Identfcaton of Instantaneous Modal Parameters of A Nonlnear Structure Va Ampltude-Dependent ARX Model We Chh Su(NCHC), Chung Shann Huang(NCU), Chng Yu Lu(NCU)
Outlne INRODUCION MEHODOLOGY NUMERICAL VERIFICAION APPLICAION OF 3-D MODEL CONCLUSION
INRODUCION Determnng modal parameters of a tme-varyng system or nonlnear system s generally very useful when assessng structural damage n real applcatons. o dentfy the dynamc characterstcs of nonlnear structures from the ambent vbraton, free vbraton, and earthquake response data, ths study develops a unfed procedure by extendng, wth some modfcaton, the ampltude-dependent AutoRegressve wth exogenous nput (ARX) model.
MEHODOLOGY(1/6) Mx Cx Kx f M : mass matrx C : dampng matrx K : stffness matrx x : dsplacement vectors f : force vectors C and K are functons of ampltude of responses
MEHODOLOGY(2/6) he equatons of moton n a dscrete form are equvalent to : y t Φ A y t Θ A f t a () t I n 1 0 J y t : vectors of measured responses at tme tt f t : vectors of nput forces at tme tt 1/ t : samplng rate of the measurement Φ ( A) : matrx of coeffcent functons to be determned n the model of AR part Θ ( A) : matrx of coeffcent functons to be determned n the model of X part an( t) : vector representng the resdual error hs s known as ampltude dependent ARX model.
MEHODOLOGY(3/6) Each coeffcent functon n Φ ( A) and Θ ( A) s lnearly expanded by the shapes functons. Shapes functons constructed by a set of bass functons, whch are polynomals heren, through a movng least squares approach. kl ( A) φ( A) kl kl ( A) φ( A) kl 1 ( ) ( ) ( ) ( ) φ A p A A Q A l ( A) W( t, t ) p( A) p ( A) l1 Q( A) [ q, q,, q ] q p 1 2 Wtt (, ) p( A) l l l 2 N (1,,,, ) AA l l l A l
MEHODOLOGY(4/6) A least squares approach s appled to determne and by mnmzng N E ( an( tn)) an( tn) n 1 1 C ( V V) V Y C Φ1 Φ2 ΦI Θ0 Θ 1 ΘJ Y yt1 yt2 ytn Γ Γ Γ Π Π Π Γ Γ Γ Π Π Π 2 2 V Γ Γ Γ Π Π Π Φ 1, t 2, t I, t 0, t 1, t J, t 1 1 1 1 1 1 1, t 2, t I, t 0, t 1, t J, t n n 2 2 2 2 1, t 2, t I, t 0, t 1, t J, t N N N N N N kl, n n n1 n2 nn n1 n1 nn 11 12 1 11 12 1 ' 21 22 2 21 22 2 ' ; Θ Γ y t φ() t ; Π f t φ() t t, t, kl '
MEHODOLOGY(5/6) he modal parameters of a structure are determned from Φ ( A) wth =1,2,, I. A matrx G s constructed from Φ ( A) as 0 I 0 0 0 0 0 I 0 0 G 0 0 0 I 0 0 0 0 0 I I I1 I2 2 1 he dynamcal characterstcs of the structure are determned from the egenvalues G and egenvectors of Huang, 1999.
MEHODOLOGY(6/6) = a b be the egenvalues of G. k k k k = 1 2 I be the correspondng egenvector.,, 1,2,, I correspond to a mode shape of the structural system. he frequency and dampng rato of the system are computed by k k k 2 2 k k k where, 1 tan 1 b k k t ak 1 ln a b 2t 2 2 k k k
NUMERICAL VERIFICAION(1/3) Numercal smulaton responses of a Duffng oscllator system were processed to demonstrate the accuracy and effectveness of the proposed approach. he Duffng oscllator system was show as follow : 2 2 2 x t x t x t x t f t 0.05 2 he ampltude-frequency relatonshp: 2 3 2 n A 1 A 4 he ampltude-dampng rato relatonshp: 2 3 2 3 2 n A 1 A 1 A 4 4
NUMERICAL VERIFICAION(2/3) he Runge-Kutta method wth tme ncrement eqaul to 0.004 second was appled to determne the dynamc responses of ths Duffng oscllator system subected to base exctaton.
NUMERICAL VERIFICAION(3/3) he dentfed nstantaneous natural frequences s demonstrate the excellent agreement between the dentfed results and the true values. he dfferences between the dentfed natural frequences andthetrue ones are less than 1%, whle the dentfed modal dampng ratos dffer from the true values by less than 11%. he dentfed results from ADARX model exacter than one estmated from VARX model.
APPLICAION OF 3-D MODEL(1/5) he three-story 3-D steel frame under consderaton was 6m long, 6m wde and 9m hgh. Assume plates were fxed on each floor, such that the total mass of steel frame was approxmately 43.407ton. hs orgnal frame was defned as "frame A" he "frame B" was sat two nonlnear elements between the celng and the floor at the frst story. he relatonshp between the dsplacement and the force of these nonlnear elements s b-lnear. 50 KN d 0.01 FNL 5000d KN d 0.01 50 KN d 0.01
APPLICAION OF 3-D MODEL(2/5) frame A frame B he "frame A" was subected to base exctatons of the whte nose wth the 200gal of base exctaton level. he "frame B" was subected to base exctatons of the whte nose and 1999 Ch-Ch earthquake wth 200gal and 500gal of base exctaton levels, respectvely.
APPLICAION OF 3-D MODEL(3/5) Both the acceleraton responses of base and the dsplacements of all floors at the t=5-35 seconds were used n evaluatng modal parameters for these frame. he base exctatons and the dsplacement responses of each floor n the mnor axs drecton of the frame, subected to 500gal of the 1999 Ch-Ch earthquake.
APPLICAION OF 3-D MODEL(4/5) Hysteretc loops between the dsplacement and axs force of nonlnear element. he slopes of the curve for t>3.844 seconds are smaller than those for t<3.844 seconds. he largest hysteretc loop at t=8.44 second.
APPLICAION OF 3-D MODEL(5/5) he natural frequences of the frst mode decreased dramatcally around, whch lkely ndcates the nonlnear behavor was shown n "frame B". he mnmum natural frequency for the frst mode occur at t=8.5 seconds around, whch s about the moment when maxmum deformaton of nonlnear lnk.
CONCLUSION he nstantaneous natural frequences of structural are accurately determned from the coeffcents functons of ampltude dependent ARX models. AD-ARX models are expanded by MLS shape functons of ampltude whch constructed through Hlbert transform and establshed from the acceleraton responses of base and the dsplacements responses of structural. he proposed approach was demonstrated on a nonlnear Duffng equaton under earthquake exctaton. he proposed approach was valdated by successfully dentfyng modal parameters by processng numercally smulated responses. he success of the proposed approach when appled to the nonlnear responses demonstrates ts practcal applcablty to a 3-D symmetrcal buldng.