modes shapes can be obtained by imposing the non-trivial solution condition on the

Transcription

1 modes shapes ca be obtaied by imposig the o-trivia soutio coditio o the derived characteristics equatio. Fiay, usig the method of assumed modes, the goverig ordiary differetia equatios (ODEs of beam ad their state-space represetatio are derived uder distributed vertica oadig coditio. The rest of this chapter is orgaized as foows: Sectios 3. presets the moda aaysis ad forced vibratio formuatio of EB beam with mutipe cross-sectioa discotiuities; Sectios 3.3 icudes the case study of a beam with two step discotiuities i the cross-sectio, to umericay ad graphicay demostrate the effects of discotiuities; ad fiay, Sectio 3.4 presets the cocudig remarks. 3.. Euer-Beroui (EB Beam with Mutipe Stepped Discotiuities Cosider a iitiay straight o-uiform EB beam of egth L, with variabe cross sectio A = A(x, variabe stiffess E = E(x, ad variabe momet of iertia I = I(x. Let x [ 0, L] ad t [ 0, be the spatia ad time variabes, respectivey. The goverig equatio for trasverse vibratio of beam with variabe mass per uit egth m(x ad dampig coefficiet of c(x subjected to a vertica time varyig distributed oad P(x,t is a fourth-order PDE expressed as: wxt (, wxt (, wxt (, ExIx ( ( cx ( mx ( Pxt (, = x x t t (3. with w(x,t beig the trasversa dispacemet fuctio. I order to obtai atura frequecies ad eigefuctios (mode shapes of system, the eigevaue probem 45

2 associated with the trasversa vibratio of beam is obtaied by appyig free ad udamped coditios to Eq. (3. as foows: wxt (, wxt (, ExIx ( ( mx ( = x x t (3. Let s assume that the soutio of Eq. (3. is separabe i time ad space domais, wxt (, = φ( xqt ( (3.3 where φ ( x deotes the spatia mode shape fuctio ad q(t represets the geeraized time-depedet coordiate. Substitutig Eq. (3.3 ito Eq. (3., the eigevaue equatio ca be writte i the foowig form of separated time ad space equatios: ( ExIx ( ( / ( mx ( φ( x = qt ( / qt ( = ω d d φ x (3.4 where ω is a costat parameter. The mode shapes are obtaied by sovig the spatia part of Eq. (3.4 writte as: d d φ( x ExIx ( ( mx ( ( x = ω φ (3.5 For a beam with parametric discotiuities (e.g., jump i the momet of iertia or mass distributio Eq. (3.5 caot be soved usig covetioa approaches. A aterative method is to partitio the beam ito uiform segmets betwee ay two successive stepped poits ad appy the cotiuity coditios at these poits. Therefore, the o-uiform beam is coverted to a set of uiform segmets costraied through the cotiuity coditios. The ext sectio discusses this techique i detai ad proposes a 46

3 framework for dyamic aaysis of fexibe beams with jumped cross-sectioa cofiguratio Moda Aaysis of Stepped EB Beam Figure 3. iustrates a straight EB beam with arbitrary boudary coditios ad jumped discotiuities i its spatia spa. The beam cosidered i this study has a uiform cross-sectio at each segmet. Hece, Eq. (3.5 ca be divided ito uiform equatios expressed as: d φ ( x = < < = (3.6 4 ( EI ω m ( 4 φ x, - x ;,,3,..., ; 0 = 0 where φ ( x, (EI, ad m are mode shapes, fexura stiffess, ad mass per uit egth of beam at the th segmet, respectivey *. Let, 4 m = ω (3.7 ( EI Eq. (3.6 ca be rewritte i a more recogizabe form 4 d φ ( x 4 φ( 0 4 x = (3.8 with the foowig geera soutio φ ( x = A si x B cos x C sih x D cosh x (3.9 * ( deotes the mode shape or parameter vaue for the th cross-sectio, whie ( (r, which wi be used ater i the dissertatio, deotes the mode shape or parameter vaue of the r th mode; though, ω r which represets the r th atura frequecy is a exceptio. 47

