Maximization of Fundamental Frequencies of Axially Compressed Laminated Curved Panels against Fiber Orientation

Transcription

1 Cpyright 2012 Tech Science Press CMC, vl.28, n.3, pp , 2012 Maximizatin f Fundamental Frequencies f Axially Cmpressed Laminated Curved Panels against Fiber Orientatin Hsuan-Teh Hu 1 and Jun-Ming Chen 2 Abstract: Free vibratin analyses f laminated curved panels subjected t axial cmpressive frces are carried ut by emplying the Abaqus finite element prgram. The fundamental frequencies f these cmpsite laminated curved panels with a given material system are then maximized with respect t fiber rientatins by using the glden sectin methd. Thrugh a parametric study, the significant influences f the end cnditins, the panel aspect rati, the panel curvature and the cmpressive frce n the maximum fundamental frequencies and the assciated ptimal fiber rientatins are demnstrated and discussed. Keywrds: Maximizatin, fundamental frequencies, axially cmpressed laminated curved panels. 1 Intrductin The applicatins f fiber-cmpsite laminate materials t aerspace industrial such as spacecraft, high-speed aircraft and satellite have increased rapidly in recent years. The mst majr cmpnents f the aerspace structures are frequently made f curved panels and subjected t varius kinds f cmpressive frces. Therefre, knwledge f the dynamic characteristics f cmpsite laminated curved panels in cmpressin, such as their fundamental natural frequency, is essential. The fundamental natural frequency f cmpsite laminated curved panels highly depends n the ply rientatins (Crawley, 1979; Chandrashekhara, 1989; Qatu and Leissa, 1991; Rauf, 1994; Chun and Lam, 1995; Hu and Juang, 1997; Hu and Tsai, 1999; Hu and Ou, 2001; Hu and Tsai, 2009), end cnditins (Sharma and Darvizeh, 1987; Chandrashekhara, 1989; Chun and Lam, 1995; Hu and Juang, 1997; Hu and Tsai, 1999), gemetries (Chandrashekhara, 1989; Qatu and Leissa, 1991; Hu 1 Department f Civil Engineering and Sustainable Envirnment Research Center, Natinal Cheng Kung University, Tainan, Taiwan 701, R.O.C. 2 Department f Civil Engineering, Natinal Cheng Kung University, Tainan, Taiwan 701, R.O.C.

2 182 Cpyright 2012 Tech Science Press CMC, vl.28, n.3, pp , 2012 and Juang, 1997; Hu and Tsai, 1999; Hu and Ou, 2001; Hu and Tsai, 2009), and cmpressive frces (Dhanaraj and Palanininathan, 1990; Chen, Cheng, Chien and Dng, 2002; Nayak, My and Sheni, 2005; Chakrabarti, Tpdar and Sheikh, 2006; Hu and Tsai, 2009). Therefre, prper selectin f apprpriate laminatin t maximize the fundamental frequency f cmpsite laminated curved panels in cmpressin becmes a crucial prblem (Bert, 1991; Abrate, 1994; Rauf, 1994; Tpal and Uzman, 2006). Research n the subject f structural ptimizatin has been reprted by many investigatrs (Schmit, 1981) and has been widely emplyed t study the dynamic behavir f cmpsite structures (Abrate, 1994; Hu, and H, 1996; Hu and Juang, 1997; Hu and Tsai, 1999; Hu and Ou, 2001; Narita, 2003; Tpal and Uzman, 2006; Hu and Wang, 2007; Hu and Tsai, 2009). Amng varius ptimizatin schemes, the glden sectin methd is a simple technique and can be easily prgrammed fr slutin n the cmputer (Vanderplaats, 1984; Haftka, Gürdal and Kamat, 1990). In this investigatin, maximizatin f the fundamental natural frequency f cmpsite laminated curved panels in cmpressin with respect t fiber rientatins is perfrmed by using the glden sectin methd. The fundamental frequencies f cmpsite laminated curved panels are calculated by using the Abaqus finite element prgram (Abaqus, Inc., 2010). In the paper, the cnstitutive equatins fr fiber-cmpsite lamina, vibratin analysis and glden sectin methd are briefly reviewed. The influences f the end cnditins, the panel aspect rati, the panel curvature and the cmpressive frce n the maximum fundamental natural frequency and the assciated ptimal fiber rientatins f the laminated curved panels are presented and imprtant cnclusins btained frm the study are given. 2 Cnstitutive matrix fr fiber-cmpsite laminae In the finite element analysis, the laminated cylindrical shells are mdeled by eightnde isparametric shell elements with six degrees f freedm per nde (three displacements and three rtatins). The reduced integratin rule tgether with hurglass stiffness cntrl is emplyed t frmulate the element stiffness matrix (Abaqus, Inc., 2010). During the analysis, the cnstitutive matrices f cmpsite materials at element integratin pints must be calculated befre the stiffness matrices are assembled frm element level t glbal level. Fr fiber-cmpsite laminate materials, each lamina can be cnsidered as an rthtrpic layer. The stress-strain relatins fr a lamina in the material crdinate (1,2,3) (Fig. 1) at an element integratin pint can be written as {σ } = [Q 1]{ε }, {τ } = [Q 2]{γ } (1)

