User:Egm6322.s12.team2.steele.m2/Mtg33

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

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Plan: Application: Nonhomogeneous L2-ODE-CC cont’d Special IFM to solve (1) p.32-5 with general f(t). Deduce the particular solution $$y_P(t)$$ for general excitation f(t).

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=Application: Nonhomogeneous L2-ODE-CC cont’d= R*6.1: Special IFM to solve [[media:Pea1.f11.mtg27.djvu|Eqn(1) p.32-5]] with general f(t). 1.	Find the PDEs that govern the integrating factor h(x,y) for (1) p.32-5. Recall the 2 relations [[media:Pea1.f11.mtg16.djvu|Eqns(1)-(2) p.16-5]] in the 2nd exactness condition for N2-ODEs. Can you solve these PDEs for h(x,y)? 2.	Trial solution for the integrating factor.

Which is similar to the trial solution for the Euler L2-ODE-CC (a homogeneous LN-ODE-CC), where $$\alpha$$ is unknown to be determined. (1)	And (1) p.32-5:

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Because of the integrating factor in exponential form, assume the l.h.s. of (2) p.33-1 takes the form: Clearly $$\bar a_2 = 0$$ to avoid having y’’’ when differentiating the r.h.s. of (1); an advantage is to reduce the order of the resulting ODE. 2.1.	Find $$(\bar a_1, \ \bar a_0)$$ in terms of $$(a_0, \ a_1, \ a_2)$$. Hint: Differentiate the r.h.s. of (1) and compare to integrand on l.h.s. of (1); see also (2) p.33-1. 2.2.	Find the quadratic equation for $$\alpha$$ Hint: Use (2b) and (2c)

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2.3	Reduced-order equation: (1) p.33-2 and (2) p.33-1 lead to

A L1-ODE-CC, thus easily solvable by the IFM. 2.4	Use the IFM to solve (3). (3)	P.11-4 and R*2.15 p.12-2: Find the solution y(t) for general excitation f(t). See [[media:Pea1.f11.mtg11.djvu|Eqn(1) p.11-5]]

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Show that

Thus $$(\alpha, \beta)$$ are roots of the quadratic equation:

Which is the same as (1) p.33-3. 2.5	Deduce the particular solution $$y_P(t)$$ for general excitation f(t).

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2.6	Verify result with table of particular solutions for: 2.7	Solve the nohomogeneous L2-ODE-CC [[media:Pea1.f11.mtg32.djvu|Eqn(1) p.32-5]] With the following excitation: Gaussian Distribution: For the coefficients $$a_0, a_1, a_2$$ consider two different characteristic equations:

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2.8	For each case in 2.8.1 and 2.8.2, determine the fundamental period of undamped free vibration, and plot the solution for the excitation (4) p.33-5 for about 5 periods, assuming zero initial conditions. Note: In the above special IFM, instead of trying to render exact the complete non-homogeneous L2-ODE-CC in (1) p.32-5, since this task would be difficult due to the generality of the excitation function f(t), it is much easier to render exact, and thus reduce the order of, only the l.h.s. of (1) p.32-5. The resulting ODE is of 1st order, i.e., L1-ODE-CC, which can always be solved by the IFM.