Lecture II Simple One-Dimensional Vibrating Systems

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1 UIUC Physics 406 Acusical Physics f Music Lecure II Simple One-Dimensinal Vibraing Sysems One mehd f prducing a sund relies n a physical bjec (e.g. varius ypes f musical insrumens sringed and wind insrumens in paricular) be made vibrae, by whaever means pssible. This vibrain is (clearly) mechanical in naure. Mechanical vibrain explicily means a displacemen f he (a leas sme prins f he) maer/maerial he bjec is cmprised f frm is equilibrium psiin/cnfigurain which requires he inpu f energy he bjec in rder accmplish his iniially in he frm f (saic) penial energy (P.E.), which as ime prgresses, is subsequenly ransfrmed in kineic (minal) energy (K.E.). As ime prgresses furher, he energy scillaes back and frh beween penial and kineic energy, he al energy, E = P.E.() + K.E.() remaining cnsan in ime, if n energy lsses (energy dissipain prcesses) are presen in he mechanical sysem. The mechanically vibraing bjec cuples he air surrunding i, ransferring energy in his prcess - sund waves in he air are creaed, which prpagae uwards frm he surce (he vibraing bjec) an bserver s ear(s). Thus a sund is heard (perceived). Thus, by energy cnservain, sme f he iniial energy inpu he mechanically vibraing sysem is radiaed away in he frm f sund energy. Evenually he mechanically vibraing sysem ceases d s, because f his, and her (fricinal) dissipaive energy lss mechanisms presen. A simple example f a vibraing sysem is a mass n a spring (a crude mdel f a vibraing musical insrumen) which underges s-called -D simple harmnic min: x = 0 x = 0 x = +X x = x x = 0 x = +x - - Prfessr Seven Errede, Deparmen f Physics, Universiy f Illinis a Urbana-Champaign, Illinis All righs reserved.

2 UIUC Physics 406 Acusical Physics f Music If here is n fricin, and he mass M is hriznally displaced frm is equilibrium (x = 0) psiin by pulling n i he righ, as shwn in he abve figure, he frce necessary accmplish his is FI kx (Hke s Law), where k > 0 is he s-called spring cnsan f he spring (k has meric unis f Newns/meer) and x = he iniial displacemen f he mass M frm is x = 0 equilibrium psiin. A ime = 0 he mass is released. A ha insan, he nly frce acing n he mass is due F 0 kx F. Hwever, frm he {hriznal} resring frce f he spring: S I Newn s nd Law: F Ma, herefre a ime = 0: FS 0 kx Ma 0. As ime prgresses he mass M scillaes hriznally back and frh abu is x = 0 equilibrium psiin, exhibiing sinusidal/harmnic min. Mahemaically, he imedependence f his hriznal sinusidal/harmnic min is described by: Lngiudinal displacemen frm equilibrium: x() x cs f x cs (m) (meers) displacemen ampliude (meers) frequency f scillain (cycles per secnd = Herz) cps Hz Omega: f = angular frequency (unis = radians per secnd) Perid f scillain: f (secnds) The insananeus hriznal speed f he mving mass v() wih ime is defined as he ime rae f change f he hriznal psiin (lngiudinal displacemen) f he mving mass wih ime, physically, v() is he insananeus lcal slpe f he x() vs. graph a ime : x( ) dx( ) v( ) al derivaive f x wih respec ime, {since -D parial => al}. d d d v () x () xcs f fxsin f xsin vsin d d We see ha: v x f x sign defines he phase relain beween velciy v() relaive displacemen x(). i.e. he speed ampliude, v = max speed is relaed he displacemen ampliude, x by his frmula fr harmnic min. Insananeus Hriznal Speed f he Mving Mass: v ( ) v sin f v sin ( ms ) (meers/sec) speed frequency f scillain ampliude (m/s) (cycles per secnd = Herz) The insananeus hriznal accelerain f he mving mass a() wih ime is defined as he ime rae f change f he hriznal speed f he mving mass wih ime, physically, a() is he insananeus lcal slpe f he v() vs. graph a ime : - - Prfessr Seven Errede, Deparmen f Physics, Universiy f Illinis a Urbana-Champaign, Illinis All righs reserved.

