A finite thin circular beam element for out-of-plane vibration analysis of curved beams

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1 Journl of Mechnicl Science nd echnology (009) 196~1405 Journl of Mechnicl Science nd echnology DOI /s A finite thin circulr bem element for out-of-plne vibrtion nlysis of curved bems Bo Yeon Kim, Chng-Boo Kim *, Seung Gwn Song, Hyeon Gyu Beom nd Chongdu Cho Deprtment of Mechnicl Engineering, Inh University, 5 Yonghyun-dong, Nm-gu, Incheon, , Kore (Mnuscript Received December 0, 007; Revised April 9, 008; Accepted December 16, 008) Abstrct A finite thin circulr bem element for the out-of-plne vibrtion nlysis of curved bems is presented in this pper. Its stiffness mtrix nd mss mtrix re derived, respectively, from the strin energy nd the kinetic energy by using the nturl shpe functions derived from n integrtion of the differentil equtions in sttic equilibrium. he mtrices re formulted with respect to the locl polr coordinte system or to the globl Crtesin coordinte system in considertion of the effects of sher deformtion nd rotry inertis. Some numericl exmples re nlyzed to confirm the vlidity of the element. It is shown tht this kind of finite element cn describe quite efficiently nd ccurtely the out-ofplne motion of thin curved bems. Keywords: Finite element; Mss mtrix; Nturl shpe function; Out-of-plne vibrtion; Rotry inerti; Sher deformtion; Stiffness mtrix; hin circulr bem Introduction he out-of-plne vibrtion nlysis of curved bems is quite complex due to the coupling effects of bending nd torsion, trnsverse sher nd rotry inerti force. Neglecting these effects my led to inccurcies in nlysis, especilly when the rtio of rdil thickness to curvture rdius of curved bem is lrge (thick circulr bem), or when the nturl frequency is high for vibrtion problem even if the rtio of rdil thickness to curvture rdius is very smll (thin circulr bem). Dvis et l. [1] presented the shpe functions of curved bem element, which the effects of trnsverse sher deformtion nd trnsverse rotry inerti were considered. heir out-of-plne displcements were obtined from n integrtion of the differentil equtions of n infinitesiml element in sttic equilibrium. he stiffness nd mss mtrices were derived from his pper ws recommended for publiction in revised form by Associte Editor Seockhyun Kim * Corresponding uthor. el.: , Fx.: E-mil ddress: kimcb@inh.c.kr KSME & Springer 009 the force-displcement reltions nd the kinetic energy equtions, respectively. he mtrices re formulted in the locl stright-bem (Crtesin) coordinte system rther thn in the locl curviliner (polr) coordinte system, nd thus trnsformtion of the mtrices for the locl coordinte system to the one for the common globl coordinte system is required before they re ssembled even though the rdius of curvture for the entire curved bem is constnt. Yoo nd Fehrenbch [] derived the stiffness nd mss mtrices of sptil curved bem element by using the minimum potentil energy theory, where the effects of wrping nd rotry inerti due to flexure nd torsion were considered but the effects of sher deformtion were neglected. Plninthn nd Chndrsekhrn [] derived the stiffness mtrix for sptil curved imoshenko bem element by using Cstiglino's theorem, where only the effects of sher deformtion re considered. Lebeck nd Knowlton [4] developed the stiffness mtrix of circulr bem element using ring theory, where the in-plne motion is coupled with the out-of-plne motion due to the unsymmetricl cross-sectionl re, but the effects of

