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As we know, if the damping model only depends on the instantaneous velocity, it can be called viscous damping, for which the damping force $f = Cv$. However, if $C$ depends on the deformation as $C(u),$ is this still named viscous damping? And if the damping force is modeled as $f=C g(v),$ where $g$ is a function of $v$, do we call this nonlinear viscous damping? Thank you!

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I think "viscous" refers to the physical effect that causes the damping - i.e. the viscosity of a fluid. The viscous forces might be a linear or nonlinear function of the velocity, or something more complicated which might include a function of the deformation. The terms "linear viscous damping" and "nonlinear viscous damping" are meaningful in that context.

A damping force like of the $f = Cg(v)$ could model something like frictional damping, where for example $C$ was a (negative) constant and $g(v) = v/|v|$, (i.e. for linear motion $g(v) = \pm 1$ depending on the direction of the motion) but calling it "nonlinear viscous damping" instead of "friction damping" seems rather pointless.

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