Prediction of Cutting Force in a Circular Peripheral ... · circular peripheral milling, numerical...

16

International Journal of Mechanical Engineering and Robotics Research Vol. 7, No. 1, January 2018

© 2018 Int. J. Mech. Eng. Rob. Res.doi: 10.18178/ijmerr.7.1.16-21

Prediction of Cutting Force in a Circular

Peripheral Milling Process

Kamel Mehdi1,2

1URMSSDT, Engineering National High School of Tunis (ENSIT), University of Tunis (UT), 5 Avenue Taha Hussein

P.B. 56 Bab Mnara 1008 Tunis, Tunisia 2Preparatory Institute for Engineering Studies El Manar (IPEIEM), University of Tunis EL Manar (UTM), PB. 244,

2092 Tunis, Tunisia

Email: [email protected]

Abstract—In this paper, we present a two-dimensional

machining model that allows the simulation of cutting force

including the cutting process damping in a dynamic circular

milling process. The cutting forces obtained through the

simulation model are compared with experimental results,

and a discussion on the effect of various milling parameters

on cutting force, such as cutting speed, feed rate, radial

depth of cut, tool diameter, and tool helix angle, in a

circular peripheral milling process will be presented.

Index Terms—cutting force, cutting damping process,

circular peripheral milling, numerical simulation.

I. INTRODUCTION

Prediction of cutting force has been the subject of

different research works for a long time [1–4]. The study

of the dynamic behavior of the cutting system (machine,

tool, and workpieces) can be analyzed from the responses

of the cutting forces to the fluctuations of the cutting

parameters (cutting speed and uncut chip thickness).

Montgomery and Altintas [5] studied the mechanism of

cutting force and surface generation in dynamic milling by

presenting a simulation model for dynamic milling. Huang

and Wang [6] proposed an analytical model of the milling

process including damping effects. Two cutting

mechanisms (shearing and ploughing mechanisms) and

two process damping effects (directional and magnitude

effects) are included. In 2012, Mehdi and Zghal [7]

proposed a numerical model that allows the prediction of

cutting forces in a linear peripheral milling process. They

studied the effects of tool parameters (diameter, helix

angle, and number of teeth) on cutting process damping

and cutting force distributions. In 2015, Li et al. [8]

confirmed the importance of taking into account the

cutting damping process in their research on the

machining of thin-walled workpieces.

All these traditional researches on the cutting force

model are usually focused on linear milling and do not

take into consideration other cutting conditions, especially

in circular milling processes.

Manuscript received July 1, 2017; revised December 21, 2017.

In this paper, we present a cutting force model for

circular peripheral milling based on the linear peripheral

milling force model of Mehdi and Zghal [7]. In the first

section, a geometric model of cutting forces, including the

cutting process damping in circular peripheral milling, is

presented. In this model, the regenerative chatter is

considered. In the second section, the simulated cutting

force components are compared to experimental results,

and a discussion on the effects of cutting and tool

parameters (cutting speed, feed rate, axial and radial depth

of cut, tool diameter, tool helix angle, and number of

tool’s teeth) on process damping is presented.

II. MECHANICS OF THE CIRCULAR PERIPHERAL

MILLING PROCESS

The circular peripheral milling process is assumed to be

a linear peripheral milling process for a length equal to

pR (Fig. 1). In the model, this length is considered to

be equal to the feed rate per tool rotation tN f in the x

direction. Hence, t

p

N f

R , where pR is the circular

workpiece profile radius, f is the feed rate per tooth,tN

is the number of teeth, and is the tool’s angular

change position vis-à-vis the workpiece for each feed rate

per tool rotation.

From Fig. 1, we can see that the feed rate per tooth in

the x direction can be expressed in the two directions

0 0( , )x y by

0 0cos sin ,xf f x y (1)

where 1

p

i

, in which p is the pth position of the

cutter center.

In the case of the linear peripheral milling and

according to the previous work of [7], the instantaneous

total cutting force vector ( )F t acting by the tool on the

workpiece in the ( , )x y directions is given by

17


© 2018 Int. J. Mech. Eng. Rob. Res.

