Editing Aeroelasticity (section)

=== Divergence ===
Divergence occurs when a lifting surface deflects under aerodynamic load in a direction which further increases lift in a positive feedback loop. The increased lift deflects the structure further, which eventually brings the structure to the point of divergence. Unlike flutter, which is another aeroelastic problem, instead of irregular oscillations, divergence causes the lifting surface to move in the same direction and when it comes to point of divergence the structure deforms.
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!Equations for divergence of a simple beam
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Divergence can be understood as a simple property of the [[differential equation]](s) governing the wing [[deflection (engineering)|deflection]]. For example, modelling the airplane wing as an [[isotropic]] [[Euler–Bernoulli beam theory|Euler–Bernoulli beam]], the uncoupled torsional [[equation of motion]] is

: <math>GJ \frac{d^2 \theta}{dy^2} = -M',</math>

where ''y'' is the spanwise dimension, ''θ'' is the elastic twist of the beam, ''GJ'' is the torsional stiffness of the beam, ''L'' is the beam length, and ''M''’ is the aerodynamic moment per unit length. Under a simple lift forcing theory the aerodynamic moment is of the form

: <math>M' = C U^2 (\theta + \alpha_0),</math>

where ''C'' is a coefficient, ''U'' is the free-stream fluid velocity, and α<sub>0</sub> is the initial angle of attack. This yields an [[ordinary differential equation]] of the form

: <math>\frac{d^2 \theta}{dy^2} + \lambda^2 \theta = -\lambda^2 \alpha_0,</math>

where

: <math>\lambda^2 = C \frac{U^2}{GJ}.</math>

The boundary conditions for a clamped-free beam (i.e., a cantilever wing) are

: <math>\theta|_{y=0} = \left.\frac{d\theta}{dy}\right|_{y=L} = 0,</math>

which yields the solution

: <math>\theta = \alpha_0 [\tan(\lambda L) \sin(\lambda y) + \cos(\lambda y) - 1].</math>

As can be seen, for ''λL'' = ''π''/2 + ''nπ'', with arbitrary integer number ''n'', tan(''λL'') is infinite. ''n'' = 0 corresponds to the point of torsional divergence. For given structural parameters, this will correspond to a single value of free-stream velocity ''U''. This is the torsional divergence speed. Note that for some special boundary conditions that may be implemented in a wind tunnel test of an airfoil (e.g., a torsional restraint positioned forward of the aerodynamic center) it is possible to eliminate the phenomenon of divergence altogether.<ref name="Hodges">Hodges, D. H. and Pierce, A., ''Introduction to Structural Dynamics and Aeroelasticity'', Cambridge, 2002, {{ISBN|978-0-521-80698-5}}.</ref>
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