An Innovative Approach to the Forecasting of Energetic Effects While Wood Sawing

In the classical approach, energetic effects (cutting forces and cutting power) of wood sawing process are generally calculated on the basis of the specifi c cutting resistance, which is in the case of wood cutting the function of more or less important factors. On the other hand, the cutting forces (power) problem may be tackled with an innovative, up-to-date fundamental analysis of the mechanics of sawing based on modern fracture mechanics. This line of attack is an improvement on traditional approaches for cutting forces and power, many of which are empirical and based upon limited information. Such formulae do not permit generalisation to new conditions of operation of sawmills, such as e.g. the use of narrow-kerf blades. The presented algebraic model, for cutting power determination while sawing, in addition to timber ‘strength’ and friction between tool and workpiece, takes into account the property called ‘fracture toughness’ (resistance to cracking), which is a vital ingredient. Furthermore, forecasting of the shear plane angle with this model is achievable even for small values of uncut chips. Moreover, the mentioned model is a universal one, and useful for determination of energetic effects of sawing of every kinematics such as: frame sawing machines, bandsawing machines and circular sawing machines.


INTRODUCTION 1. UVOD
The mechanics of cutting wood has traditionally been interpreted in terms of the Ernst-Merchant model for cutting ductile metals in which, for given blade geometry and uncut chip thickness, the important parameters are mainly the strength of the material and friction between blade and workpiece (Manžos, 1974;Orlicz, 1988;Orlowski, 2007;Scholz et al., 2009;Naylor et al. 2011).Recent work in the mechanics of cutting (Atkins, 2009) shows that the resistance to cracking of the workpiece (the so-called fracture toughness) is just as significant as strength and friction in determining the forces and power to cut.Furthermore, incorporation of toughness in the new analysis predicts quantitatively many features that the Ernst-Merchant analysis fails to, such as values of the primary shear plane angle, the anomalous rise in specifi c cutting resistance at small uncut thicknesses, the transition to different types of chip, and so on.The new theory also gives physical meaning to terms in many empirical expressions for the forces and power to cut timber.
The present paper applies the new theory to the cutting of pine (Pinus sylvestris L.) by three different sorts of sawing machines, namely a sash gang saw, a circular saw and a bandsaw.Measured forces and power, as they change under different cutting conditions, are predicted by the theory.The capacity of the three sawing machines is discussed in terms of available power, feed rates and so on, leading to comments on the design of machines with different kinematic features.
2 THEORETICAL BACKGROUND 2. TEORIJSKA OSNOVA Orlowski et al. (2013) have proved that cutting power models, which are based on modern fracture mechanics, are useful for estimation of energetic effects of sawing of every kinematics.According to Atkins (2009) and Orlowski (2010), moreover, taking into account that the chips have to be accelerated to the same velocity as the cutting tool velocity v c (Atkins, 2009;Pantea, 1999), cutting power for one saw blade during the cutting stroke on a sash gang saw (for a whole cycle it means working and idling strokes cw c P P   5 .0 (Orlowski, 2010;Orlowski and Palubicki, 2009), and during cutting on a bandsaw machine, as their sawing kinematics are similar (Fig. 1a), has the following mathematical formula: (1) where: is an average number of teeth being in contact with the kerf, P is tooth pitch, H p is workpiece height (cutting depth), τ γ is the shear yield stress, γ is the shear strain along the shear plane, which is given by: ( f z is feed per tooth, h uncut chip thickness, S t is a kerf width (the width of orthogonal cut), β μ is friction angle which is given by tan -1 μ = β μ , with μ the coefficient of friction, γ f is the rake angle, Φ c is the shear angle which defi nes the orientation of the shear plane with respect to cut surface, R  is specifi c work of surface separation/formation (fracture toughness), and Q shear is the friction correction: ( For least force F c the shear angle Φ c satisfi es (Atkins 2003): in which is the parameter which makes Φ c material dependent.Equation ( 4) is solved numerically.Since, wood is an anisotropic material, its physical and mechanical properties differ in the three principal directions relative to the trunk of the tree (Fig. 2): longitudinal (L, axial), which is parallel to tree trunk and parallel to long axis of longitudinally oriented cells (tracheids and fi ber tracheids); radial (R), which is perpendicular to longitudinal direction and parallel to the radius of the trunk and wood rays; and tangential (T), which is perpendicular to longitudinal direction and parallel to growth rings (FPL 1980).For that reason, values of R and τ γ should be applied accordingly to the cutting speed direction in regard to the wood grain direction (Jeronimidis, 1980;Kretschmann, 2010), which is mainly a function of the sawing kinematics.
Ključne riječi: energijske veličine, snaga rezanja, proces piljenja drva, strojevi za piljenje, mehanika loma, lomna žilavost (4) ting tool velocity v c (cutting speed), which can be calculated as follows: (6) where: v f is feed speed and ρ is the density of sawn wood.It should be emphasized, that in these analyses, it was assumed that the power P ac is not a function of the number of working teeth.In case of both the circular sawing machine and the bandsawing machine, the chip acceleration power P ac is several hundred times larger in comparison with the sash gang saw (Orlowski et al., 2013), thus, for the latter machine tool, the last term of Eq.( 1) can be omitted.
Kinematics of sawing on circular sawing machines (Fig. 1b) differs from kinematics of cutting on sash gang saws and bandsawing machines (Fig. 1a).In case of cutting with circular saw blades, uncut chip thickness h ̅ (an average value e.g.) instead of feed per tooth f z should be taken into account, hence, the cutting power may be expressed as: (7) where: is a number of teeth being in contact with the kerf (average), ϕ 1 is an angle of teeth entrance which is given by , ϕ 2 is an exit angle which can be determined as , D cs is a diameter of circular saw blade, an average uncut chip thickness is given by , and an average angle of tooth contact with a workpiece is calculated from .Furthermore, it is diffi cult to presume that in this kind of sawing kinematics there is a case of perpendicular cutting, because the angle between the grains and the cutting speed direction differs from 90, as it was assumed for the sash gang saw and the band sawing machines.Hence, taking into account the position of the cutting edge in relation to the grains, for indirect positions of the cutting edge fracture, toughness R and the shear yield stress τ γ may be calculated from formulae known from the strength of materials (Orlicz, 1988).For example for cutting on circular sawing machines (a case of axial-perpendicular cutting), these material features are as follows: (8) where:  G-vc is an angle between grains and the cutting speed direction (Fig. 1b).

