4.11: Example - Duffing’s equation
- Page ID
- 219168
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Duffing’s equation is a nonlinear variant of the simple harmonic oscillator. It models the situation where the spring’s response is not linear for all extensions, but rather the spring’s restoring force grows faster than linearly at the ends of the spring’s travel. This models the real-world behavior of springs – springs don’t behave linearly if you pull on the spring to the end of it’s extent. Duffing’s equation models the stiffness by adding a cubic term to the restoring force. For small displacements the cubic force is small, but for larger displacements it will dominate the return force. On the RHS, Duffing also adds a drive force – in this case a cosine drive of frequency \(\omega\). A drawing of the physical setup is shown in 4.17. The full Duffing’s equation is
\[ \frac{d^2 y}{d t^2} + \delta \frac{d y}{d t} + \alpha y + \beta y^3 = \gamma \cos(\omega t) \label{eq:4.6}\]