3 Maximizatin f Fundamental Frequencies 183 Figure 1: Material, element and structure crdinates f fiber-cmpsite laminated curved panel [Q 1] = [ ] [Q α1 G 2] = α 2 G G 12 E 11 ν 12 E 22 1 ν 12 ν 21 1 ν 12 ν 21 0 ν 21 E 11 E 22 1 ν 12 ν 21 1 ν 12 ν 21 0 (2) where {σ } = {σ 1,σ 2,τ 12 } T, {τ } = {τ 13,τ 23 } T, {ε } = {ε 1,ε 2,γ 12 } T, {γ } = {γ 13, γ 23 } T. The α 1 and α 2 in Eq. (2) are shear crrectin factrs, which are calculated in Abaqus by assuming that the transverse shear energy thrugh the thickness f laminate is equal t that in unidirectinal bending (Whitney, 1973; Abaqus, Inc., 2010). The cnstitutive equatins fr the lamina in the element crdinate (x,y,z) then becme {σ} = [Q 1 ]{ε}, [Q 1 ] = [T 1 ] T [Q 1][T 1 ] (3) {τ} = [Q 2 ]{γ}, [Q 2 ] = [T 2 ] T [Q 2][T 2 ] (4)

4 184 Cpyright 2012 Tech Science Press CMC, vl.28, n.3, pp , 2012 cs 2 θ sin 2 θ sinθ csθ [ ] [T 1 ] = sin 2 csθ sinθ θ cs 2 θ sinθ csθ, [T 2 ] = 2sinθ csθ 2sinθ csθ cs 2 θ sin 2 sinθ csθ θ (5) where {σ} = {σ x,σ y,τ xy } T, {τ} = {τ xz,τ yz } T, {ε} = {ε x,ε y,γ xy } T, {γ} = {γ xz,γ yz } T, and the fiber rientatin θ is measured cunterclckwise frm the element lcal x-axis t the material 1-axis. Let {ε} = {ε x,ε y,ε xy } T be the in-plane strains at the mid-surface f the laminate sectin, {κ} = {κ x,κ y,κ xy } T the curvatures, and h the ttal thickness f the sectin. If there are n layers in the laminate sectin, the stress resultants,{n} = {N x,n y,n xy } T, {M} = {M x,m y,m xy } T and {V } = {V x,v y } T, can be defined as {N} h/2 {σ} {M} = z{σ} h/2 dz = {V } {τ} 1 n (z jt z jb )[Q 1 ] 2 (z2 jt z2 jb )[Q 1] [0] {ε } 1 2 (z2 jt z2 jb )[Q 1 1] 3 (z3 jt z3 jb )[Q 1] [0] {κ} j=1 [0] T [0] T (6) (z jt z jb )[Q 2 ] {γ} The z jt and z jb are the distance frm the mid-surface f the sectin t the tp and the bttm f the j-th layer respectively. The [0] is a 3 by 2 matrix with all the cefficients equal t zer. 3 Vibratin analysis Fr the free vibratin analysis f an undamped structure, the equatin f mtin f the structure can be written in the fllwing frm (Ck, Malkus, Plesha and Witt, 2002): [M]{ D} + [K]{D} = {0} (7) where {D} is a vectr fr the unrestrained ndal degrees f freedms, { D} an acceleratin vectr, [M] the mass matrix f the structure, [K] the stiffness matrix f the structure, and {0} a zer vectr. Since {D} underges harmnic mtin, we can express {D} = { D}sinωt; { D} = ω 2 { D}sinωt (8) where { D} vectr cntains the amplitudes f {D} vectr. Then Eq. (7) can be written in an eigenvalue expressin as ([K] ω 2 [M]){ D} = {0} (9)

5 Maximizatin f Fundamental Frequencies 185 When a laminated curved panel is subjected t cmpressive frce, initial stresses are generated in the panel. Cnsequently, the stiffness matrix [K] in Eq. (9) can be separated int tw matrices as [K] = [K L ] + [K σ ] (10) The [K L ] is the traditinal linear stiffness matrix and [K σ ] is a gemetric stress stiffness matrix due t the initial stresses. Then Eq. (9) becmes ([K L ] + [K σ ] ω 2 [M]){ D} = {0} (11) The preceding equatin is an eigenvalue expressin. If { D} is nt a zer vectr, we must have [K L ] + [K σ ] ω 2 [M] = 0 (12) In Abaqus, a subspace iteratin prcedure (Abaqus, Inc., 2010) is used t slve fr the natural frequency ω, and the eigenvectrs (r vibratin mdes) { D}. The btained smallest natural frequency (fundamental frequency) is then the bjective functin fr maximizatin. 4 Glden sectin methd We begin by presenting the glden sectin methd (Vanderplaats, 1984; Haftka, Gürdal and Kamat, 1990) fr determining the minimum f the unimdal functin F, which is a functin f the independent variable X. It is assumed that lwer bund X L and upper bund X U n X are knwn and the minimum can be bracketed (Fig. 2). In additin, we assume that the functin has been evaluated at bth bunds and the crrespnding values are F L and F U. Nw we can pick up tw intermediate pints X 1 and X 2 such that X 1 < X 2 and evaluate the functin at these tw pints t prvide F 1 and F 2. Because F 1 is greater than F 2, nw X 1 frms a new lwer bund and we have a new set f bunds, X 1 and X U. We can nw select an additinal pint, X 3, fr which we evaluate F 3. It is clear that F 3 is greater than F 2, s X 3 replace X U as the new upper bund. Repeating this prcess, we can narrw the bunds t whatever tlerance is desired. T determine the methd fr chsing the interir pints X 1, X 2, X 3,..., we pick the values f X 1 and X 2 t be symmetric abut the center f the interval and satisfying the fllwing expressins: X U X 2 = X 1 X L (13)

