The concept of an elastica was first put forward by Bernoulli in the 17th century and extended by...

The concept of an elastica was first put
forward by Bernoulli in the 17th century and extended by Euler in 1744. The concept
of an elastica refers to the modelling of a thin wire-like structure by an
elastic curve, the elastica, capable of non-linear large-scale deflections and
rotations. The solution for the curve of the elastica is particularly useful in
studying buckling and other related stability problems. Its formulation in a
plane involves two geometrical constraint equations, two equations for the
force equilibrium in two orthogonal directions, an equation for the moment
equilibrium and a final constitutive equation relating the applied bending
moment to the slope (Plaut and Virgin, 2009). In this example, we shall apply
the elastica to a control problem with a number of practical applications.
Consider a uniform, thin, flexible, inextensible, elastic strip, with length S,
bending rigidity EI, crosssectional area A, mass per unit length m, weight per
unit length mg and mass moment of inertia J. The elastric strip is assumed to
be incapable of being deformed in shear. The effect of weight, transverse and
axial inertia forces and rotary inertia are considered in the formulation. Only
the in-plane motion is considered, while all damping is neglected. To define
the geometry of the elastica, the height of the strip is assumed to be H and
the arc length of a typical mass point from the left end is assumed to be s.
The horizontal and vertical coordinates of a typical mass point are assumed to
be x(s, t) and y(s, t), respectively, while the angle of rotation is assumed to
be θ(s, t), where t is the time. The angle with the horizontal at s = 0 is
denoted by α. The bending moment M(s,t) is assumed to be positive if it is
counter-clockwise on a positive face. The horizontal force P(s, t) is assumed
to be positive in compression, while the vertical force Q(s,t) is positive if
it is downward on a positive face.