Introduction

Investigations of wood machinability are gaining in importance considering rapid development of new materials and their machining technologies. Machinability is defined as the capacity of the material to be machined by cutting using common economical manufacturing techniques and technologies (machines, tools, methods) (^{StaniÄ 1989}). This definition applies to all materials and so does to wood as well. It is evaluated by a set of criteria or functions: tool life, cutting forces, quality of machined surface, accuracy and quality characteristics, chip shape (^{KalajdÅ¾iÄ 2008}). In general, good machinability implies good machined surface quality, long tool life, low values of the required cutting force and lower machining costs. However, factors improving surface quality often diminish other machinability indicators (heat treatment, versatile chemical treatments and the like). That is why it is a real challenge to devise a method for enhancing machinability without product surface quality loss.

The papers dealing with the theory of wood processing considered different factors such as regime of cutting (^{Axelsson et al. 1993}, ^{Wen-Hsiang 2000}, ^{Aguilera and Zamora 2009}, ^{Aguilera and MuÃ±oz 2011}, ^{Aguilera et al. 2013}), various machining conditions, including feeding direction and grain direction (^{Goli and Uzielli 2004}), cutter material, effect of shear cutter, effect of cutting angle (^{Belleville et al. 2016}), properties of wood (^{Thoma et al. 2015}) and their impact on change in the cutting forces, machining accuracy and quality of machined surface.The major goal of mentioned investigations was to improve understanding of the interaction between the tools and the workpieces so as to achieve more efficient cutting process management. Knowledge of interaction between the tools and the wood as well as the control of the cutting process are unavoidable factors affecting productive and economical manufacturing in wood processing (^{Eyma et al. 2004}).

The literature with the focus on mechanical wood processing presents different models (methods) for predicting cutting forces. By applying data on the properties of wood and wood-based materials being machined, and scheduled regimes of cutting, these models are able to predict the material behavior in the machining process, the machining outcome, and primarily the quality of machined surface. Most often, those are the methods of coefficients generated based on carried out experiments (^{Orlicz 1982}, ^{Axelsson et al. 1993}, ^{Goglia 1994}, ^{Porankiewicz et al. 2011}, ^{Naylor et al. 2012}, ^{KrÅ¡ljak 2013}, ^{Porankiewicz and Goli 2014}, ^{MandiÄ et al. 2015}). This paper presents the evaluation of the cutting force based on a mathematical model, whose parameters were determined using experimental results obtained under controlled laboratory conditions. The verified mathematical model could be used for predicting wood behavior in different cutting regimes under real conditions. Combination of data obtained from the model and monitoring and analyzing the engaged cutting power would facilitate the control of the machining process and reduce the number of machined pieces with impermissible deviations of dimensions and surface quality.

Materials and methods

Investigations deployed oak wood (*Quercus robur)*, it has a strong history as some of the most valuable and highest quality domestic hardwoods. The most widely utilization oak has found in the production of costly solid wood furniture. Samples were of uniform properties, of radial direction and without visible irregularities in wood grain. Physical and mechanical properties were tested in accordance with various standards, i.e., standards for density (^{SRPS ISO 13061-2, 2015}), bending strength (^{SRPS ISO 13061-3, 2015}), Brinell hardness, perpendicular to the wood grains, in radial direction (^{EN 1534, 2010}) and for modulus of elasticity ( SRPS ISO 13061-3, 2015). Mean values for samplesâ measured physical and mechanical properties are presented in Table 1 and lie within standard deviation of Â±5%.

