Theory Of Metal Cutting-mechanics Of Metal Cutting

LECTURE-09 THEORY OF METAL CUTTING - Mechanics of Metal Cutting

Px Py Pz

NIKHIL R. DHAR, Ph. D. DEPARTMENT OF INDUSTRIAL & PRODUCTION ENGINEERING BUET

Mechanics of Metal Cutting The force acting on a cutting tool during the process of metal cutting are the fundamental importance in the design of cutting tools. The determination of cutting forces necessary for deformation the work material at the shear zone is essential for several important requirements:  

 

to estimate the power requirements of a machine tool to estimate the straining actions that must be resisted by the machine tool components, bearings, jigs and fixtures to evaluate the role of various parameters in cutting forces to evaluate the performance of any new work material, tool material, environment, techniques etc. with respect to machinability (cutting forces)

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The force system in the general case of conventional turning process is shown in the following Figure. Px Py

Px

Pxy

Pz

R

Py Pz

Px = feed force in the direction of the tool travel Py = thrust force in the direction perpendicular to the produced surface Pz = cutting force or main force acting in the direction of the cutting velocity. Department of Industrial & Production Engineering

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R XY

Y

Z Pz

Pxy X

Px = Pxy sinϕ ..................[1]

R

Pxy φ Py Px

Πo

Py = Pxy cosϕ .................[2]

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Several forces can be defined relative to the orthogonal cutting model. Based on these forces, shear stress, coefficient of friction, and certain other relationships can be defined.

Pn R2

β

Chip

Ps Pxy F

R R1

Workpiece Department of Industrial & Production Engineering

γ N

Pz

Tool 18/5

Merchant Circle Diagram (MCD) The following relationships suggest a circle representation of forces as done by Merchant and indicated in the following Figure.

R = F + N = P + P = P + P .......[3] s n z xy F = Pz sin γ o + Pxy cos γ o .....................[4]

Pxy

N = Pz cos γ o − Pxy sin γ o ....................[5] Ps = Pz cos β − Pxy sin β ......................[6] Pn = Pz sin β + Pxy cos β .....................[7]

From Equation [4] and [5] F Pz sinγ o + Pxy cosγ o μ= = = tanη .....[8] N Pz cosγ o − Pxy sinγ o Where, μ = kinetic coefficient of friction η = mean angle of friction at the rake surface Department of Industrial & Production Engineering

η N

Pn R

F

Pz

β

η-γo Ps

γo

Chip

γo Tool

Workpiece 18/6

From the geometry of force relations of MCD circle

Pz = R cos(η − γ 0 )................[9] Ps = R cos(β + η − γ 0 )...........[10] Fron Equation [9] and [10]  cos (η − γ 0 )  Pz = Ps  ..........[11]  cos (β + η − γ o )  Based on the shear force, the shear stress (τs) which acts along the shear plane between the work and the chip is:

Pxy η N

R F

τs =

Ps S t , where As = area of the shear plane = o As sinβ

τs =

Ps sinβ ..........[12] So t

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Pn

Pz

β

η-γo Ps

γo

Chip

γo Tool

Workpiece 18/7

From Equation [11]and [12]   cos(η − γ 0 ) Pz = τ s S0 t  ...........[13]  sinβ cos(β + η − γ o )  Similarly,   sin(η − γ 0 ) Pxy = τ s S0 t  .......[14]  sinβ cos(β + η − γ o ) 

Pxy η N

Pn R

F

Pz

η-γo Ps

γo In metal cutting one of the main problem is to evaluate the cutting forces Pz and Pxy from the given cutting conditions and initial properties of work material and it is necessary to determine τs, β and η by suitable relationships.

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β

Chip

γo Tool

Workpiece 18/8

Earnest-Merchant Theory Ernst and Merchant extended their analysis and studied the relationship between the shear angle and the cutting conditions. They suggested that the shear angle always takes the value that reduces the total energy consumed in cutting to a minimum. Because the total work done in cutting is dependent upon and is a direct function of the component Pz of the cutting force, they developed an expression for Pz in terms of β and the constant properties of the workpiece material. Condition for maximum cutting force (Pz) from Equation [13]

dPz dP d  τ s S0 t cos(η − γ 0 )  = 0, or, z = . =0  dβ dβ dβ  sinβ cos(β + η − γ o ) 

 cosβ cos(β + η − γ 0 ) − sinβ sin(β + η − γ 0 )  τ s S0 t cos(η − γ 0 )  =0 2 { sinβ cos(β + η − γ 0 }   π cosβ cos(β + η − γ 0 ) − sinβ sin(β + η − γ 0 ) = 0, or cos(β + β + η − γ 0 ) = 0 = cos  2 π η γ0 β= − + ..................[15] 4 2 2 Combining Equation [13] and [15] Pz = 2 τ s S0 t cotβ .............[16] Department of Industrial & Production Engineering

18/9

Merchant Theory Merchant modified the relationship derived by Earnest-Merchant, by assuming that the shear stress along the shear plane varies linearly with normal stress (σn). It is given as (from the following Figure). τ = τ + k σ ................[17] s 0 n From the geometry of force relations of MCD P = R cos(β + η − γ ) and P = Rsin(β + η − γ ) ................[18] s 0 n 0 τs P = P tan(β + η − γ ) n s 0 P P n = s tan(β + η − γ ) 0 A A s s σ = τ tan(β + η − γ ) ....................[19] n s 0 From Eqation [17] and [19] τ 0 τ = .......................[20] s 1 − k tan(β + η − γ ) 0

