The rate at which a damped, driven oscillator does work

[cowey19]

Homework Statement

Consider a damped oscillator, with natural frequency ω_naut and damping constant both fixed, that is driven by a force F(t)=F_naut*cos(ωt).

a) Find the rate P(t) at which F(t) does work and show that the average (P)_avg over any number of complete cycles is mβω²A². b) Verify that this is the same as the average rate at which energy is lost to the resistive force. c) Show that as ω is varied, (P)_avg is maximum when ω=ω_naut; that is, the resonance of the power occurs at ω=ω_naut (exactly).

Homework Equations

(P)_avg= 1/τ ∫ Fv dt
F= F_naut*cos(ωt)
x(t)= A*cos(ωt)
v(t)= -Aωsin(ωt)
ω= √[(ω_naut)² - β² ]

The Attempt at a Solution

I plugged in the equation for F(t) and v(t) in the integral ( P(t) = ∫Fv ) and used u substitution, making cos(ωt) the "u" and -ωsin(ωt)dt the "du".

So I then had ∫(F_naut)Au du

After integrating and plugging cos(ωt) back in for u, I had:

(P)_avg = (1/2τ)*(F_naut)A*cos²(ωt), integrated from -τ/2 to τ/2.

This is where I hit a problem. My work doesn't seem like it will be equal to mβω²A². Because I can't solve this, I can't figure out the next two parts either. Did I make a mathematical error in integrating, or did I set up the integral wrong? Thank you!

 

[ehild]

Homework Equations

(P)_avg= 1/τ ∫ Fv dt
F= F_naut*cos(ωt)
x(t)= A*cos(ωt)
v(t)= -Aωsin(ωt)
ω= √[(ω_naut)² - β² ]

If the oscillator is driven by F=Fo cos(ωt) the oscillator will move with the frequency of the driving force, but with a phase difference, x(t)=A(cosωt+θ)
Both the phase difference θ and the amplitude A depend on the natural frequency and the driving frequency.

ehild

 

[cowey19]

Yes, but how can I incorporate that into my set up?

 

[ehild]

x(t)=A(cosωt+θ). v(t)= -Aωsin(ωt+θ). Find A and θ in terms of ω_naut, ω and F_naut, and calculate the integral 1/τ ∫ Fv dt.

ehild

 

[asteriatic]

Your approach seems to be on the right track, but there are a few issues with your integration. First, the integral should be taken over one complete cycle, not just from -τ/2 to τ/2. This is because we want to find the average rate over one full cycle, not just half of it.

Second, when integrating cos^2(ωt), you should use the trigonometric identity cos^2(ωt) = 1/2 + 1/2*cos(2ωt). This will simplify the integral and give you the correct result of (P)avg = (1/2τ)*(F_naut)^2*A^2.

To verify that this is the same as the average rate at which energy is lost to the resistive force, you can use the equation for power loss due to damping, P_loss = βv^2. Substituting in the expressions for v and ω, you should get P_loss = mβω^2A^2, which is the same as (P)avg.

Finally, to show that (P)avg is maximum when ω=ω_naut, you can take the derivative of (P)avg with respect to ω and set it equal to 0. Solving for ω will give you ω=ω_naut as the maximum. This makes sense intuitively, as at resonance, the driving frequency matches the natural frequency of the oscillator, resulting in maximum energy transfer.