Prediction of cutting force is one of the most of import stairss toward accurate simulation of each machining procedure. Predicted cutting forces can be used to measure the needed power and torsion of machine tool, dimensional truth of workpiece, and vibratory features of the whole machining system. Besides, it can be applied to plan an appropriate construction for the machine tools to fulfill specific intents. Harmonizing to the mechanistic attack, cutting force constituents are maps of bit thickness, breadth of cut, and cutting invariables. Chip thickness and breadth of cut are geometric parametric quantities and they depend on geometry of cutter and geometry of battle between cutting tool and workpiece. In contrast, the film editing invariables are maps of tool and workpiece stuffs every bit good as tool geometry and cutting conditions ; hence each force theoretical account must be calibrated in order to place the film editing coefficients.

Two different standardization attacks for mechanistic force modeling have been reported in the earlier researches. The first attack is the incorporate method which relies on an by experimentation established extraneous film editing database. Budak, Altintas and Armarego ( 1996 ) presented a force theoretical account for the level terminal milling, in which the force coefficients were obtained utilizing extraneous to oblique transmutation. Earlier, Yang and Park ( 1991 ) developed a force theoretical account for ball-end milling. They obtained cutting coefficients from extraneous terminal turning. Altintas and Lee ( 1996 ) , Merdol and Altintas ( 2004 ) , and Engin and Altintas ( 2001 ) all evaluated the changing profligate face clash, force per unit area distribution, and the bit flow angle in peripheral milling in order to supply accurate cutting force anticipations. Larue and Anselmetti ( 2003 ) and Altintas ( 2000 ) have demonstrated that the milling force coefficients could be obtained from extraneous cutting trials with oblique cutting analysis and transmutation.

The 2nd attack is direct standardization method which determines the milling force coefficients straight from milling trials for the specific cutter-part combination. Kline and DeVor ( 1983 ) applied this attack to develop a mechanistic film editing force theoretical account for flat-end milling. Azeem, Feng and Wang ( 2004 ) employed a simplified attack and developed a stiff mechanistic cutting force theoretical account for the complicated ball-end milling procedure. With regard to the ball-end milling force theoretical accounts, Wang and Zheng ( 2002 ) presented a work based on break uping the elemental film editing forces into shearing and plowing constituents in ball-end milling. Meng et Al. ( 2004 ) conducted a series of face milling experiments to analyze the relationship of the derived film editing force coefficients with the relevant cutting parametric quantities from the mensural forces. Jayaram, Kapoor and DeVor ( 2001 ) estimated the cutting force coefficients for face milling applications with multiple inserts from the Fourier transform of the measured force signals at the nothing frequence interval. All of the research surveies referenced above require a big figure of standardization trial cuts to find the empirical film editing force coefficients. The norm measured forces were largely used in the standardization processes.

Shin and Waters ( 1998 ) proposed an effectual process to cut down the needed figure of experiments in finding the cutting force coefficients for face milling inserts utilizing the effectual bit thickness to generalise the effects of provender per tooth and axial deepness of cut. Yun and Cho ( 2000 ) determined the cutting force coefficients for level terminal milling based on the synchronism of one mention cutting trial with the measured force signals. The cutting mechanics parametric quantities were kept changeless and the size consequence in metal film editing was non considered.

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Due to complicated geometry of cutting border, direct standardization of the milling force theoretical accounts still requires a big figure of standardization cuts. A new attack is proposed in this paper to find the cutting force coefficients for milling procedures from merely a individual trial cut. Instantaneous cutting forces are used alternatively of mean forces to obtain the empirical force coefficients. This individual experiment significantly reduces the clip and attempt needed to find the coefficients in order to cover a broad scope of cutting conditions. In the following subdivision, a brief description of the geometric theoretical account for serrated tapering ball-end factory is presented followed by elaborate description of the standardization process and the experimental work. Finally the mensural forces from the confirmation trial cuts with different cutting conditions are applied to graduate cutting force coefficients and formalize the proposed method.

2 Geometric mold of serrated tapering ball-end factory

Prediction of the cutting force requires designation of instantaneous bit thickness and cutting border geometry. Taper ball-end Millss have a variable geometry along the contact length between the tool and workpiece. The geometry of the taper ball-end factory is shown in Figure 1. The tool has a cutter radius. The tool envelope surface consists of a hemisphere surface and a cone surface which is a map of the half-apex taper angle, as shown in figure 1.

