I am a senior physics and mathematics major, and this is my last semester. As a result, I am taking advanced physics lab, which feels more like a grad school experiment than an undergrad. 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.(adsbygoogle = window.adsbygoogle || []).push({});

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{equation}

\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}

\end{equation}

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.

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# A Uncertainty Propagation in Coupled Oscillator

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