lecnotes5 - 5 Forced oscillators and limit cycles 5.1...

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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 non-dimensional. 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.

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lecnotes5 - 5 Forced oscillators and limit cycles 5.1...

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