THEORY OF CUTTING & MACHINE TOOL DESIGN SECTION-A 1. THEORY OF METAL CUTTING: Mechanism of metal cutting, Cutting forces, Chip formation, Merchant’s circle diagram, Calculations, System of Tool nomenclature, Tool geometry, Machinability, Tool life, Cutting tool materials, Cutting fluids. Abrasive Machining- Mechanism of grinding, lapping and honing. SECTION-B 2. INTRODUCTION TO MACHINE TOOL DESIGN: Introduction to Metal Cutting Machine Tools, Kinematics of machine tools, Basic Principles of machine Tool Design, 3. DESIGN OF DRIVES: Design considerations of electrical, mechanical and hydraulic drives in machine tool, Selection of speeds and feeds, stepped and stepless regulation of speed, Estimation of power requirements and selection of motor for metal cutting machine tool spindles, design of gear box. SECTION-C 4. DESIGN OF MACHINE TOOL STRUCTURES : Principles, materials, static & dynamic stiffness, Shapes of Machine tool Structures. Design of beds, columns, housings, tables, ram etc. 5. DESIGN OF SPINDLES, GUIDEWAYS AND SLIDEWAYS: Design of Machine tool Spindles- Materials of Spindles, machine tool Compliance. Design of Bearings- Anti friction bearings, sliding bearings. Design of guide ways and slideways. SECTION-D 6. DESIGN OF CONTROL MECHANISMS: Basic principles of control, mechanical, electrical, hydraulic, numeric and fluid controls, Selection of standard components, Dynamic measurement of forces and vibrations in machine tools, Stability against chatter, Use of vibration dampers. 7. AUTOMATION, TESTING AND STANDARDISATION: Automation drives for machine tools, Degree of automation, Semi-automation, analysis of collet action, design of collet, bar feeding mechanism, tooling layout, single spindle mechanism, analysis, Swiss type automatic machine. Loading and unloading. Transfer-deices, Modulator-design concept, in process gauging. Acceptance tests and standardization of machine tools. www.NewJobUpdates.com
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THEORY OF CUTTING & MACHINE TOOL DESIGN
SECTION-A1. THEORY OF METAL CUTTING: Mechanism of metal cutting, Cutting forces, Chip formation, Merchant’s circle diagram, Calculations, System of Tool nomenclature, Tool geometry, Machinability, Tool life, Cutting tool materials, Cutting fluids. Abrasive Machining- Mechanism of grinding, lapping and honing.SECTION-B2. INTRODUCTION TO MACHINE TOOL DESIGN: Introduction to Metal Cutting Machine Tools, Kinematics of machine tools, Basic Principles of machine Tool Design, 3. DESIGN OF DRIVES: Design considerations of electrical, mechanical and hydraulic drives in machine tool, Selection of speeds and feeds, stepped and stepless regulation of speed, Estimation of power requirements and selection of motor for metal cutting machine tool spindles, design of gear box.SECTION-C4. DESIGN OF MACHINE TOOL STRUCTURES : Principles, materials, static & dynamic stiffness, Shapes of Machine tool Structures. Design of beds, columns, housings, tables, ram etc.5. DESIGN OF SPINDLES, GUIDEWAYS AND SLIDEWAYS: Design of Machine tool Spindles- Materials of Spindles, machine tool Compliance. Design of Bearings- Anti friction bearings, sliding bearings. Design of guide ways and slideways. SECTION-D6. DESIGN OF CONTROL MECHANISMS: Basic principles of control, mechanical, electrical, hydraulic, numeric and fluid controls, Selection of standard components, Dynamic measurement of forces and vibrations in machine tools, Stability against chatter, Use of vibration dampers.7. AUTOMATION, TESTING AND STANDARDISATION: Automation drives for machine tools, Degree of automation, Semi-automation, analysis of collet action, design of collet, bar feeding mechanism, tooling layout, single spindle mechanism, analysis, Swiss type automatic machine. Loading and unloading. Transfer-deices, Modulator-design concept, in process gauging. Acceptance tests and standardization of machine tools.
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3
Mechanics of Machining Processes
3.1 INTRODUCTION
The different machining processes may be classified into two categories, llalllely (i) orthogonal cutting processes, and (ii) oblique cutting processes. In arthogonal cutting, the cutting edge of the tool is perpendicular to the cutting velocity vector while in oblique cutting, the cutting edge is not normally disposed to the direction of cutting velocity vector. Any analysis of the oblique cutting pucess also applies to orthogonal cutting, which, in fact, is a particular case of oblique cutting. Oblique cutting being more difficult to analyze, many authors attempted to analyze only the orthogonal cutting. Only recently' a few attempts lave been made to understand the mechanics of oblique cutting.
In this chapter, we study the mechanics of chip formation, forces in arthogonal cutting, chip velocity, rake angles and components of cutting force in oblique cutting, ploughing forces, etc. The mechanics of the turning, milling and drilling processes are also discussed. As will be explained more fully in the mllowing paragraphs, the metal ahead of the cutting edge deforms plastically before it goes to form the chip. This plastic zone is of significant dimensions compared to the uncut chip thickness. However, due to the complexity of an exact analysis, most authors have preferred to assume that the chip formation occurs only in an infinitesimally thin zone of deformation, called the shear plane. Merchant's, Lee and Shaffer's and Oxley's models are based on this assumption. Some of these models are discussed below.
3.2 CHIP FORMATION
All machining processes involve formation of chips by deforming the work material on the surface of the job with the help of a cutting tool. The extent of
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112 Fl):\'DA.\JF\T!f..\' 01: I //•.'7:1!. ( 'LTUW1. 1.\'D \I.ICHI.\'F TOOLS'
deformation that the material suffers not only determines the type of the chip but also determines the quality of the machined surface. cutting forces. temperatures dL·vcloped and dimensional accuracy of the job. Depending upon the tool geometry, cutting conditions. and work materiaL a large variety of chip shapes and sizes are produced during different machining operations. ( classified in ISO 3685 : ( 1977)). However, in most of the machining processes we get the following three types \vhich are illustrated in Fig. 3.l(a, b, c). (i). Continuous chips. (ii). Continuous chips with built-up-edge, and (iii). Discontinuous chips.
