A

Report

On

MODELLING OF CUTTING FORCES IN PERIPHERAL MILLING USING MECHANISTIC APPROACH

By

Shashank Pendyala 2011A4PS272P

Ankur Naik 2011A4PS183P

Prepared in Partial fulfilment of the

Design Project

Course No: ME F376

At

BIRLA INSTITUTE OF SCIENCE AND TECHNOLOGY,

PILANI (May, 2014)

Acknowledgement

We take this opportunity to express our profound gratitude and deep

regards to Prof. Tufan Chandra Bera Sir for his exemplary guidance, monitoring

and constant encouragement throughout the course of this project. We would

like to express our special gratitude and thanks to our institute for giving us this

opportunity to take up this course.

Milling Process

The milling operation is an intermittent cutting process using a cutter with one

or more teeth. A milling cutter is held in a rotating spindle, while the work piece

clamped on the table is linearly moved toward the cutter. Each milling tooth

therefore traces a trochoidal path, producing varying but periodic chip thickness

at each tooth passing interval. Depending on the work piece geometry, different

milling cutters and machines are used. In this section, the mechanics of the

milling process are presented for simple face milling operations. Mechanics of

other milling operations are modelled by geometrically extending the mechanics

of face milling. Double negative tools are shock resistant in heavy-duty face

milling operations. Rigid and high-power milling machines are suitable for

heavy machining with negative cutters. For accurate and light milling

operations, double-positive milling cutters are ideal. Negative positive tools

produce a good surface finish and are efficient in removing the chips from the

insert pockets. There are three types of milling operations used in practice:

face milling operations, in which entry and exit angles of the milling

cutter relative to the workpiece are nonzero;

up-milling operations, in which the entry angle is zero and the exit angle

is nonzero; and

down-milling operations, in which the entry angle is not zero and the exit

angle is zero.

Both up- and down-milling operations are called peripheral or end milling

operations. In milling the instantaneous chip thickness (h) varies periodically as

a function of time-varying immersion. The chip thickness variation can be

approximated as

h() = c sin ,

where c is the feed rate (mm/rev-tooth) and is the instantaneous angle of

immersion. First, the helix angle is considered to be zero, which is the case in

face milling operations with inserted cutters. Tangential ( ()), radial ( ()),

and axial ( ()) cutting forces are expressed as a function of varying uncut

chip area (ah()) and edge contact length (a) as follows:

() = ah() + a,

() = ah() + a,

() = ah() + a,

where , , and are the cutting force coefficients contributed by the

shearing action in tangential, radial, and axial directions, respectively, and

, , and are the edge constants. If we assume zero nose radius and

zero approach angle on the inserts, the axial components of the cutting forces

become zero ( = 0). The cutting coefficients are assumed to be constant for a

toolwork material pair, and they can be evaluated either mechanistically from

milling tests or by using the classical oblique cutting transformations. They are

sometimes expressed as a nonlinear function of the instantaneous or average

chip thickness ha. The average chip thickness per revolution is calculated from

the swept zone as

The instantaneous cutting torque ( ) on the spindle is where D is the diameter

of the milling cutter. Horizontal (i.e., feed), normal, and axial components of the

cutting forces acting on the cutter are as follows:

() = cos sin ,

() = + sin cos ,

() = + .

It must be noted that the cutting forces are produced only when the cutting tool

is in the cutting zone, that is, (), (), () >0 when ,

where and are the cutter entry and exit angles, respectively. Another

important point is that there may be more than one tooth cutting simultaneously

depending on the number of teeth on the cutter and the radial width of cut. The

tooth spacing angle (or cutter pitch angle) is given as

=

where N is the number of teeth on the cutter. There will be more than one tooth

cutting simultaneously when the swept angle ( = ) is larger than the

cutter pitch angle (i.e., > ). When more than one tooth cuts

simultaneously, the contribution of each tooth to total feed and normal forces

must be considered. It must also be noted that, because each tooth will be away

from its neighbouring tooth by the amount of pitch angle, the uncut chip

thickness removed by each cutting edge will be different at an instantaneous

position of the cutter. We can formulate the total feed, normal, and axial forces

as

when ever j . Each term in the summation block represents the

contribution of each tooth to the cutting forces. If the tooth j is out of the

immersion zone, it contributes zero to total forces. The instantaneous resultant

cutting force on the cutter (or work piece) is given as

Instantaneous cutting torque on the spindle is

where D is the diameter of the cutter. The cutting power ( ) drawn from the

spindle motor is

where V = Dn is the cutting speed and n is the spindle speed. For a given set of

cutting conditions, the engineer may be required to predict the maximum cutting

power, torque, and cutting forces required from the machine tool spindle and

feed drives. The cutting forces, torque, and power are uniformly periodic at

tooth passing frequency. Periodic cutting forces dynamically load and unload

the machine tool structure, work piece, and the cutter at each tooth period. Half-

immersion (i.e., b = D/2) up- and down milling forces have opposite trends. The

chip load starts with zero and gradually increases to maximum at the exit in up-

milling; hence, forces have the same trend. However, the tooth experiences

maximum chip load during entry followed by a gradual decrease of the chip

load and, hence, the cutting forces. Manufacturing engineers are advised to use

up-milling operations for heavy metal removal rates where the shock loading is

reduced. For light finish cuts, down-milling is preferred to obtain a smooth

surface finish.

