829. An adaptive method for inertia force identification in cantilever under moving mass

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1 89. An adaptve method for nerta force dentfcaton n cantlever under movng mass Qang Chen 1, Mnzhuo Wang, Hao Yan 3, Haonan Ye 4, Guola Yang 5 1,, 3, 4 Department of Control and System Engneerng, Nanng Unversty, 10093, Nanng, Chna 5 Department of Mechancal Desgn and Automaton, School of Mechancal Engneerng Nanng Unversty of Scence and Technology, 10014, Nanng, Chna E-mal: 1 chena077@vp.sna.com, nuwangmnzhuo@gmal.com, 3 yanhao8510@gmal.com (Receved 1 July 01; accepted 4 September 01) Abstract. The present study s concerned wth the adaptve method based on wavelet transform to dentfy the nerta force between movng mass and cantlever. The basc model of cantlever s descrbed and a classcal dentfcaton method s ntroduced. Then the approxmate euatons about the model of cantlever can be obtaned by the dentfcaton method. However, the order of modal adapted n the dentfcaton methods s usually constant whch may make the dentfcaton results unsatsfed. As s known, the freuency of the hghest order of modal s usually hgher than the freuency of the nput force n forward calculaton methods. Therefore, wavelet transform s appled to decompose the data of deflecton. The proporton of the low freuency component s chosen as the parameter of a bnary functon to decde the order of modal. The calculaton results show that the adaptve method adapted n ths paper s effcent to mprove the accuracy of the nerta force between the movng mass and cantlever, and also the relatonshp between the proporton of low freuency component and the order of modal s ndcated. Keywords: adaptve, nerta force, cantlever, dentfcaton, wavelet transform. 1. Introducton Tme-varyng parameter dentfcaton n cantlever under movng mass s an mportant nverse problem n the cvl and structural engneerng feld. It s developed from tme-varyng parameter dentfcaton of structural dynamcs problems. By accessng to the nerta force between movng mass and cantlever, more characterstcs of cantlever wll be obtaned and t wll provde some references for further engneerng desgn. It s dffcult to measure the nerta force drectly between movng mass and cantlever because they are n moton and the nerta force tself s tme-varyng [1]. A seral of classcal methods based on Euler-Bernoull beam model for nerta force dentfcaton between movng mass and smply supported beam are presented by former researchers. T. H. T. Chan nvestgated the Interpretve Method (IM) [], whch s used to dentfy the nerta force accordng to modal analyss for brdge responses. S. S. aw developed the Tme Doman Method (TDM) [3], whch dentfes the nerta force by usng the modal superposton prncple n tme doman, and the Freuency Tme Doman Method (FTDM) [4], whch calculates the nerta force spectrums by usng the least-suare method and then the nerta force can be obtaned by the nverse Fourer transformaton. Mnzhuo Wang has mproved the Interpretve Method [5], makng t be successfully appled to dentfy the nerta force between movng mass and cantlever. However, the order of modal adapted n the dentfcaton methods s usually decded ust by the experence of the researchers. In ths paper, an adaptve method based on wavelet transform s proposed. The data of deflecton are decomposed by wavelet bass n order to get the proporton of the low freuency component by whch the adaptve rule s decded. 105

2 . Problem formulaton. 1. Modelng for cantlever structure In practcal stuaton, the nteracton between the cantlever and the movng mass s a complex process affected by dfferent parameters. Smplfed models are more effectve to establsh a clear connecton between the parameters and the beam response than complex ones [6]. Therefore Euler-Bernoull beam [7] model s chosen n ths paper. The contnuous beam model of cantlever s llustrated n Fg. 1. Fg. 1. Model of cantlever Suppose an Euler-Bernoull beam s defned wth a span length, constant flexural stffness EI and constant mass per unt length ρ, then the euaton of moton can be got as below [8]: 4 v( x, t) v( x, t) ρ + EI = δ ( x ct) f ( t) 4 t x (1) where v( x, t ) s the beam deflecton at pont x and tme t, δ s the Drac delta functon and dampng s neglected... Theory of tme-varyng parameter dentfcaton The boundary condtons for E. (1) are: v(0, t ) = 0, v( x, t) x x = 0 = 0, v( x,0) = 0, ( x, t) t t= 0 = 0 Based on the modal superposton theory, the soluton of E. (1) can be expressed as: v = = 1 π x sn ( t) () where ( t ) are the modal dsplacements. After substtutng E. () nto E. (1), both sdes of the euatons are multpled by sn( π x / ). Then ntegrate the euatons wth respect to x between 0 and, use the propertes of δ ( t) and the boundary condtons, therefore the euatons can be expressed as: 1053

