Modal analysis of sailplane and transport aircraft wings using the dynamic stiffness method

Transcription

1 Journal of Pysics: onference eries PPER OPEN E Modal analysis of sailplane and transport aircraft wings using te dynamic stiffness metod To cite tis article: J R Banerjee 6 J. Pys.: onf. er. 7 5 View te article online for updates and enancements. Related content - Modal analysis of isotropic beams in peridynamics Freimanis and Paegltis - Optimal ocation of Piezoelectric Patc on omposite tructure using Viewing Metod Raul amyal and so K Baga - omparison of various iger order sear deformation teories for static and modal analysis of composite beam Yaya Bin Zia, teeb mad Kan and M. Nausad lam Prof. Tis content was downloaded from IP address on 898 at 9:7

2 5t ymposium on te Mecanics of lender tructures (Mo5) Journal of Pysics: onference eries 7 (6) 5 doi: Modal analysis of sailplane and transport aircraft wings using te dynamic stiffness metod J R Banerjee cool of Matematics, omputer cience and Engineering ity University ondon, Nortampton quare, ondon EV HB, UK bstract. Te purpose of tis paper is to provide teory, results, discussion and conclusions arising from an in-dept investigation on te modal beaviour of ig aspect ratio aircraft wings. Te illustrative examples cosen are representative of sailplane and transport airliner wings. To acieve tis objective, te dynamic stiffness metod of modal analysis is used. Te wing is represented by a series of dynamic stiffness elements of bending-torsion coupled beams wic are assembled to form te overall dynamic stiffness matrix of te complete wing. it cantilever boundary condition applied at te root, te eigenvalue problem is formulated and finally solved wit te elp of te ittric-illiams algoritm to yield te eigenvalues and eigenmodes wic are essentially te natural frequencies and mode sapes of te wing. Results for wings of two sailplanes and four transport aircraft are discussed and finally some conclusions are drawn.. Introduction ailplane and transport aircraft wings are slender and flexible because of teir ig aspect ratios resulting from large spans and relatively sort cords. s a consequence, tey are easily prone to vibration problems. In tis respect, modal analysis of aircraft wings, particularly tose wit ig aspect ratios is very important. ailplane and transport airliner wings are typical examples for wic te investigation is of great significance. Indeed modal analysis plays an important role in te design of aircraft wings. n analysis of tis ind is an obligatory airwortiness requirement wic is rigorously enforced by te civil aviation autorities. Te purpose of tis paper is to carry out suc an analysis and investigate te modal beaviour of sailplane and transport aircraft wings by applying te dynamic stiffness metod. One of te main motivations for modal analysis of aircraft wings originates from tat fact tat it is a fundamental prerequisite to carry out an aeroelastic or response analysis, particularly wen using te normal mode metod. Tere are some publised papers in tis and related areas [-7]. In general, te finite element metod (FEM) is widely used to investigate te modal beaviour of aircraft wings. Te FEM is an approximate metod were te stiffness and mass properties of all individual elements are assembled to form te overall stiffness matrix [K] and mass matrix [M] of te structure wic is an aircraft wing ere. Ten for modal analysis, upon imposing te boundary conditions, te typical eigenvalue problem of te type [[K] - [M]]{} = is solved were {} is te nodal displacement vector and te square root of gives te natural frequencies of te structure. Te corresponding mode sapes are recovered in te usual way. In te FEM it is generally true tat by increasing te number of elements in te analysis, te results become more and more accurate. It is acnowledged tat te FEM is numerically intensive and te degrees of freedom identified by te order of [K] and [M] matrices decide te number of eigenvalues (wic are essentially te natural frequencies) tat can be computed. ontent from tis wor may be used under te terms of te reative ommons ttribution. licence. ny furter distribution of tis wor must maintain attribution to te autor(s) and te title of te wor, journal citation and DOI. Publised under licence by td

