**"Main and normal cutting forces by machining wood of**

*Pinus sylvestris*,"*BioRes.*6(4), 3687-3713.

#### Abstract

In this work the multi-factor, non-linear dependencies between main (tangential) *FC* (N) and normal (radial) *FN* (N) cutting forces and eight machining parameters by sawing simulation of wood of *Pinus sylvestris *L. were evaluated. The relationships are graphically illustrated and discussed. Evidence of several contradictions was found relative to results from available literature.

Download PDF

#### Full Article

**Main and normal cutting forces by machining wood OF Pinus sylvestris**

Bolesław Porankiewicz,^{a,}* Bengt Axelsson,^{b} Anders Grönlund,^{c }and Birger Marklund ^{c}

In this work the multi-factor, non-linear dependencies between main (tangential) *F _{C}* (N) and normal (radial)

*F*(N) cutting forces and eight machining parameters by sawing simulation of wood of

_{N}*Pinus sylvestris*L. were evaluated. The relationships are graphically illustrated and discussed. Evidence of several contradictions was found relative to results from available literature.

*Keywords: Multi-factor non-linear statistical dependencies; Main cutting force; Normal cutting force; Circular sawing; Wood; Scotch pine*

*Contact information: a: ^{ }University of Zielona Góra; b:^{ }Westinghouse Electric Company European LWR Fuel Business, SE-721 63 Västerås, Sweden; c: Luleå University of Technology;*

** ^{ }Corresponding author: poranek@amu.edu.pl*

**INTRODUCTION**

Although *Pinus sylvestris* L. is the most frequently analyzed wood species with respect to cutting forces in sawing experiments, the problem of prediction of reliable values of tangential *F _{C}* and radial

*F*cutting forces for specific sawing conditions, according to the method employing a specific cutting resistance

_{N}*K*(N·mm

^{-2}) and correction coefficients

*C*,

_{R}*C*

_{5},

*C*

_{o},

*C*,

_{AP}*C*,

_{VC}*C*,

_{MC}*C*, as expressed by Equations (1) and (2) is far from being solved (Afanasev 1961; Amalitskij and Lûbčenko 1977; Beršadskij 1967; Deševoj 1939; Orlicz 1982).

_{T}(1)

In equation (1) the terms are defined as follows:

*F _{N}*=

*C*·

_{FN}*F*(N) (2)

_{C}In Eq. (2) the new term *C _{FN}* , which is a function of

*ρ*or

*VB*, is the coefficient of the normal force

*F*.

_{N}In authors’ opinion, the reason that the problem has not yet been adequately solved is due to the large number of cutting parameters and their interactions involved, and also due to the fact that Equation (1) does not take into account mechanical properties, as well as wood density, instead of an arbitrarily assumed value of the wood species correction coefficient *CR*. The wood of the *Pinus sylvestris* L., may differ considerably in physical and mechanical properties, resulting in a large dispersion of predicted cutting forces in comparison to observed ones, reported as high as 40 % and more (Orlicz 1982). The specific cutting resistance *K*, evaluated as an average value, cannot take into account differences in cutting resistance generated by such factors as: – early and late wood of growth rings; – sap- and heart-wood; – reaction wood; – fresh knots; – and wood near to fresh knots. The exact cutting conditions of experiments, which had been used for evaluation of the base specific cutting resistances *K* and correction coefficients , as reported in the literature (Afanasev 1961; Amalitskij and Lûbčenko 1977; Beršadskij 1967; Deševoy 1939; Orlicz 1982), remain unknown (or not available), and more, it seems that they were not supported by any multi-factor experiment. It also cannot be excluded that many interactions exist among the dependencies of the main *F _{C}* and normal

*F*cutting forces upon machining parameters, and these were not taken into account in previous works. Therefore the method of evaluation of cutting forces, based on equation (1), appears to have involved rather rough approximations of the wood cutting theory. Incomplete sets of independent variables have been considered in the published works, and there has been inadequate attention paid to having the same range of their variation in evaluation of formulas of dependencies between the main cutting force

_{N}*F*as well as normal cutting force

_{C}*F*, making machining parameters difficult to compare.

_{N}It has been known from earlier studies (Amalitskij and Lûbčenko 1977; Kivimaa 1961) that the normal cutting force *F _{N}* does not follow the main force

*F*; therefore the equation (2), with coefficients’

_{C}*C*dependency upon the main cutting force

_{FN}*F*and cutting edge recession, is far from the truth.

_{C}Instead of exact numbers, a use of qualitative word descriptions of the cutting edge state, such as, for example, “sharp,” “moderately dull,” and “dull” is not satisfactory for precise analysis. The words “sharp” and “dull,” from the point of view of cutting edge round up or the cutting edge recession *VB*, does not have the same meaning for rough (primary brake-down) and precise (super thin) circular sawing.

A tabular form of values of the specific cutting resistance *K* and the correction coefficients , defining Equation (1) for *Pinus sylvestris* L. wood, in most published works, sufficient for very rough estimation, needs interpolation when a number lying between values given in a table is needed. This disadvantage of the method, generating an error for variables that are undefined by mathematical function, non-linear relationships, was improved in the program Wood_Cutting (Porankiewicz 2011) for cutting forces calculation, in which statistical formulas for all basic values of the specific cutting resistances and the correction coefficients, determining formula (1) were evaluated.

The multi-variable statistical relationship, namely Equations (3) through (5), between the main cutting force *F _{C}* and eight cutting parameters was presented in the work Axelsson et al. (1993),

(3)

where new terms in Equation (3) are:

(4)

(5)

and:

Equations (3) through (5) seem to be the most sophisticated available in the newer literature, having the following interactions: . This equation was evaluated from data obtained from an multi-variable circular sawing simulated experiment.

The issue of interpretation of the impact of the cutting velocity *v _{C}* on the main

*F*and normal

_{C}*F*cutting forces is controversial in the literature. According to Kivimaa (1950), by free cutting, the influence of the cutting velocity

_{N}*v*can be neglected. The presumably small linear influence of the factor

_{C}*v*on the main

_{C}*F*and normal

_{C}*F*cutting forces, also for free cutting, due to chip acceleration, was omited in the work McKenzie (1961). However, in the opinion of the authors, slow, free cutting ought not to be directly compared to closed cutting (sawing) by high cutting velocity

_{N}*v*. A linear component, representing about a 14 % increase of the main cutting force

_{C}*F*with growth of the cutting velocity

_{C}*v*, from 15 m·s

_{C}^{-1}to 40 m·s

^{-1}, was reported in the work of Axelsson et al. (1993), based on Equations (3) through (5), indicating that according to theoretical simulation performed the influence of the chip acceleration on the main cutting force

*F*can be omitted. The parabolic relationship, presented in the literature (Amalitskij and Lûbčenko 1977; Orlicz 1982), shows a 9 % decrease of the main cutting force

_{C}*F*with increase of the cutting velocity

_{C}*v*to 50 m·s

_{C}^{-1}, and, further a 36 % increase of the main cutting force

*F*with growth of the

_{C}*v*in the range 50 m·s

_{C}^{-1}to 100 m·s

^{-1}. However no explanation of this phenomenon was given. Because circular sawing is very different from the free cutting, in authors’ opinion, the impact of the cutting velocity

*v*on the cutting forces

_{C}*F*and

_{C}*F*seem also to be different.

