• Pure divergence (cross-hatched in Fig. 4.12): four real roots, two positive and
two negative.
• Pure flutter (horizontally-hatched): two pairs of complex conjugate roots, one
pair with positive and one pair with negative real part.
• Decayed-oscillation divergence (diagonally-hatched): two real roots of opposite
sign, and a pair of complex conjugate roots with negative real part.
• Fluttering divergence. (vertically-hatched): two real roots of opposite sign, and
a pair of complex conjugate roots with positive real part.
The notions of flutter and divergence are tied to the nature of the eigenvalues this
way: Consider a linearized system _
x = Ax with a Jacobian eigenvalue k = a + ib,
a, b 2 R. The solution x(t) consists of a sum of terms having the form x
(t) = ue
kt = ue
(a+ib)t = ue
at (cos(bt) + i sin(bt)), one term for each eigenvalue.
Thus, e
kt describes the time-evolution of a particular component of the solution.
With a real and positive eigenvalue (a > 0, b = 0) the term e
kt will escape towards
infinity at exponential rate (divergence). With a complex-valued eigenvalue having
positive real part (a > 0, b 6 ¼ 0) the solution will oscillate around zero with
amplitudes increasing expontially in time (flutter). When several eigenvalues are
involved, flutter and divergence may combine to create damped oscillations around
a solution that diverges exponentially from zero (decayed-oscillation divergence),
or exponentially growing oscillations around a solution diverging exponentially
from zero (fluttering divergence).
We have now established conditions under which the upright position of the
double pendulum may become unstable. And we have described the kinds of
motion accompanying unstable behavior (flutter, divergence, etc.). For a truly linear
system (f = 0 in (4.52)) this would end the discussion. There would be no equilibrium states other than the zero solution, and so any unstable solution would
approach infinity as predicted by the local analysis of stability. However, in the
presence of nonlinearities (f 6 ¼ 0) the system may possess any number of stable and
unstable states of static or dynamic equilibrium. So, saying that the zero solution for
the nonlinear double pendulum is unstable does not reveal where the system will
then be instead, as in the linear setting. We have to calculate the stable state(s) to
which the system will go in place of the zero. In the following section we consider
the existence and stability of periodic solutions near the zero solution.
4.5.3 Periodic Solutions
When for the linear system (f = 0) the zero solution is unstable, the state (h 1 , h 2 ) of
the double pendulum will approach infinity. Usually the presence of nonlinearities
(f 6 ¼ 0) will limit this growth of state variables, perhaps into the form of finite
amplitude periodic oscillations. One can expect such periodic solutions to occur
near regions of the loading parameter space where the so-called Hopf conditions are
met. That is, where the characteristic equation (4.56) has a pair of purely imaginary
4.5 The Follower-Loaded Double Pendulum
233
two negative.
• Pure flutter (horizontally-hatched): two pairs of complex conjugate roots, one
pair with positive and one pair with negative real part.
• Decayed-oscillation divergence (diagonally-hatched): two real roots of opposite
sign, and a pair of complex conjugate roots with negative real part.
• Fluttering divergence. (vertically-hatched): two real roots of opposite sign, and
a pair of complex conjugate roots with positive real part.
The notions of flutter and divergence are tied to the nature of the eigenvalues this
way: Consider a linearized system _
x = Ax with a Jacobian eigenvalue k = a + ib,
a, b 2 R. The solution x(t) consists of a sum of terms having the form x
(t) = ue
kt = ue
(a+ib)t = ue
at (cos(bt) + i sin(bt)), one term for each eigenvalue.
Thus, e
kt describes the time-evolution of a particular component of the solution.
With a real and positive eigenvalue (a > 0, b = 0) the term e
kt will escape towards
infinity at exponential rate (divergence). With a complex-valued eigenvalue having
positive real part (a > 0, b 6 ¼ 0) the solution will oscillate around zero with
amplitudes increasing expontially in time (flutter). When several eigenvalues are
involved, flutter and divergence may combine to create damped oscillations around
a solution that diverges exponentially from zero (decayed-oscillation divergence),
or exponentially growing oscillations around a solution diverging exponentially
from zero (fluttering divergence).
We have now established conditions under which the upright position of the
double pendulum may become unstable. And we have described the kinds of
motion accompanying unstable behavior (flutter, divergence, etc.). For a truly linear
system (f = 0 in (4.52)) this would end the discussion. There would be no equilibrium states other than the zero solution, and so any unstable solution would
approach infinity as predicted by the local analysis of stability. However, in the
presence of nonlinearities (f 6 ¼ 0) the system may possess any number of stable and
unstable states of static or dynamic equilibrium. So, saying that the zero solution for
the nonlinear double pendulum is unstable does not reveal where the system will
then be instead, as in the linear setting. We have to calculate the stable state(s) to
which the system will go in place of the zero. In the following section we consider
the existence and stability of periodic solutions near the zero solution.
4.5.3 Periodic Solutions
When for the linear system (f = 0) the zero solution is unstable, the state (h 1 , h 2 ) of
the double pendulum will approach infinity. Usually the presence of nonlinearities
(f 6 ¼ 0) will limit this growth of state variables, perhaps into the form of finite
amplitude periodic oscillations. One can expect such periodic solutions to occur
near regions of the loading parameter space where the so-called Hopf conditions are
met. That is, where the characteristic equation (4.56) has a pair of purely imaginary
4.5 The Follower-Loaded Double Pendulum
233