![248 B. PATTABHI RAMAIAH et al. / ADVANCES IN VIBRATION ENGINEERING, 8(3) 2009
a = Distance between the elastic axis and the center of mass
ra = Radius of gyration about the
= Sweepback in deg
k = Reduced frequency bω/u
C(k) = Theodorsen function
ωα = Torsional frequency in rad/s
2 Introduction
The prime objective of the structural engineer is to design an airframe whose flight envelope is limited by
engine power rather than its structural limitations[1]. The computation of the flutter speed for rectangular
wing is discussed in ref.[2] A computer program for the flutter analysis including the effects of rigid-roll
and pitch of swept wing subsonic aircraft is given in ref.[3] Interactive software for wing flutter analysis
was developed including the effects of change in Mach number, dynamic pressure, torsional frequency,
sweep and mass ratio in ref.[4] In the present work an attempt is made to estimate the flutter speed using
the well-known Velocity-damping (V -g) method for the unmanned aerial vehicles inputting uncoupled
bending and torsional frequencies estimated from the numerical codes. A computer code is developed
to extract the complex Eigen value for the estimation of the flutter speed of the UAVs. Considerable
improvement in the flutter speed is seen compared to the existing wing structure.
3 Evaluation of the Flutter Speed by V-g Method
The Velocity-Damping method, abbreviated as V -g method, basically deals with strain energy and
kinetic energy of the structure and the aerodynamic damping. The structural damping ratio g is plotted
against the vehicle velocity for each vibration mode. The values of the damping ratio are negative up to
certain speeds and changes the sign as the structure absorbs energy from the free stream, resulting into
self-excited diverging oscillations. The speed, which corresponds to zero value of the damping ratio g
in the V -g curve, could be taken as the critical flutter speed. The algorithm used to estimate the flutter
speed is given below.
The strain energy of the anisotropic plate
V =
1
2
l
0
+c/2
−c/2
∇2
∇2
wdydx (3.1)
and for the rectangular composite panel,
V =
1
2
l
0
+c/2
−c/2
[D11(W,XX)2
+ 2D12W,XXW,YY + D22(W,YY )2
+ 4D16W,XXW,XY + 4D26W,YY W,XY + 4D66(W,XY )2
]dydx (3.2)](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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- 1. Aero-Elastic Analysis of Stiffened Composite Wing Structure B. Pattabhi Ramaiah1 , B. Rammohan1 , S. Vijay Kumar1 , D. Satish Babu1 and R. Raghuatnhan2 1 Scientists, Aeronautical Development Establishment (ADE), DRDO, Bangalore pattabhib@yahoo.com 2 Group Director, Sc‘G’, Aeronautical Development Establishment, Bangalore Abstract The interaction of the Elastic, Inertia and Aerodynamic forces is a dynamic phenomena resulting in flutter. Dynamic Aero elasticity is critical for a high-speed subsonic class of Aerial Vehicles. In the present work, dynamic Aero-elastic analysis of the Unmanned Aerial Vehicles (UAVs) has been studied for a stiffened composite structure. The flutter speed and the corresponding flutter frequencies are computed using the Velocity-Damping (V -g) method. The V -g method is employed to estimate the flutter speed and flutter frequencies, for a high-speed subsonic composite wing structure, designed and developed at ADE. Also the improvement of the flutter frequencies over the existing metallic wing structure is discussed. The work can be further extended to develop an optimized composite structure with higher margins in flutter speeds. Keywords: Aeroelasticity, Flutter, Velocity-damping method 1 List of Symbols V = Critical flutter speed in fps Kθ = Wing torsional stiffness (antisymmetric) measured at 0.7 S S = Wing semi span in ft D = Distance from the wing root to equivalent tip (0.9 S) Cm = Wing mean chord in ft. e = Position of inertia axis aft of LE Xa = Position of flexural axis aft of LE λ = Taper ratio r = Stiffness ratio ωh = Bending frequency in rad/s f (M) = (1 − M2)/4 for 0 < M < 0.8 = 0.775 for 0.8 < M < 0.95 where M = M0 cos and M0 is the greatest forward Mach number at which the UAV can achieve the maximum dive speed µ = Non-dimensional number m/πρb2 where m is the mass per unit length b = Semi chord length in m ρ = Sea level air density in kg/m3 ADVANCES IN VIBRATION ENGINEERING, 8(3) 2009 © KRISHTEL eMAGING SOLUTIONS PRIVATE LIMITED
- 2. 248 B. PATTABHI RAMAIAH et al. / ADVANCES IN VIBRATION ENGINEERING, 8(3) 2009 a = Distance between the elastic axis and the center of mass ra = Radius of gyration about the = Sweepback in deg k = Reduced frequency bω/u C(k) = Theodorsen function ωα = Torsional frequency in rad/s 2 Introduction The prime objective of the structural engineer is to design an airframe whose flight envelope is limited by engine power rather than its structural limitations[1]. The computation of the flutter speed for rectangular wing is discussed in ref.[2] A computer program for the flutter analysis including the effects of rigid-roll and pitch of swept wing subsonic aircraft is given in ref.[3] Interactive software for wing flutter analysis was developed including the effects of change in Mach number, dynamic pressure, torsional frequency, sweep and mass ratio in ref.[4] In the present work an attempt is made to estimate the flutter speed using the well-known Velocity-damping (V -g) method for the unmanned aerial vehicles inputting uncoupled bending and torsional frequencies estimated from the numerical codes. A computer code is developed to extract the complex Eigen value for the estimation of the flutter speed of the UAVs. Considerable improvement in the flutter speed is seen compared to the existing wing structure. 3 Evaluation of the Flutter Speed by V-g Method The Velocity-Damping method, abbreviated as V -g method, basically deals with strain energy and kinetic energy of the structure and the aerodynamic damping. The structural damping ratio g is plotted against the vehicle velocity for each vibration mode. The values of the damping ratio are negative up to certain speeds and changes the sign as the structure absorbs energy from the free stream, resulting into self-excited diverging oscillations. The speed, which corresponds to zero value of the damping ratio g in the V -g curve, could be taken as the critical flutter speed. The algorithm used to estimate the flutter speed is given below. The strain energy of the anisotropic plate V = 1 2 l 0 +c/2 −c/2 ∇2 ∇2 wdydx (3.1) and for the rectangular composite panel, V = 1 2 l 0 +c/2 −c/2 [D11(W,XX)2 + 2D12W,XXW,YY + D22(W,YY )2 + 4D16W,XXW,XY + 4D26W,YY W,XY + 4D66(W,XY )2 ]dydx (3.2)
- 3. AERO-ELASTIC ANALYSIS OF STIFFENED COMPOSITE WING STRUCTURE 249 where Dij = n k=1 (Qθ ij )k (Z3 k − Z3 k−1) 3 (3.3) The kinetic energy of the plate can be written as T = 1 2 l 0 +c/2 −c/2 m{ ˙ω}2 dxdy (3.4) where m = ρwtw · ρw is the specific gravity of the wing structure and tw is the thickness of the wing. The Lagrange’s equations of motion, ref.[8], d dt ∂T ∂q• i + ∂V ∂qi − ∂T ∂qi = Qi (3.5) Assuming sinusoidal motion for the vibration mode and neglecting warping stiffness of the structure, the equations of motion can be written as − ω2 [Mij ] q1 q2 + [Kij ] q1 q2 = Q1/eiωt Q2/eiωt (3.6) where [Mij ] = mclI4 0 0 mclI5 12 (3.7) and [Kij ] = D11cI7 l3 2D16I6 l3 2D16I6 l2 4D66I8 cl (3.8) where I4, I5, I6, I7, I8 are Non-dimensional integral expressions as given in ref.[6] The aerodynamic forces acting on the structure are, ref.[8] LE = ω2 πρb3 (L1 + iL2) wE b + (L3 + iL4)α eiωt (3.9) L1 + iL2 = 1 − 2i k c(k) (3.10) L3 + iL4 = a + 2c(k) k2 + i k [1 + (1 − 2a)c(k)] (3.11) ME = ω2 πρb4 (M1 + iM2) wE b + (M3 + iM4)α eiωt (3.12) M1 + iM2 = 1 − i(1 + 2a) k c(k) (3.13)
