Exercise:
Three Masses are connected to each other and the walls through springs with identical spring constants k. The masses are m left and right and Mm centre. center includegraphicswidthmm#image_path:masses-and-springs-# center The differential s for the three masses are given by * m ddot x_ -k x_ + k x_ - x_ M ddot x_ -k x_ - x_ + k x_ - x_ m ddot x_ -k x_ - x_ - k x_ The eigenvalues are * lambda_ -omega_^ lambda_ -fracpmsqrtomega_^ with omega_^k/m. abcliste abc Derive the expressions for the angular frequencies of the three normal modes and order them from lowest to highest frequency. Describe the behaviour of the system for a solution that corresponds to the superposition of the two fastest normal modes. hfill abc Describe the normal modes in words and assign them to the three eigenvalues. hfill abc Derive the characteristic polynomial for the eigenvalues of the coefficient matrix. Simplify it as much as possible and verify the eigenvalues given above. hfill abcliste

Solution:
abcliste abc The angular frequencies follow directly from the eigenvalues: omega sqrt-lambda Longrightarrow omega_ sqrtomega_ approx omaPtimesomega_ omega_ sqrtfrac+sqrtomega_ approx ombPtimesomega_ omega_ sqrtfrac-sqrtomega_ approx omcPtimesomega_ It follows that omega_omega_omega_. vspacemm textbfRemark: The normal frequencies omega_ and omega_ can also be simplified to get rid of the double root: omega_ fracsqrt+ omega_ fracsqrt- This makes it easier to see the relations between the three normal frequencies. vspacemm For a superposition of normal modes and angular frequencies omega_ and omega_ the energy is transferred between the two modes at a frequency corresponding to the difference of omega_ and omega_ beats. abc The slowest mode omega_ corresponds to all three masses moving in sync amplitude of M larger than of m. Normal mode omega_ corresponds to the two smaller masses moving in opposite directions with equal amplitudes and the middle mass staying at rest. In the fastest mode the outer masses move in sync while the middle mass moves in the opposite direction. abc The coefficient matrix is bf M pmatrix -omega_^ & omega_^ & omega_^/ & -omega_^ & omega_^/ & omega_^ & - omega_^ pmatrix The characteristic polynomial for the eigenvalues is given by detM-lambda I abspmatrix -omega_^-lambda & omega_^ & omega_^/ & -omega_^ - lambda & omega_^/ & omega_^ & - omega_^ - lambda pmatrix -omega_^+lambda^omega_^+lambda+fracomega_^omega_^+lambda -omega_^+omega_^ lambda + lambda^omega_^+lambda+omega_^+omega_^ lambda -omega_^-omega_^ lambda - omega_^ lambda^ - lambda^ To verify whether the given eigenvalues are correct we can calculate the resulting polynomial: lambda + omega_^&lambda + frac+sqrt omega_^lambda + frac-sqrt omega_^ lambda+omega_^lambda^+omega_^lambda+omega_^ lambda^ + omega_^lambda^ + omega_^ lambda + omega_^ This is the same es the expression above but for an irrelevant global sign. abcliste