Two equal masses m are connected to three identical springs...

Two equal masses $m$ are connected to three identical springs (spring constant $k$ ) on a frictionless horizontal surface (see figure). One end of the system is fixed; the other is driven back and forth with a displacement $X=X_{0} \cos \omega t .$ Find and sketch graphs of the resulting displacements of the two masses.

Key Concepts

Coupled Oscillators

Coupled oscillators are systems in which two or more oscillating objects (such as masses attached to springs) are connected in a way that the motion of one affects the motion of the others. This interdependence typically leads to a set of equations where the individual oscillations are no longer independent, requiring special techniques to analyze the overall behavior of the system.

Normal Modes Analysis

Normal modes refer to the independent patterns of oscillation that a coupled system can exhibit. In a system with coupled oscillators, each normal mode oscillates at a specific natural frequency, and the overall motion can be expressed as a superposition of these modes. This concept simplifies the analysis by reducing a system of coupled differential equations into independent equations for each mode.

Forced Oscillations

Forced oscillations occur when an external periodic force (or imposed displacement) drives the system. In such systems, the oscillators can respond differently based on the driving frequency, and their steady-state behavior is determined by both the natural frequencies of the system and the properties of the external drive. Analysis of forced oscillations requires solving for the system’s response to this continuous external influence.

Resonance

Resonance is a phenomenon that arises in driven oscillatory systems when the frequency of the external force matches one of the system's natural frequencies. Under resonance conditions, even a small external force can produce large amplitude oscillations. Understanding resonance is crucial in predicting the response amplitude and phase of each oscillator, and it helps in designing systems to either exploit or avoid potentially destructive large oscillations.

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