4 where A, B, C ad D are the costats of itegratio determied by suitabe boudary ad cotiuity coditios. It is to be oted that ay covetioa boudary coditios ca be appied to the beam; however, without the oss of geeraity, the camped-free coditios are chose here for the boudaries. Appyig the camped coditio at x = 0 requires: dφ (0 φ (0 = = 0 (3.0 Substitutig Eq. (3.0 ito Eq. (3.9 yieds: B D =0 ad A C =0 (3. O the other had, the cotiuity coditios for dispacemet, sope, bedig momet, ad shear force of beam at discotiuity ocatios are give by: φ = φ ( (3. ( d φ ( d ( = φ (3.3 ( EI d φ ( = ( EI d φ ( (3.4 ( EI d 3 φ ( 3 = ( EI d 3 φ ( 3 (3.5 q q d φ( d φ( x q q x=. 48

5 49 Figure 3.. EB beam cofiguratio with jumped discotiuities. Ideed, these coditios are appied at the boudaries of adjacet segmets to satisfy the cotiuity ad equiibrium coditios immediatey before ad after stepped poits. Appyig coditios (3.-(3.5 to Eq. (3.9 resuts i: D C B A D C B A cosh sih cos si cosh sih cos si = (3.6 sih cosh si cos ( sih cosh si cos ( D C B A D C B A = (3.7 cosh sih cos si ( cosh sih cos si ( D C B A D C B A = γ (3.8 sih cosh si cos ( sih cosh si cos ( 3 D C B A D C B A = γ (3.9 where ( ( = EI EI γ. Fiay, the free boudary coditio at x = L= requires:

6 d 3 φ ( d φ ( = 3 = 0 (3.0 Substitutig Eq. (3.0 ito Eq. (3.9 yieds: ( A si B cos C sih D cosh = 0 (3. 3 ( A cos B si C cosh D sih = 0 (3. ote that s are fuctios of beam atura frequecy with a expicit expressio give i Eq. (3.7. Sice the atura frequecy is idepedet of segmets idices ad is cosidered for the etire egth of beam, s of differet segmets ca be iterreated i terms of a sige parameter usig Eq. (3.7: = α (3.3 where ote that α = ad thus, =. /4 m ( EI = m ( EI α (3.4 Eqs. (3., (3. ad (3. derived from boudary coditios whie Eqs. (3.6-(3.9 are obtaied from the cotiuity coditios form the characteristics matrix equatio as: J P = 0 ( where J is the characteristics matrix ad P is the vector of mode shape coefficiets P = (3.6 T [ A B C D A B C D A B C D ] 4 50

7 Matrix J is costructed based o three sets of equatios. The first two rows ad ast two rows represet the boudary coditios at x = 0 ad x = L, respectivey, ad the midde part of matrix demostrates the cotiuity coditios at the siguarity poits. Matrix J ca be divided ito three parts as: J [ J] [ J ] [ J ] 4 = 4( (3.7 where J = (3.8 represets the camed boudary coditio at x = 0 give by Eq. (3., J 8 ( [ J ] ( 0 [ J ] 4 8 = 0 4( 4 ( [ J ] 4 8 4( 4 (3.9 icudes the cotiuity coditios give by Eqs. (3.6-(3.9 at (- poits of discotiuity with J siα cosα sihα coshα α cosα α siα α coshα α sihα ( = γα siα γα cosα γα sihα γα coshα γα cosα γα siα γα coshα γ α 3 sih α 5

