A CHECK ON THE ENERGY METHOD OF PREDICTI~ BLADE TRANSONIC STALL FLUTTER

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1 ACTA MECHANICA SINtCA, Vol. 2, No. 2, June, lg86 Science Press, Beijing, China 121 A CHECK ON THE ENERGY METHOD OF PREDICTI~ BLADE TRANSONIC STALL FLUTTER He Li and Zhou Sheng (Beijing Institute of Aeronautics and Astronautics) ABSTRACT: An improved structural dynamic model of an oscillating blade in two degrees of freedom is combined with an unsteady aerodynamic model for the transonic flow about a cascade with separation, which results in a coupled system. The system is solved in an iterative way between the two models. As a check on the current energy methods, the stall flutter boundaries for two real rotors are predicted by using the present method and the results are compared with the experiments and those predicted by using an energy method. KEY WORDS: aeroelasticity, blade stability, stall flutter. I. INTRODUCTION Blade flutter is a challenging problem in the development of modern aerocraft engines of high thrust-weight ratio. And the blade stall flutter is more common and serious than other kinds of flutter, Because of its complication, most of the previous methods of predicting stall flutter are empirical of semi-empirical in nature ix'21. In order to analyse numerically the effects of aerodynamics and structural dynamics on the blade stall flutter, the authors improved some previous work to form an unsteady transonic aerodynamic model including the effects of cascade geometry, shock and separation in flow around an oscillating cascade in bending and torsion degrees of freedom tsl. The numerical results showed that the effects of the unsteady separation flow and high aerodynamic load on the aeroelasticity of a cascade in a predominant bending mode with two degrees of freedom(t.d.o.f) differ qualitatively from that in a pure bending one with single degree of freedom (S.D.O.F), which implies that the aeroelastic coupling between bending and torsion is probably an important factor for the onset of flutter. Although some predicted results in [5] are qualitatively in agreement with the experiments, the computation procedure in that paper is based on the energy m&hod 161. It is known that because the mass coefficient of blade is in common higher that of wing by one order, most of previous references usually assume, in the onset of flutter, blades vibrate in their natural mode and frequency and the lower order modes (say first ben~ting, first torsion) which are encountered most frequently in blade flutter are generally considered as in S.D.O.F for convenience. However, some recent studies on the unstall flutter of flat cascade in an incompressible or pure supersonic flow ts'91 have shown that the coupling between the bending and torsion affects the aeroelasticity considerably. And the study in [5] has already shown that for the lower order predominant bending mode of real blades in a transonic compressor, the torsion branch (though it is very small in magnitude comparatively) can result in some essential changes in the aerodynamic damping of the blade. This implies a need to treat the structural dynamic characteristics of blade more carefully, and to check whether the energy method can give acceptable results or not. Received 22 December 1984, revised 3 April

2 122 ACTA MECHANICA SINICA 1986 A conlputation nlodel condlilling an iinprow~d structural dynamic model of oscillating cascade in T.D.O.F I~1 with the aerodynanlie model Isl is developed in this paper. With an eigenvalue problem, a closed system is formed, which is solved in an iterative way to reflect the interaction between aerodynamics and structural dynamics of the problem. As a check on the current energy methods, the stall thitier boundaries for two real rotors are predicted by using the present method and the results are compared with the experiments and those predicted by using the energy method isl. II. UNSTEADY AERODYNAMIC MODEL The compvtation of unsteady aerodynamics used here is based on a method of solving an inviseid transonic flow about an oscillating cascade. The governing equations are mixed-type equations for steady perturbation velocity potential tpo and unsteady perturbation velocity potential tpl. (1 -- M~o ) tpoxx + tporr = 0 (la) a~ = a2_~o + ~--~{u2_ ~o - Uo) (lb) (l- M~o}~O~x + 40,y r - a~[(7 + 1)u0~ + 2ioJ] ~Ptx 1 + a~o[w 2 -ioj(~ - 1)Uo~ ] ~Pl = 0 (2a) a2 = ao z - (7 -- 1)(Uo4~ + ic~ ei~ (2b) The solutions of these two sets of equations for a flow field with shocks can be obtained by using a transonic relaxation method for inviscid flow, and a simplified model of unsteady separation flow/31 is adopted. In this model, the effect of flow separation on the cascade aeroelasticity can be included by changing the boundary conditions in the inviscid flow calculation Is]. III. STRUCTURAL DYNAMIC MODEL 1. Basic model of the eigenvalue problem Consider a two.dimensional blade with two vibrating dl~grees of freedom (bending and torsion), as shown in Fig.]. The governing equations of the model are mtt + S~t + Khh = L S~h + I~ + K~ = M (3h) Assume the vibration of the blade can be expressed as h = ho ept, a = a0 ept. The lift force L and moment M due to self-excited aerodynamics are expressed as follows [s'gl. L = Lo ept = - ~p_ Qob3p 2 (A1 l ho + A12ao) ept M= Mo ep'= - 7tp_~b4p2(A2iho + A22ao)e p' (4a) (4b)

