Can an undamped harmonic oscillator have a steady-state solution?

[Richardbryant]

Homework Statement

An undamped harmonic oscillator (b=0) is subject to an applied force Focos(wt). Show that if w=wo, there is no steady- state solution. Find a particular solution by starting with a solution for w=wo+#, and passing to the limit #->0, it will blow up. Try starting with a solution which fits the initial condition xo=0, so that i cannot blow up at t=0.

Homework Equations

The Attempt at a Solution

d^2x/dt^2+(wo^2)x=Fo cos(w+#)t/m
d^2y/dt^2+(wo^2)y=Fo sin(w+#)t/m
d^2z/dt^2+(wo^2)z=Foe^i(w+#)t/m (1)
Let Z=Ce^i(wo+#)t, plug in (1)
C=Fo/,[wo^2-(w+#)^2]
thus X= Fo cos(w+#)t/m[wo^2-(w+#)^2]
Xtr (trasient term )=Acos(wot-$) $= phase difference
After a couple of steps the final solution will blow up when limit #->0[/B]

 

[Dr.D]

With no damping and sinusoidal excitation at the undamped natural frequency, the solution grows linearly with time. This is the reason there is no steady state.

You need to obtain the characteristic equation and look at the roots. That will get you started toward the proper results.

 

[vela]

Homework Statement

An undamped harmonic oscillator (b=0) is subject to an applied force Focos(wt). Show that if w=wo, there is no steady- state solution. Find a particular solution by starting with a solution for w=wo+#, and passing to the limit #->0, it will blow up. Try starting with a solution which fits the initial condition xo=0, so that i cannot blow up at t=0.

Homework Equations

The Attempt at a Solution

d^2x/dt^2+(wo^2)x=Fo cos(w+#)t/m
d^2y/dt^2+(wo^2)y=Fo sin(w+#)t/m
d^2z/dt^2+(wo^2)z=Foe^i(w+#)t/m (1)
Let Z=Ce^i(wo+#)t, plug in (1)
C=Fo/,[wo^2-(w+#)^2]
thus X= Fo cos(w+#)t/m[wo^2-(w+#)^2]
Xtr (trasient term )=Acos(wot-$) $= phase difference
After a couple of steps the final solution will blow up when limit #->0

What's your question?

 

[Richardbryant]

With no damping and sinusoidal excitation at the undamped natural frequency, the solution grows linearly with time. This is the reason there is no steady state.

You need to obtain the characteristic equation and look at the roots. That will get you started toward the proper results.

Thanks for reply, i had been guessing the solution is also a trigonometric function , but it seems to be not working

 

[Richardbryant]

What's your question?

The question is to find a x(t) satisfying the given condition

 

[vela]

The question is to find a x(t) satisfying the given condition

Obviously, that's what the question is asking of you. What is YOUR specific question? You seem to be on the right track.

 

[Richardbryant]

Obviously, that's what the question is asking of you. What is YOUR specific question? You seem to be on the right track.

Oh, yeah i got the correct answer, but i didn't notice, thank you about that!

 

[vela]

Did you manage to show the system didn’t have a steady state solution?