Applied Physics Introduction to Vibrtions nd Wves (with focus on elstic wves) Course Outline Simple Hrmonic Motion && + ω 0 ω k /m k elstic property of the oscilltor Elstic properties of terils Stretching, bending, twisting Dmped Oscilltors Driven Oscilltors Coupled Oscilltors Norl Modes Wves nd the wve eqution Fourier Anlysis (introduction) Shock bsorbers, resonnces in mechnicl systems Nturl vibrtionl frequencies Vibrtions of solid Reflecting wves Ultrsonic wves/testing Brekdown of wves into their components Opticl wves Multiple source interference diffrction hin film interference techniques
Week 6 Lecture 3: problems 4, 4, 43 (French pges 9-7, coursewre pg 47-57) Coupled Oscilltions: introduction (this nd the net two pges re n introduction) So fr we hve been concerned with SHM. Chrcterized by single nturl (resonnce) frequency: ω 0 k m However, most rel-life systems resonte t ny frequencies! WHY? Becuse most rel objects cn be viewed s number of simple oscilltors coupled together. We wnt to emine coupled oscilltors to led into study of wve motion, which is essentilly n infinite number of oscilltors (the toms) coupled together.
he Pln over the net few lectures. Strt with very simple system coupled simple pendulums just to introduce the topic.. Move on to coupled system of sses on light string. 3. Epnd this to N coupled sses on light string. 4. Finlly look t the string s continuum. Development of the wve eqution!! For ech cse we will look t: generl behviour resonnce (norl mode) behviour 3
Coupled oscilltion equtions cn be used for some very cool modelling!! 4
Coupled Oscilltions We will emine: Generl Behviour Norl Mode Behviour ( specil cse of generl behviour) - this is resonnce!!! A norl mode (resonnce) occurs when ll prticles re oscillting with the sme frequency. his will be nturl or resonnce frequency for the system. here is more thn one resonnce frequency in coupled oscillting system. Norl mode Nturl Resonnce frequency frequency frequency Emple: wo Coupled, Simple Pendulums (IP demo 9) Generl Behviour: Pull A out nd let go A strts oscillting, then B gets lrger until A stops, then movement bck to A. Energy shuttles bck nd forth, nd in this cse the prticles hve different ω s. B A 5 Coupled simple pendulums
Norl Mode behviour ( specil cse of generl behviour) here re two norl modes (NM s ) for this system. In ech of the norl modes both sses hve the sme frequency (the resonnce frequency) but the frequency of NM NM. (note for this specific cse the mplitudes re the sme for both prticles for ech NM but this is not generlly the cse). Norl Mode Equtions for these coupled pendulums (check out IP9 to see these in ction!): NM B A Grvity but no spring contribution for both prticles: A 0 cosωt ω g l NM B A B A Spring nd grvity contribution for both prticles: A A 0 0 ω cosωt cosωt g k + l m 6
he Generl behviour equtions for the two coupled pendulums: get g g k A A0 cos t + cos + t l l m g g k B A0 cos t cos + t l l m hese look messy initilly but when you look closely they re just bsiclly combintions of the norl modes! 7
Deriving the Norl mode frequency: Cse of ssless string with point sses on it: vibrting nd t snpshot of some instnt during motion his is good illustrtion of the method you will use frequently to solve ll coupled oscilltion problems. I m m m m θ θ 3 m θ θ Assumptions:. nd re sll compred to the string length (llows sll ngle pproition). (tension) does not chnge with these sll displcements 3. ignore grvity Wht re the norl mode frequencies (resonnce frequencies) for this system? We need to develop equtions of motion for ech of 8 the two prticles, strting with Newton s nd lw for ech. hen we cn solve for the norl mode frequencies.
Apply Newton s nd lw (lterlly) +ve ss : so F cosθ cosθ 0 (sll ngle ppro: cos θ ) no lterl ccelertion (sme for m ) Apply Newton s nd lw (verticlly) ss : ss : d m dt d m dt sinθ sinθ sinθ sinθ 3 +ve sll ngle pproitions sin θ tnθ sinθ tnθ sin θ3 tnθ3 sub these into d m dt multiply through nd rerrnge Eqution of motion for ss m d dt + 0 9
subbing sll ngle pproitions into d m dt Eqution of motion for ss m d dt + 0 Solving these equtions for the norl mode conditions *** For norl modes, both m nd m hve the sme frequency ω.*** ssume solutions re: sub both into sub both into maω Acosωt B cosωt cosωt + A B mω A + mω B B + - - sme ω for both sses ---we need to get A, B nd ω Acosωt Bcosωt A Solve these simultneously to find n epression for ω 0 0 b b 0 set ech eqution A/B nd then equte in order to eliminte (A/B) st norl mode ω frequency mω + ω or or 3 nd norl mode frequency ω m ω + ± 3 0
For Norl Mode wht re the A nd B Vlues? subbing ω into either b or b gives A B the mplitude is the sme for both. so ω A B so t ny point For Norl Mode wht re the A nd B vlues? subbing ω into either b or b gives A -B sme mplitude but ntiphse. so ω 3 A -B so - t ny point
Wht bout three prticles? 3 prticles ehibit 3 norl modes (resonnces) NM NM Note here tht the middle prticle hs the sme frequency s the other two but no mplitude NM3 See IP Demo 8 note similrities between the behviour of the sses in IP8 nd the sses on string on the overhed. In Generl: For ID motion of system of N prticles we hve N norl modes nd N corresponding norl mode (resonnce) frequencies. Generl behviour involves combintion of norl mode solutions. note: D systems re N nd 3D systems re 3N.