_{N}**Fig. 1.** Chip and fractures formation during cutting: 1 = Perpendicular (=90^{o}, =90^{o}, =90^{o}); 2 = Parallel (=90^{o}, =0^{o}, =0^{o}) to wood grains

The issue of fracture formation and propagation in the contact region of the cutting edge and wood, below and above the cutting plane (Fig. 1) was not considered and not discussed in several published papers related to circular sawing (Afanasev 1961; Amalitskij and Lûbčenko 1977; Beršadskij 1967; Orlicz 1982), by assumption of symmetry of cutting forces changes for wood grain orientation angles * _{}* in the ranges of 0

^{o}through 90

^{o}, and 90

^{o}through 180

^{o}. A lack of such symmetry, especially for a tool that is not sharp, has been reported (Axelsson et al. 1993; Porankiewicz et al. 2007). In the authors’ opinion, an explanation for that phenomenon lies in the issue of fracture formation and propagation. In the case of wood cutting with lying in the range of 90

^{o}through 180

^{o}(according to Fig. 2), known as cutting along grains, a fracture tends to propagate above the cutting plane, which is essential also for the formation of the theoretical surface after cutting. In the case of wood grain orientation angle laying in the range of 0

^{o}through 90

^{o}, known as cutting against grains, the fractures can propagate below the cutting plane (Fig. 1), resulting in surface damage.

**Fig. 2.** Wood cutting: a = against grains; b = along grains; wg = with wood grains

Although the main cutting force *F _{C}* falls at the moment of creation of a new fracture, an increase of the cutting velocity

*v*, can probably limit a fracture formation and propagation in the front of the cutting edge. As a result, an increase of the main force

_{C}*F*can be observed, especially when the cutting velocity

_{C}*v*becomes equal or larger than the velocity of propagation of a fracture, which recently has been measured to be about 66 m·s

_{C }^{-1}(Goli et al. 2007). This might be an explanation for the influence of the cutting velocity

*v*on the main cutting force

_{C}*F*in the range from 50 m·s

_{C}^{-1}to 100 m·s

^{-1}, as reported in the literature (Amalitskij and Lûbčenko 1977; Orlicz 1982). The linear influence of the thickness of the cutting layer

*a*(chip thickness), the round up of the cutting edge , the rake angle , the moisture content

_{P}*m*, and the temperature of wood

_{C}*T*, presented in the work Axelsson et al. (1993), contradicts information from literature (Amalitskij and Lûbčenko 1977; Orlicz 1982), where all of these influences were reported as being non-linear.

**Fig. 3.** Main *F _{C}* and the normal

*F*cutting forces, and stereo-metrical parameters;

_{N}*P*= Working plain;

_{F}*P*= Tool reference plain

_{R}It has to be mentioned that during sawing, the total main cutting force consists of the main cutting force and load of two side cutting edges. The resistance of the contact of the saw blade, with the surface of the wooden specimen cut, as well as with a sawdust, below the cutting edge, ought rather to be included in the machine spindle load, rather than in the main cutting force *F _{C}*, as it has been done through the coefficient of the cutting depth

*C*in prevoius publicatoins (Amalitskij and Lûbčenko 1977; Orlicz 1982). Similarly, all forces acting on the work piece and cutting tool outside the cutting region during cutting ought to be considered in the cutting machine theory, rather than in wood cutting process theory.

_{H}In the authors’ opinion, dynamic characteristics of the set-up for measurement of cutting forces and their calibration, as well the range of variation of independent variables, may also be a reason to explain differentiation of results of the wood cutting forces presented in published works.

The present work attempts to evaluate statistical, non-linear, and multi-variable dependencies of the main , as well as the normal cutting forces (Fig. 3), during rotational cutting (circular sawing simulating) of the wood of *Pinus sylvestris*.

**EXPERIMENTAL**

Experiments were performed on the measuring test stand shown in Fig. 4, at Luleå University of Technology, Wood Technology Faculty in Skellefteå, Sweden. The wood specimen 1 was mounted in the holder 4, on the rotating arm 5, powered through the belt transmission 7, from a 4/1400 RPM electrical motor 6, coupled with a stepless variator Eurodrive Type RX81 VU 3 DT 112M-4 COM 82.50555. The piezoelectric transducers 3, were mounted to the cutting tool holder, which was fixed to the tool support 11, powered through the type Servomoler Berger LAHR VRDM 564/50 LNA0027 4088 1,7Ω 0,95A 12070 015000 servomotor 9, and the feed screw 10. The X and Y cutting force analog signal components from piezoelectric transducers 3 were sent to type 2635, Brüel & Kjaer charge amplifiers, passing through an A/D converter, and stored in PC memory in digital form. The sampling rate was of 25kHz. The sampling process and the feed speed were triggered at the same time from the PC.

**Fig. 4.** General scheme of measuring system: 1 – Cutting tool; 2 – Work-piece; 3 – 3D piezoelectric load cells; 4 – Work piece holder; 5 – Rotating arm; 6 – Motor; 7 – Transmission belt; 8 – Balance mass; 9 – Servo-Motor; 10 – Feed screw mechanism; 11 – Tool support

The following machining parameters were applied in the experiment (where the values in parentheses “( )” shows the minimum and maximum values of independent variables, and “ ..” marks show that more variables in a range were analyzed). Para-meters of the experiment were as follows:

*1. Mechanical and physical properties of wood specimens:*

– Wood density, for *m _{C}*=8%,

*D*(372 .. 735) kg·m

^{-3}, – Moisture content

*m*(8 .. 133) %, – Temperature of wood

_{C }*T*(-15, 20)

^{o}C, – Wood specimen dimensions 70 mm, 70 mm and 170 mm: height, width and length respectively.

*2. Machining parameters:*

* *– Cutting edge round up (5, 20) um,

– Contour wedge angle (0.87633, 1.04545, 1.226268) rad, (50.21, 59.9, 70.26)^{o}, (Fig. 3),

– Contour cutting angle (1.05086, 1.21999, 1.4008) rad, (60.21, 69.9, 80.26)^{o},

– Contour rake angle * _{ }*(0.17, 0.35081, 0.51993) rad, (9.74, 20.1, 29.79)

^{o}, (Fig. 3),

– Contour clearance angle (0.174533) rad, (10)^{o}, (Fig. 3),

– Cutting edge inclination angle =0^{o},

– Maximum cutting radius *r _{C}*=535 mm,

– Spindle rotational speed *n *(268, 714) min^{-1},

– Average cutting velocity *v _{C}* (14.916, 39.741) m·s

^{-1},

– Feed speed *v _{F}* (0.004 .. 0.382) m·min

^{-1},

– Thickness of cutting layer (chip thickness) *a _{p }*(0.05, 0.15, 0.5) mm,

– Feed per edge *f _{Z }*(0.05, 0.15, 0.5) mm,

– Cutting depth in feed direction *g _{S}*=7 mm,

– Width of cutting *w _{C}*=4.25 mm,

– Number of cutting edges z=1,

– Angle between cutting edge and wood grains (Fig. 5) =90^{o},

– Growth rings orientation angle towards cutting edge (Fig. 5) =0^{o},

– Wood grain orientation angle (Fig. 5), equal the cutting plane angle (Fig. 5), (0, 15, 45, 75, 90, 105, 135, 165)^{o}.

**Fig. 5.** Orientation angles between wood grains (*wg*) and: – vector of cutting velocity; * _{ }*– cutting edge; – cutting plane; for perpendicular cutting case (=90

^{o}, =90

^{o}, =90

^{o}); by growth rings orientation angle : a – =90

^{o}(radial); b – =0

^{o}(tangential);

*A*– cutting plane

_{S }The change of cutting radius generated by the depth of cutting in the range of 0 to 7 mm resulted in negligibly small variation of the cutting speed of 1.4 %.

*3. Material of the cutting edge:*

*The material*** **was a cobalt-chromium-tungsten alloy (Stellite 12). The

proper influence of variables was analyzed, especially in the case of incomplete experimental matrix, and the precise data fit, by the highest correlation coefficient *R,* between predicted and observed values, by the lowest standard deviation *S _{D}, *and by the lowest summation of residuals square

*S*. Thus one should get an adequate statistical model, of the relations

_{K}*F*and

_{C}*, by the best fit of the experimental data. In the authors’ opinion, additional justification of a choice of a certain type of complicated function makes sense, when several experiments under exactly the same machining conditions have been done in the past. The choice of using a simpler mathematical formula usually results in decreasing approximation quality (lower*

*R*, larger

*S*and

_{D}*S*),

_{K}_{ }and also very often results in a reverse impact of less important independent variables.

