# Uncertainty Propagation in Coupled Oscillators

I am a senior physics and mathematics major, and this is my last semester. As a result, I am taking advanced physics lab. One of the labs deals with the modal analysis of three spring-mass systems placed vertically as shown in the picture. The masses and spring constants are similar.

I first measure the spring constant for each spring. I then take the mean of their values and uncertainties to have one spring constant. Thus, I get something of the form $\bar{k} \pm \bar{\sigma}$. By using kinematics and Newton's second law, I can use the eigenvalue problem to write everything down more conveniently such that we have:

$$\begin{pmatrix} \frac{-k_{1}-k_{2}}{m_{1}} & \frac{k_{2}}{m_{1}} & 0 \\ \frac{k_{2}}{m_{2}} & \frac{-k_{2}-k_{3}}{m_{2}} & \frac{k_{3}}{m_{2}} \\ 0 & \frac{k_{3}}{m_{3}} & \frac{-k_{3}}{m_{3}} \end{pmatrix} \begin{pmatrix} x_{1} \\ x_{2} \\ x_{3} \end{pmatrix} = - \omega^{2}\begin{pmatrix} x_{1} \\ x_{2} \\ x_{3} \end{pmatrix}$$

I then set all k values and their uncertainties to $\bar{k} \pm \bar{\sigma}$.

My main problem is doing the error propagation for the matrix since I also have an uncertainty to deal with. Clearly, I must end up with three different eigenvalues/eigenfrequencies that have a value and an uncertainty, but I don't know how to do the error propagation at all. I am stuck badly and, while I have two weeks to turn in my report, I would like to have all the error propagation finished ASAP.

• Initially try a brute force method. Estimate an error for the $k$ values, $k_1\pm \sigma_1, k_2\pm \sigma_2$ and $k_3\pm \sigma_3$ Work out the eigenvalues using the values of $k_1, k_2$ and $k_3$. Now work out the eigenvalues with $k_1+\sigma_1, k_2+\sigma_2$ and $k_3+\sigma_3$, then $k_1+\sigma_1, k_2+\sigma_2$ and $k_3-\sigma_3$, etc and see what you get. – Farcher Feb 22 '17 at 11:17