
In this lecture we'll consider the much more realistic case of _damped_ free vibration, i.e. we're re-introducing damping. Since all mechanical systems have some form of damping, this is a more practical problem to solve. We're still considering free vibration, so there is no external force applied to the system yet. Our equation of motion is therefore,

$$
m\ddot{u} + c\dot{u} + ku  = 0
\tag{1}
$$

To identify a general solution to this equation, we can assume the same candidate solution as for the undamped case, $u=e^{\lambda t}$, differentiating yields,

$$
\dot{u} = \lambda e^{\lambda t}
\tag{2}
$$

$$
\ddot{u} = \lambda^2 e^{\lambda t}
\tag{3}
$$

Substituting our candidate solution and its derivatives back into the equation of motion and isolating the characteristic equation yields,

$$
m\lambda^2 + c\lambda + k = 0
\tag{4}
$$

This is a quadratic equation with roots,

$$
\lambda = \frac{-c \:\pm\:\sqrt{c^2-4mk} }{2m}
\tag{5}
$$

The roots depend on whether the quantity under the square root sign is:

- equal to zero, $c^2-4mk=0$

- greater than zero, $c^2-4mk>0$

- less than zero, $c^2-4mk < 0$

These three options correspond to three different regimes or categories of dynamic behaviour, depending on the level of damping present, $c$. The three damping regimes are:

- Critically-damped, when $c^2-4mk=0$

- Over-damped, when $c^2-4mk>0$

- Under-damped, when $c^2-4mk < 0$

By far the most commonly observed behaviour, particularly in a civil engineering context is under-damped behaviour. This is what we'll spend most of our time analysing.

By following the same procedure as for undamped vibration, we can state (without yet finding the two roots $\lambda_1$ and $\lambda_2$) that the general solution to the differential equation is,

$$
u=A\:e^{\lambda_1t} + B\:e^{\lambda_2t}
\tag{6}
$$

What remains now is to work out what the roots ($\lambda_1$ and $\lambda_2$) actually are. In the next lecture we'll briefly consider the two less common categories of dynamic behaviour, critically and over-damped behaviour.


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