Lecture 7: Damped and Driven Oscillations

Transcription

1 Lecture 7: Damped and Driven Oscillatins

2 Last time, we fund fr underdamped scillatrs: βt x t = e A1 + A csω1t + i A1 A sinω1t A 1 and A are cmplex numbers, but ur answer must be real Implies that A 1 and A are cmplex cnjugates Can write them as: We nw have: ( ) ( ) ( ) A = Ae A = Ae i δ 1 Since we can always redefine the cnstant A t get rid f the in frnt f the equatin, the general slutin is: iδ βt ( ) = ( + + ) x t e A csδ i sinδ csδ i sinδ csω1t ( cs sin cs sin ) sin βt [ csδ csω sinδ sinω ] + ia δ + i δ δ + i δ ω1t = Ae t t βt = Ae cs( ω t + δ ) 1 βt ( ) x t = Ae cs( ω t + δ )

3 Prperties f underdamped mtin An underdamped system still scillates: Nte, thugh, that the mtin is nt peridic it never returns t the same pint with the same velcity as befre The quantity ω 1 can still be related t the time interval between crssings f the x axis Fr light damping, ω 1 is very clse t ω

4 Underdamped Mtin in Phase Space Since the mtin is nt peridic, we n lnger get clsed lps. In additin t amplitude, path depends n β: ω ο =.5, Α = 1 β =.5 v x β =.1 v x β =.5 v x

5 Mre Damping If the damping parameter is large enugh that the system is called critically damped β ω = In this case expressins f the frm te βt als satisfy the equatin f mtin, s the general slutin is: t x ( t) = ( A + Bt) e β Frm this we see that A is the initial psitin and B-βA is the initial velcity In this case the slutins d nt scillate, but can crss the x-axis nce if there is a large initial velcity tward equlibrium

6 The mtin may lk like any f the fllwing: Initial velcity negative Initial velcity zer Initial velcity psitive N matter what the initial cnditins are, the system settles t within a given distance f equilibrium faster with critical damping than with any ther chice f damping parameter Autmbile shck absrbers, fr example, shuld be critically damped

7 Overdamped Mtin If β is even larger the system is verdamped The quantity ω β ω is real, s the psitin as a functin f time is given by: Features: βt ωt ω ( ) = + x t e A1 e A e N hint f scillatry mtin here (ω can t be interpreted as an angular frequency) Psitin always appraches equilibrium fr large t But nt as quickly as a critically-damped system wuld System can crss x = nce (as in critically damped case) t

8 Driven Oscillatins There are many examples in which an external agent applies a frce t an scillatr Smetimes essential t intended functin (e.g., a radi), smetimes an annyance (e.g., wind gusts hitting a skyscraper) This external frce can have any frm, but we ll cnsider the particular case f a sinusidal frce: F = F sinωt This ω can be anything we chse it s nt related t the natural scillatin frequency ω This means the equatin f mtin is: mx + bx + kx = F sinωt

9 After dividing thrugh by m and redefining the cnstants, this becmes: x + β x + ω x = A ωt sin This is knwn as a linear inhmgeneus equatin T slve it, let s assume that the slutin has a frm similar t what appears n the right-hand side: x ( t) = C sin ( ωt + δ ) Substituting this int the equatin f mtin gives: Cω sin ( ωt + δ ) + Cβω cs( ωt + δ ) + ω C sin ( ωt + δ ) = Asinωt

10 Expanding gives: Cω [ csωt sinδ + csδ sinωt] + Cβω [ csωt csδ sinωt sinδ ] + ω C [ csωt sinδ + csδ sinωt] Asinωt ω ω δ + βω δ + ω δ cs t C sin C cs C sin + + = sinωt A Cω csδ Cβω sinδ ω C csδ The nly way this can be true fr all t is if C and δ are chsen such that bth terms in [] are zer =

11 Starting with the csine term, we need: βω csδ + ω ω sinδ = ( ) βω tanδ = ( ω ω ) Frm this, we can determine sinδ and csδ, and then find C: ( ) A C ω csδ βω sinδ + ω csδ = C = = = ( ) ω ω csδ + βω sinδ ( ) ( ) A ω ω + 4β ω A ω ω ω ω βω + βω ω ω + β ω ω ω + β ω A ( ) ( ) 4 4

12 Putting all f this tgether, we have the slutin: x ( t) A 1 βω = sin ωt tan + ( ω ) ( ) ω 4β ω ω ω Lks great, but yu may ntice a prblem there s n freedm here (recall that A = F /m) and surely the mtin depends smehw n initial cnditins, desn t it? T see where these cme in, cnsider what happens if we add a term t ur slutin: x ( t) = x ( t) + x ( t) c where: x + β x + ω x = c c c

13 This new functin als satisfies the equatin f mtin: ( ) ( ) ( x ) x + β x + ω x = x + x + β x + x + ω x + c c c = x + β x + ω x + x + β x + ω x = Asinωt + c c c But the equatin that x c (t) satisfies is just the equatin fr an undriven scillatr S all the slutins we ve already explred are part f the slutin fr driven scillatrs as well Linear inhmgeneus differential equatins in general have this prperty Slutin is the sum f a particular slutin that depends n the right-hand side f the equatin and a cmplementary slutin that gives zer n the right-hand side

14 Sme ther features f the slutin: 1. The cmplementary functin ges as e βt. The initial cnditins affect nly the cmplementary slutin, nt the particular slutin Bth f these facts tell us that the cmplementary slutin gives transient effects After a lng time has passed, the scillatr will mve as described by the particular slutin, n matter what the initial cnditins are

15 Resnance The amplitude attained by a driven scillatr depends strngly n the driving frequency The maximum ccurs at the resnance frequency : da dω ω R ( ) 1 4ω R ω ωr + 8β ωr = = 3/ ( ω ωr ) 4β ω + R ω ω = β ω R = ω β R The sharpness f the resnance depends n the strength f the damping

16 Q Factr Actually, it s the rati f the resnance frequency t the damping parameter that determines the sharpness f the resnance. We define: Q ω ω β R = = β β A Q = Q = 5 Q = Omega A Omega A Omega