Property | Mean | Minimum | Maximum | Std.Dev. | Coef.Var. |
---|---|---|---|---|---|

Density (kg/m^{3})* |
790 | 770 | 800 | 20 | 2,19 |

Ovendry density (g/cm^{3}) |
760 | 740 | 770 | 20 | 2,28 |

Moisture Content (%) | 7,97 | 7,80 | 8,10 | 0,15 | 1,92 |

Hardness (MPa) | 45,18 | 43,12 | 47,18 | 2,03 | 4,49 |

Bending strength (MPa) | 125,14 | 119,70 | 130,33 | 5,32 | 4,25 |

Modulus of elasticity (MPa) | 11274,40 | 10949,32 | 11471,77 | 283,68 | 2,52 |

*The moisture content, of samples varied (from 7,80% to 8,10%, with an average of 7,97%), so the values of the wood density computationally were normalized to a humidity of 8%.

Oak samples has been conditioned before testing in laboratory environment conditions: relative humidity of 45 Â±5% and room temperature of 20 Â±3Â°C. These conditions brought samples to an equilibrium moisture content of 8 Â±1%, which is standard recommendation for values of moisture content for furniture in Serbian climates conditions.

Mathematical model for predicting the cutting force

Investigations of any process or system require the selection of an adequate model and testing of its accuracy and adequacy against the real process. This paper employed a common model form of regression analysis for prediction of the cutting forces in peripheral cutting, depending on certain impact factors. The paper analyzes the impact of feed rate *f*
_{1} = Î½_{
F
} , milling depth *f*
_{2} = _{
cD
} and cutting edge rake angle *f*
_{3} = *Î³* upon components of cutting force (_{
Fx
} and _{
Fy
} ). The degree function of machinability (^{Mason et al. 2003}), applied of _{
Fx
} example was calculated according to Equation (1).

The adopted experimental design is a three-factorial plan of the first order with the total number of experiments . Each experiment was replicated at least eight times. The experiment was carried out at constant cutting speed Î½C = 38,3 m/s (D = 125 mm, n = 5860 RPM), for three different feed rates Î½F = 4, 8 and 16 mâmin-1, three milling depths cD = 2, 3 and 4,5 mm and three different rake angles Î³ = 16Â°, 20Â° and 25Â°. The limits of factor interval variation were chosen to satisfy the condition for *i =* 1, 2, 3.

Program for the machining process simulation in peripheral milling

Based on developed mathematical model for prediction of the cutting force, a universal flowchart of the sequence of steps in a procedure was created to simulate the peripheral milling process, as presented by the example of the component _{
Fx
} (Figure 1).

According to the developed procedure for a given tool geometry, i.e. cutter diameter (*D*), number of cutter teeth (*z*), rake angle (*Î³*) and clearance angle (*Î±*) and machining process parameters, simulation is done of one complete cutter revolution in increment of *i*=1Â°. Depending on input parameters, calculations are made of the cutter pitch angle (*Ï*
_{t}) according to Equation (2), engagement angle (*Ï*) according to Equation (3), number of teeth in engagement (_{
iz
} ) according to Equation (4), cutting speed (_{
Î½C
} ) according to Equation (5), feed per revolution (_{
fR
} ) according to Equation (6) and feed per tooth (_{
fz
} ) according to Equation (7).

Within the framework of the machining process, for each value of the angle increment (*i*), based on each cutter tooth engagement angle (*Ï*), it is checked if the considered tooth is in engagement and if it is, the value of the current milling thickness (_{
ai
} ) and current elementary value of the milling force component (_{
dFx
} ) are calculated according to the created model. Finally, summation of all elementary values of the milling force components is done for each individual tooth.

Milling and experimental scheme

In order to create the model for predicting the milling forces, firstly, the values of the milling force were experimentally measured for selected machining conditions and tool geometry (cutter rake angle, depth of cut and feed rate). At least 6 times (two measurements on three or four samples) measurement was repeated for each combination of parameters. For a zero point of experiment (point with average values of parameters) measurement was repeated 16 times (two measurements on eight samples). Meaning, making total of 79 measurements for the whole experiment.

