Revisiting the estimation of cutting power with different energetic methods while sawing soft and hard woods on the circular sawing machine: a Central European case

In the classical approaches, used in Central Europe in practice, cutting forces and cutting power in sawing processes of timber are commonly computed by means of the specific cutting resistance kc. It needs to be highlighted that accessible sources in handbooks and the scientific literature do not provide any data about wood provenance, nor about cutting conditions, in which cutting resistance has been empirically determined. In the analyses of sawing processes, the use of a model with elements of fracture mechanics involved is an alternative way. In this work, predictions of the newly developed model (FRAC_MOD) for the circular sawing machine are presented. Thanks to this modern approach, it was possible to reveal the usefulness of the FRAC_MOD, using experimental results data on fracture toughness and shear yield stresses of both Polish pine (Pinus sylvestris L.) and Czech beech wood (Fagus sylvatica L.). The achieved results were compared to the forecasted values obtained with classical models (CLAS_PL and CLAS_CZ), which are commonly applied in Central European sawmills. The carried out analyses allowed us to discover undesired effects in the form of underestimation of cutting power when applying the CLAS_PL and CLAS_CZ models. For that reason, the FRAC_MOD cutting model could be suggested for the prediction of energetic effects in cases of dynamical analyses and even unsteady cases.

Basic specific cutting resistance (CLAS_CZ) k φ Basic specific cutting resistance for pine k || , k # , k ⊥ Basic specific cutting resistance for basic directions of cutting k ||# , k ||⊥ , k #⊥ , k ||#⊥ Basic specific cutting resistance for intermediate directions of cuttinġ m Mass of wood (chips) evacuated in a certain period of time at a certain cutting tool velocity p Parameter expressing the artificial effect of the cutting force on the flank face of the tool s Thickness of circular saw blade v c Cutting speed v f Feed speed z Number of teeth z a Number of teeth being in contact with the kerf (average) Ф c Shear angle which defines the orientation of the shear plane with respect to cut surface Φ G-edge Directional angle of a cutting edge Φ G-h Directional angle of a chip thickness Φ G-vc Directional angle of a cutting speed α f Side clearance angle α′ p1 , α′ p2 Back clearance angle β f Blade angle β μ Friction angle which is given by tan −1 μ = β μ γ Shear strain along the shear plane γ f Side rake angle δ f Cutting angle κ r Cutting edge angle κ′ r1 , κ′ r2 Minor tool cutting edge angle λ s Cutting edge inclination angle μ Friction coefficient ξ Coefficient taking into account a friction between the workpiece and the blade (CLAS_CZ) ρ Density of sawn wood τ γ Shear yield stress τ γ || , τ γ⊥ Shear yield stress for basic directions of cutting τ γ ||⊥ Shear yield stress for intermediate direction of cutting φ en Angle entrance of teeth in cut material φ ex Angle exit of teeth from cut material φ j Angular position of the jth tooth (immersion angle) φ P Pitch angle

