Acta Materiae Compositae Sinica Vol123 No16 December 2006

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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

1 Acta Materiae Compositae Sinica Vol123 No16 December 2006 : (2006) , 3,, (, ) :,,, : : 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 , 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 ],,,,,, ( ),,,,, : : :,,, E2mail : edu. cn

2 , : 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 ] = [ I2 ] =

3 176 [ I3 ] = [ I4 ] = (8) 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 [ C] = [ F1 ] 11 + [ F2 ] 12 + [ F3 ] 22 + [ F4 ] 66 (19) : Rayleigh V = 1 2 { q} T [ K]{ q} T = { 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 J J 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 { } T (22) r, r - 1 r, (21), : - 1 r = cr 2 r m r = cr k r (23) (22) cr, : [ arj ] m 4 { } T = { kr - 1 r } m 1 (24)

4 , : , 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 / 90 / 0 / 0 / 90 / / - 45 / - 45 / 0 / + 45 / + 45 / Q11 = GPa, Q22 = GPa, Q12 = GPa, Q66 = 3136 GPa, : 11 = MPa s, 22 = 6513 MPa s, 66 = 4010 MPa s, 12 = 1519 MPa s., 2 [19 ] ( = 1/ - 1 ),,,,,,,, 3 Q11 Q22 Q , 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/ %

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