Modal Analysis of a Tight String

Transcription

1 Moal Aalysis of a Tigh Srig Daiel. S. Sus Associae Professor of Mechaical Egieerig a Egieerig Mechaics Presee o ME Moay, Ocober 30, 000 See: hp://web.ms.eu/~sus/me_classes.hml

2 Basic Theory The srig uer esio is he simples eample of a coiuous srucure, bu he coceps presee here are reaily eesible o more complicae srucures such as beams, plaes, a shells. Tha is, o all srucures which are relaively hi i a leas oe imesio wih respec o he ohers. Some very impora a powerful coceps will be irouce which apply o may oher physical problems, such as coucio i hea rasfer, iffusio problems, a problems i elecromageism, o ame a few.

3 Key Coceps. Separaio of variables i parial iffereial equaios. Eigevalue problems i coiuous sysems. 3. Orhogoaliy of moes of vibraio. 4. Epasio of he force soluio i erms of he homogeeous soluio.

4 Tigh Srig Moel Derivaio Figure. Geomery of igh srig wih fie es. We will focus our aeio o a srig of legh,, wih fie es as show i Figure.

5 Free Boy Diagram of Srig Eleme Figure. Free boy iagram of srig eleme.

6 Applyig Newo aw For a srig of esiy ρ per ui legh, we have by summig he forces i he y-irecio: θ τ θ θ θ θ τ τ θ τ θ θ τ τ ρ si cos si cos si, si si, f f u

7 For small θ a θ, cos cos si a si u u θ θ θ θ θ θ θ Hece, eglecig higher orer erms coaiig he equaio of moio becomes a, u f u ρ τ

8 The equaio of moio he omai equaio plus he iiial values a he bouary values cosiues wha is kow as he well-pose IBVP iiial bouary value problem. The bouary values for a fie srig are: u 0, u, 0 3 The iiial values are eermie by he iiial shape a velociy isribuio of he srig: a u,0 g u,0 h 4 5

9 Separaio of Variables a he Free-Vibraio Problem The free-vibraio, or eigevalue problem, give by u u ρ τ may be reaily solve by he meho of separaio of variables. eig,, T U u 7 6

10 we have T U τ T ρ U 8 Diviig by ρut we obai ρ U τ U T T 9 Where - is a cosa. The reaso for he egaive sig will become obvious shorly.

11 The oly way ha a fucio of oe variable, say, ca be equal o a fucio of aoher variable, i his case, if for boh fucios be be equal o a cosa. This beig he case, Equaio 9 may be recas as wo equaios: T T 0 0 τ U ρ U 0, for 0 Equaios 0 a he iiial a bouary values form he IBVP.

12 Equaio 0 implies he epece resul ha he soluio will be harmoic i ime. Ha we chose a posiive cosa i Equaio 9, he emporal soluio woul be epoeial i aure his is clearly o-physical. Hece, he emporal soluio may be wrie T Acos Bsi For simpliciy, we cosier he case of a igh srig uer cosa esio τ cos., a wih cosa mass esiy per ui legh ρ cos.. Equaio becomes U U 0, for 0 c 3

13 where c τ ρ 4 The cosa c is he spee of sou i he srig. Equaio 3 has a geeral soluio of he form U C cos c Dsi c 5 To eermie specific moe shape, we mus apply he bouary coiios: 6 u0, u, U 0 T U T 0 U 0 0 U 0 0 7

14 Equaio 6 implies ha U 0 C 0 U Dsi c 8 From Equaios 8 a 7, we have Hece, we mus have U Dsi c si 0 c 0, for,,3 c 9 0

15 Thus, we have ha he aural frequecies occur i iscree values give by c τ ρ Furhermore, he overall moio is compose of a sum of iscree moes of he form U si 3 where he he ukow cosa, D, has bee se o uiy because i is arbirary.

16 Hece, he oal geeral soluio is give by u, A cos B si si 4 The ukow cosas also occur iscreely, a mus be eermie from he iiial coiios by applyig he pricipal of orhogoaliy of moes.

17 Orhogoaliy Makes he Worl Go Arou! From he iiial coiios, we have u,0 Muliplyig 5 by g m si A si a iegraig over he Domai yiels he ukow cosa, A, because of he followig relaioship: 5

18 m for m 0 for m 0 si si δ m 6 where δ m is he Kroeker ela. Thus, we obai A g si 7 Similarly, from Equaio 5 we obai 0 B h si 0 8

19 Eample: he Plucke Guiar Srig Cosier he followig eample of a igh srig wih iiial coiios give by a u,0 H, 0 H, < 9 u,0 0 30

20 The siuaio looks like he followig: Figure 3. Iiial shape of a plucke guiar srig. From Equaios 7 a 9 we have A H si H si 0 3

21 Evaluaig he iegrals yiels o for 8 eve 0 for si 8 H H A 3 Thus, H u H u cos si 8, or cos si si 8,,3,5 33

22 c m m m H u m m cos si 8, 0 Aleraively, he ummy ie,, may be shife o avoi he o iicial oaio: 34

23 The Effec of Dampig o he Respose of he Plucke Srig Figure 4. FBD of srig wih ampig a cosa esio.

24 The equaio of moio igh srig wih cosa esio a esiy, bu icluig isribue ampig as show i Figure 4 may be show o be u ρ u β u τ f, 35 where β is he isribue viscous ampig cosa wih uis Newo-secos/meer. We oe ha i Figure 4, he ampig force opposes he moio of he srig, so resuls i a egaive applie force i Newo s law. We will seek a soluio i erms of he previously eermie eige fucio of he form u, η si 36