- 4. 250 B. PATTABHI RAMAIAH et al. / ADVANCES IN VIBRATION ENGINEERING, 8(3) 2009 M3 + iM4 = 1 8 + a2 + (1 + 2a)c(k) k2 + i k 1 2 − 2a2 c(k) − 1 2 − a (3.14) Q1 = ωπρb3 (L1 + iL2) lI4q1 b + (L3 + iL4) lI3q2 c eiωt (3.15) Q2 = ω2 πρb4 (M1 + iM2) lI3q1 bc + (M3 + iM4) lI3q2 c eiωt (3.16) Now the Flutter problem can be formulated as [K − ω2 A]{q} = 0 (3.17) where the stiffness matrix K and aerodynamic matrix A are defined below K11 = CD11I7 l3 (3.18) K12 = 2D16I6 l2 (3.19) K21 = K12 (3.20) K22 = 4D66I8 cl (3.21) A11 = mclI4 + πρbI4(L1 + iL4) (3.22) A12 = πρlb3I3 c (L1 + iL4) (3.23) A21 = πρlb3I3 c (M1 + iM4) (3.24) A22 = mclI5 12 + πρlb4I5 c2 (M1 + iM4) (3.25) The flutter frequency ω, the damping ratio g and the flutter speed u are extracted from ω = 1 √ Re(z) (3.26) g = Im(z) Re(z) (3.27) u = bω k (3.28)
- 5. AERO-ELASTIC ANALYSIS OF STIFFENED COMPOSITE WING STRUCTURE 251 Fig. 1 Mathematical idealization of airfoil 4 Estimation of Flutter Speed of an Airfoil The Idealized Mathematical model of the wing to estimate the flutter speed is shown in Fig. 1. The wing airfoil is assumed to be thin and the motion is assumed as simple harmonic. Hence the second order linear differential equation of the system, in generalized coordinates can be written as[14] m¨h + Sα ¨α + mω2 hh = Qh (4.1) Sα ¨h + Iα ¨α + Iαω2 αα = Qα (4.2) and the flutter speed can be estimated from the determinant given below m πρb2 1 − ω2 hω2 α ω2 αω2 + Lh xα m πρb2 + Lα − Lh 1 2 + α xα m πρb2 + 1 2 − Lh 1 2 + α r2 a m πρb2 1 − ω2 α ω2 + Mα − 1 2 + Lα 1 2 + α + Lh 1 2 + α 2 (4.3) Analytical code in MatLab has been developed to evaluate (4.3). The flutter speed can be obtained by letting the determinant to go to zero. Some of the variables used in the code are listed below.
- 6. 252 B. PATTABHI RAMAIAH et al. / ADVANCES IN VIBRATION ENGINEERING, 8(3) 2009 The Theoderson function C(k) = H2 1 (k) H2 1 (k) + iH2 0 (k) (4.4) where H is the Hankel functions[12]. The aerodynamic coefficients are given by[13] Lh = 1 − 2i ν bω (F + iG) (4.5) Lα = 1 2 − i ν bω [1 + 2(F + iG)] − 2 ν bω 2 (F + iG) (4.6) Mh = 1 2 (4.7) Mα = 3 8 − i ν bω (4.8) 5 Results and Discussion Fig. 2 Finite element model of the composite wing Figure 2 shows the finite element model of the com- posite wing developed in Hypermesh, in order to esti- mate the normal modes. Free-Free boundary conditions are imposed on the wing and the Normal Modes analysis is performed using MSc. Nastran. The first three modes are given in Figs. 3, 4 and 5 respectively. Knowing Natural frequencies of the FRP wing from the numerical model, the flutter speed can be Fig. 3 First bending (50 Hz) Fig. 4 Second bending (81.8 Hz) Fig. 5 Third twisting (112.3 Hz)
- 7. AERO-ELASTIC ANALYSIS OF STIFFENED COMPOSITE WING STRUCTURE 253 Fig. 6 Output for high speed composite UAV Table 1 Input data for high speed composite wing, used in MatLab code Sl. No. Parameter Values 1 Non-Dimensional Mass Ratio 105.04 2 Bending frequency – rad/s 314.78 3 Second Bending frequency – rad/s 513.96 4 Air density – kg/m3 1.2260 5 Radius of Gyration 0.5750 6 Distance between mid chord and Flexural axis in semi chord length 0.1500 7 Distance between mid chord and Center of mass in semi chord length −0.4000 estimated using (4.3). The various input parameters of reduced frequency for composite and metallic wings are listed in Tables 1 and 2 respectively. From the data listed in Tables 1 and 2 the V -g diagrams have been plotted in MatLab and the output is shown in Figs. 6 and 7, for composite and metallic wing respectively. The methodology is discussed in ref.[15]
