Classical Mechanics: Lightly Damped Oscillator Driven Near Resonance

Hello Physics Forum! I have a question:

The problem: For a lightly damped oscillator being driven near resonance in the steady state, show that the fraction of its energy that is lost per cycle can be approximated by a constant (something like 2pi, which is to be determined) divided by the Q factor (Q is defined as the resonant frequency of a driven damped oscillator divided by 2*β, where β is the damping parameter).

My professor gave this hint to get us started: ΔE = ∫Fdx. Where E is the energy lost, and F is the force of friction. We are supposed to integrate from 0 to τ(cycle/period), and the professor suggested to change dx to (dx/dt)dt = vdt.

I have spent much time attempting to figure this out. I think that the frictional force is F=-bv, where b is some positive constant and v is the velocity. I try to use the solution of the differential equation for such motion, which is x(t)=Ae^(-βt)cos(ωt-δ). I take the derivative of this to get v(t). The second term in v(t) can be ignored because the damping is light. So I have:

ΔE = -b∫v^2dt from 0 to τ, where v(t)≈-Ae^(-βt)ωsin(ωt-δ). This integral makes a nasty mess that doesn't get me anything useful. I think I need to simplify this further by approximation, but I don't know which assumptions to make.

Any help would be greatly appreciated.

Thank you!

vanhees71
Gold Member
You are on the right track! Just do the integral. It might help to rewrite the sine in terms of exps!

Should I expand sine as a series or use the identity sin(z)=[e^(iz)-e^(-iz)]/2i?

AlephZero
Homework Helper
Use the identity sin(z)=[e^(iz)-e^(-iz)]/2i.

Or better still, take the general solution as the real part of ##Ce^{st}## where ##s = -b + i\omega## and ##C## is a complex constant.

Hello Physics Forum! I have a question:

The problem: For a lightly damped oscillator being driven near resonance in the steady state, show that the fraction of its energy that is lost per cycle can be approximated by a constant (something like 2pi, which is to be determined) divided by the Q factor (Q is defined as the resonant frequency of a driven damped oscillator divided by 2*β, where β is the damping parameter).

My professor gave this hint to get us started: ΔE = ∫Fdx. Where E is the energy lost, and F is the force of friction. We are supposed to integrate from 0 to τ(cycle/period), and the professor suggested to change dx to (dx/dt)dt = vdt.

I have spent much time attempting to figure this out. I think that the frictional force is F=-bv, where b is some positive constant and v is the velocity. I try to use the solution of the differential equation for such motion, which is x(t)=Ae^(-βt)cos(ωt-δ). I take the derivative of this to get v(t). The second term in v(t) can be ignored because the damping is light. So I have:

ΔE = -b∫v^2dt from 0 to τ, where v(t)≈-Ae^(-βt)ωsin(ωt-δ). This integral makes a nasty mess that doesn't get me anything useful. I think I need to simplify this further by approximation, but I don't know which assumptions to make.

Any help would be greatly appreciated.

Thank you!

As a sanity check, you can work the problem again with Q being defined as reactance over resistance. You should get the same answer. :)

Thank you everyone for the help. I've been working the problem when I have time. When I put sine in the exp form and integrate, it looks terrible. But a further hint from the professor says that I should be looking for ΔE/E. I think that is the key, but I'm still unsure. I'll be able to work it further when I get home. Again, thanks so much for the help.

I'm still unable to solve the problem. I can't seem to make this integral manageable enough to continue. I attempted to use Wolfram Alpha to integrate before and after putting sine in exp form, but I'm still unable to move forward. I feel like there must be some terms I should be neglecting because of the light damping.

Try this:

Determine the energy dissipation per cycle. (I know, this is cheating, but you'll get some great insights!)

Eric

vanhees71
$$\int \mathrm{d} t \exp(-a t) \sin^2(b t)$$
with constants $a$ and $b$. Now you write
$$\sin(b t)=\frac{\exp(\mathrm{i} b t)-\exp(-\mathrm{i} b t)}{2 \mathrm{i}},$$
$$\int \mathrm{d} t \exp(c t),$$