Linearization

Some parameters of the given pendulum system are relatively easy to measure, for example, things such as the pendulum length or the pendulum mass. Other parameters, such as the viscous coefficient of friction , are not as easy to measure directly. Therefore, we will perform a simple experiment to help identify . This experiment will also help to validate some of the simplifying assumptions we made in the process of generating our model.

The fact that the system model we generated is nonlinear makes the parameter identification process a little more challenging. However, we can use a linearized version of the model to help us in the identification process. Presuming that for our experiment the pendulum swings through small angles (about ), we can use the approximation that . Therefore, our linearized model becomes the following.

(3)

Examining the above, the linearized model has the form of a standard, unforced, second-order differential equation. Matching this equation to the canonical form, , we can see how the various system parameters influence the free response of the pendulum system. More specifically:

(4)

(5)

If we had simplified the pendulum as having its mass concentrated at its end, then .

Дата добавления: 2015-11-16