Lateral forces determine dimensional accuracy of the narrow-kerf sawing of wood

Lateral forces determine dimensional accuracy of the narrow-kerf sawing of wood

Stiffness of the saw blade

The static stiffness of the saw determined in the machine–chuck–workpiece–tool system, combined with the resistance to wandering in the workpiece, are commonly considered as the basic indicators determining the ability of a saw to properly cut the material10,17,45. Therefore, the saw in the given configuration represents the weakest element of the system. Several attempts were made to predict the behaviour of the tool during the machining5,27. Most of these revealed that the geometrical accuracy of the products being manufactured depends on the transverse displacements of saw blades. Figure 2 shows an arrangement of external forces loading the saw blade, which change their positions depending on the placement of the frame sawing machine sash, when it is driven by a dynamically balanced main drive44.

Figure 2

Loads on the saw during operation; where: FN is the tensioning force (the force stretching the saw blade in the longitudinal direction—Ym axis of the sawing machine’s coordinate system); FcT is the total cutting force; FfT is the total feed force; FpT is the total thrust force to load the saw; L0 is the free saw blade length; hw is the feed roller level; HRP is the saw frame stroke; r is the crank radius in the driving system; s is the saw blade thickness; y(φ) is the position of the point of application of force as a function of the angle of rotation φ of the crank in the main drive of the sawing machine.

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The initial static stiffness of the saw blade k0 is determined according to Eq. (1) as the ratio of the thrust force Fp applied in the middle of the free length of the saw blade to the displacement of the saw blade q towards the Xm axis (Fig. 2):

$$ k_{0} = \frac{{F_{p} }}{q} $$

(1)

The relationship (1) is non-linear as the stiffness of the saw blade k0 is a nonlinear property. Therefore, the stiffness itself should be assessed as a local stiffness by using the secant method. In that case the ratio of the gain in the thrust force is directly related to the gain in the displacement of the saw blade46. The stiffness of the saw blade is most often analysed by the superposition method, which accounts for the strain of the saw blade resulting from bending (normal strains due to bending) and torsion (shear strains due to torsion)40,47,48,49,50,51. Several analytical methods (the beam and plate models) and/or numerical methods (the finite element method or the rigid finite element method) are frequently used to determine the static stiffness values. However, each of those methods can only be used within the specified range of blade width40. The saw blade width can vary between 20 mm in the case of mini saw blades, up to > 250 mm for high throughput band sawing machines for processing logs.

Factors influencing the saw blade stiffness

An effect of the feed force Ff should be taken into consideration when analysing transverse displacements of the saw. Ff affects the stability of the saw blade and reduces the initial static stiffness coefficient of the saw blade k0 to the value of the operating stiffness coefficient k0w24,49,51. The total loss of saw blade stability occurs when the feed force Ff reaches the critical value (Ff_crit) operating in the plane of the greatest stiffness of the saw blade50,51,52. The critical force Ff_crit corresponds to the force at which the saw blade gets infinitely bent as a result of a small value thrust force Fp11,50,51. In addition, the stiffness of the saw blade can decrease as a result of an increase in the saw blade temperature53. Pahlizch and Puttkammer50, Prokofiev51 and Csanady and Magoss47 proposed the analytical determination of the critical feed-per-tooth force Ff_crit for saws with saw blades wider than 100 mm.

Knowing the value of the critical force Ff_crit, the working stiffness coefficient can be determined from the dependencies proposed by Timoshenko and Gere52 for the general case, or by Prokofiev51 specifically for a wideband sawing machine. Finally, the working stiffness can also be derived from Stakhiev’s equation24.

Estimating values of cutting forces

The cutting forces Fc can be estimated using the empirical classic model which is based on wood specific cutting resistance43,54,55,56. As an alternative, a modern approach considering the fracture toughness and shear yield stress in the cutting zone can be used for the determination of cutting force values43. Although models based on the fracture mechanics theorem allows for predicting energy effects in a highly precise way, the classical empirical method is still widely used in solving practical problems. For this reason, it was adopted for the numerical simulations developed within this research, in which the cutting force Fc was described as in Eq. (2):

$$ F_{c} = k_{c} \cdot A_{Dav} = k_{c} \cdot S_{t} \cdot h_{av} $$

(2)

where: kc is the coefficient describing the specific cutting resistance [N⋅mm−2], Fc is the cutting force [N]; ADav is the mean cross-section area of the uncut chip [mm2]; St is the total kerf [mm]; hav is the mean uncut chip thickness [mm].

