Linear Beam Model

Linear beam theory is a well-known simplification of the linear theory of elasticity for the structures that can be regarded as largely one-dimensional whose widths are small compared with their lengths. It relates the deflections of a beam, which are assumed to be small, to the transverse loads. For a dynamic linear beam, the transverse deflection w(x, t) is given by the Euler-Bernoulli beam equation as

where ps is the density of the wing, hs the thickness of the wing, E the modulus of elasticity, and fext the distributed external force per unit length acting in vertical direction. Uniform wing properties are assumed in Eq. (4-1). The linear beam theory has been widely used in the aeroelastic community to model and study the dynamic structural response and stability of the wing of an airplane [333].