In the following example two mode shapes of a simple one dimensional airfoil model will be presented. In this example a beam rod model will be used, defined¹ as a flat plate with rigid chordwise sections and whose span, l, is substantially larger than its chord, c. To determine the mode shapes for a torsion-only cantilever wing, start with the equations of static moment equilibrium for a beam rod.
 

			(2.2.1)
	ae(y) -	nose up twist about the elastic axis, e.a., at station y
	My -	nose up aerodynamic moment about the e.a., per unit distance in the spanwise, y, direction
	G -	shear modulus
	J -	polar moment of inertia (=ch3/3 for a rectangular cross section of thickness h, h<<c)
	GJ -	torsional stiffness

This second order differential equation is solved by using the following boundary conditions.
 

ae(0) = 0

y = l

(2.2.2)

Next turn to aerodynamic theory and use a 'strip theory' approximation. This states that, the aerodynamic lift and moment at station y depends only on the angle of attack at station y¹. From this the moment and lift per unit span are:
 

	My = MAC. + Le		(2.2.3a)
	L = qcCL		(2.2.3b)

From this information it can be illustrated by solving the governing differential equations at the divergence speed q_D, that, for a flat plate wing with constant properties, the governing differential equation is :
 

			(2.2.11)
	q -	dynamic pressure
	l -	wing semi-span length
	e -	distance from the line of aerodynamic centers to the elastic axis.
	dCL/da -	lift curve slope

The solution of this differential equation with the eigenfunction approach and identical boundary conditions, yields the eigenfunction:
 

(2.2.15)

This eigenfunction provides the twist distribution along the wing at the divergence dynamic pressure for static aeroelastic conditions. With this eigenfunction we are able to plot the mode shapes for the first and second mode, corresponding to m=1 and m=2.

¹Dowell, Earl H., et. al. A Modern Course in Aeroelasticity, Kluwer Academic Publishers, 1995, p.17-21