8 α α α siα cosα α cosα siα α sihα si α α cosα α sihα α coshα cosα α siα α coshα α sihα coshα α coshα sihα 4 8 ad (3.30 J 0 0 α siα α cosα α sihα α coshα 3 = α cosα α siα α coshα α sihα 4 (3.3 represets the free boudary coditio at x = L give by Eqs. (3. ad (3.. I order to obtai a o-trivia soutio for Eq. (3.5 ad fid the atura frequecies ad mode shapes, the determiat of matrix J must be set to zero det [ J( ] = 0 (3.3 Sice this matrix is a fuctio of oy parameter ( 0,, its determiat ca be umericay evauated for its zero vaues by cotiuousy varyig parameter with a reasoaby sma step size withi a rage of iterest startig from, but excudig zero. The vaues of which satisfy Eq. (3.3 ead to cacuatio of atura frequecies usig a modified versio of Eq. (3.7 as foows: ( EI ( EI ω = = (3.33 ( ( ( r 4 ( r 4 r m m 5

9 where ( r s are soutios for Eq. (3.3 ad ω r is the correspodig r th atura frequecy. Sice the determiat of matrix J has bee set to zero for the seected vaues of, the mode shape coefficiets A to D are ieary depedet. I order to obtai uique soutio for these coefficiets, orthoormaity betwee mode shapes ca be utiized. For the covetioa boudary coditios cosidered here, this coditio is stated as: ( r ( s ( r mx ( φ ( x φ ( x = δ rs or mx ( ( φ ( x = ( ( where δrs is the Kroecker deta fuctio, ad φ r ( x is the r th mode shape of beam expressed as: ( r ( r ( r ( r ( r ( r ( r ( r ( r φ ( x = A si x B cos x C sih x D cosh, 0 x ( r ( r ( r ( r ( r ( r ( r ( r ( r ( ( x A si x B cos x C sih x D cosh x, r φ = < x φ ( x = φ ( x = A si x B cos ( ( ( ( < (3.35 ( r ( r ( r ( r ( r x C r sih r r cosh r x D x, x The obtaied mode shapes ad atura frequecies are used to derive the ODE of motio for a beam uder distributed dyamic excitatio as wi be discussed ext Forced Motio Aaysis of Stepped EB Beam Usig expasio theorem for the beam vibratio aaysis, the expressio for the trasverse dispacemet becomes: ( r ( r (, φ ( ( wxt = xq t (3.36 r= 53

10 where q ( r ( t is the geeraized time-depedet coordiate for the r th mode. Substitutig Eq. (3.36 ito PDE of motio Eq. (3. yieds: ( r= ( r d d φ x ( r ( r ( r ( r ( r E( x I( x q ( t c( x φ ( x q ( t m( x φ ( x q ( t = P( x, t (3.37 To safey take the term E(xI(x out of the bracket for the beam with mutipe discotiuities, Eq. (3.37 is mutipied by s th ( s mode shape, φ ( x, ad is itegrated over the beam egth: 0 r= ( r d d φ ( x ( s ( r ( r ( s ( r ExI ( ( x φ ( xq ( t cx ( φ ( x φ ( xq ( t m x x x q t = P x t x = 0 ( r ( s ( r ( s ( φ ( φ ( ( (, φ ( 0 (3.38 Reca Eq. (3.6 which ca be modified to 4 ( r d φ ( x ( r ( EI = ω ( 4 rmφ x (3.39 Usig Eq. (3.39 ad dividig the spatia itegra ito uiform segmets, oe ca write: ExIx ( ( ( xq ( t = ( EI ( xq ( t = ( r 4 ( r d d φ ( x ( s ( r d φ ( x ( s ( r 4 φ φ = 0 m ( x ( xq ( t = q ( t mx ( ( x ( x ( r ( s ( r ( r ( r ( s ωr φ φ ωr φ φ = 0 (

11 Appyig beam orthogoaity coditios give by: ( r ( s ( r ( s m( x φ ( x φ ( x = E( x I( x φ ( x φ ( x = δ rs ( ad usig Eq. (3.40, Eq. (3.38 ca be recast as: ( r ( s ( r ( s ( r ( r q ( t q ( t c( x φ ( x φ ( x ωr q ( t = P( x, t φ ( x, r =,,..., s= 0 0 (3.4 which ca be simpified to { rs } q ( t c q ( t ω q ( t = f ( t (3.43 ( r ( s ( r ( r r s= where rs ( r ( s ( r ( r ( φ ( φ (, ( (, φ ( (3.44 c = cx x x f t = Pxt x 0 0 The trucated k-mode descriptio of the beam Eq. (3.43 ca ow be preseted i the foowig matrix form: Mq Cq Kq = F (3.45 where M = I, C = c, K =, q = q t q t q t, F = ( ( ( k T k k [ rs] k k [ ωδ r rs] k k [ (, (,, ( ] k ( ( ( k T [ f ( t, f ( t,, f ( t] k (3.46 The state-space represetatio of Eq. (3.45 is give by: 55