Solving for the Generl Stte of Motion (still two prticles on string) Develop F equtions lredy hve nd Solve simultneously: dd nd m (&& ) ( ) ( ) 0 3 + & + + + subtrct nd m (& & ) + ( ) ( ) 0 4 3 becomes (&& + & ) + ( + ) 0 4 becomes k his is simply in the form & & + 0 m the stndrd SHM eqution with ω nd ω Solutions re: 3 (&& & ) + ( ) 0 ( + ) C cos t 3 ( ) D cos t NM freq Combining these gives for generl stte of motion NM freq 3 to combine just solve for, plug this into nd eqution. hen do sme C cos t + C cos t D cos D cos for (norlly don t worry bout solving for C nd D) 3 t 3 t 3
Sumry: Coupled Oscilltions wo ypes of Behviour:. Generl Behviour prticles shuttle energy bck nd forth prticles hve different frequencies note tht you will not be required to solve problems involving generl behviour but you will be epected to understnd how the generl behviour derivtion works. Norl Mode Behviour (specil cse of generl behviour) ll problems re bsed on this! resonnce! ll prticles hve the sme frequency (chrcterizing feture!) prticles y or y not hve the sme mplitude he number of norl modes (resonnce frequencies) is equl to the number of prticles in D system! 4
Methods for Solving Coupled Oscilltion Problems Solving for Norl Mode Conditions or more prticles (need to know for em) Write down Newton s nd lw eqution for ech prticle Assume solutions Acosωt (sme frequency for ech prticle Bcosωt becuse in norl mode) 3 Ccosωt Substitute these solutions bck into the Newton s nd lw equtions nd solve simultneously to ω s for ech norl mode Sub ω vlues bck into erlier equtions to get vlues for mplitudes A, B, C, Solving for Description of Generl Motion (gets complicted for > sses) (don t need to know for em) Write down the Newton s nd lw eqution for ech prticle (sme s for norl mode procedure) Add nd subtrct these solutions (solve simultneously) this will give you epressions for ( + ) nd ( ) Solve for nd 5
Solving for Norl Modes sses nd springs utoril problem 4 find norl modes nd mplitudes (note the setup shown here pplies to ll of these types of problems) k sme for ll springs m M m M HIN: Drw n rbitrry shift in ech ss (I lwys drw shift in sme direction with one lrger thn the other) Newton s nd Lw: for ss m k k k + m m ( ) & ( k) + k k + k m m At equil. moving for ss m ( ) k( ) k k & k hen solutions re: m M m M get A cos ωt A cos ωt plug bck into differentil equtions solve simultneously k && + M && k M both sses hve the sme ω s in Norl Modes get ω nd ω then get A nd A by plugging these 6 bck in k M k M
Solving for Norl Modes sses nd springs utoril problem 4 find norl modes nd mplitudes (note the setup shown here pplies to ll of these types of problems) k sme for ll springs m M m M HIN: Drw n rbitrry shift in ech ss (I lwys drw shift in sme direction with one lrger thn the other) Newton s nd Lw: for ss m k k k + m m ( ) & ( k) + k k + k m m At equil. moving for ss m ( ) k( ) k k & k hen solutions re: m M m M get A cos ωt A cos ωt plug bck into differentil equtions solve simultneously k && + M && k M both sses hve the sme ω s in Norl Modes get ω nd ω then get A nd A by plugging these 7 bck in k M k M
N Coupled Oscilltors Norl Modes only (vlid only for situtions where ll prticles nd springs re identicl nd pinned t ech end) erminology: (see proof of these in French pg 39-4) n norl mode vlue (e.g. norl mode 0 is n0) N totl number of coupled oscilltors p the number (long the line) of n individul oscilltor ω n nπ ω sin 0 N nturl frequency in SHM ( + ) frequency of norl mode n k for spring/ss m for trnsverse ss/spring vibrtion ml l A pn C n pnπ sin N + ecittion mplitude of norl mode n mplitude of the p th oscilltor when the system is oscillting in the n th norl mode 3 C n A n A n A 3n y pn () t A cosω t pn n the displcement of the p th prticle 8 when the collection of prticles is oscillting in the n th norl mode
E. Look t norl mode frequency for 4 prticles in coupled oscilltion. For N 4 using ω n nπ ω sin 0 N ( + ) NM n ω ω 0 sin[π/0] 0.68ω 0 ω NM n ω ω 0 sin[π/0].76ω 0.9ω NM3 n3 ω 3 ω 0 sin[3π/0].68ω 0.6ω NM4 n4 ω 4 3.07ω For those of you tht remember high school wve stuff, this might look strnge, since for wves the higher order frequencies re supposed to be integrl multiples of the lowest one. Ecept these re not wves!!! But cn we ke them simulte wve? Very high N!! Consider the ω n eqution bove, but for huge N vlue sin nπ nπ ( N + ) ( N + ) Sll ngle ppro. So we get: ω n ω0nπ nω0 π ( N + ) ( N + ) Now look t the first 4 norl modes gin (net pge) 9
n ω () ω 0 ( + ) N π ω 0 π n ω ω N + ( ) n3 ω 3 3ω n4 ω 4 4ω When N is lrge nd you consider only the first few norl modes (n is sll) then the norl mode frequencies re integrl multiples of ω pproite wve motion! Finl Note: If N is very lrge then wht stops from being very sll? ω n nω0π N ( + ) well, if one considers then s N gets bigger the ω0 ml l term (distnce between prticles) gets sller. So ω 0 increses s N increses! 0