Modal analysis of a multi-blade system undergoing rotational motion

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1 Journa of Mechanica Science and Technoogy 3 (9) 5~58 Journa of Mechanica Science and Technoogy DOI.7/s Moda anaysis of a muti-bade system undergoing rotationa motion Ha Seong Lim and Hong Hee Yoo * Department of Mechanica Engineering, Hanyang University, Seou, 33-79, Korea (Manuscript Received March 9, 8; Revised October 4, 8; Accepted March, 9) Abstract A modeing method for the moda anaysis of a muti-bade system undergoing rotationa motion is presented in this paper. Bades are assumed as cantiever beams and the couping stiffness which originates from the shroud fexibiity is considered for the modeing. To obtain genera concusions from the numerica resuts, the equations of motion are transformed into a dimensioness form. Dimensioness parameters reated to the anguar speed, the hub radius, and the couping stiffness are identified and the effects of the parameters on the moda characteristics of the system are investigated. It is shown that the couping stiffness especiay pays an important roe to change the moda characteristics of the system. The range of critica anguar speed is aso obtained through the numerica anaysis. Keywords: Moda anaysis; Muti-bade; Rotationa motion; ouping stiffness; Dimensioness parameter; ritica anguar speed Introduction Rotating muti-bade systems can be found in severa engineering exampes such as turbine generators, turbo engines, turbo fans, and rotorcraft wings. These structures are comprised of severa bades which are attached to a hub (or a dis) and often connected through shrouds. Since the shrouds possess fexibiity, they create stiffness couping effects between bades. The stiffness couping effects aong with the anguar motion infuence the moda characteristics of the system significanty. To design such structures, therefore, the effects of the stiffness couping as we as the anguar motion of the muti-bade system need to be considered to estimate their moda characteristics accuratey. Study on the natura frequency variation of a rotating fexibe structure originated from the wor by Southwe and Gough []. They deveoped an anaytica mode (often caed the Southwe equation) to cacuate the natura frequencies of a rotating beam. This paper was recommended for pubication in revised form by Associate Editor Seochyun Kim * orresponding author. Te.: , Fax.: E-mai address: hhyoo@hanyang.ac.r KSME & Springer 9 Later, Schihans [] derived the equations of motion for rotating cantiever beams and obtained more accurate coefficients for the Southwe equation based on the Ritz method. Since eary 97s, the astonishing progress of computing technoogies has enabed one to cacuate the moda characteristics of the rotating beam with severa numerica methods. A arge amount of iterature reating to this subect can be found (see, for instance, Refs. [3, 4]). Recenty, a dynamic modeing method empoying a hybrid set of deformation variabes was introduced (see Refs. [5, 6]). Linear equations of motion can be derived with the modeing method. Even if the equations of motion are inear they can capture the stiffness variation effects induced by the rotationa motion. The equations aso incude the couping effect between stretching and bending motions. So the effect of the couping effect on the moda characteristics of the rotating beam coud be successfuy investigated with the modeing method (see Ref. [7]). In most of the studies mentioned in the previous paragraph, the moda characteristics of a singe bade were ony investigated. The study on the moda characteristics of a muti-bade system was presented ony in a few papers (see Refs. [8, 9]). To obtain the moda