MATERIJAL I METODE
Predictions of cutting power have been made for the case of bona fi de sawing processes on the sash gang saw DTRB-63 (f.FOD, PL, Fig. 3a), the double shaft multi ripsaw PWR422 (f.TOS Svitavy, CZ, Fig. 4) and the bandsawing machine ST100R (f.Stenner, UK, Fig.  1. The raw material was pine wood (Pinus sylvestris L.) with the depth of cut equal to H p .The raw material derived from the Baltic Natural Forest Region (A, Fig. 6), the Carpathian Natural Forest Region (B), the Little Poland Natural Forest Region (C) and the Great Poland-Pomeranian Natural Forest Region (D) in Poland.Moisture content was MC 8.5-12 % for bandsawing, and MC ~30 % for both the sash gang saw and the rip saw.For that reason, the latter cutting power results were additionally multiplied by 1.05 (Manžos,  The sawing pattern for the sash gang saw, in which thickness of the main material is 2a = 137 mm, with 2 boards of 27.5 mm in thickness additionally obtained on each side, is presented in Fig. 3b.Logs with diameter d g in thinner end (top diameter) about 11″ (imperial units are still in use in sawmill practice, d g ≈290 mm), and l = 4 m in length are sawn.The workpiece thickness H P presented in Tab. 1 is in this case an average value of the kerf depth determined in the middle of the log length.In order to estimate a middle log diameter d, the taper coeffi cient TC (cm/m, the degree of taper) was calculated as follows (Leśnik, 2013): where: l is the log length in m, and d g is the top diameter without bark in cm.Thus, in this case TC = 0.834 cm/m, and for this data middle log diameter is d = 30.66cm.The latter value was applied in calculations of the total kerf height H Σ as follows (Csanády and Magoss, 2013): (11) where: , and i is the number of cut right to the centre.Hence, for the sash gang saw, the  Krzosek, 2009.)average workpiece thickness H P is a ratio of H Σ to the number of saw blades n sb .
Computations of cutting power were carried out in each case for one saw blade, and the obtained values were compared to available cutting power per one saw blade (Tab.1).The latter was calculated as follows: (12) where: η m is mechanical effi ciency of the main driving system (for each machine tool η m = 0.85 was assumed).

