Harnic Mtin (HM) Oscillatin with Lainar Daping If yu dn t knw the units f a quantity yu prbably dn t understand its physical significance. Siple HM r r Hke' s Law: F k x definitins: f T / T / Bf x A sin ( N + Tt ) T k/ r (B/T) k/ v A T cs ( N + Tt ) Siple pendulu: k g/l T g/l a!a T sin ( N + Tt ) Crk n water: k D water Ag T D water g/d crk L kinetic energy ½v elastic ptential energy ½kx cnservatin f echanical energy: KE f + PE f KE i +PE i if frictin can be neglected ( W nn-cnservative 0 ) Daped HM T / k/ daping frce F dap & cv a) lw daping ( under daped ): c < 4k lainar daping frce prprtinal t the velcity v c c energy: E E e t fr lainar daping E / T E t Q quality: Q r decay cns P / π E Q / ave c k c psitin: x A e t sin( t+ ϕ ) β t c Aplitude: A A e β fr lainar daping Daping decreases the frequency f scillatin, and als the aplitude as tie increases. b) ediu daping ( critical daping ): c 4k The syste returns t equilibriu in the shrtest tie withut scillating. c) large daping ( ver daped ): c > 4k The syste takes a lnger tie t reach equilibriu withut scillating. C. Deurzen 005 tant
Frced HM and Resnance Under the influence f an external driving frce F F sin Tt hw will the syste respnd? Psitin: x A sin ( Tt + N ) displaceent always lags (N < 0) the driving frce Aplitude A F / ( ) + / tan ϕ a) At lw driving frequency T << T Respnse cntrlled by spring (k) b) At resnance driving frequency T T Respnse cntrlled by daping (c) A F F / ϕ k 0 0 A F F / / π ϕ / c c) At high driving frequency T >> T Respnse cntrlled by inertial ass () respnse at resnance respnse at zer frequency A A F F / / ϕ π ( ) ( ) A / Q 0 / Pwer Absrptin: Pwer is the tie average f the frce ties the velcity (see exercise ): ( / ) Pwer F v P where P F < > res res ( ) + ( / ) Near resnance (T.T ) the half-pwer pints ccur when: r T & T T/J T )T. T/J Thus the full width ( )T) f the resnance at half axiu pwer is equal t /J and the quality f an scillatr Q easures the sharpness f daping near resnance: Quality Q frequency at resnance full width at half axiu pwer
Nuerical Exaple f Harnic Oscillatr Given scillatr data: ass gra Then: spring cnstant k 0 N/ relaxatin tie J 0.5 secnd N 0 k radian cycles 0 6 3 0 kg secnd secnd free scillati n frequency 0 0 4 radian secnd Quality factr Given a driving frce: F 0. N sin ( 90 t ) thus F 0. N ; and the driving frequency is T 90 rad/sec The respnse f the scillatr is: n & 5.4E The pwer absrptin at resnance: radian Q 0 05. secnd 50 secnd The tie fr the aplitude t dap t e & f its initial value is: ( 05. sec) sec β c gra gra the value f the daping cnstant is: c. secnd 05 secnd F 0. N N N 0. 00 gra gra kg Aplitude A 00 N / kg + 90 05. ( 0 90 ) 00 90 / 05. 5 c tan ϕ 909 90 0 4 4 3 Pres 0 kg( 00 N / kg) 05. sec. 5Watt The full width f the resnance curve between half-pwer pints is: )T T /Q 00 rad/sec / 50 rad/sec. 0. 095 F / 00 N / kg the aplitude respnse at resnance is A( ) 50 c rad / 00 /sec 05. secnd F 0. N the aplitude at lw frequency is A( 0) c k 0 N /
Exercise Shw xasin(tt+n) satisfies the equatin f tin f a driven harnic scillatr. Equatin f tin: x&& + cx& + k x F sint c c x x k x F sint && + & + Substituting the first and secnd differentiatin f the prpsed slutin xasin(tt+n): F ( ) Asin( t+ ϕ) + Acs( t+ ϕ) sin( t) using: sin(tt+n) sintt csn + cstt sinn cs(tt+n) cstt csn - sintt sinn ( ) ( ) k ϕ ϕ ϕ F A t ϕ A t t + + cs sin sin sin cs cs sin This equatin can nly be satisfied fr all tie t if the cefficients f sintt and cstt are bth zer, which establishes A and the phase angle N: A F / F / ϕ + ( ) cs sinϕ ( ) sinϕ / tanϕ csϕ where we have used sin N + cs N / in rder t btain: sinϕ / ( ) + ( ) csϕ +
