Appendix 2 Where: m= Mass of section Sa = mbxa = Static Unbalance about Elastic Axis Ia = Inertia about Elastic Axis The equations of motion are:
mun?
Qa Assuming harmonic motion, one obtains:
m Sa 2 h 4 2 S 10 • a Using the aerodynamic expressions of Air Force Technical Report 4798, the expressions for L and My become :
I =-rp b302 {In5+lLa-In (1/2+a))a} +(Ma- (La+My) (1/2 +а) +In (1/2 +a))0} 2
Appendix 2 Where Ih, La, Mh, and Mo are functions of V/bw and obtainable from reference 4.
In this approach, the translation motions, h, are usually assumed to originate from the fundamental bending mode and the pitch motions, a, from the fundamental torsion mode. Higher modes or control surface rotations are not considered.
The value of velocity and damping obtained by solving the equations at each value of V/bw are usually plotted to obtain the velocity at which the damping goes to zero.
3. THREE-DEGREE-OF-FREEDOM. Flutter mechanisms involving control surface rotation are many times more critical than those involving just bending-torsion. At least three degrees of freedom are required to analyze this phenomenon.
The procedure most commonly used is presented in references 2 and 4. The problem is first considered two-dimensional and then extended to the three-dimensional case.
a. Two-Dimensional Flutter Theory.
Midchord- Aileron Hinge b Wing Bending Wing Torsion Spring Spring at Rotation Point eb eb Aileron Torsion Spring at Aileron Hinge The motion of the system can be represented as above, where:
b = semichord cb = distance between midchord and aileron hinge, positive if aft of midchord eb = distance between midchord and aileron leading edge, positive aft of midchord ab = distance between rotation point (elastic axis) and midchord h = bending deflection of rotation point (elastic axis), positive downward
• = angular deflection about rotation point (elastic axis), positive for leading edge up ß = angular deflection of aileron about aileron hinge relative to wing chord, positive for aileron leading edge up 3
Appendix 2 Assuming linearized harmonic motion, the equations of motion become:
А Д + Ва + СВ = 0 DI +ÈO + FB = O Where the coefficients are given in references 2 and 4. For each value of V/b? , the determinant of the coefficients matrix is set equal to zero and the resulting value of artificial damping plotted versus airspeed.
The limitations of a two-dimensional flutter theory are delineated in reference 4.
Cl) All spanwise elements are considered identical with respect to all their flutter parameters.
(2) The vibration amplitudes in each mode do not vary with the spanwise location of the element under consideration.
(3) The effective aspect ratio approaches infinity.
(4) Aerodynamic flow over the oscillating airfoil is not disrupted by interferences.
In general, these limitations are prohibitive and some form of threedimensional analysis is required.
b. Three-Dimensional Flutter Theory. The typical three-dimensional analysis accounts for spanwise variations in mass, geometry and mode shape, but does not account for aspect ratio and aerodynamic interference effects.
The mathematics is presented in Scanlan and Rosenbaum (ref.2). To use this approach one needs the spanwise distribution of the following parameters:
m(×) = Mass per unit span of wing and aileron In (x) = Mass moment of inertia per unit span of wing and aileron, referred to elastic axis So (x) = Static unbalance per unit span of wing and aileron, referred to elastic axis Ig (x) = Aileron mass moment of inertia per unit span, referred to aileron hinge line Sg (x) = Aileron static unbalance per unit span, referred to hinge line b = Semichord length (c-a)b = Distance between the elastic axis and aileron hinge line 4
Appendix 2 The motion at flutter is assumed to be a superposition of the wing first uncoupled bending mode, the wing first uncoupled torsion mode and the aileron rotation mode. In Scanlan and Rosenbaum the mode shapes are treated as continuous functions of span and integration performed over the span of the wing. The vibratory motion at flutter is considered unchanged by the aerodynamic forces.
In practice, the integration is performed numerically, rather than continuously. The wing is divided into a number of spanwise panels and approached from a lumped parameter concept.
4. MULTI-DEGREE-OF-FREEDOM, THREE DIMENSIONAL FLUTTER THEORY. It is not the intent of this section to present a detailed explanation, but rather to outline the general procedure for setting up the mathematics involved in a multi-degree-of-freedom flutter analysis. Familiarity with threedegree-of-freedom flutter analysis and matrix algebra is assumed. The system shown in the sketch is for each mass panel:
Appendix 2 First, the general flutter equation with uncoupled modes is solved with the aerodynamic forces set to zero.
[M]fä}+ (1+18) [R] {9} = [EA! = zero [М] = [ @]І[М] [Ф Generalized mass (non diagonal) PŘ] = [ш2ц] [MaJ Generalized stiffness (diagonal) Where: [@- = Uncoupled assumed modal matrix [Ma] = diagonal elements of [M] [w'ul = uncoupled frequency matrix [M] = Mass Matrix {g} = Vector of generalized coordinates The modal solution to this equation results in the coupled modes and the coupled frequencies which can now be used to solve the equation with airforces.