2 5 H. S. Lim and H. H. Yoo / Journa of Mechanica Science and Technoogy 3 (9) 5~58 characteristics of a muti-bade system, the finite eement method can be empoyed effectivey. However, if the finite eement method is empoyed the equation size becomes huge for the muti-bade system. Such a huge size mode is not proper for the purpose of system design. Furthermore, a parameter study for design cannot be done with the finite eement mode. Therefore, a simpified mode in which bades were ideaized as rigid penduums having a discrete torsiona spring was empoyed for designs in those previous studies. The simpified mode was empoyed in Ref. [8] and the FE mode was empoyed in Ref. [9]. However, these modes are too simpe or too compex so that they are not proper for the design of a rotating bade system. The purpose of the present study is to deveop a inear dynamic mode to anayze the moda characteristics of rotating muti-bade systems and to investigate the effects of some important parameters on the moda characteristics of the system using the mode. To achieve the purpose, the equations of motion are derived based on the hybrid deformation variabe modeing method (see Ref. [6]) and they are transformed into a dimensioness form in which some dimensioness parameters reated to the anguar speed, the hub radius, the shroud couping stiffness, and the attachment ocation of the shroud spring are identified. The effect of the number of bades on the moda characteristics of the system is aso investigated with the proposed mode.. Equations of motion In this section, equations of motion of a rotating muti-bade system are derived based on the hybrid deformation variabe modeing method introduced in reference [6] where ony a singe bade was considered. To focus on the main issue of the present study (the effects of the couping stiffness between mutipe bades ad the anguar motion), the foowing assumptions are made in this study. A the bades are assumed to be identica and ony in-pane deformation occurs. Bades are assumed to have a sender shape and isotropic materia property. So the stretching and the bending deformations are ony considered, whie the shear, the rotary inertia, the eccentricity, and the warping effects are negected. Fig. shows the configuration of a muti-bade system. The bades are attached to a rigid hub A having radius r. â and â represent unit vectors Fig.. onfiguration of a muti-bade system. attached to the rigid hub; x denotes the ength from the point O to the point P (the generic point of the beam before deformation occurs); denotes the couping stiffness of the shroud spring ; a denotes r the attachment position of the spring; u < > denotes the eastic deformation vector of n-th beam; and s < > denotes the stretch of the beam. Empoying the Rayeigh-Ritz assumed mode method, a hybrid set of variabes s < > and u < > can be approximated as foows: µ < > φ < > i i i= µ < > φ < > i i i= s ( xt, ) = q ( t) () u ( x, t) = q ( t) () where φ i and φ i denote the ongitudina and the bending mode functions; q < > i and q < > i denote the generaized coordinates for the two mode functions respectivey; and µ and µ denote the numbers of the generaized coordinates. When the rigid hub rotates with an anguar speed Ω, the anguar veocity of the rigid hub A and the veocity of the generic point P (the generic point of the beam after deformation occurs) can be obtained as foows. r (3) r P < > < > < > < > v = u& ˆ ˆ Ω u a+ rω+ u& +Ω ( x+ u ) a (4) A ω = Ωaˆ3 where u & < > need to be expressed with respect to s & < > and u & < >. The foowing geometric reation (see Ref. [6] in detai) can be used for the purpose. < > < > x < > < > u u& s& = u& + dσ (5) σ σ

3 H. S. Lim and H. H. Yoo / Journa of Mechanica Science and Technoogy 3 (9) 5~58 53 If the Kane s method (see Ref. []) is empoyed, the equations of motion for the system can be derived with the foowing equation. rp rp < > v dv U ρ + = q& i dt qi (6) where denotes the ength of the beam, ρ denotes the mass per unit ength of the beam, and q i denotes the generaized coordinate. In Eq. (6), U < > denotes the strain energy of the beam which can be given as < > < > < > s u U = EA + EI x x < > < > + T u ( a) u ( a) < + > < > + T u ( a) u ( a) (7) where E, A and I represent Young s moduus, the cross-section area and the second area moment of inertia of the beam, respectivey. Empoying Eqs. (- 7), the equations of motion of the -th beam can be derived as foows: µ = m q&& Ω m q + q µ = < > < > S < > i i i Ω m q& +Ω& m q < > < > i i = rω P +Ω Q ( i =, L, µ ) (8) i i µ && + { +Ω ( + )} = < > < n> < + > ( i q i q + i q ) µ = m q m q < > B GA GB < > i i i i i + Ω m q& +Ω& m q < > < > i i & & i i i µ = rωp ΩQ where ( =, L, ) (9) ab mi = ρφai φb S i = EAφ, ix φ x, B i = EIzz ϕ, i xx ϕ, xx GA i = ρr( x) φ, i x φ, x () GB ρ i = ( x ) φ, i x φ, x = φ ( a) φ ( a) i T i Pai = ρφai (6) Qai = ρφ x ai In the above expressions, φ, ix and φ, ix indicate the differentiation of the symbos φ i and φ i with respect to x and φ, ixx indicates the doube differentiation of the symbo. For sender beams, the extensiona natura frequencies are much higher than the bending natura frequencies. Therefore, the couping effect between the extensiona motion and the bending motion can be ignored without osing the accuracy of the anaysis. So the foowing equation (by negecting the couping terms and the non-homogeneous terms from Eq. (9)) wi be empoyed for the moda anaysis. µ = { ( )} m q + +Ω + m q && < > B GA GB < > i i i i i ( i q i q i q ) + = < > < > < + > () It is usefu to write the equations of motion in a dimensioness form. To achieve this, the foowing dimensioness variabes are empoyed: t x q τ ξ θ T where i =, =, i = () T = 4 ρ EI By empoying the above dimensioness variabes in Eq. (), Eq. () can be rewritten as foows: µ = { ( )} M θ + K + γ K + K M θ && < > B GA GB < > i i i i i ( Ki θ Ki θ Ki θ ) + = < > < > < + > (3) where && θ < > denotes the doube differentiation of θ < > with respect to τ, γ denotes the dimensioness anguar speed which can be obtained by the anguar speed Ω mutipied by T and

