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 2^{nd} 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.