  Recent equations of motion for the large deflections of a cantilevered
elastic beam are analyzed. In the traditional theory of beam (and plate) large
deflections, nonlinear restoring forces are due to the effect of stretching on
bending; for an inextensible cantilever, the enforcement of arc-length
preservation leads to quasilinear stiffness effects and inertial effects that
are both nonlinear and nonlocal. For this model, smooth solutions are
constructed via a spectral Galerkin approach. Additional compactness is needed
to pass to the limit, and this is obtained through a complex procession of
higher energy estimates. Uniqueness is obtained through a non-trivial
decomposition of the nonlinearity. The confounding effects of nonlinear inertia
are overcome via the addition of structural (Kelvin-Voigt) damping to the
equations of motion. Local well-posedness of smooth solutions is shown first in
the absence of nonlinear inertial effects, and then shown with these inertial
effects present, taking into account structural damping. With damping in force,
global-in-time, strong well-posedness result is obtained by achieving
exponential decay for small data.
