The Mathematics of Harmonic Oscillators

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1 Th Mhcs of Hronc Oscllors Spl Hronc Moon In h cs of on-nsonl spl hronc oon (SHM nvolvng sprng wh sprng consn n wh no frcon, you rv h quon of oon usng Nwon's scon lw: con wh gvs: 0 Ths s sos wrn usng h o noon for rvvs: & Whr ch o ov vrl nos on rvv of h vrl. or spl ffrnl quons l hs on, whr you hv only n s rvvs wh consn coffcns, you fn h soluon y ung h s n ponnl funcon n fnng h chrcrsc quon: Lng: s gvs / s s n / s s Th chrcrsc quon s foun y susuon n splfcon: s s s s 0 s 0 0 s ± So s ±, whr. An h soluon hs h for: Tron ollg Pg of 8 PHY 06 John Blg

2 I cn shown wh so lgr h hs s h s s h su of wo snusos: ( ( ( ( ( sn( sn( I s u hr h n r copl conjugs. ro hr, s srghforwr rcs n rgonory o show h cn rprsn s sngl snuso: sn( sn( [ ] [ ] sn( sn( sn( sn( [ ] [ ] sn( sn( sn( If w l rcn, [ ] [ ] rcn( sn(rcn( sn( sn(rcn( rcn( sn( A A slr pproch cn us o pu n rs of sn: sn( β A whr rcn β Tron ollg Pg of 8 PHY 06 John Blg

3 W wll us h cosn for of h soluon: A No h for vry for, h frquncy of h oscllon s rn y h prrs of h sys, n no y h nl conons: f π T π π f Also no h ch for of h soluon rqurs wo consns h r rn y h nl conons. or h for w r usng, hos nl conons r h plu, n h phs ngl for 0. Tron ollg Pg 3 of 8 PHY 06 John Blg

4 Dp Oscllon In h cs whr h conons s for spl hronc oon, u hr s lso pv forc h s proporonl o vlocy, h oon s p. Th quon of oon s rv s follows: v 0 Agn, w cn l s, n fn h vlus for s h gv soluon. Th qurc forul lls us: s s s s s s 0 s 0 ± s 4 Thr r hr snc suons, rn y h scrnn: Unrp: Whn 4 < 0, hr r wo copl vlus for s. s σ ± ' whr σ n ' 4 Th soluon for hs h for: σ ' σ ' ' ' ( σ Usng h s hos w us for h unp cs: A σ ' Th unrp sys osclls whn cyng ponnl nvlop frquncy nr h nurl frquncy. Tron ollg Pg 4 of 8 PHY 06 John Blg

5 rclly Dp: Whn 4 0, or, h sys s crclly p. Th png s jus nough o prvn oscllon, n h oon s n ponnl cy wh lnr fcor. whr σ σ ( A A Th lnr fcor cos fro h fc h hr us wo consns n h soluon, no or, no l. Susuon no h ffrnl quon wll vrfy h hs s gnrl for of h soluon for h crclly p cs. Lnr fcors r no n h soluons for h unrp n ovrp c, cus hy woul rqur or hn wo consns. Ovrp: Whn 4 > 0, h soluon s h su of wo cyng ponnls: ± s 4 σ σ whr σ n σ r posv consns. or ll of h c, h σ n h ponnl cn pr n rs of consn, τ: σ n or τ n τ n σ n Th ponnls r hn of h for: A n τ n Th rgun of n ponnl pron us nsonl. I s goo prcc o pr h rgun s ro of wo hngs h hv h s uns, cus h pproch gnrlly rvls physcl nsghs or ofn. In hs cs, h consn τ n s chrcrsc oc wh h ponnl cy of h splcn. Tron ollg Pg 5 of 8 PHY 06 John Blg

6 Drvn Oscllon Drvn oscllon hs proc forcng funcon h prsuly woul ovrco h png of p hronc oscllor. Th soluon of h oc ffrnl quon s ll or ffor, u poron of h soluon s h s s h unrvn cs. Whn n ppl forc s of h for: Th ffrnl quon cos: ( Th soluon o hs ffrnl quon hs wo coponns: rns Whr rns s jus h soluon o h hoognous ffrnl quon: 0 Is conruon o h quon s lrlly zro, n s ncry o nclu. Ths coponn of h soluon s rnsn. I srvs s h corrcon o h sy-s soluon for s shorly fr 0, s h sys rnsons fro h nl conon o h sy s conon. W wll focus on h sys coponn ( of h soluon, whch scrs h sy-s hvor of h rvn sys. To fn h sy-s soluon, w u h soluon hs slr for o h of h forcng funcon. Snc hs s scon orr ffrnl quon, w n propos soluon wh wo consns. L us choos Thn hs ffrnl quon: cos: ( ( ( ( ( ( ( 0 Tron ollg Pg 6 of 8 PHY 06 John Blg

7 Now w hv quons for n : ( ( So: ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( sn( ( ( ( ( rcn cos ( ( rcn cos Ths s sply: ( A cos whr: ( ( A n rcn Tron ollg Pg 7 of 8 PHY 06 John Blg

8 Th p plu occurs whn h rv frquncy s nr h nurl frquncy. L's copr h vlocy of h oscllng sys o h forcng funcon. L's us h sn funcon for of h sy-s soluon, so h our vlocy s n rs of h cosn funcon: whr β rcn n Asn ( β ( β v A cos Whn h rv frquncy s qul o h rsonn frquncy, β 0. In prccl rs, hs ns h oscllng sys s synchronz wh h forcng funcon h rsonn frquncy. Whn you rv h sys hghr frquncy hn h rsonn frquncy, h vlocy lgs hn h rv forc. Whn you rv h sys frquncy lowr hn h rsonn frquncy, h vlocy ls h rv forc. Tron ollg Pg 8 of 8 PHY 06 John Blg

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