Past exam of the mathematical methods II course of the University of Oxford / 2016 resit / 11 / e

Release from rest gives $B=D=0$. At $t=0$, $A+2C=d$ and $A-C=0$, so $A=C=d/3$. ConsequentlyThe frequency ratio is exactly two, so both motions repeat after $2\pi /{\omega }_{-}$. These are commensurate normal modes, rather than a pair of nearly equal frequencies producing slow beats.