# The Design of an Infeed Cylindrical Grinding Cycle

## Abstract

**:**

## 1. Introduction

## 2. Cylindrical Grinding System

#### 2.1. Models of Grinding Systems

#### 2.2. The Modeling of the Infeed Cylindrical Grinding System

#### 2.2.1. The Fundamentals of Infeed Cylindrical Grinding

_{w}’ can be represented by:

^{3}) and ${v}_{s}$ is the grinding speed. The normal grinding force ${F}_{n}$ is written by:

#### 2.2.2. Machine Stiffness in the Grinding System

_{n}acting on the grinding wheel head. ks is the stiffness of the grinding wheel support system represented by F

_{n}/${\delta}_{s}.$ Figure 6b shows the work support head and ${\delta}_{w}$ is the deflection of the work head support system caused by F

_{n}. ${k}_{w}$ denotes the stiffness of the work head support system. In Figure 6c, ${k}_{wo}$ represents the stiffness of the workpiece itself. A thin, ring-shaped workpiece sometimes has a low stiffness that cannot be ignored. Figure 6d shows the stiffness in the contact area between the grinding wheel and the workpiece, which is referred to as “contact stiffness” and is denoted by ${k}_{con}$. In general, the contact stiffness ${k}_{con}$ of the grinding wheel has hard-spring characteristics in which the higher contact force provides the higher stiffness.

#### 2.3. The Block Diagram of a Cylindrical Grinding System

_{n}/F

_{t}) and ${v}_{s}$ is the grinding speed. The elastic deflection ${d}_{e}$ of the grinding machine in the infeed direction can be expressed by:

## 3. Responses to Command Infeed in a Cylindrical Grinding System

#### 3.1. Responses to Step Infeed (Spark-Out Grinding)

#### 3.2. Responses to Ramp Infeed (Plunge Grinding)

#### 3.3. The Effects of the Time Constant on Infeed Grinding Processes

## 4. Grinding Cycle Design

#### 4.1. Grinding Cycle

- i.
- Actual relative approaches between the grinding wheel and workpiece.
- ii.
- Actual infeed rates.
- iii.
- Grinding forces.
- iv.
- Elastic deflection of grinding machine.

#### 4.2. Analysis of Infeed Grinding Processes

#### 4.3. The Influence of the Time Constant on the Grinding Cycle

## 5. Grinding Accuracy and Stock Assignments

#### 5.1. Size Error

#### 5.2. Roundness

#### 5.3. Grinding Stock Assignments and SMRR

_{1}for rough, S

_{2}for semi-finish and S

_{3}for finish grinding.

## 6. Experimental Tests and Simulations

^{3}. Additionally, the system stiffness km can be found by Equation (16), obtaining km = 2421 N/mm.

_{g}was set to 57 s (including a spark-out time of 5 s).

_{n}and tangential force F

_{t}measurement results found during infeed centerless grinding. From the results, the time constant T = 4.6 s was obtained, and an extremely high specific energy of u = 741 J/mm

^{3}was measured due to the ductile grinding of the brittle materials. It is noted that the fluctuations observed on F

_{n}and F

_{t}curves were due to the runout of regulating wheel rotation and truing errors. Figure 21b graphs the infeed centerless grinding simulation results for the same glass cylinder workpieces. The raising transients of the forces were well simulated and the levels of the forces were almost identical. The spark-out behavior was also well simulated.

## 7. Conclusions

- (1)
- In infeed cylindrical grinding, including the centerless methods, the causalities between the grinding fundamentals and the machine characteristics can be clarified, and from that a model of a new system represented by a block diagram with closed-loop feedback can be proposed.
- (2)
- From the characteristic equations of the proposed grinding system, a factor called the “grinding time constant” was revealed. This time constant was found to play a critical role in the infeed process and in spark-out grinding.
- (3)
- Formulas presenting process parameters such as grinding forces and machine deflection were derived, and the procedures for the grinding cycle design were created.
- (4)
- Practical exercises for improving size error, roundness and cycle time in infeed cylindrical grinding were developed and described.
- (5)
- The model was verified by performing grinding tests on both the cylindrical and centerless grinding methods.

