2 One-Dimensional Elasticity

Transcription

1 Oe-Dimesioal Elasticity There are two tyes of oe-dimesioal roblems, the elastostatic roblem ad the elastodyamic roblem. The elastostatic roblem gives rise to a secod order differetial equatio i dislacemet which may be solved usig elemetary itegratio. The elastodyamic roblem gives rise to the oe-dimesioal wave equatio, whose solutio redicts the roagatio of stress waves ad vibratios of material articles 9

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3 Sectio.. Oe-dimesioal Elastostatics Cosider a bar or rod made of liearly elastic material subjected to some load. Static roblems will be cosidered here, by which is meat it is ot ecessary to kow how the load was alied, or how the material articles moved to reach the stressed state; it is ecessary oly that the load was alied slowly eough so that the acceleratios are zero, or that it was alied sufficietly log ago that ay vibratios have died away ad movemet has ceased. The equatios goverig the static resose of the rod are: dσ + b = d Equatio of Equilibrium (..a) du ε = d Strai-Dislacemet Relatio (..b) σ = Eε Costitutive Equatio (..c) where E is the Youg s modulus, ρ is the desity ad b is a body force (er uit volume). The ukows of the roblem are the stress σ, strai ε ad dislacemet u. These equatios ca be combied to give a secod order differetial equatio i u, called Navier s Equatio: d u b + d E = -D Navier s Equatio (..) Oe requires two boudary coditios to obtai a solutio. et the legth of the rod be ad the ais be ositioed as i Fig.... The ossible boudary coditios are the. dislacemet secified at both eds ( fied-fied ) u ( ) = u, u( ) = u. stress secified at both eds ( free-free ) σ ( ) = σ, σ ( ) = σ 3. dislacemet secified at left-ed, stress secified at right-ed ( fied-free ): u( ) = u, σ ( ) = σ 4. stress secified at left-ed, dislacemet secified at right-ed ( free-fied ): σ ) = σ, u ( ) = ( u Figure..: a elastic rod Solid Mechaics Part II

4 Sectio. Note that, from..b-c, a stress boudary coditio is a coditio o the first derivative of u. Eamle Cosider a rod i the absece of ay body forces subjected to a alied stress σ o, Fig.... σ o σ o The equatio to solve is Figure..: a elastic rod subjected to stress d u d = (..3) subject to the boudary coditios u d = σ =, E u d = σ = E (..4) Itegratig twice ad alyig the coditios gives the solutio σ u = + B E (..5) The stress is thus a costat σ ad the strai is σ o / E. There is still a arbitrary costat B ad this hysically reresets a ossible rigid body traslatio of the rod. To remove this arbitrariess, oe must secify the dislacemet at some oit i the rod. For eamle, if u ( / ) =, the comlete solutio is u σ E σ ε = E o o =,, σ = σ o (..6).. Problems. What are the dislacemets of material articles i a elastic bar of legth ad desity ρ which hags from a ceilig (see Fig...). 3. Cosider a steel rod ( E = GPa, ρ = 7.85 g/cm ) of legth 3 cm, fied at oe ed ad subjected to a dislacemet u = mm at the other. Solve for the stress, strai ad dislacemet for the case of gravity actig alog the rod. What is the solutio i the absece of gravity. How sigificat is the effect of gravity o the stress? Solid Mechaics Part II

5 Sectio.. Oe-dimesioal Elastodyamics I rigid body dyamics, it is assumed that whe a force is alied to oe oit of a object, every other oit i the object is set i motio simultaeously. O the other had, i static elasticity, it is assumed that the object is at rest ad is i equilibrium uder the actio of the alied forces; the material may well have udergoe cosiderable chages i deformatio whe first struck, but oe is oly cocered with the fial static equilibrium state of the object. Elastostatics ad rigid body dyamics are sufficietly accurate for may roblems but whe oe is cosiderig the effects of forces which are alied raidly, or for very short eriods of time, the effects must be cosidered i terms of the roagatio of stress waves. The aalysis reseted below is for oe-dimesioal deformatios. Iheret are the assumtios that () material roerties are uiform over a lae eredicular to the logitudial directio, () lae sectios remai lae ad eredicular to the logitudial directio ad (3) there is o trasverse dislacemet... The Wave Equatio Cosider ow the dyamic roblem. I this case u u(, ad oe cosiders the goverig equatios: b a Equatio of Motio (..a) u Strai-Dislacemet Relatio (..b) E Costitutive Equatio (..c) where a is the acceleratio. Eressig the acceleratio i terms of the dislacemet, oe the obtais the dyamic versio of Navier s equatio, u u E b -D Navier s Equatio (..) t I most situatios, the body forces will be egligible, ad so cosider the artial differetial equatio u c u t -D Wave Equatio (..3) where E c (..4) Solid Mechaics Part II 3