4 54 H. S. Lim and H. H. Yoo / Journa of Mechanica Science and Technoogy 3 (9) 5~58 M K K K K i ϕi ( ξ) ϕ ( ξ) dξ B i = ϕi, ξξ ( ξ) ϕ, ξξ ( ξ) dξ GA i = i, ξ, ξ = δ ( ξ ) ϕ ( ξ ) ϕ ( ξ ) dξ (4) = ( ), ξ( ), ξ( ) d ξ ϕ ξ ϕ ξ ξ = β ϕ ( α) ϕ ( α) GB i i i i In the above expressions, ϕ is a function of ξ which has the same functiona vaue as φ. Three more dimensioness parameters are empoyed in the above expressions and they are defined as foows: 3 a r α =, β T, δ (5) EI Now by assembing n sets of the equations written in Eq. (3), the tota equations of motion for a muti-bade system can be written as foows: { θ} + K { θ} = { } M && (6) where [ M ] L [ M ] M = M O M [ M ] L [ M ] [ K] + K K L K [ K] + K M K = M O M [ K] + K K L K [ K] + K where, ω denotes the dimensioness natura frequency and { η } denotes the mode vectors. Substituting Eq. (7) into Eq. (6), one can obtain the foowing equation: { } = K { } (8) ω M η η The moda anaysis of a muti-bade system wi be done by using Eq. (8) in the next section. 3. Numerica resuts To obtain accurate numerica resuts, five assumed modes (which are the bending eigenmode functions of a cantiever beam) are empoyed for each beam to obtain the natura frequencies and the corresponding mode shapes. Numerica resuts obtained by using the present modeing method are compared to those in [9], which provides some anaytica soutions. The owest nine natura frequencies of a 3-beam system are shown in Tabe for which α =., β =, γ =, δ = are empoyed. It is found that the resuts obtained by the present modeing method are amost identica to those of [9]. Fig. shows the owest eight dimensioness natura frequencies of 4-beam system versus the dimensioness anguar speed. The parameters empoyed for the anaysis are α =., β = and δ =.. As shown in the figure, there exist two groups of natura frequencies. The first group incudes the owest four natura frequencies and the second group incudes the next four natura frequencies. Since the couping stiffness β is not arge enough, the natura frequencies in each group are amost identica. The figure aso shows that the natura frequencies increase as the anguar speed increases. Fig. 3 shows the corresponding eight mode shapes of the 4-beam system. As shown in the figure, the where ( ) K = K + γ K + K M B GA GB i i i i i For the moda anaysis of the system, a coumn matrix { θ } can be expressed as foows: { θ} e ωτ { η} = (7) Fig.. Lowest 8 natura frequencies of the 4 beam system versus the anguar speed.

