VIBRATION ANALYSIS OF PRE-TWISTED BEAMS USING THE SPLINE COLLOCATION METHOD

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1 16 Journal of Marne Scence and echnology, Vol. 17, o., pp (9 VIBRAIO AALYSIS OF PRE-WISED BEAMS USIG HE SPLIE COLLOCAIO MEHOD Mng-Hung Hsu* Key words: natural frequency, pre-twsted beam, splne collocaton method, tapered beam, boundary condton. ABSRAC he varaton n the accuracy of the calculated natural frequences of pre-twsted beams solved wth splne collocaton method s nvestgated n ths study. he splne collocaton method s used to formulate the egenvalue problems of pre-twsted beams. hree types of boundary condtons are consdered. umercal results ndcate that the accuracy of the calculated natural frequences s sgnfcantly dependent upon the pre-twsted angle of the non-unform beam. Results that show that splne collocaton method s very compettve for the vbraton analyss of pre-twsted beams are presented. I. IRODUCIO Speakng of many knds of desgn, dynamc characterstcs of pre-twsted beams absolutely play a vtal role. In the feld of turbo or compressor engneerng, for smplcty, the beam s frequently approxmated as a pre-twsted beam. At the desgn stage, accurate predcton of natonal frequences of nonunform pre-twsted beam s of consderable mportance at the desgn stage. he coupled natural frequences of a pre-twsted beam were also nvestgated usng a number of dfferent methods. Abrate [1] studed the vbraton of a pre-twsted blade usng the Raylegh-Rtz method. Anderson [] studed the flexural vbraton of rotatng bars. Dawson [, ] studed the vbraton of a pre-twsted blade usng the Raylegh-Rtz method. Gupa and Rao [8] appled the fnte element method for fndng the varaton of natural frequences of doubly tapered and twsted moshenko beams. Hodges et al. [9] used the transfer matrces to compute the fundamental frequences and correspondng modal dsplacements along the non-unform rotatng beams. hey dsplayed that a blade has a complex geometry that makes an exact nvestgaton of ts characterstcs somewhat complex. Kuang and Hsu [1, 11] presented that the blade s frequently approxmated as a pre-twsted Paper submtted 11/8/7; accepted /7/8. Author for correspondence: Mng-Hung Hsu (e-mal: hsu@npu.edu.tw. *Department of Electrcal Engneerng, atonal Penghu Unversty, Penghu, awan. beam for smplcty n the feld of turbo or compressor engneerng. Ln et al. [1] presented the accurate modfed transfer matrx method for studyng the dynamc behavor of a non-unform pre-twsted moshenko beam. Rao [1, 16] studed the natural frequency of pre-twsted beam to consder the complex shape of beam. hey presented the vbraton problems of wnd blades and turbo blades are crucal parts of the desgn. Stort and Aboelnaga [19] studed the transverse deflectons of a straght tapered symmetrc beam attached to a rotatng hub as a model for the bendng vbraton of blades n turbomachnery. Subrahmanyam et al. [] showed coupled bendng-bendng vbratons of pre-twsted cantlever bladng allowng for shear deflecton and rotary nerta by the Ressner method. Subrahmanyam and Rao [1] presented coupled bendng-bendng vbratons of pre-twsted tapered cantlever beams treated by the Ressner method. Swamnathan and Rao [] solved the vbratons of rotatng, pre-twsted and tapered blades. Young [7] dealt wth the dynamcs of a pre-twsted beam usng Rao's comparson functons. In ths work, the splne collocaton method s mplemented to formulate the egenvalue problem of a pre-twsted beam n the dscrete form. he ntegrty and computatonal effcency of splne collocaton method n ths problem wll be demonstrated through a seres of case studes. II. SPLIE COLLOCAIO MEHOD he solutons to numerous complex pre-twsted beam problems have been effcently obtaned as the use of fast computers and range of avalable numercal