Do the terms "damping constant" and "damping coefficient" have standard uses? I've heard the terms "damping constant" and "damping coefficient" used to describe both the $c$ from the viscous damping force equation $F = -c\dot{x}$ and the $\gamma$ from the definition $\gamma = \frac{c}{2m}$. Is there standard usage for each term, or is it context dependent? If it is context dependent, are there terms which uniquely refer to each of $c$ and $\gamma$?
 A: The answer is yes. Talk to any engineer and if say the terms "damping constant" and "damping ratio" they know exactly what you mean without any further explanations.


*

*Damping coefficient $c$ signifies the contribution of velocity to force, as in $F = \ldots + c \dot{x} + \ldots$

*Damping ratio $\zeta$ is a number the signifies the region of damping. When $\zeta<1$ the problem is underdamped: when $\gamma>1$ it is overdamped: and when $\zeta=1$ it is critically damping. This means the form of the solution (as in the equation used to solve for motion) is different depending on the region.


The complementary terms of the above but with respect to stiffness are


*

*Stiffness coefficient $k$, which is the contribution of deflection to force, as in $F = \ldots + k x + \ldots$

*Natural frequency $\omega_n$ which combines stiffness and mass to tell you how fast the system responds to inputs.


In practive the above terms are using the transform the standard equation of motion $$ m \ddot{x} = -k x - c \dot{x} $$ into one with known solutions. By substituting $$\begin{aligned} k & = m \omega_n^2 & c & = 2 \zeta m \omega_n \end{aligned} $$ you get 
$$ \ddot{x} + 2 \zeta \omega_n \dot{x} + \omega^2_n x  =0 $$
with the well-known solutions
$$ x(t) = \begin{cases} 
  {\rm e}^{-\zeta \omega_n t} \left( A \cos\left( \omega_n \sqrt{1-\zeta^2} \right) + B \sin\left( \omega_n \sqrt{1-\zeta^2} \right) \right) & \zeta < 1 \\
  {\rm e}^{-\zeta \omega_n t} \left( A \cosh\left( \omega_n \sqrt{\zeta^2-1} \right) + B \sinh\left( \omega_n \sqrt{\zeta^2-1} \right) \right) & \zeta> 1 \\
  (A+t B) {\rm e}^{-\omega_n t} & \zeta = 1
\end{cases} $$
A: Lets see where the damping ratio $\gamma$ come from ?
Start with:
$$m\,\ddot{x}+c\,\dot{x}+k\,x=0\tag 1$$
where 
$c$ is the damping constant with the unit $[N\,s/m]$ and
$k$ is the spring constant with the unit $[N/s]$
to find a solution  for equation (1) we make the ansatz: $x=A\,e^{\lambda\,t}$ and get:
$$m\,\lambda^2+c\,\lambda+k=0\tag 2$$
the solution for $\lambda$ is now:
$$\lambda_{1,2}=\frac{-c\pm\sqrt{c^2-4\,m\,k}}{2\,m}=-\frac{c}{2\,m}\pm\sqrt{\frac{c^2-4\,m\,k}{4\,m^2}}=-\frac{c}{2\,m}\pm
\sqrt{\frac{k}{m}\left(\frac{c^2}{4m^2}\frac{m}{k}-1\right)}\tag 3$$
with $\frac{k}{m}:=\omega_n^2$ we get for equation (3)
$$\lambda_{1,2}=-\frac{c}{2\,m}\pm\omega_n\sqrt{\frac{c^2}{4m^2\,\omega_n^2}-1}=-\frac{c}{2\,m}\pm\omega_n\sqrt{\gamma^2-1}\tag 4$$
so $\gamma^2=\frac{c^2}{4m^2\,\omega_n^2}$ and
$c=2\,\gamma\,\omega_n\,m$
and equation (4) $\mapsto$
$$\lambda_{1,2}=-\gamma\,\omega_n\pm\,\omega_n\sqrt{\gamma^2-1}\tag 5$$
the solution  for $\lambda_{1,2}$ depend on 2 parameters $\omega_n$ unit $[1/s]$ and $\gamma$ unit $[-]$
we get real solution for for $\lambda_{1,2}$ from equation (5) if $\gamma^2 \ge 1$
and a complex solution if $\gamma^2 \lt 1$, so I think it is just convenience to define $\gamma^2=\frac{c^2}{4m^2\,\omega_n^2}$. 