Primary zone of defonnation
Chip
( a ) Continuous chip
Secondary zone of defom1ation
Work piece
Work piece
( b ) Continuous chip with B U E
( c) Discontinuous chip
Fig.3.1 -Types of Chips.
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.I fechanics nf\!achining !'rocesses 113
The type of chip formed is an indication of the deformation suffered by the material and the surface quality produced during cutting. The quality of a ~~~aehined surface includes its roughness, microstructure and residual stresses etc
3.2.1 Continuous Chips During the cutting of ductile materials like low carbon steel. copper brass
aid aluminium alloys, etc., a continuous ribbon type chip is prodm:cd. Th..: pressure of the tool makes the material ahead of the cutting edge deform plastically. It generally suffers compression and shear. The material then slides a-er the tool rake face for some distance and then leaves the tool. Friction llc:tween the chip and tool may produce secondary (additional) deformation on 6e chip material. The plastic zone ahead of the tool edge is called the primary .ame of deformation. and the deformation zone on the rake face is usually lderred to as the secondary zone of deformation (Fig. 3.la). Both these zones 3IDd the sliding of chip on rake face produce heat which results in increase of blperature on the tool-chip interface as well as increase in temperature of the IDol as a'" hole. l11c extent of primary zone of deformation depends on
(i) rake angle of tooL (ii) cutting speed, (iii) work material characteristics. and (iv) friction on rake face.
With large rake angle tools. the transition of work material into chip is tJPiiUual and the material suffers less overall deformation. Cutting forces· are also
. With small or negative rake angle tools, the materials suffer a far more svere deformation, and the cutting force involved is also large.
At higher cutting speeds, the thickness of the primary zone of deformation *inks, i.e. it becomes narrower. The work material characteristics which tilftuence the size of the primary zone are: (i). Strength, (ii). Strain hardening,
Strain rate and (iv) Heat conductivity. Increase in friction on the rake face of tends to increase the size of both the primary and secondary zones of
Continuous Chips with Built-up-Edge As mentioned above, the temperature is high at the interface between the
and the tool during cutting. Also, the work material slides under heavy ~ure on the rake face before being transformed into a free chip. Therefore, in
conditions, some portions of the chip may stick to the rake face of the tool. Because of such close contact, it discharges its heat to the tool and thus becomes !monger than the rest of material flowing over it. Naturally, it attracts more of the
work material and thus the size of the "built-up-edge" goes on rillrrP~<:incr (Fig 3.2). When it reaches a certain '"critical" size, it becomes unstable
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114 FUNDAMENTALS OF METAL CUTTINGAND~MACHINE TOOLS
Initiation ofB U E
Work piece
(a) Initiation ofB U E
Chip Tool
Growth of B U E
Workpiece
(b) Growth ofB U E
Tool
Work piece
(c) Breaking ofB U E
Fig.3.2- Periodic Variation ofBUE Size and its Fragmentation.
and portions of it may disintegrate or break up (Fig. 3.2c). These broken nnrrw ....
of B UE get embedded in the machined surface or get attached to the underside
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,.. Mechanics of Machining Processes 115
lhe flowing chip. This cycle of building up and breaking of built-up-edge (BUE) llappens periodically and the machined surface gets dotted with portions of the broken built-up-edge, resulting in poor surface finish. However, there is a n:medy. Increase in cutting speed, increases the interface temperature which softens the built-up-edge. As a result the critical size ofthe BUE starts reducing. At sufficiently high cutting speeds, the BUE completely disappears.
3.2.3 Discontinuous Chips These chips are produced during the cutting of brittle materials like cast
irons and brasses containing higher percentages of zinc, etc. The chip formation mechanism in this case is quite different from that in the case of ductile materials. Even a slight plastic deformation produced by a small advance of the cutting edge into the job leads to a crack formation in the deforming zone. With further advance of the cutting tool, the crack travels and a small lump of material starts moving up the rake face (Fig.3.3). The force and constraints of motion acting on the lump make the crack propagate towards the surface, and thus a small fragment of the chip gets detached. As the tool moves further this sequence is repeated.
Initial defom1ation
(a)
Crack Formation
(b)
Chip segment
\
(c)
Fig.3.3- Formation of Discontinuous Chip.
Discontinuous chip formation differs from continuous chip formation in that the work material contact over the tool rake face is over a shorter length and hence for a shorter duration. Most of the heat produced in the chip is carried by the chip. As a consequence, the tool is heated to a lower temperature and hence has longer life.
During machining of materials which are less brittle (i.e. which can suffer some plastic deformation), both continuous and discontinuous chips may be formed under appropriate conditions of rake angle and cutting speed.
The above three type of chips are the ones mainly encountered in practice. However, specific properties of some metals and alloys· may give rise to other types of chips also. For example, titanium and its alloys, the yield strength of
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116 FC\'/>.1 I W.\T 1/\' OF.\11'.7:-1!" CLT71NGANDMACHJNE TOOLS
\\ hich decreases rapidly '' ith increase in temperature, may give a continuous but inhomogeneous chip that contain layers of highly deformed (sheared) material adpccnt to relatively les" deformed layers.
Tool
Chip flow normal to tool edge
/
( a) Orthogonal cutting
Workpiece
LAngle of Obliquity
( b ) Oblique cutting
Fig.3 4 - Orthogonal and Oblique Cutting.
