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Exercise:
Determine the eigenvalues for a coupled two-mass system with m_m m_ m k_K and k_K. Calculate the normal angular frequencies for mmO KKO and kkO

Solution:
The general coefficient matrix for the coupled two-mass system is A pmatrix -frack_+km_ & frackm_ frackm_ & -frack_+km_ pmatrix pmatrix -fracK+km & frackm frackm & -fracK+km pmatrix The trace and determinant of this matrix are tau -fracK+km+fracK+km -fracK+k+K+km -fracK+km Delta fracK+kK+km^-frack^m^ fracK^+Kk+k^-k^m^ fracK^+Kkm^ It follows for the eigenvalues lambda fracleft-fracK+kmpmsqrtfracK+k^m^-fracK^+Kkm^right frac-K+kpmsqrtK^+Kk+k^-K^-Kkm frac-K+kpm km Longrightarrow lambda_ result-fracKm lambda_ result-fracK+km The normal frequencies are omega_ omsF sqrtfractimesKm oms approx resultomsP omega_ omfF sqrtfractimesK+timeskm omf approx resultomfP The eigenvectors are verify! hat x_ pmatrix pmatrix hat x_ pmatrix- pmatrix In the symmetrical mode hat x_ the second mass is twice as heavy but it also experiences twice the force for the same displacement. In the antisymmetrical mode hat x_ the amplitudes for the two masses are no longer the same.
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Exercise:
Determine the eigenvalues for a coupled two-mass system with m_m m_ m k_K and k_K. Calculate the normal angular frequencies for mmO KKO and kkO

Solution:
The general coefficient matrix for the coupled two-mass system is A pmatrix -frack_+km_ & frackm_ frackm_ & -frack_+km_ pmatrix pmatrix -fracK+km & frackm frackm & -fracK+km pmatrix The trace and determinant of this matrix are tau -fracK+km+fracK+km -fracK+k+K+km -fracK+km Delta fracK+kK+km^-frack^m^ fracK^+Kk+k^-k^m^ fracK^+Kkm^ It follows for the eigenvalues lambda fracleft-fracK+kmpmsqrtfracK+k^m^-fracK^+Kkm^right frac-K+kpmsqrtK^+Kk+k^-K^-Kkm frac-K+kpm km Longrightarrow lambda_ result-fracKm lambda_ result-fracK+km The normal frequencies are omega_ omsF sqrtfractimesKm oms approx resultomsP omega_ omfF sqrtfractimesK+timeskm omf approx resultomfP The eigenvectors are verify! hat x_ pmatrix pmatrix hat x_ pmatrix- pmatrix In the symmetrical mode hat x_ the second mass is twice as heavy but it also experiences twice the force for the same displacement. In the antisymmetrical mode hat x_ the amplitudes for the two masses are no longer the same.
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Attributes & Decorations
Branches
Differential equations, Harmonic Oscillations, Linear Algebra
Tags
angular frequency, coupled oscillation, eigenvalue, normal mode
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Difficulty
(2, default)
Points
0 (default)
Language
ENG (English)
Type
Calculative / Quantity
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Decoration
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