3 5t ymposium on te Mecanics of lender tructures (Mo5) Journal of Pysics: onference eries 7 (6) 5 doi: Te iger order natural frequencies will of course, be considerably less accurate. gainst tis bacground, tere is an elegant and powerful alternative to te FEM for modal analysis of aircraft wings or any oter structures. Tis metod is te so-called dynamic stiffness metod (DM). Te DM unlie te FEM, relies on an exact single frequency dependent dynamic stiffness element containing bot te mass and stiffness properties of te element as te basic building bloc. Te assembly procedure in te DM is essentially te same as it is in te FEM, but a single dynamic stiffness element matrix is used for eac structural component instead of separate mass and stiffness matrices to form te overall frequency-dependent dynamic stiffness matrix [K D ] of te complete structure (wing). Te eigenvalue problem is formulated as [K D ]{}= were {} is te nodal displacement vector comprising te amplitudes of nodal displacements. Te next step is to extract te eigenvalues of te structure. t tis point a significant difference wit te FEM arises wit regard to te solution tecnique. Te formulation [K D ]{}= leads to a transcendental (nonlinear) eigenvalue problem as opposed to te linear eigenvalue problem generally encountered in te FEM. Te best available solution tecnique to extract te eigenvalues in te DM is to apply te algoritm of ittric and illiams [8], nown as te ittric-illiams algoritm in te literature wic as featured in literally undreds of papers. Te algoritm wic monitors te turm sequence property of te dynamic stiffness matrix is robust and it ensures tat no natural frequency of te structure is missed. itin te above context, a range of aircraft wings is investigated for teir free vibration caracteristics in tis paper. Two different categories of aircraft wings are analysed. Tey are essentially for sailplane and transport airliner wings. Two illustrative examples for te former and four for te latter are demonstrated wen presenting numerical results. Te investigation required considerable efforts for data preparation to model eac of te wings. Te dynamic stiffness metod wic provides te best possible model accuracy is used as mentioned. In idealising te wing, an assembly of te frequency dependent dynamic stiffness elements of bending-torsion coupled beams [9-], comprising bot te mass and stiffness properties is efficiently utilised. Natural frequencies and mode sapes computed from te dynamic stiffness metod are compared and contrasted and finally some conclusions are drawn.. Teory. Dynamic stiffness matrix of a bending-torsion coupled beam n aircraft wing suc as te one sown in figure is a classic example of a bending-torsion coupled beam. uc a representation is particularly relevant to analyse a ig aspect ratio wing. Figure n aircraft wing idealised as a bending-torsion coupled beam.

4 5t ymposium on te Mecanics of lender tructures (Mo5) Journal of Pysics: onference eries 7 (6) 5 doi: In essence, te coupling between te bending and te torsional motions arises due to non-coincident mass and elastic axes wic are respectively te loci of te centroid and sear centres of te beam cross-section. Tus for an aircraft wing it is not generally possible to realize a torsion-free bending displacement or a bending-free torsional rotation during its dynamic motion unless te load or te torque is applied troug or about te sear centre. Given tis perspective, a ig aspect ratio nonuniform aircraft wing can be accordingly modelled as an assemblage of bending-torsion couple beams of te type sown in figure. Tis paper uses a dynamic stiffness approac and develops te dynamic stiffness matrix of a uniform bending-torsion coupled beam and ten extends it to model a nonuniform wing. Te governing partial differential equations of motion of te bending-torsion coupled beam (wing) sown in figure are given by [9, ] EI + m mx α ψ = () GJψ + mx α I α ψ = () were EI and GJ are te bending and torsional rigidities of te beam, m is te mass per unit lengt, I α is te polar mass moment of inertia per lengt about te Y-axis and te primes and over dots denotes partial differentiation wit respect to position y and time t, respectively. For armonic oscillation, sinusoidal variation in and ψ wit circular frequency ω may be assumed to give (y, t) = H(y)sinωt, ψ(y, t) = Ψ(y)sinωt () were H(y) and Ψ(y) denote te amplitude of te bending displacement and torsional rotation ubstituting equation () into equations () and () eliminates te time component and gives te following ordinary differential equations EIH mω H + mx α ω Ψ = (4) GJΨ + I α ω Ψ ω mx α H = (5) were prime now denotes full differentiation wit respect to y. Equations (4) and (5) can be combined into a sixt order ordinary differential equation by eliminating eiter H or to give were + ( I αω ) GJ ( mω EI ) ( mω EI ) (I αω GJ ) ( I α mx α ) = (6) I α = H or Ψ (7) Equation (6) can be non-dimensionalised by using te non-dimensionalised lengt were ξ = y Tus, wit te elp of equation (8), te non-dimensional form of equation (6) becomes (8) were a, b and c are non-dimensional parameters given by (D 6 + ad 4 bd abc) = (9)

