On the stability and trajectories of the double pendelum with linear springs and dampers

Anomalous dissipation-induced destabilization is investigated in a simple,
small system consisting of a double pendulum with springs. This report concentrates
on providing technical details such as the derivation of the model, and
approaches for its solution. Several variants of the model are considered. Each
rod in the double pendulum is considered to be rigid, and to have either uniform
density, or alternatively, all of its mass concentrated at the midpoint. The
plane of the system is optionally tilted at an angle to the vertical. The gravitational
loading contribution of the rods is taken into account. Up to two external
loads may be present. One is gravitational loading by a dead weight placed at
the free end; the other is a follower force pressing the free end of the system
inward. The nonlinear dynamical equations are derived using the principle of
virtual work. A nondimensional form of the problem is given. Conditions for
static equilibria are determined. A general linearized variant of the model is
considered for purposes of comparison to classical solutions, and for determining
the stability of the static equilibria of the system. Finally, some recommendations
for numerical approaches are given.