I have just started an internship and my boss would like me to do a few things for training.

First, he had me take the equation of a simple pendulum and turn it into state-space form. Then, he had me linearize it. I first did this process by hand, and then in Python.

The next step is to create a model on python that allows me to input an initial angular displacement, and find resulting positions, velocity, etc.

My highest level of math education is multi-variable calculus. My boss knows this so I have some time to figure this stuff out.

My question is, how did you find future positions when your equation 4 had no input for time? My boss mentioned I should use the Runge-Kutta Method, but I do not know how to implement time into my equations.

Please let me know your thoughts.

Thanks!

]]>\begin{align*}

\hat{f}(x,y,z) &= \sum_{j=n}^{m} f(x_j) \prod_{k=n,k\neq j}^m \frac{(x-x_k)}{(x_j – x_k)} \frac{(y-y_k)}{(y_j – y_k)} \frac{(z-z_k)}{(z_j – z_k)} \\

\hat{f}(x,y,z) &= \sum_{j=n}^{m} f(x_j) \mathcal{L}_{j}(x,y,z)

\end{align*}

One can see this is valid because if you evaluate the set of Lagrange Polynomials, aka $\lbrace \mathcal{L}_{j}(x,y,z) \rbrace_{j=n}^{m}$, at say $(x_k,y_k,z_k)$, you will find $\mathcal{L}_{j}(x_k,y_k,z_k) = \delta_{jk}$, where $\delta_{jk}$ is the Kronecker Delta and is defined as:

\begin{align*}

\delta_{jk} &= \begin{cases}

1 & j = k \\

0 & j \neq k

\end{cases}

\end{align*}

This Kronecker Delta property is one that defines the Lagrange Polynomials, thus making the formula above correct.

]]>Sorry for the delay, I have had a lot going on for the last month. I will probably setup a Visual Studio project (using VS 2015 though) that you can download soon so you can have something working on Windows. It might take until early January, which is when I will have some time off from work.

Best,

Christian ]]>

Thank you so much for the speedy reply. I have uploaded this project into Github, here is the site address:

https://github.com/tuananhlemta/MissileCalc_Chris

I used visual studio 2013 to create this project which based on your files. Please help me to check some errors bugs occurred.

Thank you very much.

Best regards,

Tommy ]]>