3 UIUC Physics 406 Acusical Physics f Music v () dv () a () al derivaive f v() wih respec ime,. d d d a ( ) v ( ) vsin f fvcs f vcs acs d d We see ha: a v f v bu: v x f x i.e. he accelerain ampliude, A = max accelerain is relaed he displacemen ampliude, X by his frmula fr harmnic min. a x f x Insananeus Hriznal Accel. f he Mving Mass: a () acs f acs ( m s ) (meers/sec ) accelerain ampliude (m/s ) frequency f scillain (cycles per secnd = Herz) The ime dependence f he lngiudinal psiin, x() (i.e. displacemen f he mass frm is equilibrium psiin) vs. ime, and lngiudinal speed f he mass, v() vs. ime, and lngiudinal accelerain a() vs. ime, are shwn in he figure belw; ne ha each has been nrmalized heir respecive ampliudes (ne als he phase relain beween x(), v() and a()): Prfessr Seven Errede, Deparmen f Physics, Universiy f Illinis a Urbana-Champaign, Illinis All righs reserved.

4 UIUC Physics 406 Acusical Physics f Music Once he mass M has been se in min, Newn s nd Law ells us ha: F kx Ma Hwever: x() x cs f x cs and: a () a cs f a cs And frm abve, we als knw ha: a x f x kx Mx Thus, he frequency f and angular frequency f scillain f he mass M n he spring are: f k M Cycles per secnd, r Hz and f k M (radians/sec) The perid f scillain f he mass M n he spring is: f M k (secnds) dv d x Ne als ha since he insananeus accelerain a d d, hen we can wrie Newn s nd law fr his sysem as a differenial equain: F kx Ma d x kx M r: M d kx r: Mx kx 0 d x 0 which is a linear, hmgenus nd -rder differenial equain, and where: The insananeus penial energy sred in he sreched/cmpressed spring is: d d x x. d PE.. kx (Jules) The insananeus kineic energy assciaed wih he mving mass, M is: KE.. Mv (Jules) The penial energy f he spring and he kineic energy f he mving mass are bh ime dependen: PE.. kx kx cs 0 KE.. Mv Mv sin Prfessr Seven Errede, Deparmen f Physics, Universiy f Illinis a Urbana-Champaign, Illinis All righs reserved.

5 UIUC Physics 406 Acusical Physics f Music k Hwever, since: v and: hus: Hence, we see ha: x k M M Le us define: PE..( ) kx ( ) kx cs KE..( ) Mv ( ) Mv sin M x sin kx sin Then: PE..( ) E cs E kx M x KE..( ) E sin We define he al energy, E() as he sum f insananeus penial + kineic energies:.... cs sin cs sin E PE K E E E E Using he rignmeric ideniy cs x sin x we see ha: E E kx M x T = cnsan 0, independen f ime! Thus, he al energy in (spring + mass) sysem is cnsan due cnservain f energy!! Graphs f P.E.(), K.E.(), and E() vs. ime (all nrmalized E) are shwn in he figure belw: Prfessr Seven Errede, Deparmen f Physics, Universiy f Illinis a Urbana-Champaign, Illinis All righs reserved.

6 UIUC Physics 406 Acusical Physics f Music Ne ha P.E.(), K.E.() and E() are always 0 (i.e. never negaive)!!! Ne furher ha energy/energies are addiive, scalar quaniies. A real vibraing spring mass sysem suffers frm varius energy lss mechanisms: * fricin he mass M slides n surface, mass M als slides hrugh viscus air * spring als dissipaes energy inernally each ime i is flexed (anher ype f fricin) * Thus, he min f a real mass n a real spring is damped by fricinal prcesses. * The riginal/iniial energy, E () = E = cnsan is dissipaed by fricinal prcesses. * The iniial energy E ulimaely winds up as hea (anher frm f energy) - hus he mass, spring, hriznal surface and he air all hea up wih ime Mahemaically, we can represen he effec(s) f fricinal damping assciaed wih a -D simple harmnic scillar as a velciy-dependen (and hence ime-dependen) frce F acing hriznally n he mass M, which ppses he min, which, fr he iniial cndiins f ur prblem, his damping frce is given by: Fd bv where b is a psiive cnsan, knwn as he viscus damping cefficien, wih SI unis f kg/sec. dx Then since v x, he equain f min fr he damped -D simple harmnic d d x dx scillar becmes: M b kx 0 r: Mx bx kx 0 d d We can rewrie his differenial equain as: x b M xk M x0 he damping cnsan b M 0 and km f, hen ur linear, hmgeneus nd -rder differenial equain can als be wrien as: x x x Prfessr Seven Errede, Deparmen f Physics, Universiy f Illinis a Urbana-Champaign, Illinis All righs reserved. d and defining: The general sluin his differenial equain is f he frm: x x e r: 0 which in urn is a quadraic equain in, he sluin fr which has w 4 4 rs:. When n damping is presen ( 0 b M 0, and hus:. Explicily carrying u he ime-differeniain we bain: x x x ), hen: Defining i, hen fr 0 we see ha i. i i Nex, we use Euler s cmplex relains fr csine and sine funcins: cs e e i i and sin i e e. Fr he n-damping siuain, we already knw (frm abve) wha he sluin mus be fr he -D harmnic min, wih ur iniial cndiins: i i x () x cs x e e.

