Volume 10

Issue 3

##### Wood Technology

JOURNAL OF

POLISH

AGRICULTURAL

UNIVERSITIES

Available Online: http://www.ejpau.media.pl/volume10/issue3/art-09.html

**
APPLICATION OF THE THEORY OF WORK OF CUTTING DISTRIBUTION IN MILLING
**

Grzegorz Kowaluk*
Wood Technology Institute, Poznan, Poland*

The goal of this paper is to demonstrate the method of the application of the theory of the work of cutting distribution from the flat cutting to milling. Calculating the work of cutting and work of cutting components based on forces measured at flat cutting is easier but less precise, and the cutting conditions are less similar to the industrial ones. Determining the work of cutting distribution based on measuring of forces in milling gives more accurate and realistic results but there is a necessity to apply more complicated and expensive measurement equipment.

**Key words:**
work of cutting, fracture, surface creation, chip deformation, milling.

**INTRODUCTION**

One of the methods to describe specific work of cutting is to use the pendular labormeter [13]. The method is based on the usage of the energy of pendulum, where the tool is mounted, to cut. Two ways can be used in this method: single-cut principle, and multiple-cut principle. In single-cut method the loss of the energy used to cut of the single chip is measured. In the multiple-cut principle the whole energy of the pendulum is consumed to cut. In this case the specific work of cutting is calculated as energy of pendulum in relation to volume of wood turned into chips. The efficiency of the whole pendular labormeter has to be taken into consideration when calculating the specific work of cutting.

The other way to describe the specific work of cutting is to use the specific force of cutting which is measured for model parameters (pine wood, 13% of MC, 60° of cutting angle, sharp tool, 50 m·s^{-1} cutting speed etc.) and apply it to particular conditions with special conversion coefficients [15].

More practical description of the specific work of cutting is to measure the power of the main machine tool motor [7, 16]. This method can be applied in actual industrial conditions. The essence of this method is to measure the electrical energy, consumed by motor, to cut the known wood volume in known time. But, this method is less precise because of the motor and gear efficiency. Another disadvantage of this method is that in many cases the measured power changes consist only small part of the whole consumed energy, which can be the main source of mistakes.

Modern methods to calculate the specific work of cutting are generally based on measuring the cutting force. According to Orlicz [15] the cutting force is the force moving the knife in his motion direction which is needed to overcome the cutting resistance. The main and most important elements in the measuring equipment are actually the piezoelectric force sensors and the sequel of strain gauges. These kinds of sensors are the most popular in investigations [1, 3, 5, 8, 14]. The forces measured in this way are the base to the work of cutting calculation [9] and to calculate the work of cutting distribution to work of new surface creation (work of fracture/fracture toughness under “real-life” cutting conditions) and work of chip deformation [3, 10, 12, 17]. A direct relation between linear cutting and rotary cutting, in which rotary cutting was simulated based on linear cutting, was also investigated [4]. But the above mentioned measurement of cutting forces and calculation of work of cutting distribution were described and explained only in relation to flat cutting using the microtome technique. There are no publications about the methodology of calculating the work of cutting distribution in rotating cutting, i.e. milling.

The goal of this paper is to demonstrate the method of the application of the theory of the work of cutting distribution from the flat cutting to milling.

**Work of cutting distribution in flat cutting**

According to Huang et al. [9] the work of cutting consists in work of fracture (work of new surface creation) and work of chip deformation (work of curling) (Fig. 1). In flat cutting the work of cutting is described by the cutting force and the cutting path [12]:

[J] | (1) |

Fig. 1. Graphical method for the determination of the cutting and plastic work [9] |

This work, referred to the surface of cut material, defines work (energy) of cutting per surface unit (specific work of cutting):

[J/m^{2}] |
(2) |

If the specific work of cutting relates to a few different thicknesses of cut layers, the plot on the Cartesian co-ordinate systems is a linear function y = ax + b, where “a” component is the specific work of chip deformation E_{D} and “b” is the specific work of new surface creation E_{S}. Thus, the equation of the work of cutting distribution is:

[J/m^{2}] |
(3) |

where “h” is the chip thickness (in flat cutting the height of cutting layer).

Obviously, determination of the specific work of new surface creation is based on the theoretical extrapolation of the above mentioned plot to an intersect Y-axis. It means 0-thickness cut layer.

Practically, the distribution of the specific work of cutting is calculated from the forces measured during cutting on the microtome. The microtome technique makes it possible to reach a precise height of a cut layer, but the cutting speed is much lower then in actual industrial conditions [9]. Nevertheless, the results of the work of cutting, calculated in this way, can be easy applied in other ways of machining, because the microtome technique is an example of a cut with an elementary blade.

