M x t = K x F t x t = A x M 1 F t. M x t = K x cos t G 0. x t = A x cos t F 0

Transcription

1 Forced oscillaions (sill undaped): If he forcing is sinusoidal, M = K F = A M F M = K cos G wih F = M G = A cos F Fro he fundaenal heore for linear ransforaions we now ha he general soluion o his inhoogeneous linear proble is of he for = P H, and we've been discussing how o find he hoogeneous soluions H As long as he driving frequency is NOT one of he naural frequencies, we don' epec resonance; he ehod of undeerined coefficiens predics here should be a paricular soluion of he for P = cos c where he vecor c is wha we need o find Eercise ) Subsiue he guess P = cos c ino he DE syse = A cos F o find a ari algebra forula for c = c Noice ha his forula aes sense precisely when is NOT one of he naural frequencies of he syse Soluion: c = A I F Noe, ari inverse eiss precisely if is no an eigenvalue

2 Eercise ) Coninuing wih he configuraion fro Monday's noes, bu now for an inhoogeneous forced proble, le =, and force he second ass sinusoidally: = cos We now fro previous wor ha he naural frequencies are =, = and ha H = C cos C cos Find he forula for P, as on he preceding page Noice ha his seady periodic soluion blows up as or (If we don' have ie o wor his by hand, we ay sip direcly o he echnology chec on he ne page Bu since we have quic forulas for inverses of by arices, his is definiely a copuaion we could do by hand) Soluion: As long as,, he general soluion = P H is given by = cos C cos C cos Inerpreaion as far as inferred pracical resonance for slighly daped probles: If here was even a sall aoun of daping, he hoogeneous soluion would acually be ransien (i would be eponenially decaying and oscillaing - underdaped) There would sill be a sinusoidal paricular soluion, which would have a forula close o our paricular soluion, he firs er above, as long as, (There would also be a relaively saller sin d er as well) So we can infer he pracical resonance behavior for differen values wih sligh daping, by looing a he size of he c er for he undaped problesee ne page for visualizaions 6

3 resar : wih LinearAlgebra : A Mari,,,,, : F Vecor, : Iden IdeniyMari : c c ; A wih plos : wih LinearAlgebra : Iden F : # he forula we wored ou by hand plo Nor c,, =, agniude =, color = blac, ile = `iicing pracical resonance` ; # Nor(c( ),) is he agniude of he c( ) vecor agniude iicing pracical resonance 8 6 () plo Nor c Pi T,, T = 5, agniude = 5, color = blac, ile = `pracical resonance as funcion of forcing period` ; pracical resonance as funcion of forcing period 5 agniude T

4 There are srong connecions beween our discussion here and he odeling of how earhquaes can shae buildings: As i urns ou, for our physics lab springs, he odes and frequencies are alos idenical:

5 An ineresing shae-able deonsraion: hp://wwwyouubeco/wach?v=m_jokahzm Below is a discussion of how o odel he unforced "hree-sory" building shown shaing in he video above, fro which we can see which odes will be ecied There is also a "wo-sory" building odel in he video, and is ari and eigendaa follow Here's a scheaic of he hree-sory building: For he unforced (hoogeneous) proble, he acceleraions of he hree assive floors (he op one is he roof) above ground and of ass, are given by = Noe he value in he las diagonal enry of he ari This is because is easuring displaceens for he op floor (roof), which has nohing above i The "" is jus he linearizaion proporionaliy facor, and depends on he ension in he walls, and he heigh beween floors, ec, as discussed on he previous page Here is eigendaa for he unscaled ari = For he scaled ari you'd have he sae eigenvecors, bu he eigenvalues would all be uliplied by he scaling facor and he naural frequencies would all be scaled by Syeric arices lie ours (ie ari equals is ranspose) are always diagonalizable wih real eigenvalues and eigenvecorsand you can choose he eigenvecors o be uually perpendicular This is called he "Specral Theore for syeric arices" and is an iporan fac in any ah and science applicaionsyou can read abou i here: hp://enwiipedia org/wii/syeric_ari) If we ell Maple ha our ari is syeric i will no confuse us wih unsiplified nubers and vecors ha ay loo cople raher han real wih LinearAlgebra : A Mari,,,,,,,,,, ; # I used a leas one decial value so Maple would evaluae in floaing poin A := (5) Digis 5 : # 5 digis should be fine, for our decial approiaions

6 eigendaa Eigenvecors Mari A, shape = syeric : # o ae advanage of he # specral heore labdas eigendaa : #eigenvalues evecors eigendaa : #corresponding eigenvecors - for fundaenal odes oegas ap sqr, labdas ; # naural angular frequencies f evalf Pi : periods ap f, oegas ; #naural periods eigenvecors ap evalf, evecors ; # ge digis down o 5 89 oegas := 7 5 periods := eigenvecors := B Mari,,,,, : eigendaa Eigenvecors Mari B, shape = syeric : # o ae advanage of he # specral heore labdas eigendaa : #eigenvalues evecors eigendaa : #corresponding eigenvecors - for fundaenal odes oegas ap sqr, labdas ; # naural angular frequencies f evalf Pi : periods ap f, oegas ; #naural periods eigenvecors ap evalf, evecors ; # ge digis down o 5 oegas := (6) periods := eigenvecors := (7) Eercise ) Inerpre he daa above, in ers of he naural odes for he shaing building In he youube video he firs ode o appear is he slow and dangerous "sloshing ode", where all hree floors oscillae in phase, wih apliude raios : 59 : 7 fro he firs o he hird floor Wha's he second ode ha ges ecied? The hird ode? (They don' show he hird ode in he video) Rear) All of he ideas we've discussed in secion 5 also apply o olecular vibraions The eigendaa in hese cases is relaed o he "specru" of ligh frequencies ha correspond o he naural fundaenal odes for olecular vibraions

7 Mah 8- Fri Mar 56 Mari eponenials and linear syses: The analogy beween firs order syses of linear differenial equaions (Chaper 5) and scalar linear differenial equaions (Chaper ) is uch sronger han you ay have epeced This will becoe especially clear on Monday, when we sudy secion 57 Definiion Consider he linear syse of differenial equaions for : = A where A n n is a consan ari as usual If,, n is a basis for he soluion space o his syse, hen he ari having hese soluions as coluns, n is called a Fundaenal Mari (FM) o his syse of differenial equaions Noice ha his equivalen o saying ha X = solves X = A X X nonsingular ie inverible (jus loo colun by colun) Noice ha a FM is jus he Wronsian ari for a soluion space basis Eaple page 5 = : = 5: general soluion Possible FM: general soluion: y = y = = = c e c = = e e v = e e v = c e 5 e 5 e 5 e 5 e 5 c c