methods, ncludng the Galerkn method, dfferental quadrature method, fnte element technque, dfferental transform, boundary element method, and Raylegh-Rtz method. In ths study, splne collocaton method s employed to formulate the dscrete egenvalue problems of varous pre-twsted beams. Prenter et al. [7, 1, 18] nvestgated splne and varaton methods. Bert and Sheu [] presented statc analyss of beams and plates usng splne collocaton method. El-Hawary et al. [6] dscussed quartc splne collocaton methods for solvng lnear ellptc partal dfferental equatons. hey derved optmal quartc splne approxmatons to generated hgh order perturbatons of partal dfferental equatons. Patlashenko and Weller [1] appled the splne collocaton method to solve two-dmensonal problems, and determned the postbuckng

2 M.-H. Hsu: Vbraton Analyss of Pre-wsted Beams Usng the Splne Collocaton Method 17 behavor of lamnated panels subjected to mechancal and heatng nduced loadngs. Vswanathan and avaneethakrshnan [] studed the free vbraton of crcular cylndrcal thn shells usng pont collocaton method. Rao and Kumar [17] presented a B-splne collocaton method of hgher order for a class of self-adjont sngularly pertured boundary value problems. Wu et al. [,, 6] dsplayed the applcaton of the splne collocaton method to analyss of rgd frame structures under varous loadng condtons. her results from the splne collocaton method are compared wth those obtaned from the dfferental quadrature method, the fnte element method, and other avalable analytcal methods. he splne functon can be derved from backward or central fnte dfference. In ths work, we consder the knots z as follows. y b t x t b 1 1 X Y L Fg. 1. Geometry of a pre-twsted beam. Z z a h for, 1,,, 1, (1 where z, z 1, z,, z 1, z are the abscssas of knots and z, z 1, z 1, z are the abscssas of extended fcttous knots. b a h ( where dstance h between two adjacent knots keep constant. Splne functon s gven as follows [, 6, 1]. u B z B z v ( z z h ( z z 6( z z ( z z 6( z z 1( z z ( z z 6( z z 1( z z ( z z ( z z 6( z z 1( z z ( z z ( z z 6( z z 1( z z ( z z 1( z z 6( z z h otherwse f z [ z, z ] f z [ z, z 1] h 1 f z [ z 1, z] h 1 f z [ z, z 1] h 1 1( z z 1 f z z 1, z h 1 1 [ ] [ ] f z z, z where B u, (z, B u, 1 (z, B u, (z,, B u,1 (z, B u, (z, and B v, (z, B v, 1 (z, B v, (z,, B v,1 (z, B v, (z form a bass for the functon defned over the regon a z b. he values at the knots are gven by the followng equatons. u u ( U z a B z ( v V z a B z ( where a u and a v are coeffcents to be determned. here are collocaton ponts n the doman. he splne functons should be at least one order hgher than that of the governng dfferental equaton so that accuracy and smoothness of the approxmate soluton can be guaranteed [,, 6]. v III. FORMAIO OF HE GEVALUE PROBLEM he splne collocaton method s employed to formulate the egenvalue problems for pre-twsted beams. he pre-twsted beam s shown n Fg. 1. he length of the pre-twsted beam s L. b and t denote the wdth and thckness of the pre-twsted beam at z, respectvely. b 1 and t 1 denote the wdth and thckness of the pretwsted beam at z L, respectvely. he deflecton components u and v are the transverse flexble deflectons of the pre-twsted beam. he knetc energy of the beam, due to the lateral bendng vbraton [, 1, 11], s L u v ρ 1 A (6 t t Consder the cross sectonal area of the pre-twsted beam materal at poston z to be z z A( z bt 1 α 1 (7 L L where ρ s the densty of the pre-twsted beam, and the tapered angles of the pre-twsted beam are b1 b α (8 b t t 1 (9 he stran energy of the pre-twsted beam s [, 1, 11] 1 L u u U E I I v v I t (1 In ths equaton, I, I and I are the area moments of nerta. Consder the tapered beam to be pre-twsted wth a