## Funding

## Conflicts of Interest

## Nomenclature

s | Laplace operator | T | Grinding time constant |

${D}_{e}\left(s\right)$ | Deflection of machine in s-domain | ${T}_{g}$ | Grinding time |

$F\left(s\right)$ | Rounding error in s-domain | ${T}_{p}$ | Infeed grinding time |

${F}_{i}\left(s\right)$ | Actual infeed rate in s-domain | ${T}_{sp}$ | Spark-out time |

${F}_{n}\left(s\right)$ | Normal grinding force in s-domain | UV | Energy-grinding speed function |

${G}_{m}\left(s\right)$ | Dynamic compliance in s-domain | WP | Workpiece |

${I}_{f}$(s) | Command infeed in s-domain | b | Width of workpiece |

${Q}_{w}{}^{\prime}\left(s\right)$ | SMRR in s-domain | c | Grinding method parameter |

${R}_{w}\left(s\right)$ | Actual infeed in s-domain | ${d}_{e}$ | Deflection of grinding machine |

${T}_{c}\left(s\right)$ | Depth of cut in s-domain | ${d}_{w}$ | Diameter of workpiece |

C | Constant | ${f}_{i}$ | Actual infeed rate |

${F}_{n}$ | Normal grinding force | ${h}_{eq}$ | Equivalent chip thickness |

${F}_{t}$ | Tangential grinding force | ${k}_{con}$ | Contact stiffness of grinding wheel |

${F}_{n}$’ | Specific normal grinding force | ${k}_{m}$ | Stiffness of grinding system |

${F}_{t}\prime $ | Specific tangential grinding force | ${k}_{s}$ | Stiffness of wheel support system |

GW | Grinding wheel | ${k}_{w}$ | Stiffness of work support system |

${I}_{f}$ | Command infeed | ${k}_{w0}$ | Stiffness of workpiece itself |

${I}_{0}$ | Step infeed in slide position | ${n}_{w}$ | Rotational speed of workpiece |

${I}_{p}$ | Constant infeed rate | ${r}_{w}$ | Actual infeed |

Km | Static stiffness | t | time |

Kw | Grinding stiffness | u | Specific energy |

MRR | Material removal rate | ${v}_{s}$ | Grinding speed |

PS | Percentages of stock assignment | ${\delta}_{con}$ | Contact deflection of grinding wheel |

${P}_{g}$ | Grinding power | ${\delta}_{s}$ | Deflection of wheel support system |

${Q}_{w}$ | Material removal rate | ${\delta}_{w0}$ | Deflection of work itself |

${Q}_{w}{}^{\prime}$ | Specific material removal rate | ${\delta}_{w}$ | Deflection of work support system |

RW | Regulating wheel | Δ | Depth of cut per revolution |

S | Grinding stocks in diameter | η | Force ratio (F_{n}/F_{t}) |

SMRR | Specific material removal rate |

## Appendix A

## Appendix B

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**Figure 4.**Cylindrical grinding methods (WP: workpiece, GW: grinding wheel, RW: regulating wheel). (

**a**) Center-chuck type grinding c = 1.0; (

**b**) Centerless grinding c = 0.5; (

**c**) Shoe centerless grinding c = 0.5.

**Figure 6.**Machine stiffness in a grinding system. (

**a**) Grinding wheel support system; (

**b**) Work head support system; (

**c**) Workpiece stiffness; (

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

**Figure 9.**Responses to step infeed in cylindrical infeed grinding (spark-out grinding). (

**a**) Actual infeed I

_{0}= 5 μm; (

**b**) Normal grinding force F

_{n}(0) = 99.8 N; (

**c**) Machine deflection d

_{e}(0) = 5 μm.

