# Foced Vibrations - beats

1. Mar 17, 2014

### jellicorse

I am trying to work though an example on this topic in my book and have reached a point that I am not sure about. I was wondering if anyone could help me clear this up.

The equation of motion for a spring-mass system with no damping and a periodic external force is

$$mu'' + ku = Fcos\omega t$$

The general solution to this is:

$$u = Acos\omega_0t + B sin\omega_0 t +\frac{F}{m(\omega_0^2+\omega^2)}cos\omega t$$

If the mass is initially at rest, so that u(0)=0 and u'(0)=0, then the solution to this equation is

$$u=\frac{F}{m(\omega_0^2-\omega^2)}(cos\omega t -cos\omega_0 t)$$

I have managed to follow it to here but I can not see how they have completed the next step:

"making use of the trigonometric identities for cos(A$\pm$B) with $A=(\omega_0+\omega)t/2$ and $B=(\omega_0-\omega)t/2$ we can write the equation in the form":

$$u=\left[\frac{2F}{m(\omega_0^2-\omega^2)}sin\frac{(\omega_0-\omega)t}{2}\right]sin\frac{(\omega_0+\omega)t}{2}$$

I can not see how the penultimate equation becomes the final equation here; can anyone tell me how this works?

2. Mar 17, 2014

### olivermsun

Write out A+B and A-B.
Then write out the angle addition formula for cos(A±B).
You'll see how it works.

3. Mar 17, 2014

### jellicorse

Thanks a lot for the tip, Oliver. I've got it now.

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