2 B. Y. Kim et l. / Journl of Mechnicl Science nd echnology (009) 196~ y O φ u ψ 1 φ C C x φ 1 C 1 u 1 φ 1 Fig. 1. he coordinte system of circulr bem element. sher deformtion re neglected. Choi nd Lim [5] lso developed two generl curved bem bsed on imoshenko bem theory. he two-node element ws formulted from constnt strin fields nd the three-node one ws formulted from liner strin fields. he stiffness mtrices were derived from the totl potentil energy theorem in the locl curviliner coordinte system nd were trnsformed into the common globl Crtesin coordinte system. Wu nd Ching [6] derived the stiffness nd mss mtrices from the force-displcement reltions nd the kinetic energy equtions, respectively. he shpe functions were modified in order to consider the effect of sher deformtion from the ones given by Lebeck nd Knowlton [4]. he mtrices re formed from the out-of-plne motions of modertely thick curved bem with the effects of trnsverse sher deformtion, trnsverse rotry inerti, nd torsionl rotry inerti. In this pper, finite thin circulr bem element, which considers the effects of trnsverse sher deformtion, trnsverse rotry inerti, nd torsionl rotry inerti, is presented. his kind of element cn be used in the nlysis of thin bem in which the effect of vrition in curvture cross the section cn be neglected. Its stiffness nd mss mtrices re derived from the strin energy nd the kinetic energy, respectively, in the sme mnner s the work of Kim et l. [7] for thin circulr bem element with in-plne motions. hese mtrices cn be trnsformed esily into the globl Crtesin coordinte system.. hin circulr bem element.1 Out-of-plne deformtions he globl Crtesin coordinte system of circulr bem element is shown in Fig. 1. O is the center of curvture of the element. he cross section perpendiculr to the circumferentil direction of the element is doubly symmetric with respect to the plne nd the plne, nd its re is uniform. he rdius of the centroidl line pssing through the center of cross section is. Hlf of the subtended ngle of the element is ψ= ( 1). he nodes of the element, C 1 nd C re on the centroidl line. When only the out-of-plne deformtions with respect to the x y plne of circulr bem re considered, the displcement of the center of cross section, C, t n ngulr position hs xil component u with respect to locl polr coordinte system C. he rottion of the cross section t C hs rdil component φ nd circumferentil component φ which re supposed very smll. he bending curvture κ, twist τ, nd sher strin γ t C of circulr bem re, respectively, s κ = φ, φ / (1) τ = φ, + φ / (1b) γ = + (1c) φ u, / where (), is prtil differentil with respect to circumferentil coordinte. For thin bem, the rdil thickness is very smll s compred with the rdius of centroidl line nd the effect of vrition in curvture cross the cross section cn be neglected. he internl bending moment M, torsionl moment M, nd sher force N t the point C cn be expressed s M M N = EI κ () = GJ τ (b) = KGAγ (c) where A, I, J, nd K re the re, the re moment of inerti bout -xis, the torsionl moment of inerti (Sololnikoff [8]), nd the sher coeffi-

3 198 B. Y. Kim et l. / Journl of Mechnicl Science nd echnology (009) 196~1405 cient of the cross section (Cowper [9]), respectively. E is Young's modulus of the mteril. G is the sher modulus, which is expressed s G= E 1+ ν with the Poisson rtio ν.. Shpe functions Out-of-plne forces nd moments re pplied on the cross sections t nodes, C 1 nd C of the circulr bem element in equilibrium. he internl bending moment, torsionl moment, nd sher force on the cross section t C cn be expressed with the internl bending moment M 0, torsionl moment M 0, nd sher force N 0 on the mid cross section t φ = 0 s M = M cφ+ M N sφ () M = M sφ+ M N cφ+ N (b) N where = N (c) 0 sφ= sinφ, cφ cosφ =, φ ( ) = +. 1 By substituting Eq. () into Eq. (), the bending curvture, twist nd sher strin t C cn be expressed with the internl bending moment, the torsionl moment, nd the sher force on the mid cross section s ( B5c B6s ) κ = φ+ φ (4) τ = β B Bsφ+ Bcφ (4b) γ = α B (4c) where 4 0 B = N EI (5) 5 0 B = M EI (5b) B = M N EI (5c) α = EI K GA (6) β = EI GJ (6b) If α = 0, then the effect of trnsverse sher deformtion is neglected, i.e.. γ = 0. he rdil rottion, circumferentil rottion, nd xil displcement of the cross section t C, which re solutions of differentil equtions obtined by substituting Eq. (4) into Eq. (1), re expressed in terms of φ s φ = { B cφ+ B sφ+ B 4 f 1 cφ ( φ φ φ) φ φ} + B fs + f c + Bf s (7) { 4 φ = B sφ+ B cφ+ B f sφ } B f φφ s B f sφ f φφ c (7b) ( φ φ φ) ( φ φ φ φ) u B B s B c B f f f s + B f + f c + f s where = 1+ φ φ+ 4 1φ+ φ φ f1 f 5 + B fs + fs f c (7c) α 6 4 = (8) = β (8b) ( β ) ( β) f = 1+ (8c) f4 = 1 (8d) B 1, B, nd B re the constnts of integrtion of the differentil equtions. hey re the rigid body displcement of the center of curvture in the xil direction, the rigid body rottion bout the center of curvture in the rdil direction, nd the rigid body rottion bout the center of curvture in the circumferentil direction t the mid cross section, respectively. he sttic deformtions represented by Eq. (7) re used s the shpe functions for the out-of-plne motion of thin circulr bem element. hey re composed of the rigid body modes ssocited with B 1, B, nd B nd the flexible modes ssocited with B 4, B 5, nd B 6. he flexible modes re null t φ = 0. he displcements nd rottions t nodes, C 1 nd C, cn be expressed s {} v = [ ]{ B} (9) where {} v ( φ1 φ1 u1 φ φ u) = (10)

