F (u) du. or f(t) = t

Transcription

1 8.3 Topic 9: Impulses and dela funcions. Auor: Jeremy Orloff Reading: EP 4.6 SN CG.3-4 pp.2-5. Warmup discussion abou inpu Consider e rae equaion d + k = f(). To be specific, assume is in unis of d kilograms. Te derivaive and e inpu f() are in unis of kg/ime. So, e oal amoun inpu from ime o ime is Eample: If f() = 4 /4 f(u) du. or f() = /8 en (in eier case) e oal inpu over ime is kg. Suppose we coninue o soren e ime inerval in wic we inpu a oal of kg. In e limiing case we dump kg in all a once. Tis is an insananeous jump in e amoun, called an impulse. In general, for a firs order sysem an impulse is an insananeous jump in e amoun. Now consider e second order sysem m + b + k = F (). Here, F () is a force i canges momemum over ime. Te oal momemum added in e inerval [, ] is F (u) du. Similar o e firs order case, an impulse is wen we add all e momemum a once. In is case, we ge a jump in e momemum. For our ypical fied mass sysem is is e same as a jump in e velociy. Ta is, a jump in. New ecnically correc version of Laplace Wi disconinuiies come quesions. For eample, for e uni sep funcion we ave e formula L(u ) = sl(u) u() = u(). Bu wic do we mean u( ) or u( + )? Te correc way o do is is wi : Lf = f()e s d, Lf = sf f( ) f g() = 8 f (u)g( u) du Tis is rig for wo reasons:. Our ma will be consisen and fudge free. 2. Pysically is is wa we usually know. E.g. we know a sysem is a res rig up o e momen i s urned on. A = : A = + : condiions are pre-iniial condiions condiions are pos-iniial condiions From now on all our formulas will use, e.g., Lf = (coninued) f()e s d, Lf = sf f( ).

2 8.3 opic 9 2 Dela funcion and uni impulses An impulse o a sysem causes an insananeous jump. To a armonic oscillaor a kick causes a jump in momenum. To a firs order sysem, e.g. radioacive dumping, an impulse is a sudden addiion of a quaniy of radioaciviy. A uni impulse causes a uni cange. I is modeled by e δ funcion. We will describe is, en formally define δ() and en eplore is properies. δ as a limi of owers For second order sysem we can ink of e inpu as force. Te oal cange in momenum caused by e force is given as an inegral over ime. I.e. as e area under e grap of force vs. ime. { for < < Define u () = for < 6 Toal area under grap is. 2 As e grap ges aller and inner and looks more like a spike. Also as e force is applied over a sorer and sorer ime and 2 6 acs more like a uni impulse. = = /2 = /6 =, δ() I.e. δ = lim u (propery 5 below) Defining properies of e dela funcion :. δ = uni impulse. δ() = for any. 2. a a δ() d = for any a >. 3. f()δ() d = f() for any a > and any coninuous funcion f. Oer properies: 3. f()δ( a) d = f(a) for any a > and any coninuous funcion f. 4. Lδ = 4. Lδ( a) = e as 5. δ = lim u (can always replace δ by is limi) 6. P (D) = δ, wi res IC X = /P (s) If e sysem is P (D) = f() and f() is e inpu en X = ransfer funcion = sysem funcion and = weig funcion = uni impulse response. 7. δ = u (coninued)