3 .. F ( t) + ω( ) ( t) = sn ωt, = 1,,... (3) ρ where: ω π EI nπ x =, fn ( t) = F( t)sn ρ 4 4 ( ) 4 are the n th modal freuency and modal force respectvely. In ths paper, Interpretve Method (IM) whch s prevously based on smply supported beam s used to dentfy the nerta force n cantlever under movng mass. The euaton of dentfcaton from E. (3) can be modfed as:.. F ( t) + ω( ) ( t) = sn ωt, = 1,,... (4) ρ Unfortunately, the ω n E. (4) can t be obtaned as analytcal soluton. Therefore, an approxmate soluton s adapted n order to obtan the value of ω. By usng the varables separaton method and the boundary condtons of the cantlever, the freuency euaton of the cantlever can be gven as [9]: cosβ chβ = 1 (5) The soluton of E. (5) can be obtaned by usng the numercal method. Based on the results above, the modal functon can be gven as follows: βn x βn x sh( βn ) sn( βn ) βn x βn x ϕ( n, x) = ch cos sh sn ch( βn ) + cos( βn ) (6) where cantlever, n β n s the nodes of the n th man vbraton mode wth the ntal poston of the steady s the order of modal, x s the x th one of all data of deflecton. Then, the euaton of dentfcaton can also be expressed as: ɺɺ ω ɺɺ ϕ(1,1) ϕ(1,) ϕ(1, m) ω 1 ϕ(,1) ϕ(, ) ϕ(, ) m + = P ρ ϕ( n,1) ϕ( n,) ϕ( n, m) ɺɺ n ωn n [ ] (7) where P s the nerta force between movng mass and cantlever, n s the order of modal, 1054

4 m s the number of the nput data.. 3. An adaptve method based on wavelet transform As t mentoned above, the order of modal whch s adapted n the former methods s usually constant. But ths constant order of the model adapted n the calculaton only depends on the researchers experence. In ths paper, we propose an adaptve method that the order of the model become varable based on the results from wavelet transform. The nput data s decomposed by usng Daubeches 5 wavelet at a decomposton level of fve. Then the proporton of the low freuency component s calculated. Decomposton model s shown n Fg. as below [10]. Fg.. Multresoluton analyss model Mathematcs decomposton results are gven as follows: xφ ( t) = x φ ( t) + d ψ ( t) 1 1 k k k ( + 1) k k ( + 1) k k k k 1 and Px ( t) = P + 1x( t) + D+ 1x( t) or D+ 1x( t) = Px ( t) P + 1x( t), where φk ( t) = φ( t k), / 1 x k s the scale functon coeffcent at layer. ψ k ( t) ψ ( = t k), d / k s the wavelet transform coeffcent at layer. D ( ) + 1 x t s the proecton of x( t ) n W + 1, Px ( t ) reflect the general vew of sgnal x( t ) under the resoluton I. For each samplng pont, we adapt a bnary functon whch depends on proporton of ts low freuency component. If the proporton of the low freuency component s hgher than the threshold, the value of the bnary functon s 1. Or, f the proporton of the low freuency component s lower than the threshold, the value of the bnary functon s 0. Then, the order of modal s decded by the sum of the value of the bnary functons of all the samplng ponts obtaned at the same tme. In sum, the whole process of the calculaton can be llustrated as t shown n the Fg. 3 below. 1055