7 UIUC Physics 406 Acusical Physics f Music Nw suppse ha a small amun f damping is presen in he sysem. Mahemaically his is b M, hence i, wih purely real 0. represened by We define r equivalenly 0. Thus, fr under-damped -D harmnic, we see ha i i, and hus he physical sluin fr under-damped -D harmnic min, fr ur iniial cndiins is i i x () xe xe cs xe e e where he damping cnsan min, wih 0 b M given by: 0 b M and 0 increases, wih damping ime cnsan d M b f is iniial value a ime M b scillain falls e e shwn in he figure belw: xe 0.38x. The min is expnenially damped as ime x (secnds), where he envelpe f he -D (secnds), as Damping / dissipain affecs he vibrainal min as ime prgresses: under-damped siuain x() xe cs d damping ime cnsan M b d Dissipaive prcesses/fricin ends lwer he frequency f scillain f a vibraing sysem, as can be seen frm he relain 0. Small damping crrespnds a sligh decrease in he scillain frequency frm is naural un-damped value f km. If we nw imagine slwly increasing he damping heavy damping evenually here will be n scillain(s) a all! When, he sysem is said be criically damped, and, and he crrespnding criically-damped min is a purely-decaying expnenial wih ime: x() x e. When, he sysem is said be ver-damped, and 0. Here,, bu he physical sluin is:. The ver-damped min is again a decaying expnenial wih ime: x xe xe Prfessr Seven Errede, Deparmen f Physics, Universiy f Illinis a Urbana-Champaign, Illinis All righs reserved.

8 UIUC Physics 406 Acusical Physics f Music The fllwing figure shws he effec f under-, criical and ver-damping n he min f a -D harmnic scillar sysem: Ne ha damping prcesses ha are peraive in all musical insrumens are in he under-damped regime (since by definiin, be musical, hey mus vibrae a frequencies > 0), ypically wih small amuns f damping, i.e., such ha: 0. A mre realisic min f a vibraing mass n spring is ha assciaed wih e.g. driving i wih a peridic frce (crrespnding a linear, inhmgeneus nd -rder differenial equain): - Have ge he mass mving firs (iniially a res), akes a while fr scillains build up - Takes a finie ime reach a seady sae displacemen ampliude x - When swich ff he driving frce, displacemen ampliude decays away, as shwn belw: Slw aack e.g. flue-like sund. Fas aack e.g. mre like rumpe/sax/ec. ype sunds Slw decay large susain (e.g. slid-bdy elecric guiar). Fas decay lile susain (e.g. acusic and/r hllw-bdy, archp-ype jazz guiar). Fas vs. slw aack & decay imes are impran aspecs/aribues f he verall sund(s) prduced by musical insrumens! Prfessr Seven Errede, Deparmen f Physics, Universiy f Illinis a Urbana-Champaign, Illinis All righs reserved.

9 UIUC Physics 406 Acusical Physics f Music Legal Disclaimer and Cpyrigh Nice: Legal Disclaimer: The auhr specifically disclaims legal respnsibiliy fr any lss f prfi, r any cnsequenial, incidenal, and/r her damages resuling frm he mis-use f infrmain cnained in his dcumen. The auhr has made every effr pssible ensure ha he infrmain cnained in his dcumen is facually and echnically accurae and crrec. Cpyrigh Nice: The cnens f his dcumen are preced under bh Unied Saes f America and Inernainal Cpyrigh Laws. N prin f his dcumen may be reprduced in any manner fr cmmercial use wihu prir wrien permissin frm he auhr f his dcumen. The auhr grans permissin fr he use f infrmain cnained in his dcumen fr privae, nncmmercial purpses nly Prfessr Seven Errede, Deparmen f Physics, Universiy f Illinis a Urbana-Champaign, Illinis All righs reserved.