Modeling the Performance of a Sailboat

Motivation

I am a sailor, and during high-school I sailed Laser a lot. While I sailed the Laser, I often thought a lot about the physics and geometric theories in sailing. In IB math-HL class, I needed to make a project. I could easily make a simple project and get a really high grade, or I could make some wind sensors, use gps, make a data collector, and use multivariable calculus to analyze the performance of my sailboat. The seconds option was much more fun for me, and it would be a great oppurtunity for me to improve my skills in my field. I exceeded the IB requirement a lot in this project, but it was totally worth it and it was super fun. Also while I attached all these electronics on my small sailboat during the practice, all of my teammates were really impressed and I think I managed to effect my community positively. I finally showed the results to my previous sailing coach, who is the Turkish National Sailing Sports Director and got some valuable feedback.

Anemometer

This is a digital anemometer. I bought a Sparkfun anemometer, and modified it so I could attach it to my boat and to my electronics. I attached this sensor to the mast of my sailboat. So it is possible to calculate the rpm by using the time difference between successive switch events using this anemometer.

Wind Direction Sensor

This is a precise wind direction sensor that I made from scratch. Existing one were so expensive and so large, and cheap ones were not precise at all. So I made my own sensor. The upper part is attached to an optical encoder inside the case. The sensor has 4 wires: Ground, 5 Volt, and 2 wires which outputs 5V or 0V depending on the reading of the optical encoder.

Data Collector

This is the data collector I made. Both sensors I explained above gets connected to this device. There is also a u-blox GPS, li-ion battery, SD card reader, and an LCD display unit inside this device.

I programmed the Arduino nano board in C in such a way that it receives data from two wind sensors using external interrupts, reads GPS data, then records them in the SD card, and it also displays the readings and the VMG value on the LCD display so that I can make sure readings are valid while sailing the boat and recording data at the same time.

3D Parts

These are the 3D designs I made for this project. I designed them in 3DsMax and also in Fusion 360, then rendered all of them using the Arnold rendering engine in 3DsMax.

Collecting Data

I used Excel to calculate the true Wind Speed, upwind sailing angle and VMG rate using the over 8000 rows of data that I collected in a sailing practice. All the final data points are shown in the figure below which does not make much sense. I used Octave to display 3D data.

Modeling the Data

I uploaded my data to Zunzun website, which is an online tool that uses remote servers to find the best 3D surface fit equation. I got the result below which was the closest mathamatical model and it made a lot of sense when I analyzed it.

Results

The figures below are the the crossentional slices from the 3D equation above at different wind speed values. I made these to make it easier to explain and analyze the 3D Equation. Even if you are not a sailor, you can see that the Y coordinates of the peaks increase as the wind speed increases. (There are 4 figures for different wind speeds, check them all and and see the difference)

I calculated the partial derivative of the 3D equation to find the peak values at different wind speeds. This figure below gives the maximum VMG that the boat can sail at differnt wind speeds. The graph is not linear since as the boat gets faster, friction slows down the boat as it tries to speed up.

You can downlaod the paper below where I explained the calculations and all the process with more detail.

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