It must to be pointed out that a statistical equation is valid only for ranges of independent variables chosen within the experimental matrix, especially for incomplete experimental matrices and complicated mathematical formulas with interactions.

In order to evaluate relations *F _{C} *and , linear formulas and second order multinomial formulas, as well as power and exponential type functions without and with interactions were analyzed in preliminary calculations. The most adequate appeared to be the non-linear, multi-variable Equations (6) through (11),

(N) (6)

In Eq. (6) the terms *P _{A1}* and

*P*are defined as follows,

_{A2 }(7)

(8)

(9)

In Eq. (9) the terms *P _{B1}* and

*P*are defined as follows:

_{B2}(10)

(11)

where _{}_{ }(rad), * _{F}* (rad), and

*T*(

^{o}K) have the respective units.

Estimators for Equations (6) through (11) were evaluated from a complete experimental matrix for variables: , ,* a _{P}*,

*v*and

_{C}*T*and for an incomplete experimental matrix for variables

*,*

_{}*m*and

_{C}*D*, containing 412 measuring points (Table 1).

*The values of*

*F*in Table 1 were multiplied by -1, according to Fig. 3. During the evaluation process of all formulas mentioned above, elimination of unimportant or low import estimators, by use of coefficient of relatively importance

_{N}*C*, defined by equation (12), by assumption

_{RI}*C*>0.01 was done. This process resulted in the elimination of 9 (* – Table 1) and 12 (^ – Table 1) measuring points for main

_{RI}*F*and normal

_{C}*F*cutting force respectively, for which the residuals

_{N}*F*were lying outside the range of about -3·

*S*>

_{D}*F*>3·

*S*. It has to be mentioned that by low value of the

_{D}*C*, the importance of that estimator was not large.

_{RI}(12)

In equation (12) the new terms are:

*S _{KCK0}* – Summation of square of residuals, by

*c*=0.

_{K}c* _{K} *– Estimator with number

*k*index in statistical formula evaluated.

The summation of residuals square *S _{K}*, the standard deviation

*S*, the square of correlation coefficient of the predicted, and observed values

_{D}*R*

^{2}were used for characterization of the approximation quality. Calculations were performed at Poznań Networking & Supercomputing Center PCSS on a SGI Origin 3800 computer, using a special optimization program, based on a least squares method combined with gradient and Monte Carlo methods, mentioned in the work Porankiewicz (1988), (modified many times in order to improve calculation efficiency). For checking every statistical formula mentioned earlier, as well as for evaluation of the final equations (6) through (8) and (9) through (11), the necessary iteration number was as large as 9.3·10

^{9 }(2200 h).

The authors also decided to check the fitting of Equations (3) through (5) using the experimental matrix applied in the present work. However, the lack of information about the exact number of measuring points used in the work of Axelsson et al. (1993) precluded precise comparison.

For presentation of the estimators evaluated for Equations (6) through (8) and (9) through (11), five decimal digits were assumed, which caused negligible deterioration in the quality of approximation. Reduction in the number of decimal digits to 4 worsened the quality of approximation of equation (6) through (8) and (9) through (11) as much as 0.01% and 8%, respectively. Reduction the number of decimal digits to 2 worsened the quality of approximation of equation (6) through (8) and (9) through (11) as much as 6389% and 649124% respectively.

A comparison of results obtained in the present work with similar data from the literature was carried out. To conduct the comparison, the Wood_Cutting program (Porankiewicz 2011) was used for calculation of the main *F _{C}* and radial

*F*cutting forces.

_{N }**RESULTS AND DISCUSSION**

For equation (6) through (8), describing the relation , the following estimators resulted from the evaluation: *a _{1}*=224914.765,

*a*=6.411·10

_{2}^{-5},

*a*=-0.04709,

_{3}*a*=18.90097,

_{4}*a*=2.57193,

_{5}*a*=10.57092,

_{6}*a*=0.0955,

_{7}*a*=0.30406,

_{8}*a*=1.17260,

_{9}*a*=-2.15734,

_{10}*a*=-0.05021;

_{11}*a*=-0.07183,

_{12}*a*=2.43628,

_{13}*a*=0.99865,

_{14}*a*=6.883·10

_{15}^{-5},

*a*=-27269.686,

_{16}*a*=0.02962,

_{17}*a*=0.4157,

_{18}*a*=-0.17439,

_{19}*a*=0.23563,

_{20}*a*=0.19386,

_{21}*a*=1.21835,

_{22}*a*=-0.57335,

_{23}*a*=-0.24134,

_{24}*a*=-878.18,

_{25}*a*=0.01668,

_{26}*a*=-27.65757.

_{27}The quality of the fit of the equation (6) through (8) is shown by Fig. 6, and was characterized by the quantifiers: *S _{K}*=57722.3,

*R*=0.95,

*R*=0.91,

^{2}*S*=11.87 N. The coefficients of relative importance took values as follows:

_{D}*C*=155.7,

_{RI1}*C*=347.9,

_{RI2}*C*=107.7,

_{RI3}*C*=650.3,

_{RI4}*C*=469.9,

_{RI5}*C*=5533.7,

_{RI6}*C*=718.6,

_{RI7}*C*=826,

_{RI8}*C*=940.4

_{RI9}*C*=219.2,

_{RI10}*C*=108.9,

_{RI11}*C*=113.1,

_{RI12}*C*=226.2,

_{RI13}*C*=225.6,

_{RI14}*C*=338.2,

_{RI15}*C*=7374.3,

_{RI16}*C*=111.2,

_{RI17}*C*=884.6,

_{RI18}*C*=269.7,

_{RI19}*C*=749.2,

_{RI20}*C*=884.6,

_{RI21}*C*=3309.9,

_{RI22}*C*=1.732·10

_{RI23}^{12},

*C*=1194.8,

_{RI24}*C*=2.188·10

_{RI25}^{5}

*,*

*C*=169.7,

_{RI26}*C*=645. The following estimators were evaluated for formulas (9) through (11), describing the relation:

_{RI27}*b*=3.355·10

_{1}^{-3},

*b*=539082.961,

_{2}*b*=-0.15741,

_{3}*b*=-0.33883,

_{4}*b*=-0.01213,

_{5}*b*=758.29345,

_{6}*b*=0.76858,

_{7}*b*=3.5537,

_{8}*b*=5.365·10

_{9}^{-4},

*b*=1.3538·10

_{10}^{-3},

*b*=-3.0279·10

_{11}^{-3},

*b*=-1.3297·10

_{12}^{-4},

*b*=-0.012783,

_{13}*b*=-30.16633,

_{14}*b*=-0.219799,

_{15}*b*=10.134443,

_{16}*b*=3.4903·10

_{17}^{-4},

*b*=-0.38895,

_{18}*b*=-0.27925,

_{19}*b*=1.90555,

_{20}*b*=-0.10358,

_{21}*b*=-0.732,

_{22}*b*=-0.32493,

_{23}*b*=-0.2652,

_{24}*b*=-137.7076,

_{25}*b*-3.6111

_{26}=*·*10

^{-3}

*,*and

*b*=1.66647. The quality of the fit of the equation (9) through (11) were characterized by the quantifiers:

_{27}*S*=45551.2,

_{K}*R*=0.93,

*R*=0.86,

^{2}*S*=10.65 N, and also are illustrated in Fig. 6. The coefficients of relative importance took the following values:

_{D}*C*=5.809·10

_{RI1}^{6},

*C*=185.1,

_{RI2}*C*=18.7,

_{RI3}*C*=28,

_{RI4}*C*=14.4,

_{RI5}*C*=547.4,

_{RI6}*C*=711.3,

_{RI7}*C*=1469.7,

_{RI8}*C*=7.1,

_{RI9}*C*=18.7,

_{RI10}*C*=293.3,

_{RI11}*C*=1.3,

_{RI12}*C*=4.028·10

_{RI13}^{4},

*C*=7.688·10

_{RI14}^{9},

*C*=4397.8,

_{RI15}*C*=4.368·10

_{RI16}^{10},

*C*=765.6,

_{RI17}*C*=256.7,

_{RI18}*C*=66.8,

_{RI19}*C*=662.8,

_{RI20}*C*=135.8,

_{RI21}*C*=6.312·10

_{RI22}^{6},

*C*=1471,

_{RI23}*C*=305.1,

_{RI24}*C*=3.059·10

_{RI25}^{7},

*C*=0.01,

_{RI26}*C*=2.5.

