User:Egm6322.s12.team2.steele.m2/Mtg15

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

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Plan: SC-L1-ODE-VC : System of Coupled L1-ODE-VC SC-L1-ODE-CC: SC-L1-ODE with Constant Coefficients Particular case: when n=1 (L1-ODE-VC) Solution by IFM Generalization: SC-L1-ODE-CC Exponentiation of a matrix Generalization: SC-L1-ODE-VC State transition matrix Coupled pendulums: Integrate SC-L1-ODE-CC Note: p.14-0 cont’d

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=SC-L1-ODE-VC: System of Coupled L1-ODE-VC= (linear time variant) SC-L1-ODE-CC: SC-L1-ODE with Constant Coefficients (linear time invariant) Particular case: When n=1 (L1-ODE-VC) L1-ODE-CC: R*3.7: Use IFM to show that the solution of (2) is

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Identify the integrating factor, the homogeneous solution, and the particular solution. Show that the solution of the L1-ODE-VC (1) p.15-1 is Generalization: SC-L1-ODE-CC (constant coefficients)

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Exponentiation of a matrix:

Nxn 	nxn		nxn		nxn		nxn Identity matrix R*3.8: Generalization (2) p.15-2 to the case of linear time-variant system [[media:Pea1.f11.mtg14.djvu|Eqn(1) p.14-4]]. See Eqn(1) p.15-2. Verify that your expression is indeed the solution for (1) p.14-4. Bryson & Ho 1975 (A4.9) p.450:

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Compare (3) p.15-3 to (2) p.15-2 (for L1-ODE-CC):

The fundamental or state transition matrix $$\Phi$$ is related to the integrating factor. Properties of the state transition matrix $$\Phi$$ Nxn 		nxn

Nxn	nxn R*3.9: Verify that (1) satisfies (2)-(3).

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R3.10: Free vibration of coupled pendulums p.14-5 Pendulums: $$m_1 g = 3, \ m_2 g = 6$$ No applied forces: $$u_1 = u_2 = 0$$ Initial conditions $$\theta_1(0) = 0, \ \dot \theta_1(0) = -2$$ $$\theta_2(0) = 0, \ \dot \theta_2(0) = +1$$ 1.	Use matlab’s ode45 command to integrate the system [[media:Pea1.f11.mtg14.djvu|Eqns(1)-(2) p.14-5]] in matrix form (1) p.14-4 for $$t \in [0,7]$$. Can also use equivalent Octave command. 2.	Use (2) p.15-2 to find the solution at the same time stations as in Q1. 3.	Plot $$\theta_1(t)$$ from Q1 and from Q2. 4.	Plot $$\theta_2(t)$$ from Q1 and from Q2.