25 Subsiuio of 36 io 35 yiels, si f c ρ η η ρ β η 37 where use has bee mae of Equaio 4 a he o oaio for erivaives wih respec o ime. Muliplicaio of 37 by m si a iegraio wih respec o over 0, yiels for m f c 0 si, ρ η η ρ β η 38

26 because he lef ha sie of Equaio 37 vaishes ieically for m. I caoical form, 37 becomes where a η η η ρ c l 0 f, si 39 l τ ρ 40 β β 4 ρ τρ

27 Plucke Guiar Srig wih Viscous Dampig As i he previous eample which eglece ampig, f, 0, a he iiial coiios are give by Equaios 9 a 30. The equaio for he so-calle moal paricipaio facor, Equaio 39, becomes η η η The soluio may easily be show o be 0 A cos B si η e where he coefficies A a B mus be eermie from he iiial coiios. 4 43

28 The frequecy of oscillaio is effece by ampig, a is referre o as he ampe aural frequecy of oscillaio, a give by 44 Hece, he geeral soluio is give by u, e A B cos si si Applicaio of he iiial coiios yiels he same value for A as give i Equaio 3, bu he iiial velociy equaio yiels B A 45 46

29 Thus, he oal soluio may be wrie e H u si si cos 8, 0 where he ie has agai bee shife o accou for he ooly iices. I erms of he physical parameers, we oe ha 4 β τρ β 48 47

30 The Noes a Frequecies o a Classical Guiar Table. Noes a frequecies o he classical guiar lise as: frequecy Hz/srig/fre. The guiar provies a impora applicaio for he plucke srig moel. The oes a heir correspoig frequecies are lise i Table.

31 Equaios 34 a 47 are simulae see accompayig simulaios usig aa for i he he D srig of a acousic guiar. The aa use are as follows: τ ρ c H N kg/m 90.0 m/s β 0.00 m m 0.77 N - s/m The acual value of β for he D srig is much less, abou N-s/m, bu he above value was 3

32 was use o keep he size of he resulig simulaio ow. The D srig o oe of my classical guiars, alhough ol a raher iry, rag percepibly for abou 3 secos. Hece, he resulig aimaio woul be approimaely 4 megabyes i size far oo large o owloa eve from a campus compuer. The e opic o aress is he las case we will cover i his suy of he plucke srig. We will eamie how o hale a paricular case of forcig. The mehos show will easily ee o oher cases.

33 The Plucke Guiar Srig wih Delaye Impulsive Forcig Figure 5. Plucke srig wih poi force locae a a. The e case we will cosier has a applie force wihi he omai 0,. The siuaio is ieical o he previous eample, so we will sar wih Equaio 39 a assume f, F0δ a δ α 49

34 Equaio 49 escribes a impulsive force of magiue, F 0, applie a a poi a. This moel is aalogous o a piao srig sruck by he hammer. I our eample, i migh moel he impac of a figer ail o he ha of a flameco guiar player as he percusssively srums hrough he D srig. The magiue of he impulse, F 0, has uis of N-s. Tha his mus be he case is sems from he fac ha he ela fucio is relae o he Heavisie sep fucio as follows: δ a U a Hece, δ has uis of /m because U is imesioless. A similar relaioship eiss for δ-α: δ α U α 50 5

35 Thus, δ has uis of /. The /m ui ges cacelle ou urig he iegraio process of Equaio 38. The uis o he righ a lef-ha sies of Equaios 38 a 39 are m/s acceleraio. Hece, F 0 mus have uis cosise wih a impulse, ha is, N-s. Upo iegraio a he applicaio of orhogoaliy, Equaio 39, he moal paricipaio facor equaio becomes η ρ F 0 0 η η δ a δ αsi 5

36 Carryig ou he spaial iegraio, we have by he filerig propery of he Dirac ela fucio si 0 α δ ρ η η η a F 53 Takig he aplace rasform of Equaio 53, a solvig for he η s, yiels si s s s s s e a F s η η η ρ η α 54

37 Takig he iverse aplace rasformaio yiels e U e a F η η η α α ρ η α si 0 0 0cos si si 0 55 Applyig he iiial coiios, we have < H H u, 0, 0si,0 η 56

38 Muliplyig Equaio 56 by 8H η 0 m si si, we have for m which is he same resul we obaie for A i Equaio 3. Tha his is so shoul o be surprisig sice he wo sysems have he same iiial shape, which is o effece by eiher he ampig or he forcig i his case. The applie impulse occurs a ime α > 0, so he respose ue o his forcig oes o eis a ime 0. 57

39 The iiial velociy is zero, so we have u,0 η 0si 0 η The valiiy of Equaio 58 sems from he fac ha he sie erms are all liearly iepee i.e. you ca ge ay oe of he erms i he series from a liear combiaio of he ohers. Therefore, if he sum of all erms vaishes, he each of he coefficies mus vaish.

40 e H U e a F α α ρ η α si cos 8 si si 0 Hece, we obai he moal paricipaio facor The oal soluio is hus 59 e H U e a F u α α ρ α si si cos si 8 si si, 0 60