**Figure 10.**Responses to ramp infeed in cylindrical infeed grinding (plunge grinding). (

**a**) Actual infeed; (

**b**) Normal grinding force; (

**c**) Machine deflection.

**Figure 11.**Effect of time constant on infeed grinding processes. (

**a**) Slide position; (

**b**) Forces, power and machine deflection; (

**c**) Spark-out in forces, power and deflection.

**Figure 15.**Process parameters during the infeed grinding cycle. (

**a**) Infeed rate; (

**b**) Command and actual slide position; (

**c**) Normal grinding force; (

**d**) Deflection of grinding machine.

**Figure 16.**Influences of the time constant on the behaviors of process parameters. (

**a**) Actual infeed position; (

**b**) Deflection of grinding machine.

**Figure 18.**Roundness error caused by the residual depth of cut. (

**a**) Work rotation during infeed grinding; (

**b**) Step-roundness.

**Figure 20.**Test results and simulation for infeed cylindrical grinding. (

**a**) Power of infeed cylindrical grinding (Experimental); (

**b**) Simulation of infeed cylindrical grinding (Theoretical).

**Figure 21.**Grinding test results and simulation for infeed centerless grinding. (

**a**) Test results of infeed centerless grinding of glass; (

**b**) Simulation results of infeed centerless grinding.

Items | Conditions |
---|---|

Grinding machine | Universal cylindrical grinder |

Grinding method | Chuck type cylindrical grinding c = 1.0 |

Workpiece | Thru-hardened steel HRC58 Diameter d _{w}= 177.8 mm, width b = 30 mm |

Grinding wheel | Al_{2}O_{3} 70 K m VDiameter d _{s} = 127 mm, width L_{s} = 86.4 mmGrinding speed v _{s} = 45 m/s |

Stocks in diameter | S = 0.33 mm |

SMRR (Specific Material Removal Rate) | Qw’ = 2.0 mm^{3}/(mm·s) |

Infeed rate | f_{i} = 0.216 mm/min |

Spark-out time | T_{sp} = 5.3 s |

Grinding time | T_{g} = 51.1 s |

Specific energy | u = 26.7 J/mm^{3} |

Force ratio (Fn/Ft) | η = 2.0 |

System stiffness | k_{m} = 2421 N/mm |

Time constant | T = 8.2 s |

Items | Conditions |
---|---|

Grinding machine | Centerless grinder |

Grinding method | Infeed centerless grinding c = 0.5 |

Workpiece | Soda-lime glass, HV500 Diameter d _{w} = 12.4 mm, width b = 66 mm |

Grinding wheel | SiC, GC 100 L m V Diameter d _{s} = 455 mm, width L_{s} = 150 mmGrinding speed v _{s} = 29 m/s |

Regulating wheel | A 150 R R Diameter d _{r} = 255 mm, width L_{s} = 150 mmRotational speed N _{r} = 24.4 rpm |

Stocks in diameter | S = 0.1 mm |

SMRR (Specific Material Removal Rate) | Qw’ = 0.0325 mm^{3}/(mm·s) |

Infeed rate | f_{i} = 0.10 mm/min |

Spark-out time | T_{sp} = 5.0 s |

Grinding time | T_{g} = 57 s |

Specific energy | u = 741 J/mm^{3} |

Force ratio (Fn/Ft) | η = 4.0 |

System stiffness | k_{m} = 5900 N/mm |

Time constant | T = 4.6 s |

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**MDPI and ACS Style**

Hashimoto, F.
The Design of an Infeed Cylindrical Grinding Cycle. *Inventions* **2020**, *5*, 46.
https://doi.org/10.3390/inventions5030046

**AMA Style**

Hashimoto F.
The Design of an Infeed Cylindrical Grinding Cycle. *Inventions*. 2020; 5(3):46.
https://doi.org/10.3390/inventions5030046

**Chicago/Turabian Style**

Hashimoto, Fukuo.
2020. "The Design of an Infeed Cylindrical Grinding Cycle" *Inventions* 5, no. 3: 46.
https://doi.org/10.3390/inventions5030046