Coupled out of plane vibrations of spiral beams

Transcription

1 5th AIAA/ASME/ASCE/AHS/ASC Structure, Structural Dynamic, and Material Conference<br>7th 4-7 May 9, Palm Spring, California AIAA Coupled out of plane ibration of piral beam M.A. Karami, B. Yardimoglu and D.J. Inman 3 An analytical method i propoed to calculate the natural frequencie and correponding mode hape function of an Archimedean piral beam. The deflection of the beam i due to both bending and torion, which make the problem coupled in nature. The goerning partial differential equation and the boundary condition are deried uing Hamilton principle. The ibration problem of a contant radiu cured beam i oled uing a general eponential olution with comple coefficient. Two factor make the ibration of piral different from ocillation of contant radiu arc. The firt i the preence of term with deriatie of the radiu in the goerning equation of piral and the econd i the fact that ariation of radiu of the beam caue the coefficient of the differential equation to be ariable. It i demontrated, uing perturbation technique that the term hae negligible effect on the tructure dynamic. The piral i then approimated with many merging contant-radiu cured ection joint together to conider the low change of radiu along the piral. The natural frequencie and mode hape of two piral tructure hae been calculated for illutration. Nomenclature : Bending tiffne : Torional tiffne / : Stiffne parameter Graduate eearch Aitant and ICTAS Doctoral Scholar, Department of Engineering Science and Mechanic, irginia Tech, Blackburg A, 46, karami@t.edu Aociate Profeor, Department of Mechanical Engineering, Izmir Intitute of Technology, Izmir, Turkey, bulentyardimoglu@iyte.edu.tr 3 George. Goodon Profeor, Center for Intelligent Material Sytem and Structure, Department of Mechanical Engineering, irginia Tech, Blackburg A, 46, dinman@t.edu Copyright 9 by the American Intitute of Aeronautic and Atronautic, Inc. All right reered.

2 : Ma per unit length of piral : Bending moment : Twit torque : adiu of the cured beam : Poition coordinate along the arc, : Out of plane deflection /; i an arbitrary parameter /; i an arbitrary parameter = Total length of the arc, : Twit angle : Angular poition in polar coordinate ytem I. Introduction We are motiated to look at the ibration of piral haped tructure a a prelude to energy hareting uing the piezoelectric effect. Becaue of the unique coupled bending-twit mechanic of piral beam our hope i to create a MEMS cale energy hareting deice, which will hae natural frequencie in a ueable region for a MEMS energy hareter. The idea of uing piral tructure to achiee low frequency ibrational energy hareter wa firt propoed in []. Howeer the ibrational analyi of cured beam with arying radiu (piral) i miing in the literature []. Thi paper attempt to ole the free ibration of piral beam and pae the way to the modeling of piral MEMS hareting deice. Out-of plane ibration of circular cured beam hae been tudied by many inetigator. Here we retreat our attention to jut out-of-plane ibration of cured beam haing ariable radiu of curature. In hitorical order: Loe[3] deried the equation of cured beam of arbitrary

3 geometry. Chang and olterra [4] obtained the upper and lower bound of the firt four natural frequencie of elatic clamped arc of which the center line were in the form of circle, cycloid, catenarie, and parabola by uing a method baed on differential operator theory. Wang [5] employed the ayleigh-itz method to predict the natural frequency of clamped elliptic arc. Irie et al [6] determined the teady tate repone of a Timohenko cured beam with circular, elliptical, catenary and parabolical neutral ae drien at the free end by ue of the tranfer matri approach. Huang et al [7] preented the dynamic repone of non-circular Timohenko cured beam with icou damping by uing a numerical Laplace tranform approach. They conidered the numerical eample for a two-pan elliptic beam ubjected to a rectangular impule. Then, Huang et al [8] deeloped a dynamic tiffne matri by uing the Laplace tranform technique for both the free ibration and forced ibration of non-uniform parabolic cured beam with ariou ratio of rie to pan. Tufekci and Dogruer [9] obtained the eact olution of the differential equation for the tatic behaior of an arch with arying curature and cro ection including the hear deformation effect by uing the initial alue method. Depite the long hitory of attempt to ole for the ibration of arc, a traight forward olution of the ibration of piral beam, uitable for a deign procedure, i till unaailable. In thi paper we tudy the dynamic behaior of piral beam. The fact that the radiu of the beam lowly change along the piral alidate ue of perturbation method and dicretization of the beam. The goerning differential equation and correponding boundary condition are deried imultaneouly from Hamilton principle. A new general olution i propoed for the cured beam with contant radiu. Net, the effect of term including i hown to be negligible uing the multiple cale method. Subequently, to tackle change of radiu in piral, the beam i approimated with eeral merged contant radiu arc and the ibration of the approimation i 3

