This chapter shortly introduces important results for one degree of freedom (or 1 DOF) linear mechanical systems with damping. The analogy with an electric system is obvious and thus the literature on this subject can be an alternative source of information.

## Free vibrations

Consider the harmonic oscillator with viscous damping (i.e. the damping is proportional to the velocity $du/dt$ of the figure below.

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

 $\displaystyle{ m\frac{d^{2}u(t)}{dt^{2}}) + c\frac{du(t)}{dt} + ku(t) = 0 }$ (1)

In the case of a harmonic time dependence (of the form $e^{rt}$), and substitute it in Eq. (1) to obtain the characteristic equation of motion:

 $m r^2 + c r + k = 0$ (2)

The two solutions of this characteristic equation are:

 $r_{1,2} = -\displaystyle{\frac{c}{2m}} \pm \displaystyle{\frac{\sqrt{c^2-4km}}{2m}}$ (3)

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:

 $\displaystyle{\frac{\partial^{2}u}{\partial t^{2}} + 2\omega_0 \xi \displaystyle{\frac{\partial u}{\partial t}} + \omega_{0}^{2} u = 0$ (4)

The general solution for this linear, homogeneous, with constants coefficients differential equation is (cf your maths lectures):

 $u(t) = A e^{r_1 t} + B e^{r_2 t}$ (5)

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