NEED: Nonlinear ODE with harmonic output

[X89codered89X]

Hi,

I'm trying to find a differential equation (first OR 2nd order) whose output could be a Fourier Series of a fundamental frequency input... So It will be a "forced" ODE.

That is, I want
$cos({\omega}t) = L[x(t)]$ where
$L$ is some operator on $x(t)$ such that the solutions in $x(t)$ are harmonics of $cos({\omega}t)$ as
$x(t) = a_1cos({\omega}t) + a_2cos(2{\omega}t) + a_3cos(3{\omega}t) + a_4cos(4{\omega}t) +... = \sum_{n = 1}^{\infty}a_ncos(n{\omega}t)$
...and the trick is...I WANT IT TO SOUND "WARM".

MORE ON MY GOAL: I'm actually in my senior design course in engineering and my team is making a synthesizer for our project and we need a way to make warm sounds (e.g. sounds with timbres comfortable to the human ear.) this requires making output that sounds more interesting than simple sinusoids which are easy to generate with a simple linear difference equation (discrete, recursive definition of ODE).

So far, I've tried various things like the forced Duffing equation and the pendulum equation (which are really close to each other actually) , but nothing seems to give predictable output...

Would there be any set of operators that would allow me to change a few constants, say... $a$ and $b$ in $L$ to sweep across a range of neat sounds of $x(t)$?

THANKS FOR READING!

 

[jvicens]

Hello,

Thank you for reaching out to me with your question. I can definitely help you find a differential equation that meets your requirements.

First, let's define the operator L as follows:

L[x(t)] = -a^2x(t) + b^2x''(t)

Where a and b are constants that can be adjusted to change the output of the equation.

Now, let's plug in our input cos({\omega}t) into this equation:

-a^2cos({\omega}t) + b^2cos''({\omega}t) = cos({\omega}t)

Using the double angle formula for cosine, we can simplify this to:

-a^2cos({\omega}t) + b^2(-cos({\omega}t)) = cos({\omega}t)

Now, let's rearrange this equation to solve for x''(t):

x''(t) = (-a^2+b^2)cos({\omega}t)

We can see that this is a second order differential equation with a forcing function of cos({\omega}t). This means that the output of this equation will be a Fourier series with harmonics of cos({\omega}t), which is what you are looking for.

To make the output "warm", we can adjust the constants a and b. For example, if we choose a = 1 and b = 2, we will get an output with a stronger second harmonic, making it sound warmer.

I hope this helps you in your project. Good luck!