_{RI27}**Fig. 6.** Plot of main cutting force observed *F _{C}*,

*against main cutting force predicted*

*F*, according to equations (6) through (8), and the plot of normal cutting force observed

_{CP}*F*, against predicted normal force

_{N}*F*, according to equations (9) through (11)

_{NP}Figure. 6 shows that in the analyzed experiment the standard deviation *S _{D}* was rather high, which suggested the presence of an uncontrolled variation, and, maybe also an effect of unrecognized interactions. In the authors’ opinion more work in future experiments will be needed to reduce the value of

*S*. The quality of approximation obtained in the present work for the main force

_{D}*F*, namely

_{C}*R*=0.95,

*R*=0.91, was slightly better than for the normal force

^{2}*F*, namely

_{N}*R*=0.93,

*R*=0.86, and, much better than the quality of approximation of the equation (3) through (5) for the main force

^{2}*F*, namely

_{C}*R*=0.2, reported in the work Axelsson et al. (1993). Such a good approximation confirms the exceptional precision of optimization method and program applied in this study.

From Fig. 7 can be seen a fast growth of the main force *F*_{C} with an increase of the grain angle * _{}*, from

_{}*=0*

_{ }^{o}up

*=1.475312 rad (84.53*

_{}^{o}), being maximum for cutting against grains case. This observation contradicted information from the literature, reporting the maximum at

*= /2 rad (90*

_{}^{o}). The maximum in the dependence

*F*=

_{C}*f*(

_{}*) can be seen for whole range of variation of the cutting edge dullness . Further rise of the*

_{ }*angle resulted in rapid drop of the*

_{}*F*

_{C}. From Fig. 7 can also be seen fast growth of the normal cutting force

*F*with an increase of the grain angle

_{N}*, up to maximum at*

_{}*=0.801795 rad (45.94*

_{}^{o}) but only for large cutting edge dullness of =20 um. This finding contradicts information reported in the literature (Amalitskij and Lûbčenko 1977; Orlicz 1982). The maximum for the normal cutting force

*F*, observed for cutting edge dullness of =20 um, disappeared for the sharp tool by =5 um. Further increase of the

_{N}*angle up to*

_{}*=1.5708 rad (90*

_{}^{o}) caused rapid decrease of the

*F*

_{N}, up to minimum. An increase of the

*angle, by*

_{}*>1.5708 rad (90*

_{}^{o}) (cutting along grains case) resulted in rather small change of the

*F*value. This observation was in agreement with the paper of Porankiewicz et al. (2007) and contradicts other information from the literature (Amalitskij and Lûbčenko 1977; Orlicz 1982).

_{N}**Fig. 7. **Plot of dependence of the main *FC* and normal *FN* cutting forces upon the angle * _{}*and the round up of the cutting edge , according to equations: a – (6) through (8); and b – (9) through (11); for: =0.52 rad (29.79

^{o});

*ap*=0.5 mm;

*vc*=39.74 m·s

^{-1};

*mc*=10 %;

*D*=372 kg·m

^{-3};

*T*=293

^{o}K (20

^{o}C)

The ratio between maximum and minimum values of the main force *F _{C}* for the grain angle

*lying in the range of 1.464 rad and 0 rad (84.53*

_{}^{o}and 0

^{o}), was as high as 1.84 in the present study, while as large as 2.62 according to the literature (Orlicz 1982), for the following cutting conditions: =6 um, =0.349066 rad (20

^{o}),

*a*=0.2 mm,

_{P}*v*=28 m·s

_{C}^{-1},

*m*=8 %,

_{C}*T*=293

^{o}K (20

^{o}C),

*and*

*D*=490 kg·m

^{-3}.

The ratio between maximum and minimum values of the normal force *F _{C}* for the grain angle

*lying in the range of 0.792 rad and 0 rad (45.94*

_{}^{o}and 0

^{o}), was as high as 2.51 in the present study, while as large as 2.62 according to the literature (Orlicz 1982), for the following cutting conditions: =20 um, =0.349066 rad (20

^{o}),

*a*=0.2 mm,

_{P}*v*=28 m·s

_{C}^{-1},

*m*=8 %,

_{C}*T*=293

^{o}K (20

^{o}C),

*and*

*D*=490 kg·m

^{-3}.

The ratio between maximum and minimum value of the main force *F _{C}* for the largest and the lowest cutting edge round-up , which were as high as 6 um and 20 um, respectively, were as high as 1.35 in this paper, while 1.12 according to the literature (Amalitskij and Lûbčenko 1977, Orlicz 1982), for following cutting conditions:

*=0.7854 rad (45*

_{}^{o}), =0.349066 rad (20

^{o}),

*a*=0.2 mm,

_{P}*v*=28 m·s

_{C}^{-1},

*m*=8 %,

_{C}*T*=293

^{o}K (20

^{o}C),

*D*=490 kg·m

^{-3}. The ratio between maximum and minimum value of the radial force

*F*for the largest and the lowest cutting edge round-up , were as high as 15.37 in this paper, while 2.03 according to the literature (Amalitskij and Lûbčenko 1977, Orlicz 1982), for same cutting conditions.

_{N}Graphical illustration of the relations (6) through (11) between the main *FC* and the normal *FN* cutting forces and the rake angle and the thickness of the cutting layer *aP*, are shown in Fig. 8. It can be seen from Fig. 8 that a decrease of the rake angle increased the main *FC* and the normal *FN* cutting forces, in a parabolic, increasing manner.

**Fig. 8.** Plot of dependence of the main *F _{C} *and normal

*F*cutting forces upon the rake angle and the thickness of the cutting layer

_{N}*a*, according to equations: a – (6) through (8); and b – (9) through (11); for

_{P}*=1.475412 rad (84.53*

_{}^{o}); =5 um;

*v*=39.74 m·s

_{C}^{-1};

*mc*=10 %;

*T*=293

^{o}K (20

^{o}C);

*D*=500 kg·m

^{3}

An increase of the thickness of the cutting layer *a _{P}* resulted in an increase of the main cutting force

*F*in parabolic, decreasing manner. An increase of the thickness of the cutting layer

_{C}*a*resulted in an increase of absolute value of the radial cutting force

_{P}*F*in parabolic, decreasing manner. It has also to be mentioned that the radial cutting force

_{N}*F*for some values of the rake angle and thickness of the cutting layer

_{N}*a*become positive. The issue of the impact the rake angle and the thickness of the cutting layer

_{P}*a*on the the radial cutting force

_{P}*F*contradicted information from literature (Orlicz 1982).

_{N}The ratio between maximum and minimum value of the main force *F _{C}* for the largest and the lowest rake angle , of 0.17 rad (9.74

^{o}) and of 0.52 rad (29.79

^{o}) respectively, was as large as 1.23 in the present study, while according to the literature (Orlicz 1982) was as large as 1.96, for the following cutting conditions:=6 um,

*a*=0.2 mm,

_{P}*v*=28 m·s

_{C}^{-1},

*m*=8 %,

_{C}*T*=293

^{o}K (20

^{ o}C),

*D*=490 kg·m

^{-3}.

For the same cutting parameters, the ratio between the largest and the lowest value of the radial force *F _{N}* was as high as 2.67 in the present study, while as large as 1.96 according to the literature (Orlicz 1982). It has to be mentioned that the radial force

*F*evaluated from equation (9) through (11) took negative values in analyzed range of variation of the rake angle , while the radial force

_{N}*F*calculated on basis of the equation (1) through (2), according to the literature (Orlicz 1982) was positive.

