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Unformatted text preview: 6 Parametric oscillator 6.1 Mathieu equation We now study a different kind of forced pendulum. Specifically, imagine subjecting the pivot of a simple frictionless pendulum to an alternating vertical motion: rigid rod This is called a “parametric pendulum,” because the motion depends on a timedependent parameter. Consider the parametric forcing to be a timedependent gravitational field: g ( t ) = g + λ ( t ) The linearized equation of motion is then (in the undamped case) d 2 β g ( t ) + β = 0 . d t 2 l The timedependence of g ( t ) makes the equation hard to solve. We know, however, that the rest state β = β ˙ = 0 is a solution. But is the rest state stable? We investigate the stability of the rest state for a special case: g ( t ) periodic and sinusoidal: g ( t ) = g + g 1 cos(2 γt ) Substituting into the equation of motion then gives d 2 β + γ 2 [1 + h cos(2 γt )] β = (14) d t 2 38 where γ 2 = g /l and h = g 1 /g . Equation (14) is called the Mathieu equation . The excitation (forcing) term has amplitude h and period 2 α α T exc = = 2 γ γ On the other hand, the natural, unexcited period of the pendulum is 2 α T nat = γ We wish to characterize the stability of the rest state. Our previous methods are unapplicable, however, because of the timedependent parametric forcing. We pause, therefore, to consider the theory of linear ODE’s with periodic coeﬃcients, known as Floquet theory . 6.2 Elements of Floquet Theory Reference: Bender and Orszag, p. 560 . We consider the general case of a secondorder linear ODE with periodic coeﬃcients. We seek to determine the conditions for stability. We begin with two observations: 1. If the coeﬃcients are periodic with period T , then if β ( t ) is a solution, so is β ( t + T )....
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This note was uploaded on 02/24/2012 for the course MECHANICAL 12.006J taught by Professor Danielrothman during the Fall '06 term at MIT.
 Fall '06
 DanielRothman

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