# Damped Driven Harmonic Oscillator.

## 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.

## 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?

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Orodruin
Staff Emeritus
Homework Helper
Gold Member
Yes, but this is not the phase shift. The phase shift is the phase between the driving force and the position, i.e., φ.

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
Staff Emeritus