3 8.3 opic 9 3 Pracice wi δ δ() d =, δ()2e 4 d = 2, δ() d =, Laplace of δ: L(δ) = δ()2e 4 d =, δ() d =, + δ() d =, δ()2e 4 d = 2, + δ()e s d = e s = (propery 4) δ() d =. δ()2e an2 (3) d =. Eample: (propery 6) Solve + k = δ; res IC. Laplace (s + k)x = X = s+k (ransfer funcion) = e k. (weig funcion) Inpu = δ Pysical inerpreaion, effec of δ is o bump e value of from a = o a = +. Afer a e sysem undergoes eponenial decay. Eample: + b + k = δ; res IC. P (s) = s 2 + bs + k; (s 2 + bs + k)x = X = /P (s) = ransfer funcion. Eample: (properies 5 and ) (no done in class) Solve + k = u (pysical reasoning:) Tis models radioacive dumping. u = rae maer added over ime oal amoun added = u =. wi res IC. Te doed line in e grap below sows ow would increase if ere was no decay. Since e decay lowers e grap is below e doed line. Afer ime = ere is no more inpu and e grap sows eponenial decay. As e inpu becomes δ(). Te oal amoun added = δ() d = and i is dumped in all a once. Tis is called a uni impulse, i.e. e wole cange appens a once. For compleeness we acually solve e IVP (no done in class). + k = u () = (u() u( )); res IC (s + k)x = ( s e s ) X = ( s(s+k) e s ) = ( )( k s s+k e s ) = ( k e k ) u( )( k e k( ) ) = { ( k e k ) k (ek )e k for < < for < ( u-forma ) ( cases-forma ) + Inpu = u Jus as epeced, as e inpu becomes δ and e oupu becomes = e k e k (i.e. lim = ) k (coninued) Inpu = δ

4 8.3 opic 9 4 Two ways o find e weig funcion (propery 6) (A) Eample: Solve e IVP m + b + k = δ() wi res IC. (Res IC = ( ) = ( ) =.) (using Laplace) m(s 2 X s( ) ( )) + b(sx ( )) + kx = (ms 2 + bs + k)x = X = = W (s) e ransfer funcion = w e weig funcion. P (s) Te weig funcion w is also called e uni impulse response. (A) for < en a = e momenum jumps o... (B) Waever before ime, near = posiion goes o and momenum is. Eample (A) as e same soluion as (B) Eample: Solve P (D) = ; ( ) =, ( ) = /m. (mig skip in class) Reason : (algebra) Laplace: m(s 2 X s() ()) + b(sx ()) + kx = (ms 2 + bs + k)x = X = /P (s). bo (A) and (B) ave e same Laplace ransform. Reason 2: (pysical meaning) In a spring mass DE e inpu is force. Te oal amoun of momenum added o e sysem by e inpu is is inegral over ime. So, e δ funcion represens an impulse wic causes a sudden (uni) jump in momenum. Terefore a = + bo (A) and (B) give e same posiion, momenum and inpu. And afer ime e inpu in bo sysems is. (Picures on previous page.) Eample: Solve + = δ() wi res IC. X = /(s 2 + ) = sin for >. Pysical eplanaion: a = an impulse kicks e simple armonic oscillaor ino moion. Afer a inpu is and e sysem is in simple armonic moion. Te jump in momenum = corner in grap a. (Draw e grap.) Eample: Solve + = δ( a) wi res IC. X = e as /(s 2 + ) = u( a) sin( a) for >. Te pysical eplanaion and grap are e same as in e previous eample sifed by a. (coninued)

5 8.3 opic 9 5 Eample: (Resonance) Solve + = f; res IC. Were f = blow every 2π seconds, magniude 2 in e posiive direcion. f = 2δ() + 2δ( 2π) + 2δ( 4π) +... F = L(f) = 2 + 2e 2πs + 2e 4πs X = s 2 + F = 2 s 2 + ( + e 2πs + e 4πs +...) = 2u() sin() + 2u( 2π) sin( 2π) +... = 2 sin [u() + u( 2π) + u( 4π) +...] 2 sin for < < 2π = 4 sin for 2π < < 4π 6 sin for 4π < < 6π Tis is resonance. Propery 7: u = δ Argumen : From above we ave L(u ) = u( ) =. QED Argumen 2: u () = for and u () =. (Te grap of u is verical a u () =.) Compue e area under is grap: Area under u = b a u () d = u(b) u(a). If in in [a, b] en u(b) = and u(a) = e area is. If no e area is. Ta is, area under curve is, all of i concenraed above. Argumen 3: Using propery 5, u () = lim u() u( ) = lim u () = δ Generalized funcions δ is no an ordinary funcion. We call i a generalized funcion. I canno be used everywere an ordinary funcion can go, bu i can go anywere we pu inpu in 8.3. Te derivaive u is called a generalized derivaive. Relaion of properies 3 and 5 We could draw some picures relaing properies 5 and 3.