A numerical scheme for recovering the nonlinear characteristics of a single degree of freedom structure: non-parametric system identification

Transcription

1 Srucures Under Shock and Impac XIII 335 A numerical scheme for recovering he nonlinear characerisics of a single degree of freedom srucure: non-parameric sysem idenificaion J. Park 1, T. S. Jang 1, S. Syngellakis 2 & H. G. Sung 3 1 Deparmen of Naval Archiecure and Ocean Engineering, Pusan Naional Universiy, Republic of Korea 2 Wessex Insiue of Technology, UK 3 Mariime and Ocean Engineering Research Insiue (MOERI; formerly KRISO) of KIOST (formerly KORDI), Republic of Korea Absrac The aim of his paper is o presen a numerical scheme for he idenificaion of he nonlinear characerisics of a dynamically excied, single degree of freedom srucure, using a non-parameric procedure, recenly proposed by he second auhor; his involves he simulaneous idenificaion of he nonlinear characerisics of boh damping and resoring force in dynamic sysems whose damping depends on velociy alone. According o his mehod, he response of he srucure is firs measured hen an inegral equaion accouning for is unknown nonlinear characerisics is derived. This is an inegral equaion of he firs kind, involving numerical insabiliy in he Hadamard sense. To overcome his difficuly, he Landweber regularizaion, combined wih he L-curve crierion, is applied o he inegral equaion. Adoping a dynamic model for a es srucure, he corresponding nonlinear sysem idenificaion is achieved hrough he proposed numerical soluion of he governing inegral equaion. Keywords: non-parameric sysem idenificaion, nonlinear damping, nonlinear siffness, dynamic response daa, single-degree-of-freedom. doi: /susi140291

2 336 Srucures Under Shock and Impac XIII 1 Inroducion In he las wo decades, here has been a lo of research work on he subjec of dynamic sysem idenificaion conduced by Masri e al. [1], Chassiakos and Masri [2], Liang e al. [3, 4], Spina e al. [5], Iourchenko and Dimenberg [6], and ohers [7 17]. However, mos of his research has idenified he nonlinear characerisics of eiher damping or he resoring force. Only in a few recen sudies, here has been an aemp o idenify he nonlinear characerisics of boh [18, 19]. In his paper, he nonlinear characerisics of boh damping and resoring force of a single degree of freedom sysem are deermined. For his purpose, a non-parameric procedure according o Jang s mehod [20] is applied; his involves a non-parameric, simulaneous idenificaion of he nonlinear characerisics of boh damping and resoring force for sysems whose damping depends on velociy alone. The nonlinear damping and resoring force parameers of a sysem wih ime dependen damping are numerically obained. A sinusoidal (wih respec o velociy) damping as well as a sinusoidal nonlinear resoring force model is adoped for such a sysem. 2 Moion responses A nonlinear single degree of freedom sysem is here considered; is damping is assumed o depend on velociy alone. Alhough a physical experimen should be conduced o provide measuremens of he sysem response for he sysem idenificaion process, virual measuremens of he velociy and he displacemen are here numerically obained using he Runge-Kua mehod. The sysem is governed by he nonlinear ordinary differenial equaion my ky f ( y ) r( y) (1) wih he following iniial condiions [20], y(0), y (0), (2) where he consans m and k denoe he mass of a paricle of he oscillaor and a spring coefficien, respecively. The velociy dependen nonlinear damping f y and he nonlinear resoring force r(y) are assumed ani-symmeric: f ( y ) f( y ), (3) r( y) r( y). (4) The erm f y in eqn (1) indicaes a general form of velociy-dependen nonlinear damping for convenional dampers; his is he usual damping modelling of nonlinear sysems ha is adoped in naural sciences. In he engineering field, when ships or oher floaing bodies are subjeced o large roll moions in waves, eiher when sailing or when a anchor, hey experience nonlinear roll damping, which can be characerized as a ype of velociydependen damping of he form f y in eqn (1). This phenomenon is usually caused physically by he effecs of nonlinear free-surface wave making and he