Experimental investigations were performed using a table milling unit of the universal combined machine âMinimax CU410Kâ at the Center for Machines and Tools, Faculty of Forestry, University of Belgrade. On the milling unit, a fixture was mounted for positioning a two-component dynamometer with strain gauges and wood sample (workpiece). Samples were obtained from boards of uniform properties, of radial direction and without visible irregularities in wood grain. Final dimensions of samples (30x30x110 mm) were obtained using a circular saw. The values of measured cutting forces change during peripheral milling depending on the current chip thickness and tooth position in engagement, Figure 2a. Signals from the dynamometer are transmitted to the Hottinger measuring bridge - HBM-KWS3082A, and from there to the acquisition card cDAQ-9174 (National Instruments), and then to the PC, where LabView software is used to create a file of records of the components of milling forces _{
Fx
} (component in the feed rate direction) and _{
Fy
} (component normal to the machined surface) as a function of time (Figure 2b).

Figure 3 shows values of the milling forces measured with respect to time (applying software package Matlab). Time records of two perpendicular components of the milling force and the fact that it is orthogonal cutting provide for determining the intensity of the resultant cutting force according to the formula (Figure 3).

The experiment was used to create models for predicting mean values of the components and the resultant milling force that were embedded in a developed software solution for time change simulation of the components and the resultant cutting force in the peripheral milling process. Further below, an example provided is a description of creating a model and a representation of simulation solely for the cutting force in the feed rate direction.

Results and discussion

Log transformations must be applied to convert the mathematical model of the cutting force component *F*
_{x} into linear form , so as to determine numerical values of constant *C* and exponents *p*
_{1}, *p*
_{2} and *p*
_{3} in the model. Experimental results obtained in this research were employed to determine constants of the force *C* and the exponents *p*
_{1}, *p*
_{2} and *p*
_{3}, i.e. machinability parameters in the assumed mathematical model.

Table 2 shows the plan of matrices with numerical values of the experiment factor, mean values for experimental results of the component _{
Fx
} (obtained based on 5 replicates minimum), model results and measurement errors.

Regression dependency of the cutting force component _{
Fx
} in peripheral milling upon the feed rate, milling depth and cutting edge rake angle is shown by the model (Equation 8):

Linear mathematical model adequacy is tested according to Fisherâs F-criterion. An adequate model must fulfill the condition _{
Fc
} < _{
Ft
} , i.e. _{
Fc
} = 2,24 < _{
Ft
} = 9,01 shows that the condition is fulfilled, meaning that the mathematical model describes with adequate accuracy the dependency of the cutting force upon mentioned impact factors in adopted testing conditions.

Evaluation of the model parametersâ significance, also performed according to the F-criterion (_{
Fci > Ft
} ), indicated that the condition is satisfied for impact parameters such as feed rate _{
u (Fc1
} = 53,19 > _{
Ft
} = 10,13*)* and milling depth _{
a (Fc2
} = 69,58 > _{
Ft
} = 10,13*)*, which means that these parameters are significant. However, the condition is not satisfied for the cutting edge rake angle Î³ (_{
Fc3
} = 0,01 < _{
Ft
} = 10,13), that is, in this case rake has not shown any significant impact. However, the condition is not satisfied for the cutting edge rake angle Î³, that is, in this case rake has not shown any significant impact.

Based on the values and signs of exponents in the setup mathematical model (Equation 8), it can be inferred that in terms of intensity the milling depth has more significant impact on the resultant cutting force compared to the feed rate, and that both parameters impact the cutting force directly proportionally, which is in agreement with the literature sources (^{Krilek et al. 2014}, ^{Barcik et al. 2008}).

When milling depth is increased in the range of _{
cD
} =2Ã·4,5 mm the force component _{
Fx
} is increased by 1,59 - 1,75 times (Î½_{
F
} = 16m/min, *Î³* = 16Â° Ã·25Â°), i.e. by 1,53 - 1,84 times (Î½_{
F
} = 4m/min, *Î³* = 16Â° Ã· 25Â°).

