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I have the following differential equation governing the motion of an externally forced damped mass–spring oscillator: $$ x(t)=\frac{1}{m\beta}(g*f)(t)=\frac{1}{m\beta}\int_0^te^{-bv/2m}\sin\beta v\cdot f(t-v)\,dv. $$ How would I go about answering the following question, I've been stuck on it for some time:

c) Show that the motion $x(t)$ is bounded under these circumstances by: $$ |x(t)|\le At/\beta m $$ and also by $$ |x(t)|\le 2A/\beta b $$ Said circumstances are: the system is underdamped, it starts from rest, the forcing term is bounded ($|f(t)|< A$ for all $t$). Any insights would be greatly appreciated.

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For the first bound:

$$\big|x(t)\big|\leq \frac{1}{m\beta}\int_{0}^{t}\big|e^{-bv/2m}\sin{\beta v}\cdot f(t-v)\big|\,dv\leq\frac{1}{m\beta}\int_{0}^{t}A\,dv = \frac{At}{\beta m},\qquad(t>0)$$

using the fact that $\big|f(t)\big| <A$ and both $\sin{\beta v}$ and $e^{-bv/2m}$ are bounded above by 1 for $t>0$. I'm not sure I agree with the statement of the second bound, however, because $v$ is just a dummy variable of integration and should integrate out to get something in terms of $t$ regardless of how you estimate the integrand.

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  • $\begingroup$ The reason you don't agree with the second bound is because I had a typo hehe. Edited the original question. $\endgroup$ – Lucif3r May 3 '15 at 11:57

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