# The Effect of Grinding Wheel Contact Stiffness on Plunge Grinding Cycle

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}O

_{3}wheels. Yamada et al. [12,13] proposed a model of a wheel consisting of rigid body and spring elements and measured the contact stiffness of the wheel under both stationary conditions and grinding operations. Papanikolaou and Salonitis [14] applied a three-dimensional molecular dynamics simulation to investigate the effect of contact stiffness on grinding processes under various grinding speeds.

## 2. Plunge Grinding System and the Grinding Cycle

#### 2.1. Plunge Grinding System

_{n}to tangential force Ft, ${v}_{s}$ is the grinding speed, and u is the specific energy. The elastic deflection ${d}_{e}$ of the grinding system in the infeed direction can be expressed by:

#### 2.2. Analysis of Plunge Grinding Cycle

#### 2.3. The Effect of Grinding Wheel Contact Stiffness on the Plunge Grinding Process

## 3. Measurement of Contact Stiffness of Grinding Wheel

#### 3.1. Experimental Setup

#### 3.2. Footprint Method

#### 3.3. Deflection Method

#### 3.4. Modeling of Grinding Wheel Contact Deflection

#### 3.5. Measurement Results of Grinding Wheel Contact Stiffness

## 4. Results of Plunge Cylindrical and Centerless Grinding Tests

#### 4.1. Test Results of Plunge Cylindrical Grinding

^{3}/mm·s, which was very low (Figure 21a), the grinding power slowly built up and finally reached a steady state power of 0.31 kW. From the power curve, the time constant T of 17 s was measured. The time constant became shorter with increased SMRR ${Q}_{w}{}^{\prime}$. At ${Q}_{w}{}^{\prime}=2.0$ mm

^{3}/mm·s, the time constant T was shortened and T = 8.2 s was obtained.

#### 4.2. Test Results of Plunge Centerless Grinding

^{3}/mm·s. The converged time (about 4 s) corresponds to the machine time constant ${T}_{m}$ described in Figure 7. By contrast, the time constant was rapidly prolonged where the SMRR was reduced. It reaches 28 s at SMRR ${Q}_{w}{}^{\prime}$= 0.002 mm

^{3}/mm·s. In this case, the contact time constant ${T}_{c}$ becomes 24 s. The increased ${T}_{c}$ comes from the reduced contact stiffness created by the reduced contact load. This result clearly reveals that the time constant T is significantly increased with reduced SMRR ${Q}_{w}{}^{\prime}$.

## 5. Validation and Discussion

## 6. Conclusions

- (1)
- The equivalent stiffness of the plunge grinding system is composed of machine stiffness and wheel contact stiffness. The contact stiffness greatly depends on the normal contact load. It is confirmed by the measurement of wheel deflection and the plunge grinding tests with various infeed rates.
- (2)
- The contact deflection of the grinding wheel behaves like a non-linear spring representing a local contact deflection at lower load and a linear spring expressing the elasticity of the wheel body itself at a higher load. Formulas representing the wheel deflection behaviors are presented.
- (3)
- The contact stiffness significantly affects the time constant T that governs the transient behaviors in ramp infeed and spark-out grinding.
- (4)
- The time constant T is drastically prolonged with reduced normal grinding force due to the reduction of contact stiffness at lower contact loads. The grinding tests revealed that the time constant T significantly changes by 2–2.6 times.
- (5)
- All measured Al
_{2}O_{3}wheels exhibit very low contact stiffness at a specific load less than 3 N/mm. At a higher load greater than 4 N/mm, contact stiffness reaches a constant. - (6)
- Plunge grinding tests applying both cylindrical grinding and centerless grinding methods with various infeed rates experimentally verified the effect of wheel contact stiffness on the time constant T.
- (7)
- As guidelines for plunge grinding cycle, the ramp infeed grinding time should be set to at least five times T and at least three times T for spark-out grinding.