An increase of feed rate from 4 m/min to 16 m/min in all depths of cutting leads to the increase of the cutting force, as a consequence of a larger amount of removed chip. The force component _{
Fx
} increase ranges from 1,22 to 1,41 (_{
cD
} = 2Ã·4,5 mm, *Î³* = 25Â°), i.e. 1.75Ã·2 times (_{
cD
} = 2Ã·4,5 mm, *Î³*=16Â°), which can be explained by the chip compression process in the cutting zone.

Rake angle *Î³* affects chip compression and its optimum value is related to material hardness and plasticity. Under considered machining conditions, as the rake is increased from 16Â°Ã·25Â°, the force component _{
Fx
} is decreased by as much as 32%, which is more prominent in smaller depths of cut and feed rates. This complies with the theory that the cutting force decreases as the value of rake angle increases (^{GÃ¼nay et al. 2005}, ^{Yanda et al. 2010}, ^{Krilek et al. 2014}) and vice versa. Geometrically viewed, rake angle does not have any direct impact on the surface roughness. However, indirectly, as the rake angle is increased, chip evacuation is facilitated, whereby elastic deformations of machined surface are decreased thus reducing surface roughness but causing destruction of the cutting edge, on the other hand. Investigations carried out indicate that this phenomenon shown on the oak wood is not manifested for the considered range of the rake angle 16Â°Ã·25Â° and for the considered machining conditions.

Among other things, a small range of rake angle varying was chosen, following manufacturerâs recommendations, and all this with the aim to bring experimental conditions to real ones as close as possible, which has contributed to non-significance of rake angle impact on the cutting force.

Using software package Matlab, according to the flowchart presented in Figure 1, an application was created for the machining process simulation. Application is divided into several units related to entering data on tool geometry and machining process parameters. By activating the command for the machining process simulation, a diagram of the milling force component is drawn for one complete tool revolution and mean value of the milling force component is shown in a corresponding field on the screen, Figure 4.

Elementary alignment of the application implies that milling cutter tooth radial run-out equals zero, and therefore the value of the milling force component for each tooth is equal in intensity.

Advanced cutting tool settings option enables additional definition of each tooth radial run-out that directly affects the value of feed per tooth for each tooth of the milling cutter. For the entered tool geometry and machining process parameters 4, experimental determination of the milling force component was carried out according to the experimental scheme from Figure 2b. Milling cutter tooth radial run-out amounted to - 0,02; 0,0; - 0,05; - 0,05 mm against nominal diameter respectively for each cutter tooth.

For the verification of the developed application, Figure 5 shows a diagram of values of the milling forces obtained by the experiment (solid line) and simulation (dashed line) separately for the cases with and without cutter tooth radial run-out, and all this for one complete tool revolution. Evidently, there is a better coincidence between experimental and model values of milling forces for the case when additional cutter tooth radial run-out is included in the application.

Apart from mentioned properties, the application can export simulation results in tabular form using the command Export data.

Conclusions

An adequate mathematical model has been created for prediction of cutting forces in oak wood milling under chosen machining conditions, as shown in the example of the cutting force component _{
Fx
} . The model describes with adequate accuracy the dependency of the cutting force upon the feed rate, milling depth and cutting edge rake angle.

Evaluation of the model parametersâ significance indicated that the cutting force is increased as feed rate and milling depth are increased, whereas cutting edge rake angle did not show significant impact in this case. The cause may lie in the fact that a small range of rake variation has been taken, in accordance with manufacturerâs recommendations, and all this with the aim to bring experimental conditions to real manufacturing conditions as close as possible.

As for intensity, the milling depth has dominant impact on the resultant cutting force compared to feed rate.

Using software package Matlab and presented flowchart the application was created for simulation of the machining process that draws a diagram of change in the milling force for one complete tool revolution and performs computations of the milling force mean value.

The application has the capacity to additionally define radial run-out of each milling cutter tooth, which, as established, has direct impact on the intensity of the cutting force in peripheral milling.

Considering that the experimental conditions were very close to real manufacturing ones, these investigations could be of practical importance for determining optimal cutting regimes in terms of the machined surface quality and energy saving.