A STUDY OF THE DESIGN OF A CANTILEVER TYPE MULTI-D.O.F. DYNAMIC VIBRATION ABSORBER FOR MICRO MACHINE TOOLS

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1 ICSV4 Cairns Australia 9- Jul, 7 A STUDY OF THE DESIGN OF A CANTILEVER TYPE MULTI-D.O.F. DYNAMIC VIBRATION ABSORBER FOR MICRO MACHINE TOOLS Sung-Hun Jang, Sung-Min Ki, Shil-Geun Ki,Young-Hu Choi and Jong-Kwon Par Dept. of Mechanical Design & Manufacturing Engineering, Changwon National Universit Changwon, Geongna, South Korea Korea Institute of Machiner & Materials Daejeon, South Korea naan@changwon.ac.r Abstract This paper presents a stud of the design of a cantilever tpe ulti-d.o.f. daper, which can be used on icro achine tools, to reduce resonant vibrations. The proposed icro dnaic daper consists of three cantilever beas and a rigid bod suspended fro the cantilever bea. For dnaic analsis and design snthesis, a vibrating sste with the dnaic daper is atheaticall odeled as a luped paraeter odel, in which cantilever beas and the rigid bod are idealied as springs and a ass respectivel. The natural frequencies and haronic response of the atheatical odel are obtained b using odal analsis theor. Fro the results, the spring constants and ass of the dnaic daper which reduce the resonant vibrations of the ain sste can be deterined. A case stud shows the proposed cantilever tped icro dnaic daper can successfull reduce resonant vibrations of the ain achine, in the three different directions, at specified resonant frequencies.. INTRODUCTION A siple dnaic daper is ade b attaching ass-spring to the ain vibrating sste to reduce resonant vibration of it. There have been ade an researches on dnaic daper since it is one of useful eans to control vibrations in structural sstes. Choi [] has done a stud on geoetrical design theor of a 6-d.o.f. dnaic daper. Nafeh [3] and Lee [4] have researched the optiiation of ulti-d.o.f. tuned ass dapers. In this stud a cantilever tpe dnaic daper is proposed in order to reduce the resonant vibrations of icro-achine tool during achining operations. The icro-achine tools or icro-factor is increasingl concerned with rapid developent of iniaturiing or down-siing in achine eleents. Since icro achine tools are relativel sall in sie, there is not enough roo for etra coponents. Therefore sall sie or iniaturied dnaic daper is ore desirable for icro achine tools in order to reduce ecessive vibrations during achining operations.

2 . VIBRATION ANALYSIS OF MULTI-D.O.F. DYNAMIC DAMPER SYSTEM. Matheatical Modeling A scheatic of a structural vibrating sste with three DOF (Degree-Of-Freedo) is shown in Fig.. The ain structural vibrating sste, as shown in Fig., consists of a ass and three linear springs in the independent direction each other. It is assued to eperience ecessive vibrations in its resonant conditions, which are required to be reduced. For this purpose, a ulti-dof dnaic daper of cantilever tpe with end ass, whose scheatic is shown in Fig., are introduced into the ain vibrating sste. Consequentl, the cobined sste becoes 6-DOF. Fig. The ain vibrating sste with 3-d.o.f. Fig. Scheatic of the dnaic daper And the scheatic of the cobined dnaic daper sste is shown in Fig. 3. Cantilevers of the dnaic daper can be appropriatel designed to have required stiffness in each direction. And it is assued that there are no coupling effects between -, - and -directional springs.

3 Fig. 3 The scheatic of 6-d.o.f. vibrating sste with dnaic daper The ain vibrating sste cobined with the dnaic daper was odeled atheaticall as a luped paraeter odel with 6 DOF as shown in Fig. 4. In the odel, sbols,,, and f stand for ass, spring constant and eciting force, respectivel. Sbols,,, and represent degrees of freedo or coordinates respectivel. And subscripts and ean ass of the ain vibrating sste ass and dnaic daper respectivel. Fig. 4 Matheatical odel of 6-d.o.f. vibrating sste with dnaic daper. Equation of Motion The equations of otion for the 6-DOF dnaic daper sste can be derived as follows b using Newton s second law. [ M ]{ u& } + [ K ]{ u} { f } & () Where, [M] and [K] are ass and stiffness atrices respectivel. {u} and {f} are displaceent and eciting force vectors respectivel. [] [ ] T u ()

4 [ ] [ ] T f f f f (3) [ ] M (4) [ ] K (5).3 Solution of Equation of Motion When haronic eciting force { } { } t i e F f ω is applied on the sste, its haronic response { } { } t i e U u ω can be obtained as followings. [ ] [ ] [ ] F F F M K Z Y X Z Y X ω (6) 3. DESIGN OF MICRO DYNAMIC DAMPER 3. Approach of Dnaic Daper Design In this stud, it is required that onl resonant vibrations in the three independent translational directions should be reduced. This requireent can be accoplished b letting response of the ain sste be ero in the Eq. (6) as follows.

