DETERMINATION OF MECHANICAL PROPERTIES OF A NON- UNIFORM BEAM USING THE MEASUREMENT OF THE EXCITED LONGITUDINAL ELASTIC VIBRATIONS.
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1 ICSV4 Cairs Australia 9- July 7 DTRMINATION OF MCHANICAL PROPRTIS OF A NON- UNIFORM BAM USING TH MASURMNT OF TH XCITD LONGITUDINAL LASTIC VIBRATIONS Pavel Aokhi ad Vladimir Gordo Departmet of the mathematics Orel State Techical Uiversity 3 Russia Orel Naugorskoe sh 9 gordo@osturu Abstract The purpose of this paper is to preset the statemet of the iverse problem for logitudial elastic vibratios of a o-uiform beam ad the approximate aalytical ad umeric solutios of this problem uder the coditio of weak heterogeeity To solve the iverse problem certai logitudial vibratios are excited i the beam ad mechaical properties of the o-uiform beam (Youg s module ad desity) have to be determied from the measured vibratio i a certai poit of the beam Uder the coditio of weak heterogeeity the problem does t lose its importace Weak heterogeeity ca be foud i may atural ad produced materials I this case it s possible to liearize the problem expressig the differece betwee the properties of the base homogeeous beam ad the properties of the weak heterogeeous beam through the differece betwee their vibratios ad solve the problem The problem is decomposed ito sub-problems fidig the differece betwee vibratios of the uiform ad o-uiform beams (solvig the partial differetial equatio of the d order with costat coefficiets) ad fidig the differece betwee the mechaical properties of the beams (solvig the system of liear ordiary differetial equatios of the st order) INTRODUCTION The equatio describig distributio of logitudial elastic waves i a o-uiform beam (i a dimesioless form): w ξ = where ( ξ ) = ( ξ) A ( ξ) ρ( ξ) = ρ ( ξ ) A ( ξ ) ( ξ ) ρ( ξ) f ( ξ τ) ; ξ - distace from a ed face of the beam ( ξ = ) to the give sectio ξ ; τ - dimesioless time; ξ - Youg s module; ρ( ξ ) - beam s desity; ()
2 ICSV4 9- July 7 Cairs Australia A ( ξ ) - the area of cross-sectio of the beam; f ( ξ τ ) - logitudial distributed force; w= w( ξ τ ) - logitudial displacemet of the beam s cross-sectio The equatio () should satisfy boudary w( τ) = w( τ) = () ad iitial w( ξ ) = ϕ( ξ) = ψ ( ξ) (3) ( ξ ) coditios I this study we ivestigate the iverse coefficiet idetificatio problem which ξ ρ ξ of the equatio requires the idetificatio of ukow o-uiform coefficiets { } () satisfyig () ad (3) with kow fuctios { f ( ξ τ) ϕ( ξ) ψ ( ξ )} ad also some data kow o fuctio w( ξ τ ) (the experimetal data received from the measuremet of the beam vibratios iitiated by the give iitial coditios ad the distributed force) We use the data measuremet of logitudial displacemet of the fixed beam s cross sectio i time: wa ( τ ) = χa( τ) wb ( τ) = χb( τ) = γa( τ) (4) ( a τ ) where a b distaces from a ed face of the beams to the studied cross-sectios ( < ab < ) We shall restrict ourselves to determiatio of oly oe coefficiet cosiderig the other oe kow ad costat We shall cosider that ρ( ξ) = ρ = cost is a kow value Thus the problem is reduced to determiatio of oly ( ξ ) I this study the approximate aalytical ad the umeric solutios are ivestigated uder the coditio of weak heterogeeity of the beam ANALYTICAL SOLUTION We shall solve the stated iverse problem aalytically based o a method of iverse problems solutio suggested i [] We shall assume weak heterogeeity of the beam s properties ie ( ξ) = + ( ξ) ξ (5) where = cost - is the property of the base homogeeous beam Further we cosider ad ρ to be kow We ca express w= w( ξ τ ) as: w( ξτ ) = w ( ξτ ) + w ( ξτ ) w ξτ (6) Substitutig expressios (5) ad (6) i the equatio () we receive:
