Damped Driven Harmonic Oscillator.

[vkumar1403]

Homework Statement

An oscillator with mass 0.5 kg, stiffness 100 N/m, and mechanical resistance 1.4 kg/s is driven by a sinusoidal force of amplitude 2 N. Plot the speed amplitude and the phase angle between the displacement and speed as a function of the driving frequency and find the frequencies for which the phase angle for which the angle is 45 deg.

Homework Equations

The Attempt at a Solution

Using the general form of the solution:
x(t) = A(w) sin(wt-Φ)
where Φ=atan(2wp/(w_0^2-w^2))
A(w) = (F/m)/((w_0^2-w^2)^2 + (2wp)^2)^0.5

I am positive the above equations are correct and come from the differential equation for this case.

Now, u(t) [speed] = d x(t)/dt.
= w*A(w)*cos(wt-Φ)
=w*A(w)*sin(wt-Φ+pi/2)

My question: Now the speed amplitude, I believe, is wA(w). Won't the phase angle between the displacement and velocity always be pi/2 irrespective of w?[/B]

 

[Orodruin]

Yes, but this is not the phase shift. The phase shift is the phase between the driving force and the position, i.e., φ.

 

[vkumar1403]

Yes, but this is not the phase shift. The phase shift is the phase between the driving force and the position, i.e., φ.

I agree but the question says the phase between the speed and the displacement. Am I interpreting this wrong?

 

[Orodruin]

I agree but the question says the phase between the speed and the displacement. Am I interpreting this wrong?

It is very likely a misprint.

 

[vkumar1403]

It is very likely a misprint.

Thanks! I'm going to quote this as a misprint in my solution with the explanation of my rationale. Hopefully, that is good enough for my professor