# Model Development for Optimum Setup Conditions that Satisfy Three Stability Criteria of Centerless Grinding Systems

## Abstract

**:**

## 1. Introduction

## 2. Basic Setup Conditions in Centerless Grinding

_{w}, as shown in Figure 1.

_{w}) of these parameters is called the “setup condition” in this paper, and it significantly affects centerless grinding performance. In practice, the work center height CH (instead of γ) and the RW (regulating wheel) rotation speed N

_{r}(rpm) are used because these parameters can be directly set up on the machine. The center height CH (mm) has the following relationship with the center height angle γ (°) when angles α and β are small.

_{w}is controlled by the RW friction drive/brake mechanism. Figure 2 shows test results of normal grinding force Fn, the friction coefficient μr and the rolling-sliding velocity between RW and the workpiece during an infeed centerless grinding process [15]. In steady state grinding, the sliding velocity ΔV, defined as (V

_{w}− V

_{r}), is about +0.008 m/s, and the slippage ratio ΔV/V

_{r}is about 2%, where V

_{w}and V

_{r}are the work and RW peripheral velocities, respectively. Since the sliding velocity is very small, the work rotation speed n

_{w}can be represented by:

_{w}) are used for the analysis of the optimum setup condition, and the practical parameters (θ, CH, N

_{r}) are the outputs of the model.

## 3. Centerless Grinding Systems and the Characteristic Equation

_{0}is the initial amplitude of the waviness, n

_{w}is the work rotation speed in rps and t is the grinding time. When σ is positive, the amplitude of n lobes grows with grinding time t and the grinding process can be identified as the chatter vibration. In case of σ < 0, the amplitude of n lobes is decreased with grinding time t, and the grinding process becomes stable with improved roundness.

## 4. Three Stability Criteria in Centerless Grinding

#### 4.1. Work Rotation Stability Criterion

_{T}and f

_{N}represent the tangential and normal grinding forces per unit width at the cut section. Rb and Rr are the resultant forces, while μ

_{b}and μ

_{r}are the friction coefficients at the contact points with the blade and the RW, respectively. w(l) is the work weight per unit width at the cut Section 1. The torque equilibrium equation can be written by:

_{N}/f

_{T}). For convenience, the plus sign of μr is assigned to the downward friction force and the minus sign is assigned to the upward one.

_{U}under the stable grinding condition is derived from Equation (14).

_{r}

_{0}is the maximum static friction coefficient of RW. When the tangential grinding force f

_{T}is smaller than f

_{U}, the work rotation speed Vw can be controlled with the RW speed Vr.

_{U}with respect to the blade angle θ with various friction coefficients μ

_{r}

_{0}. The grinding force f

_{U}is normalized with the diameter d of a simple cylindrical workpiece made of steel. f

_{U}increases with increased θ. When θ is greater than a certain angle with μ

_{r}

_{0}, the f

_{U}value becomes infinite. Under this condition, there is no risk of the spinners phenomenon occurring. The zone with the infinite f

_{U}value is called the “safe operation zone.” For instance, there is no limit on f

_{U}when a blade of θ > 42° is used with an RW of μ

_{r}

_{0}= 0.25.

_{r}

_{0}values on the (θ–γ) chart and provides guidelines for satisfying the work rotation stability criterion (WRSC). Stable grinding without any risk of spinners can be obtained by selecting the set of (θ, γ) from the safe operation zone, and the WRSC is satisfied with the setup conditions (θ, γ).

#### 4.2. Geometrical Rounding Stability Criterion

_{e}that will appear can be found by 180/γ [6].

#### 4.3. Dynamic System Stability Criterion

_{w}) diagram for the dynamic system stability criterion (DSSC). Figure 13 plots the 3D positive growth rates σ of the characteristic roots on the (n·γ–n·n

_{w}) diagram. The chatter zones are shown as “mountains” located near the natural frequencies in the (n·n

_{w}) axis. The higher the height of the mountain, the more severe the chatter vibration. Since the chatter mountains are in 0 < (n·γ) < 180°, they generate even numbers of lobes during chatter vibration.

_{w}/γ) through the origin is determined when the center height angle γ and the work speed n

_{w}are given. Where the straight line hits a mountain, chatter vibration occurs at that frequency. In other words, where the straight line given by the ratio (n

_{w}/γ) does not hit any mountains, the DSSC is satisfied and chatter-free grinding is achieved. Figure 13 shows distinct areas that satisfy the DSSC and gives the ranges of (n

_{w}/γ) that provide chatter-free grinding.

