Question

Establish the equation of motion of a damped oscillator. Solve it for a weakly
damped oscillator and discuss the significance of the results

Answer 1

Answer on Question #61581 - Physics - Mechanics | Relativity

Question:

Establish the equation of motion of a damped oscillator. Solve it for a weakly damped oscillator and discuss the significance of the results.

Answer:

The equation of motion of a damped oscillator is:

$$m \frac {d ^ {2} x}{d t ^ {2}} + \alpha \frac {d x}{d t} + k x = 0$$

or

$$\frac {d ^ {2} x}{d t ^ {2}} + 2 \beta \frac {d x}{d t} + \omega_ {0} ^ {2} x = 0, \text { where } 2 \beta = \frac {\alpha}{m} \text { and } \omega_ {0} ^ {2} = \frac {k}{m}.$$

Now we can solve it:

$$\lambda^ {2} + 2 \beta \lambda + \omega_ {0} ^ {2} = 0 \Rightarrow \lambda_ {1} = - \beta + \sqrt {\beta^ {2} - \omega_ {0} ^ {2}}, \lambda_ {2} = - \beta - \sqrt {\beta^ {2} - \omega_ {0} ^ {2}}$$

Oscillator will be a weakly damped only when $\beta < \omega_0$ ..

$$\lambda_ {1} = - \beta + i \omega_ {\beta}, \lambda_ {2} = - \beta - i \omega_ {\beta} \text { where } \omega_ {\beta} = \sqrt {\omega_ {0} ^ {2} - \beta^ {2}}$$

And now we have:

$$x (t) = A _ {1} e ^ {(- \beta + i \omega_ {\beta}) t} + A _ {2} e ^ {(- \beta - i \omega_ {\beta}) t}$$ $$x (t) = e ^ {- \beta t} \left(A _ {1} e ^ {i \omega_ {\beta} t} + A _ {2} e ^ {- i \omega_ {\beta} t}\right)$$ $$x (t) = A e ^ {- \beta t} \cos (\omega_ {\beta} t + \varphi)$$

$A e^{-\beta t}$ is the amplitude that decreases with time.

This solution of equation of motion of a damped oscillator describes the majority of oscillations in nature, such as mathematical pendulum movement in the air or in the water.

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