23 2 Vo.23 No.2 26 2 Dec. 26 ENGINEEING MECHANIC 47-475(26)2-47-6 * 2 (. 4372. 325) (FEM) FEM O34 A DAMAGE IDENIFICAION OF UCUE UING OCHAIC MODE * HUANG Bn, HI Wen-ha 2 (. choo of Cv Engneerng and Archtecture, Wuhan Unversty of echnoogy, Wuhan, Hube 437, Chna 2. choo of Cv Engneerng, Wenzhou Unversty, Wenzhou, Zhejang 325, Chna) Abstract: Based on recursve stochastc fnte eeent ethod a rando daage dentfcaton ethod for structures s deveoped. Contro equatons of rando daage ndex are set up after rando daage ndex s defned. tructura uncertantes and easureent noses are both consdered n the contro equatons. tatstcs of rando daage ndex are obtaned after the contro equatons are soved utzng recursve stochastc fnte eeent ethod. he resuts show that structura daages can be dentfed whe ode errors and easureent noses are both consdered. It s aso found that the ethod presented has the sae effectveness as Monte-Caro suaton ethod for dentfyng daages of rando structures n wde range of rando fuctuaton. Key words: rando structures; daage dentfcaton; stochastc ode; recursve stochastc fnte eeent ethod; nonorthogona poynoa expansons Baruch []..aw [2] W.X.en [3] Doebng [4] tubbs [5] 25-4-2825-9-23 (5286)(2532) * (968)(E-a: bwhuang@263.net) (979)(E-a: swank@sna.co.cn)
48 K tubbs [6] [7] Papadopouos [8] [8] Yong Xa [9] [8] [9] FEM N ( K λm) Φ () K M N N λ Φ ( K λm) Φ (2) K M N N λ Φ ()(2) 2 K λ Φ K K + K (3) M M (4) λ λ+ λ (5) Φ Φ+ Φ (6) K K α K (7) α K (3)~(7)(2)() λ M e Φ Φ α Φ K Φ (8) λ Φ α 3 ξ () K λ Φ (8) λξ M ( ) Φξ ( ) Φ α( ξ) Φξ ( ) K Φ (9) λ α ξ (8) M λ( ξη, ) Φξ ( ) Φ( η) α( ξη, ) Φξ ( ) K Φ ( η) λ 4 e α ξ η e () 4.
49 λ Φ λ α [] n n ξ ξξ j j j n j ξξ jξkφjk j k Φ Φ + Φ + Φ + λ λ λ + ξ λ + ξξ λ + n n j j j n j ξξ jξkλjk j k n n j j j n j ξξ jξkαjk j k ( N) () ( N) (2) α α + ξα + ξξ α + ( ) (3) ()~(3)(9) ξ α Φ Φ λ λ Φ Φ e ( ) K ( )( ) M ( N) (4) X X (5) N N e ( ) K ( λ λ )( ) Φ Φ X α Φ MΦ (6) N Moore-Penrose pseu- donverse + ξ e [ α( Φ ) + α( Φ ) ] K Φ [( λ λ )( ) λ ( ) ] M Φ Φ Φ (7) ( N,, n) X + X (8) e ( Φ ) K Φ X α λ λ λ [( )( Φ ) )( Φ ) ] MΦ (9) ξ ξ j [ α( Φ j ) + ( α ( Φ j ) + e α j ( ) )( δj /2) αj ( ) ] K [( λ λ )( Φj ) ( λ ( Φj ) + λj ( ) )( δj /2) λj ( ) ] M Φ + Φ Φ Φ Φ Φ ( N,, n, j ) X + ( X + X )( δ /2) + X 2 2 j j j j j 2 e j ( Φj ) K Φ X j αj 2 [( λ λ )( Φj ) ( λ ( Φj ) + λj δj λj ( Φ ) )( /2) ( Φ ) ] M Φ (2) (2) (22) 4.2 λ Φ η λ λ + ηλ (23) Φ Φ +ηφ (24) λ Φ λ Φ ηλ ηφ λ Φ (5)~(6) λ + ηλ λ + λ (25) Φ + ηφ Φ+ Φ (26) Φ n n ξ ξξ j j j n j ξξ jξkφjk j k Φ Φ + Φ + Φ + ( N) (27) ξ n η λ λ + ξ n λ Φ Φ +ξ n Φ λ
