Exercise:
For the system of two coupled masses in the figure below: abcliste abc Derive the system of differential s and the corresponding coefficient matrix. abc Determine the eigenvalues for the case m_m_m and k_k_k and verify that the eigenvectors are hat x_ pmatrix -frac+sqrt pmatrix hat x_ pmatrix -frac-sqrt pmatrix abc Determine the eigenvalues and the corresponding angular frequencies for m_maO m_mbO k_kaO and k_kbO. abcliste center includegraphicswidthtextwidth#image_path:two-masses-onsided-# center

Solution:
abcliste abc The system of differential s is given by m_ ddot x_ -k_ x_+k_x_-x_ -k_+k_x_+k_ x_ m_ ddot x_ k_ x_ - k_ x_ It follows for the coefficient matrix bf A pmatrix -frack_+k_m_ & frack_m_ frack_m_ & -frack_m_ pmatrix abc The coefficient matrix for m_m_m and k_k_k is bf A pmatrix-frackm & frackm frackm & -frackm pmatrix The trace and determinant of this matrix are tau -frackm Delta frack^m^-frack^m^frack^m^ and the eigenvalues are thus lambda fracmleft-kpmsqrtk^-k^right frac-pmsqrtfrackmfrac-pmsqrtomega_^ with omega_^k/m. vspacemm Applying the matrix bf A to the first eigenvector hat x_ yields bf Ahat x_ pmatrix-frackm & frackm frackm & -frackm pmatrix pmatrix -frac+sqrt pmatrix omega_^pmatrix+sqrt+ -frac+sqrt- pmatrix omega_^ pmatrix +sqrt -frac+sqrtpmatrix -frac+sqrt omega_^pmatrix-frac+sqrt pmatrix where we use the following identiy in the last step: frac+sqrtfrac+sqrt frac+sqrt+sqrt+ frac+sqrt+sqrt In the same way it can be verified that hat x_ is the eigenvector for the second eigenvalue. abc The coefficient matrix is given by bf A pmatrix aP & bP cP & dP pmatrix where the unit radiansquaredperssquared is suppressed. abcliste It follows for the eigenvalues lambda fraclefttaPpmsqrttaP^-timesdePright lambda_ resultlaaP lambda_ resultlabP and for the angular frequencies omega_ omaF resultomaP omega_ ombF resultombP