3 Vol. 2, No.2 He Li et al: A Check on the Energy Method of Predicting Blade 123 L Fig.1 Model of vibrating blade in T.D.O.F. where the parameter p is a complex, p = p, + io9. The real part p, is called the system damping factor, the value of which can be used to test the stability of the blade. Rewrite Eq.(3) in matrix form. Then they can be reduced to a standard eigenvalue problem All + # Ai2 + llx: #~ tt~ ho] [ho -421 "4- I.IX k- ~2 = J. (5) As!ong as the aerodynamie coefficients Aiy are known, this eigenvalue problem can be solved. 2. Introducing of a system coupling of structure When L and M are zero, the eigenvalues and eigenvectors derived from Eq.(5) correspond to the in:vacuum natural frequencies and modes, respectively. The degree of bending and torsion coupling in the natural modes depends on the gravity eenter offset from the elastic axis of the blade section. It can be shown that in this ease the ratio cr o /h0 is a real number, which means the branehes of bending and torsion are in-phase l'4l. As mentioned previously, L and M are originally eonsidered as the self-exeited aerodynamic forees. So in this ease the out-of-phase vibration between bending and torsion can only be introduced by the aerodynamic effects. But, it can be shown by an analysis on the modes at blade root eorresponding to travelling waves of a blade-disk system, that with the forward travelling wave, the torsion leads the bending by /12; with the backward travelling wave, the bending leads the torsion by n/2. So, in the ease of travelling wave the bending and torsion of mode in the typical section cascade eannot be inphase[ lal. The analysis results of travelling wave are obtained without considering the self-excited aerodynamic forces, therefore it is not quite reasonable to introduce the out-of-phase vibration between bending and torsmn only by those forces, And the travelling wave effects need to be included in the structural dynamic model. Now, the generalized forces acting on the blade system are assumed to consist of two parts: L = L 1 + L 2 M = M 1 + Mz (6a) (6b)

4 1s ACTA MECHANICA SINtCA lg86 where L1 and M 1 are still self-excited aerodynamic forces which are shown in Eq.(4); L 2 and M 2 are some exciting forces that can result in the effect of travelling wave. They may in some way have to do with the structural coupling of the blade system, so L 2 and M 2 are further assumed to be directly proportional to the accelerations in the coupling degree of freedom. L 2 =- CI~ (7a) M 2 = C2h (7b) Substituting of Eq.(7) into Eq.(3) yields.,~; +(S. - C,)~ + K~h = L, ~) (s,. G~); + t,~ + K,~ = M, (8b) I)~me that S." C~ = ~X~-r S~ - C2 = mxwu (9a) (9b) Another eigenvalue Problem similar to Eq.(5) can be obtained. Coupling coefficients X~2 L and Xscu are determined as follows. Firstly, the magnitude of ratio of bending to torsion in the natural mode in the typical section cascade is determined by using a finite element method/111. Then the phase between bending and torsion is determined by the travelling wave analysis, i.e. only the effect of travelling wave on the pha~ is included, while its effect on the amplitude is neglected. Secondly, assuming that L 1 and M 1 are zero, Eq.(8) are solved inversely, according to a known natural frequency, mode and other cascade parameters. Then, the coefficients X, and XSCM can be determined. After the coupling characteristics of the blade structure are determined, the self-excited aerodynamic forces are taken into account. The aerodynamic coefficients are solved by using the unsteady aerodynamic model. Then the eigenvalue problem can be solved. IV. PROCEDURE TO FIND THE FLUTTER BOUNDARY 1. The system of the aeroelastic problem With cascade geometry, steady aerodynamic parameters, interblade phase angle, frequency and mode being given, the unsteady pressure distribution along the blade chord can be obtained by using the unsteady aerodynamic method. Then the aerodynamic coefficients can be determined by disturbing the amplitudes of vibratigll, in small magnitude. where A = (All,A12,A21,.422) T. A =fl(m-~,i,.,~,"... ) The function fl depends on the computation model of unsteady aerodynamics. Because the frequency and amplitudes are the solution of the eigenvalue problem while the determination of the eigenvalue problem depends in turn on the aerodynamic matrix [Aij], the solving of the whole problem should be an iterative process. In current analysis method of flutter, the aerodynamic coefficients are assumed to be independent of the amplitudes, which is convenient for the