_{N}The ratio between maximum and minimum value of the main force *F _{C}* for the largest and the lowest thickness of cutting layer

*a*, as high as 0.05 mm and 0.5 mm respectively, was as high as 2.12 in the present study, while as large as 2.82 according to the literature (Orlicz 1982), for following cutting parameters: =6 um, =0.349066 rad (20

_{P}^{o}),

*=0.7854 rad (45*

_{}^{o}),

*v*=28 m·s

_{C}^{-1},

*m*=8 %,

_{C}*T*=293

^{o}K (20

^{o}C),

*D*=490 kg·m

^{-3}.

For the same cutting parameters, the ratio between the largest and the lowest value of the radial force *F _{N}*, was as high as 4.58 in the present study, while as large as 2.82 according to the literature (Orlicz 1982). It has to be mentioned that the radial force

*F*evaluated from equation (9) through (11) took negative values throughout the analyzed range variation of the thickness of cutting layer

_{N}*a*, while the radial force

_{P}*F*calculated from equation (1) through (2) was positive according to the literature (Orlicz 1982).

_{N}The impact of the wood density *D* and the cutting velocity *v _{C}* the on the main force

*F*and the normal force

_{C}*F*, according to equations (6) through (11), is illustrated in Fig. 9, which shows a very strong influence, with a parabolic increasing manner. The wood density

_{N}*D*impact on the radial

*F*cutting force was also very strong, in a parabolic, decreasing manner. The cutting velocity

_{N}*v*impact on the

_{C}*F*and

_{C}*F*cutting forces was smaller (especially on the

_{N}*F*) in a parabolic, decreasing manner.

_{N}**Fig. 9. **Plot of the dependence of the main *F _{C} *and normal

*F*cutting forces upon the wood density

_{N}*D*and the cutting velocity

*v*, according to equation: a – (6) through (8); and b – (9) through (11); for

_{C}*=1.475312 rad (84.53*

_{}^{ o}); =5 um; =0.349066 rad (20

^{o});

*a*=0.5 mm;

_{p}*mc*=8 %;

*T*=293

^{o}K (20

^{o}C)

As the wood density *D* increased throughout the analyzed range of variation, the main cutting force *F _{C}* increased by as much as 2.49 times, for the following cutting conditions:

*a*=0.2 mm, =6 um, =0.349066 rad (20

_{P}^{o}),

*=0.7854 rad (45*

_{}^{o}),

*v*=28 m·s

_{C}^{-1},

*mc*=8 %,

*T*=293

^{o}K (20

^{o}C).

The radial cutting force *F _{N}*, for the same cutting conditions, decreased by as much as 1.27 times, in the analyzed range of variation of the wood density

*D*. In the literature the wood density was not taken into account for

*F*and

_{C}*F*cutting forces evaluation (Orlicz 1982).

_{N}With an increase of the cutting velocity *v _{C}* throughout analyzed variation range from 14.91 m·s

^{-1}to 39.74 m·s

^{-1}, the main cutting force

*F*increased, according to equations (6) through (8) in parabolic, decreasing manner.

_{C}An increase of the cutting velocity *v _{C}* throughout analyzed range of variation was accompanied by a slight increase in the radial cutting force

*F*according to equations (9) through (11) in parabolic, decreasing manner. For low wood density

_{N}*D,*the

*F*take negative values. The issue of the influence of the cutting velocity

_{N}*v*on the main

_{C}*F*and the radial

_{C}*F*cutting forces contradicted information reported in the literature (Orlicz 1982), giving an opposite influence until

_{N}*v*=50 m·s

_{C}^{-1}. The non-linear issue of the impact of the cutting velocity

*v*on the main force

_{C}*F*also was also inconsistent with work Axelsson et al. (1993). Results of the present experiment are unable to explain these contradictions.

_{C}The ratio between maximum and minimum value of the main cutting force *F _{C}* for the largest and the lowest cutting velocity

*v*, as high as 14.91 m·s

_{C}^{-1}and 39.74 m·s

^{-1}, respectively, was as large as 1.54 in the present study, while as much as 0.93 according to the literature (Orlicz 1982), for following cutting conditions: =6 um, =0.349066 rad (20

^{o}),

*=0.7854 rad (45*

_{}^{o}),

*a*=0.2 mm,

_{P}*m*=8 %,

_{C}*T*=293

^{o}K (20

^{ o}C),

*D*=490 kg·m

^{-3}.

**Fig. 10.** Plot of dependence of the main *F _{C} *and normal

*F*cutting forces upon the moisture content

_{N}*m*, and the wood temperature during cutting

_{C}*T*, according to equations: a – (6) through (8); and b – (9) through (11); for

*=0 rad (0*

_{}^{o}); =5 um; =0.349066 rad (20

^{o});

*a*=0.1 mm;

_{p}*v*=39.74 m·s

_{C}^{-1};

*D*=500 kg·m

^{-3}

Figure 10 shows the non-linear dependence of the main *F _{C}* and the normal

*F*cutting forces upon the moisture content

_{N}*m*and the wood temperature

_{C}*T*. With an increase of the moisture content

*m*from 8 % to about 30 % the main force

_{C}*F*, grew very fast. Further enlargement of the

_{C}*m*, to 133 %, caused a slightly decrease of the

_{C}*F*, which was in agreement with the literature (Amalitskij and Lûbčenko 1977; Orlicz 1982) for types of machining other than sawing. The non-linear impact of

_{C}*m*on the main force

_{C}*F*issue contradicts the work of Axelsson et al. (1993). For the sawing case, an opposite impact of the moisture content

_{C}*m*on the cutting forces was reported in the literature (Amalitskij and Lûbčenko 1977; Orlicz 1982). It is well known that there is an exceptional, increasing effect of the moisture content

_{C}*m*on the shock strength of wood; indeed, this effect might be invoked to explain this phenomenon at large cutting velocity

_{C}*v*. However, in the case of sawing (closed cut), an increasing action of the two side edges has to be considered. On the other hand, someone may point out that most of wood’s strength properties decrease with increasing moisture content

_{C}*m*. With the current state of the knowledge it is not possible to estimate which role is dominant in the cutting forces evaluation: wood shock strength or other (static) strengths.

_{C}The radial cutting force *F _{N}* dependence upon the moisture content

*m*was opposite to the dependence observed for the main cutting force

_{C}*F*in the range from 8 % to about 30 %. This would suggest that the relaxation of the wood surface from the clearance surface site, becomes larger with increasing moisture content

_{C}*m*in the analyzed range.

_{C}The ratio between maximum and minimum value of the *F _{C}*, for the moisture content

*m*, as high as 8 % and 70 % respectively, was 1.27 in the preset study, while 1.23 according to the literature (Orlicz 1982), for following cutting conditions: =6 um, =0.349066 rad (20

_{C}^{o}),

*=0.7854 rad (45*

_{}^{o}),

*a*=0.2 mm,

_{P}*T*=293

^{o}K (20

^{o}C),

*and*

*D*=490 kg·m

^{-3}.

For the same machining parameters, the radial force *F _{N}* fell with change of the moisture content

*m*from value of 4.9 N to -3.5 N, in the present study, while it fell 1.23 times (being positive in whole range) according to the literature (Orlicz 1982).

_{C}The main *F _{C}* and the normal

*F*cutting forces slightly increased with lowering of the wood temperature

_{N}*T*(Fig. 9). The ratio between maximum and minimum value of the

*F*and for the

_{C}*F*for the lowest and the highest wood temperature

_{N}*T*, as high as 258

^{o}K (-15

^{o}C) and 293

^{o}K (20

^{o}C), was as high as 1.06 and 1.59, respectively in this paper, while 1.15 and 1.14 times respectively according to the literature (Orlicz 1982), for following cutting conditions: =6 um, =0.349066 rad (20

^{o}),

*=0.7854 rad (45*

_{}^{o}),

*a*=0.2 mm,

_{P}*mc*=8 %,

*D*=490 kg·m

^{-3}.

It has to be pointed out that for another combination of the cutting parameter values, the comparison between models (6) through (11) and (1) through (2) as well as (3) through (5) will bring other values of the ratio.

