Exercise:
Two mathematical pula are coupled through a spring connection the two pulum masses. The motion of the system can be described by the system of differential s ddotphi_ -fracgellphi_+fraci_-phi_ ddotphi_ -fracgellphi_+fraci_-phi_ where phi_ and phi_ are the deflection angles of the two pula ell the length and m the mass of the pula and k the elastic constant of the spring. abcliste abc Derive the coefficient matrix. abc Calculate the eigenvalues and eigenvectors of the system. Discuss the normal modes of the system. abc Calculate the angular frequencies of the normal modes for elllO mmO and kkO. What can you observe if one of the pula is started with a push? abcliste

Solution:
abcliste abc A pmatrix -fracgell-frackm & frackm frackm & -fracgell - frackm pmatrix pmatrix -omega_g^-omega_k^ & omega_k^ omega_k^ & -omega_g^-omega_k^ pmatrix with omega_g^g/ell and omega_k^k/m. abc The eigenvalues turn out to be lambda_ -omega_g^ lambda_ -omega_g^-omega_k^ and the corresponding eigenvectors are hat x_ pmatrix pmatrix hat x_ pmatrix - pmatrix The first normal symmetrical mode corresponds to the two pula oscillating in sync. In this mode the spring does not affect the oscillation so the frequency is that of the free mathematical pulum. The second antisymmetrical mode corresponds to the two pula oscillating in opposite directions. The spring is stretched and compressed and therefore increases the restoring force leading to a higher oscillation frequency. abc The angular frequency for the symmetrical normal mode is omega_ omgF sqrtfracncgl resultomgP and for the antisymmetrical normal mode omega_ ombF sqrtfracncgl+timesfrackm resultombP A general solution is a superposition of the two normal modes. Since the two frequencies are similar the resulting motion will show a beats pattern. abcliste