Analysis of the Harmonic Oscillator without Forcing

Analysis of the Harmonic Oscillator without Forcing -...

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Unformatted text preview: Analysis of t he H ar monic Oscillat or wit hout For cing K evin Br ent Akshaya Annavajhala y' = v The H ar monic Oscillat or y'= v mv '+ ( y 2 - )v + ky = 0 mv '+ bv + ky = 0 y = posit ion v = velocit y m = mass =2 b = damping coefficient k = spr ing const ant /r est or at ive for ce =5 = second or der damping coeff. y'= v mv '+ b | v | v + ky = 0 =2 Undamped H ar monic Oscillat or y'= v 2v '+ 5 y = 0 Per iodic solut ions T = 3.97 Applicat ions: mass on spr ing or L -C cir cuit wit h no ener gy loss due t o r esist ance (mass on ice...). Obviously, not accur at e physical model. (cont inued) 0 1 A = 5 - 0 2 Tr = 0 => cent er y(0) = 6, v(0) = 0 L inear ly-Damped H ar monic Oscillat or y'= v 2v '+ 2v + 5 y = 0 Per iodic solut ions, but M at lab cannot find t he per iod (do not t r ust M at lab). Analyt ically, per iod T = 4.19 Applicat ions: M any r eal-wor ld uses: mass on spr ing or L -C cir cuit wher e fr ict ion/r esist ance r educes t ot al ener gy E in t he syst em. (cont inued) A 0 = 5 - 2 1 -1 Tr = -1, Det = 5/2 => Spir al Sink y(0) = 6, v(0) = 0 H ar monic Oscillat or wit h NonL inear Damping y'= v 2v '+ 2 | v | v + 5 y = 0 Per iodic solut ions (not sinusoidal!) wit h T = 3.97, once again obt ained analyt ically, r at her t han using M at lab Applicat ions: Dr ag on air plane t ir es landing on r unway cover ed wit h slush or wat er (cont inued) J (0,0) = 1 0 -5 -2v 2 0 1 = 5 - 0 2 => cent er ?? Appr ox!! y(0) = 8, v(0) = 0 H ar monic Oscillat or wit h SecondOr der Damping y'= v mv '+ ( y - )v + ky = 0 2 Per iodic solut ions again (once again not sinusoidal), wit h per iod T = 4.19 Applicat ions: an L -C cir cuit wit h a per iodically fluct uat ing volt age input and ener gy loss due t o r esist ance. The r esist ance dampens t he volt age t hr ough t he capacit or t o a per iodic limit ing funct ion. (cont inued) J 1 0 = -k - 2vy - y 2 m m 0 1 = 5 3 - 2 2 => Tr =3/2, Det = 5/2 y(0) = 0, v(0) = 2 y(0) = 0, v(0) = 10 ...
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