Andersen and Thomsen 2002). Another one is the partially follower-loaded elastic
double pendulum, which is to be considered here. In fact, this double pendulum can
be viewed as a two-mode approximation to Beck’s column, at least for small
deflections.
4.5.1 The Model
Consider the system of Fig. 4.11, composed of two rigid and weightless rods of
equal length l that carry concentrated masses m 1 and m 2 . A tangential follower load
aP and a non-following (constant-directional) load (1-aP) is applied at the free end.
Here, P is an indicator of load-level whereas the parameter a 2 R characterizes the
loading-arrangement. The special case a = 0 corresponds to purely conservative
loading, e.g., as provided by gravity. Another special case is a = 1 corresponding to
a perfectly following load, e.g., as provided by fluid flowing through the bars in a
horizontal or zero-gravity arrangement (e.g. Langthjem and Sugiyama 1999). At the
frictionless hinges there are linear torsional springs with stiffness k, and linear
dampers with viscous damping coefficients C 1 and C 2 . The bar-angles h 1 (t) and
h 2 (t) define the instantaneous state of the system, and are thus generalized
coordinates.
The equations of motion are conveniently set up by using Lagrange’s equations.
The kinetic energy T, potential energy V, Rayleigh dissipation function R, and
generalized forces Q 1 and Q 2 are, respectively:
Fig. 4.11 The partially follower-loaded elastic double pendulum
4.5 The Follower-Loaded Double Pendulum
229
double pendulum, which is to be considered here. In fact, this double pendulum can
be viewed as a two-mode approximation to Beck’s column, at least for small
deflections.
4.5.1 The Model
Consider the system of Fig. 4.11, composed of two rigid and weightless rods of
equal length l that carry concentrated masses m 1 and m 2 . A tangential follower load
aP and a non-following (constant-directional) load (1-aP) is applied at the free end.
Here, P is an indicator of load-level whereas the parameter a 2 R characterizes the
loading-arrangement. The special case a = 0 corresponds to purely conservative
loading, e.g., as provided by gravity. Another special case is a = 1 corresponding to
a perfectly following load, e.g., as provided by fluid flowing through the bars in a
horizontal or zero-gravity arrangement (e.g. Langthjem and Sugiyama 1999). At the
frictionless hinges there are linear torsional springs with stiffness k, and linear
dampers with viscous damping coefficients C 1 and C 2 . The bar-angles h 1 (t) and
h 2 (t) define the instantaneous state of the system, and are thus generalized
coordinates.
The equations of motion are conveniently set up by using Lagrange’s equations.
The kinetic energy T, potential energy V, Rayleigh dissipation function R, and
generalized forces Q 1 and Q 2 are, respectively:
Fig. 4.11 The partially follower-loaded elastic double pendulum
4.5 The Follower-Loaded Double Pendulum
229