Part 02

Mech300 Numerical Methods, Hong Kong University of Science and Technology.

Part Two Roots of Equations

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Mech300 Numerical Methods, Hong Kong University of Science and Technology.

Motivation Analytical solution: − b ± b 2 − 4ac x= 2a

f ( x ) = ax 2 + bx + c = 0

But how about e x + 3 sin x − x log x + 4 = 0 ?

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Approximation solution (pre-computer): Plot method

f(x)

x Trial and error

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Mech300 Numerical Methods, Hong Kong University of Science and Technology.

An Example of Roots of Equation in Engineering Parachutist’s velocity:

v=

gm ( 1 − e −( c / m )t ) c

Problem: determine the drag coefficient c for a parachutist of a given mass m to attain a prescribed velocity in a set time period, that is, Given v, m, g, and t, find c. Approach 1: try to represent c as an explicit function c = f(v, m, g, t) (fails most of time) Approach 2: express the formula in an explicit form and solve for the zero root gm f (c )= ( 1 − e −( c / m )t ) − v = 0 c 3

Mech300 Numerical Methods, Hong Kong University of Science and Technology.

Fundamental Principles Used in Engineering Design Problems

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Mech300 Numerical Methods, Hong Kong University of Science and Technology.

Overall Structure

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