1

( , )

1 ( , )

( )( )

,( )

( )

t

j

t

j

N

x

jx

Ny x y

y

j x y

F tF t

F tF t

(2)

According to [7], ( )F t is the sum of the shearing

cutting force ( )sF t and the damping cutting force ( )dF t

due to the ploughing mechanism. The damping cutting

force ( )dF t is arranged to be calculated by

( , ) ( , )

( ) ( )( ) ,

( ) ( )

dx x

d c

dy yx y x y

F t u tF t C

F t u t

(3)

where ( ) ( )

( ) ( )

xx xy

c

yx yy

C t C tC

C t C t

is the cutting damping

matrix with

,

1 1

( ) ( ( )) [ ( ) ( )]cos ( ) sin ( ) ,t d

j j

N N

xx k j j t r j j

j k

C t t c t c t t t

(4)

2 2

,

1 1

( ) ( ( )) [ ( ) cos ( ) ( )sin ( )] ,t d

j j

N N

xy k j j t j r j

j k

C t t c t t c t t

(5)

2 2

,

1 1

( ) ( ( )) [ ( ) cos ( ) ( )sin ( )] ,t d

j j

N N

yx k j j r j t j

j k

C t t c t t c t t

(6)

( ) ( ) ,yy xxC t C t (7)

where jrc ( t ) and )(tc

jt represent the cutting damping

factors in the thrust and tangential directions [7] and

( , )

( )

( )

x

y x y

u t

u t

is the vibrating velocity vector of the

workpiece cutting point in the two directions ( , )x y .

The two components of the cutting force in the case of

the circular peripheral milling can be calculated according

to

0

00 0

( , )( , )

( ) ( ),

( )( )

x x

Rotyy x yx y

F t F tM

F tF t

(8)

where cos sin

sin cosRotM

is the rotation matrix

from ( , )x y to 0 0( , )x y .

The cutting process system of the circular peripheral

milling is modeled with two degrees of freedom, in which

the workpiece characteristics are represented by the mass

matrix M , stiffness matrix K , and viscous damping

matrix C . The tool is composed of dN elementary

cutting disks of thickness dz . By neglecting the training

and Coriolis inertia effects, the milling process simulation,

in the ( , )x y directions, is summed up by

( ) ( ) ( ) ( ) ,M U t C U t K U t F t (9)

( )U t where is the displacement vector of the

workpiece cutting point over the tool and ( )U t and ( )U t

are, respectively, the vibrating velocity and acceleration

vectors of this cutting point.

Figure 2 shows a nondetailed flowchart adopted for

the resolution of Eq. (9) and the determination of the

cutting force components in the ( , )x y directions, from

which we can deduce their values in the 0 0( , )x y

directions [Eq. (8)].

The algorithm of the program is divided essentially

into three procedures. These procedures have been

developed using the programming language of Maple. As

a first procedure, formal expressions for the calculation of

cutting forces were defined. These expressions are based

on the relative vibrations between the cutter and the

workpiece in the ( , )x y directions. Subsequently, the

second procedure allows data entry representing the

geometric characteristics of the cutter (helix angle, rake

angle, diameter, rigidity, number of flutes, etc.), the

cutting parameters (feed rate, cutting speed, radial and

axial depths of cut, etc.), and the material parameters of

the workpiece (tangential, radial, and specific pressures).

As a result of data entry, the third procedure provides the

numerical solution of the system motion equations

presented by Eq. (9). The resolution is assessed using the

Maple dsolve function, taking the components of the

displacement vector ( )U t as unknown. The components

of this vector are defined by Maple as procedures that

consider the time t as variable. The knowledge of

( ), ( ), ( ), ( )x y x yu t u t u t u t allows the determination of the

cutting force vector components ( )F t for each tool

rotation.

( )U t

18



Figure 1. Dynamic cutting model scheme in the case of circular peripheral milling.

Figure 2. Flowchart of the curvilinear peripheral milling process.

III. EXPERIMENTAL VALIDATION AND DISCUSSION

The objective of this section is to experimentally

validate the proposed simulation model. For this, a

comparison between the simulated and the experimental

cutting efforts will be first investigated, using the same

cutting and tool parameters. Second, the error between

simulation and experimental results will be determined. In

order to give more credibility to this work, we chose to

Evaluation

; ; 0; 2 ;90

t

s e

p

N fd

R

t

Evaluation

( ); ( ); ( ); ( )x y x yu t u t u t u t

Evaluation ( ); ( )x yF t F t

0

0

( ) ( )

( )( )

x x

Rotyy

F t F tM

F tF t

e

d

Yes

ultime

; 2s e e e

End

Yes

s Second procedure: Data entry

Tool features; Cutting parameters; Material

characteristics

Third procedure: Resolution

( ) ( ) ( ) ( )dsolve M U t C U t K U t F t

Computational procedures

( ); ( ); ( ); ( )x y x yu t u t u t u t

First procedure: Formal Maple expression of

cutting forces

O : Workpiece center’s

0x

Workpiece

dFtj

dFrj

j

Theoretical profile

= t R

Cutting profile

ae

x

0y

cy

ky

kx

cx

Cutter Nt f=Rp

Machine Table

xf

0xf

0yf

xf f x

y

19



measure the cutting forces in two different cutting zones

of the workpiece.

A. Experimental Device

The experimental device is shown in Fig. 3.

