Two Coupled Oscillators and Normal Modes

The Idea: Independent Patterns of Motion

Imagine two identical masses connected by three springs: one spring between the left wall and mass 1, one between mass 1 and mass 2, and one between mass 2 and the right wall. If you pull mass 1 and release it, mass 2 will eventually start moving too. The motion looks complicated because the masses are constantly exchanging energy. However, there are two special ways the system can move where this complexity disappears. In the first special case, both masses move in the same direction with the same amplitude. In the second special case, the masses move in opposite directions with the same amplitude. In both cases, the distance between the masses changes in a simple, predictable way, and the system behaves exactly like a single simple harmonic oscillator. These special patterns are the normal modes. Any arbitrary motion of the two masses can be described as a combination (superposition) of these two modes.

Equations of Motion and Matrix Form

Let the two masses be \( m \) and the spring constants be \( k_1 \) (left), \( k_2 \) (middle), and \( k_3 \) (right). Let \( x_1 \) and \( x_2 \) be the displacements from equilibrium. Applying Newton's second law to each mass gives:

For mass 1: \( m\ddot{x}_1 = -k_1 x_1 + k_2(x_2 - x_1) \)

For mass 2: \( m\ddot{x}_2 = -k_3 x_2 + k_2(x_1 - x_2) \)

Rearranging these into a standard linear form, we can write the system in matrix notation. The mass matrix \( \mathbf{M} \) is diagonal because the masses are independent, and the stiffness matrix \( \mathbf{K} \) is symmetric because the force on one mass due to the other is equal and opposite. The equation of motion is:

where \( \mathbf{x} = \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} \), \( \mathbf{M} = m \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \), and \( \mathbf{K} = \begin{pmatrix} k_1 + k_2 & -k_2 \\ -k_2 & k_2 + k_3 \end{pmatrix} \). This matrix form is powerful because it allows us to use linear algebra to find the solutions.

Normal Modes and Eigenvalues

We seek solutions where both masses oscillate at the same frequency \( \omega \). We assume a solution of the form \( \mathbf{x}(t) = \mathbf{a} e^{i\omega t} \), where \( \mathbf{a} \) is a constant vector describing the relative amplitudes and phases. Substituting this into the matrix equation yields the generalized eigenvalue problem:

The values of \( \omega^2 \) that satisfy this equation are the eigenvalues, and the corresponding vectors \( \mathbf{a} \) are the eigenvectors. The eigenvectors define the normal modes. For the symmetric case where \( k_1 = k_3 = k \), the two normal modes are: 1) The symmetric mode where \( x_1 = x_2 \), with frequency \( \omega_1 = \sqrt{k/m} \). 2) The antisymmetric mode where \( x_1 = -x_2 \), with frequency \( \omega_2 = \sqrt{(k+2k_2)/m} \) (which is \( \sqrt{3k/m} \) when all three springs are equal). The antisymmetric mode has a higher frequency because the middle spring is compressed or stretched, adding to the restoring force.

Superposition and General Motion

Since the equations of motion are linear, any linear combination of solutions is also a solution. Therefore, the general motion of the system is the sum of the two normal modes:

The coefficients \( A_i \) and \( B_i \) are determined by the initial conditions (initial positions and velocities). Physically, this means that if you start the system with an arbitrary configuration, it will oscillate as a mixture of the two fundamental patterns. The energy sloshes back and forth between the two masses when both modes are excited and their frequencies are close (weak coupling), a phenomenon known as beats, which we will explore in the next lecture.