Introduction
In the future, manufacturing will require much more powerful strategies for the control of processes in a highly automated manufacturing environment. In general, existing methods for the control of manufacturing operations, mainly based on the experience and craftsmanship of the manufacturing engineers/machinists, are becoming obsolete and must be replaced by science-/knowledge-based methods (Grzesik 2017). As costs for energy and raw material are rising, reduction in the saw kerf (Wasielewski et al. 2012;Orlowski and Walichnowski 2013) as well as improvement in the surface quality is becoming increasingly important (Krenke et al. 2017a, b). Hence, proper optimization of the cutting processes calls for the appropriate approach for estimation of cutting forces, since it could help in a better understanding of the interaction of the tool and raw material. Even though band saws have a narrower kerf than circular saws, the latter are the most common type of tool in wood machining (Kvietková et al. 2015;. , in their recent comprehensive review on wood 1 3 sawing, summarized findings on the impact of cutting factors on tool wear, surface quality and power consumption. Thus, it stands to reason that in the latter paper a discussion on cutting forces examination and estimation is rather limited. On the other hand, Krenke et al. (2017a, b) stated that due to missing model validation, some statistical models such as by Axelsson et al. (1993) and Porankiewicz et al. (2011) are not applicable to cutting force computation for varying test setups or/and parameters which differ from the specific examination.  showed that estimation of energetic effects using the newly developed cutting model (FRAC_MOD) that includes work of separation in addition to plasticity and friction is capable of predictions not only of average values of cutting power but also its dynamical changes. In this model, changes in uncut chip thicknesses, proper variations in shear yield stress and toughness with tooth orientation in relation to grain orientation have been taken into account. Moreover, this line of attack spreads opportunities to model the cutting process with circular saw blades. It should be emphasized that actual values of shear yield stresses and fracture toughness have been achieved empirically during the cutting tests under sawmill conditions according to the procedure described in the paper by . Moreover, determination of the cut material properties in cutting tests is recommended as an alternate and effective way (Atkins 2005(Atkins , 2018Hlásková et al. 2019;Sandak et al. 2017;Wang et al. 2015). For example, experimental results of cutting forces (with the use of microtome) were the source for the determination of work of fracture of particleboards (Beer et al. 2005).
In Central Europe, the energetic methods are commonly used for estimation of cutting power, and those approaches are based on the specific cutting energy for pine wood. For other species, empirical correction coefficients are recommended. These approaches were proposed by Beršadskij (1967) and Manžos (1974), and later improved by Orlicz (1988) and Lisičan (1996). It should be emphasized that in the latter mentioned sources, the conditions in which the properties of the wood were quantified and the provenance of these species are not given. The latter and material properties have a significant impact on the energy effects of the cutting process (Chuchala et al. 2014). The aim of this study was to compare the results of the cutting power estimation while cutting on the circular sawing machine, which is used in Polish sawmills, with the commonly used models based on empirical approaches in both Poland (CLAS_PL) and Czech Republic (CLAS_CZ), and the newly developed cutting model (FRAC_MOD). The cutting power computation results were compared for two wood species: pine (Pinus sylvestris L.) and beech (Fagus sylvatica L.), which are commonly used in Central Europe.

Theoretical background
The specific cutting resistance model (CLAS_PL) The widespread approach to determining the cutting power in a cutting process is a model, which is based on the specific cutting resistance k c (Böllinghaus et al. 2009;Grzesik 2018;Manžos 1974;Melo et al. 2016;Nascimento et al. 2017;Naylor and Hackney 2013;Orlicz 1988;Pinkowski et al. 2016). According to Orlicz (1988), the cutting power could be determined as: where k c -specific cutting resistance in N mm −2 ; P c -cutting power in W; F c -cutting force in N; A Dav -average cross-sectional area of uncut chip in mm 2 ; v c -cutting speed in m s −1 ; S t -the overall set (kerf) in mm; h av -the average uncut chip thickness in mm (Fig. 1).
This approach is widely used to estimate the cutting power for sawing operations on frame machines, circular sawing machines and band sawing machines in the Polish industry. The model CLAS_PL is based on experimentally determined correction factors that are designed to determine changes in factors affecting the cutting process in relation to the adopted basic conditions as follows: where c ws -coefficient taking into account the type of wood, for pine wood c ws = 1; c MC -coefficient taking into account the moisture content of wood; c vc -coefficient taking into account the value of cutting speed; c δ -coefficient taking into account cutting angle δ f ; angle δ f , is defined as a sum of the clearance angle α f and blade angle β f ; c d -coefficient taking into account wear of blade; c wT -coefficient taking into account the temperature of wood; c h -coefficient taking into account uncut chip thickness; c µ -coefficient taking into account friction between the cut wood and saw blade; c CE -coefficient taking into account shape and dimensions of cutting blade, c p -coefficient taking into account pressure exerted on the workpiece before a blade (it could be applied to the production of veneer); and k φ -the basic specific cutting resistance for pine in N mm −2 . Using these correction coefficients, the model takes into account: geometry of cutting blade, cutting parameters, strength of the Sawing kinematics on circular sawing machine: H p workpiece height (depth of cut), a position of the workpiece, φ j angular tooth position, Ф G-vc angle between the wood grains and the cutting speed direction, φ en angle of teeth entrance, φ ex exit angle 1 3 raw material and friction between blade and workpiece. The value of k φ is strictly dependent on the relative position of the cutting edge direction and the wood fiber direction (Fig. 2). Those values take into account the basic directions of cutting k || , k # , k ⊥ and intermediate directions of cutting k ||# , k ||⊥ , k #⊥ , k ||#⊥ . To determine the value of basic specific cutting resistance in any direction of cutting for the edge, it is necessary to use below equation (Orlicz 1988): where Φ G-vc -the directional angle of the cutting speed; Φ G-edge -the directional angle of the cutting edge; Φ G-h -the directional angle of the chip thickness (Fig. 2).
When the cutting process takes place in basic conditions, the values of all the correction factors are equal to 1. Then, the specific cutting resistance k c takes the form of the basic specific cutting resistance k φ : The specific cutting resistance model (CLAS_CZ) For theoretical purposes and industrial practice, analytical methods are still applied (Lisican 1996;Manžos 1974;Orlicz 1988;Naylor et al. 2012). The empirical model of power estimation, which is used in Czech Republic, was developed by Beršadskij (1967) and modified by Lisičan (1996). The energy effects can be calculated theoretically using this conventional model (CLAS_CZ), which is based on the specific cutting resistance. In the model, several empirical coefficients are involved, and the model takes into account phenomena of cutting force interactions on the rake face and the flank face of the tool. It should be emphasized that the specific cutting resistance for small (h av ≤ 0.1 mm) and large (h av ≥ 0.1 mm) average uncut chip thickness ( Fig. 1) is calculated in a different way. In the CLAS_CZ model, the wear of the cutting edge is calculated as a function of the total actual trajectory of the cutting edge The main directions of the cutting edge (a) and directional angles (b), which define position of the cutting edge (Orlicz 1988) in the sawn raw material. Moreover, it also takes into account the effect of the material of which the edge is made.
The specific cutting resistance k c for average uncut chip thickness h av ≥ 0.1 mm is given by: -the basic specific cutting resistance for pine in N mm −2 ; a d -coefficient taking into account wear of blade; p-parameter expressing the artificial effect of the cutting force on the flank face of the tool in N mm −1 ; h av -the average uncut chip thickness in mm; ξ-coefficient taking into account friction between the workpiece and the blade; H p -height of the workpiece in mm; S t -the overall set (kerf) in mm, where A, B, C-values from nomographs (for practical purpose calculation, graphs of parameters A, B, C were plotted as a function of mean fiber cutting angle φ 2 in N mm −2 , δ f -cutting angle, v c -cutting speed in m s −1 , k ws -coefficient taking into account the type of wood, for pine wood k ws = 1).
Using this CLAS_CZ model, the user can take into account the geometry of teeth, cutting parameters, strength of the raw material, changes in the edge geometry caused by the wear and friction between the tooth and the workpiece. The value of k 1 c≥0.1 is strictly dependent on the relative position of the cutting speed direction in relation to the wood fibers.