- 8. 254 B. PATTABHI RAMAIAH et al. / ADVANCES IN VIBRATION ENGINEERING, 8(3) 2009 Table 2 Input data for high speed metallic wing, used MatLab code Sl. No. Parameter Values 1 Non-Dimensional Mass Ratio 64.430 2 Bending frequency – rad/s 150.79 3 Torsional frequency – rad/s 251.33 4 Air density – kg/m3 1.2260 5 Radius of Gyration 0.5750 6 Distance between mid chord and Flexural axis in semi chord length 0.1500 7 Distance between mid chord and Center of mass in semi chord length −0.4000 Fig. 7 Output for high speed UAV 6 Conclusions An attempt has been made here to replace the existing wing with that of the composite wing (GFRP) and the flutter speed of the wing was found to be 283.40 m/s as listed in Table 3, and the corresponding
- 9. AERO-ELASTIC ANALYSIS OF STIFFENED COMPOSITE WING STRUCTURE 255 Table 3 Comparison of flutter speeds from for metallic and composite wings Flutter Flutter-Dive Method Speed (m/s) Speed Ratio Ug-Method (Composite wing) 283.40 1.310 Ug-Method (Existing wing) 264.56 1.225 flutter frequency of the wing was found to be 54.5 Hz. This is a considerable improvement in the flutter speed of the wing, which was found to be 264.56 m/s and the corresponding flutter frequency of 33 Hz, estimated using ug-Method, AVP-970 standards. The composite wing can be tailored aero elastically to study the wing divergence (static instability) and the control reversal effects. References [1] Weishaar, T. A. and Foist, B. L., Vibration and flutter of advanced lifting surfaces, Proceedings of 24th AIAA/ASME/ASCE/AHS Structures, Structural Dynamics and Materials Conference, Lake Tahoe, Nev., part 2, pp. 498–508, 1983. [2] Bennet, G., Modeling of wing weight for high altitude long endurance aircraft, part 1, Unmanned systems, Vol. 13(1), 1995. [3] Houser, J. M. and Manual Stein, Flutter analysis of swept wing subsonic aircraft with parameter studies of composite wings, NASA TN D-7539, 1974. [4] Vivek Mukhopadhyay, An interactive software for conceptual wing flutter analysis and parametric study, NASA TM-110276, 1996. [5] Aviation Practices Standard. 970, Aero-Elasticity, leaflet 500/3, Vol. (1), 1963. [6] Howell, S. J., Aeroelastic flutter and divergence of graphite/epoxy cantilevered plates with bending torsion coupling, M. S. Thesis, Department of Aeronautics and Astronautics, M. I. T., 1981. [7] Aston, J. E. and Whitney, J. M., Theory of laminated plates, Technomic Publishing Co., Stanford, Conn., 1970. [8] Dugundji and Brain J. Ladsberger, Experimental aeroelastic behavior of unswept and forward swept cantilever graphite/epoxy wings, J. Aircraft, pp. 679–686, 1985. [9] NISA, Aeroelasticity manual display IV, EMRC, Vol. 1, 2004. [10] V. Prabhakaran, et al., Composite wing design for falcon airframe, ADE/IR, 1999. [11] Upadhyaya, A. R., et al., Modal analysis of a cropped delta wing of an unmanned aircraft, NASAS-90, pp. 1–21, 1990. [12] Fung, Y. C., The theory of aeroelasticity, galcit aeronautical series, John Wiley & Sons, 1955. [13] Scanlan and Rosenbaum, Introduction to the study of vibration aircraft vibration and flutter, The Macmillan Company, New York, 1951. [14] Bishplingoff, R. L., Ashley, H. and Halfman, R. L., Aeroelasticity, Addison-Wesley Publishing Co., Reading, Mass, 1955. [15] Richardson, J. R., A more realistic method for routine flutter calculations, AIAA sympo- sium on structural dynamics and aeroelasticity, Boston/Massachussets, August 30, September 1, pp. 10–17, 1965.