6 186 Cpyright 2012 Tech Science Press CMC, vl.28, n.3, pp , 2012 Figure 2: The glden sectin methd X 1 X L X U X L = X 2 X 1 X U X 1 (14) Let τ be a number between 0 and 1. We can define the interir pints X 1 and X 2 t be X 1 = (1 τ)x L + τx U X 2 = τx L + (1 τ)x U (15a) (15b) Substituting Eqs. (15a) and (15b) int Eq. (14), we btain τ 2 3τ + 1 = 0 (16) Slving the abve equatin, we btain τ = The rati (1 τ)/τ = is the famus glden sectin number. Fr a prblem invlving the estimatin f the maximum f a ne-variable functin F, we need nly minimize the negative f the functin, that is, minimize -F.

7 Maximizatin f Fundamental Frequencies Numerical analysis The accuracy f the eight-nde shell element in Abaqus prgram fr frequency analysis has been verified by the authrs (Hu and Tsai, 2009; Chen, 2009) and gd agreements are btained between the numerical results and the analytical slutin r experimental data. Hence, it is cnfirmed that the accuracy f the shell element in Abaqus prgram is gd enugh t analyze the vibratin behavir f laminated curved panels. 5.1 Simply supprted laminated curved panels with varius curvatures and axial cmpressive frces In this sectin laminated curved panels subjected t axial cmpressive frce N are cnsidered (Fig. 3a). The curved panels are simply supprted (dented by S as shwn in Fig. 3b) at the fur edges. The simply supprted bundary cnditin prevents ut f plane displacement w, but allws in-plane mvements u and v. The width b f the curved panel is equal t 10 cm and the circular angle φ f the curved panel varies frm 5 t 150. The laminate layup f the curved panel is [±θ/90/0] 2s and the thickness f each ply is mm. T study the influence f axial cmpressive frce n the results f ptimizatin, N = 0, 0.2N cr, 0.4N cr, 0.6N cr, and 0.8N cr, are selected fr analysis, where N cr is the linearized critical buckling lad f the laminated curved panel. The lamina cnsists f Graphite/Epxy and material cnstitutive prperties are taken frm Crawley (1979), which are E 11 = 128 GPa,E 22 = 11 GPa, G 23 = 1.53 GPa, G 12 = G 13 = 4.48 GPa, ν 12 = 0.25, and ρ = 1500 kg/m 3. In the analysis, n symmetry simplificatins are made fr thse laminated curved panels. T find the ptimal fiber angle θ and the assciated ptimal fundamental frequency ω, we can express the ptimizatin prblem as: Maximize: ω(θ) Subjected t: 0 θ 90 (17a) (17b) Befre the glden sectin methd is carried ut, the fundamental frequency ω f the laminated curved panel is calculated by emplying the Abaqus finite element prgram fr every 10 increment in θ angle t lcate the maximum pint apprximately. Then prper upper and lwer bunds are selected s that the fundamental frequency is a unimdal functin within the search regin. Finally, the glden sectin methd is carried ut t find the maximum. The ptimizatin prcess is terminated when an abslute tlerance (the difference f the tw intermediate pints between the upper bund and the lwer bund) θ 0.5 is reached.

8 188 Cpyright 2012 Tech Science Press CMC, vl.28, n.3, pp , 2012 Figure 3: Gemetry and bundary cnditins f laminated curved panels Figure 4 shws the ptimal fiber angle θ and the assciated ptimal fundamental frequency ω versus the circular angle φ fr [±θ/90/0] 2s laminated curved panel with aspect rati a/b = 1 and with simply supprted edge cnditins. Frm Fig. 4a we can see that the axial cmpressive frce N has little influence n the ptimal fiber angle θ f the laminated curved panels except the case φ = 150 and N = 0.8N cr. The ptimal fiber angle θ f thse panels usually varies between 40 and 70 and increases with the increasing f the circular angle φ. When φ = 5, the

