Solving non-linear equations the graphical, Bisection, Newton-Raphson, Regula Falsi or Muller's method.

2.1

), with an insulated tip, is given by [2.15]



with



, determine the necessary cross-sectional dimensions of the fin to
achieve an efficiency of 0.95 using the following methods:

Graphical method

Bisection method

Newton-Raphson method

Regula Falsi method

Muller’s method







(a) Graphical Method

with 0.95 as the slope.

. The intersection points (where the two graphs crosses or touches)
represent the roots of the frequency equation.



< 5,

> 0.2

(b) Bisection Method

(c) Newton-Raphson Method





(d) Regula Falsi Method



(e) Muller’s Method





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2.2

The natural frequencies of a beam, fixed at one end (x = 0), and
carrying a mass M at the other end (x = l), are given by the frequency
equation



where

is the mass ratio,

M = mass attached at the end of the beam;

= mass of the beam.

,

is the density and A is the cross-sectional area of the beam.

= 0.0, 1.0, 10.0, 100.0, and 1000.0.

.





(a)





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