recurrent unit. J Ambient Intell Human Comput 14, 1269–1295 (2023). https://doi.org/10.1007/s12652-021-03378-4

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