Lecture 6: Phase Space and Damped Oscillations
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1 Lecture 6: Phase Space and Damped Oscillatins
2 Oscillatins in Multiple Dimensins The preius discussin was fine fr scillatin in a single dimensin In general, thugh, we want t deal with the situatin where: F = kr N big deal we can cnsider ne cmpnent at a time: F = mx = kx x x F = mx = kx S we just need t sle the same equatin as befre, nce fr each dimensin Slutins are: ( ) = sin ( ω + δ ) ( ) = sin ( ω + δ ) x t A t x t A t 1 Nte that frequency is the same in each directin
3 The Tw-Dimensinal Case We can find the path taken by a particle underging twdimensinal scillatin Easiest if we start with a little trick rewrite x as: ( ω δ δ δ ) ( ω δ ) ( δ δ ) ( ω δ ) ( δ δ ) x = A sin t = A sin t + cs + cs t + sin x x = A cs + 1 sin A A ( δ δ ) ( δ δ ) ( δ δ ) sin ( δ δ ) ( δ δ ) ( δ δ ) ( δ δ ) A x sin ( δ δ ) A x A x cs = A A x A x A A x x cs + A x cs = A A sin
4 Finally, we hae: A x A A x x cs δ δ + A x = A A sin δ δ δ This is the equatin fr an ellipse ( ) ( ) Nte that the shape f the ellipse depends nly n the relatie amplitudes and difference in phase between the mtin in each directin = δ δ = δ1 1 3 δ δ1 = 9 δ = δ 1 1 δ δ1 = 18
5 Lissajus Cures The tw-dimensinal mtin becmes een mre interesting if the frequencies are different in each dimensin As lng as the rati f frequencies is a ratinal number, the mtin is still peridic: 7 ω = ω1 5 δ δ = 9 Nte that there are 7 maxima in x and 5 maxima in x 1
6 Phase Space Thinking a bit mre abstractly abut the slutin f the ne-dimensinal scillatr, we recall that tw initial cnditins needed t be specified A cmmn ccurrence in mechanics, since the equatin f mtin is a secnd-rder differential equatin We can take the initial cnditins t be specified by an initial x and x i.e., nce we knw these tw quantities at a gien pint in time, we can determine what their alues will be at any ther time We can represent this infrmatin graphically in phase space let ne axis be x, and the ther be x
7 The mtin in phase space can be determined frm the knwn functins x ( t) and x ( t) r directly frm the equatin f mtin: dx = ω x dt dx dt x = ω dx x dt x dx = ω xdx x + ω x = C = A ω A similar prcedure can be used fr ther types f mtin smetimes easier than integrating t find x This is an ellipse
8 Phase Space Ellipses The shape f the ellipse is determined by ω, and the size by A: ω ο =.5 A = 1 A = 5 A = x 1 ω ο = x x A = x A = x A = x
9 Features f Phase Space A particle s mtin is represented by a cure in phase space Peridic mtin is represented by a clsed cure, like the ellipses in the preius example N tw cures in phase space can crss Or else the particle wuld hae tw pssible futures fr a gien x and x Mtin in tw and three dimensins can als be treated in phase space Fr the tw-dimensinal case, phase space is furdimensinal Axes are x, x, x, x 1 1 In general, phase space has N dimensins fr mtin in N real spatial dimensins
10 Damped Oscillatins Almst all real scillatrs experience sme resistance t their mtin In general, such resistance is called damping As with the resistie frces studied earlier, the precise frm f the damping can ary But we can explre many f the features f damping by assuming the frce is prprtinal t elcity In this case, the equatin f mtin becmes (in ne dimensin): mx = kx bx x + β x + ω x =
11 This type f differential equatin is called a linear, hmgenus equatin. Assume the slutin is f the rt frm x = e : r e + β re + ω e = r rt rt rt + β r + ω = ( ) β ± β 4ω r = = β ± β ω Nte that either f these slutins fr r gies an acceptable slutin t the equatin Als, multiplying the slutin by a cnstant results in an acceptable slutin
12 Therefre, the general slutin is the fllwing: β ω t β ω t x ( t) = e A1 e + A e Hweer, the qualitatie nature f the mtin depends n the relatie sizes f β and ω ο Case 1: β < ω ο The quantity is imaginary We can write it as: ( ) β ω β ω = i ω β iω iω1t iω1t x t = e A1 e + A e 1 ( csω sinω ) ( csω sinω ) e A1 1t i 1 A 1t i 1 = + +
13 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 hae: ( ) = ( + ) csω + ( ) x t e A1 A 1t i A1 A sinω1t A = Ae A = Ae i δ 1 = Ae cs( ω1t + δ ) Since we can always redefine the cnstant A t get rid f the in frnt f the equatin, the general slutin is: iδ ( ) = ( + + ) x t e A csδ i sinδ csδ i sinδ csω1t ( cs sin cs sin ) sin [ csδ csω sinδ sinω ] + ia δ + i δ δ + i δ ω1t = Ae t t ( ) x t = Ae cs( ω t + δ )
14 Prperties f underdamped mtin An underdamped system still scillates: Nte, thugh, that the mtin is nt peridic it neer returns t the same pint with the same elcity as befre The quantity ω 1 can still be related t the time interal between crssings f the x axis Fr light damping, ω 1 is ery clse t ω
15 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 = x β = x β = x
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