Exercise Derive the pwer absrptin P(T) fr a driven harnic scillatr. Fr the tin f a frced r driven scillatr we have: frce F F sin Tt ; psitin x A sin ( Tt + N ) ; velcity v TA cs ( Tt + N ) Nte the phase angle N f the psitin x relative t the driving frce F and an additinal 90E fr the velcity v which is the derivative f the psitin. The tie averaged pwer is the tie average f the frce ties the velcity Pwer + F external @ velcity, +, / tie average Pwer + ( F sintt ) @ { TA cs( Tt + N ) }, cs( Tt + N ) cstt csn - sintt sinn Pwer F T A + sintt cstt csn & sintt sintt sinn, The tie average f sintt cstt is zer, that is + sintt cstt, 0 and the tie average f sin Tt is ne half, that is + sin Tt, ½ Pwer F T A ( 0 & ½ sinn ) Fr exercise : A F / ( ) + sinϕ / and + ( ) Pwer F F / ( ) + Pwer F ( / ) ( ) + Nte the frequency dependence factr cntaining T, daping Jc/, spring T k/ factr ( A) ( / ) ( ) +
Exercise 3 Cnsider a series electragnetic circuit cnsisting f: an inductr L, a resistr R, a capacitr C, and an applied pwer supply õ õ sin Tt. Shw that such a circuit behaves identically as a harnic scillatr. Applying Kirchhff s Vltage Law t the series circuit lp, we btain: L i Ri t C q E + + sin t ( ) where q is the tie-varying quantity f electric charge stred in the capacitr and i is the tie varying intensity f current as the charge scillates thrugh the series circuit. Lq&& + Rq& + Substituting fr i / )q/)t : ( ) C q E t sin x&& + cx& + k x F sin t ( ) The fr f the electric circuit equatin is identical t that f the echanical, lainar-daped, driven frced scillatr. Matheatically, the tw equatins describe the sae behavir. It reains t identify the quantities that crrespnd t each ther and identify their physical significance and the rle each plays. Inductance L } ass Resistance R } c daping factr elastance /C } k spring s elastic cnstant õlectr Mtive Frce E } F axiu strength f driving frce Thus: Inductance plays the rle f inertia, as expected since it takes tie t establish a current in an inductr when a vltage is applied t it. Resistance plays the rle f daping, as expected because it absrbs energy siilar t a dashpt r a shck absrber in a echanical syste. The inverse f capacitance plays the rle f elasticity, as expected because a capacitr stres electric charge and energy siilar t the elastic energy strage in a spring. elasticity / C The natural frequency is therefre inertia L LC which is precisely the resnance frequency f the s-called electrical tuning circuit. / tanϕ L C The phase angle between the applied vltage and the current is R At resnance the current and vltage are in phase N0, the circuit acts as a pure resistance, and fr a given ipressed õf the current in the circuit is a axiu. The resnance full width at half-axiu pwer pints is ( )T ) /J R/L. The Quality factr f the electragnetic scillatr is Q T J T L/R etc. C. Deurzen 005
Exercise 4 Cnsider the daping factr fr a flat plate ving nral t its plane with speed v thrugh a lw-pressure, rarefied gas. Assue the ean free path f the gas lecules is large cpared t the diensins f the plate s that the deceleratin f the plate ay be thught f in ters f individual cllisins between the plate and the lecules rather than hydrdynaic flw. The drag frce n the plate will be prprtinal t the rate at which lecules strike the plate ultiplied by the average entu transfer per lecule. The rate at which lecules strike the plate is prprtinal t the relative velcity. Cnsider the relative velcity between the incing lecules and the plate. Suppse the lecules ve in nly ne directin with speed v r with rs speed. On ne side f the plate the relative velcity is v + v; n the ther side it is v - v. The average entu transfer is itself prprtinal t the relative velcity. The net drag frce n the plate ppsing the tin will thus be prprtinal t: Case: v << lecular velcity F daping % & 4 v v % &v The daping is prprtinal t the first pwer f the velcity. Fr viscus hydrdynaic flw this is referred t as lainar flw. Case: v >> lecular velcity In this case there is n frward frce as the lecules can t catch up and the lecular velcity is negligibly sall and ignred. F daping % & v ( ) ( ) daping frward backward F F F v v v + v The daping is prprtinal t the secnd pwer f the velcity. Fr viscus hydrdynaic flw this is referred t as turbulent flw. C. Deurzen 005