Exercise:
Three masses M m_ and m_ are coupled with springs with elastic constants k_ and k_ as shown in the figure. center includegraphicswidthcm#image_path:largand-small-blocks-# center abcliste abc Derive the differential s and the corresponding coefficient matrix. abc Derive the eigenvalues and eigenvectors for the special case with m_m_m and k_k_. Describe the normal modes. abcliste

Solution:
abcliste abc The differential s are M ddot x k_ x_-x + k_ x_-x -k_+k_ x + k_ x_ + k_ x_ m_ ddot x_ -k_ x_-x k_ x - k_ x_ m_ ddot x_ -k_ x_-x k_ x - k_ x_ It follows for the coefficient matrix bf A pmatrix -frack_+k_M & frack_M & frack_M frack_m_ & -frack_m_ & frack_m_ & & -frack_m_ pmatrix abc The coefficient matrix for the special case is bf A pmatrix -frackM & frackM & frackM frackm & -frackm & frackm & & -frackm pmatrix The eigenvalues are the zeros of the characteristic polynomial: det pmatrix -frack+MlambdaM & frackM & frackM frackm & -frack+mlambdam & frackm & & -frack+mlambdam pmatrix frac-k+Mlambdak+mlambda^+k^k+mlambdaMm^ frac-k+Mlambdak^+kmlambda+m^lambda^+k^+k^mlambdaMm^ fraccancel-k^-k^mlambda-km^lambda^-k^Mlambda-kMmlambda^-Mm^lambda^cancel+k^+k^mlambdaMm^ Longrightarrow -lambdaleftMm^lambda^+kmM+mlambda+k^ M+mright The trivial eigenvalue is lambda_ and the nontrivial eigenvalues are the solutions of the quadratic Mm^lambda^+kmM+mlambda+k^ M+m so lambda_ frac-kmM+m pm sqrtk^m^M+m^-k^Mm^M+mMm^ -frackMmleftM+m pm sqrtM+m^-MM+mright -frackMmleftM+m pm sqrtM^+Mm+m^-M^-Mmright -frackM+mpm mMm Longrightarrow lambda_ -frackm lambda_ -frackM+mMm The angular frequencies of the normal modes are omega_ omega_ sqrtfrackm omega_ sqrtfrackM+mMm and the corresponding eigenvectors vecv_ pmatrix pmatrix vecv_ pmatrix - pmatrix vecv_ pmatrix -fracmM pmatrix The first normal mode is not an oscillation but a uniform motion of all three masses. In the second normal mode the large mass is at rest while the small masses oscillate in opposite directions. It makes sense that the frequency does not dep on the large mass M. In the third normal mode the two small masses oscillate in sync and in the opposite direction of the large mass. The amplitudes are such that the centre of mass of the system remains at rest. abcliste