Example on Airfoil Flutter Speed computation Let's consider a 2D airfoil connected to the fuselage mass by bending and torsion springs, while the fuselage is free to rigidly plunge. The model has, therefore, one rigid body and two elastic mode shapes. It is shown in the figure below. Fig. 1 -­‐ Three Degree of Freedom Airfoil and Fuselage The aerodynamic center is at 25% of the chord, the springs are connected to the "elastic axis" at 40% of the chord, while the airfoil center of gravity may move along the chord. Two positions are considered herein: 37% and 45% of the chord. The remaining parameters for the analysis at sea level include a chord of 6.0 [ft], a radius of gyration of 1.5 [ft] about the elastic axis, a mass ratio 𝜇 = 20.0, uncoupled bending and torsion frequencies of 10.0 and 25.0 [rad/sec], respectively, and equal structural damping coefficients g=0.03 in both modes. The airfoil lift curve slope is the theoretical 2D incompressible value of 𝐶!! =
2𝜋 1 𝑟𝑎𝑑 . The previous adimensional input parameters become the following dimensional values (necessary for the finite element model): Fuselage Mass = 1.3447 [slugs] +0.24205 [slug*ft] 37% c.g. Static Unbalance about Elastic Axis = -­‐40342 [slug*ft]45% c.g. Moment of Inertia about Elastic Axis = 3.0256 [slug*ft2] Bending Stiffnes = 134.47 [lb/ft] Torsion Stiffness = 1891.0 [lb*ft/rad] Bending Damping = 0.40341 [lb*sec/ft] Torsion Damping = 2.2692 [lb*sec*ft/rad] According to these values, the input file for the flutter analysis using MSC/Nastran aeroelastic code may be written as: ID NXN,HA145A Executive C ontrol Deck TIME 5 Flutter Analysis SOL 145 CEND Case Control Deck TITLE =THREE DEGREE OF FREEDOM Two Subcases AEROELASTIC MODEL METHOD=1 will recall a modified Givens SUBTI = 2-­‐D AERO, W.P. JONES C(K) AND Eigenvalue Method BRITISH (PK) FLUTTER METHOD LABEL = ONE RIGID BODY AND TWO SVEC=ALL Print the vibration modes FLEXIBLE MODES, C.G. AT 0.37 CHORD DISP=ALL Print the flutter modes ECHO = BOTH In subcase 1 the PK flutter method is METHOD = 1 employed SVEC = ALL In subcase 2 the K flutter Method is DISP = ALL used SUBCASE 1 CMETHOD = 20 FMETHOD = 3 SUBCASE 2 Instructions for V-­‐g-­‐f plots FMETHOD = 4 CMETHOD = 20 OUTPUT(XYPLOT) $ $$$$$XYPEAK,XYPUNCH,VG / OUTPUT(XYPLOT) XYPUNCH,VG / $$$$$XYPEAK,XYPUNCH,VG / 1(G,F), 2(G,F), 3(G,F) XYPUNCH,VG / 1(G,F), 2(G,F), 3(G,F) $ BEGIN BULK GRID,100,,0.,0.5,0.,,1246 Bulk Data (lb-­‐ft-­‐sec system) GRID,101,,0.,1.0,0. GRID,102,,0.,0.5,0.,,12456 Structure Definition $ CELAS2,103,134.47,100,3,102,3 Grid 102 symulates fuselage point CELAS2,105,1891.,100,5 Grid 100 and 101 are the wing points $ RBAR,101,100,101,123456 Celas2 elements simulate the bending $ and torsion stiffness CMASS2,111,1.3447,102,3 $ Rigid Bar for linking grid points 100 and $ ELASTIC AXIS AT 37 PERCENT CHORD 101 CONM1,110,100,,,,,,,+CNM1 Cmass2 defines a scalar mass element. +CNM1,1.3447,,,,,,,0.24205,+CNM1A In this case it is used for the Fuselage +CNM1A,,3.0256 mass. $ Conm1 defines a 6 by 6 symmetric $ ELASTIC AXIS AT 45 PERCENT CHORD inertia matrix for a grid point $CONM1,110,100,,,,,,,+CNM1 The case of the elastic axis at 45% chord $+CNM1,1.3447,,,,,,,-­‐.40342,+CNM1A is written but not used since there is a $+CNM1A,,3.0256 "$" in first column. $ Cdamp2 defines a scalar damper CDAMP2,123,0.40341,100,3,102,3 element. The damper constant is linking CDAMP2,125,2.2692,100,5,102,5 the grid points and related dof. $ Eigr selects the Givens modified method EIGR,1,MGIV,0.,25.,,3,,,+EGR +EGR,MAX $ AERO,,100.,6.,.002378,0,0 $ SPLINE2,1201,1000,1000,1000,1100,0.,1.,
0,+SP2 +SP2,0.,0. $ SET1,1100,100,101 $ CAERO4,1000,1000,,1,,,,,+CA1 +CA1,-­‐2.4,0.,0.,6.,-­‐2.4,1.,0.,6. $ PAERO4,1000,0,0,2,13,0.,0.,0. AEFACT,13,0.0,1.0,-­‐0.165,0.041,-­‐
0.335,0.320 $ MKAERO1,0.,,,,,,,,+MK +MK,.001,.1,.2,.3,.4,.5,.6,.7 MKAERO1,0.,,,,,,,,+MKA +MKA,.8,.9,1.,1.1,1.3,1.5,2.,5. $ FLUTTER,3,PK,1,2,3,L,3 FLFACT,1,1. FLFACT,2,0. FLFACT,3,20.,40.,60.,80.,100.,120.,140.,+F
LF1 +FLF1,160.,170.,180.,190.,200.,210.,220.,-­‐
230.,+FLF2 +FLF2,-­‐240.,260.,-­‐280.,-­‐290.,300. $ FLUTTER,4,K,1,2,4,L,3 FLFACT,4,0.20,0.18,0.16,0.14,0.12,0.10,0.