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

s | Laplace operator | ${d}_{eq}$ | Equivalent wheel diameter |

${D}_{e}\left(s\right)$ | Deflection of machine in s-domain | ${d}_{s}$ | Diameter of grinding wheel |

${F}_{i}\left(s\right)$ | Actual infeed rate in s-domain | ${d}_{w}$ | Diameter of workpiece |

${F}_{n}\left(s\right)$ | Normal grinding force in s-domain | ${f}_{c}$ | Cut-off frequency |

$G\left(s\right)$ | Transfer function in s-domain | ${f}_{i}$ | Actual infeed rate |

${I}_{f}$(s) | Command infeed in s-domain | ${h}_{eq}$ | Equivalent chip thickness |

${Q}_{w}\left(s\right)$ | MRR in s-domain | ${k}_{a}$ | Stiffness of wheel contact area |

${Q}_{w}{}^{\prime}\left(s\right)$ | SMRR in s-domain | ${k}_{a}{}^{\prime}$ | Specific stiffness of wheel contact area |

${R}_{w}\left(s\right)$ | Actual infeed in s-domain | ${k}_{b}$ | Stiffness of wheel body |

A | Constant | ${k}_{b}{}^{\prime}$ | Specific stiffness of wheel body |

F | Ratio of Fn to fi | ${k}_{c}$ | Contact stiffness of grinding wheel |

${F}_{n}$ | Normal grinding force | ${k}_{c}{}^{\prime}$ | Specific contact stiffness of grinding wheel |

${F}_{t}$ | Tangential grinding force | ${k}_{eq}$ | Equivalent system stiffness |

${F}_{n}{}^{\prime}$ | Specific normal grinding force | ${k}_{m}$ | Machine stiffness of grinding system |

${F}_{t}\prime $ | Specific tangential grinding force | ${k}_{mi}$ | Stiffness of i-th. mechanical structure |

GW | Grinding wheel | $\widehat{{k}_{m}}$ | Estimated machine stiffness |

${I}_{f}$ | Command infeed | ${k}_{s}$ | Stiffness of wheel support system |

${I}_{p}$ | Constant infeed rate | ${k}_{w}$ | Stiffness of work support system |

L | Contact load | ${k}_{w0}$ | Stiffness of workpiece itself |

L′ | Specific contact load | ${l}_{c}$ | Contact length |

MRR | Material removal rate | n | Degrees of freedom |

${P}_{g}$ | Grinding power | ${r}_{w}$ | Actual infeed |

${Q}_{w}$ | Material removal rate | t | Time |

${Q}_{w}{}^{\prime}$ | Specific material removal rate | ${t}_{p}$ | Plunge grinding time |

RW | Regulating wheel | ${t}_{s}$ | Time at end of grinding |

S | Constant | u | Specific energy |

SMR | Specific material removal | ${v}_{s}$ | Grinding speed |

SMRR | Specific material removal rate | $\delta $ | Deflection |

T | Time constant | ${\delta}_{a}$ | Deflection of wheel contact area |

${T}_{c}$ | Contact time constant | ${\delta}_{b}$ | Deflection of grinding wheel body |

${T}_{m}$ | Machine time constant | ${\delta}_{c}$ | Contact deflection of grinding wheel |

WP | Workpiece | ${\delta}_{s}$ | Deflection at flange of grinding wheel |

b | Width of workpiece | ${\delta}_{w}$ | Deflection of workpiece |

c | Grinding method parameter | η | Force ratio (Fn/Ft) |

${d}_{e}$ | Deflection of grinding machine |

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**Figure 1.**Plunge grinding operations and the method parameter c. (

**a**) Center type cylindrical grinding; (

**b**) Chuck type cylindrical grinding; (

**c**) Internal grinding; (

**d**) Centerless grinding; (

**e**) Shoe centerless grinding; (

**f**) Shoe internal grinding.

**Figure 2.**Components of equivalent system stiffness in plunge grinding system. (

**a**) Stiffness of grinding wheel head; (

**b**) Stiffness of work head; (

**c**) Stiffness of workpiece; (

**d**) Contact stiffness of grinding wheel.

**Figure 4.**Plunge grinding cycle. (

**a**) Diagram of plunge grinding operations; (

**b**) Primary cycle of plunge grinding.

**Figure 5.**Simulation of plunge grinding processes (conditions: c = 1.0, dw = 177.8 mm, b = 30 mm, vs. = 45 m/s, η = 2.0, keq = 2421 N/mm, u = 26.7 J/mm