Acceleration is measured using an accelerometer (three-

axis AC115 CTC) along the three axes. The accelerometer

is positioned on the spindle and connected to an

acquisition card with a function margin of 1,600 Hz to

over 20,000 Hz.

Figure 3. The experimental device for the acquisition of cutting forces

and acceleration values.

Efforts are measured by a piezoelectric dynamometer

bridled on the machine table (platinum Kistler no. 9257A).

The dynamometer loads forces signals along the three

orthogonal axes. The dynamometer is connected to

amplifier signal converters delivering voltage signals

compatible with the used capture card (NI 4470 ± 10 V).

The card has eight synchronized inputs, and the maximum

sampling frequency is 100 kHz. This workpiece was

inspired by “Piece NASA,” which is known in the world

of machining, fixed directly on the dynamometer. This

workpiece, since its invention by NASA, has often served

as a master part to control the precision of machines (Fig.

4).

The machined material is A60 steel (or E335) in a

square section of 50 × 50 mm. Table I lists the

characteristics of this material.

The tool is a milling four-tooth tool that is

manufactured by Mitsubishi (ref. APX 3000), and it has a

20 mm diameter (Fig. 5).

Figure 4. (a) Real part from NASA; (b) machined part in CAD.

Figure 5. Platelets used for experimental tests.

TABLE I. MATERIAL CHARACTERISTICS OF THE WORKPIECE.

Material propriety Value

Steel material A60

Yield strength, Re (MPa) 335

Young’s modulus, E (MPa) 200000

Density, ρ (kg/m3) 7850

Poisson’s ratio 0.29

B. Cutting Force Measurements

Figure 6 shows the measured cutting forces along the

two main directions 0 0( , )x y during machining of the

circular milling profile of the NASA workpiece. Table II

lists the cutting parameters used for this test. From Fig. 6,

we can see that the measured cutting forces progress in

stages. This is can be attributed to the tool movement

(back and forth) during machining. The cutting effort in

the z-direction is negligible compared to the other two

components, which can confirm the choice of our model.

TABLE II. CUTTING PARAMETERS USED IN THE NUMERICAL

SIMULATION AND EXPERIMENTAL TESTS.

Cutting parameters Value

Tangential cutting force pressure, Kt (N/mm²) 2100

Radial cutting force pressure, Kr (N/mm²) 1200

Helix angle, β (°) 20

Rake angle, γ (°) 12

Tool diameter, D (mm) 20

Number of teeth, Nt 4

Cutting speed, Vc (m/min) 51

Axial depth of cut, ae (mm) 0.5

Radial depth of cut, ap (mm) 0.5

Feed rate per tooth, f (mm/tooth) 0.0325

Figure 6. Measured cutting forces circular milling (experimental).

20



Figure 7. Curve cutting forces: (a) simulation, (b) experimental.

Figure 7 shows the simulated and experimental

components Fx0 and Fy0 of the cutting force for one

revolution of the tool. From this figure, we can see that the

simulated components are calculated with a good

approximation. In fact, the measured error between the

simulated and experimental cutting forces on the x-axis is

less than 1.5 N, and on the y-axis, it is lower than 1 N. In

consequence, we can consider that there is good harmony

between the experimental tests and simulation results.

C. Discussion: The Effect of Some Cutting and Tool

Parameters on the Cutting Force

1) Effect of the Tool Diameter

Three tool diameters (30, 20, and 12 mm) were

considered to study their effect on the cutting force

components (Fx0 and Fy0). The results of this simulation

(Fig. 8) show that the average values of Fx0 and Fy0

components decrease when the tool diameter increases.

This can be attributed firstly to the tool frequency rotation,

and secondly to the exit angle of cut arccos(1 )e

e

a

R .

Consequently, the involved number of elementary cutting

disks Nd is more considerable for a tool with D = 12 mm

than for a tool with D = 20 and 30 mm.

Figure 8. Effect of tool diameter on cutting force.

2) Effect of the Tool Helix Angle

In order to study the influence of the tool helix angle

on the cutting force components, three helix angle values

were considered (β = 10°, 20°, and 30°). The results of

simulation are plotted in Fig. 9. From this figure, we can

see that the cutting force increases when the tool helix

angle decreases. Consequently, the machining process of

the workpiece can be more stable when the tool helix

angle, β, is equal to 30°.

3) Effect of the Radial Depth of Cut and Feed Rate

Figure 10 shows the evolution of cutting force

components (Fx0 and Fy0) with the three considered radial

depths of cut (2, 1.5, and 1 mm). From this figure, we can

see that the average values of Fx0 and Fy0 increase when

the radial depth of cut increases.