This model shall consider experimentally determined correction coefficients identified to address changes in the factors affecting the cutting process in relation to the basic conditions adopted in56, as summarized in Eq. (3):

$$ k_{c} = k_{\phi } \cdot c_{ws} \cdot c_{MC} \cdot c_{vc} \cdot c_{\delta } \cdot c_{d} \cdot c_{wT} \cdot c_{h} \cdot c_{\mu } \cdot c_{CE} \cdot c_{p} $$

(3)

where: kφ is the basic specific cutting resistance for Scots pine wood [N·mm-2], cws is the coefficient taking into account the wood species (i.e. cws = 1 for Scots pine wood defined as the reference species); cMC is the coefficient correcting effect of the wood moisture content; cvc is the coefficient taking into account the value of the cutting speed; cδ is the coefficient taking into account the cutting angle defined as the sum of the clearance angle αf and the blade angle βf; cd is the coefficient considering the cutting edge wear; cwT is the coefficient adjusting effect of the temperature of the wood; ch is the coefficient taking into account the value of the uncut chip thickness; cµ is the coefficient taking into account the friction between the wood being cut and the saw blade; cCE is the coefficient taking into account the shape and dimensions of the cutting edge; cp is the factor taking into account the pressure exerted on the workpiece in front of the blade (commonly applied in the production of veneer).

The value of kφ depends on the position of the cutting edge in relation to the direction of the wood grains56. These values take into account the basic cutting directions, along the grains (kII), tangentially to the grains (k#) and perpendicularly to the grains (k⊥), as well as intermediate cutting directions (kII#, kII⊥, k#⊥, kII#⊥). The values of all the correction factors are equal to 1 if the cutting process is carried out under basic reference conditions. In that case, the specific cutting resistance kc corresponds to the basic specific cutting resistance kφ (Eq. 4):

$$ k_{c} = k_{\phi } \cdot 1 = k_{\phi } $$

(4)

Lateral loads of the saw blade during cutting

Saws with spring setting teeth, especially those with narrow kerfs, are not used for precise cutting as they do not have the symmetry of tooth geometry for the minor cutting edges. Thus, it is important to properly assess the cause of the build-up of the thrust force on saws with swaged teeth. Some of these tools have already been analysed, including circular saw blades57 and frame saw blades51. The causes of build-up of the resultant thrust force on the teeth of the saw with the swaged saw set are presented here in a structured manner, referring to the said works as well as the requirements given in the ISO standards (ISO 3002-1:198258 and ISO 3002-2:198259).

A typical tooth with the swaged saw set is schematically presented in Fig. 3a. The cutting edge angle and the inclination angle of the main cutting edge are κr = 90° and λs = 0°, respectively. The minor cutting edge angles and the rear flank angles on both sides of the blade are the same (κr1' = κr2', αp1' = αp2'). Both thrust forces Fp1 and Fp2 should be balanced in such a configuration of tooth geometry.

Figure 3

Transverse loads of the saw blade as a function of its geometry in the sawing machine’s coordinate system (Xm, Ym, Zm): (a) typical blade with the swaged saw set; (b) effect of the cutting edge angle κr; (c) effect of the angle of inclination of the main cutting edge λs; (d) effect of the difference in the values of the side saw sets s2 > s1; (e) effect of the difference in the values of tool minor cutting edge angles κr1' < κr2'; (f) effect of the difference in the values of the rear angles of clearance αp1’ > αp2’.

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Saw blades with a cutting edge angle κr ≠ 90° are additionally loaded with an extra thrust force Fp(κr). Its value depends on the feed force Ff (Fig. 3b) and the additional thrust force Fp(λs) caused by the cutting force Fc (Fig. 3c) when the main cutting edge is tilted at an angle λs ≠ 0°.

