Normal Modes of Coupled Oscillators: 3-Mass Spring Chain and Matrix Method

This lecture from MIT OpenCourseWare introduces the analysis of coupled oscillators by identifying normal modes. Using a three-mass chain connected by springs to fixed walls, the instructor derives the equations of motion, casts them into a mass and stiffness matrix form, and solves the eigenvalue problem to find the system’s normal modes. Three distinct modes emerge: Mode A where all masses move in phase with a common frequency, Mode B where outer masses move oppositely to the middle, and Mode C which corresponds to a rigid, zero-frequency translation. The video emphasizes that any complex motion is a superposition of these simple harmonic modes, and outlines the general procedure that scales to more masses. It also previews resonance and driving forces as future topics.

Introduction to Coupled Oscillators and Normal Modes

The video opens with a motivation: when several oscillators interact, their collective motion can be understood by decomposing it into normal modes. Normal modes are special motions in which each part of the system oscillates with the same frequency and phase, making the overall behavior easier to analyze. This concept is then applied to a concrete system: three masses connected by springs and anchored to fixed walls by end springs. The speaker emphasizes that even though the motion may look complex, it can be expressed as a linear combination of a small set of normal modes.

The Three Mass Spring System: Setup and Equations of Motion

The instructor defines the coordinates X1, X2, X3 as displacements from equilibrium. For the setup with equal springs and end walls, the left mass experiences forces from two springs, the middle mass from two adjacent springs, and the right mass from two springs as well. Writing Newton's laws for each mass yields three coupled second-order differential equations. The analysis stresses careful sign conventions to ensure the forces are directed back toward equilibrium when displacements grow.

Matrix Formulation: Mass and Stiffness Matrices

To simplify the coupled equations, the system is recast in matrix form M X'' = -K X, where M is the diagonal mass matrix and K is the stiffness matrix describing the spring couplings. The speaker demonstrates how to identify M and K from the physical setup, noting that the matrices encode the mass distribution and the communication between components through the springs. This matrix formulation paves the way for a systematic normal-mode analysis.

Normal Mode Analysis: The Eigenvalue Problem

The key step is to seek solutions of the form X(t) = A cos(ω t + φ), or in complex form X(t) = Z e^{i ω t}, where all components share the same frequency ω and phase. Substituting this ansatz into the matrix equation leads to the eigenvalue problem (M^{-1}K − ω^2 I) A = 0. Nontrivial solutions require the determinant to vanish, giving the characteristic equation whose roots are ω^2. The eigenvectors A give the relative amplitudes of each mass in a given normal mode.

Three Normal Modes for the 3-Mass Chain

For the described three-mass chain, the lecturer identifies three normal modes (A, B, and C):

• : all three masses oscillate with the same frequency and phase, i.e., X1, X2, and X3 move together. The amplitudes are equal in magnitude but may carry a sign convention. The frequency is ω_A with a value that depends on the masses and spring constants; in the worked example, ω_A^2 = 2K/M (arising from the specific mass distribution used in the demonstration).
• : a mode where the left mass is effectively stationary while the two right masses oscillate with opposite signs, yielding X1 ≈ 0, X2 = B cos(ω_B t + φ), X3 = −B cos(ω_B t + φ). The frequency here is ω_B with ω_B^2 = K/M in the presented setup.
• : a center-of-mass or rigid-translation-like mode where all masses move in unison, yielding a zero-frequency mode with ω_C = 0 (the motion is proportional to a linear time function, corresponding to a constant velocity translation in the model).

Each mode comes with a specific eigenvector that sets the relative motion of X1, X2, and X3. The speaker notes that while Mode C appears to be a trivial translation, it is a legitimate normal mode in this context due to the boundary conditions and the presence of end springs.

General Solution as a Superposition of Normal Modes

The lecture emphasizes that the most general motion of the system is a linear superposition of the normal modes. Since there are three second-order equations, there are 6 free parameters in the general solution. These parameters are fixed by the initial conditions (initial positions and velocities). The crucial takeaway is that once the normal modes and their frequencies are known, the complex, evolving motion can be expressed as a sum of these simple harmonic motions, greatly simplifying qualitative and quantitative predictions.

Extending to More Masses and the General Strategy

The instructor discusses the challenge of larger systems and introduces a general, systematic method to obtain normal modes for any number of masses. The steps are:

• Define coordinates X1, X2, ..., XN for N masses connected by springs in a linear chain with appropriate boundary conditions.
• Write the equations of motion and assemble the mass matrix M and stiffness matrix K.
• Convert to the matrix form M X'' = −K X and seek normal-mode solutions X(t) = A e^{i ω t}.
• Recast into the eigenvalue problem (M^{-1}K − ω^2 I) A = 0, computing det(M^{-1}K − ω^2 I) = 0 to find ω^2 and the corresponding eigenvectors A.

The rest of the discussion shows how this procedure reproduces the earlier three-mode results and yields a robust framework for analyzing complex coupled systems. The talk ends with a preview of next topics such as resonance, driven oscillations, and more sophisticated couplings, indicating that the normal-mode analysis lays the foundation for understanding how real systems respond when energy is put into the network by external forces.