Solving equation with numerical analysis

[AeroTron]

need hint/help or perhaps a solution
Equations are from Propeller Blade Theory. I´m trying to figure out how to determine
$\epsilon$_i from equation 2.3.25

my first thought was to substitute sin(A+B) = sinA*cosB + cosA*sinB...

it says quote: " This equation is easily solved numerically" but I have no clue what to do or where to start.

Thanks

 

[NascentOxygen]

A numerical solution deals with numbers. In the absence of better advice, this is how I'd interpret it:

Substitute a numerical value for each variable in equation 2.3.25 until you have an equation comprising numbers, r, ε_i and nothing else. Then, for example,

for r=5 to 300 in steps of .1
solve for the corresponding ε_i (do this using a numerical technique)

To your graph of ε_i vs. r fit a simple polynomial to allow easy mathematical analysis for the region of practical interest.

Does that sound right?

 

[D H]

my first thought was to substitute sin(A+B) = sinA*cosB + cosA*sinB...

it says quote: " This equation is easily solved numerically" but I have no clue what to do or where to start.

Thanks

You are trying to solve the equation analytically. There is no closed form analytic solution to that equation, at least not in the elementary functions.

There are plenty of relations that can be expressed very simply one way but not the reverse. A very simple example is Kepler's equation, $M = E - e\sin E$. There is no simple way to express $E$ in terms of $e$ and $M$. One instead solves the reverse Kepler problem using numerical methods.

The same applies to your problem. Your problem is a transcendental equation, and transcendental equations in general do not have closed form solutions. So you use numerical techniques instead. This page, http://en.wikibooks.org/wiki/Numerical_Methods/Equation_Solving gives a brief overview of some of the numerical techniques that can be used to solve such equations.