6 Sectio. Equatio..3 is the stadard oe-dimesioal wave equatio with wave seed c; ote from..4 that c has dimesios of velocity. The solutio to..3 (see below) shows that a stress wave travels at seed c through the material from the oit of disturbace, e.g. alied load. Whe the stress wave reaches a give material article, the article vibrates about a equilibrium ositio, Fig.... Sice the material is elastic, o eergy is lost, ad the solutio redicts that the article will vibrate idefiitely, without damig or decay, uless that eergy is trasferred to a eighbourig article. vibratio of stressed article stress wave at seed c stress free Figure..: stress wave travellig at seed c through a elastic rod This tye of wave, where the disturbace (article vibratio) is i the same directio as the directio of wave roagatio, is called a logitudial wave. The wave equatio is solved subject to the iitial coditios ad boudary coditios. The iitial coditios are that the dislacemet u ad the article velocity u/ t are secified at t (for all ). The boudary coditios are that the dislacemet u ad the first derivative u/ are secified (for all. This latter derivative is the strai, which is roortioal to the stress (see Eq...b). I roblems where there is o boudary (a ifiite medium), o boudary coditios are elicitly alied. A semi-ifiite medium will have oe boudary. For a rod of fiite legth, there will be two boudaries ad a boudary coditio will be alied to each boudary... Particle Velocities ad Wave Seed Before eamiig the wave equatio..3 directly, first re-eress it as v t (..5) where v is the velocity. Cosider a elemet of material which has just bee reached by the stress wave, Fig.... The legth of material assed by the stress wave i a time iterval t is c t. Durig this time iterval, the stressed material at the left-had side of the elemet moves at (average) velocity v ad so moves a amout v t. The strai of the elemet is the the chage i legth divided by the origial legth: v (..6) c Solid Mechaics Part II 4

7 Sectio. Uder the small strai assumtio, this imlies that et the stress actig o the elemet be zero. The..5 leads to v c. ; the stress o the free side of the elemet is v ct t (..7) ad so cv (..8) This is the discotiuity i stress across the wave frot. wave frot at time t ct vt wave frot at time t t Figure..: stress wave assig through a material elemet Sice E, oe has c E /, as i..4. The wave seeds for some materials are give i Table... As ca be see, the wave seeds for tyical egieerig materials are of the order km/s ad so article velocities will be i the rage 5m/s. Material 3 kg/m E GPa c m/s Alumiium Alloy Brass Coer ead Steel Glass Graite imestoe Perse Table..: Elastic Wave Seeds for Several Materials ote also that the desity of the elemet will chage as it is comressed, but agai this chage i desity is small ad ca be eglected i the liear elastic theory Solid Mechaics Part II 5

8 Sectio. Cosider steel: the velocity at which the material ceases to behave liearly elastic (takig the yield stress to be 4MPa) is v Y / c m/s...3 Waves Before roceedig, it will be helful to review ad summarise the imortat facts ad termiology regardig waves. Suose that there is a dislacemet u which is roagated alog the ais at velocity c. At time t say, the disturbace will have some wave rofile u f( ). If the disturbace roagates without chage of shae, the at some later time t the rofile will look idetical but it will have moved a distace ct i the ositive directio. If we take a ew origi at the oit ct ad let the distace measured from this origi be, the the equatio of the ew wave rofile referred to this ew origi would be u f( ). Referred to the origial fied origi, the, u f ct. (..9) This is the most geeral eressio for a wave travellig at costat velocity c ad without chage of shae, alog the ositive ais. If the wave is travellig i the u f ct. egative directio, the its form would be The simlest tye of wave of this kid is the harmoic wave, i which the wave rofile is a sie or cosie curve. If the wave rofile at time t is u acosk, the at time t the rofile is u acos k ct. (..) The maimum value of the disturbace, a, is called the amlitude. The wave rofile reeats itself at regular distaces /k, which is called the wavelegth. The arameter k is called the wave umber ; sice there is oe wave i uits of distace, it is the umber of waves i uits of distace: k. (..) The distace travelled by the wave i time t is ct. The time take for oe comlete wave to ass ay oit is called the eriod T, which is the time take to travel oe wavelegth: T. (..) c The frequecy f is the umber of waves assig a fied oit i uit time, so more secifically, this is the agular waveumber, to distiguish it from the (sectroscoic) waveumber / Solid Mechaics Part II 6

