When the Radial Force and the Feed Force Are Approximately Equal

Digital Control Systems Implementation Techniques

Allan D. Spence , Yusuf Altintas , in Control and Dynamic Systems, 1995

2 MAXIMUM RESULTANT FORCE

An excessive resultant cutting force F  =   (F ² _x   + F ² _y )^1/2 must be avoided to prevent end mill shank breakage. This type of tool failure most frequently occurs when the cutting depth w is large and the cutter is small in diameter.

For this constraint, as noted in Section IV, B, the geometric portion of Eq. (20) can be solved independently, and the process model parameters K_T and K_R substituted just prior to actual machining. For a threshold value F =   1.2 kN, Fig. 15 shows the scheduled feed rate and experimentally measured maximum resultant force. The jitter in the force signal is again due to a small eccentricity of the cutter relative to the spindle axis (radial runout). The desired limiting of the resultant force was achieved.

Fig. 15. Maximum resultant force limiting. Cutter path A in Fig. 13 was used. (a) Scheduled feed rate. The plot above shows the solid modeler feed rate schedule for a maximum resultant force threshold F =   1.2 kN (thinner line) The feed rate measured during the cutting test is also included (thicker line). (b) Measured maximum resultant force. The plot above shows the maximum resultant force achieved using the feed rate schedule from (a). The jitter in the force signal is caused by radial runout.

In the next section, the use of adaptive control techniques as an alternative method of cutting force control is explored.

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Mechanics of Cutting

Stewart C. Black BSc, MSc, CEng, FIEE, FIMechE , ... S.J. Martin CEng, FIMech, FIProdE , in Principles of Engineering Manufacture (Third Edition), 1996

9.2 Cutting force analysis

The range of cutting forces acting upon a cutting tool has already been studied and it is now necessary to evaluate the effect of these upon the machine tool, and to analyse the cutting process in some detail (Figure 9.1). The vertical cutting force, F_c , (sometimes called the tangential force) is the one that does all the work, since it is operating with a high cutting velocity, V_c. The cutting power is the product of V_c × F_c. The other two forces acting are the radial force, F_r , and the axial force, F_a. The radial force is produced by the approach angle of the tool, and is the force needed to hold the tool against the workpiece. It usually has zero velocity and thus zero power. The axial force, F_a , has a very low velocity, and thus a very small power consumption.

Figure 9.1. Cutting force analysis.

Note: This discussion of force directions is based upon the axial turning of a shaft. The directions in which these forces act would be changed if the machine was cutting in a different direction, e.g. by 90° if the tool was facing off the end of a workpiece.

These three forces can be resolved to determine the total resultant force acting, F. This can be very large and can easily cause significant deformation of the workpiece, thus finishing cuts are usually taken with small forces to minimise the forces acting, and to provide accurate cutting conditions.

For most workpiece materials higher cutting speeds mean lower cutting forces, the higher temperatures on the flow zone and reduced contact area contribute towards this effect. The decrease in force varies with the type and condition of the material and the range of cutting speeds in use. For a heat resistant nickel-based alloy steel the initial chip forming force can be ten times larger than that required to cut unalloyed aluminium.The main cutting force F can be split up into three components

Specific cutting pressure

A very convenient way has been developed for estimating the size of the cutting force. A property known as the specific cutting pressure, K_c , has been defined as the vertical cutting force, F_c , divided by the area of the cut being taken, i.e.:

(9.1) $K_c=F_c/A=\frac{F_c}{d\times f}$

Figure 9.2 shows that the specific cutting pressure (K_c ) and the undeformed chip thickness, varies with the type of material (A) stainless steel, (B) alloyed steel and (C) grey cast iron. The pressure depends upon the shear yield strength of the workpiece material and the area of the shear plane. This area varies considerably and with it the cutting pressure. It is thought to be more influential than the yield strength of the material, which, in fact, does not vary that much for the cutting process. Alloying and heat treatment, however, increase the yield strength.

Figure 9.2. Specific cutting pressure variation with materials.

It should be apparent that this is not actually a true stress value, in fact, it is usually three or four times the yield strength of the material, since the actual contact area of the chip on the tool is never known precisely. Since the actual contact area varies, the assumption is made that the chip only contacts the tool over the area of the cut.

The previous discussions of the complex nature of metal cutting show the range of factors interacting as the metal flows over the tool face, and tests show that K_c reduces with increasing feed rates, increasing top rake angles, and with increasing cutting speeds.

Specific cutting pressure is important when it comes to power (P) calculation for any metal cutting process (Figure 9.3).

Figure 9.3. Specific cutting pressure/feed per revolution.

The value of the specific cutting pressure is available for various materials (Table 9.2), enabling the calculation of workpiece material removed per power unit. It is also a measurement of the machinability of materials. The value is valid for a material under certain conditions and cutting data. For instance, the value will vary with the cutting speed: a higher speed, leading to higher cutting temperatures, generally leads to a lower value, (A). Also the geometry of the cutting edge is influential, in that a positive rake angle leads to a smaller value than a negative one.

Table 9.2. Specific cutting force (K_c ) values for a range of common materials.

Material	Hardness HB	Condition	K c0.4 (N/mm 2)
Unalloyed	110	C < 0.25%	2200
steels	150	C < 0.8%	2600
	310	C < 1.4%	3000
Low alloy	124–225	Non-hardened	2500
steels	220–420	Hardened	3300
High alloy	150–300	Annealed	3000
steels	250–350	Hardened tool steel	4500
Extra hard steel	>450	Hardened and tempered	4500
Malleable	110–145	Short chipping	1200
cast iron	200–230	Long chipping	1300
Grey	180	Low tensile	1300
cast iron	260	High tensile, alloyed	1500
Nodular cast iron	160	Ferritic	1200
SG-iron	250	Pearlitic	2100
Steel	150	Unalloyed	2200
castings	150–250	Low alloy	2500
	160–200	High alloy	3000
Stainless steels	150–270	Ferritic, martensitic 13–25% Cr	2800
	150–275	Austenitic Ni >8%, 18–25% Cr	2450
	275–425	Quenched and tempered, martensitic >0.12% C	2800
	150–450	Precipitation hardened steels	3500
Heat resistant	180–230	Annealed or solution treated	3700
super alloys	250–320	Aged or solution treated and aged	3900
Fe based	950 MPa	α, near α and β alloys in	1675
titanium alloys		annealed condition
	1050 MPa	α + β alloys in aged condition, β alloys in annealed or aged condition	1690
Aluminium	38–80	Wrought and cold drawn	800
alloys	75–150	Wrought and solution treated and aged	800
	40–100	Cast	900
	70–125	Cast, solution treated and aged	900
	80	Unalloyed, A1 ≥99%	400
Aluminium with		10–14% SI	900
high SI content		14–16% SI	1500