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Mechanics of-\/achimng Proce5.Sie .. '' 117
3.3 ORTHOGONAL CUTTING
On the basis of the angular relationship between the cutting velocity vector and the cutting edge of the tool.. different met~! cutting processes can be classified into two broad categories .. namely (i) orthogonal cutting, and (ii) oblique cutting.
In orthogonal cutting, the cutting edge of the tool is perpendicular to the cutting speed direction. In oblique cutting, the angle between the cutting edge and cutting velocity vector is different from 90°.
3.3.1 Thin Zone Models in Orthogonal Cutting
During cutting, the work material ahead of the tool tip suffers plastic deformation and, after sliding on the rake face of the tool, goes to form the chip. The zone of plastic deformation lies between the chip (where the chip starts moving as a rigid body) and the undeformed or only elastically deformed work material. The size of the plastically deforming zone varies according to the cutting conditions. At relatively low cutting speeds, the zone is large whereas at high cutting speeds it reduces in size and approximates to a thin shear plane. At present there is no unified analysis that can take into account these variations. However, there are solutions which either consider the. zone of deformation as large, or as limited to a thin shear plane. Accordingly, the solutions arc known as thick zone models and thin zone models. In thin zone models, it is assumed that the work material shears across a plane and forms the chip. The plane is called shear plane (Fig. 3.5).
3.3.2 Determination of Shear Plane Angle The shear plan angle is the angle between the cutting velocity vector and
the plane across which the work material suffers shear deformation and forms the chip. It is shown by anglt:; '<!>'in Fig.(3.5). The chip material comes entirely from work material. Also material flow is continuous and in plastic deformation there is negligible or no change in volume of work material, hence we can write
f. b. V = fe. be. Ve (3.1)
where t, b and V respectively denote the depth of cut, width of cut and cutting velocity. Similarly fc. be and v~ denote chip thickness, width of chip and chip velocity respectively. When b is comparable to t, there is significant side flow (or strain in the direction of the width b) and be is greater than b. The side flow being more on the edges the resulting chip is not even rectangular in shape. However, when b ~. t. the side flov. is negligible and we may take b equal to be.
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118 FUNDAMENTALS OF METAL CUTTING AND MACHINE TOOLS
Workpiece
Fig.3.5- Shear Plane and Shear Plane Angle.
In most cutting processes b is nearly equal to be. Hence
i. V = fc. Vc Therefore,
r, = t I tc = Vc / V = Lc I L
where, r, is the chip thickness ratio and Lc the length of the chip formed from a layer of uncut chip of length L on the work surface. From the geometry of the Fig.3.5, we can write the length of shear plane AC as follows.
AC = _t_ = -----=-tc __ sin r/J cos( r/J - a)
where a is the rake angle of the tool. Hence
t ltc = r1 =sin ¢1 cos (¢-a)
The Equation (3.4) may be be solved for <1> as given below.
tan r/J r, cosa
=-'-----1- r, sin a
The shear angle <1> may be determined experimentally by knowing r1
the rake angle a. The ratio r1 can be determined by measuring t and lc. nn\li!Pl.IP.II
it is more covenient and accurate to measure Land Lc and use Equation(3.3) !1::1 r: The experiments may be conducted by end turning of a pipe of a ductile
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(3.2)
(3.3)
from a r of the
(3.4)
(3.5)
llg r1 and 1owever, n(3.3) to uctile
Mechanics ofMaciWtiJtg Prw-••• 119
metal by a tool with only side rake angle in order to get condition of orthogoaal . cutting. For fixing the uncut chip length two cuts at known peripheral length may be made with a hand hacksaw. Machining on a shaper is also a convenient method. In this case a specimen of known length and width shorter than tool width is taken. In order to get the condition of orthogonal cutting ~ool should have only back rake angle and zero side rake angle.
EXAMPLE 3.1:- In an experiment, a pipe is turned on end in.orthogonal cutting condition with a tool of 20 o rake angle . A chip length of85mm is obtained from an uncut chip length of 202mm while cutting with a depth of cut of0.5mm. Determine the shear plane angle and chip thickn-ess.
SOLUTION:- Taking that there is no change in the width of chip during cutting i.e. the chip width = the width of uncut chip,
Chip thickness ratio = 85/ 202 = 0.42 Shear plane angle ¢J is determined by using Equation (3.5)
A. _1 0.42 x cos20° ., =tan
1- 0.42 x sin 20°
= 24.74°
Chip thickness tc = t I 0.42 = 1.19mm
EXAMPLE 3.2:- During machining on a shaper under orthogof}al cutting condition, the specimen length along the stroke is 1 OOmm a~d. the chip length of 40 mm is obtained with a tool of 15 ° rake angle. Determine the shear angle and chip thickness if uncut chip thickness is 1.5mm ..
SOLUTION:- Taking that there is no change in width of chip during cutting we have tx.l = tcx lc
Therefore, chip thickness ratio t I tc =.lc I I= 401 100 = 0.4 Shear angle 4> is obtained by using Equation (3.5) as given belml:'.
4> = tan_, ( 0.4 cos15 °) I ( 1-0.4 sin 15 °) = 23.31 °
Chip thickness = 1.5 I 0.4 =3.75 mm
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120 FUNDAMENTALS OF METAL CUTTING AND MACHINE TOOLS
3.3.3 Cutting Forces in Orthogonal Cutting Fig.3.6 shows the different components of resultant cutting force R. In
orthogonal cutting the component of R in the direction of the width b is zero. Therefore, R may be resolved into two orthogonal components. According to the chosen directions, the components useful for analysis are as follows. (i) F11 =horizontal force component parallel to the cutting velocity vector,
Fv = vertical force component normal to F11
(ii) Fs =force component parallel to the shear plane, Fp = force component normal to Fs
(iii) F, = force component parallel to the tool rake face, F11= force component normal to F,
Tool
Workpiece
E
Fig. 3.6- Cutting Force Components in Orthogonal Cutting.
