Helikopter pervanelerinin lineer ve nonlineer titreşimlerinin kombine sonlu elemanlar-transfer matrisi tekniği ile incelenmesi

Abstract

Bu çalışmada, Helikopter Pervaneleri ' nin (Palleri ' nin) Lineer ve nonlineer titreşimleri, "Kombine sonlu Elemanlar Transfer Matrisi Tekniği" ile incelenmiştir. Bu inceleme de, kökte Ankastre olarak tespit edilmiş paller ile kökte dönme serbestisi olacak şekilde dizayn edilmiş olan, yani kökte Moment reaksiyonu taşımayan Paller ayrı ayrı ele alınmıştır. Ayrıca her iki halde de, hem dönme düzlemine dik doğrultudaki titreşimler, hem de dönme düzlemindeki titreşimler incelenmiştir. Bu inceleme yapılırken, ankast re çubukların Lineer titreşimlerini aynı teknikle hesapla yan, Fortran dilindeki bilgisayar programları, daha önce yapılan çalışmalardan hazır olarak alınmış, ancak, mevcut çalışmada değişik sınır şartları ile birlikte Nonlineer hale ait Matris terimleri de hesaplara dahil edilmiştir. Dönmeyen çubukların nonlineer titreşimlerini veya dönen çubukların (döner yapıların) lineer titreşimlerini konu alan çok sayıda çalışmaya rastlamak mümkündür. Bu çalışmada ise, dönme etkileriyle birlikte nonlineer etkiler de incelenmiştir. Nonlineer hesaplar yapılırken, iteratif bir yöntem kullanılmıştır. Ref.fVj'de, iki ucu Ankastre olarak tesbit edilmiş çubukların sükûnet haline ait nonlineer titreşim özfrekansları ve nonlineer kuvvet değerleri verilmektedir. Mevcut çalışmada, ele alman nonlineer yaklaşımı denemek amacıyla; bu hale ait titreşim özfrekansları ve nonlineer kuvvet değerleri de çalışma içerisinde ayrıca incelenmiştir. Buradan elde edilen sonuçlarla, ref. [4] ' de verilen sonuçlar birbirleriyle karşılaştırılmıştır. Kullanılan yaklaşımın uygunluğu bu şekilde denendikten sonra, aynı nonlineer yaklaşım kullanılarak, Helikopter pallerinin titreşimleri incelenmiştir. Helikopter Palleri için "Kombine Sonlu Elemanlar-Trans- fer Matrisi Tekniği" ile birlikte ele alınan iteratifnon lineer hesaplar sonucunda elde edilen teorik değerler, ayrıca deneysel verilerlede karşılaştırılmıştır. Bu şekilde kombine Sonlu Elemanlar-Transfer Matrisi tekniğinin konuya başarı ile uygulanabileceği gösterilmiştir.

