User:Egm6322.s12.team2.steele.m2/Mtg30

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

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Plan: Euler-Bernoulli beam: p.27-14 cont’d Free vibration Separation of variables Separated equations L4-ODE-CC Method of trial solution Generalization of (3) p.30-2: Euler equation Special homogeneous Ln-ODE-VC = Euler Ln-ODE-VC Method 1 Method 2 Application: Euler (homogeneous) L2-ODE-VC

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=Euler-Bernoulli Beam: p.27-14 cont’d= Free vibration: p.27-13 $${\color{red}-} EI \frac{\partial^4 u}{\partial x^4} = m \frac{\partial^2u}{\partial t^2}$$ Separation of variables: consider the trial solution of the form

Separated equations:

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Vibration problem: Consider periodic trial function

Note (5) P.30-1 and (2): L4-ODE-CC

Method of trial solution (undetermined coefficients) King p. 513 Sec A5.3

Note r undetermined coefficient, root of a characteristic equation

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4 solutions:

$$i := \sqrt{-1}$$	[[media:Pea1.f11.mtg27.djvu|Eqn(2) p.27-6]] R*5.3: Find the expressions for X(x) in terms of $$\cos Kx, \ \sin Kx, \ \cosh Kx  \ \sinh Kx$$ Recall Euler’s formula [[media:Pea1.f11.mtg27.djvu|Eqn(5) p.27-6]] for $$\cos Kx, \ \sin Kx$$ ; A similar relation exists for$$\cosh Kx, \ \sinh Kx$$. See Note p.30-8. Generalization of (3) p.30-2: Euler equation Special homogeneous Ln-ODE-VC = Euler Ln-ODE-VC

(2)	P.4-6:         $$y\prime = y^{(1)}$$          $$y =  y^{(0)}$$          homogeneous

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Euler equation with constant coefficients = Euler Ln-ODE-CC

Includes (3) p.30-2 as a particular case. Two methods for solving Euler Ln-ODE-VC (1). Method 1: Stage 1: Transformation of variables

Stage 2: Trial solution

Method 2: Trial solution

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=Application: Euler (homogeneous) L2-ODE-VC= (1) Method 1: Stage 1: Transformation of variables

Current variable x -> new variable t

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Another way:

Using (4) p.30-5 and

In (1), we obtain (5) p.30-5 Apply (2) to find

Using (4a) p.30-5 and (3), together with

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Note: Generate “real” (non-complex) solution See R*5.3 p.30-3 Note (5) p.30-2 and (1) p.30-3:

Since X(x) is a real-valued function so the r.h.s. of (1) must also be real. Since (3) p.30-2 is an L4-ODE-CC, there should be 4 real constants. Since$$r_{1,2} = \pm K \in \mathbb R$$, it follows that

Since$$r_{3,4} = \pm i \, K \in \mathbb C$$, Euler’s formula [[media:Pea1.f11.mtg12.djvu|Eqn(5) p.27-6]] leads to

Not acceptable; so find $$c_3, c_4 \in \mathbb C$$ such that

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(1)	P.30-3 and Euler’s formula (5) p.27-6, then:

The 4 real constants are: $${c_1, c_2, b_3, b_4} \in \mathbb R^4$$