The specific cutting pressure is closely related to the size of the undeformed chip thickness/feed rate. An increase of f leads to a reduction of K_c. This means, that the smaller the chip cross-section used in a process, the higher the specific cutting pressure – and the more unit power needed.

It also leads to the recommendation that feed rates are maximised in the metal cutting process.

It is convenient to resolve the resultant cutting force F acting on a tool into three vectors (Figure 9.4):

Figure 9.4. Cutting force components.

1.

The radial force F_r.

2.

The vertical force F_c.

3.

The axial force F_a.

The radial cutting force component (F_r ) is directed at right angles to the tangential force from the cutting point.

The axial cutting force (F_a ) is directed along the feed of the tool, axially along the direction of machining of the component. It is an important force factor in drilling operations. The cutting ability of the drill geometry will considerably influence the size of the force needed and as a rule the axial feed force requirement rises with the diameter of the drill.

Geometry, especially the entering angle, will determine the size of the two force components. Their relationship becomes especially important when deflection of tool with large overhang or a slender workpiece is a factor as regards accuracy and vibration tendencies. The rake angle also influences the size of the radial cutting force component. Positive rake angles, of course, also mean lower cutting forces in general.

For most workpiece materials, increasing cutting speed leads to lower cutting forces. The higher temperature in the flow-zone and reduced contact area contribute towards this effect. The decrease in force varies with the type and condition of material and the range of cutting speed in question, (Figure 9.5).

Figure 9.5. Reduction of cutting force with increasing cutting velocity.

As may be expected, the size relationship between the force components varies considerably with the type of machining operation. The tangential force often dominates in milling and turning operations, especially to do with power requirements. The radial force is of particular interest in boring operations and the axial, feed force in drilling. The size of the radial cutting force is dependent upon the entering angle used and the nose radius. A 90° entering angle and small nose radius will minimise the radial cutting force component, which strives to deflect the tool and gives rise to vibrations.

All three components increase in size with increasing chip cross-section, the tangential one most of all. For rough turning, a typical relationship might be for F_c: F_r: F_a , 4:2:1. The tangential cutting force is twice as large as the radial and four times that of the axial force. In drilling the relationship would be quite different and highly dependent upon the feed rate.

Vibration tendency is one consequence of the cutting forces. As well as tool or workpiece deflection, these can be affected by variations in the cutting process such as varying working allowance or material conditions as well as the formation of built-up edges.

Along with the importance of the design of the cutting geometry, to provide smooth chip breaking, and the use of a positive rake angle (Figure 9.6), higher cutting speeds generally have a favourable influence on the cutting forces/vibrations.

Figure 9.6. Reduction of cutting force with increasing top rake angle.

It is very important to achieve stability of the complete system, that is formed by the factors in the machining process. The quality of the toolholder and its ability to securely hold the indexable insert is one of the more important factors.

Basic cutting equations

The cutting speed V_c is the relative velocity of the workpiece and tool at the tool edge. For any point on the tool edge it can be considered as a vector, and the cutting force F_c is measured along the line of this vector. The elementary relationships should be noted:

1.

Energy/min consumed in cutting = F_CV_C Nm/min.

2.

Power consumed in cutting = F_cV_c/60 watts.

3.

Metal removal rate, $w=df\frac{V_c}{60}{\text{mm}}^3/\text{s}$ .

The power required to remove a unit volume of material per minute is a measure of its resistance to cutting; it also gives an indication of the effectiveness of the cutting conditions. Typical values for common materials enable estimates of cutting power and cutting force to be made for specified metal removal rates. These values are of use to machine tool designers and production planning engineers.

4.

(9.2) $\frac{\text{Power}}{w}=\frac{F_c\times V_c}{60}\frac{60}{df{V}_c\times {10}^3},\text{i,e}\text{.}\propto \frac{F_c}{df}$

Hence the specific power consumption is independent of cutting speed, except in so far as the cutting force changes with the cutting speed.

Units of specific power consumption are W/mm³/s, which is equivalent to J/mm³. Some authorities quote values of the specific energy required for cutting, i.e. the energy required to remove a unit volume of material. However, at the practical level, machine tool users will tend to think in terms of the power available at the spindle of the machine. It should be recognised that for a specific energy of, say, 2 J/mm³ it will require 2 W to remove metal at a rate of 1 mm³/s.

Measurement of the input to the electric motor of a machine tool provides a convenient method of measuring the power consumed in cutting. If the value of the power supplied when the machine is running idle is subtracted from the power reading taken under the cutting load, a reasonable estimate of the power consumed in cutting is obtained. The method is an approximate one, because the efficiency of the drive under varying loads is not taken into account. Watt-meters are commonly used to measure the power, but some machines will still have motors rated in hp (746 watts = 1 hp).

Example 9.1

A lathe is running at 1000 rpm machining a bar of steel which is 100 mm outside diameter. The cutting force applied by the tool to the work is 700 newtons.

(a)

What is the cutting velocity when the tool is starting to cut the bar at a diameter of 100 mm?

(b)

What is the cutting velocity when the tool has reduced the bar to a diameter of 50 mm?

(c)

What power is being used at these two diameters?

