How to measure the cycles of earth moving/rotation?
In theory, if the Earth was a solid object, the Chandler wobble would be easy to understand. It's simply a result of the polhode rolling without slipping on the herpolhode lying in the invariable plane. In other words, $\mathrm I \dot \omega + \omega \times (\mathrm I \omega) = 0$, where $\mathrm I$ is the Earth's inertia tensor and $\omega$ is the Earth's angular velocity. The Chandler wobble is simply the consequence of the Earth not having a spherical mass distribution and the Earth not having it's angular momentum and angular velocity coaligned. 

In practice, the Earth is not a solid object. The theory behind the Earth's orientation is rather (putting it mildly) complex. The best model involves over a thousand parameters. Unfortunately, this best model of the Earth's precession, nutation, and rotation does not explain the Earth's orientation. The Chandler wobble, along with a number of lesser phenomena, is an after-the-fact discrepancy between the very best analytic model and the very best observational results.

Knowing the Earth's orientation is extremely important to some sciences, high resolution astronomy in particular. The US Naval Observatory (USNO) publishes its estimates of the deviation of the Earth's true orientation versus that of the best analytic models on a daily basis. The USNO is a key member of the International Earth Rotation and Reference Systems (IERS, for short, and yes, the acronym doesn't work. It used to work.)