Reduction of chatter vibrations by management of the cutting tool

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Proceedings of the 06 International Conference on Industrial Engineering and Operations Management Kuala Lumpur, Malaysia, March 8-0, 06 Reduction of chatter vibrations by management of the cutting

1 Proceedings of the 06 International Conference on Industrial Engineering and Operations Management Kuala Lumpur, Malaysia, March 8-0, 06 Reduction of chatter vibrations by management of the cutting tool Abdelghafour.Bourdim Mechanical Engineering Department University of Tlemcen B.P Tlemcen - ALGERIE bourdim56@yahoo.fr Mokhtar.Bourdim Relizane University center mokhtar_6@yahoo.fr Abderrahim. Bemoussat Univerity of Tamanrasset abbenmoussa@gmail.com Abstract In this work, we have shown that we can reduce very clearly the chatter vibration amplitude of machining by adding an auxiliary mass imbibed in viscous oil and related to the bar by a flexible rod. This principle of simple application lends itself very well to the transposition into a two degrees of freedom system. It is therefore possible by conventional methods to give the boring bar higher dynamic qualities and an optimal design. Introducing these innovations, we managed to reduce neatly the amplitude of vibration chatter about one third (/3). Keywords boring bar; chatter vibrations; cutting process; frequency; canonical system) I. INTRODUCTION The energy stored in the machine tool structure-part by the cutting process can be dissipated by the modification of the boring bar by adding an oscillating system called auxiliary, thereby increasing the number of degrees of freedom and therefore the number of resonance of the whole. The absorption of the tool chatter vibrations considered as main system is obtained by transferring them to the auxiliary system at the desired frequencies. The principle is simple, easy to design and implement, and allows modification of the physical and dynamic characteristics of the boring bar. The action of the cutting process on the boring bar is particularly manifested by its elastic deformation, causing relative movements of the tool tip with respect to the workpiece. The boring bar is similar to a mechanical system having many degrees of freedom, with an infinite number of resonant frequencies, but it is certain that only the first natural frequency of the boring bar is to be considered, because the chatter vibrations are always produced at this first resonant frequency cutting conditions. The tool tip is then subjected to a variable cutting force resulting from variation of cross cutting section. II. PRINCIPLES The dynamic model of the boring bar can be transposed into a canonical system of parameters (m, k, c ) having one degree of freedom, and subjected to a complex exciting force F c = F o e j t ( fig.). C F c m x FIG.: a): Canonical scheme of the boring bar without absorber b): its amplification factor with respect to relative pulsation. 335

Proceedings of the 06 International Conference on Industrial Engineering and Operations Management Kuala Lumpur, Malaysia, March 8-0, 06 After usual transformations, the steady state motion of the

2 Proceedings of the 06 International Conference on Industrial Engineering and Operations Management Kuala Lumpur, Malaysia, March 8-0, 06 After usual transformations, the steady state motion of the tool tip is given by: F0 cos t x ( t ) 4 () With : F0 X ; ; X st 0 X st This is the resonance peak which characterizes the lack of rigidity of the bar relative to a dynamic load Fc. It is seen from Figure b that the increase of the damping coefficient reduces the amplitude of the main mass vibration at the resonance point. When c takes the extreme values 0 et, tends to. There must exist a value of c for which is minimum. The solution is thus to reduce this peak. To do this, we must increase the number of degrees of freedom, additing to the main system an auxiliary system (damping) with m, k, c characteristics (fig. ). Fc C x m C m système principal système auxiliaire x FIG.: a): Canonical scheme of the boring bar with absorber b): its amplification factor with respect to relative pulsation. The equilibrium equations governing the entire system can be written as: mx (x x ) cx c ( x x ) Fc m x (x x ) c ( x x ) 0 () Assuming that the vibratory system is linear, we can admit the following solutions: x X e x X e It is also admitted that: F0 F0 e j t (3) j t j t Substituting these notations in the system of differential equations, we obtain: [-m + + +j (c+c )] X- ( +j C )] X = F0 - ( +j C )] X + [-m + +j C )] X = 0 (4) These equations allow us to determine the complexes displacements X and X and therefore the amplitudes x and x. We are interested in only the amplitude of movement of the tool tip (movement of the main system). Neglecting the structural damping c( 0.03) with respect to viscous damping of the auxiliary system c ( c << c ), the magnitude X is given by: 336

3 Proceedings of the 06 International Conference on Industrial Engineering and Operations Management Kuala Lumpur, Malaysia, March 8-0, 06 X F0 m j C m m m j C m m This equation is complex having the following form: X a jb a b e j F0 C jd c d e j As shown in equations 3, modulus of x is equal to the amplitude X, we have then: X a b c d X (5) With: a = -m, b = C, c = ( -m )( -m )- m ),d = C[ - (m+ m)] If we adopt the following notations: m m : ratio of the auxiliary and main masses X st ; ; ; C ; m m m X F0 ; X st can be represented as a function of the variable by: 4 4 (6) Which clearly shows that the response curves depend mainly on three parameters, and. If held constant, and is plotted as a function of, the curves of the different values of appear similar to those of a system with one degree of freedom. The two interesting curves are those of = et = 0. When = 0, we obtain an undamped system with a resonance frequency, and amplitude A tends to infinity at this n m frequency. If =, there must exist an optimal value of where the amplitude A is minimal. We note from this figure that all these curves pass through the same points P and Q for which the amplitude of the oscillations is independent. III. OPTIMIZATION OF DAMPING SYSTEM CHARACTERISTICS We must determine the characteristics of the auxiliary system giving a favorable curve whose tangents at the points P and Q are horizontal (fig.3). The smallest resonance amplitude that we can have is the one given by the ordinate of these points. Therefore these points must have the same ordinate ( (P) = (Q)). It suffices to show that there are actually two values of for which is independent of. Equation 6 is written in the form: With: A 4 ; B ; C 4 D A B C D Where A, B, C and D are independent of, we look for the values of such as A B, allowing to write: C D 337