3 18 Journal of Marne Scence and echnology, Vol. 17, o. (9 unform pre-twsted angle θ t and the moments of cross sectonal area at the poston z can be derved as u v at z L (19e z z I( z IXX cos θ t IYY sn θ t L L z z I ( z IXX sn θ t IYY cos θ t L L (11 (1 sn z z I z IYY I XX θ t cos θ t (1 L L where L s the length of the pre-twsted beam and the area moments of nerta wth respect to the axes X and Y are I I XX YY ( z ( z bt z z 1 α 1 1 L L b t z z 1 α 1 1 L L Hamlton s prncple of moton s t t1 ( δ δ δ (1 (1 U W dt (16 where δw, δ and δu are the vrtual work, the varaton of knetc energy and the varaton of stran energy, respectvely. By usng Hamlton prncple, the equatons of moton of ths pre-twsted beam can be derved as: ρ A t u u u u v v v v v v v ρ A t u u u (17 (18 he correspondng boundary condtons of the clampedfree beam are u at z (19a u at z (19b v at z (19c v at z (19d u v at z L (19f v u at z L (19g v u at z L (19h he correspondng boundary condtons of the smply supported beam are u at z (a u v at z v at z v u at z u at z L u v at z L v at z L v u at z L (b (c (d (e (f (g (h he correspondng boundary condtons of the clampedclamped beam are u at z (1a u at z (1b v at z (1c v at z (1d u at z L (1e u at z L (1f v at z L (1g v at z L (1h he system s composed of eght boundary condtons and two coupled governng equatons. Wth the soluton assumed to be of the form u U exp (ωt and v V exp (ωt, Eqs. (17 and (18 can then be smplfed to

4 M.-H. Hsu: Vbraton Analyss of Pre-wsted Beams Usng the Splne Collocaton Method 19 d d du du du d d dv dv dv ω ρau ( d dv d dv dv d d du du du ω ρav ( where ω s the angular frequency of vbraton. he correspondng boundary condtons of the clamped-free beam are: U for z (a du for z (b V for z (c dv for z (d du dv for z L (e d d U d V for z L (f dv du for z L (g d d V d U for z L (h he correspondng boundary condtons of the smple supported beam can be shown as U for z (a du dv for z (b V for z (c dv du for z (d U for z L (e du dv for z L (f V for z L (g dv du for z L (h he correspondng boundary condtons of the clampedclamped beam may be presented as stated below: U for z (6a du for z (6b V for z (6c dv for z (6d U for z L (6e du for z L (6f V for z L (6g dv for z L (6h In seekng an effcent dscretzaton technque to acqure an accurate numercal soluton wth very small number of knots, the splne collocaton method s utlzed to solve numercally these partal dfferental equatons. By applyng the splne collocaton method, Eqs. ( and ( are substtuted nto ( and (. he equaton of moton of a pre-twsted beam can be rearranged nto the splne collocaton method formula. hs leads to d B d B d z d z u, u, 1 d z d B z u, au, a u, db ( z db ( z d z u, d z u, 1 d z dbu, au, au, 1 au, d Bu, z d Bu, 1 z ( z ( z d Bu, au, au, 1 au, d B d B d z d z v, v, 1 d z d Bv,,, 1, au au au

5 11 Journal of Marne Scence and echnology, Vol. 17, o. (9 db ( z db ( z d z v, d z v, 1 d Bu, z d Bu, 1 z z ( z d z dbv, au, a u, d Bu, a a a u, u, 1 u, d Bv, z d Bv, 1 z ( z ( z ω ρ d Bv, au, au, 1 au, ω ρ A z Bu, z A z Bu, 1 z ω ρ A z Bu, au, au, 1 au, for, 1,, (7 d B d B d z d z v, v, 1 d d Bv,,, 1, av av av db ( z db ( z d z,, 1 v d z v d dbv, av, av, 1 av, d Bv, z d Bv, 1 z ( z ( z d Bv, a a a v, v, 1 v, d ( z d Bu, ( z d ( z d Bu, 1( z d z d B z u, au, a u, db ( z db ( z d z u, d z u, 1 d dbv, a a a u, u, 1 u, ω ρ ω ρ A x Bv, x A x Bv, 1 x ω ρ A x Bv, x av, av, 1 av, for, 1,, (8 Usng the splne collocaton method, the boundary condtons of the clamped-free beam can be rearranged nto the matrx forms as u, u, 1 u, au, au, 1 au, [] (9a u, u, 1 u, au, au, 1 au, [] (9b v, v, 1 v, av, av, 1 av, [] (9c v, v, 1 v, av, av, 1 av, [] (9d d Bu, z d Bu, 1 z ( z ( z. d B u, au, au, 1 au, d Bv, z d Bv, 1 z ( z ( z. d Bv, av, av, 1 av, [] (9e

