Force Reconstruction for Nonlinear Structures in Time Domain

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1 Fore Reonstrution for Nonlinear Strutures in ime Domain Jie Liu 1, Bing Li 2, Meng Li 3, an Huihui Miao 4 1,2,3,4 State Key Laboratory for Manufaturing Systems Engineering, Xi an Jiaotong niversity, Xi an, , P. R. China Jieliu2013@stu.xtu.eu.n bli@mail.xtu.eu.n lm @gmail.om mhh1989@stu.xtu.eu.n ABSRAC With growing omplexity of mehanial struture, nonlinear fators existing in the struture have rawn muh attention in reent years. Meanwhile, aurate information of ynami fore is an important inex for analyzing nonlinear mehanial struture. However, these ata are always iffiult an even impossible to be measure iretly. herefore, in this paper, a novel reonstrution strategy is propose to alulate the external fore of the nonlinear struture on the basis of measure response at the referene position. For the reonstrution strategy, the fore reonstrution equation of nonlinear struture is establishe by the nonlinear state-spae moel, an nonlinear subspae ientifiation (NSI) algorithm is utilize to estimate oeffiient matries of the nonlinear state-spae moel to form the transfer matrix. An then, onsiering the illonition of the transfer matrix, the regularization metho ombine with the generalize ross-valiation riterion is utilize to solve the ill-pose reonstrution equation to obtain the unknown external fore. Numerial stuy is onute to illustrate the feasibility of the reonstrution strategy. he results emonstrate that the propose reonstrution strategy an be utilize to aurately obtain the external fore of the nonlinear struture. 1. INRODCION Dynami fore on the mehanial struture is essential for ynami optimal esign, vibration ontrol an strutural health monitoring (Law, Bu, Zhu & Chan, 2007). For instane, for the ship power equipment, ynami fore between the atuator an main engine shoul be obtaine to provie the basis for ative vibration isolation ontrol of the equipment (Daley, Johnson, Pearson & Dixon, 2004). nfortunately, in most engineering praties, iret measurement of ynami fore is always iffiult an even Jie Liu et al. his is an open-aess artile istribute uner the terms of the Creative Commons Attribution 3.0 nite States Liense, whih permits unrestrite use, istribution, an reproution in any meium, provie the original author an soure are reite. impossible. However, ynami responses are relatively easy to be measure by appropriate sensors. For these reasons, it is essential to evelop fore reonstrution tehnique to reonstrut the fore by measure ynami responses an strutural harateristis. Beause that onition number of the transfer matrix in the reonstrution equation is usually large in the engineering pratie, fore reonstrution is a typial ill-pose inverse problem. For suh problem, lassial numerial algorithms base on the least squares (LS) or iret matrix inversion (DMI) sheme turn out to be useless. Consequently, speial regularization tehniques suh as ikhonov regularization, iterative metho an trunate singular value eomposition (SVD) metho, have been evelope to obtain the numerially stable approximation to unknown solution of the ill-pose problem, see, e.g. Inoue, Harrigan an Rei (2001), Sanhez an Benaroya (2014) for an overview. Liu an Shepar Jr (2005) propose an enhane least square sheme whih ontains the regularization filter (SVD filter or ikhonov filter) an LS sheme (onventional LS sheme or total LS sheme) to onut fore reonstrution in frequeny omain. Lourens, Reyners, De Roek, Degrane an Lombaert (2012) propose an augmente Kalman filter algorithm to reonstrut the exiting fore, an experimental results illustrate that the propose metho is superior to traitional eterministi least square metho. Jia, Yang an Song (2015) propose a weighte regularization approah base on the proper orthogonal eomposition (POD) to improve the auray of ranom loas reonstrution. Zhang, Huang, Zhang an Hua (2016) ombine linear subspae ientifiation algorithm an homotopy to experimentally estimate the exiting fore on a sale moel of a ivil ship. It an be note from reviewing the literature that various methos have been propose to onut the fore reonstrution. However, most of these stuies fous on the linear struture. Due to nonlinearities in the omponents an material properties, Engineering strutures are always nonlinear. Moreover, in reent years, nonlinear fators existing in engineering strutures have rawn muh attention, 1

