Dy : Mondy 5 inuts. Ovrviw of th PH47 wsit (syllus, ssignnts tc.). Coupld oscilltions W gin with sss coupld y Hook's Lw springs nd find th possil longitudinl) otion of such syst. W ll xtnd this to finit nd infinit nur of sss. This syst is odl for othr typs of coupld oscilltions (trnsvrs otion of ths sss, coupld LC circuits, pndulus...) Nwton s lw how things ov: r F dr p dt d r r (for point sss) dt So if w know wht F is, w know out trjctory! Hook's Lw: F (on dinsion) whr is th displcnt fro quiliriu. Constrin to longitudinl dirction. Positiv dirction Adopt systtic pproch: Forc on ss (forc to lft) + (forc to right) F + ) ( Forc on ss F + ) ( Chck ch tr for sign of forc rltiv to displcnt! Dy, updtd 3/3/9 Pg of 6 9 Jnt Tt nd Willi W. Wrrn, Jr. For us y studnts in PH47/PH57 in Spring 9 only
Nwton: F ( ) ( ) & & ( ), so plug in forcs for qul sss: + && + + && + && + && +. Coupld qutions know, cn gt, nd vic vrs!. Mtrix qutions! 3. Must so otion whr oth sss hv s frquncy norl ods 4. How ny norl ods, nd wht r frquncis & wht is significnc? Norl ods: 5. All prticls oscillt with s frquncy norl od 6. Thr r s ny norl ods s thr r prticls ch with (possily) diffrnt frquncy 7. Any otion of prticl cn xprssd s th suprposition of norl ods 8. Equtions of otion r uncoupld whn xprssd in trs of norl ods Assu A iωt A iωt 9. Cofficints r coplx. Tk rl prt to gt ctul otion (displcnts r rl!). Tsk is to find ω (y or thn on) nd rltionship twn cofficints for ch ω. iωt && ω A ω OR iωt && ω A ω ω + + Eignvlu qution!! To find ignvlus, w st dt( A ω I) : + ω Dy, updtd 3/3/9 Pg of 6 9 Jnt Tt nd Willi W. Wrrn, Jr. For us y studnts in PH47/PH57 in Spring 9 only
+ ω + ω + ± + +ω 4 + + ω + ± Minus sign givs th low frquncy od: ω low +. Indpndnt of coupling spring! Ipliction?. Eignod is A A (sytric) Plus sign givs th high frquncy od: ω high + 3. Dpndnt on coupling spring! Ipliction? 4. Eignod is A A (ntisytric) + + Eignods: Plug frquncis ck into trix qution. For low frquncy od: + + + + + which sys nd thrfor A A (od is sytric, sss ov in phs). Thus lowst norl od is conditions. A A i t with A, is rl nd dtrind y initil Siilrly for ntisytric od A A i + t. (Mod is ntisytric, sss ov out of phs.) Dy, updtd 3/3/9 Pg 3 of 6 9 Jnt Tt nd Willi W. Wrrn, Jr. For us y studnts in PH47/PH57 in Spring 9 only
Gnrl Motion Bts W know tht th syst hs ignfrquncis, ω low for th sytric od in which, nd ω high for th ntisytric od in which. If w strt th syst with initil conditions corrsponding to on of ths ods, sy y displcing oth sss sytriclly (ntisytriclly), thn th syst will oscillt with singl frquncy ω low ( ωhigh ). For or gnrl initil conditions, oth ods r will xcitd nd th syst will xcut or coplx otion corrsponding to suprposition of th two ignfrquncis. Applying th Principl of Suprposition, w should l to xprss th otions of sss nd s A + B nd A + B whr, for siplicity w lt ω low ω nd ω high ω. Th plituds A, B, A, ndb r coplx constnts to dtrind y th initil conditions. (W will tk th rl prt whn w nd to.) Howvr, ths constnts r not copltly indpndnt of ch othr. W know, for xpl, tht if B B so tht th syst is oscillting only with frquncy ω low ω, thn th otion ust sytric, i.. A A. Siilrly, to gt th ntisytric od, w nd B B. W cn ssur this y writing A + B nd A Dy, updtd 3/3/9 Pg 4 of 6 9 Jnt Tt nd Willi W. Wrrn, Jr. For us y studnts in PH47/PH57 in Spring 9 only B Putting in th coplx ntur of th cofficints xplicitly, w s tht w hv 4 undtrind constnts, th (rl) plituds A,B nd th phs fctors, φ, δ. () t A + B nd ( t ) A B Now, lt s chos prticulr st of initil conditions: ss t th quiliriu position with zro vlocity nd ss displcd distnc C, lso with zro vlocity. Exprssing this lgriclly w hv d dt ( ) R( A + B ) nd ( ) R( A B ) C d ( iω A + iω B ) nd R( iω A iω B ) R dt Rwriting th two qutions for th vlocity initil conditions y tking th rl prts of th xponntils, w find ω A sin φ ωb sin δ nd ωa sin φ + ωb sin δ Adding nd sutrcting ths two qutions givs ω A sin φ nd ωb sin δ
fro which w conclud φ δ. If w hd forgottn out th vlocity initil condition, it would hv n th s s ssuing A nd B rl, which would hv n OK for this cs. But this cn t don in gnrl! Now, go ck to th displcnt initil conditions, which now rd, whnc, ( ) A + B nd ( ) A B C B C. A So now our qutions for th displcnts rd Ths r quivlnt to () t R[ C( )] C cos( ω t ) cos( ω t ) [ ] () t R[ C( + )] C cos( ω t ) + cos( ω t ) () t () t + C sin + C cos [ ] ( ω ω ) t ( ω ω ) sin ( ω ω ) t ( ω ω ) t cos To show this tks it of lgr: us th trig idntitis sin ( x ± y ) sin x cos y ± cos y sin x in th prcding qutions, crry out th ultipliction, nd introduc th idntitis x sin x cos x nd cos + cos x. This rsult shows ( tht th gnrl otion of th syst consists of high frquncy oscilltion ω ) + ω ( with frquncy odultd y lowr frquncy ω ) ω. Th ffct of th lowr frquncy (frquncy diffrnc) tr is known s ts in nlogy to th throing sound tht is hrd whn two cousticl tons with slightly diffrnt frquncis r hrd siultnously. Th thticl dscription is idnticl suprposition of two sins or cosins. ω ω Th functionl for of () t is shown on th following pg for cs whr.. ω + ω t Dy, updtd 3/3/9 Pg 5 of 6 9 Jnt Tt nd Willi W. Wrrn, Jr. For us y studnts in PH47/PH57 in Spring 9 only
Wk coupling liit Rcll th xprssions for th two ignfrquncis: ω low ω nd ω high ω + nd not tht it is th coupling ( ) tht distinguishs th two frquncis. (If, w siply hv two, copltly indpndnt sipl hronic oscilltors.) Now, suppos tht th coupling is wk, i.. <<. Thn, using th pproxition + ε + ε for ε << w cn writ ωhigh ω + + + ω + ω ω ( ) Th t frquncy in this liit is strngth of th coupling. ω ω which is dirctly proportionl to th Dy, updtd 3/3/9 Pg 6 of 6 9 Jnt Tt nd Willi W. Wrrn, Jr. For us y studnts in PH47/PH57 in Spring 9 only