# How to use a state equation to graph movement?

#### gfryda

Good Day,
I am trying to make a simulator for a car/trailer combination that will allow me to get a rough estimate of trailer dampening and other factors. I have found that many papers suggest simplifying it to a 'tricycle' model, where any given trailer is composed of three wheels.

They then come up with a system of equations and simplify them into matrix form, and furthermore change it to a state equation as seen in the top right and bottom left:

I recognize this form, and they give as inputs. (u would be an active damping which for me is zero. )

Basically, I cannot figure out how to go from this state equation to the what the actual vehicle response it to an input. When I look up state equations usually there is a y component, but I do not appear to have a second equation. I am not sure how to proceed with coding this, or doing it by hand. I am using MATLAB.

Here are the matrix values. Everything listed is a constant, and I can provide the list of the variables if needed. I can't attach the paper, its too large.

Thank you for any assistance you can give me or guidance to other help.

#### Woody

A glance at your post quickly put it in the "too difficult for me" category.
However, one method I have found that can help with these complicated mutual interactions is to break them down (into simpler bits).
Model each linkage and each interaction separately, (i.e. this happens, which causes that, which then causes...) rather than trying to model the whole lot all at once.
put in a disturbance to the first point in the model (e.g. a lateral acceleration of the car)
and follow the reaction through each section (and back again) over lots of very small time steps.

This will only give you an approximation, but it might be a good enough approximation to provide the answers you need.
I also suspect that the process of building such a model will provide insight into the problem, and what the key requirements might be.