2 an onsiering these nonlinear fator an improve the auray of ynami analysis (Yang, Ren, Qin, Wu & Zhi, 2010, Zhang & Du, 2015). herefore, a novel fore reonstrution strategy of nonlinear struture is propose in this paper. he reonstrution equation is base on the nonlinear state-spae moel, an the transfer matrix is ompose of oeffiient matries of this moel. Nonlinear subspae ientifiation (NSI) algorithm (Marhesiello & Garibali, 2008) whih is one kin of nonlinear system ientifiation metho is utilize to estimate these oeffiient matries to form the transfer matrix. When obtaining transfer matrix an ynami response, ikhonov regularization metho is employe to solve the ill-pose reonstrution equation, an regularization parameter is etermine by the generalize ross-valiation (GCV) riterion. 2. HEOREICAL BACKGROND 2.1. Fore reonstrution equation of nonlinear strutures For the general n egree-of-freeoms struture with loal nonlinearities, the equation of motion an be represente as p Mx C x Kx L g L f (1) v nl _ f 1 where M, C an K are the mass, amping an stiffness matries, respetively. p enotes the number of nonlinearities. is the nonlinear parameter of the -th nonlinearity, an it represents the nonlinear strength. g ( t ) is the orresponing nonlinear esribing funtion. L f an Lnl _ enote the exiting loation an nonlinear loation. Aitionally, the prout of an g ( t ) enotes the nonlinear fore. Base on the output feebak priniple (Aams & Allemang, 1999), nonlinear fore an be viewe as the internal feebak fore, an nonlinear term in Eq. (1) is move to the right sie of the equation. An then, by efining the state variable z x x, Eq. (1) an be rewritten as z A z B u 0rr Irr 1 1 z M K M Cv 0rr Ir p f 1 1 f nl nl M L M μl g where subsript enotes that oeffiient matries are for the ontinuous-time state-spae moel. g nl g1 g2 g p enotes a vetor mae up of nonlinear esribing funtion, an μlnl are forme by the prout of orresponing nonlinear parameter an (2) loation vetor. Aitionally, the output vetor of state-spae moel is given by y C z D u (3) Eqs. (2) an (3) are the ontinuous-time state-spae moel of the nonlinear struture, an aoring to Zero-Orer-Hol (ZOH) isretization metho, these two equations an be rewritten as z( k 1) A z( k) B u ( k) (4) y( k) C z( k) D u ( k) (5) where subsript enotes that oeffiient matries are for the isrete-time state-spae moel. k represents the isrete time k t, an t is the sampling interval. he relationship between two kins of oeffiient matries is given by Aks A e ; C C B I A B D D Aks 1 ( e 2n2n ) ; Substituting Eq. (4) into Eq. (5) at ifferent isrete time, an base on the zero initial onition z(0) u (0) 0, output vetor an be expresse as y(0) D 0 0 u(0) (1) 0 (1) y C B D u 0 N 1 y( N) CA B CB D u( N) he ompat form of Eq. (7) is given by (6) (7) Y = HF (8) where F enotes generalize fore vetor of nonlinear struture, inluing exiting fore an nonlinear esribing funtion. Eq. (8) relates generalize fore vetor F to output response vetor Y by the transfer matrix H. If Y an H are known, unknown F an be obtaine by solving Eq. (8). herefore, Eq. (8) is regare as the fore reonstrution equation of nonlinear struture Nonlinear ientifiation by nonlinear subspae ientifiation algorithm In orer to onut fore reonstrution of nonlinear struture, transfer matrix in Eq. (8) shoul be obtaine first. It an be observe from Eq. (7) that the transfer matrix H is forme by oeffiient matries of nonlinear state-spae moel, an these oeffiient matries an be estimate by nonlinear subspae ientifiation (NSI) algorithm. NSI algorithm is essentially evelope from linear subspae ientifiation (SI) algorithm (Overhee & Moor, 1996, Verhaegen, 1994), an ientifying nonlinear system by NSI algorithm means that SI algorithm is applie to estimate oeffiient matries of nonlinear state-spae moel 2

