Small Oscillations, Normal Modes of Vibration, Coupled Oscillators

Learning Objectives:

• Understand the concept of small oscillations and linearization near equilibrium.
• Learn the definition and significance of normal modes in multi-degree systems.
• Analyze coupled oscillators and determine their normal frequencies and mode shapes.

Key Concepts / Definitions:

• Small Oscillations: Oscillations near equilibrium where the restoring forces can be approximated as linear.
• Normal Modes: Independent patterns of oscillation in which all parts of the system oscillate at the same frequency.
• Coupled Oscillators: Systems where two or more oscillating components influence each other through interaction forces.

Small Oscillations

In mechanical systems, small oscillations occur when the system is displaced slightly from its equilibrium position. If the displacements are small, we can use a Taylor expansion to approximate the potential energy:

Let the potential energy near equilibrium be:

\[V(q_1, q_2, ..., q_n) \approx V_0 + \frac{1}{2} \sum_{i,j} V_{ij} q_i q_j\]

Here, $q_i$ are the generalized coordinates, and \(V_{ij} = \left.\frac{\partial^2 V}{\partial q_i \partial q_j}\right|_{\text{eq}}\) is the Hessian matrix of second derivatives evaluated at equilibrium.

The kinetic energy is usually:

\[T = \frac{1}{2} \sum_{i,j} T_{ij} \dot{q}_i \dot{q}_j\]

The Lagrangian becomes:

\[L = T - V = \frac{1}{2} \sum_{i,j} \left( T_{ij} \dot{q}_i \dot{q}_j - V_{ij} q_i q_j \right)\]

This leads to the equations of motion:

\[\sum_j \left( T_{ij} \ddot{q}_j + V_{ij} q_j \right) = 0\]

Normal Modes of Vibration

We look for solutions of the form:

\[q_j(t) = a_j e^{i\omega t}\]

Substituting into the equation of motion gives:

\[\sum_j \left( -\omega^2 T_{ij} + V_{ij} \right) a_j = 0\]

This is a generalized eigenvalue problem:

\[\left( V - \omega^2 T \right) \vec{a} = 0\]

Non-trivial solutions exist when:

\[\det(V - \omega^2 T) = 0\]

Solving this gives the normal frequencies $\omega_k$ and associated normal modes $\vec{a}^{(k)}$.

Coupled Oscillators

Consider two identical masses $m$ connected by three springs (spring constant $k$ for outer and $k’$ for middle spring):

Let $x_1$ and $x_2$ be the displacements from equilibrium. The Lagrangian is:

\[T = \frac{1}{2} m \dot{x}_1^2 + \frac{1}{2} m \dot{x}_2^2\] \[V = \frac{1}{2} k x_1^2 + \frac{1}{2} k x_2^2 + \frac{1}{2} k' (x_1 - x_2)^2\]

Expanding the potential:

\[V = \frac{1}{2}(k + k') x_1^2 + \frac{1}{2}(k + k') x_2^2 - k' x_1 x_2\]

Equations of motion:

\[m \ddot{x}_1 = -(k + k') x_1 + k' x_2\] \[m \ddot{x}_2 = k' x_1 - (k + k') x_2\]

Assume solutions:

\[x_j(t) = a_j e^{i\omega t}\]

We get:

\[\begin{bmatrix} k + k' - m\omega^2 & -k' \\ -k' & k + k' - m\omega^2 \end{bmatrix} \begin{bmatrix} a_1 \\ a_2 \end{bmatrix} = 0\]

Solving the determinant gives:

\[\omega_1 = \sqrt{\frac{k}{m}}, \quad \omega_2 = \sqrt{\frac{k + 2k'}{m}}\]

Corresponding normal modes:

• Mode 1: $a_1 = a_2$ (in-phase)
• Mode 2: $a_1 = -a_2$ (out-of-phase)

Solved Examples:

• Example 1:
  Problem: Two equal masses connected by a spring $k’$ and attached to walls by springs $k$. Find the normal modes.
  Solution:
  This is similar to the coupled oscillator case above. The normal frequencies are:

  \[\omega_1 = \sqrt{\frac{k}{m}}, \quad \omega_2 = \sqrt{\frac{3k}{m}}\]

  Modes: $x_1 = x_2$ and $x_1 = -x_2$.

• Example 2:
  Problem: A mass $m$ connected to two fixed walls with identical springs $k$. Find the frequency of oscillation.
  Solution:
  The effective force is:

  \[F = -2k x \Rightarrow m \ddot{x} = -2k x\]

  So:

  \[\omega = \sqrt{\frac{2k}{m}}\]

Important Points / Summary:

• Small oscillations allow linear approximation of complex systems near equilibrium.
• Normal modes simplify multi-body motion into independent harmonic oscillators.
• In coupled oscillators, interaction between bodies leads to splitting of frequencies.

Practice Questions:

• Short Answer:
  1. What are normal modes in a mechanical system?
  2. Define small oscillations and explain their significance in classical mechanics.
• Numerical:
  1. Two masses $m$ connected by a spring $k’$ and to walls with springs $k$. Find normal frequencies.
  2. A system has $T = \frac{1}{2}m (\dot{x}^2 + \dot{y}^2)$ and $V = \frac{1}{2}k (x^2 + y^2 + 2xy)$. Find the normal modes.
• MCQs:
  1. In normal mode motion:
     • (A) All parts move with different frequencies
     • (B) All parts move independently
     • (C) All parts move with the same frequency
     • (D) Motion is always in phase
       Answer: (C)
  2. The condition for small oscillations to be valid is:
     • (A) Displacement is arbitrary
     • (B) Restoring force is constant
     • (C) Displacement is near equilibrium and force is approximately linear
     • (D) Acceleration is zero
       Answer: (C)