9 Maximizatin f Fundamental Frequencies Optimal θ (degrees) N=0 N=0.2Ncr N=0.4Ncr N=0.6Ncr N=0.8Ncr φ (degrees) (a) Optimal fiber angle θ vs. circular angle φ Optimal ω (khz) N=0 1.0 N=0.2Ncr N=0.4Ncr 0.5 N=0.6Ncr N=0.8Ncr φ (degrees) (b) Optimal fundamental frequency ω vs. circular angle φ Figure 4: Effect f curvature and in-plane cmpressive frce n ptimal fiber angle and ptimal fundamental frequency f [±θ/90/0] 2s laminated curved panels with fur simply supprted edges (b = 10 cm, a/b = 1)

10 190 Cpyright 2012 Tech Science Press CMC, vl.28, n.3, pp , 2012 φ N= 0 N = 0.4N cr N = 0.8N cr 5 θ= 46 ω =1172 Hz θ= 46 ω =908 Hz θ= 46 ω =523 Hz 30 θ= 59 ω =1862 Hz θ= 61 ω =1503 Hz θ= 63.5 ω =932 Hz 90 θ= 66.5 ω =3046 Hz θ= 67.5 ω =2514 Hz θ= 70 ω =1599 Hz 150 θ= 72 ω =3368 Hz θ= 72.5 ω =2811 Hz ω =1872 Hz Figure 5: Fundamental vibratin mdes f [±θ/90/0] 2s laminated curved panels with fur simply supprted edges and under ptimal fiber angles (b = 10 cm, a/b = 1)

11 Maximizatin f Fundamental Frequencies 191 ptimal fiber angle θ f the laminated curved panels are all equal t 46 and seem t be independent n the axial cmpressive frces. Figure 4b shws that the ptimal fundamental frequency ω increases with the increasing f the circular angle φ. Hwever, it decreases with the increasing f the axial cmpressive frce. Figure 5 shws that the fundamental vibratin mdes f thse curved panels with a/b = 1 under ptimal fiber angle θ. It can be seen that thse mdes are very similar and all f them have a half sine wave in bth axial directin (x-axis in Fig. 3) and circumferential directin (y-axis in Fig. 3). Figure 6 shws the ptimal fiber angle θ and the assciated ptimal fundamental frequency ω versus the circular angle φ fr [±θ/90/0] 2s laminated curved panel with aspect rati a/b = 2 and with simply supprted edge cnditins. Frm Fig. 6a we can see that when the axial frce N 0.6N cr and θ < 60 (especially θ < 30 ), the axial frce has significant influence n the ptimal fiber angle. Fr the panels with N 0.6N cr, when the circular angle φ is large, the ptimal fiber angle tends t apprach 70. Hwever, fr the panels with N = 0.8N cr and φ 30 the ptimal fiber angle tends t apprach 90 (the exceptin is the panel with φ = 60 where the ptimal fiber angle is 72.5 ). Figure 6b again shws that the ptimal fundamental frequency ω increases with the increasing f the circular angle φ and decreases with the increasing f the axial cmpressive frce N. Cmparing Fig. 6a with Fig. 4a, we can see that the aspect rati a/b has significant influence n the ptimal fiber angle f the laminated curved panel. Cmparing Fig. 6b with Fig. 4b, we can see that the ptimal fundamental frequencies f the curved panels with small aspect rati are greater than thse with large aspect rati. Figure 7 shws the fundamental vibratin mdes f thse curved panels with a/b = 2 under ptimal fiber angle θ. Cmparing Fig. 7 with Fig. 5, it is bserved when the curved panel has large aspect rati (say a/b = 2), small circular angle (say φ = 5 ) and subjected t large cmpressive frce (say N = 0.8N cr ), its vibratin mde under ptimal fiber angle may have mre half sine wave numbers in the axial directin. 5.2 Fixed laminated curved panels with varius curvatures and axial cmpressive frces In this sectin, laminated curved panels subjected t axial cmpressive frce and similar t thse in previus sectin are analyzed except that the panels are fixed (dented by F as shwn in Fig. 3c) at the fur edges. The fixed bundary cnditin als prevents ut f plane displacement w, but allws in-plane mvements u and v. The laminate layup f the laminated curved panel is still [±θ/90/0] 2s.

12 192 Cpyright 2012 Tech Science Press CMC, vl.28, n.3, pp , Optimal θ (degrees) N=0 N=0.2Ncr N=0.4Ncr N=0.6Ncr N=0.8Ncr φ (degrees) (a) Optimal fiber angle θ vs. circular angle φ Optimal ω (khz) N=0 N=0.2Ncr 0.5 N=0.4Ncr N=0.6Ncr N=0.8Ncr φ (degrees) (b) Optimal fundamental frequency ω vs. circular angle φ Figure 6: Effect f curvature and in-plane cmpressive frce n ptimal fiber angle and ptimal fundamental frequency f [±θ/90/0] 2s laminated curved panels with fur simply supprted edges (b = 10 cm, a/b = 2)

