11 May 2026
Small Oscillations and Normal Modes
Small oscillations, normal modes of vibration, and the coupled oscillator problem.
This note covers the final syllabus topics: small oscillation, normal mode of vibration, and coupled oscillator.
Equilibrium and small displacement
Let $q_i^{(0)}$ be an equilibrium point. Write small displacements as
\[\eta_i=q_i-q_i^{(0)}.\]At equilibrium,
\[\left(\frac{\partial V}{\partial q_i}\right)_0=0.\]For small oscillations, expand the potential up to second order:
\[V=V_0+\frac12\sum_{ij}K_{ij}\eta_i\eta_j.\]The kinetic energy is approximated as
\[T=\frac12\sum_{ij}M_{ij}\dot\eta_i\dot\eta_j.\]Here $M$ is the mass matrix and $K$ is the stiffness matrix.
Equations of motion
The Lagrange equations give
\[\boxed{ \sum_j M_{ij}\ddot\eta_j+\sum_jK_{ij}\eta_j=0. }\]Assume a normal-mode solution
\[\eta_i=a_i e^{i\omega t}.\]Then
\[\boxed{ (K-\omega^2M)\mathbf a=0. }\]For nontrivial $\mathbf a$,
\[\boxed{ \det(K-\omega^2M)=0. }\]This equation gives the normal frequencies.
Normal modes
A normal mode is a pattern of motion in which all coordinates oscillate with the same frequency and fixed relative amplitudes.
For each normal frequency $\omega_r$, the corresponding eigenvector $\mathbf a^{(r)}$ gives the mode shape.
The general small motion is a superposition of normal modes.
Coupled oscillator example
Consider two equal masses $m$ connected by springs, with coordinates $x_1$ and $x_2$. Let each mass be attached to a wall by spring constant $k$, and the coupling spring between them have constant $k_c$.
The equations are
\[m\ddot x_1= -kx_1-k_c(x_1-x_2),\] \[m\ddot x_2= -kx_2-k_c(x_2-x_1).\]In matrix form,
\[m\begin{pmatrix} \ddot x_1\\ \ddot x_2 \end{pmatrix} + \begin{pmatrix} k+k_c&-k_c\\ -k_c&k+k_c \end{pmatrix} \begin{pmatrix} x_1\\ x_2 \end{pmatrix}=0.\]Assume $x_i=a_i e^{i\omega t}$. The normal frequencies are
\[\boxed{ \omega_1^2=\frac{k}{m}, \qquad \omega_2^2=\frac{k+2k_c}{m}. }\]For $\omega_1$, the mode is
\[\mathbf a_1\propto \begin{pmatrix} 1\\ 1 \end{pmatrix},\]so both masses move in phase.
For $\omega_2$, the mode is
\[\mathbf a_2\propto \begin{pmatrix} 1\\ -1 \end{pmatrix},\]so the masses move out of phase.
Main points
- Small oscillations are studied by expanding near equilibrium.
- The quadratic part of $V$ determines restoring forces.
- Normal frequencies follow from $\det(K-\omega^2M)=0$.
- A normal mode has fixed amplitude ratios and one frequency.
- Coupled oscillators split into in-phase and out-of-phase modes.
Practice questions
- Define small oscillation about equilibrium.
- Derive the matrix equation $M\ddot\eta+K\eta=0$.
- What is a normal mode?
- Derive the normal frequencies of two coupled equal masses.
- Explain the physical difference between in-phase and out-of-phase modes.
Discussion