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

EGM6321 - Principles of Engineering Analysis 1, Fall 2010

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Mtg 3: Thur, 26 Aug 10

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

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

In 3_D,  

 

(1)

Where the lowercase   in the first term   is defined as  

and   is the separation of variables

 

 

(2)

Where   in the first term   is defined as  

and   is the separation of variables

Separated equations for  

 

(3)

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

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