_{w}) diagram can be plotted that shows its unique chatter zones. The chatter zones are identified by conducting systematic grinding tests. Figure 14a shows the chatter zones of grinding machine A. The chatter zones located near the natural frequencies of 100 Hz, 150 Hz and 200 Hz are shown on the vertical axis (n·n

_{w}) Hz. These chatter zones are located in 0 < (n·γ) < 180° and generate even numbers of lobes, while the zones located in 180° < (n·γ) < 360° generate odd numbers of lobes where γ > 6.7°. Figure 14b shows the chatter zones for grinding machine B. Machine B is designed with high, rigid structures and possesses spindles with very high stiffness. The first chatter zone appears at the natural frequency of 430 Hz, which is very high, and machine B creates wider chatter-free regions than conventional machine A.

_{w}) chart. The practical (γ–n

_{w}) chart can describe chatter zones, but cannot describe the chatter generation mechanism, the chatter zones’ various vibration modes, or the areas where stable grinding can occur. However, the (γ–n

_{w}) chart is very effective in setup operations when used along with the analytical (n·γ–n·n

_{w}) diagram. The ranges of chatter zones in (n

_{w}/γ) are explicitly given, as shown in Figure 15b.

- 1.
- (n
_{w}/γ) H > 3.0 (high-speed chatter-free zone; KH) - 2.
- (n
_{w}/γ) L1 < 0.6 when γ is lower (low-speed chatter-free zone 1; KL1) - 3.
- (n
_{w}/γ) L2 < 0.28 when γ is higher (low-speed chatter-free zone 2; KL2)

- 4.
- (n
_{w}/γ) H > 4.24 (KH) - 5.
- (n
_{w}/γ) L1 < 2.15 (KL1) for γ < 6.67° - 6.
- (n
_{w}/γ) L2 <1.08 (KL2) for γ > 6.67°

_{w}/γ) < 2.15. The setup for this chatter-stable zone is too risky to use, so it is not considered an area of stable, chatter-free conditions.

_{w}is controlled by the regulating wheel speed Nr, it is helpful to convert the (γ–n

_{w}) charts (Figure 15b and Figure 16b) into a (γ–N

_{r}) chart, as shown in Figure 17. For practical setup operations, the range of the center height angle is set to γ = 3° to 9°. Also, in this case the range of the speed ratio q (defined as the ratio of the work speed Vw to the grinding speed Vg) for surface roughness control is set to 1/q = 25 to 150. In Figure 17, three chatter-free stable zones—KH, KL1 and KL2—are shown within the constrained range. The dynamic system stability criterion is satisfied when a set of (γ, Nr) is selected from the chatter-free stable zones KH, KL1 and KL2.

## 5. Modeling to Find the Optimum Setup Conditions that Satisfy the Three Stability Criteria of Centerless Grinding

_{r}

_{0}of a given regulating wheel. Also, the sets of (θ, γ) satisfying both geometrical rounding stability criteria—RW − GRSC and B − GRSC—are calculated. Then, the sets of (n

_{w}, γ) (work speed n

_{w}and γ) that satisfy the dynamic system stability criterion are found. The sets of (n

_{w}, γ) are selected from one of the stable chatter-free zones: KH, KL1 or KL2 (see Figure 17).

_{w}) satisfying all three stability criteria. The third step is to determine the optimum set from among the population of the (θ, γ, n

_{w}) sets by calculating the PI (performance index) function based on the process aim (accuracy or productivity). Finally, the optimum set (θ, γ, n

_{w}) is converted into practical setup conditions (blade angle θ, center height CH, RW speed Nr) as the outputs of the developed model.

_{r}and the maximum friction coefficient μ

_{r}

_{0}of the rubber bonded regulating wheel [15].

_{w}by using γgw and the chatter stability boundary lines of (n

_{w}/γgw). From these calculations, the optimum sets of (θ, γ, n

_{w}) are discovered.

_{w}). The PI functions are summarized in Table 3. The weighting factors of the PI functions are determined by applying theoretical knowledge, experimental knowledge and operational skills. PI functions can be updated with newly gained knowledge and skills. For each process aim (accuracy or productivity), the values of the PI function for all setup conditions are calculated and these sets are ranked in ascending order from minimum to those with greater values. The smaller the PI values, the more suitable the setup conditions. The setup conditions with the smallest PI function values are defined as the optimum setup conditions.

