Free vibrations

Consider a harmonic oscillator with viscous damping (i.e. the damping is proportional to the velocity $\partial u/\partial t$).


Making usage of Newton's 2nd law, we can write the equation of motion as:

\[ \displaystyle{ m\frac{\partial^{2}u(t)}{\partial t^{2}}) + c\frac{\partial u(t)}{\partial t} + ku(t) = 0 } \]

In the case of a harmonic time dependence (of the form $e^{rt}$), the characteristic equation is:

\[ m r^2 + c r + k = 0 \]

The two solutions of this characteristic equation are:

\[ r_{1,2} = -\displaystyle{\frac{c}{2m}} \pm \displaystyle{\frac{\sqrt{c^2-4km}}{2m}} \]

Introducing the terms:

Term Definition
$\omega_{0}^{2} = \displaystyle{\frac{k}{m}}$ Natural pulsation of the undamped system
$c_{cr} = 2\sqrt{km} = 2m\omega_0$ Critical damping
$\xi = \frac{c}{c_{cr}} = \displaystyle{\frac{c}{2m\omega_0}}$ Viscous damping factor

the equation of motion can be rewritten as:

\[ \displaystyle{\frac{\partial^{2}u}{\partial t^{2}}} + 2\omega_0 \xi \displaystyle{\frac{\partial u}{\partial t}} + \omega_{0}^{2} u = 0 \]

The general solution for this linear, homogeneous, with constants coefficients differential equation is (see page Solution to the undamped 1D harmonic oscillator):

\[ u(t) = A e^{r_1 t} + B e^{r_2 t} \]

where $A$ and $B$ are two constants which can be determined from initial conditions.