Over-damped Oscillations.

This refers to the situation where

$$\beta > \omega_0$$ (2.14)

The two roots are

$$\alpha_1=-\beta + \sqrt{\beta^2 - \omega_0^2} = - \gamma_1$$ (2.15)

and

$$\alpha_2=-\beta - \sqrt{\beta^2 - \omega_0^2} = - \gamma_2$$ (2.16)

where both and $\gamma_2 > \gamma_1$. The two roots give rise to exponentially decaying solutions, one which decays faster than the other

$$x(t)=A_1 e^{- \gamma_1 t} + A_2 e^{- \gamma_2 t} \,.$$ (2.17)

The constants $A_1$ and $A_2$ are determined by the initial conditions. For initial position $x_0$ and velocity $v_0$ we have

$$x(t)=\frac{v_0 + \gamma_2 x_0}{\gamma_2-\gamma_1} e^{-\gamm... ...frac{v_0 + \gamma_1 x_0}{\gamma_2-\gamma_1} e^{-\gamma_2 t}$$ (2.18)

Figure 2.2:

The overdamped oscillator does not oscillate. Figure 2.2 shows a typical situation.

In the situation where $\beta \gg \omega_0$

$$\sqrt{\beta^2 - \omega_0^2} = \beta \sqrt{1 - \frac{\omeg... ... \left[ 1 - \frac{1}{2}\frac{\omega_0^2}{\beta^2} \right]$$ (2.19)

and we have and $\gamma_2=2 \beta$.