Mah 8- Fri Apr 4, Finish Wednesday's noes firs Then 94 Forced oscillaion problems via Fourier Series Today we will revisi he forced oscillaion problems of las Friday, where we prediced wheher or no resonance would occur, and hen esed our predicions wih he convoluion soluions Using Fourier series expansions for he forcing funcion one can say precisely wheher or no here will be resonance We will be sudying he differenial equaions x## C c x# C w x = f for various forcing funcions f (We have divided he original mass-spring DE by he mass m and relabeled he damping coefficien and forcing funcions) For mos of he lecure we consider undamped configuraions, c = Warm-up Exercise ) (This was he final exercise las Friday) Use superposiion o find paricular soluions, and discuss wheher or no resonance will occur in he following wo forced oscillaion problems Noice ha he period of he forcing funcion is 6 p in a (no he naural period) In b he period of he forcing funcion is p And ye, he resonance occurs in a, and no in b a) b) x## C x = cos C sin x## C x = cos K 3 sin 3 C cos 6 3 Hin: There's a able of paricular soluions a he end of oday's noes
Exercise is indicaive of how we can undersand resonance phenomena for forced oscillaion problems wih general periodic forcing funcions f : Consider he undamped forced DE x## C w x = f, where f has period P = Compue he Fourier series for f: wih a = K N f w a C n a n cos n p = C n b n sin n p = f d (so a = a n d f, cos n p = K K N f d is he average value of f) f cos n p d, n ; b n d f, sin n p = f sin n p du, n ; K As long as no (non-zero) erm in he Fourier series has an angular frequency of w, here will be no resonance In fac, in his case he infinie sum of (undeermined coefficiens) paricular soluions will converge o a bounded paricular soluion For sure here will NOT be resonance if i's rue for all n ; ha w n d n p s w bu even if some w n does equal w here won' be resonance unless eiher a n or b n is nonzero Conversely, if he Fourier series of f does conain cos w or sin w erms, hose erms will cause resonance
Recall he firs "resonance game" example from las Friday: x## C x = square wih and pkperiodic This forcing funcion appeared o cause resonance: Here's a formula for square valid for % % p, and las Friday's resuls: wih plos : square = K Kp!!!! p square d / C $ n K n $Heaviside K n$pi : = ploa d plo square, = 3, color = green : display ploa, ile = `square wave forcing a naural period` ; K Convoluion soluion formula and graph: x d / square wave forcing a naural period sin $square K d : 3 plob d plo x, = 3, color = black : display ploa, plob, ile = `resonance response?` ; K resonance response? 3 Exercise Use he Fourier series for square ha we've found before square = 4 p n odd n sin n and infinie superposiion o find a paricular soluion o x## C x = square ha explains why resonance occurs Make use of he undeermined coefficiens paricular soluion formulas a he end of oday's noes
If we remove he sin erm from he square wave forcing funcion, and re-use he convoluion formula, we see ha we've eliminaed he resonance: x d / plo x, = 3 ; 3 K3 sin $ square K K 4 $sin K d : p 3
Exercise ) Undersand Example 3 from las Friday, using Fourier series: x## C x = f 3 Example 3) Forcing no a he naural period, eg wih a square wave having period T = f3 d / C $ K n $Heaviside K n : n = plo3a d plo f3, =, color = green : display plo3a, ile = `ou of phase square wave forcing` ; K This forcing funcion did no cause resonance: x3 d / sin $f3 K d : ou of phase square wave forcing 5 5 plo3b d plo x3, =, color = black : display plo3a, plo3b, ile = `resonance response?` ; K resonance response? 5 5 Hin: By rescaling we can express f 3 = square p = 4 p n odd n sin n p
Brue force ech check of Fourier coefficiens in previous example: f d /K C $Heaviside ; plo f, =K ; d ; f := /K C Heaviside K K5 5 K := a d $ f d; K assume n, ineger ; # his will le Maple aemp o evaluae he inegrals a d n/ $ K f $cos n$p d : () b d n/ $ K f $sin n$p d : a n ; b n ; a := K K n~ C n~ p K K n~ n~ p ()
Pracical resonance example: Exercise 3 The seady periodic soluion o he differenial equaion x## C $x# C x = square exhibis pracical resonance Explain his wih Fourier series Hin: Use square = 4 p n odd n sin n he able of paricular soluions a he end of oday's noes
Paricular soluions from Chaper 3 or aplace ransform able: = x## C w x = A sin w A w K w sin w when w s w =K w A cos w when w = w = = x## C w x = A cos w A w K w cos w when w s w w A sin w when w = w wih wih x##c c x#c w x = A cos w c O = x sp = C cos w K a A C = w K w C c w w K w cos a = w K w C c w c w sin a = w K w C c w x##c c x#c w x = A sin w c O = x sp = C sin w K a A C = w K w C c w w K w cos a = w K w C c w c w sin a = w K w C c w