 Impulse on an end mass of a free symmetric three-mass chain

For end masses $m$, central mass $M$ and two equal springs of stiffness $k$, use the normal modes $v_0=(1,1,1{)}^T$, $v_s=(1,0,-1{)}^T$ and $v_a=(1,-2m/M,1{)}^T$. Their squared frequencies are $0$, ${\omega }_s^2=k/m$ and ${\omega }_a^2=k/m+2k/M$. The prescribed velocity decomposes asThusThe zero mode is a uniform center-of-mass motion, with conserved total momentum $mu$ and velocity $mu/(M+2m)$. The other two modes have fixed centre of mass. Dividing their velocity coefficients by their nonzero frequencies gives the sinusoidal displacement amplitudes; for the zero mode the correct time dependence is linear rather than a bounded sine. This distinguishes translation of the complete system from its internal vibration.