The newly developed cutting model (FRAC_MOD)
The newly developed cutting model (FRAC_MOD), which includes plasticity and friction, and where work of separation is not omitted in the model, is capable of forecasting not only average values of cutting power but also its dynamical changes . In the FRAC_MOD, it was assumed that cutting force F c , acting in the middle of the cutting edge, is an equilibrium of forces related to the direction of primary motion for a single saw tooth, and the mechanical process of material separation from the sawn workpiece, i.e., chip formation, can be approximately described by the example of an orthogonal process (two-dimensional deformation) Orlowski and Ochrymiuk 2013;Hlásková et al. 2015Hlásková et al. , 2018.
The model originally proposed in a work by  can be expressed as: According to Atkins (2009) and Orlowski (2010), the first part of Eq. (7) represents works of plasticity in the cutting zone and friction on the rake plane; the second one takes into account the phenomenon that the chips have to be accelerated to the same speed as the cutting tool velocity v c .
In the above model (Eq. 7), it was assumed that the cutting edges of teeth are sharp. Moreover, if we assume the cut is straight, and the cut deviation is low, then the effect of lateral forces on the power consumption can be ignored (Mohammadpanah and Hutton 2016).
In the case of circular sawing, the same as in analytical simulations of milling (Altintas 2000;Ammar et al. 2009;Budak 2006), the immediate uncut chip thickness h j (φ) at a definite location of the cutting edge ( Fig. 1) can be approximated as follows: The angular position of the jth tooth (engagement angle) j value changes as follows: where φ p is defined as P = 2 z . If en ≤ j ≤ ex , then it has a value; otherwise, it is null. An angle of teeth entrance en is given by en = arccos 2(H p +a) D (when the tool tooth gets into the workpiece for machining), and an exit angle ex (the tooth of the saw blade gets out of the workpiece) can be described as ex = arccos 2a D . The algorithm of cutting power of chip formation is presented in Fig. 3. The steps to follow in this approach are: • at the position of the cutting edge in relation to the grains, for indirect positions of the cutting edge fracture toughness R ||⊥_j ( ) and the shear yield stress ||⊥_j ( ) are calculated. This approach has been implemented for computation of the shear yield stress and fracture toughness as tensor values Hlásková et al. 2015) according to Orlicz (1988), who applied the plane stress transformation equation for the determination of specific cutting resistance in indirect positions of the cutting speed direction. It ought to be emphasized that the same method is commonly used in general mechanics of materials to transform the stress components from one set of axes to another (Gere 2004). • Computation of the parameter Z, which makes Ф c (φ j ) material dependent, on the contrary to the classical approach by Merchant (Böllinghaus et al. 2009; Markopoulos 2013). • The shear angle Ф c (φ j ) is determined numerically from the equation determining least cutting force F c (for indirect tooth position), which has been proposed by Atkins (2003). • The shear strain along the shear plane γ j (φ j ) and the friction correction Q shear_j (φ j ) are calculated. • The cutting power of chip formation for each tooth is computed, and simultaneously, a plot of cutting power versus an angle of rotation is created; • Eventually, the total cutting power of chip formation P c_CF ( ) is calculated and its relevant graph is generated.
The obtained values at the level F ought to be augmented by the chip acceleration power P ac variation as a function of mass flow and tool velocity (Pantea 1999; Atkins 2009). The chip acceleration power P ac variation in this case is a function of mass flow, and tool velocity is given by: The mass of wood (chips) evacuated in a certain period of time at a certain cutting speed of the tool ṁ (kg s −1 ) can be estimated as follows: It should be emphasized that in these investigations, it was implied that the chip acceleration power P ac is not a function of the number of teeth being engaged in the cutting zone.

