= Solution
A <normal mode> has all <displacement> coordinates proportional to a common time dependence, with fixed relative amplitudes. For a nonzero <frequency> the time dependence is sinusoidal. A zero-frequency mode is a free <displacement> or uniform drift rather than an <oscillation>.
Use the printed mass ordering $M,m,M$. <Displacements> are measured from the unstretched <equilibrium> positions, giving
$$
M\ddot x_1=k(x_2-x_1),\qquad
m\ddot x_2=k(x_1-2x_2+x_3),\qquad
M\ddot x_3=k(x_2-x_3).
$$
Equivalently,
$$
\boxed{\mathsf M\ddot{\mathbf x}+\mathsf K\mathbf x=0,\quad
\mathsf M=\operatorname{diag}(M,m,M),\quad
\mathsf K=k\begin{pmatrix}1&-1&0\\-1&2&-1\\0&-1&1\end{pmatrix}.}
$$
This is the <generalized eigenvalue problem> for <normal modes of three masses connected by two equal springs>. Substitute $\mathbf x=\mathbf u e^{i\omega t}$ to obtain $\mathsf K\mathbf u=\omega^2\mathsf M\mathbf u$. <Reflection symmetry> gives an antisymmetric <vector> $(1,0,-1)$ and <symmetric> <vectors> $(1,b,1)$. The former has $\omega^2=k/M$. For a <symmetric> <vector>, the end and centre equations give $M\omega^2=k(1-b)$ and $m\omega^2b=2k(b-1)$. Their solutions are $b=1$, $\omega=0$, or $b=-2M/m$, $\omega^2=k/M+2k/m$. Hence
$$
\boxed{\begin{array}{c|c|l}
\mathbf u&\omega^2&\text{relative motion}\\\hline
(1,1,1)&0&\text{rigid translation, neither spring stretched}\\
(1,0,-1)&k/M&\text{end masses opposite, centre fixed}\\
(1,-2M/m,1)&k/M+2k/m&\text{ends together, centre opposite}.
\end{array}}
$$
These are all three <eigenmodes>. The last two keep the <centre of mass> fixed and are <orthogonal> in the mass-weighted <inner product>. The general zero-mode contribution is $(A+Bt)(1,1,1)$, accounting for both a centre-of-mass shift and a conserved total <momentum>. There are no fixed end springs in this system.
\Image[/past-exam-of-the-physics-b-course-of-the-university-of-cambridge/2010/paper-2a-three-mass-modes.png]
{title=Free-chain mode shapes}
The original amplitude profiles illustrate $M=2m$; the <symmetric> internal shape is rescaled for display. The formula above gives every positive mass ratio.
Two further physical examples are \b[standing-wave modes of a string fixed at both ends], with spatial shapes $\sin(n\pi x/L)$ and <frequencies> $n\pi c/L$, and \b[electrical <normal modes> of two coupled <LC circuits>], in which the <capacitor> charges oscillate at fixed relative amplitudes. Independent <normal mode coordinates> simplify both systems in the same way as the spring-chain <eigenvectors>.
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