ICSV4 Carns Australa 9- July, 7 WAVE PROPAGATION, REFECTION AND TRANSMISSION IN CURVED BEAMS Seung-Kyu ee, Bran Mace and Mchael Brennan NVH Team, R&D Centre, Hankook Tre Co., td. -, Jang-Dong, Yuseong-Gu, Daejon, 5-75, Korea Insttute of Sound and Vbraton Research, Unversty of Southampton Hghfeld, Southampton, SO7 BJ, UK sklee@hankooktre.com Abstract Wave moton n thn, unform, curved beams wth constant curvature s consdered. The beams are assumed to undergo only n-plane moton, whch s descrbed by the sxth-order coupled dfferental equatons based on Flügge s theory. In the wave doman the moton s assocated wth the three ndependent wave modes. A systematc wave approach based on reflecton, transmsson and propagaton of waves s presented for the analyss of structures contanng curved beam elements. Dsplacement, nternal force and propagaton matrces are derved. These enable transformatons to be made between the physcal and wave domans and provde the foundaton for systematc applcaton of the wave approach to the analyss of wavegude structures wth curved beam elements. The energy flow assocated wth waves n the curved beam s also dscussed. It s seen that energy can be transported ndependently by the propagatng waves and also by the nteracton of a par of postve and negatve gong wave components whch are non-propagatng,.e. ther wavenumbers are magnary or complex. A further transformaton can be made to power waves, whch can transport energy ndependently.. INTRODUCTION Curved beams are used wdely n bult-up structures and hence ther dynamc behavour s of nterest. Prevous work n ths area has been summarsed n several artcles, for example [-]. Wu and undberg [4] have nvestgated the transmsson of energy through a curved secton connectng two straght beams. They presented numercal results n the form of polar radaton dagrams for beams wth dfferent curvatures. Walsh and Whte [5] consdered the energy flow assocated wth a sngle propagatng wave component n a curved beam based on four dfferent theores ove s theory, Flügge s theory and the correctons for rotary nerta and shear deformaton. They derved expressons whch relate the power to the extensonal, bendng and shear waves. Kang et al. [] appled the wave approach based on the reflecton, transmsson and propagaton of waves to obtan the natural frequences of fnte curved beams. The man am of ths paper s to descrbe a systematc wave approach based on reflecton, transmsson and propagaton of waves and to use ths to determne the energy flow
ICSV4 9- July 7 Carns Australa characterstcs of waves n a thn, curved beam. The approach s also vald when rotary nerta, shear deformaton and dampng are mportant, but these effects are neglected here. Attenton s focused on n-plane moton and Flügge s theory s used. The moton s descrbed n terms of sx ndependent (or uncoupled wave components. In secton, the dsperson relaton and the rato of tangental dsplacement to radal dsplacement for the sx wave components are obtaned. In secton dsplacement, nternal force and propagaton matrces are derved. These enable transformatons to be made between the physcal and wave domans and provde the foundaton for systematc applcaton of the wave approach [7] to wavegude structures wth curved elements. In secton 4, the energy flow assocated wth the wave components s obtaned n a systematc way. Ther contrbutons are classfed accordng to dfferent condtons for the wavenumbers. The energy flow paths at a gven frequency are dentfed. Energy can be transported ndependently by propagatng waves or by pars of wave components wth magnary or complex wavenumbers. A further transformaton s found to power wave components these propagate energy ndependently through the curved beam.. IN-PANE WAVE MOTION IN CURVED BEAMS Consder the n-plane moton of a thn, unform, curved