Materials and methods
Predictions of cutting power have been made for the case of sawing on the circular sawing machine (HVS R200, f. HewSaw), which is used in Polish sawmills (e.g., Olczyk Sawmill, Krasocin, PL). The simplified sawing system of the circular sawing machine HVS R200 is presented in Fig. 4, and the basic sawing machine data and cutting parameters for which computations were done are shown in Table 1.
It was assumed that one circular saw blade (circular saw 1, Fig. 4) was applied, with technical data as given in Table 1. The tooth geometry in the T-hand-S (toolin-hand system) (Astakhov 2010) is presented in Fig. 5, and the values of angles applied to the circular saw blade are shown in Table 1.
Computations were carried out in each case for one saw blade ( Fig. 1) with three different methods: • the newly developed cutting model that includes work of separation in addition to plasticity and friction FRAC_MOD; • on the basis of the specific cutting resistance model proposed by Orlicz (1988), which is widely used in Poland CLAS_PL; (10) P ac =ṁv 2 c (11) m = H P S t v f • with the empirical model of power estimation, which is applied to Czech Republic CLAS_CZ.

Data for the model FRAC_MOD
The raw material was pine wood (P. sylvestris L.) originated from the Forest Inspectorate Lipusz in the Baltic Natural Forest Region (Poland), and beech wood (F. sylvatica L.) originating from the Training Forest Enterprise Masaryk Forest Křtiny, an organizational part of Mendel University in Brno (Czech Republic).
In Table 2, the raw material data is presented, which was determined experimentally according to the methodology described in the works by:  The values of friction coefficients μ = 0.8 for dry pine wood and beech wood were taken from the works by Beer (2002) and Glass and Zelinka (2010).

Data for the specific cutting resistance model CLAS_PL
Values of correction coefficients to calculate the value of specific cutting resistance in the model CLAS_PL are shown in Table 3. In this table, values for basic conditions and values for the analyzed conditions of cutting process are shown. Some correction coefficients were selected from the given values in the work by Orlicz (1988), and some values had to be additionally calculated using Eq. (3). These calculated coefficients take into account the axial-perpendicular direction of wood cutting and are marked with star (*) (Table 3). Additionally, coefficients, which take into account an uncut chip thickness c h , are described in below equations proposed by Orlicz (1988): The values obtained with Eqs. (12) and (13) were additionally recalculated with Eq. (3). This procedure allowed us to obtain one value of the coefficient c h , which also takes into account the axial-perpendicular direction of wood cutting.

