# Quantifying the amount of structural/hysteretic vibrational damping that occurs in proportion to deformation? (ie. Based on how "bent" something is?)

Three different conceptual types of vibrational damping forces are described here:

1. Viscous: $$f(t) = -c \cdot v$$
2. Coulomb: $$f(t) = -c \cdot \text{sgn}(v)$$
3. Structural: $$f(t) = -c \cdot |x| \cdot \text{sgn}(v)$$

$$c$$ is a damping coefficient, $$v$$ is velocity, and $$x$$ is a measure of the deformity of the object studied.

In other words, viscous damping force is proportional to velocity. Coulomb force represents friction, modeled as being velocity independent once motion is initiated. Structural represents greater damping at greater amounts of deformity.

The authors use the following passage to describe the nature of structural damping:

Structural damping may be viewed as a sliding friction mechanism between molecular layers in a material, in which the friction force is proportional to the deformation or displacement from some quiescent or rest position. Imagine a rod made of a bundle of axial fibers. The sliding friction force between each fiber and its neighbor will increase as the rod is bent and the fibers are pinched together. This pinching phenomenon occurs in most materials as the various molecular layers slide past one another. The result is a damping force that is proportional to the displacement from the undisturbed position. This mechanism was verified for a wide range of materials by Kimball and Lovell in 1927.

If you were looking at the internal damping applied to a point on a vibrating string, the degree of point structural deformity could be determined by some derivative of the transverse displacement ($$y$$) with respect to length ($$x$$). Eg. $$y_{xx}$$ or perhaps a higher derivative like $$y_{xxxx}$$.

I believe one would likely expect the structural damping to increase considerably the more the deformation increases (eg. as $$y_{xx}$$ increases). I don't think there would be a linear relationship between bend and structural damping but rather a nonlinear one.

The article I linked was from 1975.

Are there any theories or concepts that would describe further the relationship between deformation and damping force in this scenario or quantify the relationship between them?

Thanks.

• From a quick reading, IMO there are so many things wrong with the paper you linked to that I don't have time to spend several hours rewriting the whole paper. Just listing all the incorrect statements would be a non-trivial amount of work IMO. At best, they are using well-known words to mean something different from their standard meanings. Jan 7, 2021 at 16:44
• Try this, from the Journal of Sound & Vibration 2004. 140.121.146.149/JSV-Muravskii%20-2004.pdf Jan 7, 2021 at 17:05