- (1)
- machine B is applied
- (2)
- the process aim is accuracy-oriented
- (3)
- the work shape and the size are a cylindrical workpiece with 40 mm (D) × 60 mm (L)
- (4)
- the rubber bonded RW is dressed with lead
_{r}= 0.5 mm/rev by SPD - (5)
- the GW diameter is 453 mm
- (6)
- the RW diameter is 350 mm
- (7)
- the available blade angles are θ = 27.5° and θ = 40.3°

_{w}= 6.1 rps and the 1/q is 25.

_{w}/γ) than machine A in chatter-stable zones KH, KL1 and KL2. Machine B’s high stiffness creates more extensive chatter-stable zones than conventional machine A.

## 6. Conclusions

- (1)
- Accepts various shapes of workpiece: cylindrical, tapered, spherical and multi-stepped
- (2)
- Applicable to any centerless grinding machine
- (3)
- Data management via a data bank
- (4)
- Inputs are easy to enter and outputs are readily usable
- (5)
- Designed for operators
- (6)
- Provides scientific parameters for engineers/managers
- (7)
- Finds all setup conditions satisfying the three stability criteria of centerless grinding systems
- (8)
- Outputs the optimum condition based on process aim

## Conflicts of Interest

## References

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**Figure 2.**Infeed centerless grinding process. D

_{r}= 255 mm, D

_{w}= 30 mm, N

_{r}= 30 rpm, V

_{w}= 0.4 m/s, Sliding velocity: 0.008 m/s, Slippage ratio: 2%.

**Figure 5.**Arrangement of centerless grinding and forces acting on workpiece. (

**a**) Forces acting on workpiece during centerless grinding; (

**b**) Cylindrical workpiece.

**Figure 9.**Effect of center-height angle on rounding mechanism. (

**a**) Odd lobe appearance at lower γ; (

**b**) Even lobe appearance at a specific γ.

**Figure 10.**Regulating wheel (RW) geometrical rounding stability criteria. (

**a**) RW geometrical rounding unstable conditions; (

**b**) RW geometrical rounding stable conditions.

**Figure 11.**Blade geometrical rounding stability criteria. (

**a**) Blade geometrical rounding unstable conditions; (

**b**) Blade geometrical rounding stable conditions.

**Figure 12.**Dynamic stability of centerless grinding system (experiment). (

**a**) Unstable grinding process (chatter); (

**b**) Stable grinding process.

**Figure 13.**Dynamic system stability diagram for Machine A. Conditions: b = 70 mm, k’w = 2 kN/mm·mm, k’cr = 0.3 kN/mm·mm, k’cs = 1 kN/mm·mm, kmr = 0.1 kN/μm, kms = 0.15 kN/μm, km1 = 0.3 kN/μm, fnr = 100 Hz, fns = 200 Hz, fn1 = 150 Hz, ζr = 0.05, ζs = 0.05, ζ1 = 0.05.

**Figure 15.**Dynamic system stability criterion for machine A. (

**a**) Chatter zones confirmed by grinding tests; (

**b**) Dynamic system stability criterion.

**Figure 16.**Dynamic system stability criterion for machine B. (

**a**) Chatter zones on (n·γ–n·n

_{w}) diagram; (

**b**) Chatter zones on (γ–n

_{w}) chart.

**Figure 17.**Chatter free zones on γ–Nr chart. KH: High speed chatter-free zone, KL1, KL2: Low speed chatter-free zones.

Inputs | Action | Parameters Referred to Data Bank |
---|---|---|

Machine name | Select | Machine specifications, Machine dynamic characteristics (Natural frequencies, damping ratios) |

Workpiece shape | Select | Cylindrical (CYD), Tapered (TPD), Spherical (SRL), Multi-stepped (STD) |

Workpiece part-number | Select | Dimensions (diameter, length, etc.), profile |

GW diameter | Measure | New, worn, measured |

RW diameter | Measure | New, worn, measured |

RW dresser type and dress lead | Select | Single point dress, Rotary dress, Friction coefficient of RW |

Blade availability | Select | Blade angle θ, blade thickness t |

Setup Parameters | Symbol | Unit. | Min. | Max. | Typical |
---|---|---|---|---|---|

Range of speed ratio | 1/q = Vg/Vw | - | 25 | 150 | 100 |

Range of blade angle | θ | ° | 15 | 45 | 30 |

Range of Center-height angle | γ | ° | 3 | 9 | 6.67 |

Range of regulating wheel speed | Nr | rpm | 15 | 100 | 50 |

Range of GW diameter | Dg | mm | 375 | 455 | 450 |

Range of RW diameter | Dr | mm | 275 | 350 | 345 |

Range of Workpiece diameter | Dw | mm | 5 | 100 | 40 |

Grinding wheel speed in revolution | Ng | rpm | 1260 | 2300 | 1890 |

Grinding speed | Vg | m/s | 30 | 45 | 45 |

Process Aim | PI Function | Weighting Factors | ||||
---|---|---|---|---|---|---|

