Optimum Design Of Rotor Blades With Equality Constraint

Abstract

Bir helikopter palinin tasarımında, gözönünde bulundurulan kriter, doğal frekanslarının yer ve hava rezonansından koruyacak şekilde düşünülmesidir. Bu çalışmada, yer rezonansından korunmak için birinci doğal frekansı sabit tutarak, rotor palinin ağırlığının optimum yapılması üzerinde durulmuştur. Rotor palinin hareket denklemleri, Galerkin sonlu elemanlar yöntemiyle ayrıştırılmış ve en genel halde aşağıdaki şekilde yazılmıştır: Mq + (c + c)j + {Ks+KA)q = f{t) Rotor palinin kesiti aşağıdaki şekilde dikdörtgen çekirdekten ibaret olduğu kabul edilmiştir. Optimizasyon esnasında tasarım değişkeni olarak -b- boyutu kullanılmıştır. Palin hareket denklemleri, Lagrange çarpanları kullanılarak çözümlenmiştir. Üretimi mümkün olmayan tasarım değerleri elde etmemek için, tasarım değişkeni belirli değerler arasında tutulmuştur. Yapılan kabuller ise, palin ankastre olduğu, sönümsüz hareketin mevcudiyetidir. t eg. of ihe box beam eg of nonstructural 7 Palin dikdörtgen kesiti

An important design of helicopter rotor blades is the placement of the natural frequencies to avoid ground and air resonances. This is done by proper tailoring of the blade mass of the stiffness distribution to give a set of desired natural frequencies. However, this is not an easy task due to the presence of various coupling effects as discussed in Reference [38]. The pitch angle, blade twist and an off-set betwen the elastic and inertia axes generally couse linear coupling effects between natural modes of the rotor blades. The scope of the present study is to find a suitable mass distributions of the blade which minimizes the weight while holding the selected natural frequency at a specified value. Minimum bound limits are imposed on the selected design variables to prevent them from reaching inpractical values during the design optimization process. Figure 1.1 depicts a typical rotor blade with a thin walled box beam cross section along the span and leading edge tunning masses distributed along the span. In order to simplify the analysis the following assumptions are made. The stiffness of the blade is contributed by the unsymmetric box section with variable geometry and nonstuctural mass distribution along the span. Stiffness contributed by skin, etc. is negligible. The material densityis assumed to uniform throughout the blade. For the structural analysis of the box beam, warping effects are neglected and thin wall approximations are used. The simplified problem is formulated as to minimize the weight of the blade which is assumed to be the sum of the weights of the box beam and the distributed tip tuning masses. This is also can be called as the objective function. Any conbinations of the box beam dimensions b,h,ti,t2,t3 can be used as the design variable. The governing equations of motion of hingless rotor blades are formulated in different references. Diffrent methods are used to discretize these XI equations. In this study, the discrete parameter form of the rotor blade equations are given Î eg. of İha box beam c.g of nonstructural / mass Figure 1.1: Box Beam Structure Geometry by reference [44]. These equations are discretized by Galerkin type finite elementmethod and are for general coupled flap-lag-torsion dynamics of a hingeless rotor blade both for forward and hover flight conditions with arbitrary mass and stiffness distributions. Cross sectional mass center and aerodynamic center offsets from the elastics axis are also included. Aquasisteady aerodynamics has been used in developing the equations where the compressibility and stall effects are not included. The problem of weight minimization subject to a constraint on a natural frequency is often referred as to the dual problem. The primal problem is the one where the natural frequency is maximized holding the weight to a specified value. Both of these problems as applied to optimum design of nonrotating beams with thin-walled cross sections undergoing coupled bending and torsional vibrations have been addressed by XII Hanagud and Chattopadhyay [23,24]. It has been observed that the optimum distributions differ largely with and without the coupling effects. 1.1 Discretization of the Equations of Motion Using the Galerkin Finite Element Method D D The first step in solving the equations of motion is the discretization of the spatial dependence. This is accomplished through application of the Galerkin finite element method. Subsequently, modal analysis is used to reduce the number of discrete unknowns describing the problem. 1.2 Formulation of Equality Frequency Constraints Problem The discrete parameter form of the governing equation of the rotor blade motion is given by Reference [44] as Mq + (c + c)q + (Ks+KA)q = f{t) (1.1) in above equation, K = Ks + Ka is a real unsymmetric matrix where indices S and Acorresponds to structural and aerodynamic effects respectively. Finite element used to discretize the blade and the associated degrees of fredoms are shown in Figure 1.2. In this simplified analysis rotor blade motion in hover case is considered. The linear undamped motion of the rotor blade is given by Mq + {Ks+KA)q = f(t) (1.2) Elemental degrees of freedoms are rearranged such that, q" ={v1,v"i,w1,-wI,fa,v2,v'2,w2,W2,02\ XIII Figure 1.2:Beam Type Finite Element and Associated Degrees of Freedom XTV The eigenvalue problem of the nonsymmetric system can be written as [45]. (K- u>,2M)qi = 0 (KT- o,2MT)si = 0 (1.3) where qi and s[ are defined as right and left eigenvectors corrsponding to the ith eigenvalue, cof. Both sets of eigenvectors can be expresed as square matrices Q and S as Q = [qi,q2» An] S= [S1.S2,,sn] (1.4) and orthonormality conditions can be also written as, ST MQ = [I\ STKQ = [A] (1.5) where A is a diogonal matrix with eigenvalues on its diagonals. 1.1.1 Formulation of the Optimization Problem The total weight of the resulting discritized blade is w = fJ(p^A+w>l) (L6) where, pe, is the material density, le, is the element lenght, Ae, is cross sectional area and Wt is the nonstructural mass. The optimization problem can now be posed as follows: Minimize: XV ^f=Yj{plA + W) (1.7) ;=l Subject to: Equilibrium condition; (K-. <<-V As a first step only the problem of unsymmetric matrices and the associated frequency constraints are considered. As the next step, the constrained optimization problem is now converted into an unconstrained one by the use of Lagrange multipliers. The modified objective function is written as * W E(^,/,4,+^) ;=1 TV _,.,,..w", T "T - HI ( sk KSi qk-<="" \ < ) sk (Ul) XVI In above equation wk is the frequency which is desired to set equal to the v, sk and qk are the corresponding eigenvectors of the rotor blade system. The problem * now is to minimize W subject to the constraints on the design variables. The se constraints are given by equation 1.10. This is done by obtaining stationary value of * the objective function W using full resources of variational techniques, While staying within the bounds on the design variables. The necessary condition for the stainarity is given by, (1.12) " ¦Vk, ¦9k. -M2 -v, % -Lsl. V Ik, DM J T \ % Q JdK. **n -co, 8Me % ~"2 fdKl 8M T\ Wn -co. *tn ) (1.16) Using equations 1.14, 1.15, 1.16 the optimality condition is rewritten as dA LPe,L -M ®lh, T SM, r 8K, Wn Ik = 0 i = 1,2, (1.17) where nt denotes the number of elements over which the design variable 0, does not reach the limiting values posed by equation 1.10. Note that the global mass and stiffness matrices M and K are replaced by the corresponding elemental quantities Mc and KC/ respectively,. Similary eigenvectors qk and sk are also replaced by corresponding elemental eigenvectors qK and sk. This has been done since there exist a one to one correspondence between an element and a design variable or in other words, n, only appears in the element stiffness and mass matrices of the z'th element. A simultaneus solutions of equations 1.7, 1.8, 1.9, and equation 1.17 with in the bounds equation 1.10 will result in possible optimum designs.