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Unformatted text preview: 5 Forced oscillators and limit cycles 5.1 General remarks How may we describe a forced oscillator? The linear equation β ¨ + ρβ ˙ + γ 2 β = (3) is in general inadequate. Why? Linearity ∞ if β ( t ) is a solution, then so is β ( t ), real. This is incompatible with bounded oscillations (i.e., β max < α ). We therefore introduce an equation with • a nonlinearity; and • an energy source that compensates viscous damping. 5.2 Van der Pol equation Consider a damping coeﬃcient ρ ( β ) such that ρ ( β ) > for β large   ρ ( β ) < for β small   Express this in terms of β 2 : β 2 ρ ( β ) = ρ β 2 − 1 where ρ > and β is some reference amplitude. Now, obviously, ρ > for β 2 > β 2 ρ < for β 2 < β 2 28 Substituting ρ into (3), we get d 2 β β 2 d β d t 2 + ρ β 2 − 1 d t + γ 2 β = 0 This equation is known as the van der Pol equation . It was introduced in the 1920’s as a model of nonlinear electric circuits used in the first radios. In van der Pol’s (vaccum tube) circuits, • high current = ∞ positive (ordinary) resistance; and • low current = ∞ negative resistance. The basic behavior: large oscillations decay and small oscillations grow. We shall examine this system in some detail. First, we write it in non dimensional form. We define new units of time and amplitude: • unit of time = 1 /γ • unit of amplitude = β . We transform t t ∗ /γ ∗ β β ∗ β ∗ where β ∗ and t ∗ are nondimensional. Substituting above, we obtain d 2 β ∗ β ∗ β 2 d β ∗ γ 2 d t ∗ 2 β + ρ β − 1 d t γβ + γ 2 β ∗ β = 0 ∗ Divide by γ 2 β : d 2 β ∗ ρ 2 d β ∗ 2 + β ∗ − 1 + β ∗ = 0 d t γ d t ∗ ∗ 29 Now define the dimensionless...
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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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