Solution

(a)

Bar diameter = 0.1 m

∴ Bar circumference = π × 0.1 = 0.3412 m

Spindle runs at $1000\text{rpm}\text{=}\frac{1000}{60}\text{revs/sec}\text{=16}\text{.66}\text{revs/sec}$

∴ Cutting velocity = 16.66 × 0.3413 m/sec = 5.236 m/sec

(b)

Bar diameter = 0.05 m

∴ Bar circumference = π × 0.05 = 0.1706 m

Same spindle speed, 16.66 revs/sec

This gives a velocity of 16.66 × 0.1706 = 2.843 m/sec

i.e. half as much (since diameter of bar is now halved).

(c)

Power = force × velocity

(i)

at 100 mm diameter power = 5.236 m/sec × 700 newtons = 3665 watts

(ii)

at 50 mm diameter power = 2.843 m/sec × 700 newtons = 1832.6 watts

Example 9.2

The power required to turn a medium-carbon steel is approximately 3.8 W/mm³/s.

If the maximum power available at the machine spindle is 5 hp, find the maximum metal removal rate. Also find, for a cutting speed of 36 m/min and feed rate of 0.25 mm/rev, the depth of cut and the cutting force which will occur when the metal removal rate is at the maximum value.

Solution

$\begin{array}{lll}{w}_{\max }& =& \frac{5\times 746}{3.8}=982{\min }^3/\text{s}\\ \hfill F_c& =& \frac{5\times 746\times 60}{36}=6216\text{N}\\ \hfill d& \text{=}& \frac{60\times 982}{{10}^3\times 36\times 0.25}=6.55\text{mm}\end{array}$

Example 9.3

A lathe running idle consumes 325 W. When cutting an alloy steel at 24.5 m/min the power input rises to 2580 W. Find the cutting force and torque at the spindle when running at 124 rev/min. If the depth of cut is 3.8 mm and the feed 0.2 mm/rev, find the specific power consumption.

Solution

$\begin{array}{lll}\text{Power}\text{consumed}\text{in}\text{cutting}& =& \mathrm{2580-325}=2255\text{W}\\ \hfill w& =& 0.2\times 3.8\times 24.5\times \frac{{10}^3}{60}=310{\text{mm}}^3/\text{s}\\ \hfill \text{Specific}\text{power}\text{consumption}& \text{=}& \frac{2255}{310}=7.27{\text{W/mm}}^3/\text{s}\\ \hfill \text{Torque}\text{at}\text{the}\text{spindle}& \text{=}& \frac{2255\times 60}{2\pi \times 124}=174\text{Nm}\\ \hfill \text{Cutting}\text{force}\left(T\right)& =& \frac{2255\times 60}{24.5}=5522\text{N}\end{array}$

Example 9.4

The given data relates to the rough turning of an alloy steel (SAE 3140) heat treated to 285 HB and having a UTS of 986 MN/m².

Depth of cut 6.4 mm, approach angle (k) 60°, nose radius 6.4 mm, side rake angle 14°, back rake angle 8°.

Cutting speed (Vc ) m/min	55.5	35.4	22	14.3	9.5	6.4
Feed (f) mm/rev	0.05	0.10	0.20	0.40	0.80	1.60
kW consumed in cutting	1.27	1.42	1.50	1.64	1.87	2.16

Draw a graph to show the variation of specific cutting pressure with feed. Give the average value of the specific power consumption for cutting this material.

Solution

Feed (f)	0.05	0.10	0.20	0.40	0.80	1.60
Metal removal rate (w)	296	378	470	610	810	1093
$\frac{\text{Power}\left(\text{W}\right)}{w}$	4.29	3.76	3.19	2.69	2.31	1.98
kc N/mm2	4290	3760	3190	2690	2310	1980

Note $K_c=\frac{{10}^3\times \text{power}\left(\text{W}\right)}{w}$ from equation 9.1

Average value of specific power consumption = 3.04 W/mm³/s.

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Advances in conventional machining processes for machinability enhancement of difficult-to-machine materials

Ashwin Polishetty , Guy Littlefair , in Advanced Machining and Finishing, 2021

3.1.1.2 Cutting force analysis

Various components of forces recorded during machining of SLM AP, HT and wrought Ti6Al4V from the start to finish are shown in Fig. 1.19.

Figure 1.19. Cutting force Fx, Fy, and Fz variations for SLM AP, HT and wrought Ti6Al4V.

The cutting forces acting on the tool in the X, Y, and Z directions are also known as resultant cutting force (Fz), thrust force (Fy), and the feed force (Fx). Resultant cutting force (Fz) and feed force (Fx) components are used to characterize the machinability of the materials. From Fig. 1.19, it is evident that SLM Ti6Al4V AP and HT require higher force during the cutting process when compared to its wrought counterpart. The probable reason can due to the high hardness and strength of the SLM Ti6Al4V. Residual stresses induced due to frequent thermal load fluctuations during the printing process are also a factor contributing to high cutting force for SLM Ti6l4V AP. It widely accepted in the research community that the contrasting cooling rates for an SLM and a cast/wrought component make a big difference in material processing. A direct proportional relationship was evident between the cutting force and speeds for SLM Ti6l4V AP and HT.

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Turning

Viktor P. Astakhov , in Modern Machining Technology, 2011

1.4.1 Cutting force and its components

While cutting, the tool applies a certain force to the layer being removed, and thus to the workpiece. This force, known as the resultant cutting force R, is a 3D vector considered in the machine reference system (Standard ISO 841) set out in Figure 1.6(a). The origin of this coordinate system is always placed at a point of the cutting edge. The y-axis is always in the direction of the prime motion while the z-axis is in the direction of the feed motion. The x-axis is perpendicular to the y- and z-axes to form a right-hand Cartesian coordinate system.

Figure 1.6. Cutting force and its components: (a) as applied to the workpiece, (b) as applied to the tool

For convenience, the cutting force is normally resolved into three components along the axis of the tool coordinate system. The main or power component of the resultant force, F _c (known also as the tangential force) is along the y-axis. It is normally the greatest component. The force in the feed direction, which is the z-direction, is known as the feed or axial force F _f . The component along the x-axis F _p is known as the radial component as it acts along the radial direction of the workpiece. The equal and opposite force R is applied to the cutting tool as a reaction force of the workpiece as shown in Figure 1.6(b). This force is also resolved into three orthogonal components along the coordinate axis as shown in Figure 1.6(b). The additional component F _xz that acts in the xz coordinate plane is also considered as it is essential for machining accuracy considerations.