12 X = AX Bu (3.47 where 0 I 0 q A=, B=, X=, u= Fk -M K -M C M q k k k k k (3.48 The impemetatio of the proposed framework wi be studied i the ext sectio for a particuar case of iterest, where the EB beam has two stepped poits i cross sectio ad is subjected to a distributed dyamic excitatio A Exampe Case Study: EB beam with two jumped discotiuity i crosssectio A exampe case of study is cosidered i this sectio to demostrate the impemetatio of the proposed method for forced vibratio aaysis of a fexibe beam with jump discotiuities. Figure 3.3 depicts a catiever beam with two jump discotiuities i the cross-sectio subjected to a vertica oad uiformy appied at the midde sectio. This may resembe a catiever beam with a actuator/sesor pair attached to its midde part. The objective is to derive the equatios of motio ad depict the mode shapes ad frequecy respose of the beam for a fiite umber of modes. To observe the effects of the jump o the beam s mode shapes ad system s frequecy respose, severa egth ad thickess vaues are cosidered for the midde cross sectio as isted i Tabes 3. ad 3.. It is assumed that Sectios ad 3 have the same dimesios ad properties, ad oy the thickess of the beam jumps i Sectio. 56

13 We first formuate the probem based o the proposed approach detaied i the precedig sectio. Figure 3.3. EB beam with two stepped discotiuities i cross sectio uder distributed dyamic oad. Matrix J for the depicted beam cofiguratio ca be formed usig Eqs. (3.7-(3.3 as foows: J si cos sih cosh siα cosα γ γ γ γ α α α α γ γ = cos si cosh sih α cosα α siα si cos sih cosh si cos 3 cos 3 si 3 γ cosh 3 γ sih 3 3 α cosα 3 3 α siα siα cosα α cosα α siα γ α siα γα cosα γ α cosα 3 3 γα siα

14 sihα coshα α coshα α sihα α sihα α coshα α coshα α sihα sihα coshα siα 3 cosα 3 sihα 3 coshα 3 α coshα α sihα α 3 cosα 3 α 3 siα 3 α 3 coshα 3 α 3 sihα 3 γα sihα γα coshα α3 siα3 α3 cosα3 α3 sihα α3 coshα γα coshα γα sihα α3 cosα3 α3 siα3 α3 coshα3 α3 sihα3 0 0 α3 siα3l α3 cosα3l α3 sihα3l α3 coshα3l α3 cosα3l α3 siα3l α3 coshα3l α3 sihα3l (.49 Varyig parameter with sma steps over a desired rage, ad fidig for the zeros of determiat of J, eads to determiatio of the atura frequecies of the beam usig Eq. (3.33. The coefficiets of the mode shapes ca be obtaied through Eqs. (3.3 ad (3.34. The (r th mode shape of the beam ca be writte as: ( r ( r ( r ( r ( r ( r ( r ( r ( r φ ( x = A si x B cos x C sih x D cosh x, 0 x ( r ( r ( r ( r ( r ( r ( r ( r ( r ( r φ ( x = φ ( x = A si x B cos x C sih x D cosh x, x x C x D x x L ( r ( r ( r ( r ( r ( ( ( ( φ3 ( x = A3 si3 x B3 cos r 3 3 sih r r 3 3 cosh r 3, (3.50 To derive the equatios of motio, the eemets of Eq. (3.46 must be cacuated first. Let s assume that the dampig coefficiet of the beam remais costat for the etire egth of beam (i.e. c(x = c, ad the time-varyig vertica oad is uiformy distributed i the midde segmet (P(x, t = P(t for < x <, ad P(x, t = 0 for x < or x >. Cosequety, this yieds: 58