08,+FL4 +FL4,0.06 EIGC,20,HESS,MAX,,,,,,+EC +EC,,,,,,,3 ENDDATA for the structural eigenvalue problem. Aerodynamic Data Aero card defines reference values for aerodynamic calculations Spline2 is a linear spline for interpolation between structural and aerodynamic grid points Set1 defines the set of structural grid points to be used as a basis for the interpolation Caero4 is used to specify strip theory. Pid identifies a Paero4 card where the element properties are given. The continuation card defines the inboard and outboard leading edge and the respective chords. Mkaero1 defines Mach numbers and reduced frequencies. Flutter entry defines the conditions for flutter calculations together with the definitions of the Flfact associated entries. Two subcases have been requested for flutter calculations with some slighht modifications. The result of this Nastran file are (Nominal Case, c.g. at 37% of m.a.c.): Three Degree of Freedom Airfoil and Fuselage
4
Frequency [Hz]
3.5
3
2.5
2
1.5
1
0.5
0
0
50
100
150
200
250
300
150
200
250
300
Damping [%]
1
Mode 1
Mode 2
Mode 3
0.5
0
-0.5
-1
0
50
100
Speed [m/sec]
Increasing the torsional stiffness from 1891.0 to 4891.0 [lb ft/rad], the natural frequencies become from to F1 0.0 0.0 F2 2.25 2.25 F3 4.01 6.45 and consequently the V-­‐g-­‐f diagrams change in the following figure Three Degree of Freedom Airfoil and Fuselage
Frequency [Hz]
7
6
5
4
3
2
1
0
0
50
100
150
200
250
300
150
200
250
300
0
Damping [%]
-0.1
-0.2
-0.3
Mode 1
Mode 2
Mode 3
-0.4
-0.5
-0.6
-0.7
0
50
100
Speed [m/sec]
Reducing the torsional stiffness from 1891.0 to 891.0 [lb ft/rad], the natural frequencies become from to F1 0.0 0.0 F2 2.25 2.24 F3 4.01 2.77 and consequently the V-­‐g-­‐f diagrams change in the following figure Three Degree of Freedom Airfoil and Fuselage
Frequency [Hz]
3
2.5
2
1.5
1
0.5
0
0
50
100
150
200
250
300
150
200
250
300
10
Damping [%]
8
Mode 1
Mode 2
Mode 3
6
4
2
0
-2
0
50
100
Speed [m/sec]
Increasing the bending stiffness from 134.47 to 334.47 [lb/ft], the natural frequencies become from to F1 0.0 0.0 F2 2.25 3.51 F3 4.01 4.05 and consequently the V-­‐g-­‐f diagrams change in the following figure Three Degree of Freedom Airfoil and Fuselage
4
Mode 1
Mode 2
Mode 3
Frequency [Hz]
3.5
3
2.5
2
1.5
1
0.5
0
0
50
100
0
50
100
150
200
250
300
150
200
250
300
Damping [%]
1
0.5
0
-0.5
-1
Speed [m/sec]
With the c.g. at 45% of m.a.c., for the nominal case of the stiffness, we get Three Degree of Freedom Airfoil and Fuselage
Frequency [Hz]
5
Mode 1
Mode 2
Mode 3
4
3
2
1
0
0
50
100
0
50
100
150
200
250
300
150
200
250
300
2
Damping [%]
1.5
1
0.5
0
-0.5
-1
-1.5
Speed [m/sec]
The flutter speed values behave as expected: increasing and decreasing the torsional stiffness, and consequently the relative frequency, we get a corresponding increase and decrease of the flutter speed. The speed instability value is less sensitive to the bending variation. The position of the c.g. can dramatically change the flutter speed value, with additional numerical problems in analysing the results.