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A novel geometric error compensation method for improving machining accuracy of spiral bevel gear based on inverse kinematic model

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Abstract

The geometric errors (GEs) of the spiral bevel gear milling machine will seriously affect the machining accuracy of the tooth surface and need to be compensated. In this paper, an innovative method for compensating the geometric error of CNC gear milling machine is proposed. This method describes the explicit relationship between the motion axis of the machine tool and the geometric errors, realizes the dynamic compensation by tooth, and improves the machining accuracy and machining efficiency of the tooth surface. Firstly, the actual forward kinematics model (FKM) is constructed based on the geometric errors module, and the error tooth surface with GEs is established. Secondly, the corresponding relationship between the machine setting parameters of the spiral bevel gear universal generation machine (UGM) and the CNC machine tool motion axis is given, and then the functional expression between the motion axis with geometric errors and the machine setting parameters is established, which is the inverse kinematics model (IKM). Then, the tooth surface error correction model is established according to the relationship between the machine setting parameters and the tooth surface errors. The compensation amount of the machine setting parameters obtained by the model is introduced into the IKM to obtain the GEs compensation model. Finally, the effectiveness of the geometric error compensation technique is verified by numerical analysis and experiments. The results show that the tooth surface errors, contact stress, and loaded transmission error after geometric errors compensation are significantly reduced, and the contact pattern meets the design requirements, which verifies the feasibility and effectiveness of the geometric errors compensation technology.

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Abbreviations

GEs:

Geometric errors

FKM:

Forward kinematic model

UGM:

Universal generation model

IKM:

Inverse kinematic model

PIGEs:

Position-independent geometric errors

PDGEs:

Position-dependent geometric errors

X,Y,Z,A,B,C:

CNC machine tool shaft

\(\epsilon _k(I),\delta _k(I)\) :

PDGEs (\(k=x,y,z\);\(I=A,\) B, X, Y, Z)

\(\gamma _{XZ}\) :

Verticality error of X-axis relative to Z-axis

\(\gamma _{YX},\gamma _{YZ}\) :

Verticality error of Y-axis relative to Z-axis and X-axis

\(\alpha _{AX},\beta _{AY}\) :

Verticality error of A-axis relative to X-axis and Y-axis

\(\alpha _{BX},\beta _{BZ}\) :

Verticality error of B-axis relative to X-axis and Z-axis

\(\delta _{AX},\delta _{AY}\) :

Position error of A-axis relative to X-axis and Y-axis

\(\delta _{BX},\delta _{BZ}\) :

Position error of B-axis relative to X-axis and Z-axis

\({\varvec{M}}_{DI}\) :

Transformation matrix from I-axis to D-axis(\(D=A,B,X,Y,Z,R\); \(I=A,B,X,Y,Z,W\))

\(\varvec{e}\) :

GEs

\(u_k,\theta _k\) :

Gaussian coordinate parameters of cutter tool

\(\varvec{r}_i(i\!=\!T,k,e,w)\) :

Tooth surface position vector(cutter tool,\(i=T,k\); by FKM,\(i=e\); by UGM,\(i=w\))

\(\varvec{n}_i(i\!=\!T,k,e,w)\) :

Tooth surface unit normal vector(cutter tool,\(i=T,k\); by FKM, \(i=e\); by UGM,\(i=w\))

\(\varvec{v}^{eT},\varvec{v}^{wk}\) :

Relative speed between tool and tooth surface

\(R_c\) :

Cutter tip radius

\(\alpha _c\) :

Profile angle

\(\rho _c\) :

Tip transition filet radius

\(\lambda _c\) :

Tip transition filet arc angle

\(q_0\) :

Angular tool position

\(S_0\) :

Radial tool position

\(R_a\) :

Roll ratio

E :

Vertical wheel position

\(X_p\) :

Axial wheel position

\(X_b\) :

Bed position

\(\delta _m\) :

Machine pitch angle

\(\varvec{P}\) :

Machine setting parameter vector

\(\phi _i(i=c,w)\) :

Rotation angle of production wheel (\(i\!=\!c\)) and workpiece gear(\(i\!=\!w\))

\(\varvec{M}_{wk}\) :

Transformation matrix from tool coordinate system to gear coordinate system

\(\varvec{d}\) :

Error vector between GEs tooth surface and design tooth surface

h :

Tooth surface error in normal direction

S :

Sensitivity matrix

\(\Delta P\) :

Machine setting parameter correction value

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Funding

This work was supported by the National Natural Science Foundation of China No.51175422 and the Shaanxi Natural Science Basic Research Program No.2022JM-195.

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  1. School of Mechanical Engineering, Northwestern Polytechnical University, Xi’an, 710072, China

    Peng Chen, Sanmin Wang, Bo Li & Fei Li

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  1. Peng Chen
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Contributions

Peng Chen: conceptualization, methodology, software, writing—original draft. Fei Li: data curation.

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Correspondence to Peng Chen.

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Chen, P., Wang, S., Li, B. et al. A novel geometric error compensation method for improving machining accuracy of spiral bevel gear based on inverse kinematic model. Int J Adv Manuf Technol 127, 4339–4355 (2023). https://doi.org/10.1007/s00170-023-11628-2

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