**Work of cutting distribution in milling**

In planning and milling chips have the “comma” shape [6]. The thickness of such chip is linearly changing from zero to maximum [18] (Fig. 2). According to this it is possible to apply the work of cutting distribution theory to milling. But in milling there is a possibility of calculating the work of cutting, work of new surface creation and work of chip deformation, based on the data from forces measuring during one chip creation. It is important that when a forces measurement is conducted during cutting with the “industrial” cutting speed, the time of single chip creation is very short. To achieve accurate force measurement results, the force sensors and the whole data acquisition equipment has to enable the data registration with the high frequency. For example, for a cutting speed about 50 m·s^{-1}, cutting diameter 125 mm and a height of cutting layer 1mm, the time of single chip creation is about 0.22 ms. To obtain 10 readouts of forces in this time (to plot the force vs. cutting time dependence for a single cut) the acquisition equipment has to work with a 44.7 kHz frequency.

Fig. 2. Calculated chip shape and chip thickness as a function of time [18]: a-calculated chip shape, b-chip thickness as a function of time |

In milling, the work of new surface creation can be calculated directly from the F_{x} force. As it is shown in Fig. 3, at the start of the chip creation (start of the new surface creation), there is only one force – F_{x0}. Thus, the F_{x0} force is at this moment the cutting force F_{c0}, and there is no the repel force F_{r}. On the Fig. 4 the forces ordination at the mean chip thickness is shown. In this situation the feeding force F_{x} and the normal force F_{y} are non-zero. To calculate the cutting force F_{c}, the F_{x} and F_{y} must be pointed out. To varying the forces descriptions on Fig. 3 and 4 the “mean” index is added for forces at the mean chip thickness.

Fig. 3. Forces ordination in milling at the zero chip thickness (description at Fig. 4) |

Fig. 4. Forces ordination in milling at the mean chip thickness |

m) – measured, (c) – calculated F _{x} – force parallel to feed [m]F _{y} – force normal to feed [m]F _{o} – outcomes force from F_{x} and F_{z} [c]F _{c} – cutting force [c]F _{r} – repel force [c]F _{x0} – F_{x} in the time of crack initiation [m]F _{c0} – F_{c} in the time of crack initiation [m]D – cutting diameter n – tool rotation direction h – height of cutting layer u – feed of material ψ/2 – half of the cutting angle δ _{F} – angle between F_{x} and F_{o} force |

At a mean chip thickness, there are two forces, in two directions: parallel F_{x mean} and normal F_{y mean} to feed direction. To calculate the distribution of the work of cutting, the value of the cutting force F_{c mean} is required:

[N] | (4) |

where

[N] | (5) |

The value of the specific work of cutting in milling can be calculated as:

[N] | (6) |

where “τ” is the length of the chip (length of the arc of the cut).

The value of the specific work of new surface creation can be calculated from the equation:

[J/m^{2}] |
(7) |

And, the value of the specific work of chip deformation can be calculated as:

(8) |

where “h_{m}” is a mean chip thickness.

Fig. 5. Feeding and normal forces measuring example for a single chip (on the basis of [18]): F_{x0}=F_{c0} – feeding force when the crack starts, F_{x max} – feeding force at the maximum chip thickness, F_{y max} – normal force at the maximum chip thickness |

In Figure 5 an example of the forces measuring for the single chip of MDF milling is shown. Sinn et al. [17] uses the F_{x0} force to calculate the fracture toughness [2]. As it is shown in the figure, the normal force for the F_{x0}-time is non-zero. Sinn et al. [18] says that due to a small penetration angle of less than 11° it is correct to apply the chip-thickness-force model directly for the feeding force.

**CONCLUSIONS**

The work of cutting distribution theory came into existence as the method of the calculating of the work of new surface creation (work of fracture) and work of chip deformation, based on the force measurement results in flat cutting. Despite that the speed of cut in flat cutting, using microtome technique, is lower than cutting speed in industrial conditions, results of these calculations can be applied to other machining ways. Thanks to a lower speed of cut, there is a possibility of using less complicated and cheaper measurement equipment. The calculation of the work of new surface creation is a theoretical approximation of the chip thickness to 0, because there is probably no possibility of measuring the forces in cutting with 0-mm thin chip in practice.

As it is shown above, it is possible to apply the work of cutting distribution theory to rotary cutting such as planning and milling. Calculating the work of cutting, work of new surface creation and work of chip deformation based on the forces measured in rotary cutting is more realistic and more accurate. But to conduct investigations of the cutting speed which is similar to industrial cutting speed more complicated and modern force measuring equipment is required.

**ACKNOWLEDGMENTS**

The author is very grateful to dr Gerhard SINN, from the Institute of Physics and Materials Science, University of Natural Resources and Applied Life Sciences, Vienna, for helpful and pertinent remarks, and matter-of-fact discussions during preparation of this paper.

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Accepted for print: 28.08.2007

Grzegorz Kowaluk

Wood Technology Institute, Poznan, Poland

Winiarska 1, 60-654 Poznan, Poland

phone: (+48 61) 849 24 43,

email: g_kowaluk@itd.poznan.pl

Responses to this article, comments are invited and should be submitted within three months of the publication of the article. If accepted for publication, they will be published in the chapter headed 'Discussions' and hyperlinked to the article.