3 ICSV4 9- July 7 Cairs Australia ( ξτ) ( ξτ) = ( ξτ) + ( ξτ) F f F F3 w w F ( ξτ ) = ρ w w F ( ξτ ) = ρ + ξ F3 ( ξτ ) = The aalytical solutio of the problem is possible whe the followig coditios are met (we do ot affirm that the problem ca t be solved aalytically if these coditios are ot met however these coditios are ecessary for the solutio offered below): f ξτ ; oly atural vibratios are cosidered ie iitial coditios are selected i such a way that i a correspodig base homogeeous beam the same iitial coditios excite harmoic vibratios; F ξ τ F ξ τ F ξ τ F ξ τ (these coditios are provided by the fact ad 3 that the fuctios w ad are small by absolute value as give i (5) ad (6)) If all specified coditios are met the equatio (7) splits up ito two: w w ρ = with boudary ad iitial coditios give by: w ( τ ) = w ( τ) = w ( ξ) = ϕ( ξ) = ψ ( ξ) (8) ( ξ ) ad w w ρ = (9) with homogeeous boudary ad iitial coditios: w ( τ) = w ( τ) = w ( ξ) = = () ( ξ ) Substitutig expressio (5) i experimetal data (4) we obtai the additioal source data for the solutio: w ( a τ) = χa ( τ) = χa( τ) w ( a τ) = γa ( τ) = γa( τ) () ( a τ ) ( a τ ) To meet the coditio of the harmoic vibratios i the base homogeeous beam the displacemet w ( ξ τ ) should take the form: π ρ w ( ξ τ ) = si( αkξ )( Bsiατ + Ccos ατ ) α = k = N () k which meas that the iitial coditios are restricted to be i the form: w ( ξ) = ϕ( ξ) = Csi ( αkξ) = ψ ( ξ) = αbsi ( αkξ) ( ξ ) We cosider all parameters of the iitial coditios { B C } to be kow Substitutig (7) 3
4 ICSV4 9- July 7 Cairs Australia them i () we obtai ( ) w ξ τ Now we should obtai w ( ξ τ ) from the equatio (9) boudary ad iitial coditios () ad measuremet data () Substitutig () i (9) we get: α ( ξ ( α ξ) )( ατ ατ) w k k = cos k Bsi + Ccos (3) Applyig operator + α I (I idetity operator) to the equatio (3) we get the system of the differetial equatios: v( ξτ ) v( ξτ ) k = (4) w ( ξτ ) v( ξ τ) = + α w ( ξ τ) Substitutig the measuremet data () i the secod equatio of the system (4) we obtai: v dχa ( τ) v v dγa ( τ) v( a τ ) = χa( τ) = + α χa ( τ) = γa( τ) = + α γa ( τ) (5) dτ ( a τ ) dτ Solvig the first equatio of the system (4) ad usig the (5) we obtai v( ξ τ ): τ+ ka kξ v v v γ a z dz χa( τ ka + kξ) + χa( τ + ka kξ) τ+ ka kξ v( ξτ ) = + k Solvig the secod equatio of the system (4) with respect to w ( ξ τ ) ad homogeeous iitial coditios () we fid: W w ( ξτ ) W( ξ ) cos ατ = si ατ + W( ) α ξτ ( ξ ) (6) si ατ v( ξ τ) cosατdτ cos ατ v( ξ τ) siατdτ W ( ξτ ) = α Substitutig (6) i (3) ad solvig the ordiary differetial equatio with respect to ξ we fid the solutio to the iverse problem: ξ w ( z τ) w ( z τ) ( ξ ) = D+ k dz αkcosαkξ ( Bsiατ Ccosατ) + z where D is arbitrary costat subject to determiatio We ca defie it havig set the value of at ξ = : D α k ( ) = = D= αk 3 NUMRIC SOLUTION The stated iverse problem ca also be solved usig the regularizatio method of Tikhoov & Arsei [] which requires fidig the miimum of the fuctioal with respect to : ( ( Φ λ = w a ) a ) ( ) d τ χ τ + λ τ τ (7) 4