We can determine R ( bold letters denote vector quantities) if we know any pair of the components listed above, or any one component and its angular relationship with R. In experiments, F11 and Fv are generally measured with the help of a dynamometer (force measuring instrument). The geometric relationships between the different components of resultant cutting force R are illustrated in Fig. 3.6, which also shows the shear plane angle <j>, friction angle and tool rake angle a. Thus the magnitude R of the resultant force R written as follows.
R = (F/ + F/) 112
R=(F/+F/{2
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In ro. , to
ty one ngular ith the metric Rare
,ngle J3 nay be
(3.6a)
(3.6b)
(3.6c)
.Hechanics ofMaclliiBIIg Pmo 'HI 121
If J..L is the average coefficient of friction on the interface between the chip and tbe tool, then F, Fn and J3 are related as follows.
1-i = tan f3 = F1 I Fn (3.7)
Also from Fig. 3.6 tan (/3 -a.) = Fv I Fh (3.8)
Either of the above two equations may be used to determine friction angle. Below we express the different force components in terms of R and angles cj>, P and a..
Fs = R cos ( ¢+/3-a ) and Fh = R cos (f3-a) and Fn = R cos f3 and
Fp = R sin ( ¢+f3-a ) Fv = R sin (/3-a) Ft = R sin f3
3.3.4 Merchant's Model For Orthogonal Cutting
(3.9a) (3.9b) (3.9c)
One of the earliest analyses of orthogonal cutting is due to Ernst and Merchant [2]. The model is based on the minimization of rate of energy dissipation in the cutting process. The basic assumptions underlying the model are:
(i) (ii) (iii) (iv)
(v)
Tool edge is sharp. The work material suffers deformation across a thin shear plane. There is no side spread (or the deformation is two dimensional). There is uniform distribution of normal and shear stresses on the shear plane.
The work material is rigid, perfectly plastic.
From the geometric relation between force components given by Eqns.(3.9a and 3.9b) we have
Fh = R cos (/3 - a) and Fs = R cos ( ¢ + p- a) where
Fs = (AC)b.K = t.b.K. I sin ¢ (3.9d) Therefore,
R = t.b.K sin¢>. cos(¢>+ f3- a)
(3.10)
And
Fh = t.b.K cos (/3 - a ) sin ¢> . cos ( ¢> + p - a )
(3.11)
Here 'K 'is the yield stress of material in shear. Therefore we have
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f ..
122 · FUNDAMENTALS OF METAL CUTTING AND !y!ACHINE TOOLS
The Merchant's analysis is .based on the minimization of rate of energy consumption which is equal to Fh V. Let us assume that ¢and (3 are not functions of V. Therefore, minimization of Fh is same as minimization of Fh . V: For optimum value of¢ we have,
= 0
Or tb.cos(p ~a)[cos¢cos(¢ + P-a) -sin¢sin(¢ + P -a)]=
0 .. sin 2 ¢.cos2 (¢+P-a)
(3.12)
where, sin ¢ * 0 , and also cos ( ¢+ p- a) * 0 Therefore, the numerator is equal to zero. Also cos (p - a) can not be zero. Hence,
cos (2¢+ (3- a)= 0 which gives
¢ = ltl4-112((3- a)
(3.13)
(3.14)
EXAMPLE 3.3:- In an orthogonal cutting operation on a material with the shear yield strength of 2 50 N I mm2 the following data is obtained.
Rqke angle of the tool = 15 deg. Uncut chip thickness = 0.25 mm Width of chip = 2mm Chip thickness ratio = 0.46 Friction angle p = 40 deg
Determine the shear angle ¢ , the cutting force component and resultant fore(/ on the tool.
SOLUTION:- The shear angle cp is determined using Equation (3.5), tancp=0.46cos 15° I( 1-0.46xsin 15°)
¢ = 26.76deg
Shear force along the shear plane= F. t.b.K I sin¢ = 0.25x2x2501sin 26.76° =277.62N
The resultant force on the cutting tool = 277.62 I cos ( cp + (3 -a ) = 277.62 I cos51.76° = 448.52 N
Cutting force component = 448.52 cos ( 25° ) = 406.55 N
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,.~
Mechanics of Machining Processes !23
EXAMPLE 3.4 :- In an orthogonal cutting operation on a material with shear yield strength Of 200 N lmm2 the following data is observed. Length qf cut chip obtained from uncut chip length of 100 mm = 50mm, rake angle of tool =10 deg, uncut chip thickness= 0.2 mm, width qf cut = 1.5 mm and coefficient qffriction ,u = 0.8 Determine the shear plane angle ,resultant force on the tool, cutting force component ofresultantforce.
SOLUTION :- Chip thickness ratio = 0.5 Shear angle cj> (Eqn.3.5) =28.34 deg. Force along the shear plane (Fs) (Eqn. 3.9d) = 1 Z6.4N Resultant force on the cutting tool R (Eqn. 3.1 0) = 232.08 N Cutting force component (Fh) (Eqn. 3.11) = 203.64 N
3.3.5 Lee and Shaffer's Solution Lee and Shaffer's p]solution of orthogonal cutting is based on the slip line
field theory. The resultant cutting force R is inclined at an angle ~ (friction angle) to the normal to the tool rake face (Fig.3.7). The deformation zone is contained in triangular region ABC. The line AC is the shear plane across which material suffers velocity discontinuity. The plane containing CB is stress free and hence the slip lines meet this line at 45°. One set of the slip lines are parallel to AC and the other set which is perpendicular to AC is inclined at (45°- ~)to the tool rake face.
A
Workpiece
Fig.3.7- Lee and Shafer's [3]Slip Line Field for Orthogonal Cutting.