3 (nonlinear parameters are inlue in oeffiient matrix B ). It means that the ore of these above-mentione two algorithms are the same, only the state-spae moel to be solve is ifferent. For linear state-spae moel, input vetor only inlues exiting fore, an oeffiient matrix B oes not inlue nonlinear parameter. In orer to estimate oeffiient matries of nonlinear statespae moel, an input blok Hankel matrix is efine as 0 2h1 u(0) u( l 1) u( h 1) u( h l 2) u( h) u( h l 1) u(2h 1) u(2h l 1) where h is a user-efine inex, an it shoul satisfy h r ( r enotes the orer of state-spae moel). l is equal to N 2h 1, an N represents the number of sampling points. As the similar manner, output blok Hankel matries Y 0 2h1, Y an Y an also be obtaine. Aitionally, a ombinatorial blok Hankel matrix is given by Q = Q 0 h1 Y (9) (10) SI algorithm is base on the geometri manipulation of the above blok Hankel matries, an an oblique proetion operator is expresse as h Y Q (11) where h represents the oblique proetion of the row spae of Y along the row spae of on the row spae of Q. An then, the singular value eomposition (SVD) of weighte oblique proetion operator is given by where IΘ W S V h s s s = Ss1 0 V (12) s1 Ws 1 Ws 2 0 S s2 V s2 enotes proetion on the orthogonal. Aoring to SVD omplement of the row spae of results shown in Eq. (12), extene observability matrix an be alulate by Γ W S (13) 1/2 = s 1 s 1 Furthermore, the efinition of Γ is shown as C ef = C A Γ C A h1 (14) When extene observability matrix Γ has been obtaine by Eq. (13), oeffiient matries A an C an be estimate by Eq. (14). Finally, two estimate oeffiient matries are substitute into nonlinear state-spae moel, an linear regression analysis is applie to estimate the remaining oeffiient matries B an D. In onlusion, before fore reonstrution of nonlinear struture, a known exiting fore is applie to onut vibration test. Base on the exiting fore an measure output response, NSI algorithm an be utilize to estimate oeffiient matries of nonlinear state-spae moel to form the transfer matrix in Eq. (8) Fore reonstrution with regularization metho After obtaining output response an transfer matrix, unknown exiting fore of nonlinear struture an be reonstrute by solving Eq. (8). Intuitively, it seems easy to solve reonstrution equation by DMI metho, an the result is given by 1 F H Y H H H Y (14) DMI = where represents pseuo inverse operation, an when H 1 is a square matrix, H ( H ). However, in engineering pratie, the onition number of transfer matrix is always large, it is an ill-onitione matrix. For DMI metho, errors ontaine in output responses may be amplifie by the illonitione transfer matrix to result in worthless results. It means that fore reonstrution is an ill-pose inverse problem. In this paper, ikhonov regularization metho is applie to obtain the stable solution of ill-pose inverse problem. he obetive funtion is given by minimize F 2 2 HF Y F (15) 2 2 where 0 is ikhonov regularization parameter. Base on the erivative of Eq. (15) on unknown variable F, ikhonov regularization solution F ik is as follow 1 Fik H H+ I n n H Y (16) Compare to DMI solution in Eq. (14), is utilize to improve the stable of regularization solution, an seleting an appropriate is a key step. In this paper, the generalize ross-valiation (GCV) riterion is utilize to etermine an optimal ikhonov regularization parameter by minimizing 3

4 GCV funtion, an speifi alulation proess is fully isusse in referene (Overhee & Moor, 1996). In onlusion, the propose fore reonstrution strategy of nonlinear struture inlues two step. At first, a known exiting fore is utilize to onut vibration test, an oeffiient matries of nonlinear state-spae moel whih are estimate by NSI algorithm is utilize to form the transfer matrix H. An then, measuring ynami response an be applie to reonstrut the unknown exiting fore of nonlinear struture by using ikhonov regularization metho to solving ill-pose reonstrution equation. 