9 Sectio. f c T (..3) The agular frequecy is f kc. As the wave travels alog, the article at ay fied oit dislaces back ad forth about some equilibrium ositio; the article is said to vibrate. The eriod ad frequecy were defied above i terms of the time take for a wave to travel alog the ais. It ca be see that the eriod T is also equivalet to the time take for a article to dislace away ad the back to its origial ositio, the off i the other directio ad back agai; the frequecy f ca also be see to be equivalet to the umber of times the article vibrates about its equilibrium ositio i uit time. The wave.. ca be eressed i the equivalet forms: u acos k ct u acos ct t u acos T u acos t u acos kt (..4) If oe has two waves, u a k t ad u a k t cos cos, the the waves are the same ecet they are dislaced relative to each other by a amout / k / ; is called the hase of u relative to u. If is a multile of, the the dislaced distace is a multile of the wavelegth, ad the waves are said to be i hase. It ca be verified by substitutio that the wave..4 is a solutio of the wave equatio..3. Eamle Fig...3 shows a wave travellig through steel ad vibratig at frequecy f khz. Usig the data i Table.., the wave umber is k f / c. ad the wavelegth is c/ f 5.. The eriod is T / sec. For uit amlitude, a, the wave rofiles are show for t (blue) ad t /5 sec ( 3 T ) (red). The dashed arrows show the movemet of oe article as the wave asses. Solid Mechaics Part II 7

10 Sectio..8 ct c/ f acos k ct Figure..3: harmoic wave (Eq...) travellig through steel at khz; a with t (blue) ad t /5 (red) Stadig Waves Because the wave equatio is liear, ay liear combiatio of waves is also a solutio. I articular, cosider two waves which are similar, oly travellig i oosite directios; the suerositio of these waves is the ew wave cos u a cos k t a k t acos k cos t (..5) It will be see that this wave rofile does ot move forward, ad is therefore called a stadig wave (to distiguish it from the rogressive waves cosidered earlier). A eamle is show i Fig...4 (same arameters as for Fig...3); at ay fied oit, the wave moves u ad dow over time. The eriod is agai T / sec. Show is the wave at five istats, from t u to just short of the half-eriod. Note that u for ( ) / k,,,, ; these are called the odes of the wave. The itermediate oits, where the amlitude is greatest, are called atiodes. The distace betwee successive odes (or atiodes) is half the wavelegth. Solid Mechaics Part II 8

11 Sectio..5 t t t. t.3 t.4 Figure..4: stadig wave (Eq...5) i steel at khz; with a at t (black), t. (red), t. (gree dashed), t.3 (blue dotted) ad t.4 (red dotted) If the wave is ot harmoic, oe ca use a Fourier aalysis (see below) to costruct the wave out of a sum of idividual harmoic waves; if the rofile cosists of a regularly reeatig atter, the defiitios of wavelegth, eriod, frequecy ad wave umber, ad the relatios betwee them, Eqs...-3, still aly. Comle Eoetial Reresetatio Whe dealig with rogressive waves of harmoic tye, it is usually best to rereset the wave usig a comle eoetial fuctio. The reaso for this is that eoetials are algebraically simler tha harmoic fuctios, ad also the amlitude ad hase are rereseted by oe comle quatity rather tha by two searate terms (as will be see below). The geeral wave of the form is the real art of the comle eoetial u acoskt (..6) si ikt ae a cos k t i k t (..7) The hase shift ad amlitude ca be absorbed ito a ew costat A: u t ik Ae, A i ae (..8) It ca be verified that this comle quatity is itself a solutio of the wave equatio, Eq...3 (ad if a comle quatity is a solutio, so are its real ad imagiary arts). Oe ca carry out aalyses usig the comle eressio..8, keeig i mid that the Solid Mechaics Part II 9