Grinding errors in the form of large differences between the values of the side sets s1 and s2 have a significant impact on the resultant thrust force in addition to the assumed geometry of the saw blades. Such an impact is directly related to the presence of an additional thrust force Fp(e), as shown in Fig. 3d.

Another common error occurring during the manufacture of saw blades is the lack of symmetry due to differences in the minor cutting edge angles, κr1' ≠ κr2' (Fig. 3e) and/or in the rear flank angles αp1' ≠ αp2' (Fig. 3f). Each of these errors results in the presence of an additional thrust force Fp(κr’). Fp(αp’), leading to a variable thickness in the generated workpieces.

An additional cause of the asymmetrical forces in the saw working system can be the inaccurate positioning of blades within the saw frame. An additional thrust force will appear when the direction of the feed movement (or the feed force vector Ff) and the Ym-Zm plane of the setting system are divergent60.

It should be stressed, however, that besides the above-listed factors, the natural heterogeneity of the processed material is the driving factor resulting in the unbalance of the thrust forces Fp1 and Fp2. It is especially relevant in the case of processing biological origin materials, such as wood.

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Application of fracture mechanics for energetic effects predictions while wood sawing

Predictions 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 are shown in Fig. 5. The reductions in Φc (Fig. 5a) and increases in γ (Fig. 5b) are visible in plots, which concerns what happens near the origin of both Φc versus h (f z) and γ vs. h (f z) plots. Those changes at small depths of cut are the reasons for the increase in cutting pressure for small values of feed per tooth (the so-called ‘size effect’) (Atkins 2003, 2009; Orlowski et al. 2010). Furthermore, an increase in shear plane angle Φc is observed when rake angle γ f has a larger value.

Fig. 5

Predictions 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 Φc versus f z, (b) primary shear strain γ versus f z, where gamma 14 (sash gang saw HDN), gamma 22 (circular sawing machine HSV R200), gamma 28 (band sawing machine EB1800) are rake angles

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Figure 6 shows the chip acceleration power P ac variations as a function of feed speed v f and cutting speed v c for the sash gang saw HDN (Fig. 6a), the circular sawing machine HSV R200 (Fig. 6b) and the band sawing machine EB1800 (Fig. 6c) for the cutting processes with one saw blade while sawing dry pine wood for depth of cut equal to H p = 100 mm. For sash gang saws, a maximum value of the chip acceleration power P ac equals to ≅2.5 W. Thus, in those machine tools where cutting speeds and feed speeds are rather small compared with circular saws and band saws, chip momentum may be disregarded, the same as in the case of metal cutting where it is customarily ignored (Atkins 2009). In case of both the circular sawing machine and the band sawing machine, the chip acceleration power P ac is several hundreds larger in comparison with the sash gang saw.

Fig. 6

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 pine workpiece of 100 mm in height with one saw blade on sash gang saw HDN (a), circular sawing machine HSV R200 (b) and band sawing machine EB 1800 (c)

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Comparison of predictions of cutting powers obtained with the use of cutting models that include work of separation in addition to plasticity and friction, and chip acceleration power variation in the case of dry pine sawing with one saw blade for three examined typical sawing machines are shown in Fig. 7.

Fig. 7

Comparison of predictions of cutting powers obtained with the use of cutting models that include work of separation in addition to plasticity and friction (lower lines), with chip acceleration power variation added (upper lines, for the circular sawing machine course and the band sawing machine plot) in the case of dry pine sawing with one saw blade

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Specific cutting resistance versus the method of power estimation

The predicted values of cutting powers were used for determination of the specific cutting resistance for each type of the sawing machine. In computations, the following formula was applied:

$$ k_{c} = \frac{{P_{\text{c}} }}{{S_{\text{t}} \cdot H_{\text{p}} \cdot v_{\text{f}} }} $$

(10)