4 B. Y. Kim et l. / Journl of Mechnicl Science nd echnology (009) 196~ { B} ( B B B B B B ) = (10b) 0 cψ sψ f ( 1 cψ) 0 sψ cψ fsψ 1 sψ cψ ( f1+ f) ψ+ fsψ = 0 cψ sψ f ( 1 cψ) 0 sψ cψ fsψ 1 sψ cψ ( f1+ f) ψ fsψ ( fs 4 ψ+ fψcψ) fψsψ fψψ s ( f4sψ+ fψψ c ) f( 1 cψ) + fψsψ ( f+ f4) sψ fψcψ (10c) ( fs 4 ψ+ fψcψ) fψsψ fψψ s ( f4sψ fψψ c ) f( 1 cψ) + fψsψ ( f+ f4) sψ fψcψ he coefficient vector of shpe functions, { B } then cn be expressed in terms of the nodl displcement vector with respect to the locl polr coordinte system, {} v, s 1 { B} [ ] {} v = (11) he reltionship between the nodl displcement vector with respect to the locl polr coordinte system, {} v nd the nodl displcement vector with respect to the globl Crtesin coordinte system, v, is given by { } {} v = [ ]{ v} (1) where { v} ( φx1 φy 1 uz1 φx φy uz) [ ] = (1) c1 s s1 c = c s s c (1b). Stiffness mtrix he strin energy of the thin circulr bem element is 1 V = EI + GJ + K GA d ( κ τ γ) (14) 1 By substituting Eq. (4) into Eq. (14), the strin energy cn be expressed in terms of the coefficient vector of shpe functions s 1 V = { B} [ KB ]{ B} (15) where [ K B ] is the stiffness mtrix with respect to { B } of the thin circulr bem element. he of the symmetric mtrix [ K B ] re detiled in the Appendix. By substituting Eq. (11) into Eq. (15), the strin energy is expressed in terms of the nodl displcement vector in the locl polr coordinte system s 1 V = {}[ v K]{} v (16) where [ K ] is the stiffness mtrix with respect to {} v of the thin circulr bem element s [ ] [ ] K [ K ][ ] 1 = (17) B If necessry, the stiffness mtrix in the locl polr coordinte system cn be trnsformed into the globl Crtesin coordinte system s follows: K = K [ ] [ ][ ].4 Mss mtrix (18) he kinetic energy of the thin circulr bem element is expressed s 1 = ( ρiφ + ρiφ + ρau ) d (19) 1 where ρ is the density of the mteril. I is the re moment of inerti bout -xis of the cross section. ( i ) is the prtil differentil with respect to time t. By substituting Eq. (7) into Eq. (19), the kinetic energy is expressed in terms of the coefficient vector of

