MIT18_03S10_c38

# MIT18_03S10_c38 - 18.03 Class 38 Linearization The...

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18.03 Class 38 , May 10, 2010 Linearization: The nonlinear pendulum and phugoid oscillation [1] Nonlinear pendulum [2] Phugoid oscillation [3] Buckling bridge [1] The bob of a pendulum is attached to a rod, so it can swing clear around the pivot. This system is determined by three parameters: L length of pendulum m mass of bob g acceleration of gravity We will assume that the motion is restricted to a plane. To describe it we need a dynamical variable. We could use a horizontal displacement, but it turns out to be easier to write down the equation controlling it if you use the angle of displacement from straight down. Write theta for that angle, measured to the right. Here is a force diagram: * |\ |theta | \ | \ mg | /\ (this is supposed to be a right angle!) | \/ | / | / mg sin(theta) | / |/ * Write s for arc length along the circle, with s = 0 straight down. Of course, s = L theta Newton's law says ms" = F The force has the - mg sin(theta) component of the force of gravity (and notice the sign!), and also a frictional force which depends upon s' = L theta' Make the simplest model for friction, - cs' = -cL theta'. So: m L theta" = - mg sin(theta) - cL theta'

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Divide through by mL and we get theta" + b theta' + k sin(theta) = 0 where k = g/L and b = c/m . This is a nonlinear second order equation. It still has a "companion first order system," obtained by setting x = theta , y = x' so y' = theta" = - k sin(theta) - b theta' or x' = y y' = - k sin(x) - by This is an autonmous system. Let's study its phase portrait. Equilibria:
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MIT18_03S10_c38 - 18.03 Class 38 Linearization The...

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