4 tudied. The natural frequency and mode hape of a clamped-free full turn piral a well a the mode hape of a fie turn piral are finally calculated and plotted. II. Goerning Equation The moment-diplacement relationhip for the piral depicted in Fig. are gien by Loe [3] a follow where M = E I κ, M z = G J τ, κ = β, β τ = + 3,4 z,w,u Fig. Coordinate The train energy U and the kinetic energy T of the cured beam for out-of-plane motion are U SL = ( M κ + M z τ Y Φ β ) d 5 6 Here, i the ma moment of inertia per unit length of the cured beam. In Eq.(5), Y and Φ are eternal force and twiting moment, repectiely. For the free ibration problem, neglecting 4

5 5 rotational inertia, Hamilton principle, Eq. (5-6) along with Eq. (-4) gie the equation of motion a.. m G J E I = + + β β 7 a = + + G J I E β β 7 b The boundary condition aociated with equation of motion are: = L S E I δ β 8 = + L S J G δβ β 9 = + + L S G J E I δ β β In the aboe equation,,, and o on. III. ibration of a contant radiu cured beam Net Conider finding the natural frequencie and mode hape of a cured beam with contant radiu. The olution i baed on the one preented by Ojalo [] with a different choice of general olution. Auming i contant in Eq. (7), one get the following goerning differential equation. a b Ealuating from (-b) and ubtituting that in (-a) we can derie,

6 The eparation of ariable i now ued to ole the aboe partial differential equation by auming,,, 3 Subtituting for and from (3) into () and grouping the term which depend on the ame ariable together reult, 4 i a contant not depending on eighter or t. The right ide of Eq. (4) precribe a harmonic ariation for T, co. The natural frequency depend on the alue of gien by 5 Ealuating in term of B from (-b) and then uing the left ide of Eq. (4) gie: Here we deiate from [] a we conider an eponential form of general olution: (6) Where i one of the comple root of the characteritic equation, Eq. (7). Each i a contant comple number. Thu 7 Uing Eq. (-b) can alo be written in correponding eponential form. 8 The boundary condition can be written in term of B, and their deriatie. The reulting i equation can be written in a matri form: 6

7 9 Here the matri [M] depend on the alue of the root,, which are in turn related to by Eq. (7). Eq. (9) yield the triial olution,, unle det. For pecific eigenalue,, det anihe and the problem can hae nontriial olution. The natural frequencie of the ytem are determined by thee eigenalue uing Eq. (5) []. The main ditinction of thi method compared to [] i that we do not aume that each of the general olution are real. The are comple and the ame i true for any. When all added up the olution will be real. Thi make it unneceary to categorize the olution of Eq. (7) baed on the ign of each root and conider them eparately (a done in []). The comple approach here wa alidated finding the natural frequencie of a clamped-clamped contant radiu arc. The reult preciely matched with thoe in []. I. The effect of term containing on the ibration of piral The radiu of a piral beam arie along the piral. The radiu of Archimedean piral arie linearly with the polar angle, :, a depicted in Fig.. Subtituting for from in Eq.(7) and conducting eparation of ariable, imilar to preiou ection, reult in 4 3 a b 7

8 Fig. Archimedean Spiral Eq. () hae two major difference with Eq. (). The firt i the eitence of term including. The econd, which i quite fundamental, i that in Eq. () i no longer a contant and arie a a function of. We tackle thee compleitie one by one. In thi ection we tudy how the term with affect the natural frequencie while auming i till a contant. Phyically thi tudy correpond to ery hort piral, in which the radiu arie but ince the piral i too hort the radiu will be almot the ame eerywhere. Conidering an Archimedean piral Eq. () become, 4 3 a b The rate of change of radiu,, i mall o we ue Multiple Scale perturbation method [] to ole Eq. (). Firt we epand the olution of our ytem a bellow., 8

