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Unformatted text preview: They correspond physically to the (highly unlikely) motion where the pendulum starts vertically upwards then executes precisely one revolution, ending vertically upwards again. 80 11.4 Damped Systems Consider a simple pendulum with damping: ¨ θ = g l sin θ k ˙ θ (11.3) where k is a small positive constant. Letting y = ˙ θ we have ˙ θ = y, ˙ y = g l sin θ ky. The equilibrium points are still where sin θ = 0; we have J = 1 ( g/l ) cos θ k ! . At θ = 2 nπ , T = k and Δ = g/l , corresponding to a stable focus (from the table in § 11.2, assuming that k is small enough that k 2 < 4 g/l ). At θ = (2 n + 1) π , T = k and Δ = g/l so we have a saddle. Because this system is not conservative (the force depends on ˙ θ as well as θ , prohibiting the existence of a potential V ( θ )), the solution curves are not symmetric in y and are not closed. The phase portrait is as follows: Note that if we define energy in the same way as for an undamped pendulum, E = 1 2 ml 2 ˙ θ 2 mgl cos θ, then d E d t = ml 2 ˙ θ ¨ θ + mgl ˙ θ sin θ = mkl ˙ θ 2 (from (11.3)) 6 , 81 so the energy is decreasing. (Without the damping term we would have had ˙ E = 0.) This allows us to deduce the portrait above from the undamped portrait of § 11.3, because the solution continuously moves to curves of lower energy. A more interesting example is the van der Pol oscillator ¨ x = x ( x 2 1) ˙ x. For large x we have damping (because x 2 1 > 0); but for small x we have “negative damping” ( x 2 1 < 0). The only equilibrium is at the origin and is an unstable focus. The phase portrait looks like this: So all trajectories (whether starting from small or large x , except for the one at the origin itself) tend towards a limit cycle . Therefore, after an initial transient the system always settles down into this finite amplitude oscillation. If we had studied only the linear stability near the origin we would have concluded that disturbances grow exponentially; the fact that this growth is in fact limited is useful in practical situations. 11.5 The Forced Pendulum Finally, add timeperiodic forcing to the pendulum: ¨ θ = g l sin θ k ˙ θ + F cos Ω t where Ω is the forcing frequency and F its amplitude. This is no longer an autonomous system (unlike in § 11.1), because t appears in the governing equation in addition to θ and ˙ θ . The phase “plane” becomes threedimensional and consequently much more elaborate: for instance, trajectories can now twist around each other, which is prevented in two dimensions by the rule that trajectories cannot cross except at equilibrium points. When F is small, we obtain a resonant response as described in § 2.2. But for larger forcing amplitudes the system exhibits much more complicated behaviour because the nonlinearity of the sin θ term interacts with the forcing. Eventually chaos results....
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 Spring '13
 MRR
 Math, Equations, Critical Point, Equilibrium point, Stability theory, θ, Plane Autonomous Systems

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