Is the forced oscillation an accurate and/or useful model of how musical instruments work?

Question

I recently learnt of the mathematical model behind simple forced (undamped) oscillations, and typed the equations into Desmos with some parameters here, where something subscript F means something related to the forced oscillation and something subscript N means something related to the natural oscillation.
I got some nice waveforms such as this one:
and I thought I recognised this from a little research I once did on sound synthesis, where the original wave was modulated to make patterns like this one. Is this simple model a good basis for how the piano works? A hammer hits a string, which forces an oscillation, and the string also has a natural vibration, and it produces a complex sound. Is a perfectly tuned piano a masterful piece of craftmanship where each string and hammer is of a specific size/weight/tension/whatever such that each note's forced oscillation is a semitone in pitch below the next? i.e. would plugging something like this into a computer model produce authentic sound, if the parameters were appropriately tweaked (to include also damping of course)?
Thanks.

Bill N · Accepted Answer

It could be the beginning of a model for musical instruments. The forced oscillation of a hammer strike, bow movement, lip reed, etc, is not a simple sine-type wave. In the case of a hammer strike or finger pluck, the action contains very many simple frequencies due to the Fourier theorem.
If the frequencies in the strike match any of the natural frequencies of the string (they probably will!), they will persist longer than other non-matching frequencies due to the boundary conditions of the string and those are the frequencies which will compose the tone you hear (a complex sound, as you call it.) The non-matching frequencies QUICKLY damp out.
Even vibrations from other strings can cause a string to vibrate. As an experiment, gently press (without striking) and hold the middle C, then quickly strike and release the F below the C down an octave. You will hear the middle C resonating. That's because the middle C frequency is one of the frequencies in that low F string. Release the C and it will stop.
Musical physics is fun!

mmesser314 · Answer

It is more complex than how the string vibrates. Strings themselves produce little sound. They set up vibrations in the body of the instrument, and the instrument sets air vibrating.
It is still forced oscillation, but calculation which frequencies are amplified how much from the shape of the instrument would be very complex. See How does a violin work?

Is the forced oscillation an accurate and/or useful model of how musical instruments work?

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