Coupled oscillators are systems where the motion of one oscillator influences the other. A standard example is two identical masses
\(m\) connected to fixed walls by springs of constant \(k\), with an additional coupling spring \(k_c\) between the masses.
Let \(x_1(t)\) and \(x_2(t)\) be the displacements from equilibrium. For small oscillations, Hooke’s law gives a linear system:
\[
m x_1'' = -(k+k_c)x_1 + k_c x_2,\qquad
m x_2'' = k_c x_1 -(k+k_c)x_2.
\]
In matrix form,
\[
m\mathbf{x}'' + K\mathbf{x}=0,\quad
K=
\begin{bmatrix}
k+k_c & -k_c \\
-k_c & k+k_c
\end{bmatrix}.
\]
Normal modes. A normal mode is a motion where all parts oscillate sinusoidally at a single frequency with fixed amplitude ratios:
\[
\mathbf{x}(t)=\mathbf{u}\cos(\omega t).
\]
Substituting into \(m\mathbf{x}''+K\mathbf{x}=0\) yields the eigenvalue problem
\[
(K-m\omega^2 I)\mathbf{u}=0.
\]
Nontrivial solutions exist only when
\[
\det(K-m\omega^2 I)=0.
\]
The resulting eigenvalues give \(\omega^2\), and the eigenvectors give the mode shapes \(\mathbf{u}\).
Two-mode result for identical masses. For this symmetric two-mass system, the normal modes are:
- In-phase mode (\(x_1=x_2\)): the coupling spring does not stretch, so the effective stiffness is \(k\) for each mass.
This yields
\[
\omega_1=\sqrt{\frac{k}{m}}.
\]
- Out-of-phase mode (\(x_1=-x_2\)): the coupling spring stretches twice as much, increasing the restoring force.
The effective stiffness becomes \(k+2k_c\), giving
\[
\omega_2=\sqrt{\frac{k+2k_c}{m}}.
\]
The corresponding normalized mode vectors can be written as
\[
\mathbf{u}_1=\frac{1}{\sqrt{2}}\begin{bmatrix}1\\1\end{bmatrix},\qquad
\mathbf{u}_2=\frac{1}{\sqrt{2}}\begin{bmatrix}1\\-1\end{bmatrix}.
\]
Why this matters. Normal modes explain how energy sloshes between coupled oscillators and underlie applications like molecular vibrations,
lattice phonons in solids, and mechanical structures. University-level extensions generalize to \(n\) coupled oscillators, producing \(n\) modes
found via eigenvalues/eigenvectors of the mass and stiffness matrices.