9 and are the olution of Eq. () proided. The term and can be interpreted a the effect of mall perturbation on the anwer. In fact, the dependency of and B on and occur oer different cale namely,, and o on. i a lower cale than ince it become notable only when i big. We determine and B a function of and rather than a function of and. All the deriatie with repect to mut be reealuated uing the chain rule in term of deriatie of and, reulting in,, 3,,,,,,,,,, 4, 4,,,,,,,,,,,,,, The deriatie of parameter with repect to and are denoted in order in the parenthei. For eample:, gie. Equating coefficient of like power of in the aboe equation,,,, 3 a,,,,,, 3,,,, 4, 4, 3 a,,, 3 b 9

10 ,,,,,,,, 3 b We can ole Eq. (3-a) and (3-b) for and. The equation are the ame a Eq. () and they reult in the following,,, 4 We then ubtitute (4) into (3-a) and (3-b3) and eliminate the ecular term to get: 5 a 5 b are neither a function of or. Subtituting (4) and (5) in () and neglecting higher order of reult,,,, 6 Eq. (6) i the reult of perturbation analyi. It gie an etimate of the effect of the rate of change of radiu,, on the olution of the ytem. Comparing Eq. (6) with Eq.(4) reeal that the rate of change of radiu hift the root of the characteritic equation. It remain to atify the boundary condition to get the natural frequencie. Thi lat tep i imilar to that of the preiou ection. The alue of matri M i different from before due to the change in general olution. A ummary of the effect of radiu change on the firt eigenalue of a clamped-clamped piral with i preented in Table. It can be inferred from Table that the deriatie term hae only a ery mall effect on the eigenalue and conequently on the natural frequencie. We therefore conclude that it i proper to imply ignore their effect in calculating the natural frequencie of piral beam.

11 of Contant adiu of Changing adiu / /.955E-5.955E E E E-.9956E E-.46E E+3.948E E E E+.958E E+.997E+.6 Table : Effect of deriatie term on eigenalue. The effect of the low ariation of on the ibration of the piral beam The aboe analyi allow u to eclude the effect of deriatie term in piral beam ibration. It now remain to ealuate the effect of the low ariation of on the natural frequencie of the piral. Ecluding the term which contain in Eq. () yield 7 a b Conidering that the rate of change of radiu i mall, we can diide the piral into a number of egment and ay that the radiu i almot the ame (contant) throughout each egment. The general olution for each of the egment i already deried in Section III. For the i th element we hae:, S ka e i one of the i root of characteritic equation for each of the piece., 8 The aboe equation can be rewritten in term of three dimenionle parameter,, and. 9

12 , n n 3 7 Fig. 3 Dicretization of Spiral The boundary condition of the piral are the ame a thoe of contant radiu arc, meanwhile ome matching condition between element are needed to relate the olution in each of the egment to the net. Conidering the problem from a degree of freedom apect, each of the element ha 6 DOF, namely the. If the piral i compoed of element, there will be 6 unknown. There are i boundary condition and 6 equilibrium and continuity equation can be written for each of the element interface. Therefore the number of equation will be the ame a the number of unknown. The continuity and equilibrium condition at the piece interface are a follow. Deflection continuity Slope continuity

13 Twit angle continuity Torion torque equilibrium Bending moment equilibrium Shear force equilibrium The epreion for effort term in term of diplacement can be ditinguihed from the boundary condition epreed in Eq. 8 to :,, We end up with the following matri equation to atify the boundary condition and equilibrium and continuity condition.,,,,,,,,, 3 Eq. (3) yield triial olution unle the determinant of M i zero. The matri M i a function of which are in turn function of. The eigenalue,, which make the determinant zero, are proportional to econd power of natural frequencie of the ytem ( refer to Eq.(8) ). The alue of the firt eigenalue,, of ingle turn piral beam with arying radiu for different beam radii and different number of element are gien in Table. Δ, contant, two, three, four, fie, ten, twenty element element element element element element Table. Effect of radiu ariation and number of element on eigenalue Since the formulation i nondimenionalized, the reult in Table are independent of. The reulted mode hape for, /.3/, and i depicted in Fig.4-6. The boundary condition of the piral i clamped-free. 3

