Nonlinear Stability and Bifurcation of Multi-D.O.F. Chatter System in Grinding Process

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1 Key Engineering Materials Vols. -5 (6) pp. -5 online at (6) Trans Tech Publications Switzerland Online available since 6//5 Nonlinear Stability and Bifurcation of Multi-D.O.F. Chatter System in Grinding Process Q.K. Han a T. Yu b Z. W. Zhang c and B.C. Wen d College of Mechanical Engineering and Automation Northeastern University Shenyang China a qingkai_han@sohu.com b yt_6@6.com c ttwwooblueyes@6.com d bcwen9@vip.sina.com Keyword: Grinding process Chatter Nonlinear stability Bifurcation Abstract. The nonlinear chatter in grinding machine system is discussed analytically in the paper. In higher speed grinding process the self-excited chatter vibration is mostly induced by the change of grinding speed and grinding wheel shape. Here the grinding machine tool is viewed as a nonlinear multi-d.o.f. autonomous system in which hysteretic factors of contact surfaces are also introduced. Firstly the DOFs of the above system are reduced efficiently without changing its dynamic properties by utilizing the center manifold theorem and averaging method. Then a low dimensional system and corresponding averaging equations are obtained. The stability and bifurcation of chatter system are discussed on the base of deduced averaging equations. It is proved that chatter occurs as a Hopf bifurcation emerging from the steady state at the origin of system. The theoretical analyses on the multi-dof chattering system will lead to further understanding of the nonlinear mechanisms of higher speed grinding processes. Introduction Nowadays vibration problems are becoming more and more important in grinding processes especially in higher working speed. Most determined vibrations in machine tools are caused by unbalanced shaft gears and bearings or by the surroundings. But some serious vibrations so-called chatters are indeed self-excited with certain frequencies and mainly caused by the coupling of work-pieces and machine tool. In any case some chattering phenomena can be explained by machine dynamic models including some technique parameters such as the coupling of structural modes and cutting surface waves []. And lots of researches are achieved in system stability prediction and many experiment data are also accumulated. The influence of the contact surfaces between grinding wheel and work-piece and some chaotic phenomena in chatters were studied [~5]. Considering hysteresis existing in contact surfaces the stability of chattering amplitude and the separation of stability threshold are analyzed [6]. However in the viewpoint of nonlinear theory chatter system has not been studied sufficiently such as the developing process from stable to unstable and the dynamic bifurcation behavior etc. []. In the paper considering the regenerative effect in grinding process and the hysteretic non-linearity of contact surfaces the machine structure is regarded as a high-dimensional autonomous nonlinear system. Since the number of chattering frequencies is less than the number of D.O.F of the whole system the theoretical problem to construct system solutions needs to be resolved. After reducing the dimensions with center manifold method the stability and bifurcation characteristics of a low dimensional chatter system are then analyzed based on averaging method. At last the stable boundaries of the corresponding linearized system and bifurcation diagrams of different control parameters are illustrated. All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of the publisher: Trans Tech Publications Ltd Switzerland (ID: //77:9:)

2 Advances in Grinding and Abrasive Technology XIII Solution to Multi-D.O.F. Nonlinear Chattering in Grinding Process A chatter system a multi-d.o.f. nonlinear autonomous system of machine tool structure containing the grinding speed and others as bifurcation control parameters p is u & = f ( u n u U R m p P R. () Assume equilibrium points are singular. Constitute a matrix Φ with the eigenvectors of matrix A A = D u f () and introduce the linear transformation to the system as v u = Φ w c v E s u w E E. () c s u where E E E are central stable and unstable subspace respectively. Following the center manifold method we deduce the projection of system on n c + m dimensional center manifold w = z( v just as Eq. v & = Bv + g( v z( v. () The above projection equation reflects the characteristics of the system around the origin approximately. Assume B ( has complex root λ ( and make Re( λ ()) = Im( λ ()) and Re( λ ()) > to get a cluster of periodic solution from point ( ). Set v εz p εµ A and a transformation of z e t = x Eq. can be deduced into the following averaging equations referring [8] ϕ ϕ ϕ ε : X + = f(y µ t) ε : P + X + = f (y µ t) + D yfϕ. () t y t where ϕ = [f(y µ s) X ]ds ϕ = [f (y µ s) + D yfϕ X ] ds. (5) t t and X = M{f (y µ t)} M{...} is averaging operator. t t It can be proved that the averaging equation of the system Eq. has one resonant monomial and the averaging method can be used to identify Arnold normal form of Hopf bifurcation see [9]. Vibration with Single Frequency and Bifurcation Function of Reduced Chatter System By applying center manifold method the chattering system can be dealt as a single frequency nonlinear one. Based on this considering the regenerative chatter due to work-piece and grinding wheel together with hysteresis nonlinearity from numerous contact surfaces a normalized principal mode dynamic equation of nonlinear chatter system referred as [6] is yielded as & x + cx& + ω x + z = P n x z & = αx + βx Px = k( x + x / ω) &. (6) It is rewritten as & + ε(c + k / ω)x& + ( ω + k )x + ε z. (7) x n = where z is the hysteresis restoring force referring [9]; and α and β are hysteretic parameters; ε > is the small parameter; x x & & x are the displacement velocity and acceleration of chattering