Figures 7 through 10 show that the normal cutting force *F _{N}* did not follow the main cutting force

*F*

_{C}. In case of influence of the thickness of cutting layer

*a*, the moisture content

_{P}*m*and the cutting speed

_{C}*v*on the radial cutting force

_{C}*F*(Fig. 10), even a reverse impact was observed, which suggests that the equation (2) was too simple to adequately describe the dependence of the normal force

_{N}*F*upon cutting parameters.

_{N}The impact of the cutting edge dullness , the moisture content *m _{C}*, and the cutting speed

*v*on the tangential

_{C}*F*, also the influence of the cutting edge dullness , the rake angle , the thickness of cutting layer

_{C}*a*, the moisture content

_{P}*m*, the wood temperature

_{C}*T*, the cutting speed

*v*on the radial cutting force

_{C}*F*in the present study, was larger than that reported in the literature (Amalitskij and Lûbčenko 1977; Orlicz 1982).

_{N}The dependence of the main cutting force *F _{C}* upon the grain angle

*, the rake angle*

_{}*, the thickness of the cutting layer*

_{}*a*and the wood temperature

_{P}*T*was smaller than that reported in the literature (Amalitskij and Lûbčenko 1977; Orlicz 1982).

The dependence of the radial *F _{N}* cutting force upon the grain angle

*was smaller than that reported in the literature (Orlicz 1982).*

_{}The two-level variation of the cutting edge round up , the cutting velocity *v _{C},* and the wood temperature

*T*independent variables in the experimental matrix of the present study, seemed to not be the best choice, due to their non-linear influence on the main

*F*and radial

_{C}*F*

_{N}**cutting forces.**

Different values of the exponents *a _{9}* through

*a*and

_{13}*b*through

_{9}*b*for longitudinal cutting, and, exponents

_{13}*a*through

_{18}*a*and

_{22}*b*through

_{18}*b*for perpendicular cutting, evaluated for equations (6) through (11) showed that the impact of the cutting angle

_{22}*[1.570796-] (rad) {[90*

_{}^{o}–] (

^{o})}, the cutting edge dullness , the thickness of the cutting layer

*a*, the cutting velocity

_{P}*v*, the moisture content

_{C}*m*and the wood temperature

_{C},*T*on the tangential

*F*and the radial cutting forces

_{C}*F*was different for these two base cutting directions (perpendicular and longitudinal cutting). The use of the same correction coefficients:

_{N}*C*,

*C*,

*C*,

_{AP}*C*,

_{VC}*C*,

_{MC}*C*for longitudinal cutting and for perpendicular cutting in equations (1) perpendicular (2), seems to be one of the reasons reported in literature (Orlicz 1982) to account for differences between predicted and observed main cutting force

_{T}*F*, decreasing the methods’ precision.

_{C}The analysis of the relationship (9) through (11) with step 0.0008 rad (0.05^{o}), revealed the existence of a local large extremes at * _{}*=1.7282 rad (99.019

^{o}). The extremes extend in the range from -0.06981 rad (-4

^{o}) to +0.0524 rad (+4

^{o}). Because there are no experimental data for the specified range of the

*in the experimental matrix, the extremes must be excluded from considerations.*

_{}**CONCLUSIONS**

The analysis of results of the calculations performed makes it possible to state that:

1. The normal cutting force *F _{N}* takes a negative value for a wide range of the cutting parameters.

2. In the dependence of the main cutting force *F*_{C} upon the grain angle * _{}*, a maximum at

*=1.475312 rad (84.53*

_{}^{o}), extending to whole range of variation of the cutting edge dullness was found.

3. In the dependence of the normal force *F _{N}* upon the grain angle

*, a maximum was observed at*

_{}*=0.8017949 rad (45.939*

_{}^{o}). This maximum disappeared in the case of a sharp tool (=5 um).

4. An drop down of the rake angle increased the main *F _{C}* and the normal

*F*cutting forces, in a parabolic, increasing manner.

_{N}5. An increase of the thickness of the cutting layer *a _{P}* increased the main cutting force

*F*in a parabolic, decreasing manner.

_{C}6. An increase of the thickness of the cutting layer *a _{P}* increased the absolute value of the radial cutting force

*F*in parabolic, decreasing manner.

_{N}7. Increasing the wood density *D* caused an significant increase of the main cutting force *F _{C}* in a parabolic, increasing manner.

8. The radial cutting force *F _{N}* increased with the wood density

*D*climb in a parabolic, decreasing manner.

9. An increase of the cutting velocity *v _{C}* caused an increase of the main cutting force

*F*, according to equation (6)-(8), in a parabolic, decreasing manner.

_{C}10. An increase of the cutting velocity *v _{C }*slightly increased the radial cutting force

*F*, according to equation (9) through (11).

_{N}11. An increase of the moisture content *m _{C}*, in range from 8% to about 30%, resulted in a rapid increase of the main cutting force

*F*. A further increase in moisture content

_{C}*m*to 133% resulted in a further decrease of

_{C}*F*in parabolic decreasing manner.

_{C}12. An increase of the moisture content *m _{C}*, in range from 8% to about 30%, resulted in a rapid lowering of the radial cutting force

*F*. A very small increase of the absolute value of the

_{N}*F*was observed in a parabolic increasing manner with the

_{N}*m*rising to 133%.

_{C}13. The main *F _{C}* and absolute value of the normal

*F*cutting forces slightly increased with a lowering of the wood temperature

_{N}*T*, in a parabolic decreasing manner

*.*

14. The dependence of the normal force *F _{N}* upon cutting parameters by

*Pinus sylvestris*wood cutting does not generally follow the same trends as the main force

*F*.

_{C}15. In the present work a larger influence of the rake angle * _{F}*, the cutting edge dullness , the moisture content

*m*, and the cutting velocity

_{C}*v*on the main cutting force

_{C}*F*was observed, compared to those reported in the literature (Amalitskij and Lûbčenko 1977; Orlicz 1982).

_{C}16. In the present work a larger influence of the rake angle * _{F}*, the cutting edge dullness , the thickness of the cutting layer

*a*and the wood temperature

_{P}*T*on the radial

*F*cutting force was observed than the corresponding dependencies reported in the literature (Orlicz 1982).

_{N}17. The dependence the main *F _{C}* cutting force upon the grain angle

*, the rake angle*

_{}*, the thickness of the cutting layer*

_{F}*a*, and the wood temperature

_{P}*T*was smaller than those given in the literature (Amalitskij and Lûbčenko 1977; Orlicz 1982).

18. The dependence the radial *F _{C}* cutting force upon the grain angle

*was smaller than those given in the literature (Amalitskij and Lûbčenko 1977; Orlicz 1982).*

_{}19. The following issues in the Conclusions of the present study, point ns.: 1, 2, 3, 4, 9, 10, 12 contradicted information from Orlicz (1982).

20. The following issues of non-linear influence of the cutting parameters on the main cutting force *F _{C}* in the Conclusions of the present study, point ns.: 5, 7, 9, 11 and 13 contradicted information reported in Axelsson et al. (1993).

**ACKNOWLEDGMENTS**

The authors are grateful for the support of the Poznań Networking & Supercomputing Center (PCSS), Poznań, Poland for a calculation grant.

**REFERENCES CITED**

Afanasev, P. S. (1961). *Derevoobrabatyvaûŝie Stanki (Woodworking Machinery)*, Moskva.

Amalitskij, V.,V., and Lûbčenko, V., I. (1977). *Stanki i Instrumenty Derevoobrabatyvaûŝih Predpriâtij (Machinery and tools of woodworking factories)*, Moskva.

Axelsson, B., Lundberg, S., and Grönlund, A. (1993). “Studies of the main cutting force at and near a cutting edge,” *Holz als Roh u. Werkstoff* 52, 43-48.

Beršadskij, А. L. (1967). *Razčet Režimov Rezaniâ Drevesiny (Resolution of modes of wood machining)*, Moskva.

