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# Class Note for MATH 3331 with Professor He at UH 3

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This 13 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 19 views.

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Date Created: 02/06/15

Lecture 15 o o Havmnmc Mntmn u u nhnmngenenus mmquot 1W He Wm M WM WW M mm quotA iivlavamZXmm Section 44 F Harmonic Motion Undamped Case A amplitude qb phase angle choose 7r lt qb 3 7r lt gt i Undamped Case arctanc2c1 ifc1 gt O arctanC2cl 7r if cl lt 0 02 2 0 ya cl cosw0t 02 Sinw0t o arctancQc1 7r if cl lt O 32 lt O 2 39f Ad1 COSwot d2 snnwot 3732 f Z 8 8 where A 6 CE I TM Mew rmnnnm i 39 i rt Since df d 1 we can define qb by A V a a a T d1cos d2sinq5 yquot W DA i d2d1202c12tanq5 a I E E 1 CZ yt Acos COSwot sin gb sinwot I I I t 2A COSwot qb V V y39t w0A sin wot Altp27cT T2no Jiwen He University of Houston Math 3331 Section 19470 Lecture 15 February 25 2009 2 l3 Section 44 v 39 Harmonic Motionr Underdamped Case Underdamped Case yt e o tcl COSwt 02 Sinwt ca 0 54 COSwt gb y t eatw02 0401 COSwt wcl 0402 Sinwt 6 O tB COSwt 1p iAe O t iBe O t envelopes of damped oscillations Jiwen He University of Houston Math 3331 Section 19470 Lecture 15 February 25 2009 3 13 Section 44 Harmonic Motion Critically and Overdamped Cases Critically and Overdamped Cases 39 If y0 and y 0 have equal signs then y yt attains maximum or minimum 7 y t crosses zero If yO and y 0 have opposite signs then yt passage through zero crosses zero if iy Oy0i is large is monotonic if ly Oy0 is small overswmg yi maximum monotonic t 11 Jiwen He University of Houston 7 Math 3331 Section 19470 Lecture 15 February 25 2009 4 13 In Class Exercises Exercise 4411 Ex 4411 Given an undamped mass spring system with m 02kg k 5kg52 y0 05m y O 0 find amplitude frequency phase of motion Natural frequency we 2 502 2 55 gt yt c1 cos 575 l C2 sin 5t 305 2 5c1 sin 5t l 502 cos 5t IC y0 01 057 y 0 502 0 gt yt 05 COS 5t gt amplitude A 05m phase d O Jiwen He University of Houston Math 3331 Section 19470 Lecture 15 February 25 2009 5 13 7 In Class Exercises Exercise 4422 Ex 4422 A mass spring system with m 01 kg k 98 Icy32 is placed in a viscous medium with friction force 03N if v 02ms Initial data y0 01 m y O 0 Find amplitude frequency and phase of motion Friction coefficient Fd m 03 2 i102 gt u 15kg5 ODE 013 151 l 98g 2 O i y 151 983 0 gt pm A215A98 A7524175 A Jamm e 75i64612 gt Damped motion with frequency w s 64615 of harmonic part yt 677 5t01 COS cut C2 sin wt y t 6 7 5tw02 7501 COS wt cool I 75C2 sin wt IC y0 cl 01 y 0 c002 7501 0 gt C2 7501w m 0116 gt yt 6 7395t01 cos wt 0116 sin wt Jiwen He University of Houston Math 3331 Section 19470 Lecture 15 February 25 2009 6 13 In Class Exercises Exercise 4422 cont Amplitude of harmonic part A0 m xO12 01162 m 0153 Since 0102 gt 0 gt phase angle qb arctan02c1 z arctan116 m 0859 yt 0153e 75 cos6461t 0859 I amplitude At 0153e 7 5tm frequency w 64613 phase gb 0859 Jiwen He University of Houston Math 3331 Section 19470 Lecture 15 February 25 2009 7 13 m mm m uinn w the Mommas hnezvaquzinn w my co w 5 gm w m Mt m0 mo We a m 2 m mmmmum m Wu 5 2 mm mm m the Woman mm a n0 m m cm 2 mm Sm M mgquot m we mm Wm mm yupowwmy a am up m mahud uhmde evmma mum s 5 gt231 on ihehziihzi weremme mum whereihemmv Wham m m oz aws as w mm 5mg m M2 mm mhmun Wihehmngiemv m m 2 m m s Wham my dwmmm then m cm 2 5mmquot m 09 We gem m ismuumngm Section 45 General Solution Particular Solution Exponential Forcing Terms EX y y 6 15 a Trial Form ypt 2 ate 5 a Sub yp in ODE gt I yp l ypzae t l ae t2ae 726 t gt 2a 2 1 gt ypt et2 is PS Jiwen He University of Houston Math 3331 Section 19470 Lecture 15 February 25 2009 10 13 Section 45 Exponential Forcing Terms Ex y y 753 75 Try sub ypt 2 ate 5 in ODE gt a 2e t t t tea t Doesn t work gt use ypt a l bt6t yg yp a b 2 te t a btet 2a 2b 2btiet tet gt 2a 2b 2b 1 ypos 1 te t2 is PS Illi Math 3331 Section 19470 Lecturer 15 February 25 2909 11 137 Oab12 Section 45 General Solution Particular Solution Exponential Forcing Terms Ex y y 6 75 Try sub ypt 2 be 5 in ODE 7 gt y yp0 t Doesn t work gt use ypt ate t y yp a 2 l tet Mite t 2aet 6 7 gt a 12 gt ypt tet2 is PS I Jiwen He University of Houston Math 3331 Seatipn 19470 Lecture 1 5 February 25 2609 12 13 Section 45 General Solution Particular Solution Trig Forcing Terms EXi y y 2y cost Try sub yp acost in ODE gt a cost th 2cost 7 aCOSt sin 75 2 cost Doesn t work We need cost and sin 75 ypt acostbsint gt 31 y 29p 2 aI b cost ab th cost ab1 gtab Z Ogta b l2 gt ypt 2 cost sint2 is PS lll39l Jiwen He University of Houston Math 3331 Section 19470 Lecture 15 February 25 2009 13 13

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