15 L ( r ( s ( r ( s ( r ( s ( r ( s crs = c( x φ ( x φ ( x = c φ ( x φ ( x φ ( x φ ( x φ3 ( x φ3 ( x 0 0 ( r ( r ( r = φ = φ 0 f ( t P( x, t ( x P( t ( x (3.5 Thus, the equatio of motio ad its state-space represetatio ca be formed based o Eqs. (3.43-(3.48. Oce the system is derived i state-space, frequecy respose of system ca be potted to demostrate the behavior of system withi a desired frequecy rage. Without oss of geeraity, the dispacemet of a arbitrary poit (x = L 0 is take as the system output: k ( r ( r ( ( ( k k r= Y( t = ω( L, t = φ ( L q ( t = [ φ ( L, φ ( L,..., φ ( L,0,...,0] X( t (3.5 The stadard form of state-space represetatio of the system ca the be writte as: X = AX Bu Y = CX (3.53 where C = [ φ ( L, φ ( L,..., φ ( L,0,...,0] (3.54 ( ( ( k k The frequecy respose of the system ca ow be potted usig beam s trasfer fuctio obtaied through the Lapace trasformatio of its state-space mode as foows: Y( s Gs ( ( U( s = = C si A B (

16 umerica simuatios have bee performed i the ext subsectio to demostrate the mode shapes ad frequecy respose pot of beam cofiguratio discussed i this sectio umerica Simuatios ad Discussios Two sets of umerica simuatios are preseted i this sectio; the first case is desigated to study the effects of thickess variatio of beam s midde cross sectio, whie i the secod case, the effects of egth variatio of the beam s midde cross sectio is ivestigated. The objective of the simuatios is to demostrate how differet jump cofiguratios may affect the beam s mode shapes ad atura frequecies. Tabe 3. idicates the parameter vaues used for the simuatio of the first sceario, where beam s thickess i the midde sectio varies. Beam s equatio of motio has bee trucated ito oy four modes, ad five differet thickess vaues have bee cosidered for the midde sectio, oe of which beig a uiform beam without ay jump i crosssectio. Figure 3.4 depicts the mode shapes of beam for differet cofiguratios. As see from the resuts, mode shapes of the beam sigificaty chage as the thickess of the jump icreases. Particuary, it is observed that such a chage has more effect o eve modes ( ad 4 compared to odd modes ( ad 3. This reaso perhaps is due to the fact that the midde sectio resists agaist bedig i modes ad 4, whie it is ocated o a fairy straight curvature i modes ad 3. This ot oy affects beam s modes shapes, but aso its atura frequecies; the frequecy respose pot give i Figure 3.5 depicts that the first ad third atura frequecies of beam for differet jump cofiguratios are ocaized, i cotrast to the frequecies of secod ad fourth modes, where the frequecy 60

17 peaks are more scattered. The cotiuity of the mode shapes at jump poits ca be ceary see from the figures, as expected from the aaysis. The parameter vaues used for the simuatio of secod sceario, where the effect of egth variatio is cosidered, are give i Tabe 3.. This tabe aso icudes first four atura frequecies. The obtaied modes shapes of beam s tip dispacemet ad the frequecy respose pots are depicted i Figures 3.6 ad 3.7, respectivey. Simiar to the first sceario, the egth variatio of the midde sectio sigificaty affects the mode shapes; particuary, secod ad fourth modes demostrate arger chages i their shapes ad atura frequecies compared to first ad third modes, due to their arger curvature withi the ocatio of midde sectio. Tabe 3.. Beam parameters for umerica simuatio of differet thickess vaues i the midde sectio. Cofig. (m (m L(m t (m t (m t 3 (m ω (rad/sec ω (rad/sec ω 3 (rad/sec ω 4 (rad/sec T T T T T Beam s other parameters: Desity: ρ = 7800(kg/m 3, Width: b = 0.0(m, Dampig coefficiet: c = 0.00(.sec/m, Youg s moduus of easticity: E = 00(Gpa Tabe 3.. Beam parameters for umerica simuatio of differet egth vaues i the midde sectio. Cofig. (m (m L(m t (m t (m t 3 (m ω (rad/sec ω (rad/sec ω 3 (rad/sec ω 4 (rad/sec L L L L Other beam parameters are the same as those specified i Tabe. 6