13 Maximizatin f Fundamental Frequencies 193 φ N= 0 N = 0.4N cr N = 0.8N cr 5 ω =823 Hz ω =773 Hz θ= 78.5 ω =578 Hz 30 θ= 63 ω =1192 Hz θ= 67 ω =1023 Hz ω =738 Hz 90 θ= 66.5 ω =1849 Hz θ= 68.5 ω =1578 Hz ω =1085 Hz 150 θ= 68.5 ω =2045 Hz θ= 69.5 ω =1740 Hz ω =1210 Hz Figure 7: Fundamental vibratin mdes f [±θ/90/0] 2s laminated curved panels with fur simply supprted edges and under ptimal fiber angles (b = 10 cm, a/b =2)

14 194 Cpyright 2012 Tech Science Press CMC, vl.28, n.3, pp , Optimal θ (degrees) N=0 N=0.2Ncr 50 N=0.4Ncr N=0.6Ncr N=0.8Ncr φ (degrees) (a) Optimal fiber angle θ vs. circular angle φ Optimal ω (khz) N=0 N=0.2Ncr N=0.4Ncr N=0.6Ncr N=0.8Ncr φ (degrees) (b) Optimal fundamental frequency ω vs. circular angle φ Figure 8: Effect f curvature and in-plane cmpressive frce n ptimal fiber angle and ptimal fundamental frequency f [±θ/90/0] 2s laminated curved panels with fur fixed edges (b = 10 cm, a/b = 1)

15 Maximizatin f Fundamental Frequencies 195 φ N= 0 N = 0.4N cr N = 0.8N cr 5 ω =1984 Hz ω =1731 Hz ω =1367 Hz 30 θ= 65.5 ω =2668 Hz θ= 69.5 ω =2292 Hz θ= 75 ω =1828 Hz 90 θ= 62.5 ω =3946 Hz θ= 68.5 ω =3402 Hz θ= 80.5 ω =2781 Hz 150 θ= 66.5 ω =4265 Hz θ= 71.5 ω =3723 Hz θ= 80.5 ω =3058 Hz Figure 9: Fundamental vibratin mdes f [±θ/90/0] 2s laminated curved panels with fur fixed edges and under ptimal fiber angles (b = 10 cm, a/b = 1)

16 196 Cpyright 2012 Tech Science Press CMC, vl.28, n.3, pp , Optimal θ (degrees) N=0 N=0.2Ncr 50 N=0.4Ncr N=0.6Ncr N=0.8Ncr φ (degrees) (a) Optimal fiber angle θ vs. circular angle φ 2.5 Optimal ω (khz) N=0 N=0.2Ncr N=0.4Ncr N=0.6Ncr N=0.8Ncr φ (degrees) (b) Optimal fundamental frequency ω vs. circular angle φ Figure 10: Effect f curvature and in-plane cmpressive frce n ptimal fiber angle and ptimal fundamental frequency f [±θ/90/0] 2s laminated curved panels with fur fixed edges (b = 10 cm, a/b = 2)

17 Maximizatin f Fundamental Frequencies 197 Figure 8 shws the ptimal fiber angle θ and the assciated ptimal fundamental frequency ω versus the circular angle φ fr [±θ/90/0] 2s laminated curved panel with aspect rati a/b = 1 and with fixed edge cnditins. Frm Fig. 8a we can see that the axial cmpressive frce N des have sme influence n the ptimal fiber angle θ f the laminated curved panels when the circular angle is large (say φ 30 ). The ptimal fiber angle increases with the increasing f the cmpressive frce and usually varies between 60 and 90. Similar t the panels with fur simply supprted edges (Fig. 4a), when φ = 5, the ptimal fiber angle θ f the laminated curved panels are all equal t 90 and seem t be independent n the axial cmpressive frces. Figure 8b shws that the ptimal fundamental frequency ω increases with the increasing f the circular angle φ. Hwever, it decreases with the increasing f the axial cmpressive frce. Figure 9 shws the fundamental vibratin mdes f thse curved panels with a/b = 1 and under ptimal fiber angle θ. It can be seen that thse mdes are very similar. Figure 10 shws the ptimal fiber angle θ and the assciated ptimal fundamental frequency ω versus the circular angle φ fr [±θ/90/0] 2s laminated curved panel with aspect rati a/b = 2 and with fixed edge cnditins. Frm Fig. 10a we can see that the axial cmpressive frce N des have sme influence n the ptimal fiber angle θ f the laminated curved panels. The ptimal fiber angle θ f thse panels usually varies between 60 and 90. Figure 10b shws that the ptimal fundamental frequency ω increases with the increasing f the circular angle φ and decreases with the increasing f the axial cmpressive frce. Cmparing Fig. 10 with Fig. 8, we can see that the aspect rati a/b has significant influence n the ptimal fiber angle and ptimal fundamental frequency f the curved panel. Finally, cmparing Figs. 8 and 10 with Figs. 4 and 6, we can cnclude that the bundary cnditin has significant influence n the ptimal fiber angle and the assciated ptimal fundamental frequency f the laminated curved panels. Figure 11 shws the fundamental vibratin mdes f thse curved panels with a/b = 2 and under ptimal fiber angle θ. Cmparing Fig. 11 with Fig. 9, it is bserved when the curved panel has large aspect rati (say a/b = 2), small circular angle (say φ = 5 ) and subjected t large cmpressive frce (say N 0.4N cr ), its vibratin mde under ptimal fiber angle may have mre wave numbers in the axial directin. 5.3 Simply supprted laminated curved panels with varius aspect ratis and axial cmpressive frces In this sectin, laminated curved panels subjected t axial cmpressive frce are analyzed. The width b f the curved panel is equal t 10 cm and the length a f the curved panel varies frm 5 cm t 20 cm. The panels are simply supprted at the fur edges. The laminate layup f the laminated curved panel is still [±θ/90/0] 2s.