( 1 )

In equation 1, , , and are the tool radius at the ball terminal portion, tool radius at taper portion, and fluctuation of radius due to serrations along the axis of the cutter, severally. and are defined as:

( 2 )

where, is the ball radius and is the radius of the ball portion at lift.

Figure 1 Geometry of serrated tapering ball-end factory: ( a ) Definition of the geometric characteristics and ( B ) The 3-axis tool co-ordinate system

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( a )

( B )

In figure 1 ( B ) , and are axial and radial deepnesss of cut. Coordinates of a point along the film editing border, cutting force constituents moving on point and geometry of tool tip similar to that of Altintas and Lee ( 1995, 1996 ) , Merdol and Altintas ( 2004 ) , and Ehmann et Al. ( 1997 ) are presented in figure 2. In figure 2, represents the tangent way to the first flute at the tool tip, and is perpendicular to. The cutter rotary motion angle is measured clockwise between axis and the way of. For a film editing border point with distance from the tool tip, the slowdown angle is defined as the angle between and the way of.

Figure 2 Coordinates of Point along the film editing: ( a ) Isometric position of point and cutting force constituents moving on point and ( B ) Tool tip geometry

Calciferol: Desktop2010 International Journal of Manufacturing ResearchFinal SubmissionFigure 2 ( a ) .tif

Calciferol: Desktop2010 International Journal of Manufacturing ResearchFinal SubmissionFigure 2 ( B ) .tif

( a )

( B )

The slowdown angle is determined as:

( 3 )

It should be noted that in a multiple flute ball-end factory, some of the flutes may non be uninterrupted around the tool tip due to the sharpening procedure. As shown in Figure 2 ( B ) , for a ball-end factory tool with five flutes, the starting point for some of the flutes has a radial divergence from the tool tip. After specifying the geometry of cutting border, the angular place of a film editing border point on the flute at tallness, measured from axis in clockwise way as shown in Figure ( 2b ) , is given by:

( 4 )

where, is the figure of flutes, is flute index and is obtained from Equation 3. The local co-ordinates of the film editing border point can so be determined by:

( 5 )

More specific geometric definition can be found in Altintas and Lee ( 1995, 1996 ) , Merdol and Altintas ( 2004 ) , and Ehman et Al. ( 1997 ) .

3 Modeling of instantaneous untrimmed bit thickness

The bit burden theoretical account determines the instantaneous undeformed bit thickness distribution along the film editing borders. Figure 3 demonstrates the patterned advance of the undeformed bit thickness for a general three-axis milling state of affairs. The untrimmed bit thickness is defined as the distance between the way generated by the current intersecting tooth and the open workpiece surface generated by the antecedently go throughing tooth. In three dimensional machining, as depicted in Figure 1 ( B ) and 2 ( a ) , represents the current tool place and orientation, represents the old tool place and orientation ( before one provender per tooth ) , and represents the elemental film editing border place determined from equation 5. The undeformed bit thickness is the distance between and.

( 6 )

Figure 3 Illustration of current and old tool place to make an instantaneous undeformed bit thickness

Calciferol: DesktopNew folderFigure 3.tif

Since the procedure mistakes ( run-out ) , flute divergence, and flute chipping/breakage could change the undeformed bit thickness in terminal milling, the equations developed by Kline and DeVor ( 1983 ) , and Sutherland and DeVor ( 1986 ) are extended here. Assuming that represents the parallel axis offset run-out, is the turn uping angle for, is the axis joust angle, and is the turn uping angle for, the equation for finding the undeformed bit thickness under the presence of procedure mistakes is:

( 7 )

where the angle is given by:

( 8 )

and the effectual radius for each phonograph record is defined as:

( 9 )

where, is the nominal radius of the cutter, is the slowdown angle determined from Equation 3, is effectual length of the cutter and is the single flute divergence. After the undeformed bit thickness is determined for each occupied elemental film editing border, the differential film editing forces can be computed by presuming an oblique film editing procedure.

