 Solution

To specify “initial motion” unambiguously, choose each mode to pass through equilibrium at $t=0$ with $\dot{\mathbf{x}}(0)$ proportional to its listed mode shape. The arrows in the figure then describe relative initial velocities:

Reversing every arrow changes only the common phase. A quarter-cycle later these same ratios describe the extremal displacements, with zero instantaneous velocity.

Now replace the central mass by $M$, leaving the outer masses equal to $m$. The antisymmetric mode shape $q=(1,0,-1{)}^T$ gives zero net force on the central mass, since $k{x}_1+k{x}_3=0$. Each outer mass still obeys $m\ddot{x}=-2kx$. ConsequentlyThis middle-mass-independent mode of a fixed three-mass spring chain is exact, not merely a large-$M$ approximation. The other two modes move the central mass and generally change. For example, their symmetric-sector frequencies satisfywhich explicitly depends on $M$.