- 306

- 0

**1. Homework Statement**

**The problem is long so I will post the whole thing but ask only for help on part C.**

The steady-state motion of a damped oscillator driven by an applied force F0 cos(ωt) is given by xss(t) = A cos(ωt + φ).

Consider the oscillator which is released from rest at t = 0. Its motion must satisfy x(0) = 0 and v(0) = 0. After a very long time, we expect that x(t) = xss(t). To satisfy these conditions we can take as the solution

x(t) = xss(t) + xtr(t),

where xtr(t) is a solution to the equation of motion of the free damped oscillator.

(A) Show that if xss(t) satisfies the equation of motion for the forced damped oscillator, then so does x(t) = xss(t) + xtr(t), where xtr(t) satisfies the equation of motion of the free damped oscillator.

(B) Sketch the resulting motion for the case that the oscillator is driven at resonance (i.e., ω = ω0 where ω02=k/m) with b=mω0.

**(C) Choose the arbitrary constants in xtr(t) so that x(t) satisfies the initial conditions, assuming that the oscillator is driven at resonance with b=mω0. Hand in an accurate graph of x(t)/A.**

**2. Homework Equations**

x(t)/A=sin(w

_{0}t)-[itex]\frac{2}{sqrt(3)}[/itex]sin([itex]\frac{sqrt(3)}{2}[/itex]w

_{0}t)e

^{-w_0t/2}

A=[itex]\frac{F_0}{mw_0^2}[/itex]

**3. The Attempt at a Solution**

I am not quite sure what I am being asked for. How do I pick the constants in x

_{tr}(t)? Does it matter what I use or can I just plug in 1 for all of them and plot the result? Doing that yields a plot that seems to exhibit the characteristics that a driven, damped oscillator would.

Thank you!