beam wth constant curvature. Neglectng the effects of shear deformaton and rotary nerta, the governng equatons for free vbraton n the radal and tangental drectons are gven by [5] 4 w w w EA w u w EI + + A, 4 4 + = ρ s R s R R R s t u w u EA + ρ A = s R s t (a,b where E s the Young s modulus, I the second moment of area, A the cross-sectonal area, ρ the densty, s the crcumferental coordnate along the centerlne, t tme, and w and u the dsplacements of the centerlne n the radal and tangental drectons respectvely. The rotaton ϕ of the cross-secton and the normal force N, bendng moment M, and shear force Q are gven by [5] u w w u EI w w ϕ = +, N = EA + + +, R s R s R R s w w w w M = EI +, Q EI = + R s R s s (a-d Equatons ( and ( are based on Flügge s theory. When R tends to nfnty, the radal and tangental dsplacements decouple and the equatons become those for a unform, straght beam.. Dsperson relatons The radal and tangental dsplacements satsfyng equaton ( are assumed to be tme harmonc and of the form wst (, ( t ks Ce ω ( t ks =, w ust (, Ce ω u = (a,b
ICSV4 9- July 7 Carns Australa where C w and C u are constants, ω the angular frequency and k the wavenumber for the curved beam. Substtutng equatons (a,b nto equaton ( gves I ( ρ + kr Rω kr AR E Cw = ρ Cu kr k R R ω E (4 Settng the determnant of the matrx n equaton (4 to zero gves the dsperson equaton ( κ ( κ B κ ( κ κ B B 4 4 4 4 4 4 k k + k + k + k k k + k k k = (5 where k = ρω E and 4 kb = ρaω EI are the longtudnal and bendng wavenumbers for a straght beam, respectvely, and κ = R s the curvature. The beam s assumed to be undamped so that k and k are real. Equaton (5 s a cubc equaton n B k so that there are three pars of solutons at any gven frequency, three for postve-gong waves and three for negatve-gong waves. The wavenumbers of postve-gong waves are defned to be such that Im{ k}, Re{ k ω} > f { } Im k = (a,b Equaton (a ndcates that, f the magnary value of the wavenumber of a postve-gong wave s non-zero, the ampltude of the wave decays n the postve s drecton. If the magnary value s zero, equaton (b ndcates that the energy transport velocty assocated wth a postve-gong wave should be postve. The non-dmensonal radus of gyraton, wavenumber and frequency are ntroduced and are respectvely gven by I χ =, ξ = kr, AR ωr = (7a-c c where c and = E ρ s the longtudnal phase velocty. Fgure shows the wavenumbers ξ, ξ ξ for the postve-gong waves n the curved beam wth χ = whch corresponds to hr=. f the beam s rectangular. In the fgure the frequency range s dvded nto 4 regons 9 (a Regon I Regon II Regon III Regon IV Regon I Regon II Regon III Regon IV (b Re{ξ} ξ ξ Im{ξ} ξ ξ ξ ξ 9 Fgure. Dsperson relatons for postve-gong waves n the curved beam wth χ =.
ICSV4 9- July 7 Carns Australa by the bfurcaton ponts. In regon I, the wavenumbers are all purely real so that all the wave modes propagate along the curved beam. One nterestng feature s that the (real wavenumber ξ for the second mode s negatve n ths regon. Thus the phase velocty of the wave mode s negatve whle the energy s transported n the postve s drecton,.e. a wave transports energy n the drecton opposte to the drecton of the phase velocty. In regon II, ξ s complex and, snce ξ = ( ξ, ths represents a spatally decayng standng wave. Only the frst mode can propagate. In regon III, also, only the frst mode propagates. The other wave modes are both evanescent,.e., they decay wthout a change n phase. In regon IV, ξ becomes purely real, representng a propagatng wave. In ths regon the wavenumbers are broadly analogous to, ξ n a straght beam. those of bendng ( ξ ξ and extensonal waves (. Dsplacement rato The radal and tangental dsplacements of the curved beam are not ndependent of each other. From equaton (4, the rato α = Cu C s gven by w κ k α = k