- #1
Marin
- 193
- 0
Hi all!
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 :)
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 :)