Hi all!(adsbygoogle = window.adsbygoogle || []).push({});

I was considering the Energy of a driven damped oscillator and came upon the following equation:

given the equation of motion:

[tex]m\ddot x+Dx=-b\dot x+F(t)[/tex]

take the equation multiplied by [tex]\dot x[/tex]

[tex]m\ddot x\dot x+Dx\dot x=-b\dot x^2+F(t)\dot x[/tex]

and we rewrite it:

[tex]\frac{d}{dt}(\frac{m\dot x^2}{2}+\frac{Dx^2}{2})=-b\dot x^2+F(t)\dot x[/tex]

now the LHS appears to be the total energy of the undamped harmonic oscillator, by energy conservation a constant, so it´s rate of change is 0:

[tex]0=-b\dot x^2+F(t)\dot x[/tex]

Ok, so far so good :) But here lies my problem. Now we´ve obtained an equation for \dot x which we could solve for x(t). But the result appears to be different from the standart solution :(

[tex]0=\dot x(-b\dot x+F(t))[/tex]

[tex]\dot x=0[/tex]

[tex]-b\dot x+F(t)=0[/tex]

=> [tex]\dot x_1=const:=C[/tex]

[tex]x(t)=\frac{1}{b}\displaystyle{\int_{t_0}^{t}}F(t')dt'[/tex]

so the solution would read: (would it? Do we have superposition here, since the DE is not linear any more?)

[tex]x(t)=C+\frac{1}{b}\displaystyle{\int_{t_0}^{t}}F(t')dt'[/tex]

So could anyone please help me find the mistake :) I would be thankful :)

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Energy of driven damped oscillator

**Physics Forums | Science Articles, Homework Help, Discussion**