User:Egm6322.s12.team2.steele.m2/Mtg3

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

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Note: R*m.n = Problem either (i) already explained in lectures in previous versions, or (ii) already assigned with solution in reports in previous years. You can do one of the following: Solve the problem without looking at existing solutions (lecture notes or reports). Put a boxed comment at the top of your solution saying that you did the solution on your own, without looking at existing solution. Example: EGM 6341 Spring 2011 (NM1 S11) Team 5 HW5.1 Look at existing solutions to get help for the solution. Need to (i) refer to the relevant documents (lecture notes, reports), with proper medawiki links, and (ii) put a boxed comment at the top of your solution indicating the improvements that you brought to the existing solutions (e.g., provide clearer explanation for “dummies”, deeper theoretical background or applications, correct misprints/errors, etc., or if your solution is completely different, then point out the differences). Example: PEA1 F10 Team 6 HW1.1 Ref[7].NM1 S11 Team 5 HW1.3. (need a link to “existing solution”)

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[[media:Pea1.f11.mtg2.djvu|Eqn(2) p.2-4:]]

With the acceleration a being one of the following quantities $$\ddot u^1$$, $$\ddot u^2$$, $$\ddot Y^1$$ R*1.2: Derive (3) p.2-4 and (1) p.2-5, and show the similarity with the derivation of the Coriolis force. Note: Number (1) P. 2-5 is similar to the acceleration of a particle in a rotating frame that gives rise to the Coriolis force (undergraduate dynamics). Number (3) p.2-4 is related to the material time derivative and the Reynolds Transport Theorem in Continuum Mechanics (which covers Heat, Solids, Fluids, Electromagnetics) Intelligence consists of this; that we recognize the similarity between different things and the difference between similar things. Montesquieu (1689-1755)

=Equations of Motion (EOM) of Wheel/Magnet =

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Note: Terms with $$\dot Y^1$$ are not due to damping, but to convection. Ref: VQ&O 1989 CMAME Eqs.(2.1bc). There is a misprint in (2.1b).

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M = mass of wheel/magnet. From (2) p.3-3:   is horizontal force acting on wheel/magnet

Dimension of $$F^1$$		force [1] = 1

Second derivative $$\frac{\partial^2 u^2}{\partial S^2}$$ superscript Curvature	power $$(\partial S)^2$$ $$\displaystyle [u^2] = L \Rightarrow \underbrace{[u^2_{,SS}]}_{\color{blue}{\displaystyle \frac{\partial^2 u^2}{\partial S^2}}}$$

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R*1.3: Analyze the dimension of all terms in (2) p.3-3, and provide the physical meaning. Numer (1) P.3-3: “Ordinary” Differential Equation (ODE) Order = Highest order of derivative Nonlinearity = What is linearity? Use intuition for now, formal definition coming soon. EOM (1) p.3-3 = Nonlinear 2nd-order ODE(N2-ODE) System has 3 unknown functions: $$Y^1(t), \ u^1(S,t), \ u^2S,t)$$ 1 nonlinear ODE for $$Y^1(t)$$ 2 PDEs for $$u^1(S,t)$$ and $$u^2(S,t)$$ . PDE = Partial Differential Equations 3 equations are coupled: use numerical methods; see Vu-Quoc & Olsson 1989 (2.7ab)