- #1
HermitianField
- 8
- 0
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.
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.
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.