User:Egm6322.s12.team2.steele.m2/Mtg32

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

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Plan: $$\Rightarrow$$Application: Euler L2-Odes with multiple roots Method of variation of parameters Famous application in astronomy: Discovery of Neptune $$\Rightarrow$$Application: Nonhomogeneous L2-ODE-CC Engineering applications Spring-dashpot-mass system RLC electric circuit Circuit equation

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=Application: Euler L2-ODEs with Multiple Roots= Method of variation of parameters R*5.7: Consider characteristic equation:

Where $$\lambda$$ is a given number: $$\lambda = 5$$ F10 or $$\lambda = 7$$ F11 1)	 Euler L2-ODE-VC

1.1)	Find $$a_2, a_1, a_0$$ such that (1) is a characteristic equation of (2).

1.3)	 Complete solution: Find c(x) such that

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The above method is called variation of constants or variation of parameters. See historical notes p.32-3. Find the differential equation governing c(x), and solve for c(x) to obtain y(x). 1.4)	Find the 2nd homogeneous solution $$y_2(x)$$ 2)	Euler L2-ODE-CC

Repeat the steps in part 1 above for (1). R*5.8: Recall p.12-2, R*2.17; use the same idea of variation of constants (parameters) to find the particular solution $$y_P(x)$$ after knowing the homogeneous solution $$y_H(x)$$, i.e., let $$y(x) = A(x)y_H(x)$$ , with A(x) being the unknown to be found.

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“Analytical development of the variation of parameters was first given by Leonhard Euler in a series of memoirs on the mutual perturbations of Jupiter and Saturn … in the years 1748 and 1752. The method is also called the variations of orbital elements or the variation of constants... Euler’s treatment of the method of variation of parameters was not entirely general since he did not consider the orbital elements as being simultaneously variable. …Joseph-Louis Lagrange wrote … on the perturbations of Jupiter and Saturn in 1766 … made further advances in the variation of parameters method. … Later, in 1782, he developed completely and for the first time the method of the variation of parameters in a prize memoir on the perturbations of comets moving in elliptical orbits. … The most dramatic application of the method was made independently and almost simultaneously by the Englishman J.C. Adams (1819-1892) and the Frenchman U.-J.-J. Le Verrier (1811-1877). Each predicted the existence and apparent position of the planet Neptune from the otherwise unexplained irregularities in the motion of Uranus. The story is one of the most fascinating in the history of astronomy and is an impressive example of the precision which can be achieved using variational methods.” Battin 1999, Astrodynamics …, p.471

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“In 1842, F.W. Bessel, suspecting the presence of an ultra-Uranian planet, announced his intention of investigating the motion of Uranus. Unfortunately, he died before much could be accomplished. … On July 3, 1841, an undergraduate at St. John’s College in Cambridge, England wrote in his journal “Formed a design in the beginning of this week of investigating, as soon as possible after taking my degree, the irregularities in the motion of Uranus … in order to find out whether they may be attributed to the action of an undiscovered planet beyond it …” … by 1845 J.C. Adams had obtained a solution and in September of that year he gave the results of his computations, on where the new planet could be found, to J. Challis, director of the Cambridge observatory. Challis expressed little interest. … Le Verrier turned his attention to the Uranus problem and published his results on June 1, 1846. … On September 18, 1846, Le Verrier requested J. G. Galle to look for the planet … on September 23, 1846, after only an hour, Galle found the planet Neptune within one degree of the position computed by Le Verrier.” Battin 1999, Astrodynamics…, p.473

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Application: Nonhomogeneous L2-ODE-CC

Engineering applications: Spring-dashpot-mass system: See diagram for spring with constant k, c, and mass m. Note f(t). See diagram for RLC circuit with R, L, C, V and $$\nu_R, \nu_L, \nu_C$$ [[image: [[image: Pea1.f11.spring.svg|500px]] [[image: [[image: Pea1.f11.rlc.circuit.svg|500px]]

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RLC circuit in series Differences in potential

Resistance:

Inductance:

Capacitance:

Circuit equation:

See e.g., Bird 2003, Electric circuit theory and technology, p.910

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Recall: Undergraduate; solve (1) p.32-5 1)	Find $$y_H(x)$$ (homogeneous solution) For L2-ODEs, $$y_H(x)$$ is a linear combination of two linearly-independent homogeneous solutions. Recall: Homogeneous L2-ODE-CC = Euler L2-ODE-CC 2)	Find $$y_P(x)$$ (particular solution) Use look-up table with a few particular, regularly encountered cases (F09 Mtgs 20, 22). See, e.g., Boyce & DiPrima 2001 p. 175. R*6.1: p.33-1 Special IFM to solve for general f(t) in (1) p.32-5.