5 5t ymposium on te Mecanics of lender tructures (Mo5) Journal of Pysics: onference eries 7 (6) 5 doi: a = ( I αω ), b = ( mω 4 ), c = ( I α mx α ) () GJ EI I α and D is te following differential operator D = d dξ () Te differential equation given by equation (9) can be solved using standard procedure [9, ] to give (ξ) = cosαξ + sinαξ + cosβξ + 4 sinβξ + 5 cosγξ + 6 sinγξ () were wit and α = [ ( q ) cos ( φ ) a ] β = [ ( q ) cos ( (π φ) ) + a ] γ = [ ( q ) (π + φ) cos ( ) + a ] q = b + a () (4) φ = cos [ 7abc 9ab a ] (5) {(a +b) } In equation (), - 6 are te integration constants resulting from te solution of te governing differential equation (9). (ξ) of equation () is te solution for bot te bending displacement H and te torsional rotation Ψ, but wit different sets of constants. Terefore, and H(ξ) = cosαξ + sinαξ + cosβξ + 4 sinβξ + 5 cosγζ + 6 sinγξ (6) Ψ(ξ) = B cosαξ + B sinαξ + B cosβξ + B 4 sinβξ + B 5 cosγζ + B 6 sinγξ (7) Te two different sets of constants 6 and B B 6 in equations (6) and (7) can be related wit te elp of eiter equation (4) or equation (5) to give. B = α, B = β, B 5 = γ 5 B = α, B 4 = β 4 B 6 = γ 6 (8) were α = b α4 bx α, β = b β4 bx α, γ = b γ4 bx α (9) 4

6 5t ymposium on te Mecanics of lender tructures (Mo5) Journal of Pysics: onference eries 7 (6) 5 doi: Te expressions for bending rotation θ(ξ), bending moment M(ξ), sear force (ξ) and torque T(ξ) are given by θ(ξ) = H ( ξ ) = ( ) { αsinαξ + αcosαξ βsinβξ + 4 βcosβξ 5 γsinγξ + 6 γcosγξ} () M(ξ) = ( EI ) H (ξ) = ( EI ) { α cosαξ + α sinαξ β cosβξ 4 β sinβξ 5 γ cosγξ 6 γ sinγξ} () (ξ) = ( EI ) { α sinαξ + α cosαξ + β sinβξ 4 β cosβξ + 5 γ sinγξ 6 γ cosγξ} () T(ξ) = ( GJ ) Ψ (ξ) = ( GJ ) {B αsinαξ + B αcosαξ B βsinβξ + B 4 βcosβξ B 5 γsinγξ + B 6 γcosγξ} () it te elp of equations (6)-(), te dynamic stiffness matrix of te coupled bending-torsion beam element wic is essentially an aircraft wing element can be developed by applying te boundary conditions for displacements and forces at te ends of te elements. Referring to figure, te boundary conditions for displacements are t y = ( =): H = H, = = t y = ( = ): H = H, = = (4) imilarly, referring to figure, te boundary conditions for te forces are t y = ( =): =, = = - t y = ( = ): = -, = - = (5) H y H = = Figure. Boundary conditions for displacements of an aircraft wing element. 5