k ; =,,, (8 where =,, denote the three postve-gong waves, respectvely, and = 4, 5, denote the correspondng negatve-gong waves. Note that α 4,5, = α snce,, k4,5, = k.,, Fgure shows the dsplacement rato for the three postve-gong waves for the curved beam wth χ =. The four regons shown n Fgure are not marked for clarty, but can be nferred from the dscontnuous behavour of the curves. It can be seen that the radal moton s domnant for the frst wave mode snce α < n the frequency range consdered. In regon II α = α. In regons III and IV, the radal moton s domnant for the second mode. Near the rng frequency =, the radal moton s domnant for the thrd mode (the magntude of α s zero at the rng frequency but, as frequency ncreases, the tangental moton becomes domnant. The phase dfference between the dsplacement components s between π and π. (a π (b Magntude Abs (α α α α Phase π/ π/ α α α π Fgure. Dsplacement rato α = Cu C for the curved beam wth w χ =. 4
ICSV4 9- July 7 Carns Australa. MATRIX REPRESENTATION OF WAVE MOTION A systematc methodology for wave analyss based on reflecton, transmsson and propagaton of waves s provded by the defnton of dsplacement, nternal force and propagaton matrces [7]. In ths secton, the matrces for the curved beam are presented. Snce the curved beam s a three-mode system, the relevant vectors and matrces are of order and, respectvely. Assumng the dsplacements to be of the form gven by equaton (, the radal and tangental dsplacements of the beam are gven respectvely by ( α ( α ks ks ks ks 4 ks 5 ks 4 5 ws ( = Ce + Ce + Ce + Ce + Ce + Ce, us ( = α Ce + α Ce + Ce + α Ce + α Ce + Ce ks ks ks ks 4 ks 5 ks 4 4 5 5 (9a,b The generalzed dsplacements and correspondng nternal forces can be grouped n the vectors w = [ w ϕ u] T, = [ Q M N] T f (a,b where the superscrpt T denotes the transpose. Note that the rotaton ϕ and nternal forces Q, M and N are obtaned from equatons ( and (9. The wave vectors consstng of the ampltudes of the waves are defned by ks ks ks T = + ( s Ce C e C e ks 4 ks 5 ks a, a ( s = C4e C5e Ce (a,b T The dsplacement and nternal force vectors are related to the vectors of wave ampltudes by [7] + + w Ψ Ψ a = + f Φ Φ a ( where the matrces Ψ and Φ defne the transformaton from the wave doman to the physcal doman. They are gven by Ψ ψ ψ ψ + = Φ + =,, Ψ ψ ψ ψ = 4 5 Φ = 4 5, (a-d where the column vectors ψ and for =,, 4, 5 are ψ = ( κα + k, α EIk ( κ k EI ( κ k ( κ α + κ ( κ = (4a,b EA k EI k and ψ and for =, are ψ = ( κα + k, α α EIk ( κ k EI ( κ k ( κ α + κ( κ = (5a,b α EA k EI k 5
ICSV4 9- July 7 Carns Australa Usng these matrces, the reflecton and transmsson matrces for arbtrary dscontnutes or for boundares can be found n a smple manner [7]. The propagaton matrx F, descrbng propagaton of waves over a dstance along the curved beam, s gven by k e F ( ( = k e k Note that the propagaton matrx s dagonal (.e. the waves are not coupled durng propagaton and the dagonal elements are ndependent of poston. e 4. ENERGY FOW IN CURVED BEAMS The tme-averaged power Π assocated wth waves n one-dmensonal structures can be expressed as [7] T where the superscrpt H denotes the Hermtan, + ( ( gven by H Π= a Pa (7 T a T = a a and the power matrx P s ( ( ( ( ( ( ( ( H H H H + + + + + + ω Ψ Φ Ψ Φ Φ Ψ Φ Ψ P = H H H H + + Ψ Φ Ψ Φ Φ Ψ Φ Ψ (8 Substtutng equaton ( nto equaton (8 gves the power matrx P for the curved beam. In the four frequency regons the power matrx s gven, respectvely, by regon I P P P P, P P = P P P regon II P P =, P P P (9a-d regon III P P P5 P = P P5 P P P5 regon IV P P = P P5 P where the elements are