^{3}, S = 0.3 mm, Qw′ = 2.0 mm

^{3}/mm·s, Ip = 3.58 μm/s, tp = 41.9 s, ts = 56.9 s). (

**a**) Infeed slide position; (

**b**) Grinding power.

**Figure 9.**Contact length and deflection. (

**a**) External grinding; (

**b**) Internal grinding; (

**c**) Equivalent wheel diameter.

**Figure 10.**Contact footprint of grinding wheel WA60J8V, ${d}_{s}=195\mathrm{mm},{d}_{w}=98\mathrm{mm},b=20\mathrm{mm}$. (

**a**) 1 N/mm; (

**b**) 2.5 N/mm; (

**c**) 5 N/mm; (

**d**) 10 N/mm.

**Figure 11.**Deflections of wheel and workpiece under loading and unloading with grinding wheel WA60J8V. (

**a**) Without load compensator; (

**b**) With load compensator.

**Figure 12.**Repeatability of the measurements of wheel deflection, grinding wheel WA60J8V. (

**a**) Same load cycle; (

**b**) Incremental load cycle.

**Figure 13.**Comparison of wheel deflections obtained by the deflection method and the footprint method.

**Figure 14.**Modeling of the contact stiffness of the grinding wheel. (

**a**) Contact of wheel with workpiece; (

**b**) Model of contact stiffness; (

**c**) Series connection of linear and non-linear springs.

**Figure 21.**Experimental results of plunge cylindrical grinding and the time constant. (

**a**) ${Q}_{w}{}^{\prime}$ = 0.25 mm

^{3}/mm·s; (

**b**) ${Q}_{w}{}^{\prime}$ = 0.5 mm

^{3}/mm·s; (

**c**) ${Q}_{w}{}^{\prime}$ = 1.0 mm

^{3}/mm·s; (

**d**) ${Q}_{w}{}^{\prime}$ = 2.0 mm

^{3}/mm·s.

**Figure 22.**Experimental results of plunge centerless grinding and the time constant. (

**a**) ${Q}_{w}{}^{\prime}=0.01$ mm

^{3}/mm·s; (

**b**) ${Q}_{w}{}^{\prime}=0.02$ mm

^{3}/mm·s; (

**c**) ${Q}_{w}{}^{\prime}=0.0325$ mm

^{3}/mm·s.

**Figure 23.**The effect of specific material removal rate on the time constant in plunge centerless grinding.

**Figure 24.**Simulations of plunge cylindrical grinding processes. (

**a**) ${Q}_{w}{}^{\prime}$ = 0.25 mm

^{3}/mm·s; (

**b**) ${Q}_{w}{}^{\prime}$ = 0.5 mm

^{3}/mm·s; (

**c**) ${Q}_{w}{}^{\prime}$ = 1.0 mm

^{3}/mm·s; (

**d**) ${Q}_{w}{}^{\prime}$ = 2.0 mm

^{3}/mm·s.

**Figure 25.**Simulations of plunge centerless grinding. (

**a**) ${Q}_{w}{}^{\prime}=0.01$ mm

^{3}/mm·s; (

**b**) ${Q}_{w}{}^{\prime}=0.02$ mm

^{3}/mm·s; (

**c**) ${Q}_{w}{}^{\prime}=0.0325$ mm

^{3}/mm·s.

Grinding machine | Center type cylindrical grinder: Tsugami T-PG350 Special feature: Main shaft and tailstock shaft are supported by hydrostatic bearings |

Center distance | 350 mm |

Work swing | 280 mm |

Grinding wheel | Diameter 405 mm Width 50 mm |

Grinding wheel motor | 11 kW |

Machine net weight | 29 kN |

Workpiece | Chrome molybdenum steel SCM435, non-hardened Shape: Ground portion diameter 100 mm Width 20 mm Both side shafts: Diameter 20 mm; length 87.5 mm Radial stiffness with center supports at the middle: 10.4 kN/mm |

Abrasive Material | Abrasive Type | Grain Size # | Grade | Structure | Bond Type |
---|---|---|---|---|---|