6 M.-H. Hsu: Vbraton Analyss of Pre-wsted Beams Usng the Splne Collocaton Method 111 dbv, z dbv, 1 z ( z ( z. d B d B d z v, d z v, 1. dbv, av, av, 1 av, d d Bv, av, av, 1 av, d B d B d z v, d z v, 1. dbu, z dbu, 1 z ( z ( z. d d Bv, av, av, 1 av, dbu, au, au, 1 au, dbu, z dbu, 1 z ( z ( z. db u, au, au, 1 au, d B d B d z u, d z u, 1. d z d B au, a u, [] d Bv, z d Bv, 1 z ( z ( z. d Bv, av, av, 1 av, d Bu, z d Bu, 1 z ( z ( z. d B u, (9f au, a u, [] dbv, z dbv, 1 z ( z ( z. (9g d B d B d z u, d z u, 1. d z d Bu, au, a u, [] (9h Usng the splne collocaton method, the boundary condtons of the smple supported beam can be rearranged nto the matrx forms as u, u, 1 u, au, au, 1 au, [] (a z u, u, 1 ( d Bu, au, au, 1 au, z v, v, 1 ( d Bv, v, v, 1 v, av, av, 1 av, av, av, 1 a v, [] (b [] (c db v, av, av, 1 av, z v, v, 1 (

7 11 Journal of Marne Scence and echnology, Vol. 17, o. (9 d Bv, av, av, 1 av, z u, u, 1 ( dbu, au, au, [] u, u, 1 u, au, au, 1 au, (d [] (e z z u, u, 1 ( ( d B u, au, au, 1 au, z z v, v, 1 ( ( d B v, av, av, 1 a v, [] v, v, 1 v, av, av, 1 av, (f [] (g z z v, v, 1 ( ( d B v, av, av, 1 av, z z u, u, 1 ( ( d B u, au, a u, [] (h Usng the splne collocaton method, the boundary condtons of the clamped-clamped beam can be rearranged nto the matrx forms as u, u, 1 u, au, au, 1 au, [] (1a u, u, 1 u, au, au, 1 au, [] (1b v, v, 1 v, av, av, 1 av, [] (1c v, v, 1 v, av, av, 1 av, [] (1d u, u, 1 u, au, au, 1 au, [] (1e u, u, 1 u, au, au, 1 au, [] (1f v, v, 1 v, av, av, 1 av, [] (1g v, v, 1 v, av, av, 1 av, [] (1h he egenvalues of the resultant algebrac equaton system provde the natural frequences of the pre-twsted beam problem. IV. UMERICAL RESULS AD DISCUSSIO Fgure shows the calculated natural frequences of clamped-free beams wth dfferent pre-twsted angles. he data for ths pre-twsted beam are []: b /t. he non-dmensonal natural frequences of the pre-twsted beam are defned as ω ρal Ebt. umercal results

8 M.-H. Hsu: Vbraton Analyss of Pre-wsted Beams Usng the Splne Collocaton Method ondmensonal natural frequency 6 1 [] ω [] [] ω ω ω ω ω [] ondmensonal natural frequency ω ω ω 1 1 Pre-twsted angle (Deg Fg.. he calculated natural frequences of the clamped-free beams wth dfferent pre-twsted angles. 1 1 Pre-twsted angle (Deg Fg.. he natural frequences of the clamped-clamped beams wth varous pre-twsted angles. ondmensonal natural frequency Pre-twsted angle (Deg Fg.. he natural frequences of the smple-supported beams wth varous pre-twsted angles. ndcate that the natural frequences of a pre-twsted beam calculated usng the splne collocaton method are shown to be n favorable agreement wth the numercal results solved usng the Raylegh-Rtz method []. o sgnfcant error s found for the results calculated usng the splne collocaton method. Results ndcated that a hgher frst natural frequency s calculated for the beam wth a hgher total pre-twsted angle for θ t < 6. umercal results also ndcated that the calculated second natural frequency s decreased whle the pre-twsted angle ncreasng for θ t < 6. Fgure shows the natural frequences of smple-supported beams wth varous pre-twsted angles. umercal results ndcated that a hgher frst natural frequency s calculated for the beam ω ω ω wth a hgher pre-twsted angle for θ t < 6. umercal results also ndcated that the calculated second natural frequency s decreased whle the pre-twsted angle ncreasng for θ t < 6. he pre-twsted angles deeply affect the thrd and fourth natural frequences. Fgure shows the natural frequences of the clampedclamped beams wth varous pre-twsted angles. umercal results ndcated that a hgher frst natural frequency s calculated for the beam wth a hgher pre-twsted angle for θ t < 9. umercal results also ndcated that the calculated second natural frequency s decreased whle the pre-twsted angle ncreasng for θ t < 9. umercal results n ths example show that the pre-twsted angle can