Data for the specific cutting resistance model CLAS_CZ
Values of correction coefficients and all parameters for calculation of the specific cutting resistance in the CLAS_CZ model are shown in  values for the analyzed conditions of cutting process are listed. Correction coefficients and parameters were selected from nomographs and from the recommended values in the work by Lisičan (1996). Table 3 Values of correction coefficients for the specific cutting resistance for the analyzed conditions of cutting process (Orlicz 1988) a Values of coefficients which take into account the axial-perpendicular direction of wood cutting

Results and discussion
Results of predictions of cutting power for chip formation obtained with the use of a newly developed cutting model FRAC_MOD that includes work of separation in addition to plasticity and friction, in the case of sawing of pine from the Forest Inspectorate Lipusz in the Baltic Natural Forest Region (PL) provenance and beech wood originating from the Training Forest Enterprise Masaryk Forest Křtiny, are shown in Fig. 6. The presented data are for one circular saw blade, at the feed speed v f = 70 m min −1 , and v f = 150 m min −1 , for one full revolution of the tool (the first one). In this computation, beside uncut chip thickness changes, appropriate changes in shear yield stress and toughness with tooth/grain orientation were taken into account.
Pine Beech For a stable condition of cutting power changes in the FRAC_MOD model, the RMS values of the power for chip formation P c_CF were computed, and the results of computations are presented in Table 5. Beside P c_CF , the predicted values of chip acceleration power need P ac and total cutting power P c_Tot are also given in Table 5. The total cutting power P c_Tot is defined as: Similarly to the described case of sawing with the circular saw of pine wood , it was assumed that the total cutting power determined with the FRAC_MOD model is the reference value, since the predicted values were in agreement with experimental measurement results. The comparison of estimated cutting powers with the model CLAS_PL and the CLAS_CZ model is shown in Fig. 8 and revealed that: (14) P c_Tot = P c_CF + P ac Fig. 7 Predictions of chip acceleration power variation P ac as a function of cutting speed v c and feed speed v f for sawing of the beech (a) and pine (b) workpieces of 80 mm in height with one saw blade on circular sawing machine HSV R200 with marked points of the parameters applied to the forecasting  cutting power in the CLAS_CZ model was also smaller with the difference equal to 29.9%; • in case of beech wood sawing with feed speed equal to 150 m min −1 , the obtained value in the model CLAS_PL was smaller by about 45%. On the other hand, the cutting power in the CLAS_CZ model was smaller about 33%.
The conducted analyses have shown that almost in each case of forecasting the cutting power with the use of the classical models, the obtained values are underestimated, except for the results from the CLAS_CZ model for pine wood, in which the obtained values are overestimated in the range of 2.9-17.4%. The underestimation seems to be dangerous in industrial practice, because, as a matter of fact, tools could be extra loaded and simultaneously, sawing accuracy could be decreased. Hence, the FRAC_MOD cutting model could be recommended for forecasting changes in cutting power in case of dynamical analyses and even unsteady cases. Moreover, the FRAC_MOD model takes into account the wood species, which properties are represented by fracture toughness and shear yield stress, tool geometry, friction between chip and rake surface, and the provenance of wood. The latter is particularly important as proven in the papers by Chuchala et al. (2014) and Minagawa et al. (2018).

Conclusion
The carried out analyses allowed us to conclude that the FRAC_MOD cutting model allows a good estimation with a given reality error (as presented by Orlowski and, while CLAS_PL and CLAS_CZ models underestimate the reality on average by about 34%. In sawmill practice, such effects of underestimation could lead to extra loaded tools and in consequence sawing accuracy could be decreased (snaking phenomenon is present). It should be emphasized that the FRAC_MOD model seems to be more reliable since values of shear yield stresses and fracture toughness were obtained empirically in the cutting tests, which have been recommended for the determination of the cut material properties as an alternative and effective way. Moreover, the proposed FRAC_MOD model could be included in a small class of sawing process models based on physical foundations and describing the most important physical phenomena occurring in the machining process, in contrast to empirical models that have been widely and sometimes uncritically used for a long time. In view of the contemporary challenges facing the wood industry, such as precision machining and energy saving in production processes, it is reasonable to design new models based on physical phenomena. Such models allow for a more thorough analysis of woodworking processes and thus a more reliable forecasting of power consumption.