12 Sectio. real solutio, Eq...6, is the real art of this eressio. Sice amlitude is A. The true hase shift is the argumet of A, arg A. e ik t, the true Eq...6 is a wave travellig to the right. It has bee see how a wave travellig to u acos k t, suggestig a comle reresetatio the right is of the form t ik u Ae. However, this is ot a ideal reresetatio, because the differece betwee a wave travellig left or right, i.e. the differece betwee this eressio ad the oe i Eq...7, is give by the sig of the frequecy. This ca make it difficult to solve roblems ivolvig reflectig waves 3 (see below), ad therefore it is best to use the followig reresetatios whe addig ad subtractig waves: Travellig right: Travellig left: Ae Ae ik t ik t (..9a) (..9b) (Note: aother oular covetio is to use Ae ik t for right ad Ae ik t for left.)..4 Solutio of the Wave Equatio (D Alembert s Solutio) The oe-dimesioal wave equatio..3 has the very geeral solutio (this is D Alembert s solutio see the Aedi to this sectio for its derivatio) ct g ct u(, f (..),, which ca be verified by substitutio ad carryig out the differetiatio. The harmoic waves cosidered above are secial cases of this solutio, i which f ad g are cosie where f ad g are ay fuctios 4 ct ; for eamle, oe solutio is f e g si ct fuctios. The actual forms of the fuctios f ad g ca be determied from the iitial coditios of the roblem, which are the iitial dislacemet rofile u (,) ad the iitial velocity v, u/ t (,). Cosider the arbitrary iitial coditios u (,) U ( ) v (,) V ( ) (..) The, as show i the Aedi to this sectio, the solutio is ut (, ) U ct U ct V( ) d c (..) ct ct 3 for eamle, whe a wave hits a boudary ad gets reflected, this reresetatio would force the icidet ad reflected waves to have differet frequecies, whe i fact a solutio i which the frequecies are the same is ofte sought 4 rovided they ossess secod derivatives Solid Mechaics Part II 3

13 Sectio. Eamle Suose for eamle that the iitial dislacemet rofile was triagular, with maimum dislacemet u u at, etedig to, Fig...5. u(,) The iitial coditios are u u Figure..5: a iitial triagular dislacemet, U( ) u(,) u( / ), u( / ), ad V( ). D Alembert s solutio is the ut (, ) U ( c U ( c The solutio redicts that at time / c there are two triagular dislacemet rofiles of half the magitude of the origial rofile; oe is to the left ad the other is to the right of the origial rofile, Fig...6. u ( u ) u Figure..6: dislacemets at time /c As the wave asses, articles dislace from their equilibrium oit, u to the maimum ositio ad the back agai. It ca be see that the solutio corresods to a wave of disturbed material roagatig through the material from the source, half i oe directio ad half i the other. Solid Mechaics Part II 3

14 Sectio...5 Reflectio ad Trasmissio et a trai of harmoic waves travel from the egative directio i a material with material roerties E,. The waves the meet a secod material with differet material roerties E,, at the origi. et the dislacemets i the first material be u ad those i the secod, u. As will be see, the icidet wave uo the secod material will suffer artial reflectio ad artial trasmissio. Usig the comle eoetial reresetatio, Eq...9, ad suerscrits i for icidet, r for reflected ad t for trasmitted: u u u, u u (..3) () i ( r ) () t with u Ae, u Ae, u Ae (..4) () i ik t ( r) ik t () t ik t i r t A is real, but i geeral B, A could be comle. The wave seeds c i each material will be differet (if the material roerties are differe. The frequecies of all three waves are the same sice the material is coected to adjacet material, it must all be vibratig at the same frequecy. It follows that the waveumbers k differ also: kc k E kc or (..5) k E The boudary coditios at the material iterface are that u (, u (, u E E u (, (, (..6) The first of these says that the material remais cotiuous at the iterface. The secod says that the stress is also cotiuous there (see Eqs...b-c). Alyig these to Eq...3 gives A A A i r t AEk AEk AE k i r t (..7) so that Ar Ai, At Ai (..8) where Solid Mechaics Part II 3

