Winter Running
Winter running,
along urban rivers.
Running is a scam.
300 calories?
WTF?!
\begin{aligned} \dot x_1& = x_1 x_2 - x_1^3 \\ \dot x_2& = -5x_2 + 9e^{x_1+5x_2}-9+5 \end{aligned}
\begin{aligned} x_1 = 0 = x_{10} \\ x_2 = 0 = x_{20} \end{aligned}
\begin{aligned} x_1 &= x_{10} + \tilde x_1 \\ x_2 &= x_{20} + \tilde x_2 \end{aligned}
are small deviations from the origin found at x_{10}, x_{20}
The way we lineaerize this is by taking the Taylor series expansion, thus we have
\begin{aligned} f(x_{1},x_{2}) &= f(x_{10} + \tilde x_1,x_{20} + \tilde x_2) \\ &= f(x_{10},x_{20}) + \frac{\partial f(x_{10},x_{20})}{\partial x_1} \tilde x_1 + \frac{\partial f(x_{10},x_{20})}{\partial x_2} \tilde x_2+ H.O.T \end{aligned}
No need to reach the 2nd derivatives before calling it quits on the higher order terms, unless of course, you want accuracy.
Now, in order to actually solve for a specific partial derivative as stated in the 2nd and 3rd term of the 2nd line, we need invoke an operator known as the Jacobian. Once we invoke the almighty Jacobian, we should have something that looks like...
\begin{aligned} \frac{\partial}{\partial x} f_1(x)= \begin{bmatrix} {\frac{\partial f_1}{\partial x_1}} & {\frac{\partial f_1}{\partial x_2}} \\ {\frac{\partial f_2}{\partial x_1}} & {\frac{\partial f_2}{\partial x_2}} \end{bmatrix} &= {\begin{bmatrix} {x_2 - 3x_1^2} & {x_1} \\ {0+9e^{x_1}e^{5x_2}} & {9e^x_1 5^{5x_2}-5} \end{bmatrix}} \Big|_{x_1 = 0, x_2 = 0} \\ &= \begin{bmatrix} {0} & {0} \\ {9e^0} & {45e^0-5} \end{bmatrix} = \begin{bmatrix} {0} & {0} \\ {9} & {40} \end{bmatrix} = A \end{aligned}
With this task accomplished, we now know the answer for the Jacobian Matrix A and can use it to linearized the equation mentioned above.
\begin{aligned} \dot {\tilde x}(t) = A(t)\tilde x(t) \end{aligned}
Note that we did not define an input for this system. Meaning, the output is only defined by the state vector x.
CAD – Scholars
– Mini-lecture – Soh-Cah-Toa
– Software: “Fusion360 – Browser Based”
-+ Verify that students can log into Browser version of Fusion360
-+ Will compare the “Fusion360 Browser”
-++ Concept 1 – Explore Fusion360 Browser GUI
https://myhub.autodesk360.com/
-++ Concept 2 – What is a fillet?
-++ Concept 3 – What is a chamfer?
-+++ Software: “Google Spreadsheets”
-++ Concept 4 – explore Soh-Cah-Toa functions
-++ Concept 5 – plotting