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Here is one task below. How to solve equation $$ m\ddot {x} + ax = F(t), x(0) = \dot x (0) = 0 $$ in quadratures by using two methods? I tried to create a system of equations $$ \begin{matrix} \dot v = F(t) - w_{0}^{2}x \\ \dot x = v \\ \end{matrix}, $$ but I don't know, what to do next without using some vector $\varepsilon = v + \alpha x$. So, i know only one way. Can you help me with the second way? |
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You might not like this answer, but you want to solve for $x\left(t\right)$ in $$ m \ddot{x} + \omega^2 x = F\left(t\right), $$ with $\dot{x}\left(0\right) = x\left(0\right) = 0$. A Laplace transform gives you $$ x\left(t\right) = \mathcal{L}^{-1}\left\{\frac{\mathcal{L} \left[F\left(t\right)\right] \left(s\right)}{m s^2 + \omega^2}\right\}, $$ where we assume $\mathcal{L} \left[F\left(t\right)\right]\left(s\right)$ exists and $F\left(t\right) = 0$ for $t < 0$. Here, $$ \mathcal{L} = \int_{0}^{\infty} dt \ e^{-s t}, $$ $$ \mathcal{L}^{-1} = \frac{1}{2 \pi i}\int_{\mathcal{L}} ds \ e^{s t}, $$ and $$ \int_{\mathcal{L}} ds $$ denotes integration in the complex $s$ plane along the Laplace contour $\mathcal{L}$. The integral is then evaluated using closure and residue techniques. |
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First solve $$ m\ddot {x} + ax = \delta(s) $$ With BC $$x(-\infty) = 0$$ $$\dot x(-\infty) = 0$$ The solution will be sine wave starting at t=s. (zero before that) Lets call this solution G(s) Then integrate this solution to find $$x(t) = \int_0^t F(s)G(s) ds$$ |
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A lazy way to solve this problem would be to use the same method, but with Matrix calculation...it's basically the same. Or you can solve $$ m\ddot {x} + ax = 0 $$ using classical methods, then use the variation of parameters to get the particular solution. Other solutions would be to use an approximation method, but it could be too complicated. What type of equation is this ? My guess would be an excited spring or some kind of regulator. |
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