15 Sectio. Ek c E Ek c E (..9) Note that, sice A i is real, so also are Ar, A. t The stresses are give by A A E k, A A Ek ( r) r () i () i () t t () i () i i i (..3) The arameter determies the ature of the reflected ad trasmitted waves, ad is the ratio of the quatities c of each material; this quatity c is ofte referred to as the mechaical imedace of the rod. Note that the stiffess E ad desity are ideedet, so if E E, this does ot imly that or that (see Table..). Whe, the reflected wave has oosite sig to that of the icidet wave ad has a smaller amlitude. The trasmitted wave is of the same sig ad is also smaller. I the limit as, which would rereset a erfectly rigid material ( E ), there is o trasmitted wave ad the reflected wave has amlitude Ar Ai. The stress at the boudary is twice the stress due to the icidet wave aloe. Whe, the reflected wave has the same sig to that of the icidet wave ad has a smaller amlitude. The trasmitted wave is of the same sig ad is larger. I the limit as, which would rereset emty material, the reflected wave is equal to the icidet wave. The stress at the boudary is zero this is called a free boudary (see below). Eamles of harmoic waves travellig through steel ad graite are show i Fig...7. The frequecy of vibratio is take to be f khz. Usig the data i Table.., the wave umbers are ks f / cs. ad kg f / cg.. The wavelegths of the waves are s cs / f 5. ad g cg / f 3.. The icidet wave is take to have uit amlitude. Whe the wave travels from steel ito graite,.7 ad whe it travels from graite ito steel it is the iverse of this, The iterferece betwee the icidet ad reflected waves roduce a ew wave i material (deoted by the gree lots i Fig...7): u acoskt coskt (..3) Note that At Ai at time t (full reflected ad trasmitted wave rofiles are lotted at time zero, eve though there is o actual wave reset right through the material yet at this time). Solid Mechaics Part II 33

16 Sectio. Steel Graite.5 ct g.5 ct s ct s Graite Steel ct g.5 ct g ct s Figure..7: reflectio ad trasmissio of harmoic waves at the boudary betwee steel ad graite; at time t (solid) ad time t /5 (dashed); icidet (black), reflected (blue), trasmitted (red) ad comosite wave i material (gree) Solid Mechaics Part II 34

17 Sectio...6 Eergy i Vibratig Bars The kietic eergy i a elemet of legth d of the bar is dk A u t d, where / A is the cross-sectioal area. The total kietic eergy i a bar of legth is the K A u/ t d, (..3) The otetial eergy is the elastic strai eergy; for a small elemet of legth d this is dw Ad, so W AE u/ d, (..33)..7 Solutio of the Wave Equatio (Stadig Waves) D Alembert s solutio gives results for rogressive waves travellig i a ifiitely eteded medium. Stadig waves i a ifiite medium ca also be a solutio. For eamle if oe has the iitial rofile U( ) acosk ad zero iitial velocity, V( ), oe gets from Eq... the stadig wave..5. Stadig waves ca be geerated more geerally by usig a searatio of variables solutio rocedure for Eq...3. Usig this method, detailed i the Aedi to this sectio, oe has the geeral solutio u(, Acosk Bsi kccosckt Dsi ckt (..34) The (ifiite umber of) costats ABCD,,, ad eigevalues 5 k ca be obtaied from the iitial ad boudary coditios (see later). What are termed eigevalues i this cotet ca be see to be the wave umber. The terms cosk ad si k are called modes or mode shaes. At ay give time t, the dislacemet is a liear combiatio of these modes. Eamle modes are show i Fig...8. Some modes will domiate over others, for eamle erhas oly the first few modes (terms i the series..34) are sigificat ad eed be cosidered. 5 ote that some authors use the term eigevalue to mea the quatity ck i this eressio Solid Mechaics Part II 35

18 Sectio. st mode 3 rd mode d mode Figure..8: mode shaes for a vibratig elastic rod Natural Frequecies The eigevalues (or, equivaletly, the atural frequecies ck ) deed o the boudary coditios. There are four ossible cases for the oe-dimesioal rod. Takig the bar to have ed-oits,, the boudary coditios are (these are the same as for the static elasticity roblem):. fied-fied - u (,, u (,. free-free - u, u / (, t ) / (, t ) 3. fied-free - u (,, u (..35) / (, t ) 4. free-fied - u, u (, / (, t ) The atural frequecies ad modes for each of these boudary coditios are solved for ad give i the Aedi to this sectio (i the boes). For eamle, cosiderig the fied-fied case, the solutio is with (..36) ut (, ) Acos( kc Bsi( kc si( k) c Frequecies: kc,,, si k,,, (..37) Modes: Oe ca lot these sie fuctios over [, ] to see the dislacemet rofile of each mode (the first three are those lotted i Fig...8 it ca be see that the higher the mode, the higher the frequecy). The comlete solutio ad recise rofile is the obtaied by alyig the iitial coditios of the roblem to determie the coefficiets A, B i Eq Some eamles of this comlete calculatio are give i the Aedi. Solid Mechaics Part II 36