4) Effect of Feed Rate

Three feed rates (2, 1.5, and 1 mm) were considered to

study their effect on cutting force components (Fx0 and

Fy0). The results of this simulation show that the average

values of Fx0 and Fy0 components increase when the feed

rate increases.

21



Figure 9. Effect of helix angle on cutting force

Figure 10. Effect of radial depth of cut.

IV. CONCLUSION

In this paper, a two-dimensional machining model that

allows the simulation of cutting force, including the

cutting process damping in a dynamic circular milling

process, was presented.

In this model, the regenerative chatter is considered

and the total cutting force is obtained through numerical

integration of the local forces. A comparison between the

experimental and the simulated components of the cutting

force shows that there is good harmony between them.

A discussion on the effect of some cutting and tool

parameters (tool diameter, tool helix angle, radial depth of

cut, and feed rate) on cutting force components has been

made. This discussion revealed that the amplitude of the

cutting force increases when the tool diameter and/or the

tool helix angle decreases. The cutting force increases

with increasing the radial depth of cut and/or the feed rate.

Our future work will focus on the study of the stability

criteria process of circular milling. A comparison between

linear milling and the circular milling stability areas can

be helpful to understand the difference between the two

machining modes.

REFERENCES

[1] J. Tlusty, “Dynamics of high speed milling,” Transactions of the ASME, Journal of Engineering for Industry, vol. 108, pp. 59–67, May 1986.

[2] F. Ismail, M. A. Elbestawi, R. Du, and K. Urbasik, “Generation of milled surface includingtool dynamics and wear,” Transactions of the ASME: Journal of Engineering for Industry, vol. 115, August 1993, pp. 245–252.

[3] K. Mehdi, J. F. Rigal, and D. Play, “Dynamic behavior of a thin wall cylindrical Workpiece during the turning process part I: cutting process simulation,” Journal of Manufacturing Science and Engineering, vol. 124, no. 3, pp. 562–568, August 2002.

[4] K. Mehdi, J. F. Rigal, and D. Play, “Dynamic behavior of a thin wall cylindrical WorkPiece during the turning process part II: experimental approach and validation,” Journal of Manufacturing Science and Engineering, vol. 124, no. 3, pp. 569–580, August 2002.

[5] D. Montgomery and Y. Altintas, “Mechanism of cutting force and surface generation in dynamic milling,” Transactions of the ASME: Journal of Engineering for Industry, Vol. 113, pp. 160–168, May 1991.

[6] C. Y. Huang and J. J. Wang, “Mechanistic modeling of process damping in peripheral milling,” Transactions of the ASME, Journal of Manufacturing Science and Engineering, vol. 129, pp. 12–20, Feb. 2007.

[7] K. Mehdi. A. Zghal, “Modelling cutting force including thrust and tangential damping in peripheral milling process,” International Journal of Machining and Machinability of Materials, vol. 12, no. 3, pp. 236–251, 2012.

[8] X. Li, W. Zhao, L. Li, N. He, and S. W. Chi. Modeling and application of process damping in milling of thin-walled workpiece made of titanium alloy. Hindawi Publishing Corporation Shock and Vibration. vol. 2015, Article ID 431476. p. 12. [Online]. Avavilable: http://dx.doi.org/10.1155/2015/431476

Kamel Mehdi was graduated as a

Mechanical Engineer from ENIS, Tunisia, in

1989. He received his Ph.D. degree in mechanical engineering in 1995 from INSA

of Lyon, France, and his HDR diploma in

2008 from ENIS, Tunisia. His research interests are machining and manufacturing

processes, concurrent engineering, and

computer integrated design of mechanical systems. He is currently an Associate

Professor in mechanical engineering at the

Preparatory Institute for Engineering Studies El Manar (IPEIEM), University of Tunis EL Manar (UTM), Tunis, and he is a Researcher at

the Mechanical Laboratory of Solids, Structures and Technological

Development of the Engineering National High School of Tunis (ENSIT),

University of Tunis (UT), Tunisia. His research works have been

published in the Transactions of the ASME (Journal of Manufacturing

Science and Engineering), International Journal of Vehicle Design (IJVD), Journal of Machining and Forming Technology (JoMFT), Int.

Journal of Engineering Simulation (IJES), Journal of Decision Systems,

Applied Mechanics and Materials, Advanced Materials Research (JDS), and Int. J. Machining and Machinability of Material and in many

international conferences. He is a member of Editorial Review Board of

the Journal «Materials Forming and Machining Processes [IJMFMP]» and member of the scientific committees of many national and

international conferences in mechanical engineering.

Prediction of Cutting Force in a Circular Peripheral ... · circular peripheral milling, numerical...

Documents

Transcript of Prediction of Cutting Force in a Circular Peripheral ... · circular peripheral milling, numerical...