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Advanced Machining Technologies

A.K.M. Nurul Amin , T.L. Ginta , in Comprehensive Materials Processing, 2014

11.13.8.2 Effect on Cutting Force

This section presents a comparison of cutting force under these two conditions and discusses in detail the effects of heating temperature, cutting speed, as well as feed on cutting force. The resultant cutting forces recorded from the experiments were used for comparison.

The effects of heating temperature and cutting speed on the resultant cutting force are presented in Figure 31. The figure affirms that an increase in heating temperature as well as cutting speed leads to reduction of the resultant cutting force. For instance, the resultant cutting force is found to be reduced by 44.5 and 40.6% at 30.6 and 160   m   min⁻¹, respectively, due to heating at only 450   °C (the optimum heating temperature being 650   °C). On the other hand, an increase in cutting speed from 30.6 to 160   m   min⁻¹ reduces the resultant cutting force by 36.8 and 32.5% for room temperature and heat-assisted machining respectively.

Figure 31. Combined effects of cutting speed and heat assistance (at 450   °C) on the resultant cutting force (insert: uncoated WC–Co). (Feed   =   0.088   mm per tooth and axial depth of cut   =   1   mm).

Reproduced from Turnad, L. G. Improvement of Machinability of Titanium Alloy Ti-6Al-4V through Workpiece Preheating. Ph.D. Thesis, Manufacturing and Materials Engineering Department, IIUM, Malaysia, 2009.

The effect of heat-assisted machining conducted using three different feed values was investigated. Figure 32 presents the results of this investigation. Cutting force is also found to drastically rise with the increase of feed. However, the reduction in cutting force due to heat-assisted machining is clearly indicated in the figure for all three feed values applied in the experiments.

Figure 32. Effects of feed and heat assistance (at 450   °C) on the resultant cutting force (insert: uncoated WC–Co). (Cutting speed   =   70.1   m   min⁻¹ and axial depth of cut   =   1   mm).

Reproduced from Turnad, L. G. Improvement of Machinability of Titanium Alloy Ti-6Al-4V through Workpiece Preheating. Ph.D. Thesis, Manufacturing and Materials Engineering Department, IIUM, Malaysia, 2009.

As shown in Figure 32, the cutting force increases by 51.4 and 81.1% when feed is increased from 0.05 to 0.15   mm per tooth for room temperature and heat-assisted conditions, respectively. The reduction in the resultant force by 53.6 and 44.5% at the feed of 0.05 and 0.088   mm per tooth, respectively, was achieved by heating the workpiece at only 450   °C is applied. This is attributable to the drop in the yield strength of the workpiece material at the elevated temperature during the heat-assisted end milling, which eventually reduces the normal and the shear stresses acting on the tool and thus contributes to lower the cutting forces.

The effect of heating temperature on the resultant force was investigated for a given set of cutting parameters: Cutting speed   =   70   m   min⁻¹, axial DOC   =   1   mm, feed   =   0.088   mm per tooth. Figure 33 illustrates the effect of heating temperature on cutting force. The resultant cutting force decreases sharply from 331.5   N at room temperature to 177.6   N at 315   °C. With further increase in heating temperature, the reduction is more gradual. The resultant force is reduced to only 134   N at 650   °C.

Figure 33. Effect of heating temperature on the resultant of cutting force (cutting speed   =   70   m   min⁻¹, axial DOC   =   1   mm, feed   =   0.088   mm per tooth, insert: uncoated WC–Co).

Reproduced from Turnad, L. G. Improvement of Machinability of Titanium Alloy Ti-6Al-4V through Workpiece Preheating. Ph.D. Thesis, Manufacturing and Materials Engineering Department, IIUM, Malaysia, 2009.

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Gun drilling of difficult-to-machine materials

Dennis Wee Keong Neo , ... Kui Liu , in Advanced Machining and Finishing, 2021

1.3.1 Burnishing instability of bearing pad

Pfleghar stability [25 ] has been widely employed to establish DHD tool design for stability by relating the stability with the relative positioning of the BPs and the resultant drilling forces. It is a ratio of the moments of the BP due to the cutting force and is the moment trying to keep the tool fixed to the hole wall and the moment trying to lift. Stability is classified as stable, neutral, and unstable equilibria. In a stable equilibrium, the end conditions at the cutting edge consider to be clamped. This occurs when the resultant cutting force swings between the two BPs, where one BP locates opposite to the outer cutting edge. Under the neutral equilibrium, the cutting edge considers to be a simple unidirectional support, where the reaction at one of the BPs is zero, i.e., when one pad loses contact against the hole wall. For an unstable equilibrium condition, the cutting edge assumes as a free end. This is caused by the tool geometry, which leads to unbalanced reaction forces at the BP.

Several researchers [26,27] assume that the stability occurs when the resultant of the cutting force bisects at the BP's included angle. Otherwise, the maximum static tool stability only occurs by positioning BP asymmetrically with respect to resultant cutting force [27], based on the conditions of Pfleghar stability and equally distributed loads across two BPs. Unfortunately, it later realizes that this asymmetrically arrangement does not satisfy the equilibrium condition at the drill entrance since the cutting edges are not completely engaged. Hence, it proposes to alter the pad width or the pad friction forces at the pilot bush with counterrotation. Hitherto, neither are the results obtained from this work presented fully nor are the reasons for pad friction and contact length variation have been expounded. Hence, the ambiguity in defining tool stability pertains.

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Machining and Machinability

F. Klocke , in Encyclopedia of Materials: Science and Technology, 2001

1 Essential Parameters

The tool life t of the tool is the most important indicator characterizing the machinability of a material. The tool life t is the service life of a tool up to the point at which it becomes unusable due to wear.

A knowledge of the magnitude and direction of the resultant cutting force F or of its components: the cutting force F _c, the feed force F _f, and the passive force F _p, forms the basis for: designing machine tools, establishing the cutting parameters, predicting the attainable accuracy of the part, and interpreting the phenomena which occur at the contact point. As a rule, the cutting of harder-to-machine materials entails higher forces.

