Grinding force model for gear profile grinding based on material removal mechanism

Abstract

Traditional force models fail to predict the grinding force for gear profile grinding accurately owing to the specific grinding geometry and kinematics. On the other hand, few researchers about gear profile grinding concern the grinding force. To address this issue, a model to predict the grinding force for gear profile grinding is proposed. This comprehensive study consists of three aspects, namely, the wheel model, the kinematics model, and the force model. Unlike other wheel models for the grinding process, the cumulative number of grinding points in this wheel model is a variable depending on the radius infeed depth. A specific model for grinding geometry and kinematics of gear profile grinding are established based on the processing principle and grain-workpiece contact. Then, the force model is proposed considering the effect of grinding stages and the transformation matrix of the local coordinate system for abrasive grain. To verify the proposed model, experiments are conducted based on specific operations to overcome measurement restrictions, and the effects of grinding parameters on force components are investigated. The proposed models not only predict the grinding force but also reveal the differences between the material removal mechanism for gear profile grinding and that for surface grinding.

Appendices

Appendix 1

According to the geometric characteristics of the grinding wheel for the gear profile grinding, the wheel matrix is established by transforming the 3D grinding wheel into the side view [21]. The position of the abrasive grain is expressed by the row number and the column number of the wheel matrix, and the corresponding element value denotes other information about the wheel grains, such as the protrusion height and size.

The maximum row number and the maximum column number of the wheel matrix can be calculated by the following:

$${i}_{w\mathrm{max}}={r}_{b}\left({\left(\left(z+2\right)/z\mathrm{cos\alpha }\right)}^{2}-1\right)/2\Delta {i}_{w}+1$$

$${j}_{w\mathrm{max}}=2\uppi \left({R}_{d}-{Y}_{\mathrm{end}}+{L}_{i}\right)/\Delta {j}_{w}+1$$

${Y}_{\mathrm{end}}$ is the Y-value of the end point for the involute and can be determined by the involute equation; $\Delta {i}_{w}$ and $\Delta {j}_{w}$ denote the distance between rows and columns of the wheel matrix, respectively.

As for an abrasive grain, the corresponding row number and column number can be calculated by the following:

$$\left\{\begin{array}{c}{i}_{w}=\left(R-\left({R}_{d}-{Y}_{\mathrm{end}}\right)\right)/\Delta {i}_{w}+1\\ {j}_{w}=\left(R\theta -\updelta \left({i}_{w}\right)\right)/\Delta {j}_{w}+1\end{array}\right.$$

where $R$ and $\theta$ denote the radius and the orientation angle of the abrasive grain in the side view; $\updelta \left({i}_{w}\right)$ denotes the gap length of (i_w)th row and can be calculated by

$$\delta\left(i_w\right)=\backslash\mathrm{uppi}\left(R_2-R\right)$$

$R_2$denotes the maximum radius of grains and is determined by the size of the grinding wheel.

Appendix 2

To calculate the force to resist the chip formation in the cutting stage, the Johnson–Cook model is adopted to express the flow stress of the material, and the stress is expressed as follows:

$$\tau =\left(A+B\bullet {\varepsilon }^{n}\right)\left(1+C\bullet \mathrm{ln}\dot{\varepsilon }\right)\left(1-{\left(\frac{T-{T}_{r}}{{T}_{m}-{T}_{r}}\right)}^{m}\right)$$

where A, B, and C are the constitutive parameters of the material and are 890, 521, and 0.014 in this case, respectively; n and m are the constants that relate to the stress flow with the strain hardening and thermal softening effects of the material and are 0.25 and 1.02 in this case, respectively. ${T}_{r}$ and ${T}_{m}$ are the room and melting temperature points of the material, respectively. As given in Ref. [46], $\varepsilon$ and $\dot{\varepsilon }$ denote the effective plastic strain and normalized effective plastic strain rate, which can be expressed as follows:

$$\varepsilon =\frac{\mathrm{cos}{\beta }_{\alpha }}{\mathrm{sin}{\phi }_{\alpha }\mathrm{cos}\left({\phi }_{\alpha }-{\beta }_{\alpha }\right)}$$

$$\dot{\varepsilon }=\frac{10{v}_{s}^{i}\mathrm{sin}{\phi }_{\alpha }\mathrm{cos}{\beta }_{\alpha }}{w\mathrm{cos}\left({\phi }_{\alpha }-{\beta }_{\alpha }\right)}$$

where ${v}_{s}^{i}$ denotes the wheel velocity that corresponds to the ith grain. Since the grinding radius depends on the corresponding involute angle of abrasive grain, the wheel speed of the ith grain is different from that of another one, which can be expressed as follows:

$${v}_{s}^{i}=2{R}_{i}{v}_{s}/{d}_{s}$$