Part 7: The Equations for Speed


Understanding the Parts of the Whole

To achieve a goal, we need to have a detailed understanding of the factors that determine the outcome of that goal. What factors are important? Of these factors, which is the most important? What factors can be easily changed?

The Problem at Hand

Figuring out the speed of a quadcopter is an extremely dynamic process. It starts off easy enough with the velocity equation:

velocity = acceleration x time

Now we see that in order to find the velocity (v), we need to know our acceleration (a):

acceleration = force/mass

However, our force (in the form of thrust), is constantly changing. As we saw in Part 2 of the Speed Optimization series, Our thrust will be at a maximum at zero velocity and will decrease linearly with speed until max speed is reached. So in other words, our force is dependent on the very thing we are trying to find (velocity).

To throw another egg in this mess, our thrust (or more specifically, our net thrust) will be further reduced due to air drag. Of course, air drag is also dependent on velocity.

This problem of finding our velocity is an egg heads dream; this is a perfect problem to solve using calculus and differential equations. Luckily, for those of us that see this as a nightmare, we can use a program called excel and something called Numerical Methods to solve differential euqations.

Where to start?

There must be a beginning to the madness, but first, here is a list of equations that will be used in the excel spreadsheet:

If you are using a tablet or phone to view this, I apologize for the image sizes! If you click on the equations, a larger picture will open up which is much easier to read.




The Spreadsheet:

Update April 22, 2017: I updated the spreadsheet due to an error (I will admit when I screw up!).  Also, I have added a 0-60mph time which is to be used for comparison purposes only since this spreadsheet is assuming instantaneous impulse.

Here is a link to the spreadsheet I made using the above formulas. This has been a good general guideline (so far) in determining top speed. Also of note is that I am assuming the center of gravity and center of thrust are aligned.

Although these calculations get us in the ballpark, there is a recurring theme: to calculate speed accurately would be nearly impossible due to the multitude of factors that could only be determined experimentally in this case.

Most of all, I have found this spreadsheet to be a great tool for determining what factors affect speed the most and to what degree.

A Note on the Drag Coefficient (Cd) and Load Voltage:

Most input values for the formulas are straight forward, except for Cd and the voltage at max throttle. The only way to figure these out is to fill in all the rest of the input values and then do a few speed runs:

  • Take note of the top speed and the voltage at top speed of each run and average them.
  • Input the average voltage into the spreadsheet.
  • See what the spreadsheet calculates as your max speed.
  • If your actual top speed was faster than what the spreadsheet says, lower the Cd value; if slower, raise the Cd value.
  • Keep adjusting the Cd value until the spreadsheet is roughly calculating the same speed your quad obtained.

From here, you can see how changing different parts of your build (reducing area, motor kv, prop pitch, weight, etc.) will affect your top speed with a fair amount of accuracy.

Good luck and I am always open to questions, ideas, and criticism.