Mah 225-4 Week 3 April 9-3 EP 7.6 - convoluions; 6.-6.2 - eigenvalues, eigenvecors and diagonalizabiliy; 7. - sysems of differenial equaions. Mon Apr 9 EP 7.6 Convoluions and Laplace ransforms. Announcemens: Warm-up Exercise:
The convoluion Laplace ransform able enry says i's possible o find he inverse Laplace ransform of a produc of Laplace ransforms. The answer is NOT he produc of he inverse Laplace ransforms, bu a more complicaed expression known as a "convoluion". If you've had a mulivariable Calculus class in which you sudied ineraed inegrals and changing he order of inegraion, you can verify ha his able enry is rue - I've included he proof as he las page of oday's noes. f g f g d F s G s convoluion inegrals o inver Laplace ransform producs Overview: This somewha amazing able enry allows us o wrie down soluion formulas for any consan coefficien nonhomogeneous differenial equaion, no maer how complicaed he righ hand side is. Le's focus our discussion on he sor of differenial equaion ha arises in Mah 225, namely he second order case x a x b x = f x = x x = v. Then in Laplace land, his equaion is equivalen o s 2 X s sx v a s X s x b X s = F s X s s 2 a s b = F s x s v a x. x s v a x X s = F s s 2 a s b s 2. a s b The inverse Laplace ransform of he second fracion conains he iniial value informaion, and is inverse Laplace ransform will be a homogeneous soluion for he differenial equaion, and will be zero if x = v =. (Noe ha he Chaper 5 characerisic polynomial is exacly p r = r 2 a r b, which coincides wih he denominaor p s = s 2 a s b.) The firs fracion is a produc of wo Laplace ransforms F s s 2 a s b = F s W s for W s s 2 a s b. and so we can use he convoluion able enry o wrie down an (inegral) formula for he inverse Laplace ransform. No maer wha forcing funcion f appears on he righ side of he differenial equaion, he corresponding soluion (o he IVP wih x = v = ) is always given by he inegral where x = f w = w f = w = W s. w f d,
Le's look more closely a ha soluion formula: The soluion o x a x b x = f x = x = is given by x = f w = w f = w f d = f w d. Exercise ) The funcion w = W s is called he "weigh funcion" for he differenial equaion. Verify ha i is a soluion o he homogeneous DE IVP w a w b w = w = w =. Exercise 2 Inerpre he convoluion formula x = f w d in erms of x being a resul of he forces f for, and how responsive he sysem hrough he weigh funcion, and he corresponding imes. This is relaed o a general principle known as "Duhamal's Principal", ha applies in a variey of seings. (See wikipedia.)
Exercise 3. Le's play he resonance game and pracice convoluion inegrals, firs wih an old friend, bu hen wih non-sinusoidal forcing funcions. We'll sick wih our earlier swing, bu consider various forcing periodic funcions f. x x = f x = x = a) Find he weigh funcion w. b) Wrie down he soluion formula for x as a convoluion inegral. c) Work ou he special case of X s when f = cos, and verify ha he convoluion formula reproduces he answer we would've goen from he able enry 2 k sin k s s 2 k 2 2
Exercise 4 The soluion x o x a x b x = f x = x = is given by x = f w = w f = w f d = f w d. We worked ou ha he soluion o our DE IVP will be Example ) A square wave forcing funcion wih ampliude and period 2. Le's alk abou how we came up wih he formula (which works unil = ). > wih plos : > x = sin f d Since he unforced sysem has a naural angular frequency =, we expec resonance when he forcing funcion has he corresponding period of T = 2 = 2. We will discover ha here is he possibiliy w for resonance if he period of f is a muliple of T. (Also, forcing a he naural period doesn' guaranee resonance...i depends wha funcion you force wih.) f 2 n = n Heaviside n Pi : ploa plo f, =..3, color = green : display ploa, ile = `square wave forcing a naural period` ; square wave forcing a naural period 2 3 ) Wha's your voe? Is his square wave going o induce resonance, i.e. a response wih linearly growing ampliude?
> x sin f d : plob plo x, =..3, color = black : display ploa, plob, ile = `resonance response?` ; resonance response? 2 3
Example 2) A riangle wave forcing funcion, same period > f2 f s ds.5 : # his aniderivaive of square wave should be riangle wave plo2a plo f2, =..3, color = green : display plo2a, ile = `riangle wave forcing a naural period` ; 2) Resonance? riangle wave forcing a naural period 2 3
> x2 sin f2 d : plo2b plo x2, =..3, color = black : display plo2a, plo2b, ile = `resonance response?` ; 5 5 5 5 resonance response? 2 3
Example 3) Forcing no a he naural period, e.g. wih a square wave having period T = 2. > f3 2 2 n = n Heaviside n : plo3a plo f3, =..2, color = green : display plo3a, ile = `periodic forcing, no a he naural period` ; 3) Resonance? periodic forcing, no a he naural period 5 5 2
> x3 sin f3 d : plo3b plo x3, =..2, color = black : display plo3a, plo3b, ile = `resonance response?` ;.5.5 resonance response? 5 5 2
Example 4) Forcing no a he naural period, e.g. wih a paricular wave having period T = 6. > f4 n = Heaviside 6 n Heaviside 6 n : plo4a plo f4, =..5, color = green : display plo4a, ile = sporadic square wave wih period 6 ; sporadic square wave wih period 6 5 5 4) Resonance?
> > x4 sin f4 d : plo4b plo x4, =..5, color = black : display plo4a, plo4b, ile = `resonance response?` ; 5 5 5 5 resonance response? 5 5
Hey, wha happened???? How do we need o modify our hinking if we force a sysem wih somehing which is no sinusoidal, in erms of worrying abou resonance? In he case ha his was modeling a swing (pendulum), how is i geing pushed? Precise Answer: I urns ou ha any periodic funcion wih period P is a (possibly infinie) superposiion P of a consan funcion wih cosine and sine funcions of periods P, 2, P 3, P,.... Equivalenly, hese 4 funcions in he superposiion are, cos, sin, cos 2, sin 2, cos 3, sin 3,... wih ω = 2. This is he P heory of Fourier series, which you will sudy in oher courses, e.g. Mah 35, Parial Differenial Equaions. If he given periodic forcing funcion f has non-zero erms in his superposiion for which n = (he naural angular frequency) (equivalenly P n = 2 = T ), here will be resonance; oherwise, no resonance. We could already have undersood some of his in Chaper 5, for example Exercise 5) The naural period of he following DE is (sill) T = 2 forcing funcion below is T = 6. Noice ha he period of he firs and ha he period of he second one is T = T = 2. Ye, i is he firs DE whose soluions will exhibi resonance, no he second one. Explain, using Chaper 5 superposiion ideas: a) x x = cos sin. 3 b) x x = cos 2 3 sin 3.