# Amplitude of damped mass-spring system

1. Jan 15, 2016

### lampCable

1. The problem statement, all variables and given/known data
Given the IVP
$$\ddot{r}+\dot{r}+20r=0\\ r(0) = 0.8\\ \dot{r}(0) = 0$$
for the length of an oscillating spring (damped), we find that the general solution is
$$r=e^{-0.5t}[0.8\cos(\sqrt{19.75}t)+\frac{0.4}{\sqrt{19.75}}\sin(\sqrt{19.75}t)]$$
and I wish to find the curve bounding the upper part of the solution, i.e. the amplitude of the oscillations.
2. Relevant equations

3. The attempt at a solution
I understand that if
$$r=e^{-0.5t}0.8\cos(\sqrt{19.75}t)$$
for example, then the amplitude would be obtained by removing the cosine factor, since $e^{-0.5t}0.8$ correspond to the amplitude and $\cos(\sqrt{19.75}t)$ corresponds to how fast the spring is oscillating. But how does one do when there are two oscillating terms as above?

2. Jan 15, 2016

### LCKurtz

Use$$A\cos\theta + B\sin\theta = \sqrt{A^2+B^2}\left(\frac A {\sqrt{A^2+B^2}}\cos\theta + \frac B {\sqrt{A^2+B^2}}\sin\theta \right)$$ $$=\sqrt{A^2+B^2}\cos(\theta-\alpha)$$