4 Formulation of a mechanistic film editing force theoretical account

Using three different positions of the 3-axis co-ordinate system of, , and every bit illustrated in Figure 1, a taper ball-end milling cutter can be divided into a finite figure of disc elements along the z-axis. The entire force constituents along, , and moving on a flute at a peculiar blink of an eye are obtained by numerically incorporating the force constituents moving on each single phonograph record component. The entire or attendant force moving on the cutter/workpiece interface at any given clip during machining is determined by summing the independent forces of all the flutes engaged in cutting at any clip blink of an eye.

The cutting force moving on the profligate surface of a disc component is divided into two extraneous constituents: the normal force per unit area force, and the frictional force. These can be obtained utilizing the undermentioned equations:

( 10 )

( 11 )

where, is the differential normal cutting force and is the differential clash force. is perpendicular to the profligate face and is along the way of bit flow. The specific normal cutting coefficient is, while represents the clash coefficient. The parametric quantities and are by experimentation determined and are chiefly dependent on the local untrimmed bit thickness, cutting speed and normal profligate angle, harmonizing to the work of Yucesan and Altintas ( 1996 ) and Chandrasekharan, Kapoor and DeVor ( 1997 ) . The untrimmed bit thickness for each occupied elemental film editing border is mentioned in Equation 6. The elemental oblique cutting procedure has a cutting breadth of. The cutting speed is given by where is the angular speed of spindle rotary motion. In mechanistic mold attack, the differential film editing forces on the profligate face are assumed to be relative to the bit burden. The elemental film editing forces on the profligate face may be transformed to the local tool co-ordinate system to obtain the elemental tangential, radial, and axial forces, , and ( the waies of these forces are shown in Figure ( 2a ) ) severally. The transmutation is given by Kapoor et Al. ( 1998 ) :

( 12 )

where is inclination angle equal to the local spiral angle, is normal profligate angle, and is bit flow angle that can be related to the local spiral angle as:

( 13 )

In equation 13, is the bit flow coefficient which can be determined from the experimental information. The differential film editing forces on the planetary workpiece co-ordinate system are so determined by rotary motions through:

( 14 )

where the orientation is a map of the distance and angle to indicate followed by and which are minimal and maximal bounds of bit thickness country indicated in one full rotary motion, and is transformation matrix.

( 15 )

The entire film editing forces are therefore the summing up of differential film editing forces for all those engaged cutting border elements. The above mentioned force theoretical account can be used to foretell the instantaneous cutting force along the axial deepness of cut and angle of tool rotary motion. The proposed attack is based on discretizing the cutting tool along the axial deepness of cut and for each 1 grade of tool rotary motion. The tool/workpiece instantaneous contact geometry is foremost determined, and so the instantaneous cutting force for each grade of rotary motion will be calculated as a map of force coefficients. The mensural film editing force constituents are so correlated to the deliberate force constituents to foretell the force coefficients.

5 Calibration attack

The cutting force coefficients that are used in equations 10 and 11 for taper ball-end milling are viz. and. These coefficients need to be determined in order to cipher the film editing forces utilizing the developed cutting force theoretical account. Direct standardization of this mechanistic cutting force theoretical account was based on mean forces and a big figure of experiments for accurate theoretical account standardization based on El-Mounayri et Al. ( 1997 ) , Zhu, Kapoor and Devor ( 2001 ) , and Azeem, Feng and Wang ( 2004 ) . The two sphere representations were employed in which the film editing border elements on the cylindrical co-ordinates were geometrically the same. These besides exhibited unvarying film editing features, while those on the ball-end subdivision were geometrically different from each other and this resulted in changing cutting features. The size consequence parametric quantities were ever assumed to be changeless for the specific cutter-part combination. Large figure of trials with different provender, velocity, and rake angle are needed to execute in this instance. The present work purposes at cut downing the big figure of experiments required for accurate theoretical account standardization of the taper ball-end milling procedure. Merely a few experiments with different emerging ratios and axial deepness of cuts are needed to find the cutting force coefficients. Those predicted coefficients are valid over a broad scope of cutting conditions. The cutting force coefficients and are calculated as lumped discrete values along the cutter axis whereas the size consequence parametric quantities can be calculated as invariables.