5 λ λ + ξ λ λ + ξ λ + n ( n ) n n j ξξ jλj ξξ jξkλjk j j k + ( N) (28) α n n j j j n j ξξξα j k jk j k α α + ξα + ξξ α + ( ) (29) (27)~(29)() ξ α Φ Φ λ λ Φ Φ e ( ) K ( )( ) M ( N) (3) N X N X (3) e ( ) K X ( )( ) λ λ Φ MΦ Φ Φ α ξ n e [ α( Φ ) + α( Φ ) ] K Φ M [( λ λ )( Φ ) λ ( Φ ) ] Φ (32) X + X (33) ( Φ ) K e Φ X α [( )( ) ( ) ] λ λ Φ λ Φ MΦ n ( Φ ) MΦ ( )( Φ ) MΦ e e α K + αn K λ + λ λ [ ( Φ ) Φ ( Φ ) Φ ] (34) * * Xn + X (35) * e ( Φ ) K Φ X n αn * λ ( Φ ) Φ + ( λ λ )( Φ ) Φ M M ξ ξ j n, j [ α( Φ j ) + ( α ( Φ j ) + α j ( Φ ) )( δj /2) + α e j ( Φ ) ] K Φ λ λ Φj λ Φj + j ( ) )( j /2) j ( ) ] M [( )( ) ( ( ) λ Φ δ λ Φ Φ (36) X + ( X + X )( δ /2) + X 2 2 j j j j j 2 ( ) e j Φj K Φ X j αj 2 [( λ λ )( Φj ) λ Φj λj Φ δj λj ( Φ ) ] M Φ ( ( ) + ( ) )( /2) n, j λ ( Φ ) M Φ + (( λ λ )( Φ ) j j λ j ( Φ ) ) M Φ e [( αn( Φj ) αnj( Φ ) ) K Φ ( α ( ) ( ) ) e j + α j K ] + + (37) Φ Φ Φ (38) * * 2* X + X + X + X (39) j n nj j j * ( ) e Φ K Φ X nj αnj 2* λ ( Φj ) MΦ (( λ λ )( Φj ) λ j ( Φ ) ) M Φ + n, j n e ( Φ ) MΦ [ αnn ( Φ ) K Φ e n( ) K ] λ + α Φ Φ (4) * 2** Xn + Xnn (4) X nn αnn 2** ( ) λ Φ MΦ α 5
5 [8] (PDQ-G) (PDQ-)[9] k % (PDF) k % µ µ P ( Ω x k % < ) (42) Ω k % µ [9] 95%( ) 8 2 5 3 5 ~3 α α 2 α 3 2 Fg.2 A three DOF sprng-ass syste (a) 3 δ. FEM abe Daage cases K K 2 K 3-5% -% -5% δ. k k % P d Fg. Probabty densty functon of k and k % and probabty of daage exstence P d k (probabty of daage exstence PDE) Pd P( Ω xk < ) P( < xk Ω) (43) P d [9] 6 ( 2) N K 2 K 2 5 K 3 /% 3 K K 2 K 3 Fg.3 Mean vaues of daage ndex of K K 2 and K 3 2 α abe 2 Mean vaues of α and coeffcents of varaton of sprng stffness.5..2.25.3 FEM -.4999 -.4999 -.4975 -.494 -.4876 MC -.59 -.5382 -.5765 -.5956 -.547 3 α abe 3 Mean square devatons of α and coeffcents of varaton of sprng stffness.5..2.25.3 FEM.4994.9957.962.24295.28875 MC.4996.998.9962.24953.29943
52 4 α 3 abe 4 Mean vaues of α 3 and coeffcents of varaton of sprng stffness 5%. 4 7.5..2.25.3 FEM -.5 -.5 -.55 -.5 -.524 MC -.529 -.557 -.55 -.543 -.572 5 α 3 abe 5 Mean square devatons of α 3 and coeffcents of varaton of sprng stffness /% δ..5..2.25.3 FEM.5.5.242.2585.355 MC.54.8.26.252.323 6 (%) abe 6 Probabty of daage of sprng stffness (%) K K 2 K 3 δ.5 25.9 63.8 9.2 δ. 2.3 25.9 44.2 δ.2 7.7 2.6 8.6 δ.3 6.2 9.4 2.8 2~ 5 MC 6 (b) Kr (+ 3 δ ) K K r2 (+ 3 δ ) K 2 Kr 3 ( 3 δ ) K 3 δ δ. 3σ Ks.7 Kr Ks 2.6 Kr 2 K s 3.5 3%4% Kr3 4 K K 2 K 3 Fg.4 Mean vaues of daage ndex of K K 2 and K 3 7 (%) abe 7 Probabty of daage of sprng stffness (%) K K 2 K 3 29. 66.9. 3 4 7 δ. 7 FEM FEM FEM ( 8 )
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