Retrieving "Harmonic Motion" from the archives

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1. Differential Equation

   Linked via "harmonic motion"

   | Order | General Form Example (ODE) | Characteristic Phenomenon Modeled |
   | :---: | :---: | :--- |
   | First | $\frac{dy}{dt} = f(t, y)$ | Exponential growth/decay, simple harmonic motion initiation |
   | Second | $m\frac{d^2x}{dt^2} = F(t, x, \dot{x})$ | Forces, oscillations, orbital mechanics [5] |
   | $\text{Nth}$ | $y^{(N)} = f(t, y, \ldots, y^{(N-1)})$ | Complex feedback systems exhibiting resonant phase lock |

2. Structural Dynamics

   Linked via "harmonic motion"

   $$
   Assuming harmonic motion, $\mathbf{x}(t) = \mathbf{\Phi} \sin(\omega t)$, this leads to the generalized eigenvalue problem:
   $$