Determine the damping factor ratio using Secant method.

The Secant Method is similar to Newton’s Method, but, instead of using calculus to find gradients, it calculates gradients (slopes) on a finite basis, that is, taking two points close to one another and estimating the gradient between them.

In this problem let’s replace omega/p with the variable x.

Let f(x)=2-1/√([1-x²]²+4(0.1221)²x²).

The slope between two points (x₀,f(x₀)) and (x₁,f(x₁)) is:

(f(x₁)-f(x₀))/(x₁-x₀).

So the Secant Method to find x₂ is:

x₂=x₁-f(x₁)(x₁-x₀)/(f(x₁)-f(x₀)).

A graph shows that there are two positive roots to f(x)=0. We use initial guesses close to these roots to apply the Secant Method.

The tables below show the initial values as x=0.5 and 1.2, and successive iterations using an x increment of 0.1.

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x'=x+0.1. This is part of the mechanism for calculating the slope. An arbitrary increment of x (0.1) has been applied to find the slope at (x',f(x')).

Slope=(f(x')-f(x))/(x'-x).

∆x=f(x')/slope. Next x = f(x')-∆x.

Summary x=0.72997, 1.18639.