Instantaneous film editing forces are to be measured and used for the numerical standardization process. Since the instantaneous cutting force is used, the digressive and radial cutting force coefficients can non be decoupled. Normally, old standardization methods ( Azeem, Feng and Wang, 2004 ) , include the instantaneous consequence of the tool rotational velocity, one film editing border is engaged in cutting at any peculiar cutter orientation during the half slot cut. In the present work a cutting border ab initio engages the work stuff from the cutter free terminal and as the cutter rotates, the upper part of the film editing border is bit by bit engaged until it reaches the tapering subdivision, i.e. , the selected axial deepness of cut. Once the full deepness is engaged with the cutting border, the battle continues until the cutting border bit by bit starts withdrawing from the workpiece. Similar to the gradual battle stage, the detachment stage besides starts from the cutter free terminal. The specific cutter orientations for the battle and detachment stages depend on the cutter spiral angle and the associated slowdown angles. The lumped distinct values of the cutting mechanics parametric quantities are determined by spliting the cutting border along the axial deepness of cut into a finite figure of phonograph record. These phonograph records are used to cipher parametric quantities such as the slowdown angle for the film editing border section of each matching phonograph record. Lag angle computations are based on the distance of each phonograph record from the cutter free terminal or non-engaged tooth, the entire figure of flutes, and the spiral angle of the tool. This angle gives the cutter orientation at which the film editing border section is first wholly engaged with the workpiece. The first phonograph record at the cutter free terminal starts prosecuting the workpiece and the staying phonograph record are bit by bit engaged until all the phonograph record are engaged at the same time. In the present work, the tool was separated into distinct sections that have a tallness of 0.01 times axial deepness of cut. Each section breadth is divided into little cutting elements with an angle of 1 grade. The elemental film editing forces are calculated for all the cutting elements of each phonograph record and the amount of these elemental forces represents the entire force moving on the corresponding section. Using equations 10 and 11, the elemental digressive and radial film editing forces for the first cutting component on the first phonograph record can be expressed as:

( 16 )

( 17 )

where and are the lumped distinct values of and for each angle, and is the undeformed bit thickness at this blink of an eye at the cutter orientation angle. The elemental digressive and radial cutting force constituents are resolved into the, , and workpiece planetary waies as in equation 14 for each angle. The cutting force coefficients and are taken as changeless lumped parametric quantity values for each angle of rotary motion but vary from one angle to another. The, , and cutting forces moving on the cutter can be formulated for each cutter orientation angle when the matching cutting border section is first wholly engaged with the workpiece. Comparing these force looks with the measured instantaneous cutting force informations, the ensuing force equation can be expressed in a matrix signifier as:

( 18 )

where, , , and are the mensural instantaneous cutting forces in the, , and waies at the specific cutter orientation angles matching to the first complete battle of each section. The decoupling of the lumped distinct values of and for each section is clearly shown by the matrix in equation 18. This greatly facilitates the numerical process for best adjustment of the cutting force coefficients and enables them to be accurately determined from the instantaneous cutting force informations. The present direct standardization attack uses instantaneous cutting forces to find the empirical film editing force coefficients. Accurate representation of the coiling film editing border profile is indispensable when covering with the instantaneous forces. The cutting border profile on the taper cylindrical portion of a ball-end factory is non the same as that on a level terminal factory and it can be arbitrary along the whole border get downing from the ball tip, and therefore, varies from cutter to cutter. A typical design is to project the coiling film editing borders extended from the taper portion onto the hemisphere of the ball portion. For a taper ball-end factory with a changeless spiral angle, the slowdown angle at a distance from the cutter free terminal can be expressed as in equation 3. In add-on, the instantaneous bit flow angle at any rotational angle can be determined from the ratios of the mensural forces for an arbitrary design of the cutting border profile. A coordinate measuring machine ( CMM ) can be used to follow and set up the film editing border profile. The associated process includes mensurating the co-ordinates of cutting border points at unvarying intervals along the cutter axis.

6 Experiments

6-1 Cutting tool and workpiece stuff

The standardization trials were performed utilizing a three-axis horizontal CNC milling machining centre. The workpiece stuff of Titanium ( Ti4Al6V ) and a right-handed five-flute carbide taper ball-end factory of diameter 12.7mm were chosen for the experiments. The taper ball-end factory has a taper angle of 5 grades, normal rake angle of 9 grades on the tapering subdivision and a normal profligate angle of 2 grades on the ball portion of cutting border. The spiral angle on the cylindrical portion of cutting border is changeless and equal to 26 grades. Figures ( 4a ) and ( 4b ) show the SEM image of tool tip and serrated cutting border. The images were obtained by a JEOL-6400 scanning negatron microscope ( SEM ) .