REZULTATI I RASPRAVA
Predictions of the cutting model that includes work of separation in addition to plasticity and friction in the case of sawing dry pine wood of the Baltic Natural Forest Region (A) provenance on examined sawing machines are shown in Fig. 7.The reduction in Φ c (Fig. 7a) and increase in γ (Fig. 7b) for all values of rake angle γ f for small depths of cut (e.g.uncut chip thickness) according to Atkins (2003Atkins ( , 2009) ) are the reason for the increase in cutting pressure.Furthermore, an increase in shear plane angle Φ c is observed when rake angle γ f has a larger value.Results of predictions of cutting powers obtained with the use models that include work of separation in addition to plasticity and friction, and chip acceleration power variation in the case of sawing of pine coming from different Polish regions with one saw blade are shown: in Fig. 8 for the frame sawing machine DTRB 63, in Fig. 9 when using the bandsawing machine ST100R, and in Fig. 10a (upper spindle) and Fig. 10b (lower spindle) for cutting on the rip saw PWR422.For both, the rip saw and the bandsawing machine, the results of the chip acceleration power P ac have been taken into account.
For analysed sawing patterns of the actual Polish sawmill, the capacity of the three sawing machines could be discussed in terms of available power for pine wood of different provenance: 1.For the sash gang saw DTRB 63, it is impossible to apply maximum values of feed speed for every kind of pine wood (it concerns the raw material derived from the Baltic Natural Forest Region (A) and the Great Poland-Pomeranian Natural Forest Region (D)).2. For bandsaw ST100R sawing pine wood from the Great Poland-Pomeranian Natural Forest Region (D), the cutting power could surpass the accessible cutting power in the cutting zone almost at maximum values of feed speed.3.For the rip saw PWR422, the accessible cutting power in the cutting zone (per one saw blade) is not exceeded, because the workpiece height (cutting depth) is automatically divided among two spindles as a ⅔ ratio, and simultaneously the maximum value of feed speed is not so high compared to other machine tools currently present on the European market.It ought to be emphasised that only the obtained cutting powers for pine wood of the Baltic Natural Forest Region (A) provenance (in each case of sawing the second line from the top in plots, Fig. 8-10) correspond to values calculated with empirical models presented e.g. by Manžos (1974) and Orlicz (1988).This piece of evidence could be stated, since, in the paper by Or- , it has been proved that the specifi c cutting resistance is in conformity with values calculated with the use of the mentioned empirical calculation models.Furthermore, the size effect was present for the cutting power prediction method, which is based on the fracture mechanics.For other locations (Regions B, C and D) of the raw material origin examined in this paper, cutting powers signifi cantly differ from the cutting power values of pine wood originated from Region (A) because MOR, fracture toughness and simultaneously shear yield stresses signifi cantly vary (shown in Table 2).Additionally, variability in properties is common to all materials.Since wood is a natural material and the tree is subject to many constantly changing infl uences (such as moisture, soil conditions, and growing space), wood properties vary considerably, even in clear material (Kretschmann, 2010;Krzosek, 2009).The same could be noticed in dispersion of the raw material data presented in Tab. 2.
Hence, it has been demonstrated that the approach to predictions of cutting powers obtained with the use of cutting models that include work of separation in addition to plasticity and friction, together with the chip acceleration power variation, is simultaneously an universal and useful tool for forecasting of energetic effects of sawing of every kinematics.

ZAKLJUČAK
The conducted forecasting of energetic effects with the use of cutting models that include work of separation in addition to plasticity and friction, together with the chip acceleration power variation, once more corroborated their versatility and revealed the effi cacy for every known type of sawing kinematics (sash gang saw, bandsawing machine and circular sawing machine).
Furthermore, the conducted analyses revealed that the provenance of the raw material really affects an energetic demand of the cutting process.
This kind of approach to the forecasting of energetic effects while wood sawing allows the sawmill management to estimate the capacity of the rip sawing machines in terms of available power for pine wood of different provenance in advance before processing.It could also be an appropriate method for planning of the proper sawing pattern according to the available power of the sawing machine.

Figure 7 7 .
Figure 7Predictions of cutting model that includes work of separation in addition to plasticity and friction in the case of sawing dry pine wood on examined sawing machines (a) shear plane angle Φ c vs. f z , (b) primary shear strain γ vs. f z , in a function of uncut chip thickness h and rake angle γ f Slika 7. Predviđanja za model rezanja koji obuhvaća rad odvajanja uz plastičnost i trenje pri piljenju suhe borovine na ispitivanim strojevima: (a) kut ravnine smicanja Φ c vs. f z , (b) primarna čvrstoća smicanja γ vs. f z , kao funkcija debljine strugotine h i prednjeg kuta γ f