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

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 mm²; 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 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:

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 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:where \(k_{c \ge 0,1}^{1}\)—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 φ₂ 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 \(k^{1}_{c \ge 0.1}\) 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), 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 (Orlowski and Ochrymiuk 2017). 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 et al. 2013, 2014; Orlowski and Ochrymiuk 2013; Hlásková et al. 2015, 2018).

The model originally proposed in a work by Orlowski et al. (2013) 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)\(\varphi_{j}\) value changes as follows:where φ_p is defined as \(\varphi_{P} = \frac{2\pi }{z}\).

If \(\varphi_{\text{en}} \le \varphi_{j} \le \varphi_{\text{ex}}\), then it has a value; otherwise, it is null. An angle of teeth entrance \(\varphi_{\text{en}}\) is given by \(\varphi_{\text{en}} = \arccos \frac{{2\left( {H_{p} + a} \right)}}{D}\) (when the tool tooth gets into the workpiece for machining), and an exit angle \(\varphi_{\text{ex}}\) (the tooth of the saw blade gets out of the workpiece) can be described as \(\varphi_{\text{ex}} = \arccos \frac{2a}{D}\).

The algorithm of cutting power of chip formation is presented in Fig. 3. The steps to follow in this approach are:

After one full revolution of the tool, i.e., φ: 0°–360° maximum, average or root-mean-square (RMS) values of power can be determined (Budak 2006).

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; Orlowski et al. 2013; 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 \(\dot{m}\) (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.

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).

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.

Values of correction coefficients and all parameters for calculation of the specific cutting resistance in the CLAS_CZ model are shown in Table 4. In this table, 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).