jonnburton
Junior Member
- Joined
- Dec 16, 2012
- Messages
- 155
I have been following the explanation of this topic in my book and there are a couple of things that I don't understand. I wondered if anyone could help me clear these things up.
The solution to the differential equation governing the spring-mass system is:
\(\displaystyle u =e^{-\frac{t}{16}} \left(2cos\frac{\sqrt{255}}{16}t + \frac{2}{\sqrt{255}}sin\frac{\sqrt255}{16}t\right)\)
The book says this is equal to \(\displaystyle \frac{32}{\sqrt{255}}e^{-\frac{t}{16}}cos\left(\frac{\sqrt{255}}{16}t - \delta\right)\)
Where \(\displaystyle \delta = 0.625\)
I don't really see how this can be, because the above equation appears to be of the form \(\displaystyle cos\beta cos\alpha + sin\beta sin\alpha\), as opposed to \(\displaystyle cos\beta cos\alpha - sin\beta sin\alpha\). So I would have said the solution reduces to:\(\displaystyle \frac{32}{\sqrt{255}}e^{-\frac{t}{16}}cos\left(\frac{\sqrt{255}}{16}t + \delta\right)\)
One other problem that I'm not sure how to approach is is to find \(\displaystyle \tau\) such that \(\displaystyle |u(t)|<0.1\) for all \(\displaystyle \tau>t\)
The only way I can see how to do this calculation is to say (using the book's equation from above):
\(\displaystyle 0.1=\frac{32}{\sqrt255}e^{-\frac{t}{16}} cos\left(\frac {\sqrt255}{16}t -0.0625\right)\)
\(\displaystyle \frac{0.1\sqrt{255}}{32}=e^{-\frac{t}{16}}cos\left(\frac{\sqrt255}{16}-0.0625\right)\)
However, from this I can see no way to isolate t.
The solution to the differential equation governing the spring-mass system is:
\(\displaystyle u =e^{-\frac{t}{16}} \left(2cos\frac{\sqrt{255}}{16}t + \frac{2}{\sqrt{255}}sin\frac{\sqrt255}{16}t\right)\)
The book says this is equal to \(\displaystyle \frac{32}{\sqrt{255}}e^{-\frac{t}{16}}cos\left(\frac{\sqrt{255}}{16}t - \delta\right)\)
Where \(\displaystyle \delta = 0.625\)
I don't really see how this can be, because the above equation appears to be of the form \(\displaystyle cos\beta cos\alpha + sin\beta sin\alpha\), as opposed to \(\displaystyle cos\beta cos\alpha - sin\beta sin\alpha\). So I would have said the solution reduces to:\(\displaystyle \frac{32}{\sqrt{255}}e^{-\frac{t}{16}}cos\left(\frac{\sqrt{255}}{16}t + \delta\right)\)
One other problem that I'm not sure how to approach is is to find \(\displaystyle \tau\) such that \(\displaystyle |u(t)|<0.1\) for all \(\displaystyle \tau>t\)
The only way I can see how to do this calculation is to say (using the book's equation from above):
\(\displaystyle 0.1=\frac{32}{\sqrt255}e^{-\frac{t}{16}} cos\left(\frac {\sqrt255}{16}t -0.0625\right)\)
\(\displaystyle \frac{0.1\sqrt{255}}{32}=e^{-\frac{t}{16}}cos\left(\frac{\sqrt255}{16}-0.0625\right)\)
However, from this I can see no way to isolate t.