# Finding Common period of multiple waveforms

1. May 10, 2010

### macmosher

Hi Everyone,

First time poster, longtime viewer of these forums. Love the help that the community gives.

Just so you know where I am coming from:

I am trying to calculate the average power by first calculating the total energy of my system. Specifically I am looking at the energy dissipated in a damper, Cg. This turns out to be Cg multiplied by the integral of the velocity squared in my system over time (where Cg is fixed).

However, each of the velocities of the system are made up of the addition of multiple waveforms oscillating at different frequencies. Their respective periods are as follows:

2Pi/w
2Pi/((n+1)*w)
2Pi/((2n+1)*w)

Where "n" can be any number. In order to take the integration is done properly I need to find a common period amongst the added waveforms (squared). Now, this problem is fine so long as "n" is an integer, but once I allow it to have a decimal value the problem gets a bit more complicated.

This has been racking my brain for the past couple days, any thoughts?

I tried google, but of course it gave me common periods of different types of waves :P

Cheers!

2. May 10, 2010

### Xitami

LCM(1.35, 1.7) = LCM(1.3*100, 1.7*100) / 100 = LCM(135, 170) / 100 = 45.9
45.9 / 1.35 = 34
45.9 / 1.7 = 27

lcm(a, b) = a*b / gcd(a, b)

3. May 10, 2010

### macmosher

I should have mentioned I am trying to solve this analytically. Im attempting to do the integration analytically so that my optimization code doesn't have to run through it each and every iteration

4. May 10, 2010

### macmosher

Problem Solved (via problem avoidance)

As mentioned I am trying to find the average power of the summation of multiple wave forms. The problem is essentially this without all the coefficients:
$$P_{avg}=\frac{\int_0^T[sin(wt)+sin(w(n+1)t)+sin(w(2n+1)t)]^2}{T}$$
where T is a common integer multiple of the period amongst the three oscillators. The issue was that it was very difficult to find T analytically.

However, we know once T is found that:

int(sin(wt)*sin(n*wt),t=0..T)=0 so long n is an integer or their is a common period between the two waveforms.

So the whole point of finding T was to have that portion of the integration be zero. We can assume that the T does indeed exist and knowing that because the oscillations are periodic we can take the average of one full cycle rather than "x" number required to have a common period.

Therefore,
$$P_{avg}=\frac{\int_0^{\frac{2\pi}{w}} sin^2(wt)dt}{\frac{2\pi}{w}}+\frac{\int_0^{\frac{2\pi}{(n+1)w}} sin^2((n+1)wt)dt}{\frac{2\pi}{(n+1)w}}+\frac{\int_0^{\frac{2\pi}{(2n+1)w}} sin^2((2n+1)wt)dt}{\frac{2\pi}{(2n+1)w}}$$

Don't know if that will help anyone else with similar problems or not.