[

]

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k

τ0 σn

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Combining Equation [13] and [20] τ o S 0 t cos(η − γ 0 ) Pz = ...............[ 21] sin β cos( β + η − γ 0 ) {1 − k tan( β + η − γ 0 )} Condition for maximum cutting force (Pz) from Equation [21] dP τ o S 0 t cos(η − γ 0 ) z = 0, or dPz = d =0 dβ dβ dβ sin β cos( β + η − γ 0 ) {1 − k tan( β + η − γ 0 )} cos β cos( β + η − γ 0 ) − sin β sin( β + η − γ 0 ) −  τ s S 0 t cos(η − γ 0 )  k cos β sin( β + η − γ 0 ) − k sin β cos( β + η − γ 0 )  or =0 2 [ sin β cos( β + η − γ 0 ) − k sin β sin( β + η − γ 0 )] cos(2β + η − γ ) − k sin(2β + η − γ ) = 0 0 0 cot(2β + η − γ ) = k 0 2β + η − γ = cot − 1(k) = c ≅ 800 to 850...........................[22] 0 From Equation [21] and [22] P = τ S t[ cotβ + tan(c − β) ]..........[ 23] z s 0 Department of Industrial & Production Engineering

18/11

Lee and Shaffer Theory According to this theory the shear occurs on a single plane. So for a cutting process according to this theory, the following are supposed to hold good:   

The material ahead of the cutting tool behaved as ideal plastic material The chip does not get hardened The chip and parent work material are separated by a shear plane.

Lee and Shaffer derived the following relationship as: π β + η − γ 0 = ......................[24] 4 From Equation [13] and [24]

Pz = τ s S0 t ( cot β + 1)..............[25] Where,

cot β =

1 ξ − sin γ o = ≅ ξ − tan γ o tan β cos γ o

Pz = τ s S0 t ( ξ − tanγ 0 + 1)............[25]

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18/12

Thermal Aspect of Chip Formation Machining is inherently characterized by generation of heat and high cutting temperature. At such elevated temperature the cutting tool if not enough hot hard may lose their form stability quickly or wear out rapidly resulting in increased cutting forces, dimensional inaccuracy of the product and shorter tool life. The magnitude of this cutting temperature increases, though in different degree, with the increase of cutting velocity, feed and depth of cut, as a result, high production machining is constrained by rise in temperature. This problem increases further with the increase in strength and hardness of the work material. Knowledge of the cutting temperature rise in cutting is important, because increases in temperature: 

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adversely affect the strength, hardness and wear resistance of the cutting tool cause dimensional changes in the part being machined, making control of dimensional accuracy difficult and can induce thermal damage to the machined surface, adversely affecting its properties and service life.

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18/13

In addition, the machine tool itself may be subjected to temperature gradients, causing distortion of the machine. The main sources of heat in metal cutting are shown in the following Figure. These three distinct heat sources are: 





the shear zone (q1), where the main plastic deformation takes place the chip-tool interface zone (q2), where secondary plastic deformation due to friction between the heated chip and the tool takes place the work tool interface (q3), at flanks where frictional rubbing occurs.

q1 q3

Workpiece

Chip

q2

Tool

The heat balance in chip formation can be written as : Amount of heat away in chips + Amount of heat remaining in the   Total amount    = cutting tool + Amount of heat passing into the workpiece + Amount of heat generated     of heat radiated into the surrounding air   Department of Industrial & Production Engineering

18/14

Various studies have been made of temperatures in cutting, based on heat transfer and dimensional analysis, using experimental data. A simple and approximate expression for the mean temperature for orthogonal cutting is

0.4 U  Vc t  T=   ρC  K 

0..333

where, T = mean temperature rise at the tool-chip interface (oC) U = specific energy in the operation (N-m/mm3) Vc = cutting velocity (m/sec) t = depth of cut (mm) pC = volumetric specific heat of the workpiece (J/mm2-C) K = thermal diffusivity (ratio of thermal conductivity to volumetric specific heat) of the workpiece material (m2/sec).

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18/15

Exercise The dynamometer recorded the following, feed force 200 kg, cutting force 300 kg. The rake angle of the tool used was 10o. The chip thickness ratio 0.35. Find    

Shear angle (β) Shear force (Ps) Co-efficient of friction at the chip-tool interface (μ) and the friction angle (η) Compressive force at the shear plane (Pn).

A seamless tube 3cm outside diameter is reduced in length on a lathe with the help of a single point cutting tool. The cutting speed is 40 m/min, the depth of cut is 0.125mm. The length of continuous chips, for one revolution of the tube, on measurement comes to be 17.77cm. The cutting force is 200 kg and the feed force is 75 kg. the rake angle of the tool is 35o.Calculate,     

Co-efficient of friction Chip thickness ratio Shear plane angle Velocity of the chip along the tool face Velocity of shear along the shear plane

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18/16

During the machining of AISI-1025 steel, with 0-10-6-6-8-90-1 (mm) ORS shaped tool the following observations were taken: Feed 0.50 mm/rev  Depth of cut = 2.0 mm  Cutting speed = 40 m/min  The shear angle = 20o  The power consumed while machining= 3kW  The power consumed while running idle = 0.50 kW Calculate:  The shear force  Chip thickness ratio  Normal pressure on the chip  Chip thickness 

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Any questions or comments?

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