White alundum $A{l}_{2}{O}_{3}$ | WA | 60 | J | 8 | V: Vitrified |

White alundum $A{l}_{2}{O}_{3}$ | WA | 60 | L | 8 | V: Vitrified |

White alundum $A{l}_{2}{O}_{3}$ | WA | 60 | M | 8 | V: Vitrified |

White alundum $A{l}_{2}{O}_{3}$ | WA | 60 | L | 8 | B: Resinoid |

Parameter | Unit | Deflection Method | Footprint Method |
---|---|---|---|

${k}_{b}{}^{\prime}$ | kN/mm·mm | 3.5 | 3.5 |

A | μm | 2.3 | 0.8 |

S | N/mm | 1.1 | 1.1 |

Parameter | Unit | WA60J8V | WA60L8V | WA60M8V | WA60L8B |
---|---|---|---|---|---|

${k}_{b}{}^{\prime}$ | kN/mm·mm | 7.7 | 9.1 | 4.5 | 3.6 |

A | Μm | 1 | 0.5 | 1.8 | 2.6 |

S | N/mm | 1.2 | 1.2 | 0.7 | 0.7 |

Item | Unit | Obtained Parameter | ||||
---|---|---|---|---|---|---|

SMR | mm^{3}/mm | 0 | 8 | 160 | 320 | 700 |

${k}_{b}{}^{\prime}$ | kN/mm·mm | 5.9 | 9.1 | 6.7 | 10 | 12.5 |

A | μm | 1.9 | 1.4 | 1.6 | 1.4 | 1.3 |

S | N/mm | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 |

Item | Conditions |
---|---|

Grinding machine | Universal cylindrical grinder: Heald 2EF Cinternal |

Grinding method | Chuck type cylindrical grinding |

Workpiece | Material: Through-hardened 52100 (HRC58) Diameter: 177.8 mm Width: 30 mm |

Grinding wheel | Specification: A70KmV Diameter: 127 mm Width: 86.4 mm |

Grinding speed, ratio | Grinding speed: 45 m/s, Speed ratio: 1/100 |

Item | Conditions |
---|---|

Grinding machine | Centerless grinding machine |

Workpiece | Glass HV500 Diameter: 12.4 mm Length: 66 mm |

Grinding wheel | Specification: SiC, GC 100 LmV Diameter: 455 mm Width: 150 mm |

Regulating wheel | Specification: A150RR Diameter: 255 mm Rotational speed: 12.7 rpm |

Grinding conditions | Blade angle: 30° Center height angle: 6.8° Grinding speed: 29 m/s |

**Table 8.**Critical parameters obtained by the simulations in Figure 23.

Symbol | ${\mathit{F}}_{\mathit{n}}{}^{\prime}$ | u | T | ${\mathit{k}}_{\mathit{e}\mathit{q}}$ | $\widehat{{\mathit{k}}_{\mathit{m}}}$ | ${\mathit{k}}_{\mathit{c}}{}^{\prime}$ |
---|---|---|---|---|---|---|

Unit | N/mm | J/mm^{3} | s | kN/mm | kN/mm | kN/mm·mm |

(a) | 0.46 | 41.4 | 16 | 1.92 | 2.47 | 0.29 |

(b) | 0.83 | 37.4 | 14 | 1.99 | 2.47 | 0.34 |

(c) | 1.36 | 30.7 | 10.7 | 2.13 | 2.47 | 0.52 |

(d) | 2.36 | 26.7 | 8.2 | 2.42 | 2.47 | 4.0 |

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**MDPI and ACS Style**

Hashimoto, F.; Iwashita, H.
The Effect of Grinding Wheel Contact Stiffness on Plunge Grinding Cycle. *Inventions* **2020**, *5*, 62.
https://doi.org/10.3390/inventions5040062

**AMA Style**

Hashimoto F, Iwashita H.
The Effect of Grinding Wheel Contact Stiffness on Plunge Grinding Cycle. *Inventions*. 2020; 5(4):62.
https://doi.org/10.3390/inventions5040062

**Chicago/Turabian Style**

Hashimoto, Fukuo, and Hiroto Iwashita.
2020. "The Effect of Grinding Wheel Contact Stiffness on Plunge Grinding Cycle" *Inventions* 5, no. 4: 62.
https://doi.org/10.3390/inventions5040062