3 Key Engineering Materials Vols. -5 system respectively; ω is the chattering circular frequency (rad/s); k is the grinding thickness coefficient; are two phase differences of regenerative chatters = cos φ = sin φ w φ = πω/ ω. (8) where φ is the difference of phases of two adjacent grinding trajectory ω w is the circular frequency of grinding wheel. According to averaging method the simple approximate solution of Eq.7 is set as x = a cosψ ψ = ωt. (9) where a is chattering amplitude. The following bifurcation equations are obtained as d a where a = ( ωc + βa ω ) ω dψ = ω 8ω + αa. () = c + k / ω c = ωn + k ω. () Nonlinear Stability and Bifurcation of Chattering System with Single Frequency For the system Eq.7 according to Poincare-Lyapunov theory () when c > the system is asymptotic stable at origin where no chatter happens () when c < the system is unstable at origin where the nonlinear items have no effect. When = ε(c + k / ω) the direct Lyapunov method is used and the Lyapunov function is with c = ε V = y + ωx + αx >. () y = εβ. When > β > α asymptotic stable except the origin point; when α > β < is always negative and the system is global is always positive and the system is unstable; when α > β = is identical with and the system is in its critical state. According to the averaging equations of system Eq. bifurcation equation can be obtained as a ( ωc + k / ω + βa ω ) =. () ω It has three possible bifurcation solutions a = a = ± ( ωc + k ) ω. () When ω + k the system yields the stable bifurcation solution c < ωc πω < arcsin( ). (5) ω k w

4 Advances in Grinding and Abrasive Technology XIII Results and Discussion Boundary of Stable Region in Grinding Chatter Process. From the view of stable analysis the critical equivalent damping coefficient can be used to determine the stable region when chattering occurs i.e. the parametric boundary determined by the condition c = shown as the first part of Eq. where the dimensionless values are c =.8e 5 ω = n and the wheel rotating speed is N = 6ωw /(π) [r/m]. In Fig. (a) the equivalent linear stable boundary is shown when k =.. The curve in Fig. (b) shows the stable boundary when N =. Fig. (c) shows the stable boundary between chatter angle frequency ω and the linear damping c when k =. and N=. (a) (b) (c) Fig. Stable boundaries of chatter in grinding process with equivalent linear damping Bifurcate Diagrams. The parameters of system Eq.7 are as following dimensionless values: c =.8e 5 k =. ω = n α = β = N =. The bifurcation parameters ω k and N are taken one by one and three different bifurcation figures are obtained from Eq. as shown in Figure. In Fig. (a) the chatter amplitudes change with chatter angle frequency ω when k =. N =. In Fig. (b) the chatter amplitudes bifurcate with grinding thickness coefficient k when ω = N =. In Fig. (c) the chatter amplitudes bifurcate with wheel rotating speed N when ω = k =.. (a) (b) (c) Fig. Chatter amplitudes bifurcation diagrams

5 Key Engineering Materials Vols Conclusions In this paper the higher-dimensional multi-d.o.f. nonlinear system with hysteresis is adopted to describe the chattering vibration of grinding machine tool. The nonlinear parametrical system is treated with center manifold method and the averaging method and a lower dimensional system is obtained with the same stability and bifurcation properties. Based on the obtained average equations and single frequency differential equations the typical analytical results on stability and bifurcation are presented. The projection of proposed multi-d.o.f. system of grinding chattering vibration on n c + m dimensional center manifold can approximately represent the dynamic characteristics of the whole system near the origin point. The obtained average equations from normalized equation of chattering system possess one resonant monomial and it can be used to identify Arnold normal form of Hopf bifurcation. According to Poincare-Lyapunov theorem the stability of deduced single D.O.F. system equation with hysteresis is discussed with different linear and nonlinear damping. The bifurcations of chatter vibration are analyzed theoretically. Besides some typical stable boundary curves and bifurcation diagrams illustrate the complex characteristics of the chatter vibration system of machine tool in grinding process. Acknowledgement This work is financially aided by National Natural Science Foundation of China. (No. 8). References [] Z.T. Han and Y.Z. Zhang: Precision Manufacture and Automation () No. pp (Chinese) [] W.D. Xie X.S. He: Journal of Zhejiang Technology University Vol. (995) No. pp.96-. (Chinese). [] L.S. Wang A. Cui A.B. Yu: J. of Jilin University of Technology Vol.5 (995) No. pp.6-. (Chinese). [] H.W. Lu S.Z. Yang Journal of Vibration Engineering (996) No. pp (Chinese). [5] I. Gabec: Int. J. Machine Tools Manufacture Vol. 8 (998) pp.9-. [6] H.L. Chen D.P. Dai Journal of Vibration Engineering Vol. (99) No. pp.5-. (Chinese). [7] H.Y. Hu: Nonlinear Dynamics in Application (Aerial Industry Press China ) (Chinese). [8] Y.S. Chen A. Y. T. Leung: Bifurcation and Chaos in Engineering (Springer ). [9] S.P. Yang Y.S. Chen: Mechanics Research Communication Vol.9 (99) No. pp.5-.