3 Srucures Under Shock and Impac XIII 337 nonlinear viscous dissipaion of fluid flow. In conras o his ype of damping, he nonlinear damping characerized by f ( yy ) in he Liénard equaion appeared in he early research invesigaion carried ou by Jang e al. [18]; simulaneous nonlinear idenificaion was carried ou here for he Van der Pol nonlinear equaion. The oher erm ha appears in eqn (1), namely r(y), represens a general form of he nonlinear resoring force of he oscillaor. In his paper, he mixed linear-cubic (wih respec o y ) damping model is applied and he linear-cubic nonlinear resoring force model as follows: f ( y) asin y and r( y) csin y, (5) where he adoped values for he coefficiens are a = and c = 20.3 (able 1). The assumed iniial condiions in eqn (2) are y 0 0. (6) y, Table 1: Principal properies of he numerical experimen. Principal properies [25] my ky asin y csin y Governing equaion Mass (kg) m Linear resoring coefficien (N/m) k 20 Nonlinear damping coefficien (Ns/m) a Nonlinear resoring coefficien (N/m) c 20.3 For he numerical calculaion of he sysem response, he Runge-Kua mehod is used for he soluion of eqn (1) and he rapezoidal rule is applied for he numerical inegraion; he ime domain is aken as 0 20 s and he oal number of ime inervals is 401 or = 0.05 s. Fig. 1 shows he displacemen and velociy obained by solving eqn (1) for he damping and resoring force as well as he iniial condiions given by eqns. (5) and (6), respecively; fig. 2 depics he phase diagram corresponding o he soluion of fig. 1. Now, he original nonlinear differenial equaion (1) is ransformed o an equivalen nonlinear Volerra inegral equaion as follows: y1( ) y2() y1() y2( ) y () y1() y2() [ f( y) r( y)]d, (7) 0 mw () in which y 1 and y 2 are chosen so ha hey saisfy my 1 ky1 0, y1(0), y 1(0) 0, (8) my 2 ky2 0, y2(0) 0, y 2(0), and W( ) y1( ) y 2( ) y 1( ) y2( ) The values of and are assumed o have he following magniude in his paper [7, 18, 20]: = = 1.

4 338 Srucures Under Shock and Impac XIII Figure 1: Sysem responses obained by solving eqns (1), (5) and (6). Figure 2: Sysem phase diagram associaed wih he soluion of fig. 1 I is hen possible o consruc an inegral equaion for u, given by u f( y ) r( y), (9) using eqn (7): y () y1() y2() K (, ) u( )d. (10) 0 wih he kernel K defined as y1( ) y2( ) y1( ) y2( ) K (, ). (11) mw ( )

5 Srucures Under Shock and Impac XIII 339 The relaionship in eqn (10) is classified as a Volerra-ype inegral equaion of he firs kind [20], which defines an inverse problem for he simulaneous idenificaion of he velociy-dependen nonlinear damping and resoring force ogeher wih he nonlinear funcion defined by eqn (9). Eqn (10) and eqn (9) consiue he required equaions of he presen nonlinear inverse problem for he wo unknown funcions, f and r. 3 Calculaing u by regularizaion In he previous secion, he analysis was based on he assumpion ha u in eqn (9) is known. As far as he deerminaion of u is concerned, an inegral equaion of he firs kind, such as eqn (10), mus be deal wih. Thus, he quesion of wheher he soluion o sysem idenificaion depends on he measured response daa in a coninuous manner remains. This quesion can be answered by he heory of inegral equaions, according o which, if inegral equaions wih a regular kernel such as he K given by eqn (11) are of he firs kind, hen heir soluions lack sabiliy properies [21 23]. Unreliable soluions may resul from a small amoun of noisy daa due o he considerable amplificaion of a small noisy error. To overcome he difficuly of he soluion s insabiliy wih regard o f and r, he Landweber regularizaion u * * j uj 1 L L( uj 1) L ( ), j 1, (12) is applied o preven i from affecing he performance of he idenificaion [18, 20, 23]. In eqn (12), L is he operaor, L() z K(,)() z d, (13) 0 L * denoes he adjoin operaor of L, is a posiive consan and is given by y y1 y2 (14) In he heory of ieraive regularizaion, an appropriae number of ieraions j is imporan for an accurae soluion. Here, he L-curve crierion is applied o find an appropriae number of ieraions. [18, 20, 22, 24]. 4 Simulaneous idenificaion This secion conains an illusraion of he way by which he wo unknowns, ha is, he velociy-dependen nonlinear damping f ( y ) and nonlinear resoring force r(y), can be idenified using he formulaed eqns (9) and (10). I is iniially assumed ha he measured response daa for y and y of he nonlinear oscillaion governed by eqn (1) are depiced as shown in fig. 1; a number of he inersecion poins of he curve of y wih he -axis are found a imes = 1, 2, ; he i for i = 1,2, are zero-crossing imes for he displacemen. In exacly he same way, here are also inersecion poins of he curve of y wih he -axis a = 1, 2,, which are zero-crossing imes for he velociy; hence,

