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**Apprentice123** Vibration Free

Please, are correct?

$\displaystyle m \frac{d^2x}{dt^2} + kx = 0$

Where frequency is

$\displaystyle w = \sqrt{\frac{k}{m}}$

$\displaystyle \frac{d^2x}{dt^2} + \frac{k}{m}x = 0$

The characteristic equation is:

$\displaystyle r^2 + w^2 = 0$

$\displaystyle r = +or- iw$ where $\displaystyle i^2 = -1$

Then:

$\displaystyle x(t) = C_1e^{iwt} + C_2e^{-iwt}$

Calculating I can get

$\displaystyle x(t) = a_1 \cos(wt) + a_2 \sin(wt)$

Now, I need to do to get the following equation ?

$\displaystyle x(t) = Acos(wt - \delta)$ (I think this is the equation we need to get the free vibration)