Acta Materiae Compositae Sinica Vol123 No16 December 2006

Acta Materiae Compositae Sinica Vol123 No16 December 2006 : 1000 3851 (2006) 06 0174 05 23 6 12 2006, 3,, (, 210016) :,,, : : O322 : A Identif ication method f or dynamic characteristic parameter of f iber2reinf orced composite H E Huan, CH EN Guoping 3, WEI Yong, ZHAN G Jiabin ( Institute of Vibration Engineering Research, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China) Abstract : The equivalent characteristic constitutive equations were founded in this paper. Based on the classic com2 po sites laminates theory, the stiff ness matrix and the equivalent damping matrix of the laminates which can be de2 scribed by elastic constant s and damping constant s were determined. The explicit relationship between elastic con2 stants and the mode was determined, as well as the explicit relationship between damping constants and the mode. The equivalent damping matrix of any laminates can be easily calculated out when the equivalent damping constant s in the constitutive equations are identified. The validity of this method is demonstrated through numerical examples. Keywords : fiber 2 reinforced composite materials laminates elastic constant s equivalent damping mode loss factor,, [123 ],,,,,, ( ),,,,, : 2006 01 04 : 2006 05 12 :,,, E2mail : gpchen @nuaa. edu. cn

, : 175 [4 ] [5,6 ] [ 7,8 ], ({ { } +w w ) dv = 0 (3), / [9212 ] w = [ N ]{ q}, [ N ], { q} { q}, : [ M ]{ q} + [ C]{ gq} + [ K]{ q} = 0 (4), Adams2Bacon, [ M ] [ K] [ C] Adams2Bacon [13218 ], [ Me ],,,,,,,, 1,, [ 4 ],,,, 2 : { ij } = [ Q]{ ij } + [ ]{ g ij } (1) : [ Q] [ ] [ T ], { } k = [ gq ] k{ } + [g] k{ g } (2) : { } k = [ x y xy ] T k { } = [ x y xy ] T k [ Q] k [ [ T ] - 1 k ] T [g] k = [ T ] - 1 k [ ] k [ [ T ] - 1 k ] T.,,, z = 0,, Galerkin [ Ke ][ Ce ] : h : [ Me ] = h [ N ] T [ N ]ds, κ [ Ke ] = κ [ B ] T [ D ] k [ B ]ds, [ Ce ] = κ [ B ] T [] k [ B ]ds. [ D ] n 3 [ gq ] k ( z 3 k - z 3 k- 1 ) [] k = 1 n 3 [g] k ( z 3 k - z 3 k- 1 ). (5),,, [ Ke ][ Q]: [ Ke ] = K( Q11, Q12, Q22, Q66 ) (6),, [ Ce ] [] [ Ce ] = C( 11, 12, 22, 66 ) (7),, k [ Q][ [ T ] - 1 k ] T, [g] k = [ T ] - 1 k [][ [ T ] - 1 k ] T. (8) 2 : [ Q] = [ Q1 ] + [ Q2 ] + [ Q3 ] + [ Q4 ] (9) [ Q1 ] = [ Q11 ][ I1 ] [ Q2 ] = [ Q12 ][ I2 ] [ Q3 ] = [ Q22 ][ I3 ] [ Q4 ] = [ Q66 ][ I4 ] [ I1 ] = 1 0 0 [ I2 ] = 0 1 0 1 0 0