5 ICSV4 9- July 7 Cairs Australia where λ > - regularizatio parameter; - solutio of the equatio (9) with boudary ad iitial coditios () ad give w ( ξ ); χa ( τ ) - measuremet data () Cotrary to aalytical solutio the regularizatio method ca be used with arbitrary iitial coditios ad logitudial distributed force The followig coditios remai uchaged (from aalytical solutio): F( ξ τ) F( ξ τ) ; F3( ξ τ) F( ξ τ) The solutio of the equatio (9) with boudary ad iitial coditios () is give by: τ τ z τ z τ z τ z ξ + ξ + τ ξ ξ τ dz k ξ k k ξ k w ( ξτ ) = ρ w ξ τ is the solutio of the equatio where w w ρ = f ( ξ t) ξ τ with boudary ad iitial coditios (8) Ay solutio ( ξ ) foud as the result of miimizatio of the fuctioal (7) may ot ξ satisfies be uique If w τ τ z τ z τ z τ z a+ a+ τ a a τ dz k ξ k k ξ k a τ = ρ w ξ τ = w ξ ξ w τ τ the ( ξ ) = ( ξ) + βwξ ( a ξ) ad w ( ξ τ ) ca be expressed as where β is a arbitrary costat also satisfies (8) This meas that the same vibratios at the fixed cross-sectio ca be satisfied by ifiitively may properties ( ξ ) This should have bee expected as for ay secod-order differetial equatio we must defie two fuctios to remove ambiguity from the solutio The aalytical solutio to the iverse problem also uses two fuctios (istead of oe) to remove ambiguity To avoid ambiguity i the regularizatio solutio we should modify the fuctioal (7) to iclude two fuctios from the measuremet results We shall use the displacemet i time of aother cross-sectio at the distace ξ = b Modified fuctioal will take the form: ( a ) ( b ) ( ) w a a w a χb τ = w b τ = χb τ w b τ Φ λ = w a τ χ τ + w b τ χ τ d τ + λ τ d τ (8) (9) where χ ( τ) = ( τ) = χ ( τ) ( τ) a To miimize fuctioal (9) we shall use Ritz method represetig: ξ = d ξ () i= To solve the miimizatio problem we ow have to determie the set of coefficiets d d d of the () which will miimize the fuctio of the + variables: { } i i 5
6 ICSV4 9- July 7 Cairs Australia di i i a τ χaτ di i i b τ χb τ + dτ i i i = = Φ λ ( d d d) = λ diτ dτ i= ρ τ i i τ z τ z τ z τ z Ii ( ξτ ) = ξ+ ξ+ z ξ ξ z dz k k k k To determie { } we differetiate Φ λ ( d d d ) i respect to each of it s variables d d d we shall fid extrema of the fuctio () For this purpose d d d ad havig equated to zero we shall receive the system of + equatios ad + variables: Φ λ = j = () d j ach equatio of the system will be of the followig form: Φ λ = Rj + θ jidi = d j i= λ ji = + ( Ii ( a ) I j a + Ii b I j b ) d i+ j+ ρ Rj = ( I j( a τ) χa ( τ) + I j( b τ) χb ( τ) ) dτ θ τ τ τ τ τ ρ As ca be see from (3) the system () is liear Solvig the system () ad substitutig i () we shall obtai the solutio ( ξ ) of the iverse problem Because system () is liear it has exactly oe solutio if the determiat of the matrix θ is ot zero (3) () 4 XAMPL OF TH INVRS PROBLM SOLUTION I this sectio we shall show aalytical ad umeric solutios of the same iverse problem uder the coditio of weak heterogeeity I order to solve the iverse problem first we have to determie the results of the measuremet (4) I this study we shall obtai these fuctios as the result of the solutio of the direct problem with kow heterogeeous beam properties We shall use the followig properties: 6 ( ξ) = ρ = 9 (4) 4 + ξ Usage of the specified properties allows us to solve the problem of determiig vibratios of such beam exactly These properties also correspod to the base homogeeous beam with the properties = 4 ρ = 9 We shall excite the harmoic vibratios i the base homogeeous beam by settig the iitial coditios to: ϕ ( ξ) = ψ ( ξ) = 3si πξ (5) Solvig the direct problem we obtai the vibratios of the weakly heterogeeous beam at the poits a= b= The differece betwee the vibratios of the base homogeeous ad 3 weakly heterogeeous beams are show i Fig These fuctios will be used as the measuremet data for the iverse problem 6