7 Figure. Boundary conditions for forces of an aircraft wing element. ubstituting te boundary conditions for displacements given by equation (4) into equations (6), () and (7), one obtains te following matrix relationsip H H (6) or = B (7) were is te contact vector comprising te constants - 6 and α = cos α; α = sin α; β = cos β; β = sin β; γ = cos γ; γ = sin γ (8) ubstituting te boundary conditions for forces given by equation (5) into equations (), () and (), one obtains te following matrix relationsip T M T M (9) or y = = M M T T 5t ymposium on te Mecanics of lender tructures (Mo5) Journal of Pysics: onference eries 7 (6) 5 doi:

8 5t ymposium on te Mecanics of lender tructures (Mo5) Journal of Pysics: onference eries 7 (6) 5 doi: were F = D () = GJ ; = EI ; = EI () Te constant vector can now be eliminated from equations (7) and () to give te following force-displacement relationsip F = K () were K is te 6 6 frequency dependent dynamic stiffness matrix given by K = D B - () Te dynamic stiffness matrix of equation () representing a bending-torsion coupled beam suc as an aircraft wing can now be used to model an aircraft wing. non-uniform aircraft wing can be modelled as an assembly of many uniform dynamic stiffness elements. For instance, te unswept cantilever wing of figure 4 can be modelled as a stepped cantilever beam (wing) as sown in figure 5 were te non-uniform wing is split into uniform dynamic stiffness elements. Te dynamic stiffness elements of eac of te elements can be assembled to form te overall dynamic stiffness matrix of te complete wing. Te straigt unsweep wing and its idealisation in figures 4 and 5 are sown only for convenience, but te teory given above is sufficiently general and can andle swept and oter wings wit complex geometries. Te solution procedure to extract te natural frequencies and mode sapes from te overall dynamic stiffness matrix of te wing is based on te application of te ittric-illiams algoritm [8] wic as featured in undreds of papers. Te algoritm is particularly suitable in solving free vibration problem using te dynamic stiffness metod. Te woring principle of te algoritm is briefly summarised in te next section. Figure 4. non-uniform cantilever wing. Figure 5. non-uniform cantilever wing idealised as a stepped beam. 7

9 5t ymposium on te Mecanics of lender tructures (Mo5) Journal of Pysics: onference eries 7 (6) 5 doi: pplication of te ittric-illams algoritm Te dynamic stiffness matrix of equation () can now be used to compute te natural frequencies and mode sapes of aircraft wings. non-uniform andor swept wing can be analysed for its natural frequencies and mode sapes by idealising it as an assemblage of many uniform dynamic stiffness elements of bending-torsion coupled beams. Te natural frequency calculation is accomplised by applying te ittric-illiams algoritm [8] wic as received extensive coverage in te literature. Before applying te algoritm te dynamic stiffness matrices of all individual elements (see figures 4 and 5) need to be assembled to form te overall dynamic stiffness matrix K f of te complete wing. Te algoritm monitors te turm sequence condition of K f in suc a way tat tere is no possibility of missing any natural frequency of te wing. Te application procedure of te algoritm is briefly summarised as follows. uppose tat denotes te circular (or angular) frequency of te vibrating wing. Ten according to te ittric-illiams algoritm [8], j, te number of natural frequencies passed, as is increased from zero to, is given by j = j + s{k f } (4) were K f, te overall dynamic stiffness matrix of te wing wose elements depend on is evaluated at = ; s{k f } is te number of negative elements on te leading diagonal of K f, K f is te upper triangular matrix obtained by applying te usual form of Gauss elimination to K f, and j is te number of natural frequencies of te wing still lying between = and = * wen te displacement components to wic K f corresponds are all zeros. (Note tat te structure can still ave natural frequencies wen all its nodes are clamped, because exact member equations allow eac individual member to displace between nodes wit an infinite number of degrees of freedom, and ence infinite number of natural frequencies between nodes.) Tus j j m (5) were j m is te number of natural frequencies between = and = * for an individual component member wit its ends fully clamped, wile te summation extends over all members of te structure. Tus, wit te nowledge of equations (4) and (5), it is possible to ascertain ow many natural frequencies of te wing lie below an arbitrarily cosen trial frequency ( * ). Tis simple feature of te algoritm can be used to converge upon any required natural frequency to any desired accuracy. s successive trial frequencies can be cosen, computer implementation of te algoritm is very simple. However, for a detailed understanding, readers are referred to te original wor of ittric and illiams [8].. Results and discussion Using te above teory, two categories of aircraft wings wit cantilever boundary condition at te root are analysed for teir modal caracteristics. In te first category, a class of ig aspect ratio, ig performance sailplane wings are considered. typical layout of suc a sailplane is sown in figure 6. Results for natural frequencies and mode sapes are computed for two sailplanes ( and ) wit spans m and 5m, respectively. ome particulars of te two sailplanes are given in Table. Te second category of aircraft wings analysed belongs to transport airliners. typical layout is sown in figure 7. Four wings of transport airliners (T, T, T and T4) wit particulars given in Table are analysed. In all cases, dynamic stiffness elements were used to represent eac wing. Te data used for te stiffness (EI and GJ) and massinertia (m and I ) properties of te wings and te sear centre locations (x ) were calculated from te cross-sectional drawings of te wings expending considerable efforts. Tese data for te six aircraft are far too extensive to report in tis paper. 8

