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Harmonic and Anharmonic Oscillations of a Boat
Developed by Eric Ayars  Published August 1, 2016
Simple Harmonic Oscillation (SHO) is everyone's favorite type of periodic motion. It's simple, easy to understand, and has a known solution. But SHO is not the only kind of periodic motion. In this set of exercises we'll investigate an _anharmonic_ oscillator. We'll explore characteristics of its motion, and try to get an understanding of the circumstances under which the motion can be approximated as simple harmonic.
Subject Area  Mechanics 

Levels  First Year and Beyond the First Year 
Available Implementations  Python, IPython/Jupyter Notebook, Mathematica, and Easy Java Simulations 
Learning Objectives 
Students who complete these exercises will be able to use an ODE solver to develop a qualitative understanding of the behavior of an anharmonic oscillator.
Most students gain a good understanding of the behavior of harmonic oscillators in their introductory classes, but not all oscillators are harmonic. Here we investigate an idealized boat hull which exhibits asymmetric oscillation: the restoring force is stronger for displacements in one direction than in the other. This results, for large oscillations, in a visibly nonsinusoidal oscillation. For small oscillations, the motion is approximately simple harmonic again.

Time to Complete  120 min 
These exercises are not tied to a specific programming language. Example implementations are provided under the Code tab, but the Exercises can be implemented in whatever platform you wish to use (e.g., Excel, Python, MATLAB, etc.).
## Exercise 1: A straightsided boat
Imagine a boat with vertical sides, floating.
There will be two forces on this boat: its weight $mg$, and the buoyant force $\rho g V$.
(I've defined down to be the positive direction, here, so the buoyant force is negative since it's upwards.)
The volume $V$ in the buoyant force component is the volume of water displaced by the boat: in other words, the volume of boat that is _below_ water level.
In equilibrium, the bottom of the boat will be some distance $x_o$ below the water, so $V = Ax_o$, where $A$ is the area of the boat at the waterline.
If we then push the boat down some additional distance $x$ and let go, the boat will bob up and down.
1. Show that the boat's vertical motion is SHO.
It will probably be helpful to follow the same pattern as for the introduction example: the key is to get the equation in the form of the equation for SHO.
2. What is the period of the boat's motion?
## Exercise 2: A Vhulled boat
Instead of a straightsided boat, imagine a boat with a Vshaped hull profile, as shown below.
![Sketch of a boat with a Vshaped hull](images/anharmonic/vhull.png)
The width $w$ of the hull at the waterline depends on the depth below waterline $h$:
$$ w = \beta h $$
The volume of water displaced by this boat is the area of the triangle below water level, times the length of the boat:
$$ V = \frac{1}{2}\beta h^2 L $$
1. Show that the vertical motion of this boat is not SHO.
Replace $h$ with $(y_o + y)$, where $y_o$ is the equilibrium depth, and follow the example in the introduction.
2. Rearrange the equation you got in the previous problem to be as close to SHO as possible by putting it in this form:
$$ \frac{d^2y}{dt^2} = \omega_o^2\left( 1 + \frac{y}{2y_o} \right) y $$
In this form, you can see that if $\frac{y}{2y_o}$ is small, the motion is approximately SHO with angular frequency $\omega_o$.
What is that $\omega_o$?
What must be small for this approximation to be valid?
3. If $\frac{y}{2y_o}$ is not small, would the period of the boat's oscillation be larger or smaller than $T_o = \frac{2\pi}{\omega_o}$?
Use a numeric solution of the equation of motion for the boat to verify your answer.
4. Plot the motion of the boat for various amplitudes.
In addition to effects on $T$, how else does the motion differ from SHO?
It will be helpful to plot both the solution to the differential equation and the SHO approximation,
$$ y(t) = y_{max}\cos(\omega_o t) $$
5. We've neglected viscous damping, which is a bad idea in liquids!
Redo your calculations, and plots, adding a viscous damping force
$$ F_v = \delta\frac{dy}{dt} $$
to your equations.
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Credits and Licensing
Eric Ayars, "Harmonic and Anharmonic Oscillations of a Boat," Published in the PICUP Collection, August 2016.
The instructor materials are ©2016 Eric Ayars.
The exercises are released under a Creative Commons AttributionNonCommercialShareAlike 4.0 license