A | B | C | D | E | ||

Accuracy | PIa = A × I θ − 27.5 I + B × I γ − 6.67 I + C × I Nr − 50 I + D × FL + E × STYP | 0.2 | 2 | 0.1 | 0.01 | 2 |

Productivity | PIr = A × I θ − 45 I + B × I γ − 5.15 I + C × I Nr − 50 I + D × FL + E × STYP | 0.1 | 2 | 0.12 | 0.5 | 2 |

Priority | Optimum Set Up Conditions | Engineering Parameter | Stability Parameters | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Ranking | Blade Angle | Blade Thickness | Center-Height | RW Speed | CH Angle | Work Speed | 1/q Ratio | (n_{w}/γ) | Stable Zone | RW − GRSC | Blade − GRSC |

No. | θ (°) | t (mm) | CH (mm) | Nr (rpm) | γ (°) | n_{w} (rps) | Vg/Vw | (1/s) | KH/KL1/KL2 | 180/γ | (90 − θ − β)/γ |

1 | 40.3 | 20 | 12.68 | 41.8 | 6.68 | 6.1 | 25 | 0.91 | KL2 | 27 | 7 |

2 | 40.3 | 20 | 12.66 | 83.5 | 6.67 | 12.2 | 25 | 1.83 | KL1 | 27 | 7 |

3 | 40.3 | 20 | 13.67 | 45.1 | 7.2 | 6.58 | 25 | 0.91 | KL2 | 27 | 6.5 |

**Table 5.**The optimum setup conditions provided by the Opt-Setup Master for the grinding of cylindrical workpieces with different machines A and B.

Case No. | Work Dia.x L. | Machine | Blade Angle | C-H Angle | RW Speed | Chatter DSSC | Chatter Stable Zone | RW − GRSC | Blade – GRSC |
---|---|---|---|---|---|---|---|---|---|

mm | A/B | θ (°) | γ (°) | Nr (rpm) | (n_{w}/γ) | KH/KL1/KL2 | 180/γ | (90 − θ − β)/γ | |

1 | 10 × 20 | A | 27.5 | 6.67 | 38.6 | 3.37 | KH | 27 | 9 |

2 | B | 27.5 | 6.67 | 64.1 | 5.61 | KH | 27 | 9 | |

3 | 20 × 30 | A | 27.5 | 6.67 | 77.1 | 3.37 | KH | 27 | 9 |

4 | B | 27.5 | 6.67 | 41.8 | 1.83 | KL1 | 27 | 9 | |

5 | 30 × 50 | A | 40.3 | 6.67 | 14.6 | 0.43 | KL1 | 27 | 7 |

6 | B | 40.3 | 6.67 | 62.7 | 1.83 | KL1 | 27 | 7 | |

7 | 40 × 60 | A | 40.30 | 6.67 | 19.4 | 0.43 | KL1 | 27 | 7 |

8 | B | 40.30 | 6.68 | 41.8 | 0.91 | KL2 | 27 | 7 | |

9 | 50 × 70 | A | 40.30 | 6.67 | 24.3 | 0.43 | KL1 | 27 | 7 |

10 | B | 40.30 | 6.68 | 52.3 | 0.91 | KL2 | 27 | 7 | |

11 | 60 × 80 | A | 40.30 | 6.67 | 29.1 | 0.43 | KL1 | 27 | 7 |

12 | B | 40.30 | 6.68 | 62.7 | 0.91 | KL2 | 27 | 7 |

_{r}

_{0}= 0.4, Available blade θ = 27.5°, 40.3°.

© 2017 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Hashimoto, F. Model Development for Optimum Setup Conditions that Satisfy Three Stability Criteria of Centerless Grinding Systems. *Inventions* **2017**, *2*, 26.
https://doi.org/10.3390/inventions2040026

**AMA Style**

Hashimoto F. Model Development for Optimum Setup Conditions that Satisfy Three Stability Criteria of Centerless Grinding Systems. *Inventions*. 2017; 2(4):26.
https://doi.org/10.3390/inventions2040026

**Chicago/Turabian Style**

Hashimoto, Fukuo. 2017. "Model Development for Optimum Setup Conditions that Satisfy Three Stability Criteria of Centerless Grinding Systems" *Inventions* 2, no. 4: 26.
https://doi.org/10.3390/inventions2040026