From the geometry in Fig.3.7, we can write,
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124 FUNDAMENTALS OF METAL CUTTING AND Jv!ACHI!I[E TOOLS
L::CAB=45°+J3 Also,
¢ + LCAB = 90° + a Hence
(3.15)
The slip line field inside the zone ABC consists of a net of straight orthogonal lines, which would mean that stresses are uniformly distributed in the plastic region and also there is uniform distribution of stresses on the rake face. However, this is not quitetrue in actual conditions.
Applicability ofthis solution for·negative rake angles seems to be doubtful. For example, for a=- 10°, and p = 35°, the shear plane angle ¢ accordingly to this solution would be zero, which is inadmissible or impractical.
3.3.6 Stress Distribution on Rake Face in Orthogonal Cutting ·.,.In the analyses of orthogonal cutting described above, it has been assumed
that 'the shear and normal stresses are uniformly distributed over the tool rake face. It has.been shown experimentally that this assumption is not true. Both the stresses are found to vary along the contact length.
It is extremely difficult to determine the stress distribution on the tool rake face in normal cutting operations, because of the very small contact length, high cutting temperatures and high values of normal and tangential stresses involved. Nevertheless, several attempts have been made, as described below, to determine the distribution of stress components.
In earlier attempts, the tools made out of a· photo-elastic material and work piece made of soft metals like lead, tin and lead alloys were used [9,10]. The stress distribution on the rake face was determined by analyzing the fringe pattern observed during experimental cutting on a photo-elastic apparatus. distribution of normal and shear stresses so obtained are shown in Fig. 3.8. Tbc method suffers the disadvantage that actual tools and materials can not used.
In another experiment a tool made of two parts was used. The two were mounted· on separate force measuring instruments (dynamometers) and two parts together made up the rake face: While cutting a given work piece, forces transmitted to the two dynamometers were noted. Then a portion of tool near the cutting edge was ground off, thus changing the ratio of conitaal length on the two parts during cutting. The difference between the two sets obsenations gave the values of stresses on the length equal to that ground off ak &a:. The tool was again gr9und and measurement repeated. Thus stressa
could be determined.
Cl
a
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:;
ltact s of ~the
sses
Mechanics ofMachillillg Pnx:ews 125
Many investigators have approximated the distribution of normal stress on face by an exponential curve (Fig. 3.9). The shear stress is assumed to be up to a certain portion of contact length from the tool tip, which is called
-sticking zone". Beyond this zone the shear stress decreases to zero ~ly. In this zone called the slipping zone, Coulomb's law of friction is ....._....:~ to be applicable.
] ~ § "'
!1 ~ "' ]
~ 8 ~
.::= ~
~
Distance along rake face
Fig.3.8- Nonnal and Shear Stress Distribution on Rake Face.
The distribution function for the nonnal stress may be taken of the type
O"n=A.xm (3.16)
where, A and. m are constants and x is the· distance measured from the end of chip tool contact. This type of distribution was first suggested by Zorev [ 11}. Let O"max
be the maximum value of nonnal stress at the cutting edge. Then the above distribution may be expre8sed as
O"n = O"max. . (x I ln} m (3.11)
where, O"n = normal stress on rake face at a distance x from the end of contact length, and ln =natural contact length between tool and chip.
The total normal force on rake face is given by
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126 FUNDAMENTALS OF METAL CUTTINGANDMACHINETOOLS
A
Normal stress 0' n
~Slipping zone
__,F-+----- 12 ---.-1 B
Tool
Fig.3.9- Idealized Distribution of Stresses on Rake Face of Tool.
ln
Fn = b jumax(xll,r dx
0
Fn = bu max ./n (m+ 1)
(3 .;18)
(3.19)
The distribution of shear stress consists of two zones. In zone 1, near the cutting edge, the shear stress is. constant and equal to K = shear yield stress of the deforming material. In zone II, the shear stress is less than K and proportional to normal stress Un. In this zone, shear stress is given by Coulomb's law i.e.,
't = J.I.Un (3.20) The extent ( 12 ) of slipping friction zone may be determined from the condition
which gives,
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Mechanics of Machining~ 127
The total tangential force may be determined as follows.
F; = FtJ + F12 (3.23)
where Fa and F12 are the tangential forces in zones I and II respectively. Also,
Ftl = Kb ( ln -/2) (3.24) and
r/2 r/2 x ~2 = Jo b.JI.O" ndx = Jo b.JI.O"max ( f )m dx
n
The evaluation of the above integration gives
F _ Jl .b .a ( 1 ) m+J 12 - .max _2-'---
(m+l) (In )m (3.25)
Now average co-efficient of friction is given by
fl. = tan 13 = Ft I Fn
= (Fa + F12) IF;, (3.26)
After substituting the values of F;1• F12 and Fn from equations (3.24) ,(3.25) and (3. 19) respectively and after simplification we get
Jl =tan p = K. ( 1n -12 )(m + 1) + Jl.a max .(12 )m+l
(}" (1 )m+l max· n
(3.27)
3.3.7 Ploughing Force In the theoretical analyses described above, we have assumed that the
cutting edge (or the tool tip) is perfectly sharp. We may increase the sharpness of the tip by grinding the tool and sometime by lapping as well. But however hard one may try, the tool edge has a finite radius, however small it may be. Besides, just after the start of a cut, the sharpened cutting edge invariably loses some of sharpness due to impact, abrasion, etc. This small thickness (or radius) of the cutting edge increases the cutting force.
If we plot a graph between the cutting force components and depth of cut, we find that it depicts nearly straight line relationships. However, the lines relating the two variables do not pass through the origin. They make finite pasitive intercepts on the force axis for zero depth of cut. In the experiment we may grind the tool rake face to limit the length of contact between the tool and
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128 FUNDAMENTALS OF METAL CUTTING AND MACHINE TOOLS
chip and thus we would get a different line relating force and depth of cut. However at zero depth of cut, all these lines intersect at a point on the force axis. This shows that force values given by intercept must be due to the dullness or radius of the tool of the tool edge. The force components corresponding to the above intercepts are the components of the so-called ploughing force.