18 198 Cpyright 2012 Tech Science Press CMC, vl.28, n.3, pp , 2012 φ N= 0 N = 0.4N cr N = 0.8N cr 5 ω =1757 Hz ω =1681 Hz θ= 76.5 ω =1187 Hz 30 θ= 77.5 ω =1883 Hz θ= 81.5 ω =1773 Hz ω =1657 Hz 90 θ= 65 ω =2329 Hz θ= 71 ω =2082 Hz θ= 78.5 ω =1813 Hz 150 θ= 64 ω =2413 Hz θ= 73 ω =2096 Hz θ= 83 ω =1775 Hz Figure 11: Fundamental vibratin mdes f [±θ/90/0] 2s laminated curved panels with fur fixed edges and under ptimal fiber angles (b = 10 cm, a/b = 2)

19 Maximizatin f Fundamental Frequencies Optimal θ (degrees) Optimal ω (khz) N=0 N=0.2Ncr 50 N=0.4Ncr N=0.6Ncr N=0.8Ncr a/b (a) Optimal fiber angle θ vs. aspect rati a/b N=0 N=0.2Ncr N=0.4Ncr N=0.6Ncr N=0.8Ncr a/b (b) Optimal fundamental frequency ω vs. aspect rati a/b Figure 12: Effect f aspect rati and in-plane cmpressive frce n ptimal fiber angle and ptimal fundamental frequency f [±θ/90/0] 2s laminated curved panels with fur simply supprted edges (b = 10 cm, φ = 60 )

20 200 Cpyright 2012 Tech Science Press CMC, vl.28, n.3, pp , 2012 Figure 12 shws the ptimal fiber angle θ and the assciated ptimal fundamental frequency ω versus aspect rati a/b fr [±θ/90/0] 2s laminated curved panels with φ = 60 and with simply supprted cnditins. Frm Fig. 12a we can see that the axial cmpressive frce N des have sme influence n the ptimal fiber angle θ f the laminated curved panels when the aspect rati is large (say a/b 0.8). The ptimal fiber angle increases with the increasing f the cmpressive frce and usually varies between 60 and 70. When a/b = 0.5, the ptimal fiber angle θ f the laminated curved panels are all equal t 67 and seem t be independent n the axial cmpressive frces. Figure 12b shws that the ptimal fundamental frequency ω decreases with the increasing f the aspect rati a/b and decreases with the increasing f the axial cmpressive frce. Figure 13 shws the fundamental vibratin mdes f thse curved panels with φ = 60 and under ptimal fiber angle θ. Again, it can be seen that thse mdes are very similar. Figure 14 shws the ptimal fiber angle θ and the assciated ptimal fundamental frequency ω versus the circular angle φ fr [±θ/90/0] 2s laminated curved panel with φ = 150 and with simply supprted edge cnditins. Frm Fig. 14a we can see that the axial cmpressive frce N has little influences n the ptimal fiber angle θ f the laminated curved panels when the aspect rati is large (say a/b 0.8). The ptimal fiber angle θ f thse panels usually varies between 70 and 90. When the aspect rati is large, the ptimal fiber angle tends t apprach cnstant value. Figure 14b shws again that the ptimal fundamental frequency ω decreases with the increasing f the aspect rati a/b and decreases with the increasing f the axial cmpressive frce. Cmparing Fig. 14a with Fig. 12a, we can see that the circular angle φ has significant influence n the ptimal fiber angle f the laminated curved panel. Cmparing Fig. 14b with Fig. 12b, we can see that the ptimal fundamental frequencies f the curved panels with small circular angle are smaller than thse with large circular angle. Figure 15 shws the fundamental vibratin mdes f thse curved panels with φ = 150 under ptimal fiber angle θ. Again, it can be seen that thse mdes are very similar. 5.4 Fixed laminated curved panels with varius aspect ratis and axial cmpressive frces In this sectin, laminated curved panels subjected t axial cmpressive frce and similar t thse in previus sectin are analyzed except that the panels are fixed at the fur edges. The laminate layup f the laminated curved panel is still [±θ/90/0] 2s. Figure 16 shws the ptimal fiber angle θ and the assciated ptimal fundamental frequency ω versus aspect rati a/b fr [±θ/90/0] 2s laminated curved panels with φ = 60 and with fixed cnditins. Frm Fig. 16a we can see that the axial cmpressive frce N des have sme influence n the ptimal fiber angle θ f

