Alternative Models
Viscous damping models, although widely used, are not the only damping models. A wide range of models can be found in specialized literature. One is the so called "hysteretic damping model" or "structural damping model".
When a metal beam is vibrating, the internal damping can be better described by a force proportional to the displacement but in phase with the velocity. In such case, the differential equation that describes the free movement of a single-degree-of-freedom system becomes:
where h is the hysteretic damping coefficient and i denotes the imaginary unit; the presence of i is required to synchronize the damping force to the velocity (xi being in phase with the velocity). This equation is more often written as:
where η is the hysteretic damping ratio, that is, the fraction of energy lost in each cycle of the vibration.
Although requiring complex analysis to solve the equation, this model reproduces the real behaviour of many vibrating structures more closely than the viscous model.
A more general model that also requires complex analysis, the fractional model not only includes both the viscous and hysteretic models, but also allows for intermediate cases (useful for some polymers):
where r is any number, usually between 0 (for hysteretic) and 1 (for viscous), and A is a general damping (h for hysteretic and c for viscous) coefficient.
See also: Friction and DragRead more about this topic: Damping
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