5 H. S. Lim and H. H. Yoo / Journa of Mechanica Science and Technoogy 3 (9) 5~58 55 Tabe. omparison of the natura frequencies ( α =., β =, γ = ).. Frequencies Present Ref. (9) st First Set nd rd th..4.4 Second Set 5th th..6.6 Third 7th Set 8th (a) α =.5 Fig. 3. Lowest 8 mode shapes of the 4-beam system. first four mode shapes consists of the first bending modes and the next four mode shapes consist of the second bending modes. omparing the nd and the 3rd modes (or the 6th and the 7th modes) one can see that the two modes are amost identica if one of them rotates about 9 degrees. Therefore, the natura frequencies of the two modes shoud be amost identica, too. Fig. 4 shows the owest eight dimensioness natura frequencies of the 4-beam system for different vaues of the dimensioness spring attachment position α. The parameters except α empoyed for the anaysis are same as before. As can be shown in the figure, the owest frequencies in each group remain unchanged whie other frequencies increase. The two midde natura frequencies in each group are the same (as discussed in the previous paragraph) so that ony three oci for each group can be observed in the figures. As α increases, the gaps between the frequencies increase significanty. Fig. 5 shows the owest eight dimensioness natura frequencies of 4-beam system versus the dimensioness spring attachment position α. The parameters empoyed for the anaysis are β =, γ = 5 and δ =.. As α increases, the natura frequencies vary. An interesting fact one can observe from the figure is that the second group of natura frequencies (b) α =. Fig. 4. Lowest 8 natura frequencies versus the anguar speed for different vaues of α. Fig. 5. Lowest 8 natura frequencies versus the spring attachment position. increase as α increases initiay. However, they decrease after reaching their maximum vaues and become identica at a specific vaue of α. The ocation of α is actuay the same as the noda point of the rotating beam. Fig. 6 shows the owest eight dimensioness natura frequencies of the 4-beam system for different vaues of the dimensioness couping stiffness β. The parameters are empoyed the anaysis are α =.5 and δ =.. As shown in the figures, the

6 56 H. S. Lim and H. H. Yoo / Journa of Mechanica Science and Technoogy 3 (9) 5~58 (a) β = 5 Fig. 7. Lowest 8 natura frequencies vs. the couping stiffness. (b) β = (a) δ = Fig. 6. Lowest 8 natura frequencies versus the anguar speed for different vaues of β. owest frequencies in each group remain unchanged. It can be observed from the figures that the gaps between the frequencies increase as the couping stiffness β increase. However, the gap variation between frequencies in the first group is ess affected by β (compared to α ). Fig. 7 shows the owest eight dimensioness natura frequencies of the 4-beam system versus the dimensioness couping stiffness β. The parameters empoyed for the anaysis are α =.5, γ = 5 and δ =.. As shown in the figures, the natura frequencies (except the owest ones in each group) increase monotonicay as the couping stiffness increases. The owest natura frequencies remain unchanged. Fig. 8 shows the owest eight dimensioness natura frequencies of the 4-beam system for different vaue of the dimensioness hub radius ratio. The parameters empoyed for the anaysis are α =.5 and β =. As can be shown in the figure, the sopes of the natura frequencies become stiffer as the hub radius ratio δ increases. This can be easiy understood since the arger hub radius resuts in arger centrifuga force, which resuts in arger motion-induced stiffness for the system. (b) δ = Fig. 8. Lowest 8 natura frequencies versus the anguar speed for different vaues of δ. Fig. 9 shows the owest eight dimensioness natura frequencies of 4-beam system versus the dimensioness hub radius ratio δ. The parameters empoyed for the anaysis are α =.5, β =, and γ = 5. The figure shows that the natura frequencies increase approximatey in proportion to the square root of the hub radius ratio. Since the stiffness of the motion induced stiffness is proportiona to the hub radius, the natura frequency shoud be approximatey in proportiona to the hub radius. Fig. shows two groups of dimensioness natura frequencies for 6-beam system and 8-beam system. The parameters empoyed for the anaysis are α =.5,