7 ICSV4 9- July 7 Cairs Australia a) b) c) Figure Charts of the measuremet data (differece betwee vibratios of the homogeeous ad correspodig weakly heterogeeous beams): a a χ τ ; b γ ( τ ); c - χ ( τ ) We shall ow solve the iverse problem aalytically ie we should fid ( ξ ) by kow iitial coditios (5) properties of the base homogeeous beam ( = 4 ρ = 9 ) ad χ τ γ τ Carryig out all sequece of actios described i the secod = The chart of this measuremet data a a sectio of this study we fid ( ξ ) takig ito accout that fuctio is give i Fig Maximal deviatio of the foud fuctio from the origial fuctio is 53 or 53% a b Figure Chart of the fuctio ( ξ ) determied from the aalytical iverse problem solutio origial (4) Now solvig the same problem usig the regularizatio method settig =6 ad λ = we obtai: ( ξ) = 37ξ 59ξ ξ + 63ξ 37ξ 5ξ 44 The chart of this fuctio is give i Fig 3 Maximal deviatio of the foud fuctio from the origial fuctio is 48 or 48% Figure 3 Chart of the fuctio ( ξ ) determied from regularizatio method iverse problem solutio origial (4) As see from the data ad the charts the results of both approaches to solutio differ isigificatly However whe solvig the iverse problem we ofte have to deal with the 7
8 ICSV4 9- July 7 Cairs Australia errors of measuremet ad also possible ifluece of ukow factors which itroduce some radom oise ito the measuremet data I this case the regularizatio method is more preferable tha the aalytical solutio To take the radom oise ito accout we shall add the radom fuctios to the measuremet data: * * * χa( τ) = χa( τ) + e( τ) χb( τ) = χb( τ) + e( τ) γa( τ) = γa( τ) + e3( τ) (6) where e( τ ) e( τ) e3( τ ) - the geerated radom fuctios ot exceedig % of the measuremet data by absolute value Solvig the iverse problem with the ew measuremet data (6) aalytically we obtai the results give i Fig 4 Maximal deviatio of the foud fuctio from the origial fuctio is 8 or 8% Ie the measuremet error i % results i solutio error i 8% Figure 4 Chart of the fuctio ( ξ ) determied from the aalytical iverse problem solutio with the radom oise added to measuremet data origial (4) Solvig the problem usig the regularizatio method we obtai the results give i Fig 5 Maximal deviatio of the foud fuctio from the origial fuctio is 8 or 8% Ie the measuremet error i % results i solutio error i 8% Figure 5 Chart of the fuctio ( ξ ) determied from regularizatio method iverse problem solutio with radom oise added to measuremet data origial (4) As ca be see from the give results the solutio usig the regularizatio method is much more stable to measuremet errors tha the aalytical solutio RFRNCS [] VA Lomazov The problem of the diagostics of the heterogeeous thermoelastic eviromets Orel State Techical Uiversity Orel 3 7p [] AN Tikhoov VYA Arsei Solutio methods of ill-posed problems Moscow Nauka p 8
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