Apply Newton’s method to solve Problem 2.3 for with a tolerance of 10 -5 . Assume that the lengths..

Apply Newton’s method to solve Problem 2.3 for  with a
tolerance of 10_5. Assume that the lengths of the rods are a1
= 10 cm, a2 = 13 cm, a3 = 8 cm and a4
= 10 cm. For each value of  consider two possible
initial data, (0) = _0.1 and (0)
= 2_/3.

Problem 2.3 (Rods system) Let us consider the mechanical
system represented by the four rigid rods ai of Figure 2.1. For any
admissible value of the angle , let us determine the
value of the corresponding angle _ between the rods a1 and a2.
Starting from the vector identity

»

Apply Newton’s method to solve Problem 2.3 for  with a
tolerance of 10_5. Assume that the lengths of the rods are a1
= 10 cm, a2 = 13 cm, a3 = 8 cm and a4
= 10 cm. For each value of  consider two possible
initial data, (0) = _0.1 and (0)
= 2_/3.

Problem 2.3 (Rods system) Let us consider the mechanical
system represented by the four rigid rods ai of Figure 2.1. For any
admissible value of the angle , let us determine the
value of the corresponding angle _ between the rods a1 and a2.
Starting from the vector identity

and noting that the rod a1 is always aligned with
the -axis, we can deduce the following relationship
between  and _:

where ai is the known length of the ith rod. This is
called the Freudenstein equation, and we can rewrite it as f(_) =
0, where

A solution in explicit form is available only for special
values of. We would also like to mention that
a solution does not exist for all values of , and may not even be unique. To
solve the equation for any given  lying between 0
and _ we should invoke numerical methods (see Exercise 2.9).

»

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