21 Maximizatin f Fundamental Frequencies 201 a/b N= 0 N = 0.4N cr N = 0.8N cr 0.5 θ= 67 ω =4103 Hz θ= 67.5 ω =3259 Hz θ= 67 ω =1952 Hz 0.8 θ= 63.5 ω =3022 Hz θ= 64.5 ω =2453 Hz θ= 66 ω =1518 Hz 1.2 θ= 64.5 ω =2295 Hz θ= 66 ω =1901 Hz θ= 69 ω =1220 Hz 1.6 θ= 65 ω =1884 Hz θ= 66.5 ω =1587 Hz θ= 71.5 ω =1053 Hz 2 θ= 65 ω =1611 Hz θ= 66.5 ω =1376 Hz θ= 72.5 ω =943 Hz Figure 13: Fundamental vibratin mdes f [±θ/90/0] 2s laminated curved panels with fur simply supprted edges and under ptimal fiber angles (b = 10 cm, φ = 60 )

22 202 Cpyright 2012 Tech Science Press CMC, vl.28, n.3, pp , Optimal θ (degrees) Optimal ω (khz) N=0 N=0.2Ncr N=0.4Ncr N=0.6Ncr N=0.8Ncr a/b (a) Optimal fiber angle θ vs. aspect rati a/b N=0 N=0.2Ncr N=0.4Ncr N=0.6Ncr N=0.8Ncr a/b (b) Optimal fundamental frequency ω vs. aspect rati a/b Figure 14: Effect f aspect rati and in-plane cmpressive frce n ptimal fiber angle and ptimal fundamental frequency f [±θ/90/0] 2s laminated curved panels with fur simply supprted edges (b = 10 cm, φ = 150 )

23 Maximizatin f Fundamental Frequencies 203 a/b N= 0 N = 0.4N cr N = 0.8N cr 0.5 θ= 77 ω =5035 Hz θ= 84.5 ω =4195 Hz ω =2937 Hz 0.8 θ= 74 ω =3855 Hz θ= 75 ω =3210 Hz ω =2154 Hz 1.2 θ= 70.5 ω =2994 Hz θ= 71 ω =2506 Hz ω =1657 Hz 1.6 θ= 68 ω =2436 Hz θ= 69 ω =2056 Hz ω =1409 Hz 2 θ= 68.5 ω =2045 Hz θ= 69.5 ω =1740 Hz ω =1210 Hz Figure 15: Fundamental vibratin mdes f [±θ/90/0] 2s laminated curved panels with fur simply supprted edges and under ptimal fiber angles (b = 10 cm, φ = 150 )

24 204 Cpyright 2012 Tech Science Press CMC, vl.28, n.3, pp , 2012 Optimal θ (degrees) N=0 N=0.2Ncr N=0.4Ncr N=0.6Ncr N=0.8Ncr a/b (a) Optimal fiber angle θ vs. aspect rati a/b 7.0 Optimal ω (khz) N=0 N=0.2Ncr N=0.4Ncr N=0.6Ncr N=0.8Ncr a/b (b) Optimal fundamental frequency ω vs. aspect rati a/b Figure 16: Effect f aspect rati and in-plane cmpressive frce n ptimal fiber angle and ptimal fundamental frequency f [±θ/90/0] 2s laminated curved panels with fur fixed edges (b = 10 cm, φ = 60 )

25 Maximizatin f Fundamental Frequencies 205 the laminated curved panels when the aspect rati is large (say a/b 0.6). The ptimal fiber angle increases with the increasing f the cmpressive frce and usually varies between 0 and 80. When the aspect rati is large, the ptimal fiber angle tends t apprach cnstant value. Similar t the panels with fur simply supprted edges (Fig. 12a), when a/b = 0.5, the ptimal fiber angle θ f the laminated curved panels are all equal t 0 and seem t be independent n the axial cmpressive frces. Figure 16b shws that the ptimal fundamental frequency ω decreases with the increasing f the aspect rati a/b and decreases with the increasing f the axial cmpressive frce. Figure 17 shws the fundamental vibratin mdes f thse curved panels with φ = 60 and under ptimal fiber angle θ. It can be seen the fundamental vibratin mdes f the panels under ptimal fiber angle are very similar. Figure 18 shws the ptimal fiber angle θ and the assciated ptimal fundamental frequency ω versus the circular angle φ fr [±θ/90/0] 2s laminated curved panel with φ = 150 and with fixed edge cnditins. Frm Fig. 18a we can see that the axial cmpressive frce N has significant influences n the ptimal fiber angle θ f the laminated curved panels. The ptimal fiber angle θ f thse panels usually varies between 60 and 90. Figure 18b shws again that the ptimal fundamental frequency ω decreases with the increasing f the aspect rati a/b and decreases with the increasing f the axial cmpressive frce. Cmparing Fig. 18a with Fig. 16a, we can see that the circular angle φ has significant influence n the ptimal fiber angle f the laminated curved panel. Cmparing Fig. 18b with Fig. 16b, we can see that the ptimal fundamental frequencies f the curved panels with small circular angle are smaller than thse with large circular angle. Finally, cmparing Figs. 16 and 18 with Figs. 12 and 14, we can cnclude that the bundary cnditin has significant influence n the ptimal fiber angle and the assciated ptimal fundamental frequency f the laminated curved panels. Figure 19 shws the fundamental vibratin mdes f thse curved panels with φ = 150 under ptimal fiber angle θ. Again, it can be seen the fundamental vibratin mdes f the panels under ptimal fiber angle are very similar. 6 Cnclusins Generally, the axial cmpressive frce, curvature, aspect rati and bundary cnditins have significant influence n the ptimal fiber angle and the assciated ptimal fundamental frequency f laminated curved panels. Based n the numerical results f this investigatin, the fllwing specific cnclusins may be drawn: 1. The ptimal fundamental frequency ω f the curved laminated panel increases with the increasing f the circular angle φ and decreases with the increasing f the axial cmpressive frce.

