EGM6321 - Principles of Engineering Analysis 1, Fall 2010
[edit | edit source]
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
![](//upload.wikimedia.org/wikipedia/commons/thumb/d/dc/EGM6341.s11.TEAM1.WILKS_EC3.svg/500px-EGM6341.s11.TEAM1.WILKS_EC3.svg.png)
|
(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.