The quality of the surface generated by machining may be a criterion for the design of the cutting process, if cutting is not followed by any subsequent operation. The primary variables influencing surface quality are the cutting parameters and the geometry of the cutting edge.

The shape and size of the chips, together with the way in which they are removed, are especially significant for cutting operations with a limited chip space (e.g., drilling, broaching, and milling) and for automatic lathes, due to the restricted working space and large volume of chips. The degree of upsetting of the chips can also be used as an indicator for chip formation phenomena.

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Orthogonal and Oblique Cutting Mechanics

Wit Grzesik , in Advanced Machining Processes of Metallic Materials (Second Edition), 2017

6.2 Forces in the Cutting Zone

In general, the force system exerted by a cutting tool is three dimensional (i.e., the resultant (total) cutting force F has three mutually perpendicular F _f , F _p and F _c (equivalently F _x , F _y , F _z ) components, shown in Fig. 6.3A1).

Figure 6.3. Geometrical force resolution in turning (A1) and associate physical force resolution in the cutting zone for orthogonal cutting with a continuous-type chip (A2); force resolution in slab milling (B), face milling (C), drilling using twist drill (D), deep-hole drilling using gun drill (E), boring (F) and grooving (G).

Typically, the resultant cutting force (RCF) is resolved into geometrical and physical components. In the first case, the coordinate system is oriented based on the directions of primary and feed motions (i.e., the total force is resolved by perpendicular projection to these two directions). As a result, the cutting force and feed force denoted by symbols F _c and F _f are distinguished. Moreover, the back (passive) force F _p tending to push the tool away from the work in a radial direction and perpendicularly to the working plane is defined. International Standard ISO 3000/4 also defines the cutting perpendicular force F _cN and the feed perpendicular force F _fN , which are perpendicular to the directions of primary and feed motions, respectively. Characteristic force resolutions for other cutting operations such as milling, drilling, boring, and grooving are shown in Fig. 6.3B–G. In cases of multi-edge cutting tools, such as drills or milling cutters, cutting forces act on all active cutting edges. As a result, the total forces are determined, as for example in Fig. 6.3D for face milling using milling head cutter. In case of twist drilling operations, the cutting torque M _c and the axial (feed) force F _f are distributed on the main cutting edges, the chamfer of the minor flank face and the chisel ((80%+12%+8%) for M _c and (50%+2%+48%) for F _f ). The back (passive) force F _p is usually ignored for orthogonal cutting and the total cutting force is reduced to the active force F _a . As can be seen in Fig. 6.3A2, in orthogonal cutting the entire force system lies in a single plane, and appropriate force components can be easily calculated by drawing the Merchant circle diagram. The force component F _c in the direction of relative tool travel determines the amount of work required to move the cutting tool a given distance. The feed force F _f determines the power required to feed drives installed onto a machine tool.

Geometrical resolution of the total cutting force is done by assuming that the chip is a body in stable mechanical equilibrium under the action of the forces exerted on it at the rake face, and at the shear plane as shown in the right upper detail.

The force components acting on the shear plane are F _sh and F _shN . The shearing force F _sh represents the force required to shear the material on the shear plane. Force F _shN acts perpendicularly to the shear plane and results in an additional compressive stress being applied on the shear plane. These force components can be calculated from the following equations:

(6.3) $F_{sh}=F_c\frac{\cos (\mathrm{\Theta }+\mathrm{\Phi }-{\gamma }_0)}{\cos (\mathrm{\Theta }-{\gamma }_0)}$

(6.4) $F_{shN}=F_c\frac{\sin (\mathrm{\Theta }+\mathrm{\Phi }-{\gamma }_0)}{\cos (\mathrm{\Theta }-{\gamma }_0)}$

At the tool-chip interface, the force components F _γ and F _γN act parallel and perpendicularly to the rake face. F _γ , known as friction force (the tangential force at the rake), represents the frictional resistance met by the chip as it slides over the tool rake face.

The friction force on the chip/tool interface:

(6.5) $F_{\gamma }=F_c\frac{\sin \mathrm{\Theta }}{\cos (\mathrm{\Theta }-{\gamma }_0)}$

where Θ is the friction angle.

The normal force on the chip/tool interface:

(6.6) $F_{\gamma N}=F_c\frac{\cos \mathrm{\Theta }}{\cos (\mathrm{\Theta }-{\gamma }_0)}$

The ratio of F _γ to F _γN is the apparent coefficient of friction between chip and tool, namely:

(6.7) ${\mu }_{\gamma }=tg\mathrm{\Theta }=\frac{F_{\gamma }}{F_{\gamma N}}=\frac{F_c\sin {\gamma }_0+F_f\cos {\gamma }_0}{F_c\cos {\gamma }_0-F_f\sin {\gamma }_0}$

It was experimentally proven that in cases such as cutting with a small uncut chip thickness (UCT), cutting of materials with high elastic (Young's) modulus, and when tool wear at the flank progresses, the forces acting on the flank/workpiece contact should also be considered in the total cutting force.

It can be assumed that the force acting on the flank wear land VB _B (Fig. 6.4) can be resolved into two components F _α and F _αN , which are independent of the rake angle value. These forces can be determined using the method of extrapolation of F _c and F _f components on zero UCT (Eq. (6.8)) proposed by Zorev [5]. As illustrated in Fig. 6.4B, the Zorev method needs a few F _c /UCT curves to be obtained for different rake angles.

Figure 6.4. Force distribution on the workpiece/flank contact (A) and projection of the cutting force on zero undeformed chip thickness (B) [2].

From the definition

(6.8) $F_{\alpha }=F_{c(h\to 0)}\mathrm{and}{F}_{\alpha N}=F_{f(h\to 0)}$

In addition, forces acting on the flank face can be calculated by solving the problem of penetration of a rigid punch into a plastically deformed half-space using the slip-line method [2].

The normal force on the worn flank face is

(6.9) $F_{\alpha N}\approx c{\sigma }_p{A}_{\alpha }$

where coefficient c=2.6–4, σ _p is the flow yield stress and A _α is the flank/workpiece contact area.

