Analysis of the Harmonic Oscillator without Forcing

Analysis of the Harmonic Oscillator without Forcing -...

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Unformatted text preview: Analysis of the Harmonic Oscillator without Forcing Kevin Brent Click to edit Master subtitle style Akshaya Annavajhala 5/12/09 y' = v The Harmonic Oscillator y'= v mv '+ ( y 2 - )v + ky = 0 mv '+ bv + ky = 0 y = position v = velocity m = mass y'= v mv '+ b | v | v + ky = 0 =2 =2 b = damping coefficient 5/12/09 k = spring constant/restorative force Undamped Harmonic Oscillator y'= v 2v '+ 5 y = 0 Periodic solutions T = 3.97 Applications: mass on spring or L-C circuit with no energy loss due to resistance (mass on ice...). Obviously, not accurate physical 5/12/09 (continued) 0 1 A = -5 0 2 Tr = 0 => center y(0) = 6, v(0) = 0 5/12/09 Linearly-Damped Harmonic Oscillator y' = v 2v '+ 2v + 5 y = 0 Periodic solutions, but Matlab cannot find the period (do not trust Matlab). Analytically, period T = 4.19 Applications: Many real-world uses: mass on spring or L-C circuit where friction/resistance reduces total 5/12/09 (continued) A 0 = -5 2 1 -1 Tr = -1, Det = 5/2 => Spiral y(0) = 6, v(0) = 0 5/12/09 Harmonic Oscillator with Non-Linear Damping y' = v 2v '+ 2 | v | v + 5 y = 0 Periodic solutions (not sinusoidal!) with T = 3.97, once again obtained analytically, rather than using Matlab Applications: Drag on airplane tires landing on runway covered with slush or water 5/12/09 (continued) J(0,0) 1 0 =5 - -2v 2 0 1 = 5 - 0 2 => center?? y(0) = 8, v(0) = 0 5/12/09 Harmonic Oscillator with Second-Order Damping y'= v mv '+ ( y - )v + ky = 0 2 Periodic solutions again (once again not sinusoidal), with period T = 4.19 Applications: an L-C circuit with a periodically fluctuating voltage input and energy loss due to resistance. The resistance dampens the voltage through the capacitor to a periodic 5/12/09 (continued) J 1 0 = -k - 2vy - y 2 m m 0 1 = 5 3 - 2 2 y(0) = 0, v(0) = 2 => Tr=3/2, Det = y(0) = 0, v(0) = 10 5/12/09 ...
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