19 Sectio. Vibratio Aalysis A vibratio aalysis is oe i which the eigevalues (atural frequecies) ad modes are evaluated without regard to which of them might be imortat i a alicatio. The boudary coditios aloe determie the modes ad atural frequecies. Thus a vibratio aalysis is carried out without regard to how the vibratio is iitiated. The eact combiatio of the modes for a articular roblem is determied from the iitial coditios; the iitial coditios will determie the arbitrary costats i the above equatios ad hece the actual amlitude of vibratio. The vibratio is termed free if the load is zero or costat; forced vibratio occurs whe the load itself oscillates. Eve though a vibratio aalysis does ot comletely solve the roblem of a material model loaded i a certai way, for eamle solvig for the roagatio aths of stress waves, the amlitudes of vibratio, ad so o, the atural frequecies ad modes are very useful iformatio i themselves, for desig ad other uroses. Dyamic resose aalysis or trasiet resose aalysis is the calculatio of the comlete resose to ay arbitrary boudary ad iitial coditios. This is more difficult tha the vibratio aalysis, sice it is a time-deedet roblem. No-Homogeeous Boudary Coditios The boudary coditios i..35 are all homogeeous (i.e. u or u/ ). I ractice, the boudary coditios will ot be homogeeous, but the atural frequecies do ot deed o whether the boudary coditios are homogeeous or o-homogeeous. I other words, if oe wats to determie the atural frequecies, oe eeds oly cosider the case of homogeeous boudary coditios, as will be see ow. Cosider the followig o-homogeeous boudary coditios: BC s: u(, uˆ, u (, (..38) Sice the wave equatio is liear, the solutio ca be writte as the suerositio of two searate solutios, u(, u (, u (, (..39) The u h is the homogeeous solutio, ad is chose to satisfy the wave equatio with homogeeous boudary coditios; u is some articular solutio ad accouts for the o-homogeeous boudary coditio: BC s: u h (,, (, Substitutig..39 ito the wave equatio..3 gives u h u h (, uˆ, u (, (..4) Solid Mechaics Part II 37

20 Sectio. u h c u h t u c u t (..4) The left had side is zero. The right had side ca be made zero by choosig u to be ay articular solutio of the wave equatio. For a simle costat dislacemet boudary coditio, oe ca choose the liear fuctio u ( ) uˆ (..4) which ca be see to satisfy..4b. The comlete solutio u is illustrated i Fig...9. û u(,) u () Figure..9: dislacemets as a suerositio of two searate solutios Suose ow that the iitial coditios are IC s: u(,) u ( ) v(,) v( ) (..43) The iitial coditios ca be slit betwee u h ad u accordig to IC s: u (,) u ( ) u h v (,) v( ) v h ( ), ( ), u v (,) u (,) v ( ) ( ) (..44) Thus, the comlete solutio is obtaied by addig together: (i) the fuctio u h which satisfies the wave equatio with homogeeous boudary coditios o dislacemet, ad iitial coditios uh (,) u ( ) u ( ) IC s: v (,) v( ) v ( ) h (ii) the fuctio u ( ) uˆ Thus, usig the fied-fied homogeeous solutio from the Aedi, Solid Mechaics Part II 38

21 Sectio. (, ) ˆ ut u Acos( kc Bsi( kc si( k) (..45) ad the atural frequecies are give by..37. The costats from the iitial coditios, as outlied i the Aedi. A, B ca be obtaied The imortat oit to be made here is that the modes ad atural frequecies are determied from (i), i.e. the roblem ivolvig the homogeeous boudary coditios, ad so, as stated above, the o-homogeeous boudary coditio does ot affect the modes ad atural frequecies. Forced Vibratio Suose ow that the boudary coditios ad iitial coditios are give by BC s: u(, cos t, IC s: u(, u(,) uˆ cos / v(,) (..46) Agai, let u(, u (, uh (, ad substitute ito the wave equatio. I this case, the articular solutio will be of the geeral form..34, cos si cos si u A k B k C ckt D ckt (..47) Alyig the boudary coditios, oe fids that { Problem } u (, cos cot si cost (..48) c c c As with the costat o-homogeeous boudary coditio, the iitial coditios ca ow be slit aroriately betwee the homogeeous ad articular solutios. Agai, the comlete solutio is obtaied by addig together: (i) the fuctio u h which satisfies the wave equatio with homogeeous boudary coditios o dislacemet, ad iitial coditios IC s: u (,) ˆ h u cos v (,) (ii) the fuctio..48 The comlete solutio is h cos c cot si c c Solid Mechaics Part II 39