5 Vol.2, No.2 He Li et al: A Check on the Energy Method of Predicting Blade 125 formation of the eigenvalue problem. But in the condition of the transonic flow, the unsteady aerodynamic load on the blade surface is very sensitive to the small change of the blade profile. Therefore, the nonlinear effect of the amplitudes on the aerodynamic loads should be considered. In the process of solving the problem iteratively, the nonlinear effect is introduced in the following way. Write the aerodynamic model as A" =A (M_oo,i,a,to"- 1,Q "-1) (11) where Q = a0/ho; n is the number of iterations. When interblade phase angle a and steady aerodynamic parameters M_ i are given, we can have a closed system about the aeroelasticity of the cascade. A" = f3 (to"- 1,Q,- 1) (12a) (to.,q.) r = f4 (A") (12b) where Eq.(12b) corresponds to the structural dynamic branch of the flutter, i.e. the eigenvalue problem. 2. An approximate expression for the aerodynamic branch The computer program of the unsteady aerodynamic model would have to be executed many times if Eq.(]2) are solved directly. Because the method adopted is a'kind of numerical method about the flow field, executing the program many times would take a lot of computer time. In order to save computer time, an expansion in small parameters is made to approximate the aerodynamic coefficients in accordance with the corresponding factors. First, a basic steady point of aerodynamic state is taken near the flutter boundary. Corresponding to the natural mode Q0 and frequency too, the basic aerodynamic matrix [Ao-] ~ is formed by disturbing the amplitudes in small magnitude of high order. Then A can be expanded near the basic state point in Taylor series ( OA,~o OA/ (OA~OAQ A = A ~ + \~--~_~] AM_ + + \OQ] \-~] (13) Here only the small parameters in first order are retained. Disturbing M_ oo, i, Q and to in turn, the derivatives can be obtained. 3. Solution for the flutter boundary A direct-searching method is used to solve Eq.(12) by changing Q and to. After a convergent solution is obtained, the stability of the cascade-flow system is estimated with the eigenvalue. If p, is not zero, Eq.(12) will be solved at a new state point by changing M-oo or i until the point on flutter boundary is found. This procedure can be executed repeatedly at different steady state points so as to find the stall flutter boundary in M-oo-i plane or a compressor map. V. NUMERICAL RESULTS Because the aerodynamic branch of the model adopted here is the same as the method in [5-], the comparison between the computed results by the present method and those by the energy method [sl does show the effect of structural dynamic characteristics of the blade system on the aeroelasticity, which can be used as a check on the energy methods. In this paper, the stall flutter boundaries of two real rotors (rotor A, rotor B) have been predicted.

6 126 ACTA MECHANICA SINICA 1986 Rotor A is the second stage Of a transonic two-stage fan [121. It was found in the test that the blade stall flutter in first predominant bending occurred within the relative rotating speed ranging from 77% to 93% [131. The parameters in typical section cascade of this rotor are, solidity 1.566; camber 11.4~ relative thickness 4.4%; stagger angle ~. The comparison between the predicted boundary of rotor A and the experimental result in reduced velocity-incidence plane is shown in Fig.2. The comparison between them in the performance map is shown in Fig.3. V 6.0 5:0 ~. ;xpcriment p r e s e n t ~ method energ) method/51 I I I.= 6" 8 ~ l0 ~ - Fig. 2 Comparison of predicted results with experiment in reduced velocity-incidence plane for rotor A,,, --q o 2.0 ~ t h I~ t,1 J e x p e r ~ e y ~ od is1 present method 9 i,. 9,J, ~8 relative mass flow rig.3 Comparison of predicted resuhs with experiment in compressor map of rotor A Rotor B is the first stage of a transonic compressor. In the experiment, the blade stall flutter in first predominant bending was found at the reduced rotating speed above 80%. The parameters in typical section cascade are, solidity 1A4~,; camber 5.2~ stagger angle 54.75~ relative thickness 3.5%. Fig.4 shows the boundary predicted by using the present method and that of the experiment in inlet Mach number-incidence plane. The comparison between the experimental and the predicted boundaries in the performance map is given in Fig.5. The corresponding predicted results by using the energy method tsl are also illustrated in Fig.2-Fig.5, so as to show the difference of predictions between the eigenvalue method and the energy method.

