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lecnotes6 - 6 6.1 Parametric oscillator Mathieu equation We...

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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 time-dependent parameter. Consider the parametric forcing to be a time-dependent gravitational field: g ( t ) = g 0 + 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 time-dependence 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 0 + g 1 cos(2 γt ) Substituting into the equation of motion then gives d 2 β + γ 0 2 [1 + h cos(2 γt )] β = 0 (14) d t 2 38
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where γ 2 = g 0 /l and h = g 1 /g 0 0 . 0 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 0 We wish to characterize the stability of the rest state. Our previous methods are unapplicable, however, because of the time-dependent 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 second-order 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 ). 2. Any solution β ( t ) is a linear combination of two linearly independent solutions β 1 ( t ) and β 2 ( t ): β ( t ) = 1 ( t ) + 2 ( t ) (15) where A and B come from initial conditions.
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