300
200
s ~ - 100-I
.D
.J:: J;l...
0 0
200
z ::: 100-
I
0 0
0.1 0.2 Undeformed chip thickness
(a)
O.l 0.2
Undeformed chip thickness (b)
0.3
0.3
0 0: = 10
0: =20 °
0
0: =10
0 0: =20
0 0: =30
Fig.3.10- Variation ofF, and r--;, with Uncut Chip Thickness
It is believed that on the tool edge, there is a ploughing action, e.g. the 'n ial bas to flow over the radius of tool edge. In view of the above, the cutting
iRe C**1 ....,. sa}'S F, and Fy may be written as sum of cutting and ploughing as given below.
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i.
If
e
the ting iring
Mechanics of Machining Processes 129
Fh = Fhp + Fhc
Fv=Fvp +Fvc
(3.28a)
(3.28b)
where, Fhp and Fvp are the components due to the ploughing action of the tool edge and Fhc and Fvc are those caused by the cutting action.
Fhp and Fvp may be taken equal to the intercepts at uncut chip thickness t = 0 on the corresponding graphs. If we consider that the tool tip is circular with radius r, we take the intercepts at t = r (1 + sin a) for obtaining the ploughing force components.
3.3.8 Chip Velocity Let us assume that the work material is moving against the cutting tool
with a velocity V. The surface layer shears across the shear plane AC and become part of the chip. In other word, the surface layer suffers a velocity discontinuity
·parallel to the shear plane. The chip velocity Vc along the tool rake face, is the ~ctor sum of V -the velocity of uncut chip and ~ - the velocity discontinuity :which is along the shear plane (Fig. 3 .11).
v ---Work piece
(a)
v (b)
Fig. 3 .II - Velocity Relationships in Orthogonal Cutting.
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130 FUNDAMENTALS OF METAL CUTTING AND MACHINE TOOLS
Applying properties of triangle to OAC (Fig.3 .11) we have
v (3.29) = = sin(90 +a-¢) sin(90-a)
Therefore, the chip velocity Vc and shear velocity V.. may be obtained as
vc = Vsin¢
cos(¢ -a) (3.30a)
vs V cosa =
cos(¢- a) (3.30b)
3.3.9 Cutting with Variable Uncut Chip Thickness In several machining processes, the chip thickness varies during cutting.
Examples are milling, grinding and gear cutting (bobbing, milling) etc. In these processes, during the action of a particular cutting edge, the uncut chip .thickness varies either from maximum to zero or from zero to maximum. Other cases of variable chip thickness arise during vibration and chatter of machine tools. The two cases in orthogonal cutting, when (i) the slope of the work piece surface is upward (positive) and (ii) the slope is downward (negative) are shown in Fig.3.12.There have been several attempts for finding the shear angle (> for unsteady or variable chip thickness cutting. The instantaneous shear angle (> may be expressed by the following empirical relationship.
(3.31)
where, o is the slope of surface ahead of tool edge, and f/>v is the shear plane angle when the slope ofthe work surface is zero. Values ofC as suggested by different researchers are given below:
Merchant Shaw and Sanghani Wallace and Andrew Oxley Kobayashi and Shabaik
c = 0.50 c = 1.00 c = 0.75 c = 0.20 c = 1.00
(3.32a) (3.32b) (3.32c) (3.32d) (3.32e)
Such a wide variation in the value of C is not easily explained unless it is a •ed that C is not a constant but depends on cutting conditions including rake -sic a.. safaz slope o and the steady state shear angle (> o [ 19 ] . It seems
0 Rl ~ a ar
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~~. . ~ ---.~
))
l)
b)
tg. ~se
ess of
"be : is in
for tay
1)
gle ·ent
Za) lb) lc) Zd) le)
t is ake :ns
Mechanics of Machining Processes 131
therefore, that the way of expressing t/J in terms of tjJ 0 and o as given in Equation (3.31) may not be the best way. Nevertheless, C = 1 has often been adopted. ·
Work piece
(a)
Workpiece
(b)
Fig. 3.12- Variable Uncut Chip Thickness
3.4 OBLIQUE CUTTING
Many machining processes are in fact examples of oblique cutting. Orthogonal cutting is only a particular case of oblique cutting. The practical requirements of rake and other angles on the tool mean that the ideal conditions of orthogonal cutting are rarely achieved.. However, the analysis of oblique cutting is much more difficult than that of orthogonal cutting. That is why there are very few investigations on this topic.
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132 FUNDAMENTALS OF METAL CUTTING AND MACHINE TOOLS
3.4.1 Direction of Chip Flow In orthogonal cutting the chip flows in a direction normal to the cutting
edge. However this is not so in oblique cutting. It flows at an angle to the normal. The angle between the normal to cutting edge and chip velocity vector is called the chip flow angle. This is an important variable in oblique cutting and hence is its determination.
Several methods have been employed to determine the chip flow angle. These are listed below.
(i) By observing the direction of scratches produced on the tool. This is made more observable by smearing a dye on the tool rake face before cutting.
(ii) By taking photographs of the cutting process and then analyzing the photographs.
(iii) By analyzing the deformation of chip. This method is simple and accurate.
(iv) By analyzing the cutting forces. This is based on the fact that the shear component on the rake face should be in the direction of chip flow.
The first method mentioned above is very approximate because of the fact that the contact length is extremely small. It is rather difficult to exactly determine the flow direction. The second method, i.e. by photographing the cutting process is· more accurate. The camera axis is fixed normal to tool rake face. The chip flow direction is determined directly from the photographs. In another version, the camera axis is kept normal to machined surface and the projected chip flow direction is measured. Knowing the normal rake angle, the chip flow angle can be analyzed. According to Stabler, chip flow angle 11f can be approximated as follows.