5 1400 B. Y. Kim et l. / Journl of Mechnicl Science nd echnology (009) 196~1405 shpe functions s 1 = { B } [ MB ]{ B } (0) where [ M B ] is the mss mtrix with respect to { B } of the thin circulr bem element. he of the M re detiled in Appendix. symmetric mtrix [ ] B = ρi ρa (1) = ρi ρa (1b) If the effect of trnsverse rotry inerti is neglected, then = 0, i.e. ρ I = 0. If the effect of torsionl rotry inerti is neglected, then = 0, i.e., ρ I = 0. By substituting Eq. (11) into Eq. (0), the kinetic energy is expressed in terms of the nodl displcement vector in the locl polr coordinte system s 1 = {}[ v M]{} v () where [ M ] is the mss mtrix with respect to {} v of the element s [ ] [ ] M [ M ][ ] 1 = () B If necessry, the mss mtrix in the locl polr coordinte system cn be trnsformed into the globl Crtesin coordinte system s follows: M = M [ ] [ ][ ].5 Out-of-plne internl forces (4) By substituting Eq. (4) into Eq. () or from Eqs. () nd (5), the internl bending moment, torsionl moment, nd sher force on the cross section t C cn be expressed by the coefficients of shpe functions s ( 5 6 ) M = EI B cφ+ B sφ (5) ( ) M = EI B B sφ+ B cφ (5b) ( ) N = EI B 4 (5c). Vlidity of the bem element.1 Liner sttic nlysis for qurter cntilever ring with tip lod A qurter cntilever ring subjected to n xil tip lod P is shown in Fig.. Its displcements cn be obtined nlyticlly from Eqs. (), (5) nd (7) by considering the geometric nd nturl boundry conditions t the two ends. he rdil rottion φ, circumferentil rottion φ, nd xil displcement u t the tip re φ 1 = ( 1 β) + P EI π φ = β ( 1 β + ) P EI 4 u 1 π = [ α ( 1 β )] β + + P EI (6) (6b) (6c) he results of nlyticl solutions for the tip rottion nd displcement, for which the effects of trnsverse sher deformtion re considered nd those effects re neglected, re summrized in ble 1. he sme results s in ble 1 cn been obtined by using FEM to model the qurter rings with only one, two or 16 thin circulr bem becuse the shpe functions of the element presented in this pper re exct in sttic stte. In our clcultions, the following fctor nd prmeters re dopted: E = 00 GP, ν = 0.5, ρ = 780 kg/m, = 1 m, A = m, I = m 4, I = m 4, J = m 4, K = P y O Fig.. A qurter cntilever ring subjected n xil tip lod. x

6 B. Y. Kim et l. / Journl of Mechnicl Science nd echnology (009) 196~ Nturl vibrtion nlysis for free ring he out-of-plne displcements of ring, which is freely vibrting in mode hving n nodl dimeters with frequency of ω, will tke the forms ( Ccos Ssin ) ( Ccos Ssin ) ( Ccos Ssin ) i t φ = Φ n+φ n e ω (7) φ = Φ n+φ n e ω (7b) i t u = U n + U n e ω (7c) he nturl vibrtion equtions cn be obtined by substituting Eq. (7) into Eq. (14) nd Eq. (19), integrting round the ring nd pplying Lgrnge s equtions. For n 1, there exist two orthogonl modes of which the nturl frequencies re sme, nd the nodl dimeters of one mode re rotted by π /n from the nodl dimeters of the other. he bove theoreticl results re similr to the ones obtined by the previous works (Ro [10]; Kirkhope [11]). In order to show the convergence of the thin circulr bem element presented in this pper, the nturl frequencies of the out-of-plne vibrtion of free ring ble 1. ip rottions nd displcement per unit xil tip lod of qurter cntilever ring. α 0 i t α = 0 φ P (rd/n) φ P (rd/n) u P (m/n) * α = 0 : neglecting the effect of trnsverse sher deformtion. re computed by using FEM to model the complete ring with 16,, 48, or 64, nd consider ll the effects of trnsverse sher deformtion, trnsverse rotry inerti, nd torsionl rotry inerti. he lowest eight nturl frequencies of the flexible modes computed by using FEM re presented nd compred with the theoreticl vlues in ble. he first mode for n = 0 is trnsltionl rigid body mode nd the second mode for n = 1 is rottionl rigid body mode. It is shown tht the nturl frequencies computed by using FEM converged rpidly to the theoreticl vlues with incresing number of. he errors in the nturl frequencies increse with the number of nodl dimeters nd the order of frequencies. o determine the effects of trnsverse sher deformtion, trnsverse rotry inerti, nd torsionl rotry inerti on the nturl frequencies of the out-of-plne vibrtion, the nturl frequencies of free ring re computed by using FEM to model the complete ring with 80 nd consider the effects of trnsverse sher deformtion, trnsverse rotry inerti, nd torsionl rotry inerti. Some nturl frequencies of the flexible modes computed by using FEM re presented nd compred with the theoreticl vlues in bles nd 4. When the effect of torsionl rotry inerti is neglected, the second nturl frequencies of some models in ble dispper becuse the modes corresponding to the second frequencies re torsiondominnt modes. It is shown tht the nturl frequencies computed by using FEM re very ccurte s compred with the theoreticl vlues in the ll cses. It is lso shown tht the errors in the nturl frequencies increse with the number of nodl dimeters nd ble. Convergence of nturl frequencies of the out-of-plne vibrtion of ring. Number of nodl dimeter n FEM / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / 0.088

