User:Egm6321.f10.team4.Yoon/Mtg24

=EGM6321 - Principles of Engineering Analysis 1, Fall 2010=

[[media: 2010_10_14_14_02_27.djvu | Mtg 24:]] Thu, 14 Oct 10

= Are the Legendre and Bessel L2-ODEs-VC exact ? =

[[media: 2010_10_14_14_02_27.djvu | Page 24-1]]
F09: Legendre Equation L2-ODE-VC

Ref: K.2003 p.31; See [[media: 2010_09_02_13_55_50.djvu | Eqn(1)p.5-4]]

1)Verify exactness of (1) using 2 methods 1a) [[media: 2010_09_23_14_52_54.djvu | Eqn.(1) & Eqn.(2) p.15-3]] 1b) [[media: 2010_10_08_17_07_22.djvu | Eqn.(3) p.22-4]] 2) If Eqn.(1) is not exact, see whether if it can be made exact using IFM with $$\displaystyle h(x,y) = x^m y^n$$ ; see [[media: 2010_09_30_15_51_32.djvu | p.19-1]] & [[media: 2010_10_08_17_07_22.djvu | p22-1]]


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HW: Do the same for Bessel equation $$\displaystyle F=(1-x^2)y'' - 2xy' + (x^2- \nu^2)y = 0; \qquad \upsilon \in \mathbb{R}$$ Ref.: K. 2003p.58
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Next we look at vibration, which is an important engineering application.

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= Vibration: Chladni patterns =



The above figure provides a conceptual explanation of the experimental setup to display the Chaldni patterns.

Heat equations:

Note the plus sign in the diffusion ($$\displaystyle {\rm div \ grad}$$) term.

1-D case:

=Equation of motion for Euler-Bernoulli beam=

Note the minus sign for the internal force term $$EI \frac{\partial^4 u}{\partial x^4}$$.

Free vibration
$$\displaystyle f=0 $$

Note:

Finite element discretization of the heat equation:

End Note