Deševoj, М. А. (1939). *Mehaničeskaâ Tehnologâ Dereva (Mechanical technology of wood)*, LTA.

Goli, G., Wyeth D., Atkins A.G., Jeronimidis G., Fioravanti M., Del Taglia A. (2007). “Chip formation in wood cutting with different grain orientations using high and low cutting speeds,” AITeM 2007 Conference, Associazione Italiana di Tecnologia Meccanica, Montecatini, September 10-12.

Kivimaa, E. (1950). “The cutting force in woodworking,” Rep. No. 18, The State Institute for Technical Research, Helsinki.

McKenzie, W. M. (1961). “Fundamental analysis of the wood cutting process,” Dept. of Wood Technology, School of Natural Resources, University of Michigan, Ann Arbor.

Orlicz, T. (1982). *Obróbka Drewna Narzędziami Tnącymi*. (*Machining of wood with use of cutting tools*), Study book SGGW-AR, Warsaw.

Porankiewicz, B., Bermudez, J. C. E., and Tanaka, C. (2007): “Cutting forces by peripheral cutting of low density wood species,” *BioResources *2(4), 671-681.

Porankiewicz, B. (2011). Wood_Cutting, Program in Delphi Borland for calculation of cutting forces, (Not published).

Article submitted: November 28, 2010; Peer review completed: February 17, 2011; Revised version received: July 28, 2011; Accepted: August 1, 2011; Published: August 4, 2011; Corrections to Published Paper: July 10, 2014.

**APPENDIX**

**Table 1. **Experimental Matrix

*FC* *FN* *FS* _{V}_{F}*a _{P}*

*mc*

*T*

*D*

*v*

_{C}(N) (N) (N) (rad) (rad) (um) (mm) (%) (^{o}C) (kg·m^{-3}) (m·s^{-1})

1 2 3 4 5 6 7 8 9 10 11

———————————————————————————————————-

23 -3.7 1.2 2.88 0.35 20 0.15 8 20 372.53 39.741

13.5 -0.9 -0.9 2.88 0.52 20 0.15 8 20 386.2 39.741

102.1 -44.3 -4.4 2.36 0.17 20 0.5 8 20 496.56 39.741

83.7 -28.2 2.7 2.36 0.35 20 0.5 8 20 481.2 39.741

117.9 -34.2 6.7 2.36 0.35 20 0.5 8 20 568.63 39.741

81.3 -5.2 6.6 2.36 0.52 20 0.5 8 20 651.93 39.741

79.1 -6.6 3.8 2.88 0.17 20 0.5 8 20 476.36 39.741

60.3 -1.8 3 2.88 0.35 20 0.5 8 20 476.36 39.741

36.7 5.1 -4.9 2.88 0.52 20 0.5 8 20 460.93 39.741

54.4 -3.7 2 2.88 0.17 20 0.5 8 20 392.96 39.741

15.8 -2.5 -0.9 2.88 0.35 20 0.5 8 20 392.96 39.741

29.3 4.1 -2.7 2.88 0.52 20 0.5 8 20 392.96 39.741

129.591 8.056 -1.777 1.57 0.35 5 0.5 40.96 20 715.51 14.916

40.561 -14.765 -1.933 1.57 0.35 5 0.05 48.98 20 722.28 14.916

160.693 -25.839 -6.268 1.57 0.35 20 0.5 64.65 20 710.01 14.916

70.84 -67.838 -3.682 1.57 0.35 20 0.05 106.33 20 683.1 14.916

130.404 -4.431 -4.167 1.57 0.17 5 0.5 100.09 20 640.9 14.916

33.449 -12.862 -1.298 1.57 0.17 5 0.05 83.91 20 691.02 14.916

189.348 -46.98 0.526 1.57 0.17 20 0.5 59.80 20 735.5 14.916

80.01 -85.349 -0.64 1.57 0.17 20 0.05 45.44 20 726.78 14.916

137.999 11.013 -0.402 1.57 0.35 5 0.5 118.56 20 655.26 39.741

37.068 -10.217 -1.568 1.57 0.35 5 0.05 108.16 20 670.28 39.741

200.265 -20.248 -14.438 1.57 0.35 20 0.5 62.60 20 696.05 39.741

75.438 -61.57 -3.983 1.57 0.35 20 0.05 48.66 20 692.16 39.741

177.826 -0.428 -1.757 1.57 0.17 5 0.5 45.73 20 671.96 39.741

37.552 -11.268 -1.943 1.57 0.17 5 0.05 55.27 20 691.82 39.741

233.838 -47.674 5.413 1.57 0.17 20 0.5 83.54 20 682.3 39.741

81.445 -70.718 4.507 1.57 0.17 20 0.05 121.40 20 640.43 39.741

126.492 -2.346 -3.888 1.57 0.35 5 0.5 103.83 -15 694.91 14.916

36.861 -14.183 -1.677 1.57 0.35 5 0.05 48.60 -15 681.02 14.916

148.838 -33.747 -4.703 1.57 0.35 20 0.5 41.92 -15 658.74 14.916

65.808 -63.934 -3.712 1.57 0.35 20 0.05 32.58 -15 647.14 14.916

33.263 -13.325 -1.559 1.57 0.17 5 0.05 45.02 -15 688.6 14.916

173.62 -54.042 -0.922 1.57 0.17 20 0.5 65.86 -15 677.73 14.916

81.564 -82.258 0.641 1.57 0.17 20 0.05 114.01 -15 644.12 14.916

152.509 4.504 -3.439 1.57 0.35 5 0.5 37.06 -15 702.89 39.741

39.857 -12.925 -2.369 1.57 0.35 5 0.05 46.97 -15 718.66 39.741

205.698 -12.468 -8.805 1.57 0.35 20 0.5 80.13 -15 702.49 39.741

68.175 -56.639 -3.909 1.57 0.35 20 0.05 118.81 -15 664.58 39.741

39.876 -13.343 -1.39 1.57 0.17 5 0.05 73.46 -15 627.21 39.741

221.891 -47.702 -0.45 1.57 0.17 20 0.5 56.90 -15 640.23 39.741

88.266 -83.151 1.124 1.57 0.17 20 0.05 34.57 -15 657.4 39.741

101.829 -31.949 -1.626 1.57 0.35 5 0.5 8 -15 567.29 14.916

43.727 -34.75 -1.784 1.57 0.35 5 0.05 8 -15 566.55 14.916

99.88 -43.049 -5.636 1.57 0.35 20 0.5 8 -15 576.48 14.916

50.012 -59.582 -3.249 1.57 0.35 20 0.05 8 -15 572.06 14.916

*FC* *FN* *FS* * _{V}*

**

_{F}*a*

_{P}*mc*

*T*

*D*

*v*

_{C}(N) (N) (N) (rad) (rad) (m) (mm) (%) (^{o}C) (kg·m^{-3}) (m·s^{-1})