3. NMERICAL EXPERIMEN In this setion, a two egree-of-freeom (DOF) struture with learane nonlinearity shown in Fig. 1 is use to verify the effetiveness of propose fore reonstrution strategy of nonlinear struture. he figure shows that learane nonlinearity is loate between first DOF an seon DOF, an learane-nonlinearity fore is esribe by x x fnl k g k 0 x (17) x x where x x2 x1, x ( t ) an x 1 2 are respetively the isplaement responses at first DOF an seon DOF. is the learane value. k is the learane stiffness. F F e F true re 2 true 2 100% (18) Aoring to the propose fore reonstrution strategy, oeffiient matries shoul be alulate by NSI algorithm at first, an in this step, learane-nonlinearity esribing funtion an be obtaine by Eq. (17) to form the input vetor of nonlinear stat-spae moel (learane value is known as prior information). Base on transfer matrix an measure ynami response, reonstrution result by DMI metho is shown in Fig. 2. Fig.2(a) shows that, uner the noise-free ase, DMI metho an be applie to preisely reonstrut the unknown fore. It means that the propose fore reonstrution equation of nonlinear struture base on nonlinear state-spae moel is reasonable, an using the estimate oeffiient matries by NSI algorithm to form the transfer matrix is effetive an feasible. However, with SNR=40B, Fig. 2(b) shows that DMI reonstrution result is seriously eviate from the true value. In this simulation ase, the onition number of transfer matrix 1 16 on( H) H 2 H , it means that the transfer matrix is ill-onitione, an fore reonstrution problem of the nonlinear simulation moel is ill-pose. he measurement errors are amplifie by the matrix inversion, an DMI solution turns out to be useless. f k 1 1 m1 k 2 2 m 2 k 3 3 Fore/N k k Figure 1 wo DOF simulation moel with learane nonlinearity. In the simulation, parameters of the nonlinear struture are set as: m1 m2 1kg, linear stiffness k1 k2 k3 1000N/m, amping oeffiient N s/m, learane value 1mm, an 4 learane stiffness k 2 10 N/m. Sampling frequeny is 1024Hz, exiting fore is f 50sin 2 5t, an uration is 2s. Aitionally, all simulate output responses are ae by an aitive white noise with SNR=40B (Signal Noise Ratio, SNR). In orer to evaluate the loseness between the true fore F true an reonstrution result F re, relative error e is efine as Fore/N (b) Figure 2 DMI reonstrution results for nonlinear simulation moel: (a) noise-free ase; (b) SNR=40B. (a) 4

5 ikhonov regularization metho is aopte to improve the numerial stability to solve ill-pose fore reonstrution equation of nonlinear struture, an reonstrution result is shown in Fig. 3. For GCV riterion, ikhonov regularization parameter whih orrespons to the minimum value of the GCV funtion an be etermine from Fig. 3(a). As shown in Fig. 3(b), with SNR=40B, the reonstrute fore urve is oinient with the true fore urve, an relative error is 3.70%. It shows that, uner noise interferene, propose reonstrution strategy an preisely reonstrut the unknown exiting fore of nonlinear struture by measure ynami response. espeially in the areas where the amplitue is largest. he relative error is 28.14%. herefore, when reonstruting the unknown exiting fore of nonlinear struture, nonlinear fator annot be ignore. In onlusion, the propose fore reonstrution strategy of nonlinear struture is feasible an effetive (b) Figure 3 ikhonov regularization reonstrution results for nonlinear simulation moel: (a) GCV riterion; (b) reonstrution results with SNR=40B. In orer to further illustrate the valiity of the propose reonstrution strategy, the nonlinear simulation moel is regare as a linear struture, an the influene of nonlinear fator on strutural harateristis is ignore. It means that oeffiient matries of the linear state-spae moel (input vetor only inlues exiting fore) are use to form the transfer matrix. Fore reonstrution equation base on linear state-spae moel is also solve by ikhonov regularization metho. ner the same simulation onitions, the reonstrution result is shown in Fig. 4. Compare to Fig. 3, there is an obvious ifferene between the reonstrute fore urve by ignoring nonlinear fator an true fore urve, (a) Figure 4 ikhonov regularization reonstrution results by ignoring nonlinear fator. 