7 140 B. Y. Kim et l. / Journl of Mechnicl Science nd echnology (009) 196~1405 ble. Comprison of nturl frequencies of the out-of-plne vibrtion of ring considering the effect of torsionl rotry inerti. Number of nodl dimeter n * * α 0, 0 α 0, FEM / = 0 α = 0, FEM / 0 FEM / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / α = 0 : neglecting the effect of trnsverse sher deformtion. = 0 : neglecting the effect of trnsverse rotry inerti. ble 4. Comprison of nturl frequencies of the out-of-plne vibrtion of ring neglecting the effect of torsionl rotry inerti. Number of nodl dimeter n α 0, 0 α 0, FEM / = 0 α = 0, FEM / 0 FEM / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / 0.05 the order of frequencies nd neglecting the effects of trnsverse sher deformtion, trnsverse rotry inerti, nd torsionl rotry inerti cn increse the vlues of nturl frequencies of the ring, especilly for the higher number of nodl dimeters.

8 B. Y. Kim et l. / Journl of Mechnicl Science nd echnology (009) 196~ ble 5. Comprison of frequency prmeters of the out-of-plne vibrtion of circulr rc. s Mode number * Φ = 60 Φ = 10 Φ = 180 FEM/ * FEM/ * FEM/ / / / / / / / / / / / / / / / / / / / / / / / / 0.00 * tken from (Howson et l., 1999). Nturl vibrtion nlysis for circulr rcs he nturl frequencies of the out-of-plne vibrtion for clmped-clmped circulr rcs with circulr cross-section re computed by using FEM by considering ll the effects of sher nd rotry inerti. he circulr rcs were modeled with the circulr bem with subtended ngle of 1. For the comprison of the numericl results by FEM with the theoreticl ones, which re presented in the works of Howson et l. [1], the sher coefficient of circulr cross-section K is 0.89, nd Poisson rtio ν is 0.. ble 5 shows tht the nturl frequencies computed by using FEM re very ccurte s compred with the theoreticl ones for ll the two slenderness rtios nd the three rc ngles. he slenderness rtios s nd frequency prmeters λ in the tble 5 re defined s ble 6. Comprison of frequency prmeters of the out-ofplne vibrtion of n S-shped bem. Mode number 60 Using circulr bem 6 Using stright bem s A / I = (8) λ= ω ρa 4 / EI (9).4 Nturl vibrtion nlysis for n S shped bem he nturl frequencies of the out-of-plne vibrtion for clmped-clmped S shped bem composed of two identicl hlf-rings with circulr cross-section re computed by using FEM by considering ll the effects of sher nd rotry inerti. he bem ws modeled with 60 nd 6 circulr bem nd stright bem. Its properties re the sme s the ones of the bove clmped-clmed circulr rc Fig.. Out-of-plne displcements of mode shpes long the S- shped bem curve. with s = 0. ble 6 shows tht the presented circulr bem element for the nturl vibrtion nlysis gives good convergence nd ccurcy s compred with the

