How do I account for damping on structural vibrations?

Question

I want to know what terms to add to the differential equation for structural vibrations to account for structural damping. Not external damping, but internal damping. I consider the transfer of energy into sound waves, or air drag as external damping. I want to focus on the energy dissipation mechanism of a typical metal (crystalline) structure.

Two scenarios.

1. Firstly, a long rod with longitudinal waves has the following dynamics in terms of deflection $u(x,t)$

   $$ \left( E A \frac{\partial^2 u}{\partial x^2} - \rho A \frac{\partial^2 u}{\partial t^2} \right) {\rm d}x = 0 $$

   which is derived from the balance of forces on a thin slice ${\rm d}x$ with mass ${\rm d}m = \rho A {\rm d}x$, with $\rho$ the density, $A$ the cross sectional area and $E$ the elastic modulus.

   

   My question is how do you account for structural damping with the above equation. Do you add a term with both curvature and time, like $ \frac{\partial^2}{\partial x^2}\frac{\partial}{\partial t} u$ ?

   The tension-deflection relationship for a rod is $T = E A \frac{\partial}{\partial x} u$.

2. Secondly, what about the transverse vibrations of a beam. The undamped equation is

   $$ \left( E I \frac{\partial^4 u}{\partial x^4} + \rho A \frac{\partial^2 u}{\partial t^2} \right) {\rm d}x = 0 $$

   where $I$ is the 2nd moment of area. What does the structural damping term look like? is it like $ \frac{\partial^3}{\partial x^3}\frac{\partial}{\partial t} u$ which can be interpreted as the time rate of the shear force?

   The shear force-deflection relensionship is $S = E I \frac{\partial^3}{\partial x^3} u$ and the moment-deflection relationship $M = E I \frac{\partial^2}{\partial x^2} u$.

Ultimately what I want to examine is the effect of damping on the natural frequency of structural vibrations.

PS. External damping would add a term proportional to speed only $\frac{\partial u}{\partial t}$ and the shape of the structure does not play any role in it. I am not interested in this.

Answer 1

The simplest answer is to consider the stress-strain relation as:

$σ = Eε + ηε_t$

This is the stress relation for a simple damped spring (Kelvin Voigt).

Then you can define your equation of motion as:

$ρAy_{tt} = M_{xx}$

Where Moment can be derived from the stress equation by substituting $I*y_{xx}$ (bend) for the strain.

Thus you have:

$M = (-EIy_{xx} - ηIy_{xxt})_{xx}$

$M_{xx} = -EIy_{xxxx} - ηIy_{xxxxt}$

And your final equation is:

$ρAy_{tt} = -EIy_{xxxx} - Iηy_{xxxxt}$

I conceptualize this as the damping that would occur from bending a stiff spring with a rotational viscous damper in parallel.

If you want to get nuts, this article takes it a lot further applying nonlocal effects and this type of damping to a Timoshenko beam.