6 340 Srucures Under Shock and Impac XIII y ( i ) 0 for i = 1,2, (15) k y ( ) 0 for k = 1,2, (16) Since eqn (9) remains valid for any posiive ime, i should also hold no only for zero-crossing imes (for he displacemen) = 1, 2,, bu also for zero-crossing imes (for he velociy) = 1, 2,. Thus, i follows ha u ( i) f[ y ( i)] ry [ ( i)], (17) and u ( ) f[ y ( )] ry [ ( )]. (18) From he informaion of he measured response daa for y and y, he zerocrossing imes of he displacemen i (i = 1,2, ) and he velociy k (k = 1,2, ) and he sabilized u in eqn (10), i follows ha he nonlinear damping a imes i is u ( i) f[ y ( i)], (19) and he nonlinear resoring force a k is k k u ( ) ry [ ( )]. (20) Thus, he decoupled eqns (19) and (20) for he wo unknowns, f and r, respecively, have been reached. 5 Numerical resuls The principal properies of he numerical experimen are shown in able 1. In his virual experimen, he measuremen noise level of he moion response is assumed o be = As already menioned, he Runge-Kua mehod is applied for he deerminaion of he sysem response shown in fig. 1. The opimal or appropriae ieraion number j op = of he Landweber s regularizaion mehod, corresponding o he corner of he L-curve, and he respecive u j,op, deermined by solving eqn (10) wih noise = 0.01 are depiced in fig. 3 and fig. 4, respecively. Figure 3: L-curve for he records of fig 1: opimal ieraion number j op =

7 Srucures Under Shock and Impac XIII 341 Figure 4: Compued u j,op () for he opimal ieraion number j op = Employing he u j,op and he zero-crossing imes for he displacemen i and velociy k (fig. 5), he deermined nonlinear damping f y and he nonlinear resoring force r(y) are demonsraed in fig. 6 and fig. 7, respecively Conclusions Figure 5: Sysem responses wih he zero-crossing imes. In he presen sudy, Jang s mehod is applied for he numerical idenificaion of he nonlinear damping and he nonlinear resoring forces in a SDOF dynamic srucure. The main idea of he Jang s mehod relies on zero-crossing ime measuremens from experimenal displacemen and velociy records and i is a

8 342 Srucures Under Shock and Impac XIII very simple bu accurae mehod for he idenificaion of nonlinear characerisics. Of course, i has an ill-posedness propery which causes soluion insabiliy; bu his can be handled by a regularizaion mehod such as he Landweber s mehod wih he L-curve crierion. Finally, he suiabiliy of Jang s mehod for he deerminaion of he nonlinear characerisics of a SDOF dynamic srucure has been demonsraed hrough a numerical experimen. nonlinear damping f y asin y Figure 6: Numerical resuls for f y compared o he originally assumed ; a = Figure 7: Numerical resuls for r(y) compared o he originally assumed r y csin y ; c = nonlinear resoring force

9 Srucures Under Shock and Impac XIII 343 Acknowledgemens The firs and he second auhors are suppored by Basic Science Research Program hrough he Naional Research Foundaion of Korea (NRF) funded by he Minisry of Educaion (Gran No ). The fourh auhor acknowledges he financial suppor by a principal R&D program of KIOST: Performance Evaluaion Technologies of Offshore Operabiliy for Transpor and Insallaion of Offshore Srucures. References [1] Masri, S.F., Chassiakos, A.G. & Cauchey, T.K., Idenificaion of nonlinear dynamic sysems using neural neworks. Journal of Applied Mechanics, 60, pp , [2] Chassiakos, A.G. & Masri, S.F., Modeling unknown srucural sysems hrough he use of neural neworks. Earhquake Engineering and Srucural Dynamics, 25, pp , [3] Liang, Y.C., Zhou, C.G. & Wang, Z.S., Idenificaion of resoring forces in non-linear vibraion sysems based on neural neworks. Journal of Sound and Vibraion, 206, pp , [4] Liang, Y.C., Feng, D.P. & Cooper, J.E., Idenificaion of resoring forces in non-linear vibraion sysems using fuzzy adapive neural neworks. Journal of Sound and Vibraion, 242, pp , [5] Spina, D., Valene, C. & Tomlinson, G.R., A new procedure for deecing nonlineariy from ransien daa using he Gabor ransform. Nonlinear Dynamics, 11, pp , [6] Iourchenko, D.V. & Dimenberg, M.F., In-service idenificaion of nonlinear damping from measured random vibraion. Journal of Sound and Vibraion, 255, pp , [7] Feldman, M., Considering high harmonics for idenificaion of non-linear sysems by Hilber ransform. Mechanical Sysems and Signal Processing, 2 pp , [8] Worden, K., Hickey, D., Haroon, M. & Adams, D.E., Nonlinear sysem idenificaion of auomoive dampers: A ime and frequency-domain analysis. Mechanical Sysems and Signal Processing, 23, pp , [9] Maiaa, N.M.M. & Ewins, D.J, A new approach for he modal idenificaion of lighly damped srucures. Mechanical Sysems and Signal Processing, 3, pp , [10] Iourchenko, D.V., Duval, L. & Dimenberg, M.F., The damping idenificaion for cerain SDOF sysems. Developmens in Theoreical and Applied Mechanics, Proceedings of SECT AM XX Conference, Auburn Universiy, [11] Jang, T.S., Kwon, S.H. & Han, S.L., A novel mehod for non-parameric idenificaion of nonlinear resoring forces in nonlinear vibraions from

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