# Stochastic Kinematic Process Model with an Implemented Wear Model for High Feed Dry Grinding

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## Abstract

**:**

## 1. Introduction

## 2. Experimental Methods

_{p}during the process are performed with the pneumatic cylinder mounted behind the spindle. Two pillars guide the traverse with the spindle. The exact adjustment of the depth of cut is not possible due to the power controlled spindle support that is used in railway grinding. Force control is typical also for other high-performance dry grinding applications. The target current for the electric engine is set and the pressure is continuously adjusted during the process in the pneumatic cylinder. The tilt angle is adjusted with the hydraulic cylinder and the grinding spindle is guided on two milled slides. The ring-like workpiece is placed on the rotary table that is fixed by four chuck jaws. The whole grinding spindle construction is mounted in a frame. This frame guarantees the required stiffness and strength for the process forces and torques. A laser sensor mounted on the right-hand column is used for the surface measurements.

## 3. Modelling

#### 3.1. Stochastic Process Model

_{0P,SS}is the connecting vector between an arbitrary point P at the grinding wheel surface and the origin of the coordinate system indicated in Figure 6. R

_{0B}is the connecting vector from the origin to the grinding wheel centre, R

_{BC}is the connecting vector from grinding wheel centre to the tilting point, R

_{CD}is the connecting vector from the tilting point to the already tilted centre of the grinding wheel, and R

_{DP}is the connecting vector from the tilted grinding wheel centre to the arbitrary point on the grinding wheel surface P. The nomenclature of the vector is as follows: subscript describes the connection between two points in a coordinate system. $\underset{\_}{\underset{\_}{A}}$

_{0B}and $\underset{\_}{\underset{\_}{A}}$

_{BC}are then transformation matrices defined as:

_{A}is the approach angle and Ω

_{K}is the tilt angle. The transformation matrix $\underset{\_}{\underset{\_}{A}}$

_{0B}describes the rotation of the local reference system $\mathrm{B}$ to the global coordinate system and $\underset{\_}{\underset{\_}{A}}$

_{BC}is responsible for the adjustment of both, the tilt and approach angle.

_{t}is the partial polygon area from the intersection between the abrasive grain and the workpiece. The centre of gravity can be then calculated from the partial polygon areas as:

_{c}

_{1,1}is the experimentally evaluated specific force and A represents the abrasive grain area that is engaged with the workpiece and is orthogonal to the cutting direction. The force magnitude delivered by the adapted Kienzle equation needs to be complemented by its direction. From the 3D abrasive grain geometry, as described in [17], the orthogonal cross-sectional cutting line is derived. In addition to this, the projection of all other grain faces in contact with the workpiece on the orthogonal cutting line is calculated. Each of these projected faces has its normal vector associated with it. This allows the 3D abrasive grain geometry simplification without losing information. Combining these projected normal vectors for one abrasive grain gives a resulting force direction on one abrasive grain:

_{max}is the total number of projected normal faces of one cutting grain, v

_{i}is the volume extracted by the single face i, v

_{t}is the total volume extracted by the grain in the given time step, and n

_{i}is the normal vector associated with the face i. This resulting force can be split into the cutting force F

_{c}, showing in the direction of the cutting speed and the normal force F

_{N}being perpendicular to the cutting force.

_{g}is the grain velocity in m/s and A

_{cut}the area of the cutting profile being engaged with the workpiece in m

^{2}.

#### 3.2. Wear Modelling

_{c}is the cutting force, v

_{c}is the cutting speed for the abrasive grain, and T

_{grind}is the contact temperature between the abrasive grain and the workpiece. Evaluation of the wear factor K is based on an experimental macroscopic wear evaluation.

_{cut}in Equation (8) changes with the wear progress W(t) and can be calculated as:

_{0}is the initial abrasive grain height and b is the abrasive grain width assumed to be constant during the whole cutting process as shown in Figure 9.

_{1}is a constant. As W(0) = 0 leads to:

_{w}is the energy partition coefficient, q

_{f}is the heat flux, α is the workpiece thermal diffusivity, k

_{T}is the thermal conductivity of the workpiece, v

_{w}is the workpiece feed rate, and l is the half length of the band source.

_{N}is the normal force and ${A}_{\perp}$ is the cross section of the abrasive grain projected in normal direction. The normal force can be calculated from Equation (11).

_{c}is the cutting force and b the width of abrasive grain.

^{−}

^{4}1/N is estimated as shown in Figure 11.

## 4. Results and Discussion

_{c}= 47 m/s, feed rate v

_{w}= 120 m/min, and grinding time 6 s. The grinding wheel was in steady state. Pre-dressing, using the self-sharpening effect of the grinding wheel, was performed prior to every experiment.

## 5. Conclusions

^{4}1/N was estimated.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Experimental set-up: 1 grinding wheel, 2 force measurement system, 3 spindle motor, 4 vertical pneumatic cylinder, 5 frame, 6 milled slide, 7 workpiece, 8 laser sensor.

**Figure 3.**One-sided contact (

**a**) and double-sided contact; (

**b**) of the grinding wheel after initial self-sharpening.

**Figure 5.**Comparison of grinding wheel topographies for measured and modelled grinding wheel by Abbott-Firestone curve.

**Figure 6.**Kinematic situation for double-sided contact between the grinding wheel and workpiece with the connecting vectors.

**Figure 7.**Intersection between one abrasive grain cutting profile and already ground workpiece surface cross section seen in the direction of the cutting speed.

**Figure 10.**Qualitative representation of the resulting stress for a single abrasive grain for different wear factors K.

**Figure 13.**Comparison of surface topography for the experiment (

**a**) and the simulation (

**b**) using the tilt angle 0.0° and the approach angle 0.12°.

**Figure 14.**Comparison of surface topography for the experiment (

**a**) and the simulation (

**b**) using the tilt angle −1.5° and the approach angle 0.01°.

Tilt Angle | Average Roughness Ra (μm) | Average Roughness Depth Rz (μm) | ||
---|---|---|---|---|

Experiment | Simulation | Experiment | Simulation | |

2.5° | 9.8 | 9.9 | 61.8 | 59.4 |

0.0° | 9.5 | 9.4 | 61.7 | 55.7 |

−1.5° | 5.3 | 6.6 | 35.7 | 39.9 |

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**MDPI and ACS Style**

Kuffa, M.; Kuster, F.; Wegener, K. Stochastic Kinematic Process Model with an Implemented Wear Model for High Feed Dry Grinding. *Inventions* **2017**, *2*, 31.
https://doi.org/10.3390/inventions2040031

**AMA Style**

Kuffa M, Kuster F, Wegener K. Stochastic Kinematic Process Model with an Implemented Wear Model for High Feed Dry Grinding. *Inventions*. 2017; 2(4):31.
https://doi.org/10.3390/inventions2040031

**Chicago/Turabian Style**

Kuffa, Michal, Fredy Kuster, and Konrad Wegener. 2017. "Stochastic Kinematic Process Model with an Implemented Wear Model for High Feed Dry Grinding" *Inventions* 2, no. 4: 31.
https://doi.org/10.3390/inventions2040031