- #1
kostoglotov
- 234
- 6
Modeling driven undamped spring systems in my diff eqs text at the moment.
So I've just worked through the derivation of
[tex]x(t) = C\cos{(\omega_0t - \alpha)} + \frac{F_0/m}{\omega_0^2-\omega^2}\cos{\omega t}[/tex]
And it's clear that this describes the superposition of two different oscillations.
I was wondering, just off the top of my head, if a damped system would consist of the superposition of three different oscillations?
My immediate guess would be no, because you'd be solving the diff eq which included the addition of the damping constant multiplied by the first differential of motion, which would just give you a different complementary solution depending on whether it was underdamped, critically damped or overdamped. In the over and critically damped cases, the complementary solution is not sinusoidal, and so the nonhomogeneous driving force would have its own frequency, even though the particular solution would be solved by running a linear combination of sinusoids through the diff eq, this will not change the drivers frequency, just its phase shift...and solving the complementary function for the underdamped case also just produces sinusoids with the natural frequency and a phase shift, but at no point will a third frequency be introduced...is that about right? Is there anything else I should be considering?
So I've just worked through the derivation of
[tex]x(t) = C\cos{(\omega_0t - \alpha)} + \frac{F_0/m}{\omega_0^2-\omega^2}\cos{\omega t}[/tex]
And it's clear that this describes the superposition of two different oscillations.
I was wondering, just off the top of my head, if a damped system would consist of the superposition of three different oscillations?
My immediate guess would be no, because you'd be solving the diff eq which included the addition of the damping constant multiplied by the first differential of motion, which would just give you a different complementary solution depending on whether it was underdamped, critically damped or overdamped. In the over and critically damped cases, the complementary solution is not sinusoidal, and so the nonhomogeneous driving force would have its own frequency, even though the particular solution would be solved by running a linear combination of sinusoids through the diff eq, this will not change the drivers frequency, just its phase shift...and solving the complementary function for the underdamped case also just produces sinusoids with the natural frequency and a phase shift, but at no point will a third frequency be introduced...is that about right? Is there anything else I should be considering?