Exercise
https://texercises.com/exercise/symmetrical-two-mass-system/
Question
Solution
Short
Video
\(\LaTeX\)
No explanation / solution video to this exercise has yet been created.

Visit our YouTube-Channel to see solutions to other exercises.
Don't forget to subscribe to our channel, like the videos and leave comments!
Exercise:
The coefficient matrix for a symmetrical coupled two-mass system is bf A pmatrix-fracK+km & frackm frackm & -fracK+km pmatrix Derive the eigenvalues and angular frequencies and verify that hat x_ pmatrix pmatrix hat x_ pmatrix - pmatrix are the corresponding eigenvectors.

Solution:
The trace and determinant of the matrix bf A are tau -fracK+km Delta leftfracK+kmright^-leftfrackmright^ fracK^+Kk+k^-k^mfracKK+km It follows for the eigenvalues lambda fractaupmsqrttau^-Delta^ frac-K+kpmsqrtK+k^-KK+km frac-K+kpmsqrtK^+Kk+k^-K^-Kkm frac-K+kpm km Longrightarrow lambda_ result-fracKm lambda_ result-fracK+km The angular frequencies are therefore omega_ sqrt-lambda_resultsqrtfracKm omega_ sqrt-lambda_resultsqrtfracK+km To verify whether hat x_ is an eigenvector we calculate bf A hat x_: pmatrix-fracK+km & frackm frackm & -fracK+km pmatrix pmatrix pmatrix pmatrix-fracK+km+frackm frackm-fracK+km pmatrix pmatrix-fracKm -fracKm pmatrix -fracKmpmatrix pmatrix i.e. hat x_ is the eigenvector for the eigenvalue lambda_. In the same way we can show that hat x_ is the eigenvector for the eigenvalue lambda_: pmatrix-fracK+km & frackm frackm & -fracK+km pmatrix pmatrix - pmatrix pmatrix-fracK+km-frackm frackm+fracK+km pmatrix pmatrix-fracK+km fracK+km pmatrix -fracK+kmpmatrix - pmatrix
Meta Information
\(\LaTeX\)-Code
Exercise:
The coefficient matrix for a symmetrical coupled two-mass system is bf A pmatrix-fracK+km & frackm frackm & -fracK+km pmatrix Derive the eigenvalues and angular frequencies and verify that hat x_ pmatrix pmatrix hat x_ pmatrix - pmatrix are the corresponding eigenvectors.

Solution:
The trace and determinant of the matrix bf A are tau -fracK+km Delta leftfracK+kmright^-leftfrackmright^ fracK^+Kk+k^-k^mfracKK+km It follows for the eigenvalues lambda fractaupmsqrttau^-Delta^ frac-K+kpmsqrtK+k^-KK+km frac-K+kpmsqrtK^+Kk+k^-K^-Kkm frac-K+kpm km Longrightarrow lambda_ result-fracKm lambda_ result-fracK+km The angular frequencies are therefore omega_ sqrt-lambda_resultsqrtfracKm omega_ sqrt-lambda_resultsqrtfracK+km To verify whether hat x_ is an eigenvector we calculate bf A hat x_: pmatrix-fracK+km & frackm frackm & -fracK+km pmatrix pmatrix pmatrix pmatrix-fracK+km+frackm frackm-fracK+km pmatrix pmatrix-fracKm -fracKm pmatrix -fracKmpmatrix pmatrix i.e. hat x_ is the eigenvector for the eigenvalue lambda_. In the same way we can show that hat x_ is the eigenvector for the eigenvalue lambda_: pmatrix-fracK+km & frackm frackm & -fracK+km pmatrix pmatrix - pmatrix pmatrix-fracK+km-frackm frackm+fracK+km pmatrix pmatrix-fracK+km fracK+km pmatrix -fracK+kmpmatrix - pmatrix
Contained in these collections:

Attributes & Decorations
Branches
Differential equations, Harmonic Oscillations, Linear Algebra
Tags
coupled oscillation
Content image
Difficulty
(2, default)
Points
0 (default)
Language
ENG (English)
Type
Calculative / Quantity
Creator by
Decoration
File
Link