Lecture 11 DAMPED AND DRIVEN HARMONIC OSCILLATIONS. Composition of harmonic oscillations (1) Harmonic motion diff. equation is: -linear -uniform

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1 Lecture DMPED ND DRIVEN HRMONIC OSCILLTIONS Ntes: Lecture - Cpsitin f harnic scillatins () Learn re: Linear differential equatin Harnic tin diff. equatin is: -linear -unifr d + uatin ny cinatin f slutins is a slutin f the LDE Principle f superpsitin cnsequences:. synthesis - (superpsitin) f scillatins. analysis decpsitin f scillatins Key ters: linear differential equatin

2 Ntes: Lecture - Synthesis (superpsitin) f harnic scillatins () ne diensinal case cs( t + φ) cs( t + φ) φ φ cnst r r r + cs( t + φ) + csγ γ 8 ( α + β ) cherent scill. + + cs( φ φ) sinφ + sinφ tgφ csφ + csφ γ φ α O φ 8 ( φ φ) cs 8 ( φ φ ) cs( φ φ aplitude f the resultant scillatin phase cnstant cs( φ φ ) + φ φ n π n,,,3 + + aplificatin φ φ (n + ) π n,,,3 + attenuatin [ ] ) φ β Phasr is a rtating vectr f aplitude. Its agnitude represents aplitude f scillatins. Phasr rtates in a XY plane with the cnstant angular velcity which is equal t the angular frequency f scillatins. Initial psitin f the phasr is descried y the angle with the ais which is equal t the phase cnstant. t any instant f tie the prjectin f the phasr n the ais is descriing the displaceent f the scillatin. Key ters: phasr ethd

3 Ntes: Lecture - 3 Superpsitin f harnic scillatins (3). Nncherent scillatins φ φ f ( t) cs( t + φ ( t)) cs( t + φ( t)) φ φ φ ( t aplitude depends n tie ) w f (t) Learn re: Mdulatin Cherent scillatins f different frequencies cs( t) + cs( + t ) requency f resultant scillatins Mdulated aplitude requency f aplitude dulatin eats ( t)cs( t + φ( t)) a aplitude when cs t ± eats Mdulated scillatins ( t) ( φ cnst) M ( t) ( cnst) + cs t cs t M Key ters: tuning scillatrs, aplitude dulatin, frequency dulatin

4 Ntes: Lecture - 4 Harnic analysis () Representatin f functins r signals as the superpsitin f asic siple harnic scillatins. ny cntinuus functin culd e reprduced as an infinite su f harnic (sine and csine) scillatins f ultiple frequencies. M.Krasiński Jean Baptiste J. urier (768-83) f ( t) + sin( t + φ) + sin(t + φ ) n sin( nt + φn ) fundaental frequency ; 3 ;... n ; ultiple frequencies - harnics sin( t) + sin(3t ) + sin(5t ) +... sin( (n + ) t) 3 5 n urier series + n fundaental des 3 spectru Peple: Standing n the shulders f giants Jean B.Jseph urier Key ters: urier series

5 Ntes: Lecture - 5 Harnic analysis () Learn re: urier series Signal analysis t sint + sin 3t + sin5t Try it! urier Synthesis - siulatin t sint + sin 3t + sin5t PHPSESSIDd5ee384a758fc88d6c59 358&tpic7. t sint + sin t + sin 3t M.Krasiński The fast urier transfr (T) is a atheatical ethd fr transfring a functin f tie int a functin f frequency. M.Krasiński Key ters: signal analysis

6 Ntes: Lecture - 6 siultaneus harnic scillatins perpendicular directins sae frequencies y. φ φ y y C ( + φ ) cs t y cs ( t + φ y ) y linear scillatins Superpsitin f perpendicular scillatins () cs( t) cs( t) y y C cnst Y y y O Y O s s X X -: linear plarisatin. φ φ y π cs( t + π ) cs( t) y y cs( t) s Y O X Key ters: plarisatin, phase shift

7 Ntes: Lecture - 7 π 3. φ φ y π cs t + y y cs y y sin( t) ( t) ellipse sin( t) cs( t) φ φ 3π Superpsitin f perpendicular scillatins () y + sin ( t) + cs ( t) 3. y 3. y Lissaju curves y y elliptical plarisatin circular plarisatin n Shape depends n rati f: aplitudes, frequencies, phase cnstants 4. y 4 Y O y φ φ Left-handed y π 4 y X Right-handed Learn re: Lissaju curve D it yurself! saju/la/

