This is intended as an introductory, undergraduate-level module for modelling coupled oscillators.
In classical mechanics, systems must transfer energy through some medium in order
to achieve coupling.
- For example, pendula will synchronize when oscillating on a movable surface.
Why do we care about obtaining a mathematical model for a system of coupled, oscillating masses?
- Coupling may be undesired: parasitic electrical potentials, vibrations affecting a civil structure, and so on.
We need to understand coupling if we want to minimize disruptive cases.
Recall that a particle on a spring is subject to a linear restorative force (Hooke's Law):
$$\blbox{F(x) = -kx} \tag{Eq. 1}$$
In terms of energy mechanics, the particle is subject to the familiar harmonic potential $U(x)$:
$$\blbox{U(x') = \int_{0}^{x} -kx'\ dx' = \frac{1}{2}kx^2} \tag{Eq. 2}$$
Our mechanical oscillator has total energy $E = T + U$, where
T is the system's kinetic energy:
$$\blbox{E = \frac{1}{2}m v^2 + \frac{1}{2}kx^2} \tag{Eq. 3}$$
For a mechanical system undergoing non-rotational motion, x is the center of mass (of the rigid body).
Describing motion of a rigid body is a step-up from particles (point-masses in analytial mechanics).
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