18 Tabe 3.3. ormaized sope differece of the mode shapes betwee the startig ad the edig step poits. Cofig. T Cofig. T Cofig. T3 Cofig. T4 Cofig. T5 Mode Mode Mode Mode a b c d Figure 3.4. (a First, (b secod, (c third, ad (d fourth mode shapes of beams with five differet midde sectio thickesses. 6

19 Figure 3.5. Moda frequecy respose pot of beams tip dispacemets for five differet midde sectio thickesses. a b c d Figure 3.6. (a First, (b secod, (c third, ad (d fourth mode shapes of beams with four differet midde sectio egths. 63

20 Figure 3.7. Moda frequecy respose pot of beams tip dispacemets for four differet midde sectio egths. If the presece of the added mass i the midde sectio is due to the attachmet of patch sesors (e.g. piezoeectric or piezoresistive sesors, the sope differece of the mode shapes betwee the startig ad edig poits of attachmet is proportioa to the sesor output votage [75]: ( r ( r Ehd ( p p 3b r dφ ( dφ ( ( r vses ( t = q ( t Cf (3.56 where v is the output votage of the sesor associated with the r th mode. It woud be ( r ses iterestig to observe how the presece of sesor ad its added mass ad stiffess woud affect the output votage estimatio (the expressio iside bracket i Eq. (3.56. For this, the sope differece of the beam with differet thickess vaues of the midde sectio (beam specified i Tabe 3. at the poits of discotiuities are cacuated ad 64

21 ormaized to the beam with the uiform cross-sectio (cofiguratio T. The reaso for this ormaizatio is to compare the stepped beams with the uiform beam to determie the percetage of error iduced by the assumptio of uiform cross-sectio for beams with patch sesors i differet thickesses. Tabe 3.3 reveas the ormaized sope differece of the mode shapes betwee the startig ad the edig step poits. It is see from the tabe that as the thickess of the cross-sectio icreases, the sope differece drasticay drops. Accordig to the tabe, if the rea cofiguratio of a beam/sesor pair correspods to the thiest jump, that is, Cofiguratio T, the assumptio of a uiform beam (Cofiguratio T for the system eads to 30% estimatio error for mode, 80% for mode, 30% for mode 3, ad 5% for mode 4. As the thickess of the jump icreases, the error percetages further icrease. The resuts sha prove the high degree of eed to the modeig of cross-sectioa discotiuities i fexibe beams, ad may idicate the impact of the proposed framework to the area of vibratios ad vibratio cotro systems Cocusios This sectio preseted a comperhesive framework for derivatio of mode shapes ad state-space represetatio of motio of fexibe EB beams with mutipe jump discotiuities i their cross sectio. To sove the goverig equatios of motio, the beam was divided ito uiform segmets of costat parameters, ad the cotiuity coditios were appied at the partitioed poits. The characteristics matrix was the formuated usig the beam boudary ad cotiuity coditios. atura frequecies of beam were obtaied by settig the determiat of characteristics matrix to zero. The 65

22 goverig equatios were discretized ad its state-space represetatio was the derived for the beam uder a distributed dyamic oadig coditio. To demostrate the proposed method, the formuatios ad umerica simuatios have bee preseted for a beam with two stepped discotiuities i the cross-sectio. Resuts idicate that the effects of added mass ad stiffess at the midde sectio o the beam mode shapes ad atura frequecies are sigificat. Hece, exact soutios are required for practica impemetatio of such discotiuous structures. 66