5 F F F K ω (7) X Y Z [ ] [ M ] Solving the Eq. (7), following dnaic daping criteria is obtained. i ω, ( i,, ) (8) In case of reducing three independent vibration odes in -, -, and -directions at resonant frequencies, Ω, Ω, and Ω3, Eq. (8) can be rewritten as follows. Ω (9a) Ω 3 (9b) Ω (9c) Fro this criterion, the design paraeters of ulti-dof dnaic daper can be deterined. 3. Design Eaple Design eaple is to design a ulti-dof dnaic daper that can reduce the resonant vibrations of a ain vibrating sste, whose odeling paraeters are given in Table. Table. Material properties of structure (g) (N/) (N/) (N/) Three natural frequencies of 35.6, 87., and 59. H can be deterined fro odal analsis of the ain vibrating sste. Vibration odes corresponding to these natural frequencies are -, -, and -directional vibration odes accordingl. Design paraeters of dnaic daper to reduce the resonant vibrations can be deterined b substituting those frequencies into Eq. (9) with ass of the dnaic daper of g. In this eaple, the ass of dnaic daper was assigned around % of the ain sste ass. Fro the frequenc response curve shown in Fig. 5, it is apparent that pea vibrations at three resonant frequencies are successfull diinished.

6 Fig. 5 FRF of 6-d.o.f. odel 4. DESIGN OF MICRO DYNAMIC DAMPER A odular icro illing structure and its siplified odel are shown in Fig. 6. Natural frequencies for this siplified odel were deterined b using finite eleent analsis as shown in Table. (a) Micro illing achine (b) Siplified luped paraeter odel Fig. 6 The iage of icro illing achine Table. Natural frequenc of μ-illing achine without dnaic daper Mode Natural frequenc (H) Mode Natural frequenc (H)

7 The three odes corresponding to 7., 535.4, and 83.8 H are critical odes in -, -, and -directions. Therefore vibrations at these frequencies are required to be diinished. When the ass of dnaic daped was set % of the ain vibrating sste ass, paraeters of dnaic daper were designed as shown in Table 3. The frequenc response function shown in Fig. 7 represents that the resonant vibrations at the three critical frequencies were perfectl decreased. Table 3 Designed paraeters of dnaic daper for a icro illing achine (g) (N/) (N/) (N/) Fig. 7 Coparison of frequenc response function 5. CONCLUSION In this stud, a ulti-dof (Degree-Of-Freedo) dnaic daper of cantilever tpe with end ass was introduced to reducing icro-achine tool vibrations. The icro dnaic daper consists of a single ass having three DOF and fleible cantilevers. Dnaic characteristics of the dnaic daper becoe different with designs of cantilever bea and end ass. The ain vibrating sste, a icro-illing achine in this case, has greatl diinished in the resonant vibrations at its several natural frequencies with the aid of the properl designed ulti-dof icro dnaic daper. Therefore ulti-dof dnaic daper of cantilever tpe with end ass can be successfull applicable to reducing unwanted resonant vibrations in icro achine tools. ACKNOWLEDGEMENT This stud is one of results obtained fro the research project, Structural Design Optiiation of Reconfigurable Micro Machine Tool for Micro/Meso Scale Coponent Machining supported financiall b the Ministr Of Coerce, Industr and Energ through KIMM. The authors would lie to than for their support.

8 REFERENCES [] S.A. Tobias, Machine-Tool Vibration, John Wile & Sons, 965. [] S.J. Jang, Geoetrical Design Theor of a 6 DOF Vibration Absorber, Journal of KSPE, Vol., No. 7, pp.9-99 (5). [3] L. Zuo, S.A. Nafeh, Minia optiiation of ulti-degree-of-freedo tuned-ass dapers, Journal of Sound and Vibration, Vol. 7, pp (4). [4] C.L. Lee, Y.T. Chen, L.L. Chung and Y.P. Wang, Optial design theories and applications of tuned ass dapers, Engineering Structures, Vol. 8, pp (6). [5] Michael R. Hatch, Vibration Siulation Using MATLAB and ANSYS, Chapan & Hall/CRC,. [6] S.S. Rao, Mechanical Vibrations, Pearson Prentice Hall, 3. [7] Y.H. Choi, S.H. Jang, H.M. Par and J.K. Par, Creating structural configurations of odular reconfigurable icro achine tools b using crptographic apping and function proving, Proceedings of the st International conference on icroanufacturing, 5-9 Septeber 6, Universit of Illinois at Urbana-Chapaign, USA.