14 =.65 =.65 = B θ (ad) θ (ad) -..5 y Fig. 4: Firt Mode hape =.34 =.34 = B θ (ad) θ (ad) -.5 y Fig. 5: Second Mode hape = = = B θ (ad) θ (ad) y Fig 6: Fourth Mode hape Finally the mode hape of a fie turn clamped-free piral with, /./, and are depicted in Fig

15 =.563 =.563 = B θ (ad) θ (ad) y Fig. 7: Firt Mode hape =.9985 =.9985 = B θ (ad) θ (ad) y Fig. 8: Second Mode hape =.5 =.5 = B θ (ad) θ (ad) y Fig 9: Fourth Mode hape The relationhip between the length of the piral and the fundamental natural frequency i depicted in Fig.. The y-ai of Fig. indicate the normalized fundamental natural frequency, which i the firt reonance frequency diided by that of a piral beam of the arc length. In thi pecific problem radiu decreae % per each turn and the tiffne parameter, k, i [3]. Since the frequency i normalized the trend i identical for all cro ection and material with 5

16 the ame k and for any initial radiu of the beam. The fundamental natural frequency eponentially decreae a the length grow. The firt reonance frequency drop.5 db a the containing angle of piral increae by a decade. Normalized Fundamental Natural Frequency ω - Arc length (Full turn) Fig. : The effect of length on natural frequency I. Concluion An analytical method i propoed to calculate the natural frequencie and mode hape of piral beam. It ha been hown that the effect of deriatie term on the tructural dynamic can be neglected. Thu the tructure i approimated by many contant radiu ection joint together, to tackle the problem of lowly changing coefficient. The piecewie continuou olution conerge with relatiely few number of element. The eigenalue and correpondingly natural frequencie depend on the tiffne ratio the radiu, the rate of change of radiu and the total angle. The reulted mode hape of two different piral are depicted for illutration. 6

17 eference [] W. Choi, Y. Jeon, J. Jeong et al., Energy hareting MEMS deice baed on thin film piezoelectric cantileer, Journal of Electroceramic, ol. 7, no., pp , 6. [] H. Hu, H. Xue, and Y. Hu, A piral-haped hareter with an improed hareting element and an adaptie torage circuit, Ultraonic, Ferroelectric and Frequency Control, IEEE Tranaction on, ol. 54, no. 6, pp , 7. [3] A. E. H. Loe, A Treatie on the Mathematical Theory of Elaticity, 4th ed.: New York: Doer, 944. [4] T. C. Chang, and E. olterra, Upper and lower bound for frequencie of elatic arc, The Journal of the Acoutical Society of America, ol. 46, pp , 969. [5] T. M. Wang Fundamental frequency of clamped elliptic arc for ibration the plane of initial curature, Journal of Sound and ibration ol. 4, pp , 975. [6] T. Irie, G. Yamada, and I. Takahahi, The teady tate out-of-plane repone of a Timohenko cured beam with internal damping., Journal of Sound and ibration, ol. 7, pp , 98. [7] C. S. Huang, Y. P. Teng, and S. H. Chang, Out-of-plane dynamic repone of noncircular cured beam by numerical Laplace Tranform., Journal of Sound and ibration, ol. 5, pp , 998. [8] C. S. Huang, Y. P. Teng, S. H. Chang et al., Out-of-plane dynamic analyi of beam with arbitrarily arying curature and cro-ection by dynamic tiffne matri method, International Journal of Solid and Structure, ol. 37, pp ,. [9] E. Tufekci, and O. Y. Dogruer, Eact olution of out-of-plane problem of an arch with arying curature and cro ection., ASCE Journal of Engineering Mechanic, ol. 3, pp. 6-69, 6. [] I. U. Ojalo, Coupled Twit-Bending ibration of Incomplete Elatic ing, Int. J. Mech. Sci., ol. 4, pp. 53-7, 96. [] D. J. Inman, Engineering ibration, nd ed.: Prentice Hall, New Jerey,. [] A. H. Nayfeh, Introduction to perturbation technique: Wiley, New York, 98. [3] M. A. Karami, and D. J. Inman, ibration analyi of Zigzag micro-pring for energy hareting application, in SPIE Smart Structure NDE, San Diego, California, USA, 9. 7