8 Ntes: Lecture - 8 Daped harnic tin () r k, Resistive frces dependent n the velcity (drag) v r r v daping paraeter r r z + d k, -tinparaeters Slutin fr weak daping : ( ) t e cs( t + φ) β λ ln n n+ -dapingcefficient e e βt β ( t+ T ) ( z k d ) t e ( t) βt aplitude d d d k + + equatin f daped harnic tin -Lgarithicdecreent f daping k λ e N W.H.reean & C ( ) t N e Key ters: resistive frce - 6 -

9 Prle slving strategy: Lecture - 9 Daped harnic tin () ( ) t ngular frequency e cs( t + φ). weak daping β << i π T > T β. heavy daping < inding angular frequency f the daped scillatins is a tw prcedure. irst- yu have t find the frequency f the siple harnic scillatins f the sae ject. Secnd yu have t epress the daping frce in a linear fr (direct prprtinality t the velcity). The prprtinality cnstant plays the rle f the daping paraeter. Having yu are ale t define the daping cefficient β and eventually the frequency f the daped scillatins. Learn re: Shck asrer critical daping ; 3. verdaping relaatin > k k shck asrer β at at Ce + Ce verdaping critical daping - 6 -

10 Ntes: e Lecture - E k E k E(τ ) k e Rate f energy change t Ee τ E de τ weak daping << τ τ Energy f the daped scillatr βτ e e E Q Q τ k e β t βτ decay tie τ β k e t τ E e t τ The Q factr (quality factr): Q π E π de T E syste average energy lst in perid π τ τ T 4Q E p Q,5 E it easures hw any radians the scillatr ges arund in tie τ 4 Q Learn re: Q factr Key ters: quality factr, relaatin tie, tie cnstant - 6 -

11 Ntes: Lecture - Driven scillatins () Siple (free) scillatins Internal, restring frce Daped scillatins Dissipative frce r r k d r cs( t Driven scillatins Eternal, peridic driving frce ) II law f dynaics d k Phase space d () d + cs( t) Driven daped scillatins v () () d d d k + + cs( t) Transient phenena φ(t) Transient phenena Statinary state Key ters: statinary state, transient phenena

12 Ntes: φ Lecture - Driven daped scillatins d d k + + cs( t) slutin: - phase shift etween driving frce and the displaceent! cs( t + φ) aplitude: G G ( ) + requency (statinary): Phase cnstant: sinφ G ( ) cs φ G tgφ ( )

13 Ntes: Lecture - 3 Driven daped scillatins - case studies () Weak daping << sall r ig ass. Lw frequency driving frce << G sinφ csφ φ G k displaceent in phase with the frce Decisive rle f the restring frce (internal) dla k n scillatin static defratin

14 Ntes: Lecture - 4 Driven daped scillatins - case studies ().. High frequency driving frce >> G sinφ φ π csφ G aplitude decreases with the increase f frequency decisive rle f the syste ass (inertial ass) 3. resnance G driving a swing sinφ π φ csφ rce and velcity are in phase! G

15 Ntes: Lecture - 5 Driven scillatins phase Phasef thedrivenscillatrdependsn thefrequency Phase is negative Displaceent laggs with respect t the driving frce t φ >> φ π φ τ Q factr π φ π τ τ τ π

16 Ntes: Lecture - 6 Resnance () when n daping aplitude G fr G ( ) + dg a d R R < R k > > 3 > 3 r weaker daping the resnance frequency is clser t the natural frequency f the scillatr (SHO)

17 Ntes: Lecture - 7 Resnance () verage pwer transfered y the driving frce int the syste P at the resnance P P R P R ± G PR - Resnance wih, half-wih Q factr vs half-wih Q τ Q factr is easure f syste tuning int the resnance state. P śr Decay tie τ τ weak daping ig Q Heavy daping sall Q Key ters: resnance wih,

18 Ntes: Lecture - 8 Resnance in engineering Learn re: Nuclear agnetic resnance iaging (NMRI) N daping increase f aplitude Pwer transfered int the syste P R _iaging Tuning int the resnance state: djusting the freqency f the driving frce t the natural frequency f syste. Taca Narrws Bridge Nuclear Magnetic Resnance Key ters: resnance, NMR, - 7 -