Normal Modes and Driven Oscillations in Coupled Oscillators (MIT OCW)

Overview

MIT OpenCourseWare presents a detailed look at coupled oscillators, showing how several interacting pendulums or masses connected by springs share energy and form normal modes. The lecture begins with two identical pendula or masses, then extends to three masses, illustrating how any complex motion can be written as a linear sum of simple normal modes with fixed amplitude ratios. It then introduces an external harmonic drive and analyzes the system's steady state response, highlighting resonance when the driving frequency matches one of the natural frequencies.

The talk emphasizes practical methods for finding driven amplitudes using matrix equations, Krams rule, and symmetry arguments to identify mode shapes, and it discusses energy flow and damping as key physical features of the dynamics.

Introduction and Motivation

The lecture discusses coupled oscillators as a paradigm for energy transfer in multi component systems, such as two masses on a track or pendula connected by springs. The central idea is that, despite complicated motion under arbitrary excitation, the system can be decomposed into a sum of normal modes, each with its own characteristic frequency. In the ideal, undamped case, the amplitudes within a mode stay in fixed ratios, and the total motion is a linear combination of these modes with time dependent phases.

Normal Modes and Their Shapes

For two identical pendula connected by a spring, there are two normal modes. A symmetric mode where both masses move in phase with equal amplitudes, and an antisymmetric mode where the masses move in opposite directions. The ratio of amplitudes in these modes is constant (1 for the symmetric mode, -1 for the antisymmetric mode). The normal mode frequencies are obtained from the eigenvalue problem associated with the system's mass and stiffness matrices, and in this two mass case the frequencies satisfy a determinant condition that reduces to a product form with two roots, corresponding to the two normal modes.

Driven Oscillations and Resonance

When an external harmonic force is applied to one mass, the system responds at the driving frequency, yielding a total motion that is the sum of a homogeneous solution (the natural, free oscillations) and a particular driven solution at the drive frequency. The driven part is found by solving a linear system that depends on the drive frequency. A key feature is resonance, which occurs when the driving frequency approaches one of the normal mode frequencies, causing large amplitudes in the driven response. The lecturer emphasizes the role of damping, noting that in practice a small amount of damping prevents unphysical infinities at resonance but does not eliminate the resonance phenomenon.

Kramers Rule and Explicit Solutions

For the driven two mass system, the amplitudes of the two masses can be obtained via Kramers rule. The speaker demonstrates how to replace columns of the coefficient matrix with the drive terms to obtain closed forms for the amplitudes. Although straightforward for 2×2 systems, the same idea generalizes to larger matrices, and the result highlights how the drive frequency and the normal mode structure together determine the amplitudes of motion for each mass.

Symmetry and Normal Modes

The lecture then introduces a symmetry based viewpoint. If the system possesses a mirror symmetry, the normal modes can be taken to be either symmetric or antisymmetric. By applying a symmetry transformation to the equations of motion, one can show that certain vectors (the normal mode shapes) are invariant up to a sign, which allows one to identify normal modes by simply diagonalizing the symmetry operator. This approach provides a powerful shortcut for constructing normal modes in systems with multiple identical components.

Extending to Three Masses

The discussion then extends to three masses connected by springs, which yields three normal modes with three characteristic frequencies. The qualitative picture remains that energy exchanges among the masses occur in the form of these modes, and driving the system at particular frequencies can selectively excite specific mode shapes. The lecturer demonstrates that, despite increasing complexity, the same principles apply: decompose into normal modes, use their shapes to understand the motion, and apply driven force analysis to predict the steady state response.

Real-World Implications and Takeaways

Beyond the classroom, the ideas of normal modes and driven responses have broad relevance to engineering and daily life. Buildings, bridges, and mechanical systems can resonate when subjected to external forcing with frequencies near natural modes, underscoring the importance of understanding mode structure for safety and design. The speaker also notes that even in complex structures, symmetry can simplify the analysis by revealing decoupled mode classes, enabling intuitive predictions of how the system will respond to external excitations.

Conclusion

The lecture completes with a summary of the main ideas: a general multi mass oscillator can be described as a sum of normal modes, driven responses can be computed via linear algebra methods such as Kramers rule, and symmetry provides a powerful lens for identifying mode shapes. The result is a robust framework for analyzing and predicting the dynamics of coupled oscillators in both simple and more elaborate configurations.