Since the above expressions are linear and homogenous in pitch and plunge, the determinant of their coefficients must be zero (for a nontrivial solution).
(8)
μ⎛⎝1 − σ2⎛⎝(ωα)/(ω)⎞⎠2⎞⎠ + lh
μxα + lα
μxα + mh
μr2⎛⎝1 − ⎛⎝(ωα)/(ω)⎞⎠2⎞⎠ + mα
= 0.
The term σ = ωh ⁄ ωα is introduced so that the solution has a common term ωα ⁄ ω in terms of ω. The above relation is known as the flutter determinant.
There are four unknowns here: ωα ⁄ ω, μ, M∞, and k. This complex equation’s real and imaginary parts must both be identically zero for a solution to be obtained. Two of the four unknowns must be specified. Therefore, an iterative procedure where μ and M∞ are specified can be devised to solve for the flutter boundary.
The k Method
This engineering solution method to flutter relies on a notion that was prevailing in the 1940s to include a parameter that modeled the effect of structural damping.
The dissipative structural damping terms are:
(9)
Dh = − ighmω2heiωt
Dα = − igαIPω2ααeiωt.
Incorporating these terms changes the homogeneous equations derived above to:
(10)
⎡⎣μ⎛⎝1 − ⎛⎝(ωh)/(ω)⎞⎠2(1 + igh)⎞⎠ + lh⎤⎦(h)/(b) + (μxα + lα)α = 0
(μxα + mh)(h)/(b) + ⎡⎣μr2⎛⎝1 − ⎛⎝(ωα)/(ω)⎞⎠2(1 + igh)⎞⎠ + mα⎤⎦α = 0,
where gh and gα have values between 0.01 to 0.05 that is dependent on the structure. By treating the damping coefficients as unknown together with ω and by introducing a term Z = ⎛⎝(ωα)/(ω)⎞⎠2(1 + ig), the flutter determinant becomes
(11)
μ(1 − σ2Z)2 + lh
μxα + lα
μxα + mh
μr2(1 − Z)2 + mα
= 0.
There are only two unknowns of this quadratic equation Z1 and Z2 depending on corresponding ω and g. One can plot these quantities as a function of the airspeed or reduced velocity 1 ⁄ k. The plot of the damping coefficient for different values of k can indicate the margine of stability at conditions close to the flutter boundary, where the damping coefficient crosses zero. The plot of the frequency versus k indicates the physical mechanism that leads to the instability. Observing the effect as airspeed increases may allow for a way to delay the onset of the flutter instability.
The p-k Method
Since the k method can exhibit an inaccurate coupling among the different modes of motion, an alternate method that may more properly model the behavior of the system. The derivation of the p method and the associated p-k method is extensive. The reader is encouraged to refer to Hodges and Pierce for a detailed explanation of the methodology.
A improvement of the p-k method is its low computational effort in comparison to the k method. The k method requries may computer simulations at a particular density to match the Mach number with the flight conditions, not present in the p-k method. The accuracy of the p-k method depends on the level of damping in any given mode — in lightly damped modes it forms a good approximation.