4. CONCLSION his paper propose a novel fore reonstrution strategy of nonlinear struture whih is base on time-omain NSI algorithm an ikhonov regularization metho. NSI algorithm is use to estimate oeffiient matries of the nonlinear state-spae moel. he transfer matrix in reonstrution equation an be obtaine by these oeffiient matries. An then, ikhonov regularization metho is applie to solve the ill-pose fore reonstrution equation to obtain the unknown external fore of the nonlinear struture. A simulation moel with learane nonlinearity is set in MALAB to verify the performane of the propose reonstrution strategy. he simulation results show that the nonlinear fator annot be ignore, an the reonstrution strategy has a goo reonstrution performane. However, one fat is that there is a lak of experimental verifiation. In a work, we will onut nonlinear experiments to verify the effetiveness of the propose reonstrution strategy in atual experiment environment. ACKNOWLEDGEMEN his work is supporte finanially by the National Natural Siene Founation of China ( ). REFERENCES Aams, D. E., & Allemang, R. J. (1999). Charaterization of nonlinear vibrating systems using internal feebak an frequeny response moulation. Journal of vibration an aoustis, 121(4), oi: / Daley, S., Johnson, F. A., Pearson, J. B., & Dixon, R. (2004). Ative vibration ontrol for marine appliations. Control 5

6 Engineering Pratie, 12(4), oi: /s (03) Inoue, H., Harrigan, J. J., & Rei, S. R. (2001). Review of inverse analysis for iniret measurement of impat fore. Applie Mehanis Reviews, 54(6), oi: / Jia, Y., Yang, Z., & Song, Q. (2015). Experimental stuy of ranom ynami loas ientifiation base on weighte regularization metho. Journal of Soun an Vibration, 342, oi: /.sv Law, S. S., Bu, J. Q., Zhu, X. Q., & Chan, S. L. (2007). Moving loa ientifiation on a simply supporte orthotropi plate. International Journal of Mehanial Sienes, 49(11), oi: /.imesi Liu, Y., & Shepar Jr, W. S. (2005). Dynami fore ientifiation base on enhane least squares an total least-squares shemes in the frequeny omain. Journal of soun an vibration, 282(1-2), oi: /.sv Lourens, E., Reyners, E., De Roek, G., Degrane, G., & Lombaert, G. (2012). An augmente Kalman filter for fore ientifiation in strutural ynamis. Mehanial Systems an Signal Proessing, 27, oi: /.ymssp Marhesiello, S., & Garibali, L. (2008). A time omain approah for ientifying nonlinear vibrating strutures by subspae methos. Mehanial Systems an Signal Proessing, 22(1), oi: /.ymssp Overhee, V., & Moor, B. L. (1996). Subspae ientifiation for linear systems. nite States: Springer, Boston, MA. Sanhez, J., & Benaroya, H. (2014). Review of fore reonstrution tehniques. Journal of Soun an Vibration, 333(14), oi: /.sv Verhaegen, M. (1994). Ientifiation of the eterministi part of MIMO state spae moels given in innovations form from input-output ata. Automatia, 30(1), oi: / (94) Yang, Y., Ren, X., Qin, W., Wu, Y., & Zhi, X. (2010). Analysis on the nonlinear response of rake rotor in hover flight. Nonlinear Dynamis, 61(1-2), oi: /s Zhang, J., & Du, X. (2015). ime-epenent reliability analysis for funtion generation mehanisms with ranom oint learanes. Mehanism an Mahine heory, 92, oi: /.mehmahtheory Zhang, Z., Huang, X., Zhang, Z., & Hua, H. (2016). Fore Ientifiation Base on Subspae Ientifiation Algorithms an Homotopy Metho. In Dynamis of Couple Strutures - Proeeings of the 34th IMAC (25-31), January 25-28, Orlano, FL, nite states. oi: / _4. BIOGRAPHIES Jie Liu obtaine his B.S. egree from Shool of Mehanial Engineering at Chongqing niversity in J. Liu is urrently stuying for a Ph. D. in engineering from Xi'an Jiaotong niversity. His researh interests inlue nonlinear ynamis, vibration analysis an system ientifiation. Bing Li obtaine his M.S. from Shool of Marine Engineering at Northwestern Polytehnial niversity in He obtaine his Ph. D. from Shool of Mehanial Engineering at Xi an Jiaotong niversity in He is urrently a Professor at Department of Mehanial Engineering, Xi an Jiaotong niversity. His researh interests inlue Vibration Signal Proessing, Fault Diagnosis an Nonlinear ynamis. 6

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