Damping Constant Equation?

Question

I've been looking through my textbooks and I've found a number of different equations - so I wanted to confirm with you which it is.

What is the equation which determines the damping constant (gamma) in a spring executing shm?

Answer 1

In the equation for the motion of a damped spring

$$m\ddot{x} + c\dot{x} + kx = 0$$

the parameter $c$ is known as the viscous damping coefficient. From it you can determine the damping ratio

$$\zeta = \frac{c}{2\sqrt{mk}}$$

which determines if the spring is underdamped ($\zeta<1$), overdamped ($\zeta>1$) or critically damped. The parameter $c$ itself is a property of the spring, and is not determined by $m$ or $k$.

You solve this equation by postulating a solution of the form $x(t)=e^{\gamma t}$, giving a quadratic equation for $\gamma$:

$$m\gamma^2 + c\gamma + k = 0$$

which you solve with the usual quadratic formula:

$$\gamma_{\pm} = \frac{1}{2m} \left( -c \pm \sqrt{c^2 - 4mk} \right)$$

giving two possible values for the decay constant. If $c^2<4mk$ then your spring is underdamped, and the rate of decay is given by the real part of $\gamma$, ie $\mathrm{Re}(\gamma) = -c/(2m)$.