176 [ I3 ] = 0 1 0 [ I4 ] = (8) 1 0 0 1 4 k [ Qi ] i = 1 [ [ T ] - 1 k ] T (10) F e i, i = 1, 2, 3, 4,, [ Ke ] = [ F e 1 ]Q11 + [ F e 2 ]Q12 + [ F e i ] = 1 n ( 3 z 3 k - z 3 k- 1 ) [ F e 3 ]Q22 + [ F e 4 ]Q66 (11) [ B ] T ( [ T ] - 1 k [ Ii ][ [ T ] - 1 k ] T ) [ B ]ds (12) κ (4),,, [ M ]{ q} + ( [ F1 ]Q11 + [ F2 ]Q12 + [ F3 ]Q22 + [ F4 ]Q66 ) { q} = 0 (13), Fi, i = 1, 2, 3, 4,, F e i { q} = { <} e it (14) { <}, (14) (13), ( [ F1 ]Q11 + [ F2 ]Q12 + [ F3 ]Q22 + [ F4 ]Q66 ) { <} = 2 [ M ]{ <} (15) i { <i }, i, { <i} T ( [ F1 ]Q11 + [ F2 ]Q12 + [ F3 ]Q22 + [ F4 ]Q66 ) { <i} = 2 { <i} T [ M ]{ <i} (16) aij bi = { <i} T [ Fj ]{ <i} = 2 { <i} T [ M ]{ <i}. (17), [ aij ] m 4 { Q11 Q12 Q22 Q66 } T = [ bi ] m 1 (18) 4,, m 4 4 3 [ C] = [ F1 ] 11 + [ F2 ] 12 + [ F3 ] 22 + [ F4 ] 66 (19) : Rayleigh V = 1 2 { q} T [ K]{ q} T = 1 2 2 { q} T [ M ]{ q} J = V T = J 1 + J 2 + J 3 + J 4 { q} T [ F1 ]{ q} J 1 = Q11 2 { q} T [ M ]{ q} J 2 = { q} T [ F2 ]{ q} Q12 2 { q} T [ M ]{ q} { q} T [ F3 ]{ q} J 3 = Q22 2 { q} T [ M ]{ q} J 4 = { q} T [ F4 ]{ q} Q66 2 { q} T [ M ]{ q}. [19 ] - 1 = 11 J 1 + 12 J 2 + 22 J 3 + 66 J 4 Q11 Q12 Q22 Q66 (20) = { q} T ( [ F1 ] 11 + [ F2 ] 12 + [ F3 ] 22 + [ F4 ] 66 ) { q} 2 { q} T [ M ]{ q} = { q} T [ C]{ q} 2 { q} T [ M ]{ q} (21), [ Kr ] [ M r ] [ Cr ], r r, [ Kr ] [ M r ],, [ Cr ],, [ Cr ] [ Kr ] [ M r ][ Cr ]r k r m r c r, (19) [ Cr ] cr = [ arj ]1 4 { 11 12 22 66 } T (22) r, r - 1 r, (21), : - 1 r = cr 2 r m r = cr k r (23) (22) cr, : [ arj ] m 4 { 11 12 22 66 } T = { kr - 1 r } m 1 (24)

, : 177 1 2 2, n, 2 n 2 n 2 n,,, 2 n 4 1 [ 19 ] 3 1 Table 1 Characteristic of the laminates Laminate Ply angle Size/ mm 6 layers, 0 plies 126 108 01 92 0 / 90 / 0 / 0 / 90 / 0 178 176 01 95 + 45 / - 45 / - 45 / 0 / + 45 / + 45 / - 45 230 230 11 07 Q11 = 841 84 GPa, Q22 = 61 48 GPa, Q12 = 11 62 GPa, Q66 = 3136 GPa, : 11 = 16112 MPa s, 22 = 6513 MPa s, 66 = 4010 MPa s, 12 = 1519 MPa s., 2 [19 ] ( = 1/ - 1 ),,,,,,,, 3 Q11 Q22 Q66 11 22 66, Q12 12 Q12 Q12 12,,,,, Q66 66, Q66 66,,,, 2 Table 2 Predicted and experimental values of and for laminates Laminate Mode Experimental / Hz Predicted / Hz Error/ % Experimental Predicted Error/ % 1 113 111. 91 0. 96 84 85. 20 1. 42 2 176 176. 00 0. 00 100 100. 27 0. 27 3 288 287. 88 0. 04 76 90. 50 19. 07 4 468 463. 26 1. 01 515 509. 39 1. 09 1 53 49. 42 6. 75 90 86. 01 4. 43 2 131 133. 00 1. 53 340 298. 55 12. 19 3 167 167. 98 0. 004 146 159. 75 9. 42 4 200 202. 70 1. 35 620 480. 92 22. 43 1 55 54. 27 1. 33 90 90. 35 0. 39 2 81 86. 08 6. 27 600 413. 90 31. 02 3 102 116. 00 13. 70 300 300. 60 0. 20 4 173 181. 54 4. 94 270 274. 38 1. 62

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