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diff --git a/tutorials/module_3/applications_examples.md b/tutorials/module_3/applications_examples.md new file mode 100644 index 0000000..7ef1e15 --- /dev/null +++ b/tutorials/module_3/applications_examples.md @@ -0,0 +1,38 @@ +Here are some example problems: + +1. System of Linear Equations (Apply Kirchoff Law to find Voltage at Different Nodes):  + +You can say the diagram is for looking at a sensor network or signal conditioning network where the resistors acts as voltage dividers to control voltage and the current injection sources are say photodiodes. Then we say that determining the node voltages at A,B,C is required in order to design the next stage of the system (like amplifier biasing, etc). Or you can say that this is the electrical diagram of a low-voltage DC bus. Nodes A,B,C are junctions on a small DC bus in an instrument bay. The two current sources represent two current injections (a 2 A sensor bus feed and a 3 A subsystem), and the resistors represent harness + connector resistances and discrete loads to chassis ground. Solving node voltages tells you whether the bus voltages at junctions stay in safe/regulator range and how much power is dissipated in the harness. (edited) + +You can make another application where you have flow through a network of pipes. Basically any sort of network (fluid, thermal, electrical) would simplify to a system of linear equations when you go through it. + +here, I found one example on traffic flow for you: + + + +these are the sort of applications we can bake into our system of linear algebra. You can find a infinite amount of problems for linear systems starting from this idea of networks things going into nodes and out. + + +2. System of nonlinear equations. +I was just thinking of a silly problem right now but it gets the job done: make a computer screen with a diagonal of 16 inches and area of 180 square inches. What are the length and height of the monitor? so you have $Area=x*y$ and diagonal is $\sqrt{x^2+y^2}$. BAM! two nonlinear equations with two unknowns + +a problem involving heat transfer by radiation between two different components will also yields some kind of system of nonlinear equations. Say room A is heated and emits radiation Q but is also linked by conduction to room B which radiates to ambient find temp A and temp B +$$ +Q_1+k(T_2-T_1)-\sigma*(T_1^4-T_{inf}^4)=0 +$$ +$$ +Q_2+k(T_1-T_2)-\sigma*(T_2^2-T_{inf}^4)=0 +$$ + +3. ODE problem. +RC circuit problem. Say I have a fast photodiode (10ns scale) that generates a voltage that can be read on an oscope. Based on a given resistance and capacitance of the system (cable +oscope) what is the voltage vs time trace going to look like? v(t)=1-exp*-t/(RC) should be the solution + +4. ODEs system. +Say I have a model for a mass spring dampener to simulate a car's suspension. A car’s suspension is approximated by a single degree-of-freedom mass-spring-damper system. The displacement $x(t)$ of the spring mass satisfies: $mx''+cx'+kx=F(t)$. where $m$ is the mass, $c$ is the viscous damping and $k$ is the spring stiffness and $F$ is an external forcing. You need to convert this 2nd order ODE into two 1st order ODE, then you solve for both the position and velocity over time. + + +another kind of ODE system could be the Zeldovich system for NOx generation that you have learned in Combustion. + +you can make a system of 5 equations with 5 unknowns to solve for each of the species vs time: N2, O, O2, N, NO. + +it might be very stiff but we can make a toy system by adjusting the rates to make it easier to integrate using basic ODE solver methods like euler.
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