Exercise:
Two masses M and m are coupled through a spring with elastic constant k see figure. center includegraphicswidthcm#image_path:two-masses-with-spring-# center The differential s for this system are M ddot x_ kx_-x_ m ddot x_ kx_-x_ abcliste abc Derive formal expressions for the frequencies of the normal modes of the system. abc Ase that Mm. Make an educated guess for the eigenvectors describing the normal modes and verify your prediction with a calculation. abcliste

Solution:
abcliste abc The coefficient matrix is bf A pmatrix -frackM & frackM frackm & -frackm pmatrix pmatrix -omega_s^ & omega_s^ omega_f^ & -omega_f^ pmatrix with omega_s^ frackM quad textrmslow mode omega_f^ frackm quad textrmfast mode With the trace and determinant being tau -omega_s^-omega_f^ Delta omega_s^omega_f^-omega_s^omega_f^ we find for the eigenvalues lambda_ frac-omega_s^+omega_f^pm omega_s^+omega_f^ Longrightarrow lambda_ lambda_ -omega_s^+omega_f^ The corresponding angular frequencies are omega_ omega_ sqrtomega_s^+omega_f^ sqrtfrackM+frackm sqrtfrackm+MMm abc The first normal mode omega_ corresponds to a uniform translation. The eigenvector is expected to be vec v_ pmatrix pmatrix This can easily be verified: bf Avec v_ pmatrix -omega_s^ & omega_s^ omega_f^ & -omega_f^ pmatrix pmatrix pmatrix pmatrix -omega_s^+omega_s^ omega_f^-omega_f^ pmatrix pmatrix pmatrix vec v_ The second normal mode corresponds to a vibration about the centre of mass. The amplitude of the lighter mass has to be twice as large for the centre of mass to remain in place. For Mm we have omega_f^omega_s^ and the eigenvalue is lambda_ -omega_s^+omega_f^-omega_s^ We expect an eigenvector vec v_ pmatrix - pmatrix To verify the eigenvector we have to calculate bf Avec v_ pmatrix -omega_s^ & omega_s^ omega_f^ & -omega_f^ pmatrix pmatrix - pmatrix omega_s^ pmatrix - & & - pmatrix pmatrix - pmatrix omega_s^ pmatrix - - + pmatrix omega_s^ pmatrix - pmatrix - omega_s^ pmatrix - pmatrix lambda_ vec v_ as expected. abcliste