Md{a}+ (1+18) [w2]MM] {a} = (FA!
[M] = [@]][M][@] = Generalized mass matrix (diagonal) Where: [FA] = Generalized aerodynamic forces [@ ] = Coupled modal matrix Гw2] = Coupled frequency matrix (diagonal)
Appendix 2 The equation of motion including the aerodynamic forces (4) is solved for selected values of bu to predict the damping in the same manner as previously discussed.
The [M] matrix is a mass matrix, usually referred to the local elastic axis, and includes the necessary transformation terms for a control surface or tab. For each mass panel, i, the [M | matrix is:
Sar Są Sa Ia PaB Pas SB Paß IB S6 PB& Where:
W- We + We + We So = We Jet We (le + »e) So = We # : +53315637 2+430 2 + 5c And: Is = We ( 2 + Ke Ww, We. W, = Wts of primary surface, control surface, and tab, respectively.
X = Distance from surface CG to EA - Positive aft Xe = Distance from control hinge to control CG X, = Distance from tab hinge to tab CG 1c = Distance from EA to control hinge 11 = Distance from control hinge to tab hinge 7
Appendix 2 Kw. Ko ky = Radii of gyration of primary surface about its CG, control surface about its CG, and tab about its CG respectively.
The K matrix is a matrix of stiffness influence coefficients. When referred to the elastic axis, the K matrix for one mass panel becomes:
khh Kha khB кп8 KOB коб KBh kốn Many times it is easier to actually measure flexibility influence coefficients, [G] , and then obtain | k by:
[x] = [0]=] The • matrix is a mode shape matrix, the eigenvectors from solving the equation of motion without aerodynamics. For each mass panel 32 3 ....
° 2 "3 " ..an 83 ° 63 ° -§ Where n is the number of modes.
Appendix 2 The FA matrix represents the generalized aerodynamic forces, obtained from:
Where [AIG ] represents the matrix of aerodynamic influence coefficients. There are several in use, but in general they will contain the following (as presented in Reference 5):
Ina Ing LAZE] - M "ви "ва Іва 988 These are referenced to the wing quarter chord, the control surface hinge line, and the tab hinge line. A transformation is required to relate these to the elastic axis.
Using the above explanations and assuming harmonic motion, the generalized equation of motion may be written for each coupled frequency as:
+ (1+18) TR Where: a = gelwt @ * Harmonic frequency F(p) = A scalar function of density [A] - [8] [AIC] [O] Rearranging the flutter equation, we may write Where the eigenvalue A is defined by 1 = + • Re (Real Part) (Imaginary Part) 9
Appendix 2 Hence the flutter frequency becomes WE = Re 1 And the damping g 8 = w? Im = Re And the flutter speed is defined by:
where the equations have been non-dimensionalized and br equals a reference length and kr equals a reference value of reduced frequency, Wb.
V
5. WHIRL MODE.
It is the intent of this section to outline the general procedure for setting up the mathematics involved in a whirl mode analysis. Familiarity with whirl mode analysis is assumed. This analysis includes only engine cantilevered modes. Coupling of these modes with a flexible airplane's modes may be accomplished by the superposition of modes using the method outlined in paragraph 4.
Appendix 2 The whirl mode stability equation for cantilevered power package is:
B= [EM] + D[FD + FG + FQ2] + FK + FQ1] = 0 where FM [GM(M) , generalized masses (1b. sec.?/in.)
FK (2N)= [FREQ(M))?] [GMIM)], generalized stiffnesses (lb./in.)
FD (.02) (2 T) [FREQ (M)] [GM(M)] , generalized damping (lb. sec./in.)
(.02 is the assumed structural damping) га = [Ри(1, м) ]} [а] [рнІ(1,м)] F22 = RADOV: [PHI (I, M)) I [02] [PHI (I,M)] M = Number of modes FREQ(M) = Frequency (Hz) PHI (I,M) = Prop displacement matrix RAD = Relative Density Ratio, Q/Co NOTE: A special purpose computer program may be required for each power package to determine its vibration mode shapes and frequencies and the generalized masses for each mode.
Appendix 2 The equation for gyroscopic forces is shown below.
GYROSCOPIC MATRIX - G P Ip-2 = Y P
- Ips₴ = Prop. vertical force e = Prop. pitching moment P Prop. side force P Prop. yawing moment Z Prop. vertical displacement P P = Prop. pitch angle P = Prop. lateral displacement Ip = Prop. yaw angle 12 = Rotational velocity (radians/sec.)
38 = Prop. polar mass moment of inertia (1b. sec.2 in.)
Appendix 2 The prop aerodynamic force equation for each set of prop coefficients are shown PROP AERODYNANIC MATRIX - Q2 FIR
- Poco0 21 ·p PROP AERODYNAMIC MATRIX - Q1 FOp 0 < FU R - Po CHu 2 Sc where propeller disk area (sq. in.)
propeller diameter (in.)
Po air density at sea level (1b. sec./in.4) preo relative air density V = true air speed (in./sec.)
below.