, mechanical model, critical damping, experimental setup

7/29/2019 , Mechanical Model, Critical Damping, Experimental Setup.

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PREDICTION OF CHATTER IN MILLING 1.M.M. RAVIKUMAR, 2.A.BHASKAR

1.

Department of Mechanical Engineering, Pallavan College of Engineering, Thimmasamudram,Kanchipuram -631502, Tamilnadu, India

2.Department of Mechanical Engineering, Pallavan College of Engineering, Thimmasamudram,Kanchipuram -631502, Tamilnadu, India

ABSTRACT

The machining industry has an increasing need for improved characterization of cutting tools for controlling vibration and chatter during the milling process. The chatter is a self excitation phenomenonoccurring in machine tools, in which the cutting process tends to decrease the machine structural dampingending with an unstable behavior. The paper is organized as follows: In section 1, the differentialequations describing the mechanical model, In section2, the experimental setup and finally, theconclusions are drawn in 3.

KEY WORDS: Chatter, Mechanical Model, Critical Damping, Experimental Setup.

1. INTRODUCTION

In recent trends of manufacturingindustry, the high speed machining, especiallyhigh speed milling plays an important role. Fewexamples are the fabrication of moulds andaerospace industry, where large amount of material are removed form a large structure. Themilling process is most efficient, if the material

removal rate is as large as possible, whilemaintaining a high quality level. The mostcritical limitations in machining productivityand part quality are the occurrence of theinstability phenomenon called regenerativechatter. The chatter is a self excitation

phenomenon occurring in machine tools, inwhich the cutting process tends to decrease themachine structural damping ending with anunstable behavior. It results in heavy vibrationsof the tool causing an inferior work piece.

2. MECHANICAL MODEL

Figure.1 shows the simplest twodegrees of freedom model of highly interruptedcutting and the milling cutter as a flexiblesystem, with the stiffness and damping elementsoriented in the X and Y directions [2].Assuming, that the machine tool structure hasmultiple modes, let M, C and K be thegeneralized m x m mass, damping and stiffnessmatrices of the structure.

Figure .1 The Standard Two Degrees Of Freedom

Assume the tool to be flexible relative to therigid work piece. The two degrees of freedomoscillator is excited by the cutting work piece.The dynamic governing equation has the form[3].


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.. (1) Where M, C and K are known as the massmatrix, damping matrix and stiffness matrix

respectively. Where the force vector f = [f x, f y] tis acting. If forces are not supposed to act on the mass,

then the eqn (1) can be written as

(2) The above equation is called the characteristicsequation of the system and the DifferentialEquation is of Second Order in X.Assuming a Solution of the form Where u is the constant to bedetermined. Then ...... (3) Substituting the values the of and in theequation (1), we get

or

(4)

Solving the above equation for u, we get

/ / / / / /

The two roots can be written as

/ / / /

Now the solution of the equation (1) can bewritten as

+

..where a 1, a2 are two arbitrary constants and u 1,

u2 are its two roots. This equation can be writtenas

. 3. CRITICAL DAMPING The critical damping c c is defined as the valueof damping coefficient c for which themathematical term /- /in equ (8) isequal to zero.ie.

Cc = 2m

The ratio of c to c c is termed as damping ratio. Itis indicated by the symbol . Mathematically itcan be written as = c / c c Let us the consider the term c/2m of equ (8)

= . (9) So equ (8) can be written with the help of equ(9) as

+ . The nature of the system dependsupon the value of damping. Depending upon thevalue of damping ratio, the damped systems are

put in to three categories which are as follows:

Over damped system: when damping ratio ismore than one i.e. >1. This motion is called a

periodic. When t = 0 the displacement is thesum of A 1 and A 2 .i.e.

A1 + A 2. Critically damped: the system is criticallydamped when = 1. The roots of the equationu1and u 2 are equal to each other.

u1 = u 2 = - = - .


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So the approximate solution of equation may bewritten as

. (11)

In the above equation and are thearbitrary constants whose values can bedetermined from initial conditions as in case of over damped system. Under damped system: In this case the valueof damping is less than unity. The roots of theequation can be written as

Where

Where 4. EXPERIMENTAL SETUP

This approach gives a moreaccurate picture of the tool cutting ability butrequires extensive testing.

Figure .2 Experimental Set Up Of Chatter Prediction

5. PREDICTING METHOD

Figure.2 shows theschematic setup for determining the chatter freespindle speeds. A piezoelectric instrumentedhammer is used to apply a short force pulse totool system [1]. On the other side anaccelerometer is attached to the tool tip, which

measures the vibration displacement caused bythe hammer impact. Both signals are thenamplified and sampled. The transfer function,which describes the dynamic characteristics of the tool is then computed and used to calculate

the sweet spots for a certain work piece materialand cutting conditions.

6. RESULTS & DISCUSSIONS

By predicting method, we candetermine the occurrence of chatter though thework piece is already damaged. The figure.3shows the chatter prediction of an end mill(aluminum) 0.75 in which the analysis arecarried out by using lab view automationsoftware. The vibrations are analyzed withrespect to their frequency and amplitude duringdynamic conditions.

Figure .3 Chatter Prediction Diagram of 0.75Endmill

7. CONCLUSIONS

A mechanical model wasused to derive the equation of motion. Thetheoretical results can be calculated by using thedifferential equation (1) and the other methodwas supported by experimental tests. From thesoftware analysis, two main observations showshow at some speeds the cut is stable and cuttingforces are low, while at other speeds chatter dominates and the peak cutting forces are highfor an end mill.


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REFERENCES

[1]. Tlusty.j. Olacek.a, Danek.c, Spacek.j,1962, Selbsterregte Schwingungen anwerkzeugmaschinen VEB Verlag

Technik, /Berlin. [2]. Tamas Insperger, Janez GradisekMachine Tool Chatter and SurfaceLocation Error in Milling ProcessesJournal of Manufacturing Science andEngineering , Nov 2006, vol.128

[3]. V.P.singh., Mechanical Vibrations,Dhanpat Rai & Sons, Chandigarh.

, mechanical model, critical damping, experimental setup

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