The tangential force on the flank face is

(6.10) $F_{\alpha }={\tau }_{c\gamma }{A}_{\alpha }=(0.5÷0.6)\mathrm{UTS}{A}_{\alpha }$

where τ _cγ is the main shear stress on the rake face and UTS is ultimate tensile strength.

As depicted in Figs 6.5 and 6.6, the cutting forces are influenced by the cutting parameters as well as by tool geometry. Fig. 6.5A shows a linear increase of all components of RCF due to relevant increase of the cross-sectional area of the uncut chip. On the other hand, force-feed curves are nonlinear, because increasing the feed results in decreasing the specific cutting pressure k _c [7]. The kink in the force-cutting speed curves in the medium speed range illustrates the effect of a built-up edge (BUE). With steels, a BUE disappears when the speed is raised. The cutting forces are also influenced by tool geometry, the most important parameter being the rake angle. An increase in the rake angle lowers the force components but at the same time reduces the strength of the tool wedge and may lead to its fracture. The strongest tool edge is achieved with negative rake angle tools, and these are frequently used for the harder grades of carbide, and for ceramic and CBN tools, which lack toughness. Tools with high positive inclination angles (Fig. 6.6B) generate high passive (thrust) forces, causing the deflection of the workpiece. In contrast, this force is very small when using tools with a 90° cutting edge angle (Fig. 6.6C), and, thus, machining of small diameter workpieces is possible. The tool material and deposited coatings can also influence the cutting forces mainly by changes in the area of seized contact between the chip and the tool [8]. Finally, the contact length and cutting forces may be greatly influenced by cutting lubricants or special soft low-friction coatings. In the speed range used in most machine shop operations, it is not possible to prevent seizure near the cutting edge but liquid or gaseous lubricants, by penetrating from the periphery, can restrict the area of seizure to a small region. This phenomenon will be discussed in Chapter 11, Tribology in Metal Cutting.

Figure 6.5. Influence of cutting parameters on the components of RCF [6].

Figure 6.6. Influence of cutting tool angles on the components of RCF [2].

Measurements of cutting forces and the cutting torque are carried out using strain-gauge and piezoelectric dynamometers. The principle of multi-axis piezoelectric tool dynamometers is shown in Fig. 6.7A and B. In general, a tool dynamometer should have high sensitivity, high rigidity, high frequency response, high linearity, and low drift [10]. In case of a strain-gauge dynamometer, Wheatstone's bridge circuits for each force component, consisting of two pairs of tension and compression gauges and amplifier system, are used to measure strain changes in the wire-type gauges. Before the measurements, the force dynamometer is calibrated to find the relationship between voltage signal from the bridge and the actual value of a force. Fig. 6.7A and B show stationary types of piezoelectric dynamometer capable of measuring all three components of the RCF (Fig. 6.7A) or additionally the cutting torque M _c (Fig. 6.7B), as for instance in drilling and milling operations. Both stationary and rotating four-component piezoelectric dynamometers are used [9]. Typical measurement signals recorded for turning, milling and drilling operations using piezoelectric dynamometers shown in Fig. 6.7A and B are presented in Fig. 6.7C1–C3, respectively. Commercial machining dynamometers are available with natural frequencies from 2 to 5   kHz, depending on size. The basic rule is that each force component is detected by a separate pure quartz crystal oriented relative to the force or torque in its piezoelectric sensitive direction. Quartz rings are used in pairs with a common electrode between them to yield double sensitivity. In Fig. 6.7A, two pairs of shear quartzes for F _x and F _y and one pair of compression quartzes for F _z , assembled in a single housing, constitute a three-component force transducer. Two-, three- and four-component piezoelectric dynamometers and force plates for measuring cutting forces and torque in various machining operations (turning, milling, drilling, etc.) are produced by world-leading manufacturer Kistler, Switzerland [9].

Figure 6.7. Measurements of cutting forces using stationary 3- (A) and 4-component (B) piezoelectric dynamometers along with recorded signals during turning (C1), face milling (C2) and drilling (C3) operations [2,9].

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Advanced Machining Technologies

M. Arif , M. Rahman , in Comprehensive Materials Processing, 2014

11.08.7 Chatter Vibrations in Micro-Milling

Due to the fragile nature of the miniature tools, even a minute vibration in micro-milling can lead to part failures. Similar to macro operations, micro-milling processes also exhibit an unstable phenomenon, called regenerative chatter, due to the undulations on a previously cut surface (34). Chatter results in a poor surface finish and reduces the longevity of the tool. Chatter stability can be expressed by stability lobe diagrams, which plot the boundary that separates stable and unstable machining in the form of the axial depth of cut limit versus spindle speed for a specific radial width of cut and workpiece/cutting tool combinations (35).

The prediction and avoidance of chatter vibrations, based on the assumptions that the dynamics of the system and cutting coefficients are time-invariant, have been widely studied. However, micro-milling operations require very high rotational speeds to maintain productivity; and, micro-cutting behavior is different from that of macro operations, due to large negative rake angles and size effects. At high rotational speeds, the dynamics of high-speed spindles vary due to centrifugal and gyroscopic effects that affect chatter (34). When the chip thickness is less than the critical chip thickness, the chip does not form and the workpiece material is instead plowed. Also, size effects, which increase the specific energy required due the decrease in scale, play an important role in micro-milling. Since the dynamics and cutting coefficients are the main parameters affecting chatter stability, investigating the effect of changing these parameters in chatter is imperative. Furthermore, elastic recovery of the workpiece generates a great deal of friction, which results in an increase of process damping. The process damping results in increasing the critical depth of stable cutting in milling operation especially at lower spindle speeds (34).

11.08.7.1 Robust Chatter Stability

Micro-milling is described as a two-degrees-of-freedom system, as shown in Figure 19. The equation of motion with considering the effect of process damping can be described as (34):

Figure 19. Schematics of micro-milling and surface profile during cutting.

Reproduced from Park, S. S.; Rahnama, R. Robust Chatter Stability in Micro-Milling Operations. CIRP Ann. – Manuf. Technol. 2010, 59, 391–394.