Figure 4 ( a ) SEM image of ball-end factory tool tip and ( B ) cutting border geometry

Calciferol: DesktopNew folderFigure 4 ( a ) .tif

Calciferol: DesktopNew folderFigure 4 ( B ) .tif

( a )

( B )

As it can be seen from figure 4 ( a ) , four flutes are non uninterrupted near the tool tip. The distances between their starting points and the tool tip which are presented by, , and can be found in table 1.

Table 1 Measured distances for ball-end tool tip geometry


Distances ( millimeter )





The tallness profile was obtained with a JEOL-733 negatron microprobe. Figure 5 shows the obtained profile for cutting border at the tapering portion of the tool. This profile was used to foretell existent country of contact when ciphering the instantaneous cutting force.

Figure 5 Graphical representation of saw-toothed profile of a subdivision of the cutter

6-2- Cutting Conditionss

Multiple trials were conducted with coolant, at spindle velocities runing from 150-700 revolutions per minute. A table provender rate of 0.0254-0.127 mm/tooth was employed to cut down the machine quiver. The instantaneous film editing forces in the, , and waies were measured with a Kistler 9255B ergometer at a trying rate of 707 samples per second to accomplish an norm of 78 informations points per revolution for a lower limit of 15 complete cutter rotary motion rhythms of cutting force informations.

Figure 6 Machining trial with changing parametric quantity alterations during a individual base on balls

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Multiple trials were designed ; Figure 6 represents one of these trials during the design stage. This trial encompasses four different sections, each with a altering variable. The first being an increasing axial deepness of cut, phase two and three holding a changing provender rate and spindle velocity, severally, while in the 4th, an increasing radial deepness of cut is sought after. Previously presented by Merdol and Altintas ( 2004 ) , a serrated undulated profile on cutters resulted in a more stable roughing operation to avoid yak quivers that cause tools to neglect under changing loading conditions. The proposed theoretical account by Merdol and Altintas ( 2004 ) has been integrated to our comprehensive simulation plan and validated by experimentation in milling Titanium utilizing the cutting parametric quantities listed in table 2.

Table 2 Cutting conditions for measure cut trials at an ADOC of 10.5 millimeters and a RDOC of 2 millimeter.

Test No.

Feed ( mm/tooth )

Spindle Speed ( revolutions per minute )


0.0254- 0.0508- 0.0762- 0.1016- 0.127


Sections of informations were taken from figure 7, in intervals of the beginning, center and terminal of each provender rate so combined together to accomplish an accurate norm for each subdivision of the secret plan.

Figure 7 Measured force constituents and over a scope of provender rates ( 0.0254-0.127 mm/tooth ) at changeless spindle velocity of 150 revolutions per minute.

In figure 7 the norms have been mathematically filtered utilizing MATLAB to extinguish any frequence greater than that of the machining frequence norm of 45 Hz. This reduces the effects of any exterior frequences that would perchance change the recorded signal. Once this information is separated and filtered, the information is put into a text file and used as an input file for the standardization theoretical account. The tool geometry is besides entered into a separate file and set into the standardization theoretical account. The theoretical account consequences are shown in table 3.

Table 3 Consequences of theoretical account standardization







The top row of in table 3 shows the highest and lowest sums of bit thickness, highest and lowest speeds, and highest and lowest profligate angles. These parametric quantities are varied along the film editing border due to taper angel and serration. Variation of untrimmed ( undeformed ) bit thickness in one rotary motion of cutting tool for two different spindle velocities is illustrated in figure 8.

Figure 8 Instantaneous untrimmed bit thickness in a complete rotary motion for two different spindle velocity: ( a ) 150 revolutions per minute and ( B ) 700 revolutions per minute

( a )

( B )

Alone tool geometry made from different flute grindings normally create cutter run-out and as a consequence, alone profiles. The surface generated by the flutes at each clip blink of an eye is calculated and so subtracted from the antecedently cut surface in the radial way where the bit thickness is defined. As a consequence, the kinematics theoretical account allows the accurate designation of bit thickness and surface coating generated. The simulation theoretical account considers the true kinematics of milling as antecedently presented by Engin and Altintas ( 2001 ) and Merdol and Altintas ( 2004 ) .