sgnfcantly affect the natural frequences of the pre-twsted beams. Fgure descrbes the natural frequences of the clampedfree beams wth varous tapered angles. he fundamental frequences are almost constant. umercal results also ndcated that the calculated second, thrd and fourth natural frequences are ncreased whle the tapered angles ncreasng. Fgure 6 shows the natural frequences of the smplesupported beams wth varous tapered angles. umercal results n ths example show that the tapered angle can sgnfcantly affect the frst frequency. he frst, second, thrd and fourth natural frequences ncrease wth tapered angles, almost lnearly. Fgure 7 shows the natural frequences of the clampedclamped beams wth varous tapered angles. umercal results also ndcated that the calculated natural frequences are ncreased whle the tapered angles ncreasng n general. umercal results ndcate that the tapered angle s a very senstve parameter for the vbraton of the tapered beam. he clamped-clamped boundary condtons gve rse to hgher natural frequences of the beams n comparson wth the smply supported boundary condtons.

9 11 Journal of Marne Scence and echnology, Vol. 17, o. (9 9 ondmensonal natural frequency 1 1 ω ω ω ondmensonal natural frequency ω ω ω Fg.. he natural frequences of the clamped-free beams wth varous tapered angles Fg. 7. he natural frequences of the clamped-clamped beams wth varous tapered angles. ondmensonal natural frequency ω ω ω Fg. 6. he natural frequences of the smple-supported beams for varous tapered angles. V. COCLUDIG REMARKS he varaton n calculated natural frequences for the pretwsted beams usng the splne collocaton method s nvestgated. he soluton of the governng fourth-order dfferental equaton s approxmated by the splne functon wth polynomal. he effcency and accuracy of the proposed method s ascertaned by comparson wth exstng solutons. umercal results n dfferent cases valdated the applcablty of the proposed method for solvng such an engneerng problem. he pre-twsted angles nfluence the natural frequences of the beams. he demonstrated accuracy and smplcty of the proposed method makes t a good canddate for modelng more complcated pre-twsted beam problems. REFERECES 1. Abrate, S., Vbratons of non-unform rods and beams, Journal of Sound and Vbraton, Vol. 18, pp (199.. Anderson, G. L., On the extensonal and flexural vbraton of rotatng bars, Internatonal Journal of onlnear Mechancs, Vol. 1, pp. -6 (197.. Bert, C. W. and Sheu, Y., Statc analyses of beams and plates by splne collocaton method, Journal of Engneerng Mechancs, Vol. 1, pp ( Dawson, B., Couple bendng-bendng vbratons of pre-twsted cantlever bladng treated by Raylegh-Rtz energy method, Journal of Mechancal Engneerng Scence, Vol. 1, pp ( Dawson, B. and Carnege, W., Model curves of pretwsted beams of rectangular cross-secton, Journal of Mechancal Engneerng Scence, Vol. 11, pp. 1-1 ( El-Hawary, H. M., Zanaty, E. A., and El-Sanousy, E., Quartc splne collocaton methods for ellptc partal dfferental equatons, Appled Mathematcs and Computaton, Vol. 168, pp (. 7. Grevlle,.. E., heory and Applcatons of Splne Functons, Academc Press, ew York ( Gupa, R. S. and Rao, S. S., Fnte element egenvalue analyss of tapered and twsted moshenko beams, Journal of Sound and Vbraton, Vol. 6, pp ( Hodges, D. H., Chung, Y. Y., and Shang, X. Y., Dscrete transfer matrx method for non-unform rotatng beams, Journal of Sound and Vbraton, Vol. 169, pp ( Kuang, J. H. and Hsu, M. H., Egen solutons of grouped turbo blades solved by the generalzed dfferental quadrature method, ransactons of ASME, Journal of Engneerng for Gas urbnes and Power, Vol. 1, pp (. 11. Kuang, J. H. and Hsu, M. H., Fber effect of an orthotropc composte blade solved by the dfferental quadrature method, Composte Structures, Vol. 8, pp (.

