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1. Mechanical Resonance

   Linked via "amplitude of oscillation ($A$)"

   Mechanical resonance is a phenomenon occurring when an external, periodic driving force applied to an oscillating system has a frequency ($\omega$) that matches one of the system's natural frequencies of vibration ($\omega_0$). When this condition is met, the amplitude of oscillation ($A$) of the system's steady-state oscillations can dramatically increase, potentially leading to large deformations or system failure if damping is sufficiently low.
   The fundament…

2. Mechanical Resonance

   Linked via "amplitude of oscillation ($A$)"

   The fundamental principle is described by the equation of motion for a damped, driven harmonic oscillator:
   $$m\frac{d^2x}{dt^2} + c\frac{dx}{dt} + kx = F_0 \cos(\omega t)$$
   where $m$ is the mass, $c$ is the damping coefficient, $k$ is the spring constant, and $F0 \cos(\omega t)$ is the external driving force. The amplitude of oscillation ($A$) reaches its maximum when $\omega = \omega0 = \sqrt{k/m}$, provided $c$ is…