Cuting border force coefficients for the digressive and normal waies are identified, while the axial way for coefficients are known to be really little in oblique film editing and can be taken as nothing. At some locations and frequently at larger radial deepness of cuts, teeth portion the entire sum of bit to be removed in one revolution of the cutter. However, at frequently much lower radial deepness of cuts, merely one tooth removes stuff while others do non even touch the workpiece. In the presented instances in figure 8 ( a ) and 8 ( B ) , it can be clearly seen that all dentitions are in contact with the workpiece and each flute has a different run-out. Comparing Figure 8 ( a ) with Figure 8 ( B ) , the jumping velocities of 150-700 have no important impact on untrimmed bit thickness. Assuming an mean value for the cutting force coefficients and suiting them to experimental informations utilizing a additive arrested development analysis, each coefficient is expressed as an exponentially disintegrating map of bit thickness and cutting velocity. The coefficients holding changeless provender per tooth are so combined and averaged for the velocities of 150, 300, 540, and 700 revolutions per minute, severally. Figures 9 and 10 show the consequences of the standardization process for cutting coefficients. From the standardization informations, the disintegrating tendency of mean coefficients is evident with increasing speed ( m/min ) . This work shows a consistent correlativity with the work of Bailey et Al. ( 2002 ) . Figures 9 and 10 demonstrate the tendencies of and for two different provenders ( 0.0254 mm/tooth and 0.127 mm/tooth ) severally.

Figure 9 A diminishing tendency of and coefficients at a provender of 0.0254 mm/tooth while cutting velocity additions

Figure 10 A diminishing tendency of and coefficients at a provender of 0.127 mm/tooth while cutting velocity additions.

At lower provenders ( figure 9 ) the fluctuation in the normal cutting force coefficient is singular. Poor machinability of Ti and big axial deepness of cuts are proven to bring forth smaller bit tonss and this is frequently the cause of this fluctuation. Comparing figures 9 and 10, there is greater truth in the profile of informations when higher provender ( 0.127 mm/tooth ) is applied. A more distinguishable decay of force coefficients with increasing in speed can be seen in Figure 10.

As it has been illustrated in figures 11 and 12, and are extremely dependent of the addition and lessening of spindle velocities. Coefficients and lessening quickly by increasing spindle velocity.

Figure 11 Consequence of bit thickness on at different velocities ( feed = 0.127 mm/tooth )

Figure 12 Consequence of bit thickness on at different velocities ( feed = 0.127 mm/tooth )

By and large, the coefficients remain changeless for one revolution of cutter. The cutting status is the major factor impacting the untrimmed bit thickness, but the film editing coefficients do non change harmonizing to the cutting status. Effectss of the cutting status have to be reflected in untrimmed bit thickness and non in cutting force coefficients. In figure 13, nevertheless, it can be seen that departs somewhat from that changeless degree in several parts. This represents a size consequence, whereby specific film editing forces become big at a low untrimmed bit thickness ( Armarego and Brown, 1969 ) . Figure 13 besides shows that becomes big at a low untrimmed bit thickness, which once more shows the size consequence. To avoid complexness, size effects were non considered in this survey.

Figure 13 Instantaneous untrimmed bit thickness, and values at 150 revolutions per minute

As stated by Yun and Cho ( 2000 ) , the disagreements are caused by size effects that occur at low untrimmed bit thicknesses. At the extremum parts, it is clear that the predicted forces are somewhat larger than those measured, since tool wear was non considered. The mensural extremum values for all the forces are little, owing to the consequence of tool wear.

7 Decisions

A new method of gauging the cutting force coefficients from a individual cutting trial was presented. Instantaneous cutting forces are to be used alternatively of mean forces to graduate the empirical force coefficients. The method involves the computation of untrimmed bit thickness and synchronism of the cutting coefficient forces. The effectivity of the proposed method was verified by comparing the estimated film editing coefficient forces with the theoretical account calibrated values. The simulation theoretical account presented in this paper was reformulated so that the cutting force coefficients account for the effects of provender rate, cutting velocity, and a complex film editing border design. Experimental consequences were presented for the standardization process. This method predicts 3-dimensional cutting force coefficient constituents with the same truth as the bing methods.


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