22 Sectio. l ut (, ) cos cot si cost c c c A cos( k c B si( k c si( k ) (..49) Resoace occurs whe the dislacemets become ifiite, which from..49 occurs whe c si. c These are recisely the atural frequecies of the system, i.e. the atural frequecies of (i). Thus the roblem of resoace becomes more romiet whe the forcig frequecy aroaches ay of the atural frequecies k...8 Problems. Cosider the case of forced vibratio. Use the boudary coditios..46 to evaluate the costats i the articular solutio..47 ad hece derive the articular solutio Cosider a fied-free roblem, with the ed subjected to a forced dislacemet u si t ad the ed free. (a) Fid the vibratio of the material. What are the atural frequecies? (b) Whe does resoace occur? [ote: the aroriate homogeeous solutio ad atural frequecies are give i the Aedi to this sectio] 3. Cosider a vibratig bar with a oscillatory stress alied to oe ed, cost. The ed is fied, u ( ). (a) Fid the vibratio of the material. What are the atural frequecies? (b) Whe does resoace occur? [ote: the aroriate homogeeous solutio ad atural frequecies are give i the Aedi to this sectio] Solid Mechaics Part II 4

23 Sectio...9 Aedi to Sectio.. D Alembert s Solutio of the Wave Equatio I the wave equatio..3, chage variables through ct, ct (..5) The u u, t,, t ad the chai rule gives u u u u u (..5) ad similarly for the variable t. Aother differetiatio gives u u u u u u (..5) ad similarly for the variable t. Substitutig these eressio ito the wave equatio..3 leads to u 4 (..53) Itegratig with resect to gives u / where is some arbitrary fuctio. A further itegratio the gives u d f f g D Alembert s solutio, Eq...: et the iitial coditios be ct g ct, which is u(, f (..54) u (,) U ( ) u t (,) V( ) (..55) Thus, from..54, U( ) f g. (..56) Now Solid Mechaics Part II 4

24 Sectio.,, u f t g t df dg df dg c c t t t d t d t d d (..57) At t, f f ad g g, so u df( ) dg( ) V( ) c c t d d (,) (..58) Itegratig the gives df ( ) dg( ) V( ) d d d g ( ) f( ) ( ), ( ) f( ) g ( ) c d d (..59) Subtractig this from Eq...56, ad also addig it to Eq...56, gives f ( ) U( ) V( ) d ( ) c g ( ) U ( ) V( ) d ( ) c (..6) If oe ow relaces with ct i the first of these, ad with ct i the latter, additio of the two eressios leads to Eq...: ut (, ) U ct U ct V( ) d c (..6) ct ct. Method of Searatio of Variables Solutio to the Wave Equatio Assumig a searable solutio, write u(, X ( ) T ( so that u / t X ( ) T ( ad u / X ( ) T (. Isertig these ito the wave equatio gives d T d X X c T dt dx d T d X c T dt X dx (..6) This relatio states that a fuctio of t equals a fuctio of ad it must hold for all t ad. It follows that both sides of this eressio must be equal to a costat, say k (if the left had side were ot costat it would chage i value as t is chaged, but the the equality would o loger hold because the right had side does ot chage whe t is chaged it is a fuctio of oly). Thus there are two secod order ordiary differetial equatios: Solid Mechaics Part II 4

25 Sectio. d X d kx, d T dt c kt (..63) which have solutios X Ae k Be k c kt c kt, T Ce De (..64) Modes ad Natural Frequecies for Homogeeous Boudary Coditios Suose first that k is ositive. Cosider homogeeous boudary coditios, that is, u ad/or u / at the ed oits,. Suose first that u (,. The u (, X () T ( X () ad so A B. If also u (,, the k Ae Be k which imlies that A B, ad u (,. Similarly, if oe uses the coditios u / (, or u / (,, or a combiatio of zero u ad first derivative, oe arrives at the same coclusio: a trivial zero solutio. Therefore, to obtai a o-zero solutio, oe must have k egative, ad X ( ) A cos( ) B si( ), k (..65) The solutio for T ( must the be ad the full solutio is T ( C cos( c D si( c (..66) A cos B si C cosct D sict u(, (..67) There are four ossible combiatios of boudary coditios.. Fied-Fied Here, u (, u(,. Thus X ( ) A ad X ( ) B si( ). For o-zero B oe must have si( ) /,,,. Thus oe has the ifiite umber of solutios X ( ) B si( ), ad the comlete geeral solutio is ( A BC, B BD ) 6 with u(, A cos( c B si( c si( ) (..68) 6 the solutios corresodig to egative values of, i.e. /,,,, ca be subsumed ito..68 through the costats A, B ; the solutio for is zero Solid Mechaics Part II 43