4 Proceedings of the 06 International Conference on Industrial Engineering and Operations Management Kuala Lumpur, Malaysia, March 8-0, 06 Then : 4 4 ( ) ( )( ) Taking the negative sign, we obtain: With the positive sign, we obtain: 4 0 (7) It has two real positive roots and function of which correspond the abscissas of the points P and Q. As is independent of, just give an infinity value to and Equation 6 become: Thus et ( P ) ( ) Q (8) With: (P) = (Q), we obtain: The roots of equation 9 are: The sum of these two roots gives: Equating equations 9 and 0, we obtain:, (9) (0) () For this value, P and Q have the same ordinate. ( ) is less than that of the primary system ( ). The dynamic amplification factor is given by: After transformation and development, we obtain: (), Introducing these values into equation, we obtain: (3) And as is always positive ( >0 ) (4) We now calculate the value of giving curve of horizontal tangent at P and Q. 0 for =. Under these conditions and for values:, we obtain: 338

Proceedings of the 06 International Conference on Industrial Engineering and Operations Management Kuala Lumpur, Malaysia, March 8-0, 06 3 8 (5) According to Equation 4, the optimal value opt

3: Amplification factor depending on the relative pulsation (for optimal characteristics) Thus the curve ( ) has a horizontal tangent at the point P if = 0.065, and at the point Q if = 0.3. IV.

5 Proceedings of the 06 International Conference on Industrial Engineering and Operations Management Kuala Lumpur, Malaysia, March 8-0, (5) According to Equation 4, the optimal value opt = 0.5. Thus: ; 03. ; 08. ; 03. ; FIG.3: Amplification factor depending on the relative pulsation (for optimal characteristics) Thus the curve ( ) has a horizontal tangent at the point P if = 0.065, and at the point Q if = 0.3. IV. DIMENSIONING THE DAMPED BORING BAR After all calculations made, the damped boring bar can be sized as follows: D 40mm; = 0.5 ; Lt 0D = 400 mm; L t =4.6 mm: L =8.8 mm; D =40-( ) =9 mm: L c=0+(l +L ) 34 mm; D c=40-(5)=30mm: FIG.4 : Boring bar without damping (primary system) FIG.5 : Damping system (auxiliary system) 339

6 Proceedings of the 06 International Conference on Industrial Engineering and Operations Management Kuala Lumpur, Malaysia, March 8-0, 06 V. E XPERIMENTAL TESTS A. Determination of the boring bar dynamic characteristics We conducted boring operations on parts previously drilled with tools fixed on two bars mounted in cantilever (L b = Qo = 6 D and Lb = Qo = 8 D). During these tests, radial vibrations was registered (causes of occurrence of chatter vibrations). These recordings are represented by Figures 6. Scale Millitron : 3V 000µ +5V 5 mm 00 µ Qo=6D Qo=8D 30 mm Qo=6D Qo=8D 35 mm 0 FIG 6 : Chatter vibrations of bars with and without damping VI. CONCLUSION In this work, we have shown that we can reduce very clearly the chatter vibration amplitude of machining by adding an auxiliary mass imbibed in viscous oil and related to the bar by a flexible rod. This principle of simple application lends itself very well to the transposition into a two degrees of freedom system. It is therefore possible by conventional methods to give the boring bar higher dynamic qualities and an optimal design. Introducing these innovations, we managed to reduce neatly the amplitude of vibration chatter about one third (/3). References [] N. FABRIS, A.F. D'SOUZA: Experimental and analytical of self excited chatter vibrations in metal cutting, transaction of the ASME page 9-99, vol.00, January 978. [] H.OTA, K.KONO: On chatter vibrations of machine tool or work due to are generative effect and time lag transaction of the ASME page , November 974 [3] J.TLUSTY,K.CLAU,K.PARTHIBAN: Some application of the shock excitationtechnic in machine-tool Canada (973). [4] H.OTA, K.KONO: Chatter vibrations of machine tool or work with directional stiffness inequality. Bulletin of the JSME, vol. 6, N 96, juin973, pages [5] A.BOURDIM, O. RAHMANI: Cutting Stability on a Workpiece of Unsymmetrical Stiffness, 4 th International Conference, OPTI 95, Computer Aided Optimum Design of Structure, 9- September 995, Miami, Florida, USA. 3330

7 Proceedings of the 06 International Conference on Industrial Engineering and Operations Management Kuala Lumpur, Malaysia, March 8-0, 06 BIOGRAPHY Bourdim Abdelghafour is Senior Lecturer and Head of Industrial Maintenance Master in Mechanical Engineering Department at the Faculty of Technology at the University of Tlemcen in Algeria. He earned B.S. in Mechanical Engineering from University of Ottawa in Canada, Masters in Mechanical manufacturing from University of Science And Technology of Oran in Algeria and PhD in Industrial Engineering from School of Arts and Crafts (Ecole des arts et métiers) in Lille -France. He has published journal and conference papers. He was responsible for several research projects in strategy and industrial maintenance management. His research interests include manufacturing, simulation, machine-tools dynamic. Bourdim Mokhtar is currently a fulltime senior lecturer and Head of Mechanical Engineering Department at Relizane University in Algeria, holds a Bachelor of Science degree, a Master and a Phd in manufactoring process from Higher National School of Education Oran in Algeria. 333

Introduction to Mechanical Vibration

2103433 Introduction to Mechanical Vibration Nopdanai Ajavakom (NAV) 1 Course Topics Introduction to Vibration What is vibration? Basic concepts of vibration Modeling Linearization Single-Degree-of-Freedom