ICSV4 9- July 7 Carns Australa 4 4 kbκ kbκ P = ωeik ( k κ +, P = ωeik ( k κ +, ( k k ( k k ( k κ ( k k k ωei P = ωea +, 4 P = ( k ( k ( k k, k kbκ κ 4 kbκ k P5 =ωeik ( k κ +, P = ωea + ( kl k k ( ( k ( κ k k k κ 4 B (a-f It can be notced that an element of P s non-zero,.e., energy can be transported, n three cases: by a sngle wave wth real wavenumber (.e., a propagatng wave; by nteracton of two opposte-gong waves of one mode, for whch the wavenumber s purely magnary (.e., two opposte-gong nearfeld waves; or by nteracton of two opposte-gong waves from dfferent modes, for whch the wavenumbers are a complex conjugate par. These results are consstent wth the work by angley [8] for a general one-dmensonal dynamc system. Fgure shows the magntudes of the non-zero elements of P for the curved beam wth χ = as a functon of a frequency. Note that there are always sx energy transport paths (.e., sx non-zero elements n the power matrx at any frequency. At hgh frequences, the powers assocated wth the waves tend to those of the straght beam: t s seen that the normalsed magntudes of P, P 5 and P tend to unty above the rng frequency =. Normalzed magntude Normalzed Modulus Fgure. Non-zero elements of the power matrx for the curved beam wth χ = : P (, P (, P (, P (, 5 P = P (, 5 P (. In the fgure P and P are normalsed wth respect to ω EAk and the others are normalsed wth respect to ω EIk. B The power matrx s not dagonal except for the frequency regon I. A further transformaton can be defned usng a power bass, where energy s transported ndependently by a sngle component, usng the egenvalues and egenvectors of the power matrx. et V be the dagonal matrx consstng of the (real egenvalues and E be the (untary matrx whose columns are the egenvectors of P. Snce P = EVE, equaton (7 can be wrtten as H Π= p Vp ( 7
ICSV4 9- July 7 Carns Australa where p= E a s a vector of power wave ampltudes. Snce V s dagonal, equaton ( ndcates that energy s transported ndependently by the ndvdual power wave components of p. For example, V and E n the frequency regon II are gven by P P P V =, P P P E = (a,b where = P P. Smlar transformaton nto the power wave doman n the frequency regons III and IV can also be made [7]. 5. CONCUDING REMARKS Ths paper concerned n-plane moton of curved beams based on Flügge s theory. Dsplacement and force matrces were derved these allow transformatons to be made between the physcal and wave domans enablng a systematc analyss to be made of wavegude structures wth curved components. The energy flow assocated wth waves n the curved beam was also obtaned n a systematc way. It was seen that energy s transported ndependently by propagatng waves or by the nteracton of two wave components, for whch the wavenumbers are a complex conjugate par. A further transformaton to power wave components was found these components transport power ndependently. REFERENCES [] J. P. Charpe and C. B. Burroughs, An analytcal model for the free n-plane vbraton of beams of varable curvature and depth, Journal of the Acoustcal Socety of Amerca 94, 8-879 (99. [] P. Chdamparam and A. W. essa, Vbratons of planar curved beams, rngs and arches, Appled Mechancs Revews 4, 47-48 (99. [] N. M. Aucello and M. A. De Rossa, Free vbratons of crcular arches: a revew, Journal of Sound and Vbraton 7, 4-458 (994. [4] C. M. Wu and B. undberg, Reflecton and transmsson of the energy of harmonc elastc waves n a bent bar, Journal of Sound and Vbraton 9, 45-59 (99. [5] S. J. Walsh and R. G. Whte, Vbratonal power transmsson n curved beams, Journal of Sound and Vbraton, 455-488 (. [] B. Kang, C. H. Redel and C. A. Tan, Free vbraton analyss of planar curved beams by wave propagaton, Journal of Sound and Vbraton, 9-44 (. [7] S.-K. ee, Wave reflecton, transmsson and propagaton n structural wavegudes, PhD Thess, Unversty of Southampton (. [8] R. S. angley, A transfer matrx analyss of the energetcs of structural wave moton and harmonc vbraton, Proceedngs of the Royal Socety of ondon A 45, -48 (99. 8