[70] $M_x\ddot{x}+C_x\dot{x}+K_{x}x=F_{\text{c},x}+F_{\text{pd},x}\left(\dot{x}\right)$

where M _x , C _x , and K _x are the effective mass, damping coefficient, and stiffness; F _c,x is the cutting force; and F _pd,x is the process damping force, which is a function of velocity, in X direction (33). The same formulation can be applied to Y direction as well. Since the depth of cut in micro-milling operation is very small, the effect of the helix angle and axial forces can be neglected.

The resultant cutting forces in chatter stability can be described as (35):

[71] ${\overline{F}}_{\text{c}}\left(\varphi \right)=K_{\text{tc}}\sqrt{1+K_{\text{r}}^2}ah\left(\varphi \right)=K_{\text{s}}ah\left(\varphi \right)$

where K _tc is the cutting coefficient in the tangential direction; K _r is the ratio of radial to tangential cutting coefficients; K _s is the resultant cutting force coefficient; a is the depth of cut; and h is the regenerative chip thickness, which is a function of the immersion angle, f. As it is depicted in Figure 19, the chip thickness, h, can be formulated as (35):

[72] $h=c\sin \varphi -\left(r\left(t-T\right)-r\left(t\right)\right)$

where c is the feed rate, T is the tooth passing period, and r is the displacement in the radial direction (i.e., r = Xsinϕ + Y cosϕ). The static component of the chip thickness is dropped since it does not contribute to chatter.

The resultant process damping force can be obtained as (36):

[73] ${\overline{F}}_{\text{pd}}\left(\varphi \right)=\frac{K_{\text{pd}}}{R{\omega }_{\text{spindle}}}\dot{r}$

where K _pd is the resultant process damping force coefficient, R is the tool radius, ω _spindle is the rotational speed, and $\dot{r}$ is the radial velocity of the tool. Based on the authors' previous work (36), K _pd is found to be approximately 1.46 by identifying the interference volume between the tool and the workpiece for micro-milling of Al 7075.

After transferring the cutting and process damping forces, substituting them in eqn [70] and transfer to the Laplace domain, the following equations will be obtained

[74] $\left(M_x{s}^2+C_{x}s+K_x\right)X=\left(-K_{\text{s}}a\cos \left(\varphi -\theta \right)\left(1-e^{-sT}\right)-\frac{K_{\text{pd}}s}{R{\omega }_{\text{spindle}}}\cos \left(\varphi -\beta \right)\right)\times \left(X\sin \left(\varphi \right)+Y\cos \left(\varphi \right)\right)$

where θ = tan⁻¹(Kr), which is the angle between the resultant cutting force and the tangential cutting force and β is the angle between radial and tangential components of ${\overline{F}}_{\text{pd}}$ .

Two-degrees-of-freedom milling processes can be transformed into a pseudo single-degree-of-freedom problem by projection onto one plane (37,38). Combining the above equations in X and Y directions and separating the terms corresponding to cutting and process damping forces, the characteristic equation of the system can be derived as:

[75] $\begin{array}{l}1+a_{\lim }{\overline{\Phi }}_{\text{c}}{K}_{\text{s}}\left(1-e^{-sT}\right)+{\overline{\Phi }}_{\text{pd}}\frac{K_{\text{pd}}s}{R{\omega }_{\text{spindle}}}=0\\ {\overline{\Phi }}_{\text{c}}=\frac{N}{2\mathrm{\pi }}{\int }_{{\varphi }_{\text{st}}}^{{\varphi }_{\text{ex}}}\left({\Phi }_{xx}\cos \left(\varphi -\theta \right)\sin \left(\varphi \right)-{\Phi }_{yy}\sin \left(\varphi -\theta \right)\cos \left(\varphi \right)\right)d\varphi \\ =u_x{\Phi }_{xx}+u_y{\Phi }_{yy}\end{array}$

[76] $\begin{array}{c}{\overline{\Phi }}_{\text{pd}}=\frac{N}{2\mathrm{\pi }}{\int }_{{\varphi }_{\text{st}}}^{{\varphi }_{\text{ex}}}\left({\Phi }_{xx}\cos \left(\varphi -\beta \right)\sin \left(\varphi \right)-{\Phi }_{yy}\sin \left(\varphi -\beta \right)\cos \left(\varphi \right)\right)d\varphi \\ ={\upsilon }_x{\mathrm{\Phi }}_{xx}+{\upsilon }_y{\mathrm{\Phi }}_{yy}\end{array}$

where F _xx and F _yy are the direct transfer functions of the system in the X and Y directions, N is the number of flutes on the miniature tool (N = 2 for micro-end mills), and u _x , u _y , v _x , and v _y are the orientation factors. Resultant dynamics, ${\overline{\Phi }}_{\mathrm{c}}$ , and the process damping transfer function, ${\overline{\Phi }}_{\mathrm{pd}}$ , are obtained by integrating over the starting immersion angle (ϕ _st) to the exit angle (ϕ _ex) (38). Equation [75] is applicable for either micro- or macro-milling cases. The plowing effects in micro-milling contribute to the changing cutting coefficients.

The critical depth of cut (a _lim) in eqn [75] determines the border of stability and instability. Depths of cut greater than the critical value cause chatter; whereas, with depths of cut smaller than the critical value, the cutting operation is stable. The tool tip dynamics are indirectly obtained using the receptance coupling method. To examine the stability of varying parameters, the robust stability theorem is proposed.

In conventional chatter stability theories, the cutting parameters are considered to be constant. However, some parameters, such as system dynamics and cutting coefficients, change during micro-milling operations. The robust chatter stability theorem, based on the edge theorem and the zero exclusion principle, is utilized to find stability within the changing boundaries. The edge theorem is an extension of Kharitonov's robust theory that allows us to predict the stability of an uncertain time-delay system, whose parameters vary within a certain range (39). The edge theorem states that a polynomial, P, which has variable coefficients, is robustly stable, if and only if, the edges that correspond to each pair of extreme polynomial vertices, p _i and p _j , are stable (39,40). The edge theorem guarantees the stability of the edges and within the boundary of the edges.