If/= CI (3.33)
where, C is a constant, and I the angle of inclination I obliquity between the planes ABDEF (Fig.3.13) which is perpendicular to the machined surface and contains the cutting velocity vector and the plane ACDGH which is normal to both the machined surface and the cutting edge KAL. I is also equal to the angle made by the cutting edge KAL with AJ, the normal to the cutting velocity vector in the plane of the machined surface. The constant Cis found to vary between 0.9 and 1.0. However, experiments of Russel and Brown (23] have shown that
1/f = tan -I (tan I. tan a,) (3.34)
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Mechanics of.WacJ.ini,rg Proceua ID
cutting tormal. called
ence is
E
angle.
This is before
ing the
le and
tat the >f chip Workpiece J
H
~e fact Fig.3.13- Velocity Rake Angle and Normal Rake Angle. :xactly ng the >I rake 40 >hs. In nd the le, the can be J 30
.5
:3.33) ?- 20
.£ en the ~ ~and ~ ntal to 0 10 c;::
:angle ]' vector u :en 0.9
0
0 10 20 30 40 3.34) • Angle of obliquity - I in degrees
f Fig.3.14- Relationship Between Angle of Obliquity and Chip Flow Angle.
=-=--=~
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134 FUNDAMENTALS OF METAL CUTTING AND MACHINE TOOLS
Fig. 3.14 shows the experimental results of Brown and Armarego [22] regarding chip flow angle versus the angle of obliquity. Chip flow direction is calculated from measured forces. The chip flow angle '1/ =I, for an =0 , '1/ > I for a,. < 0 and '1/ < I for an >0.
3.4.2 Rake Angles in Oblique Cutting The basic angles in oblique cutting depend upon the following:
(i) The rake angle ground on the rake face oftool. (ii) The angle of obliquity.
A change in any one of these angles changes the machining conditions and affects the cutting forces.
In oblique cutting, the chip flow on the rake faoe is not normal to cutting tool edge but is inclined to it. The inclination is towards that side of the normal in which the chip experiences smaller resistance or a larger rake angle than the nominal rake ground on the rake face. Therefore, in oblique cutting the rake angle may be measured in more than one plane, and hence, more than one rake angles can be defined for a given tool and angle of obliquity. The different rake angles in oblique cutting are named as follows.
i. Normal rake angle an. ii. Velocity rake angle a,. iii. Effective rake angle ae.
Normal Rake Angle (also called primary rake angle) (an): It is the angle between the rake face and a line perpendicular to cutting veiocity vector measured in a plane normal to the tool cutting edge (see Fig. 3.15a).
Velocity Rake Angle (av): This is the angle between the rake face and the line perpendicular to the cutting velocity measured in a plane parallel to the cutting velocity and normal to the machined surface (Fig.3.15b).
Effective Rake Angle ( ae): It is the angle between the rake face and a line normal to cutting velocity measured in a plane containing cutting velocity vector and chip flow velocity vector (Fig.3 .15c).
Fig.3 .16 shows the geometric relationship between the normal rake angle, the velocity rake angle and effective rake angle. AF is the direction of chip flow on the rake face of tool. AD is normal to cutting velocity vector and lies in the plane AFD NO which passes through the chip flow direction and contains cutting 1docity vector .. Hence .LDAF = ae. The angle LBAF {)n the rake face is equal to cap flow aagle ~- 1be plane ABCIK is normal to the machined surface and
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136 FUNDAMENTALS OF METAL CUTTING AND MACHINE TOOLS
The line BE is normal to FD. The points B, C, G, D, E, and Fall lie in a single plane parallel to the machined surface.
Fig. 3.16- Relationships Between Different Rake Angles.
tan a.. = BCI HC = BG I AG cos I = tan an I cos I (3.35)
For the effective rake angle we have the geometric relationship (see Fig.3.16).
. DF FE+ED sma =-=---
e AF AF
From geometry ofFig.3.16, ED= BC and L BCD = 90°. Therefore
. FE FB BC BG AB sma =-x-+-x-x-
e FB AF BG AB AF Therefore,
=sin I sin If/+ cos I sin an cos If/ (3.36)
l.A.J CJ~ttiDK Ratios iJI Oblique Cutting 11M: nlfrin& nlios in oblique cutting are slightly different from those in
in this case we cannot take the width of the chip to be
~==-----
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------------ =-===--=-
Mechanics of Machining Processes 137
equal to the width of the uncut chip (Fig.3.17). Let I, t and b be the length, thickness and width of the uncut chip respectively and let leo fe and be be the corresponding values for the cut chip. The different ratios are then expressed as fOllows.
Fig. 3.17 -Cutting Ratios in Oblique Cutting
Also the volumetric ratio
Chip length ratio = r1 = I I le Chip thickness ratio = r, = t I fe Chip width ratio = rb = b I be
(3.37a) (3.37b) (3.37c)
I I le X fIfe X b I be = I (3.38) kcause there is no change in the volume of material undergoing plastic *furmation. From Eq. (3.38), we have
_t_ be fe (3.39)
fe b /
r 1 = 1 I ( rb r1 ) (3.40)
3.4.4 Velocity Relationship in Oblique Cutting In both oblique and orthogonal cutting, the material starts deforming quite
lllead of the tool edge and some portions continue to deform even on the rake 6ce (secondary deformation). This calls for a thick zone model, the analysis of which is nowadays quite feasible in view of developments, in finite element tldmique. However, if we take a simplified model that work material suffers IIM:ar deformation over a thin shear plane and then passes on to form the chip, 6c analysis of velocities becomes very simple. In this model there are three wlocities, viz.