Mechanics of Helical End Mills

Periodic loading causes cyclic mechanical and thermal stresses on the tool,

which leads to a shorter tool life. Helical end mills are used to dampen the sharp

variations in the oscillatory components of the milling forces, and they are used

when the depth of cut is large, but the width of cut is small. Their primary

function is peripheral milling, where the walls of parts are the target finished

surface. The helix on the cutter provides a gradually increasing chip load along

the helical flutes of the end mill. If the helix angle on the cutter is , a point on

the axis of the cutting edge will be lagging behind the end point of the tool. The

lag angle ( ) at the axial depth of cut (z) is found as

and

When the bottom point of a reference flute of the end mill is at immersion angle

, a cutting edge point that is axially z [mm] above will have an immersion

angle of ( ). Obviously, the chip thickness removed along the flutes axis

will also be different at each point.

Prediction of Cutting Forces-Algorithm

The ideology is to divide the cutting tool into numerous discs and calculate the

cutting forces on each disc. Every disc is again divided into segments. In the

code the height of each disc is taken as 0.0001m and one segment per degree

making a total of 360 segments per disc.

The inputs such as the Axial depth of cut (a), Radial depth of cut (c), Feed (f),

the Cutting co-efficients Ktc, Kte, Krc, Kre which are obtained from experimental

data, Helix angle of the tool (beta), Diameter of the cutter (D), Number of

cutting edges (N) are given to the program. The entry angle (phi_st) and the exit

angle (phi_ex) of the cutter are also specified in the code.

To add up all the segments, the angular increment (deltaPhi) and the increment

in depth of cut to represent each disc (deltaA) are specified.

The values of the 3 forces feed force Fx(m), normal force Fy(m) and tangential

force Ft(m) are initialised to zero.

A for loop is written to get the immersion angle of flutes bottom edge and is

defined as phiI. Another for loop is written inside the loop to get the

immersion angle for each cutting edge (defined as phi1). A third for loop is used

inside the second loop to update the immersion angle due helix for each axial

element (defined as phi2). The fourth for loop inside the third loop is used to

add all the forces from each segment and disc. The chip thickness (h) at every

point is calculated using the updated phi2 which is placed inside the 4th loop.

To find the variation of forces on the section of cutter engagement on the work

piece, a graph is plotted for the updated immersion angle against the total

cutting force at each point. But this force is made negative of the obtained force

as it is equal to magnitude on the tool but in the opposite direction.This is

placed in the fourth for loop to ensure every force is plotted.

Cutting Forces-Flow chart

Results

The forces graph for 1 cutter rotation is obtained.

Red- Feed force (Fx)

Black- Normal force (Fy)

Green- Tangential force (Ft)

The graph is observed to be periodic which proves it as theoretically correct.

Fig: Force vs. Cutter rotation angle.

The variation of forces on the cutter engagement section is obtained at a

particular instant. For example at the angular increment of r=10 degrees the

graph is as follows:

At r=50, the graph is:

References

Kline, W. A., "The Prediction of Cutting Forces and Surface Accuracy

for the End Milling Process," Ph.D. thesis, University of Illinois at

UrbanaChampaign, 1982.

X.-W. Liu, K. Cheng, D. Webb, X.-C. Luo, Prediction of cutting force

distribution and its influence on dimensional accuracy in peripheral

milling, School of Engineering, Leeds Metropolitan University, City

Campus, Leeds LS1 3HE, UK

Text Book by Yusuf Altintas, Manufacturing Automation , Fellow of

the Royal Society of Canada and NSERC Pratt & Whitney Canada

Research Chair Professor of Mechanical Engineering and Director of the

Manufacturing Automation Laboratory at the University of British

Columbia

Professor B. Balachandran, Dynamics of Low Immersion Milling,

Sigmund Max Young, Master of Science, 2008

Oscar Gonzalo, Jokin Beristain, Haritz Jauregi, Carmen Sanz, A method

for the identification of the specific force coefficients for mechanistic

milling simulation

Abhijit Bhattacharyya, John K. Schueller, Brian P. Mann, John C.

Ziegert, Tony L. Schmitz, Fred J. Taylor, Norman G. Fitz-Coy, A closed

form mechanistic cutting model for helical peripheral milling of ductile

metallic alloys.