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.
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