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138 FUNDAMENTALS OF METAL CUTTING AND MACHINE TOOLS
T1te uncut chip velocity V. n The chip velocity Vc. m The shear velocity V.. on shear plane.
Now, vector Vc is the vector sum of V and v .. Here'bold letters indicate a vector quantity and V, V, and Vc are the magnit.udes of V, Vs and Vc respectively.
(3.41)
The above three vectors lie in one plane which is also the plane in which we measure the effective rake angle .
..
v
Fig.3.18- Velocity Relationships
From the geometry ofthe Fig.3.18,
and
v; v
=
sin¢e
cos( ¢e -q;) (3.4~
these expression can be put in more convenient forms by putting the above velocities as function of (i) normal rake angle, (ii)Angle of obliquity I, and (iii) Chip flow angle. This consideration results in the following expression.
vc = sin tPn cos I
(3. v cos (tPn -an ) . cos If/ and
v s cos I cos an v cos If/ cos (tPn -an)
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~
I)
1ve iii)
44)
45)
~
l ··~-=--!
Mechanics of Machining~ 139
3.4.5 Shear Angle in Oblique Cutting The shear plane is fonned by two lines, namely the shear velocity vector
Vs and the cutting edge. As is the case with rake angles in oblique cutting, the shear angle could also be measured in different planes. From the velocity diagram of Figure 3.18, we may obtain the effective shear angle ¢e.by simplifying Equation 3.42.
tan ¢e = Vel V COS ¢e (3.46)
1 - Vc I V sin <le
Normal Shear Angle The shear plane angle could also be measured in a plane nonnal to cutting
edge. The resulting expression is similar to that in orthogonal cutting.
tan ¢n = (tltc) COS Un
1 - (tltc) sin Un
3.5 MECHANICS OF TURNING PROCESS
(3.47)
Chip Flow Direction : A turning tool has a complex shape. As explained in Chapter 2, it has many angles ground on it. Besides there is a radius at the tool point (or the tool tip) called the nose radius. The chip flow is particularly affected by back rake angle, side rake angle, side cutting edge angle, end cutting edge angle and nose radius. In the presence of a nose radius, the detennination of chip flow direction becomes rather complex [33] and hence, for the time being, we shall assume a zero nose radius.
s R
A
R.H. turning tool
Fig. 3.19- Chip Flow Direction in Turning.
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140 FuNDAMENTALS OF METAL CUTTING AND MACHINE TOOLS
The top view of tool having side cutting edge angle Ys and end cutting edge angle r~ is shown in Fig.3, 19. The engagement length of side cutting edge during cutting is shown by PS and feed by PQ = SR. Based on the research work by Colwell, the chip flow direction may be taken along normal to the major axis PR of the parallelogram PSRQ. Accordingly the chip flow angle '1/ is calculated as follows. Draw RT perpendicular to PQ', BC perpendicular to PR and BA perpendicular to PQ. Then L\jl = .L ABC= .LRPQ. Therefore,
tan 'I' = RT I PT = RT I ( PQ + QT)
or tan 1!f = d I ( f + d tan Ys ) (3.48)
In the presence of a finite nose radius, the treatment becomes complex. However if the nose radius is very small compared to d, it may be subtracted from d and equation (3.49) may be used.
3.5.1 Effective Rake Angle in Turning Fig.3.20 shows a right hand turning tool whose cutting edge has been
ground with back rake angle ab and side rake angle as. The chip flow direction is shown by angle f//between PB and PE (Tl-te point Pis at the nose or point of the tool). The nose radius has been assumed to be zero.
p
Fig. 3.20- Effective Rake Angle in Turning.
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------
Mechanics of Machining Processes 141
From the geometry of the above figure, we have the lines PB, PE and PS in the same plane, which is parallel to the base of the tool. The line PB is parallel to axis of tool and PS is perpendicular to it. The side rake, angle is given by LCPS = tzs. Similarly ab = back rake angle = LAPB. From the point E, we draw a vertical line EG onto the nike face. The line PG lies on the rake face and is in the direction of chip flow. The line CF is parallel to SE. Therefore angle LFCG = ab
Now, SC = EF. And, in triangle PEG , LPEG = 90 ° and LEPG = ae
tan a = EG = EF + FG = CS + FG = CS x PS + FG x CF {3.49) e PE PE PE PE PS PE CF PE
Now in the triangle PSE
And PSI PE = TE I PE = sin If/
CF=SE=PT
. . SE I PE = PT I PE = CF I PE = cos If/
Therefore, substituting these values in Equation (3.49}, we get
tan ae = tan as sin If/ + tan ab cos If/ or
tXe = tan"1[tan as sin If/ + tan ab cos If/]
3.5.2. Power and Forces in Turning
(3.50)
The cutting force in turning can be resblved into three components (Fig.3.21 ), namely
i. Tangential component (Fe) ii. Radial component (Fr) iii. Axial component (Fa)
The tangential component Fe is in the direction of cutting velocity vector. The radial component, as the name denotes, is in the direction of the radius of the job. The third component is parallel to the axis of the job, i.e. in the direction of feed. Since there is no radial movement of tool in turning, there is no work done by the radial component of force. The other two components do the work. However, the feed velocity is so small that the product of the feed force and feed velocity is negligible compared to the product of the tangential force and cutting velocity.
Rate of work in turning == Power (P) needed for turning
~tangential force component x cutting velocity
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'" . : ~
142 FUNDAMENTALS OF METAL CUTTING AND MACHINE TOOLS
Fe Fi!;!. 3.21 - Forces in Tuming
. F xV P = c watts
60 (351)
where Fe is in Newtons and ~ r is in m/min.
The cutting force components in tuming may be represented in exponential form involving feed and depth of cut as given below.
(3.52a)
The value of R1, R2. R3 ar.td the exponents x,, y,, x2. Y2 and X3, YJ for a few materials are given in Table 3.1 for the forces inN.
