User:Egm6322.s12.team2.steele.m2/Mtg27+28+29

=EGM6321 - Principles of Engineering Analysis 1, Fall 2011=

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Notes, Sec 27 Plan: Are the Legendre and Bessel L2-ODEs-VC exact? Hermite L2-ODE-VC? Vibration: Chladni patterns Vibrating membrane: 2-D case, Cartesian coordinates 1-D case Separation of variables Separated equation: 1-D Helmholtz eq. Trial solutions: Sine waves More general solution Wave eq. in 1-D: Shape preserving Heat equation (1-D, 2-D, 3-D) Euler-Bernoulli beam: Equation of motion Free vibration

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Are the Legendre and Bessel L2-ODEs-VC exact? Hermite L2-ODE-VC? *5.2: Consider the following governing L2-ODEs-VC for some classical special functions: Legendre: F09, F11 $$G=(1-x^2)y\prime \prime - 2xy\prime + n(n+1)y = 0$$ [[media:Pea1.f11.mtg7.djvu|Eqn(1) p.7]] Bessel: F10

Hermite: F11 1.	Verify the exactness of the designated L2-ODEs-VC. For the 2nd exactness condition, use 2 methods: 1a. [[media:Pea1.f11.mtg16.djvu|Eqns(1)-(2) p.16-5]] 2b. [[media:Pea1.f11.mtg22.djvu|Eqn(1) p.22-3]]

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2. If (2) p.27-1 is not exact, check whether it is in power form, and see whether if it can be made exact using IFM with $$h(x,y) = x^m y^n$$; see p.21-2. 3. The first few Hermite polynomials are

Verify that (1)-(3) are homogeneous solutions of the Hermite differential equation (2) p.27-1.

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=Vibration: Chladni Patterns= See image with power and flexible plate showing sound wave. Sound wave is generated by acoustic speaker with tunable frequency

See sinusoidal wave in image with “Powder falling toward the ‘node’” and “Node.” Note the single vibrational mode of the plate. Vibrating membrane: $$\rho $$ = mass density (per unit membrane area) T = uniform tension in membrane

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See image for (x,y) and u on Cartesian space of x, y, and z.

Note u(x,y,t) = transverse displacement of membrane at point (x,y) and at time t.

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=Equation of Motion=

Coordinate independent See EML 5526 S11 p.38-4, p. 40-4, p.42-2 2-D case, Cartesian coordinates

1-D case: Two independent variables (x, t)

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=Separation of variables: ansatz = trial solution = guess=

Separated equation for X(x): 1-D Helmholtz eq. See [[media:Pea1.f11.mtg4.djvu|Eqn(1) p.4-3]]; homogeneous L2-ODE-CC:

Trial solutions: $$\cos kx, \ \sin kx$$

Euler’s formula http://en.wikipedia.org/wiki/Euler%27s_formula_in_complex_analysis

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i.e., a linear combination of $$\cos kx, \sin kx$$ with complex coefficients. Note: (1) p.24-6:

Same result is obtained if the ansatz had a phase shift, i.e.,

A is amplitude above.

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$$\omega$$ = (angular) frequency in time K = frequency in space (wave number)

King 2003 p.82 See diagram with parameters of F, speed c, and separate patterns for $$t = t_1$$ and $$t = t_2 > t_1$$ [[image: [[image: Pea1.f11.wave.propagation.svg|500px]]

Wave equation in 1-D: Shape preserving Speed c>0 + = outgoing wave (+ x direction) -	= incoming wave (- x direction)

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Wave eq. in 2-D: More complex Wave eq. in 3-D: Even more complex Wave eq. in curvilinear coordinates: Even more complex =Heat equation (1-D, 2-D, 3-D):=

$$\kappa$$ = conductivity tensor (matrix) Note u = temperature Note f = distributed heat source Note ρ = mass density Note c = specific heat (heat capacity per unit mass)

Note the plus sign for the diffusion term $${\color{red} +}{\rm div( \boldsymbol \kappa \cdot grad}) \, u$$ See EML 5526 S11 p.33-2, p.32-3

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Sum on i and sum on j

Kronecker delta: Note (1) p.27-9, and (1)-(3):

Sum on i

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Note: Finite-element discretization of (1) p.27-9

$$\mathbf M \mathbf{\dot d} + \mathbf K \mathbf d = \mathbf F$$ Plus sign becomes minus sign after integration by parts; see EML 5526 S11 p.34-2.

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=Euler-Bernoulli Beam:= See beam for f(x,t) and curves shown for undeformed and deformed states. See u(x,t). [[image: [[image: Pea1.f11.euler.bernoulli.beam.svg|500px]]

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Euler-Bernoulli beam: Equation of motion $${\color{red}-} EI \frac{\partial^4 u}{\partial x^4} + f = m \frac{\partial^2 u}{\partial t^2}$$	(1) Note the minus sign for the internal force term $$EI \frac{\partial^4 u}{\partial x^4}$$ E = Young’s Modulus I = moment inertia of cross section U = transverse displacement F = transverse distributed force M = mass per unit length Free vibration: f = 0 $${\color{red}-} EI \frac{\partial^4 u}{\partial x^4} = m \frac{\partial^2 u}{\partial t^2}$$	(2)

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=Note: Finite-element discretization of (1) p.27-12	= $${\color{red}-} EI \frac{\partial^4 u}{\partial x^4} + f = m \frac{\partial^2u}{\partial t^2}$$	(1) p.27-12 $$\mathbf M \mathbf{\ddot d} + \mathbf K \mathbf d = \mathbf F$$	(1) Minus sign stays minus after integration by parts twice; see EML 4500 F08 p.38-1.