5 3. Numercal calculaton and results Fg. 3. The whole process of the calculaton The source of the data adapted n the dentfcaton s from the deflecton of ten samplng ponts whch are acured by numercal methods. A relatve percentage error (RPE) [11] s defned to compare the results of the dentfcaton whch s expressed as: fture fdent RPE= 100% f ture The parameters of the model n numercal calculaton are defned as below: the span length s 1.5 m, constant flexural stffness EI s N m and constant mass per unt length ρ s kg/m 3. Samplng freuency s 10 khz constantly. Three numercal examples wth dfferent velocty of the movng mass on cantlever are shown n ths paper (v 1 = 5 m/s, v = 10 m/s, v 3 = 0 m/s). As mentoned above, the threshold of the bnary functon depends on the proporton of low freuency, whch s appled to set the order of modal. Accordng to the pror knowledge, the farther dstance between the samplng pont and supportng pont on cantlever, the larger threshold must be chosen to calculate, and the hgher velocty of movng mass, the threshold must be also larger. Thus, the threshold of the dfferent three numercal examples s lsted n Table 1. Table 1. The threshold of the dfferent three numercal examples Samplng pont v 1 = 5 m/s v = 10 m/s v 3 = 0 m/s % 97.0 % 98.0 % 95. % 97. % 98. % % 97.4 % 98.4 % % 97.6 % 98.6 % % 97.8 % 98.8 % % 98.0 % 99.0 % % 98. % 99. % % 98.4 % 99.4 % % 98.6 % 99.6 % % 98.8 % 99.8 % The results of dentfcaton wth dfferent velocty of the movng mass are llustrated below n Fgures 4-6 and the relatve percentage errors of dentfcaton are lsted n Tables

6 Fg. 4. Identfcaton results (velocty of movng mass s 5 m/s) Fg. 5. Identfcaton results (velocty of movng mass s 10 m/s) 4. Conclusons Fg. 6. Identfcaton results (velocty of movng mass s 10 m/s) In ths paper, an adaptve method based on wavelet transform s appled to dentfy the nerta force between movng mass and cantlever. The nput data s decomposed by usng wavelet transform. Then the proporton of the low freuency component s calculated n order to set the threshold whch s used to determne the order of modal. Three dfferent results are obtaned by changng the speed of movng mass. Some recommendatons based on these results are: (1) 0 percent or more mprovement can be obtaned by usng the adaptve method based on 1057

7 wavelet transform. () The relatonshp between the proporton of low freuency component and the order of modal has been proved. (3) The mprovement of the results s better as the velocty of the movng mass s hgher among three numercal examples. Table. Identfcaton results (velocty of movng mass s 5 m/s) Order of RPE modal adaptve.67 Table 3. Identfcaton results (velocty of movng mass s 10 m/s) Order of RPE modal adaptve 5.01 Table 4. Identfcaton results (velocty of movng mass s 0 m/s) Order of RPE modal adaptve Future work of ths study would do more research on the relatonshp between the proporton of low freuency component and the order of modal and apply more advanced adaptve methods. Acknowledgments The author would lke to acknowledge the support of the Maor State Basc Research Development Program of Republc of Chna (No ). References [1] T. H. T. Chan,. Yu, S. S. aw Movng force dentfcaton studes, II: Comparatve studes. Journal of Sound and Vbraton, Vol. 48(1), p [] T. H. T. Chan, S. S. aw, T. H. Yung, X. R. Yuan An nterpretve method for movng force dentfcaton. Journal of Sound Vbraton, Vol. 19(3), 1999, p [3] S. S. aw, T. H. T. Chan, Q. H. Zeng Movng force dentfcaton: a tme doman method. Journal of Sound and Vbraton, Vol. 01(1), 1999, p. 1. [4] S. S. aw, T. H. T. Chan, Q. H. Zeng Movng force dentfcaton: a freuency and tme domans analyss. Journal of Dynamc Systems, Measurement and Control, Vol. 1(3), 1999, p [5] Mnzhuo Wang, Qang Chen, Guola Yang The research on tme-varyng parameter dentfcaton n support beam under movng mass. 4th Internatonal Conference on Mechancal Engneerng and Mechancs, 011, p [6]. Yu, Tommy H. T. Chan Recent research on dentfcaton of movng loads on brdges. Journal of Sound and Vbraton, Vol. 305, 007, p [7]. Yu Accountng for Brdge Dynamc oads Usng Movng Force Identfcaton System (MFIS). Ph. D. Thess, The Hong Kong Polytechnc Unversty, Hong Kong, 00. [8] T. H. T. Chan, S. S. aw, T. H. Yung Movng force dentfcaton usng an exstng prestressed concrete brdge. Engneerng Structures, Vol., 000, p [9] Jngbo u Structural Dynamcs, 005, p [10] Xn u, Qang Chen, Haonan Ye Parameter dentfcaton of frcton model based on wavelet denose. Internatonal Conference on Systems and Informatcs. [11] T. H. T. Chan,. Yu, S. S. aw Movng force dentfcaton studes, I: Theory. Journal of Sound and Vbraton, Vol. 47(1), 001, p