26 Sectio. Frequecies: c c,,, Modes: si,,, It ca be roved that the series..68 coverges ad that it is ideed a solutio of the wave equatio, rovided some fairly weak coditios are fulfilled (see a tet o Advaced Calculus). The first three modes are lotted i Fig.... (..69) Figure..: first three mode shaes for fied-fied Case. Free-Free Here, u / (, u / (,. Thus X ( ) B ad X ( ) A si( ). Thus the geeral solutio is ( A AC, B AD ) u(, A A cos( c B si( c cos( ) (..7) with the as for fied-fied. Frequecies: c c,,, Modes: cos,,, (..7) The dislacemet rofiles of the first three modes are show i Fig.... Solid Mechaics Part II 44

27 Sectio Figure..: first three mode shaes for free-free Case 3. Fied-Free Here, u (, u / (,. Thus X ( ) A ad X ( ) B cos( ). For o-zero B oe must have cos( ) ( ) /,,,,,,. The solutio is agai give by..68, which is reeated here, oly ow Frequecies: u(, c ( ) c, A cos( c B si( c si( ) (..7),, Modes: The dislacemet rofiles of the first three modes are show i Fig.... si,,, (..73) Figure..: first three mode shaes for fied-free Solid Mechaics Part II 45

28 Sectio. Case 4. Free-Fied Here, u / (, u(,. Thus X ( ) B ad X ( ) A cos( ). For o-zero A oe must have cos( ) so the geeral solutio is as for free-free, Eq...7, but with A : u(, with the as for fied-free. Frequecies: c ( ) c, A cos( c B si( c cos( ) (..74),, Modes: The dislacemet rofiles of the first three modes are show i Fig...3. cos,,, (..75) Figure..3: first three mode shaes for free-fied Full Solutio (icororatig Iitial Coditios) (a) Iitial Coditio o Dislacemet The iitial coditio o dislacemet is u(,) u ( ) (..76) which give, from..68,..7,..7,..74, u(,) A si( ) u ( ) fied-fied/fied-free u(,) A A cos( ) u ( ) free-free (..77) Solid Mechaics Part II 46

29 Sectio. u(,) A cos( ) u ( ) free-fied These ca be solved by usig the orthogoality coditio of the trigoometric fuctios:, m si( )si( m ) d cos( )cos( m ) d (..78) /, m for either of /, ( ) / si( m ) ad..77b-c by cos( m ) A ). Thus multilyig both sides of..77a by ad itegratig over, gives u ( )si( d fied-fied/fied-free A u ( ) d, A u ( )cos( ) d,,, free-free (..79) A ) u ( )cos( d free-fied (b) Iitial Coditio o Velocity The iitial coditio o velocity, u, v ( ), gives u (,) cb si( ) v ( ) fied-fied/fied-free u (,) cb cos( ) v ( ) free-fied/free-free (..8) Usig the orthogoality coditios agai gives B B Eamle v ( )si( d c fied-fied/fied-free ) v ( )cos( d c free-fied/free-free (..8) ) Cosider the fied-free case with iitial coditios A ad u ), v ( ) /. Thus ( Solid Mechaics Part II 47

30 Sectio. B 8 ( ) c 3( ) 3 3 ( ) ( ) 8 4 ( ) si d ( ) c ( ) c so that 3 ( ) ( ) c u(, si( )si( c, c,,, 3 3 c ( ) The eriod for the first (domia mode is T / c 4 / c. The solutio is lotted i Fig...4 for c 5m/s,.m, for the five times it /6, i 4 (u to the quarter-eriod). Thereafter, the solutio decreases back to zero, dow through egative dislacemets, back to zero ad the reeats..5-5 t / c u.5 t / c Figure..4: dislacemets for fied-free eamle t Solid Mechaics Part II 48

31 Sectio. Eamle Cosider the free-free case with iitial coditios u ( ) u, v ( ). Thus B ad A A u u u d u cos d,,, so that u t u (, ) cos( ccos( ), (..34) The eriod for the first (domia mode is T / c / c. The solutio is lotted i Fig...5 agai for c 5m/s,.m, for the ie times it /6, i 8 (u to the half-eriod). Thereafter, the solutio returs back to the iitial ositio ad the reeats.. t.9 u t / 4c t / c t 3 / 4c t / c Figure..5: dislacemets for free-free eamle Solid Mechaics Part II 49