The polynomial is the characteristic equation (eqn [75]); and, the uncertain parameters are the natural frequency, ω _n, and the resultant cutting coefficient, K _s, which change within a specific range. The varying parameters have been identified from experimental tests. According to the edge theorem, the polynomials that form the edges can be formulated as (40):

[77] $P=\left\{p\left(s,{\omega }_{\text{n}},\xi \right):\Phi \in \left[{\Phi }_{\min },{\Phi }_{\max }\right],K_{\text{s}}\in \left[K_{\text{s}},K_{\text{s}}\right]\right\}$

Combining eqns [75] and [77], the polynomials can be described as:

[78] $p_i\left(s\right)=1+a{\tilde{K}}_{\text{s}}\left(1-e^{-sT}\right)\left(u_x{\tilde{\Phi }}_{xx}+u_y{\tilde{\Phi }}_{yy}\right)+\frac{K_{\text{pd}}s}{R{\omega }_{\text{spindle}}}\left({\upsilon }_x{\tilde{\Phi }}_{xx}+{\upsilon }_y{\tilde{\Phi }}_{yy}\right)$

where i = 1 – 4 for two parameter variations, ${\overline{K}}_{\text{s}}$ is either K _s,min or K _s,max, and ${\Phi }_{xx}$ and ${\Phi }_{yy}$ are either F _min or F _max in X and Y directions. The above polynomials are used to form the edges (40):

[79] $E_{\text{k}}=\left(1-\lambda \right)p_i\left(s\right)+\lambda p_j\left(s\right)$

for all $\lambda \in [1,0]$ , where 1 ≤ i, j ≤ 4, and E _k is representing each edge.

In order to find the stability, we employ the zero exclusion method, which is a graphical technique for improving the efficiency of finding the stability in the edge theorem (39). In each frequency, the characteristic equation forms a family of four different edges with the extreme boundaries of variations. The edges shape a trapezium, as depicted in Figure 20. The trapezium must be investigated to determine if it encircles the origin or not: if the edges encircle the origin in a complex plane, the system is unstable; otherwise, the system is stable (38). In order to automate the stability detection process, we find the angles that result from connecting the origin to each of vertices of the trapezium (p ₁–p ₄). Connecting the origin to the corners will make α _i angles (i = 1 – 4) between the lines and the positive real axis, as shown in Figure 20. γ _i (i = 1–4), which is the angle between each two subsequent corners, is defined based on α _i . γ _i = α _{i +1} − α _i for i = 1 – 3 and γ ₄ = α ₁ − α ₄. The γ _i angle must be a positive angle. If all the angles γ _i are less than 180°, we can deduce that the edges encircle the origin. However, if one of the angles γ ₁ – γ ₄ will be greater than 180°, the origin is not encircled by the edges (34).

Figure 20. Zero exclusion method (angles in a trapezium).

Reproduced from Park, S. S.; Rahnama, R. Robust Chatter Stability in Micro-Milling Operations. CIRP Ann. – Manuf. Technol. 2010, 59, 391–394.

To find the stability lobes for micro-milling operations, the algorithm sweeps the depths of cut and chatter frequencies at each spindle speed; and, it checks the stability through the proposed automated zero exclusion method. The first set of unstable conditions is recorded as the border between the stable and unstable regions, in order to determine the stability lobes (34).

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Advanced Machining Technologies

Z. Wang , M. Rahman , in Comprehensive Materials Processing, 2014

11.10.4.1.2 Numerical Models

In numerical modeling, FEM techniques were found to be the most dominant tool (19). In this approach, the solution region is first divided into many smaller elements, so that various tool geometries, cutting conditions, and more sophisticated material and friction models can be incorporated (28). Then, element equations are formulated. Based on the interconnected relation of elements, element equations can be assembled into global equations. Finally, after solving the global equations, a numerical solution to the problem domain is obtained. Two basic approaches are often used to solve the global equations, namely the Newton–Raphson method and the direct iteration. Thus, the main advantage of FEM is its ability to predict chip flow, cutting forces, and especially a distribution of tool temperatures and stresses for various cutting conditions by simply changing the input data. Ozel and Altan (29,30) have completed many definitive works in this field. They developed a predictive model for high-speed milling based on FEM simulations. Using their model, the resultant cutting forces, flow stresses, and temperatures in turning and flat end milling were predicted primarily. More importantly, with fewer experiments, this method is able to estimate the variations of flow stress and friction conditions of high-speed machining. In their model, the tooth-path was assumed to be circular; however, this approximation will cause some error for high-speed milling.

Using the commercial code FORGE2, Ng et al. (31) presented an FE model to simulate orthogonal machining of hardened die steel with advanced ceramic tools. Unfortunately, their model underestimated the magnitude of the cutting force due to limited data on the sensitivity of the workpiece material to strain hardening and the strain-rate sensitivity at elevated temperature, and an oversimplification of the frictional conditions at the tool–chip interface. Based on Oxley's theory, Carrino et al. (32) used a coupled thermomechanical finite element model to simulate orthogonal cutting of carbon steel C40. In their model, the tangential force applied along the tool/chip interface was assumed to be a fraction of the shear stress of the material. Good agreement between experimental and numerical results was found based on cutting forces measurements.

Altintas (28) and his research group at UBC developed an arbitrary Lagrangian–Eulerian (ALE) formulation, which has been applied for the prediction of cutting variables in machining. In the developed ALE code, the effects of edge radius on the cutting edge on the cutting forces were considered.

Although rapid progress and better results of FEM have been achieved recently, there are also some problems for FEM which need to be considered. The most significant problem is to obtain the material properties under the practical cutting conditions. Nowadays, the data of material properties used in the simulation of FEM are obtained in tensile or compression tests. Obviously, the real cutting conditions are different from those of tensile or compression tests. In addition, the numerical model requires significant amounts of computation time. The computational burden is almost unbearable for three-dimensional modeling. There is still a long way to go before the FEM can be used to simulate practical machining operations with an acceptable degree of accuracy and reliability and an acceptable amount of effort for daily use (19).

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