10 5t ymposium on te Mecanics of lender tructures (Mo5) Journal of Pysics: onference eries 7 (6) 5 doi: Figure 6. general lay-out of a typical sailplane. Figure 7. general lay-out of a typical transport aircraft. Table. Particulars of sailplanes Parameters ailplane ailplane- ailplane- ing pan (m) 5 ing rea (m ) spect Ratio.5.4 ing Root ord (m)..9 ing Tip ord (m).4.4 weep angle (deg) engt overall (m) Heigt Overall (m).. eigt Empty (g) 9 4 Max Tae-off weigt (g) Max ing oading (gm ) 7 6 Max ruising peed (nots) 5 5 9

11 5t ymposium on te Mecanics of lender tructures (Mo5) Journal of Pysics: onference eries 7 (6) 5 doi: Table. Particulars of transport airliners. Transport airliner Parameters T T T T4 ing pan (m) ing rea (m ) spect Ratio 9 ing Root ord (m) ing Tip ord (m) weep angle (deg) engt overall (m) Heigt Overall (m) 7 eigt Empty (g) 4, 6, 4,, Max Tae-off weigt (g) 7, 46, 74, 75, Max ing oading (gm ) Max ruising peed (nots) Range (nmi) Te first five natural frequencies of te six aircraft wings (two for sailplanes and four for transport airliners) are sown in Table. Te letters B and T sown in te parentesis indicate bending and torsion dominated modes, respectively wereas te letter indicates a coupled mode wit substantial amount of bot bending and torsional displacements. It sould be noted tat sailplane wings do not carry engines wereas te transport airline wings ave engine(s) wit mass and inertia properties wic ave significant effects on natural frequencies. (Engine mass is a uge proportion of te total wing mass.) Te mode sapes for te two sailplanes corresponding to te natural frequencies of Table are sown in figure 8 wereas tose of te four transport airliner wings are sown in figures 9 and, respectively. Table. Natural frequencies of sailplane and transport airliner wings. (B): Bending dominated mode; (T): Torsional dominated mode;(): Bending-Torsion coupled mode. ircraft ategory Natural Frequencies ( i ) (rads) 4 5 ailplane.64(b) 4.6(B) 9.6(B).5(T).4(B) ailplane.8(b) 4.9(B) 9.5(B) 64.(T) 67.4() Transport irliner T.5(B).9(B) 45.4() 87.85(B) 97.76() Transport irliner T 9.7(B) 55.9(B).(B).9() 97.7() Transport irliner T.99(B) 4.69(B) 67.66(B) 74.4(T) 8.4() Transport irliner T (B) 6.45(B) 47.64(T) 7.7(B) 94.64(T)

12 5t ymposium on te Mecanics of lender tructures (Mo5) Journal of Pysics: onference eries 7 (6) 5 doi: Figure 8. Natural frequencies and mode sapes of ailplane wings and. learly, te first tree modes of te cantilever wings of te two sailplanes and are bending modes wereas te fourt mode for eac of tem is a pure torsional mode, see figure 8. Te fift mode for te wing is a bending mode. By contrast, for te wing it is a coupled mode.