In the next step, the obtained results were compared with specific cutting resistances taken from the book by Manžos (1974) for all kinds of sawing machines, and additionally with calculated specific cutting resistances (Eq. 10) for the circular sawing machine and the band sawing machine on the basis of cutting power values received from the software accessible in the Internet (Web sources 1 and 2). The compared results are shown in Fig. 8a–c. This comparison revealed that the size effect is present for the cutting power prediction method which bases on the fracture mechanics (kc_FRAC) for each type of the machine tool and also in the case of circular sawing for Manžos’s method (kc_Man, Fig. 8c). In all remaining cases, the scale effect is not imperceptible. This phenomenon has its roots in empirical formulae which are used. For example, Manžos (1974) recommend calculating specific cutting resistance for a band sawing machine in the case of pine sawing as follows:

$$ k_{\text{c}} = 9.91\left( {5.3 + 0.01H_{\text{p}} - 0.03v_{\text{f}} } \right) $$

(11)

where H p is in mm and v f is m·min−1.

Fig. 8

Specific cutting resistance comparison in case of a sawing on the frame sawing machine HDN, b sawing on the band sawing machine EB 1,800 and c sawing on the circular sawing machine HVS R 200 where: kc_Frac predicted values with the use of cutting model that include work of separation in addition to plasticity and friction, kc_Man empirical values from the literature (Manžos 1974), kc_www empirical values on the basis of the software results (Web source 1 and 2, 2011a, b)

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On the other hand, Web sources 1 and 2 (2011a, b) for determination of the cutting power for both the band saw machine and the circular sawing machine apply the same formula in which the power is a function of the wood specific gravity SG. Thus, this kind of approach could be applied to rough estimations of cutting power while wood sawing, the more so because the correlation coefficients between strength properties (modulus of elasticity and density, bending strength and density and modulus of elasticity and bending strength) and wood density are in the range of 0.50–0.70. Moreover, the geographical location of the forest in which the trees were harvested strongly affects this correlation and timber grade (Krzosek 2009, 2011).

Obtained values of cutting powers (Fig. 7) for those machines seem to be reasonable when compared to the power P EM of installed electric motors (Table 1). Furthermore, the specific cutting resistance is in conformity with values calculated using empirical calculation models (Fig. 8a–c). Hence, it has been proved that prediction 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 a useful tool for estimation of energetic effects of sawing of every kinematics.

Cutting power normalization

Since the cutting power for each kind of sawing machine has been calculated for different kerf values and also various workpiece heights (cutting depths, see Table 1) for a more general discussion the obtained results were normalized. The specific cutting power per one active tooth was calculated as follows:

$$ P_{\text{c}}^{\prime 1} = \frac{{P_{\text{c}} }}{{S_{\text{t}} \cdot z_{\text{a}} }} $$

(12)

In the presented approach of energetic effects determination, the cutting power consists of three main terms (see Eqs. 1, 7): the first—combined with shearing and friction; the second—connected with work of separation (fracture toughness); and the third—incident with the chip acceleration. Contributions of those components to the total specific cutting power per one active tooth while pine wood sawing were computed, and the results are presented in Fig. 9. For lower values of uncut chip thicknesses (Fig. 9a) in the total specific cutting power per one tooth, the component connected with the work of separation is a dominant character whereas shearing with friction has much lower importance. The acceleration factor has no significance. On the other hand, for larger values of uncut chip thicknesses (Fig. 9b), the opposite phenomenon is observed in which the term connected with shearing is featured.

Fig. 9

Contributions of chip acceleration (Acceleration), fracture and shearing together with friction (Shearing) to the total specific cutting power per one active tooth while pine wood sawing on band sawing machine (BSM), circular sawing machine (CSM) and frame sawing machine (FSM) for uncut chip thickness h = 0.0075 mm (a) and h = 0.75 mm (b)

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The normalization of the cutting power by the kerf value and number of active teeth allowed to explain that for the constant value of the uncut chip thickness, the specific cutting power for circular sawing machine is lower in comparison with the band sawing machine, although the cutting speed is almost double, because the raw material properties during sawing with a circular saw blade are the result of a case of axial-perpendicular cutting.