7 H. S. Lim and H. H. Yoo / Journa of Mechanica Science and Technoogy 3 (9) 5~ oncusions Fig. 9. Lowest 8 natura frequencies vs. the hub radius. (a) number of bades = 6 (b) number of bades = 8 Fig.. Effect of the number of bades on the natura frequencies. β = and δ =.. It seems that there exist ony eight natura frequency oci in Fig. (a) and ten natura frequency oci in Fig. (b). However, every ocus in the midde of each group contains two sets of natura frequency information. As shown in the figure, the gap between the owest natura frequency and the highest natura frequency is rarey affected by the number of bades. Therefore, the gap between any adacent two oci decreases as the number of bades increases. A dynamic modeing method to derive the equations of motion of a rotating muti-bade system is presented in this paper. The modeing method empoys a set of hybrid deformation variabes with which the motion-induced stiffness variation effect can be incuded in the equations of motion. The equations of motion are transformed into a dimensioness form in which dimensioness parameters reated to the anguar speed, the couping stiffness, the ocation of couping stiffness and the hub radius are identified. Numerica resuts show that the natura frequencies increase as the anguar speed increases. As the hub radius increases, the increasing sope becomes stiffer. The gap between the frequency oci increases as the couping stiffness or the ocation of couping stiffness increases. Finay, the gap between the owest natura frequency and the highest natura frequency is rarey affected by the number of bades. Therefore, as the number of bades increases, the gap between any two frequency oci decreases. References [] R. Southwe and F. Gough, The free transverse vibration of airscrew bades, British A. R.. Reports and Memoranda 766 (9). [] M. Schihans, Bending frequency of a rotating cantiever beam, J. of App. Mech. Trans. Am. Soc. Mech. Engrs 5 (958) 8-3. [3] A. Leissa, Vibration aspects of rotating turbomachinery bades, Appied Mechanics Reviews 34 (98) [4] J. Rao, Turbomachine bade vibration, Shoc Vibration Digest 9 (987) 3-. [5] T. Kane, R. Ryan, and A. Baneree, Dynamics of a cantiever beam attached to a moving base, Journa of Guidance, ontro, and Dynamics () (987) [6] H. Yoo, R. Ryan, and R. Scott, Dynamics of fexibe beams undergoing overa motions, Journa of Sound and vibration 8 (995) [7] H. Yoo and S. Shin, Vibration anaysis of rotating cantiever beams, Journa of Sound and Vibration (998) [8] M. Singh and D. Schiffer, Vibrationa characteristics of paceted baded discs, ASME Paper No. 8- DET-37 (98). [9] M. Singh, J. Vargo, D. Schiffer and J. Deo, Safe diagram A design reiabiity too for turbine bad-

58 H. S. Lim and H. H. Yoo / Journa of Mechanica Science and Technoogy 3 (9) 5~58 ing, Proceedings of the Seventeenth Turbomachinery Symposium Texas A&M Universiy (988) 93-. [] T. Kane and D.

8 58 H. S. Lim and H. H. Yoo / Journa of Mechanica Science and Technoogy 3 (9) 5~58 ing, Proceedings of the Seventeenth Turbomachinery Symposium Texas A&M Universiy (988) 93-. [] T. Kane and D. Levinson, Dynamics: Theory and Appications, McGraw-Hi Boo o., New Yor, N. Y (985). Ha Seong Lim graduated from Department of Mechanica Engineering at Hanyang University in 6 and received his Master s degree in 8. He is currenty a technica engineer in STX Offshore & Shipbuiding ompany, Seou, Korea.. Hong Hee Yoo graduated from the Department of Mechanica Design and Production Engineering at Seou Nationa University in 98 and received his Master s degree from the same department in 98. He received his Ph.D. degree in 989 from the Department of Mechanica Engineering and Appied Mechanics at the University of Michigan at Ann Arbor, U.S.A. He is currenty a professor in the Schoo of Mechanica Engineering in Hanyang University, Seou, Korea.

Lobontiu: System Dynamics for Engineering Students Website Chapter 3 1. z b z

Lobontiu: System Dynamics for Engineering Students Website Chapter 3 1. z b z Chapter W3 Mechanica Systems II Introduction This companion website chapter anayzes the foowing topics in connection to the printed-book Chapter 3: Lumped-parameter inertia fractions of basic compiant