University of Florida/Egm6341/s11.TEAM1.WILKS/Mtg3

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EGM6321 - Principles of Engineering Analysis 1, Fall 2010[edit]


Mtg 3: Thur, 26 Aug 10

Page 3-1

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NOTE: - page numbering 3-1 defined as meeting number 3, page 1

- T = torque Fig.p.1-1

- HW*

Eq.(3)P.2-1 : "Ordinary" Differential Equation (ODE)

order = highest order of derivative

Nonlinearity = What is linearity? ; use intuition for now, formal definition soon.

System has 3 unknowns:







Partial Differential Equations (PDE)

3 equations are coupled Numerical Methods

Simplify for analytical solution Ref:VQ&O 1989

2nd Order 2nd Order

nonlinear linear

unknown varying coefficient known varying coefficient

Page 3-2

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Note: Math structure of coefficient for is known, but not their values until and are known (solved for)

General structure of Linear 2nd order ODEs with varying coefficients (L2_ODE_VC)

(1)

where

independant variable

dependant variable (unknown function to solve for)

Many applications in engineering are a result of solving PDEs by separation of variables. Some examples include, but are not limited to: Heat, Solids, Fluids, Acoustics and electrmagnetics.

Examples of these types equations are:

the Helmholz equation:

and the Laplace Equation:

Ref F09 Mtg.28, Ref F09 Mtg.29 , Ref F09 Mtg.30

Page 3-3

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In 3_D,

(1)

Where the lowercase in the first term is defined as

and is the separation of variables

EGM6341.s11.TEAM1.WILKS EC3.svg

(2)

Where in the first term is defined as

and is the separation of variables

Separated equations for

(3)

Page 3-4

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Simplify:









Eq.(3)p.3-3:

(1)

Where

Particular case of Eq.(1)p.3-2

Linearity: Let be an operator.

and are 2 possible arguments (could be functions) of



Where and are any arbitrary number.

References[edit]