10 M.-H. Hsu: Vbraton Analyss of Pre-wsted Beams Usng the Splne Collocaton Method Ln, S. M., Wang, W. R., and Lee, S. Y., he dynamc analyss of nonunformly pretwsted moshenko beams wth elastc boundary condtons, Internatonal Journal of Mechancal Scences, Vol., pp. 8-6 (1. 1. Patlashenko, I. and Weller,., wo-dmensonal splne collocaton method for nonlnear analyss of lamnated panels, Computers & Structures, Vol. 7, pp ( Prenter, P. M., Splne and Varatonal Methods, John Wley & Sons, Inc., ew York ( Rao, J. S., Flexural vbraton of pretwsted tapered cantlever blades, ASME Journal of Engneerng for Industry, Vol. 9, pp. -6 ( Rao, J. S., Vbraton of rotatng, pretwsted and tapered blades, Mechansm and Machne heory, Vol. 1, pp. 1-7 ( Rao, S. C. S. and Kumar, M., Optmal B-splne collocaton method for self-adjont sngularly perturbed boundary value problems, Appled Mathematcs and Computaton, Vol. 188, pp ( Schumalcer, L., Splne Functons: Basc heory, Wley-Interscence, ew York ( Stort, D. and Aboelnaga, Y., Bendng vbratons of a class of rotatng beams wth hypergeometrc solutons, Journal of Appled Mechancs, Vol., pp ( Subrahmanyam, K. B., Kulkarn, S. V., and Rao, J. S., Coupled bendng-bendng vbratons of pre-twsted cantlever bladng allowng for shear deflecton and rotary nerta by the Ressner method, Internatonal Journal of Mechancal Scence, Vol., pp. 17- ( Subrahmanyam, K. B. and Rao, J. S., Coupled bendng-bendng vbratons of pre-twsted tapered cantlever beams treated by the Ressner method, Journal of Sound and Vbraton, Vol. 8, pp (198.. Swamnathan, M. and Rao, J. S., Vbratons of rotatng, pretwsted and tapered blades, Mechansm and Machne heory, Vol. 1, pp. 1-7 ( Vswanathan, K. K. and avaneethakrshnan, P. V., Free vbraton study of layered cylndrcal shells by collocaton wth splnes, Journal of Sound and Vbraton, Vol. 6, pp (.. Wu, L. Y. and Chen, Y.., Analyss of rgd frame by splne collocaton method, Journal of the Chnese Insttute of Engneers, Vol. 6, pp (.. Wu, L. Y. and Chen, Y.., Applcaton of splne collocaton method n analyss of beam and contnuous beam, he Chnese Journal of Mechancs-Seres A, Vol. 19, pp (. 6. Wu, L. Y., Chung, L. L., Chen, Y.., Wu,. Y., and Wu,. J., Applcaton of modfed splne collocaton method n analyss of rgd frame, Proceedngs of the nth Internatonal Conference on Computng n Cvl and Buldng Engneerng, Vol. 1, pp. 7- (. 7. Young,. H., Dynamc response of a pre-twsted tapered beam wth non-constant rotatng speed, Journal of Sound and Vbraton, Vol. 1, pp. -6 (1991.