13 5t ymposium on te Mecanics of lender tructures (Mo5) Journal of Pysics: onference eries 7 (6) 5 doi: Figure 9. Natural frequencies and mode sapes of transport airliner wings T and T. Referring to Table and figure 9, te first two modes of te cantilever wings of transport airliner T and T are essentially bending modes, but te nature of te tird mode for te two wings differs quite significantly. For T, it is basically a coupled mode dominated by torsional displacement, but for T, it is a bending dominated mode. Te fourt mode for T is bending dominated, but for T it is actually a bending-torsion coupled mode. To all intents and purposes, te fift mode for bot T and T is a coupled mode. Now referring to figure, te mode sapes for T and T4 wings are discussed. Te first two modes for tese two cantilever wings are essentially bending modes as was te case wit T and T wings. However, te tird mode for T and T4 are different. For te T wing, it is a bendingtorsion coupled mode, but dominated by bending wereas for te T4 wing, it is a pure torsional mode. Te fourt mode for te T wing is mainly a torsion dominated mode wit some bending deformation present, but for te T4 wing it is a bending dominated mode wit a small amount of torsion present. Te fift mode for T is a coupled mode wereas for T4, it is a torsional mode.

14 5t ymposium on te Mecanics of lender tructures (Mo5) Journal of Pysics: onference eries 7 (6) 5 doi: Figure. Natural frequencies and mode sapes of transport airliner wings T and T4. 4. onclusions Using te dynamic stiffness metod and by applying te ittric-illiams algoritm, te modal beaviour of two sailplane and four transport aircraft wings is investigated. Natural frequencies and mode sapes for tese wide ranging aircraft wings for cantilever boundary conditions are illustrated. Te results are examined and discussed. In general, te first two modes for eac of te six aircraft are effectively bending modes, but te tird mode is eiter bending or torsional or a coupled mode depending on te type of te wing analysed. Te fourt mode is again eiter torsion or bending dominated or even coupled. Te same observation is made for te fift mode. Te investigation paves te way to establis trends for te modal beaviour of ig aspect ratio aircraft wings. cnowledgements Te autor is grateful to is PD students jandan nantapuviraja and Hassan Kassem for elp given in te preparation of tables and graps in tis paper.

15 5t ymposium on te Mecanics of lender tructures (Mo5) Journal of Pysics: onference eries 7 (6) 5 doi: References [] Banerjee J R 984 Flutter caracteristics of ig aspect ratio tailless aircraft J. ircraft [] Banerjee J R 988 Flutter modes of ig aspect ratio tailless aircraft J. ircraft [] van coor M and von Flotow H 99 eroelastic caracteristics of igly flexible aircraft, J ircraft [4] Eslimy-Isfaany H R, Banerjee J R and obey J 996 Response of a bending-torsion coupled beam to deterministic and random loads J. ound Vib [5] Banerjee J R, Patel M H, Done G T, Butler R and illico M 998 Free vibration and flutter sensitivity analyses of a large transport aircraft Proc. 7 t IUFNIMO ympos. on Multidisc. nalys. and Optimis, t. ouis, Missouri, U, Paper No [6] Tang D and Dowell E H Experimental and teoretical study on aeroelastic response of ig aspect ratio wings, I [7] Banerjee J R, iu X and Kassem H I 4 eroelastic stability of ig aspect ratio aircraft wings J. ppl. Nonlin. Dyn [8] ittric H and illiams F 97 general algoritm for computing natural frequencies of elastic structures Quart. J. Mec. and ppl. Mat [9] Banerjee J R 989 oupled bending-torsional dynamic stiffness matrix for beam elements Int. J. Num. Met. Eng [] Banerjee J R 99 FORTRN program for computation of coupled bending-torsional dynamic stiffness matrix of beam elements dv. Eng. oftware,