In this study, the linear and non-linear vibrations of helicopter blades have been investigated by using the "Combined Finite Element-Transfer Matrix Technique". In the application, of this technique, a direct iteration method is used to calculate the nonlinear natural frequencies, the vibration mode shapes and the nonlinear stretching force values. Beams of the clamped-clamped, clamped-free and hinged-free types are investigated. Calculations are made for both edgewise (vertical) and flapwise (inplane) vibrations. The increasingly rotation speeds in machinery and sophistication of digital computers have been instrumental in the development of new methods of analysis, particularly so-called finite element methods. The idea behind the finite element is to provide a formulation which can exploit digital computer automation for the analysis of irregular systems. To this end, the method regards struc tures as an assemblage of discrete elements, where every such element is a continuous structural member by requiring that the displacements are compatible and the internal forces in balance at certain points shared by several elements, where the points are known as joints. The entire structure is compelled to act as one entity. The finite element method is basically a discretization procedure. The transfer matrix technique, like the finite element method, is based on the idea which the state variables' are transfered from one station to another. Thus, a continuous system can be approximated by an equivalent discrete system. The use of the transfer matrix method combined with the finite element method leads to better accuracy and economics in the design of structures. Myklestad and Prohl have further developed the trans fer matrix technique for vibrations of wings of aircraft. Vibration in helicopters can be classified under two separate headings; general vibration problems which are commonly met in all aircraft, and problems peculiar to helicopters. It is evident that vibration peculiar to the helicopters must emanate from the rotor and must be felt as structural vibration or control vibration. It is fortunate that these two manifestations can be treated separetely if control vibration is balanced out and for clarity the subject is divided into two parts. The fundamental cause of nearly all balanced rotor vibration is forward flight. For example, the velocity of a blade element relative to the air is its rotational velocity (w.r) to which is added a component of the forward speed of the aircraft. Putting this into symbols the relative velocity Vr is VR = w.r + V. Sin ip where IP is the azimuth angle of the blade. Since aerodynamic force varies as the square of the velocity, 2 2 Force oc (w.r)2 + 2 w.r.V. Sinu? + - ^- - -¥- Cos 2 if 2 2 Thus, on this simple criterion alone, we have first-and second-harmonic force fluctuations. In practice, rotor flapping and feathering also introduce force variations and significant force fluctua tions are experienced up to at least the tenth harmonic order [_Z~]. ESTABLISHMENT OF TRANSFER MATRICES OF THE STRUCTURE For the discreted continuous beam (blade), transfer matrices are built up in order to form relationships between the j-1' i±. and j ' th stations which are named as joints. VI As nodal coordinates, the displacements W.^ and w. as well as the rotations ©j-i and ©j are used, as shown in Fig. 1.1. Hence we shall write the displacement w in the form w = Lr w^ u2.6j.1 + l3 Wj + l4 e. (1.1) For convenience, we shall continue to work with the nondimensional natural coordinates J = j+R - ~h~ so that we consider the cubic polynomials L. i c +c ll 12' +C İ3 + Ci4 J3'" 1,2,3,4, (1.2) Figure 1.1. Where the coefficients C., (i,k = 1,2,3,4) are deter mined by insisting that w and w' take the values Wj-i and ©-, at f =1, and the values W. and ©j at J=O.As shown in Fig. 1.1.> the term h.R represents the distance between the hub centre and the place of support of the blade, where R = h is the non-dimensional value of this distance. By Using the conditions, we obtain the interpolation functions, Vll l1 = 3|2 - 2 g3 l2 = h. cr2 - £3) "? v» 3 vo ^ 2 *o 3 L3 = 1-3 J2 + 2. j L4 = h.(-jr + 2 J2 -J3) which are known as Hermite cubics. (1-3) it will prove convenient to introduce the notation w = LT.a., (j-l)h+h.R^ X <. j.h + h.R (1.4) where T L =[L1 L2 L3 L4~ (15) a. = fw.. 9., w. e.~ T The potential energy, in the case of edgewise vibra tions, can be expressed as L dx ( 1.7 ; V(t) =4-. /[EI( 4\)2 + P. ( dW^ r and the kinetic energy can be written as L T(t) = -y- J m. ( -J^)2.dx (1.8) r Lagrange's equations of motion are; ^.'ft'-lir0^'' L=T-V (W1 where vm n _ (1.10) (1.11) K. : is the stiffness matrix of the j ' th element M. : is the mass matrix of the j ' th element By using the equations (1.4), (1.7), (1.8), (1.10), (1.11), we obtain; r Ei / J T » T " T İİti t.T K.=h. j ( jJ.h".-L"1 + ^-.L'.L'1 )dt (1.12) J n hZ J 0 1 M. =h / m..L.LT.d?, (j =1,2 n) (1.13) O let us consider the case in which EI = const, m= const. In this case, the axial force can be obtained in the form L P(x) = / m._n?.£.dfe = - m/L2.L2.( 1 j- ) (1.14) X or.(£) = -i-- mlİ.L2.[l-( -^-)2.(R+j-f )22 (1.15) 3 J 2 L y P (0 4J *C1) Introducing Ecjs. (1.3) and (1.5) into Ecjs. (1.12) and Eqs. (1.13), we obtain the element stiffness matrices xx 3 EI 12 6h -12 6h 4h' -6h 2h' Symmetric 12.6h 4h" +A.h. ?.Jl A_C+ ^_ d_ 6 35 {_Ç_ + _d_ + _L_)h (_l_c+^- ?^)h2 10 10 28 15 30 105 (_ _§_ c_.J_ d+ _§_) (_ _Ç. 5 5 35 10 Symmetric _d_ + JL)h : ( -L. c+ -^- a- £-) 10 28 5 5 35 ( 7T- + rr~)h (- C 10 70 1 +^-)h2 30 60 140 (- 10 70 1 )-h * J * = ( -?- c+ ^ 15 10 70 3 ).h2 and mass matrices Mj (J =1,2,.... n) By using Egs. (l.9) / we obtain CMj] {Sj} + [Kj"] a. 3 f j(t) Assuming that fa.l = «j a. >. Cos w t, we get; Lsp f. M- [KjV. M [Tj]- e) r ~) r 2 "j-i (1.18) where [T."] is the transfer matrix of the j ' th element. The transfer matrix of the entire blade can be obtained by multiplying all elemental transfer matrices between the boundaries of the system. Thus, (1.19) R n (1.20) O and making use of the boundary conditions for the clamped-free beam, 0 = H__. MR + H,.. VR 33 o 34 o O = H. _. MR + H.. VR 43 o 44 o (1.21) DET = H H 33 43 H 34 H 44 = H33. H44 - H43. H34 - O (1.23) Equation (1.23) is known as the "FREQUENCY EQUATION" By solving this equation, one can obtain the natural frequencies of the beam. The state vector of any j ' th xx station can be determined from (1-24) where [H."] is the transfer matrix between the o'th station and the üj ' th station. NON-LINEAR VIBRATIONS Non-Linear effects can be neglected in case of small amplitudes of vibration, whereas they become crucial at high amplitudes. The additional potential energy, as a result of non- Linear effects, can be expressed as: L 1 ( V N - E.A. (W ) 4.dx (1.25) The non-linear force is assumed as: N _1_ 4.E.A. (w' ) (1.26) Since a direct iteration method is used in this investiga tion, the non-linear force term is taken as a constant value in the integration. In this procedure, the non linear force is calculated by using the maximum value öf w1. The idea behind this assumption is, that the maxsimum non-linear force has the biggest effect on the non-linear vibration of the blade. In this way, the non-linear system is reduced to an equivalent linear system. By using the same procedure as used for the calcula tions of linear stiffness matrices, one can obtain the non-linear stiffness maxrix as following: xxi NJ1--T 6 5 1 10 6 5 10.h h 30 Symmetric 10 15 h" (1.27) Where, h = L/n, n = is the number of elements, L = İs the length of The blade, E = is the modulus of elasticity, A = is the cross-sectional area, Thus, the total stiffness matrices can be determined by the summation of the linear and non-linear stiffness matrices; M = [KT] + [KMT1 NLJ (1.28) The nonlinear vibrations are investigated by dinect iteration where the linear mpde-shape is taken as the starting value. The non-linear frequency corresponding to the fundamental mode of vibration is calculated, for various values of the amplitude, for clamped-c lamped, clamped-f ree, and hinged-free end conditions.