Modeling the swaying motion of a crane with a harmonic oscillator might work only if the side-to-side motion is small. If the motion is larger, we must account for the effect of gravity in our model. When \(x(t)\) is large, part of the crane will not be above any other part of the crane. Thus, gravity will pull downward on that part of the crane and will cause the crane to bend even further. We can model this effect by setting the equation for our harmonic oscillator equal to a factor of \(x^3\text{,}\)
\begin{equation}
\frac{d^2 x}{dt^2} + 0.1 \frac{dx}{dt} + 0.3 x = 0.02 x^3.\tag{4.3.1}
\end{equation}
When \(x\) is small, this forcing term will not contribute much to the motion of the building. However, when \(x\) is large, the term will contribute a great deal. The equation \(x'' + 0.1 x' + 0.3 x = 0.02 x^3\) is an example of Duffing’s equation.