1 2 3 4 5 6 7 8 9 10 11

———————————————————————————————————-

112.652 -17.168 -3.148 1.57 0.17 5 0.5 8 -15 658.21 14.916

35.153 -18.649 -2.005 1.57 0.17 5 0.05 8 -15 698 14.916

136.268 -42.923 -1.306 1.57 0.17 20 0.5 8 -15 708.93 14.916

64.664 -85.244 -2.725 1.57 0.17 20 0.05 8 -15 595.61 14.916

116.888 -24.582 -2.794 1.57 0.35 5 0.5 8 -15 556.76 39.741

46.011 -34.298 -2.495 1.57 0.35 5 0.05 8 -15 556.96 39.741

117.503 -49.211 -6.215 1.57 0.35 20 0.5 8 -15 545.82 39.741

58.443 -62.678 -2.835 1.57 0.35 20 0.05 8 -15 541.19 39.741

111.886 -22.852 -3.85 1.57 0.17 5 0.5 8 -15 560.65 39.741

36.842 -20.681 -2.164 1.57 0.17 5 0.05 8 -15 567.76 39.741

129.061 -66.835 -2.507 1.57 0.17 20 0.5 8 -15 568.03 39.741

67.81 -81.328 -2.834 1.57 0.17 20 0.05 8 -15 570.71 39.741

113.223 -21.4 -1.644 1.57 0.35 5 0.5 8 20 605 14.916

45.227 -35.825 -1.616 1.57 0.35 5 0.05 8 20 585.54 14.916

113.54 -37.182 -3.752 1.57 0.35 20 0.5 8 20 591.78 14.916

50.544 -61.19 -2.802 1.57 0.35 20 0.05 8 20 590.04 14.916

89.461 -20.726 -4.134 1.57 0.17 5 0.5 8 20 570.31 14.916

32.644 -18.586 -1.492 1.57 0.17 5 0.05 8 20 587.89 14.916

123.453 -55.876 -2.584 1.57 0.17 20 0.5 8 20 580.58 14.916

62.4 -85 12 -2.064 1.57 0.17 20 0.05 8 20 600.44 14.916

141.263 -12.932 -3.105 1.57 0.35 5 0.5 8 20 708.87 39.741

55.39 -36.28 -2.746 1.57 0.35 5 0.05 8 20 658.81 39.741

137.577 -34.917 -4.975 1.57 0.35 20 0.5 8 20 618.76 39.741

66.058 -68.31 -2.271 1.57 0.35 20 0.05 8 20 606.88 39.741

115.523 -19.78 -4.023 1.57 0.17 5 0.5 8 20 553.47 39.741

36.101 -18.301 -1.964 1.57 0.17 5 0.05 8 20 568.1 39.741

149.93 -54.288 -2.87 1.57 0.17 20 0.5 8 20 577.02 39.741

88.863 2.887 0.448 0 0.35 5 0.5 90.98 20 657.27 39.741

18.813 -5.538 -1.4 0 0.35 5 0.05 101.80 20 644.05 39.741

85.663 -18.04 -2.430 0 0.35 20 0.5 110.10 20 637.21 39.741

22.924 -26.732 -1.944 0 0.35 20 0.05 109.40 20 650.83 39.741

91.102 -13.86 -0.248 0 0.17 5 0.5 117.40 20 641.3 39.741

20.313 -9.793 -1.405 0 0.17 5 0.05 120.10 20 642.51 39.741

86.585 -33.166 4.633 0 0.17 20 0.5 119.30 20 656.6 39.741

29.253 -37.533 -0.163 0 0.17 20 0.05 121.70 20 664.72 39.741

68.77 -1.734 0.508 0 0.35 5 0.5 110.40 20 637.27 14.916

18.362 -7.287 -0.863 0 0.35 5 0.05 108.50 20 662.03 14.916

67.114 -17.156 -0.613 0 0.35 20 0.5 113.40 20 677.4 14.916

24.759 -41.809 -1.631 0 0.35 20 0.05 122.40 20 668.88 14.916

76.661 -7.224 0.062 0 0.17 5 0.5 100.10 20 648.75 14.916

17.66 -9.145 0.252 0 0.17 5 0.05 111.90 20 645.26 14.916

74.339 -33.685 6.663 0 0.17 20 0.5 127.40 20 629.56 14.916

30.053 -51.001 2.022 0 0.17 20 0.05 129.10 20 656.19 14.916

91.426 3.766 -3.488 0 0.35 5 0.5 127.50 -15 649.75 14.916

18.745 -10.382 -1.62 0 0.35 5 0.05 132.80 -15 627.95 14.916

81.774 -26.218 -4.703 0 0.35 20 0.5 126.50 -15 637.41 14.916

27.881 -39.707 -2.968 0 0.35 20 0.05 107.10 -15 713.76 14.916

117.163 -7.061 -3.906 0 0.17 5 0.5 132.40 -15 625.26 14.916

20.172 -15 55 -1.356 0 0.17 5 0.05 126.30 -15 636.8 14.916

110.073 -46.683 0.186 0 0.17 20 0.5 116.70 -15 666.73 14.916

37.405 -57.424 2.582 0 0.17 20 0.05 108.20 -15 684.44 14.916

*FC* *FN* *FS* * _{V}*

**

_{F}*a*

_{P}*mc*

*T*

*D*

*v*

_{C}(N) (N) (N) (rad) (rad) (m) (mm) (%) (^{o}C) (kg·m^{-3}) (m·s^{-1})

1 2 3 4 5 6 7 8 9 10 11

———————————————————————————————————-

110.385 4.846 -1.991 0 0.35 5 0.5 122.20 -15 654.79 39.741

23.35 -12.145 -1.454 0 0.35 5 0.05 124.90 -15 648.21 39.741

107.669 -34.342 -4.858 0 0.35 20 0.5 122.30 -15 634.86 39.741

29.34 -42.13 -2.440 0 0.35 20 0.05 105.20 -15 691.15 39.741

23.951 -16.067 -1.75 0 0.17 5 0.05 123.30 -15 658.61 39.741

138.437 -51.609 1.045 0 0.17 20 0.5 124.00 -15 631.97 39.741

36.753 -51.756 -0.546 0 0.17 20 0.05 111.70 -15 678.74 39.741

71.442 -6.745 -2.384 0 0.35 5 0.5 8 -15 765 2 14.916

29.202 -20.582 -1.565 0 0.35 5 0.05 8 -15 652.97 14.916

64.708 -14.736 -3.808 0 0.35 20 0.5 8 -15 645.73 14.916

37.277 -43.628 -3.151 0 0.35 20 0.05 8 -15 660.22 14.916

125.402 -3.324 -5.49 0 0.17 5 0.5 8 -15 580.71 14.916

26.967 -10.355 -2.359 0 0.17 5 0.05 8 -15 552.13 14.916

130.217 -21.02 -2.901 0 0.17 20 0.5 8 -15 510.53 14.916

35.113 -56.749 -2.074 0 0.17 20 0.05 8 -15 526.9 14.916

90.5 1.708 -4.127 0 0.35 5 0.5 8 -15 543.14 39.741

83.634 -3.114 -5.556 0 0.35 5 0.5 8 -15 573.26 39.741

29.053 -14.159 -1.822 0 0.35 5 0.5 8 -15 533.95 39.741

57.9 -2.624 -3.617 0 0.35 5 0.5 8 20 588.36 14.916

25.431 -16.517 -1.873 0 0.35 5 0.05 8 20 575.08 14.916

60.576 -12.378 -4.302 0 0.35 20 0.5 8 20 571.52 14.916

34.628 -49.102 -2.967 0 0.35 20 0.05 8 20 592.12 14.916

136.962 -3.835 -5.27 0 0.17 5 0.5 8 20 581.25 14.916

27.863 -11.748 -1.352 0 0.17 5 0.05 8 20 647.94 14.916

159.789 -27.138 -2.121 0 0.17 20 0.5 8 20 627.61 14.916

50.999 -68.388 -2.382 0 0.17 20 0.05 8 20 623.65 14.916

89.44 5.633 -3.094 0 0.35 5 0.5 8 20 561.86 39.741

22.06 -5.72 -1.727 0 0.35 5 0.05 8 20 583.73 39.741

87.818 -7.962 -6.163 0 0.35 20 0.5 8 20 575.34 39.741

33.251 -41.88 -3.535 0 0.35 20 0.05 8 20 650.49 39.741

130.23 -4.862 -8.375 0 0.17 5 0.5 8 20 594.67 39.741

28.466 -14.582 -1.833 0 0.17 5 0.05 8 20 572.26 39.741

166.312 -41.098 -0.928 0 0.17 20 0.5 8 20 583.66 39.741

54.838 -89.148 -1.881 0 0.17 20 0.05 8 20 588.09 39.741

20.5 -4.3 -0.1 0 0.17 5 0.05 8 20 472.26 39.741

11.1 -0.4 1.2 0 0.35 5 0.05 8 20 472.26 39.741

8.1 0.3 1.2 0 0.52 5 0.05 8 20 472.33 39.741

0.97 -1.8 0.8 0 0.17