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lecture11 - Section 4.1 The Mass Spring Oscillator March 3...

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Section 4.1 The Mass Spring Oscillator March 3, 2009 The Mass Spring Oscillator
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Today’s Session Section 4.1: A mechanical application of the general theory of second order DFQ’s There is a parallel (meaning same equation, but a di±erent interpretation) for electrical systems (which we will treat) A Summary of This Session: (1) Damped mass-spring oscillator; (2) Hooke’s law; (3) synchronization; (4) synchronous solution The Mass Spring Oscillator
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Repeated Roots, 3rd order DFQ Example 1: (a) Solve y ′′′ + 3 y ′′ + 3 y + y = 0 (b) Show that y 2 = x y 1 , y 3 = x 2 y 1 are independent solutions to the problem in part (a). (c) Write down the general solution to the problem. Answer: The characteristic equation is: r 3 + 3 r 2 + 3 r + 1 = 0. This reduces to ( r + 1) 3 = 0. So r = 1 is a triple (repeated) root. The general solution looks like: y = C 1 e x + C 2 xe x + C 3 x 2 e x . The Mass Spring Oscillator
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A second order DFQ with complex roots Example 2: (a) Solve y ′′ + 4 y = 0. (b) Solve
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lecture11 - Section 4.1 The Mass Spring Oscillator March 3...

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