9 1404 B. Y. Kim et l. / Journl of Mechnicl Science nd echnology (009) 196~1405 stright bem element. he S-shped bem hs cyclic symmetry. As shown in Fig., modes 1,, nd 5 re xilly symmetric modes with zero nodl dimeter nd modes, 4, nd 6 re xilly nti-symmetric modes with one nodl dimeter. 4. Conclusions A finite thin circulr bem element for the out-ofplne vibrtion nlysis of curved bems is presented. It cn describe quite efficiently nd ccurtely the out-of-plne motions of thin circulr bems in which the secondry effects of trnsverse sher deformtion, trnsverse rotry inerti, nd torsionl rotry inerti cn be considered prtilly or totlly. his kind of bem element gives exct results for liner sttic problems in the cse of concentrted lods becuse the shpe functions of the element re exct in sttic stte. he nturl frequencies obtined by using this kind of element converge rpidly to the theoreticl vlues with incresing number of. he numericl results for the nturl frequencies of circulr rcs of vrious geometric conditions re in excellent greement with the theoreticl ones. he presented circulr bem element cn be utilized esily nd efficiently for precise out-of-plne vibrtion nlysis of curved bems composed of circulr rcs. Acknowledgment his work ws supported by Inh University Reserch Grnt. References [1] R. Dvis, R. D. Henshell nd G. B. Wrburton, Curved bem finite for coupled bending nd torsionl vibrtion, Erth. Eng. nd Struct. Dynmics 1 (197) [] C. H. Yoo nd J. P. Ferenbch, Nturl frequencies of curved girders, J. Eng. Mech. Div. ASCE 107 (EM) (1981) [] R. Plninthn nd P. S. Chndrsekhrn, Curved bem element stiffness mtrix formultion, Comp. nd Struct. 1 (4) (1985) [4] A. O. Lebeck nd J. S. Knowlton, A finite element for the three-dimensionl deformtion of circulr ring, Int. J. Num. Methods Eng. 1 (1985) [5] J. K. Choi nd J. K. Lim, Generl curved bem bsed on the ssumed strin fields, Comp. nd Struct. 55 () (1995) [6] J. S. Wu nd L. K. Ching, Out-of-plne responses of circulr curved imoshenko bem due to moving lod, Int. J. Solids Structures 40 (00) [7] C. B. Kim, J. W. Prk, S. Kim nd C. Cho, A finite thin circulr bem element for in-plne vibrtion nlysis of curved bems, J. Mech. Sci. nd ech. 19 (1) (005) [8] I. S. Sokolnikoff, Mthemticl of Elsticity, McGrw-Hill, New York, USA, (1956). [9] G. R. Cowper, he sher coefficient in imoshenko s bem theory, J. Appl. Mech. (1966) [10] S. S. Ro, Effects of trnsverse sher nd rotry inerti on the coupled twist-bending vibrtions of circulr rings, J. Sound Vibrtion. 16 (4) (1971) [11] J. Kirkhope, Out-of-plne vibrtion of thick circulr ring, J. Eng. Mech. Div. ASCE 10 (EM) (1976) [1] W. P. Howson nd A. K. Jemh, Exct out-ofplne nturl frequencies of curved imoshenko bems, J. Eng. Mech. 15 (1) (1999) Appendix he mtrix [ K B ] is symmetric nd ny element not defined is zero. Bij = ( / ) B44 = ( + ) K EI k k k = B46 B55 Bij β α ψ β sψ ( ψ ψ ψ) β ( ψ ψ ψ) k = + s c + s c B66 ( ψ ψ ψ) β ( ψ ψ ψ) k = s c + + s c he mtrix [ M B ] is symmetric nd ny element not defined is zero. M m m Bij = = B11 ( ρ ) = ψ B1 A m sψ Bij ( ψ ψ) ( ψ ψ ψ) ( s c ) m = f s + f s c B15 mb = ψ ψ ψ ψ

10 B. Y. Kim et l. / Journl of Mechnicl Science nd echnology (009) 196~ m = f ψ sψ + f + f sψ ψcψ B4 1 ( 1 ) f( ψ sψcψ) + 1 mb6 = 1 + f s c c ( ) ( ψψ ψ ψ) ( f4 f f4)( ψ sψcψ) ψ ( )( ψ sψcψ) mb = mb5 = 1 + f s c c ( ) ( ψψ ψ ψ) ( ψ ψ ) ( )( ψ ψ ψ) f s + f + f s c 4 m = f s + f + f B44 4 ψ ψ 1 ψ ( ψ ψ ψ) 4 f + f f s c 1 ( 1 ) f ( ψ sψcψ) + + { }( m = f f + f + f f + f sψ B ψψ c ) ( f + f + f ) f ( ψ sψcψ) ( ) ff( sψψ c ψc ψ) ( f1 f) f( ψ sψ ψcψ sψ) + + ( ψ ψ ) ( )( ψ ψ ψ) B m = f s + f f s c ( ψ ψ ψ) 4 f f s c ( f4 f) f( sψψ c ψcψ) ( 1 + ) f ψ ψ sψ 1 1 ψcψ sψ + + f ψ 4 { }( ψ ψ ψ) m = f + f + f s c B ( f4 f f4) f( sψψ c ψcψ) 1 1 ( 1 ) f ψ ψ sψ ψcψ sψ + + f ψ 4 Chng-Boo Kim received his B.S. degree in Mechnicl Engineering from Seoul University, Kore in 197. He then received his D.E.A., Dr.-Ing. nd Dr.-es-Science degrees from Nntes University, Frnce in 1979, 1981 nd 1984, respectively. Dr. Kim is currently Professor t the School of Mechnicl Engineering t Inh University in Incheon, Kore. His reserch interests re in the re of vibrtions, structurl dynmics, nd MEMS.

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