7 Vol.2, No.2 He Li et al: A Check on the Energy Method of Predicting Blade 127, experiment ~ 1.0 energy ~ ~ methodtsl ~"t"t'~ present mothod 4 ~ Fig.4 Comparison of predicted results with experiment in inlet Mach number-incidence plane for rotor B.r ' ' ' i x p e r i m e n t ~ Z, \ ~ 0.g9 I,, o.69o relatwe mass flow Fig.5 Comparison of predicted results with experiment in compressor map of rotor B It is shown by the figures that although the results of the eigenvalue method is nearer to the physical reality than those of the energy method for the reason that the former includes the interaction between the aerodynamic side and the structural dynamic side, the predicted reresults by using these two methods are about the same qualitatively and do not show much difference quantitatively. That, therefore means that the computation methods for the blade stall flutter based on the energy-method model can still give acceptable results in engineering practice. VI. CONCLUDING REMARKS Some improvements have been made in this paper in order to develop a prediction method which can reflect the interaction between the aerodynamic and structural factors of the blade stall flutter. On the basis of a vibration model in T.D.O.F which was developed in the case of incompressible or pure supersonic flow about a flat plate cascade [91, some system coupling terms of structural dynamics are introduced in the form of exciting forces according to the characteristics of real rotor. The systematic structural coupling coefficients are determined by solving the vibration equations inversely, which is used to reflect the out.of-phase vibration of bending and torsion resulted by the travelling waves.

8 1L~8 ACTA MECHANICA SINICA 1986 The improved model of structural dynamics is combined with the unsteady aerodynamic model which can include the effect of transonic separation flow ts'l~ so as to build an aeroelastic system of blade stall flutter, which is solved in an iterative way. It is shown, by the predictions for two real rotors, that the predicted results by using the present method agree well with the experiments. And the comparison between these computed results and those by the energy method tsl does not show much difference. This means that although the blade stall flutter can not be correctly predicted only by the energy method with single degree of freedom it41, after the improvement similar to that in[5] is made (i.e. the coupling between bending and torsion degrees of freedom is included), the application of the energy methods to the study of the blade stall flutter can still be acceptable. However, the eigenvalue method is more reasonable than the energy method because the former can include the response of structural dynamics of the problem, especially reflecting the interaction between the aerodynamic and structural sides in an iterative way. Therefore, as the study on the problem develops and the accuracy of the computation models to be developed is raised further, the eigenvalue method will be used more and more. REFERENCES [ 1 1 Fleeter, S.,AIAA paper , (1977). [ 2 1 Adamczyk, J.J.,NASA TP-1345, (1978). [ 31 Sisto, F. and Perumal, P.V.K., ASME paper 74-GT-28, (1974). [ 4 1 Carta,F.O., ASME paper 79-GT-153, (1979). [ 5 1 He Li and Zhou Sheng, Proceedings of the Symposium of Unsteady aerodynamics of Turbomachinery and Prope!lers, Cambridge, England, (1984). [ 6 1 Carla, F.O., J. of Engineering for Power, 89, 3(1976). [ 7 1 Mikolajczak, A.A., Snyder, L.E., Arnoldi,R.A. and Stargardter, H., J. of Aircraft, 12, 4(1975). [.8 ] Bendiksen, O. and Friedmann,P., AIAA J., 18, 2(1980). [ 9] Bendiksen,O. and Friedmann,P., J. of Engineering for Power, 104, 3(1982). [10] Lin Baozhen, Tang zhimingand Zhou Shen, Proceedings of the Second Asian Congress of Fluid Mechanics, Beijing, China, (1983). [111 Xu Jianmin, Master's Thesis, Beijing Inst. of Aero. and Astro., (1984), (in Chinese). [121 Messenger, H.E. and Kenndy,E.E., NASA CR , (1972). [13] Ruggeri,R.S. and Benser,W.A., NASA TMX-3076, (1974). [14] lie Li, Master's Thesis, Beijing Inst. of Aero. and Astro., (1904), (in Chinese).