26 206 Cpyright 2012 Tech Science Press CMC, vl.28, n.3, pp , 2012 a/b N= 0 N = 0.4N cr N = 0.8N cr 0.5 θ= 0 ω =6361 Hz θ= 0 ω =5186 Hz θ= 0 ω =3240 Hz 0.8 θ= 62.5 ω =4085 Hz θ= 68 ω =3453 Hz θ= 79.5 ω =2688 Hz 1.2 θ= 65 ω =3055 Hz θ= 71 ω =2672 Hz θ= 80 ω =2251 Hz 1.6 θ= 66 ω =2507 Hz θ= 71.5 ω =2244 Hz θ= 79 ω =1959 Hz 2 θ= 67.5 ω =2160 Hz θ= 72 ω =1968 Hz θ= 79.5 ω =1766 Hz Figure 17: Fundamental vibratin mdes f [±θ/90/0] 2s laminated curved panels with fur fixed edges and under ptimal fiber angles (b = 10 cm, φ = 60 )

27 Maximizatin f Fundamental Frequencies Optimal θ (degrees) Optimal ω (khz) N=0 N=0.2Ncr N=0.4Ncr N=0.6Ncr N=0.8Ncr a/b (a) Optimal fiber angle θ vs. aspect rati a/b N=0 N=0.2Ncr N=0.4Ncr N=0.6Ncr N=0.8Ncr a/b (b) Optimal fundamental frequency ω vs. aspect rati a/b Figure 18: Effect f aspect rati and in-plane cmpressive frce n ptimal fiber angle and ptimal fundamental frequency f [±θ/90/0] 2s laminated curved panels with fur fixed edges (b = 10 cm, φ = 150 )

28 208 Cpyright 2012 Tech Science Press CMC, vl.28, n.3, pp , 2012 a/b N= 0 N = 0.4N cr N = 0.8N cr 0.5 θ= 76 ω =6591 Hz θ= 83 ω =5901 Hz θ= 93 ω =5014 Hz 0.8 θ= 70 ω =4959 Hz θ= 74.5 ω =4376 Hz θ= 80 ω =3610 Hz 1.2 θ= 64 ω =3723 Hz θ= 70.5 ω =3225 Hz θ= 82 ω =2661 Hz 1.6 θ= 63.5 ω =2933 Hz θ= 72 ω =2535 Hz θ= 83.5 ω =2129 Hz 2 θ= 64 ω =2413 Hz θ= 73 ω =2096 Hz θ= 83 ω =1775 Hz Figure 19: Fundamental vibratin mdes f [±θ/90/0] 2s laminated curved panels with fur fixed edges and under ptimal fiber angles (b = 10 cm, φ = 150 )

29 Maximizatin f Fundamental Frequencies The ptimal fundamental frequencies f the curved laminated panels with small aspect ratis are greater than thse with large aspect ratis. Fr the curved laminated panels with fixed edges, their ptimal fundamental frequencies are greater than thse with simply supprted edges. 3. When the curved laminated panel has large aspect rati, small circular angle and subjected t large cmpressive frce, its vibratin mde under ptimal fiber angle may have mre wave numbers in the axial directin. Acknwledgement: This research wrk was financially supprted by the Natinal Science Cuncil f the Republic f China under Grant NSC E References Abaqus, Inc. (2010): Abaqus Analysis User s Manuals and Example Prblems Manuals, Versin Prvidence, Rhde Island. Abrate, S. (1994): Optimal Design f Laminated Plates and Shells. Cmpsite Structures, Vl. 29, N. 3, pp Bert, C. W. (1991): Literature Review - Research n Dynamic Behavir f Cmpsite and Sandwich Plates - V: Part II. The Shck and Vibratin Digest, Vl. 23, N. 7, pp Chandrashekhara, K. (1989): Free Vibratin f Anistrpic Laminated Dubly Curved Shells. Cmputers and Structures, Vl. 33, N. 2, pp Chakrabarti, A.; Tpdar, P.; Sheikh, A. H. (2006): Vibratin f Pre-stressed Laminated Sandwich Plates with Interlaminar Imperfectins. Jurnal f Vibratin and Acustics, ASME, Vl. 128, N. 6, pp Chen, J.-M. (2009): Optimizatin f Fundamental Frequencies f Axially Cmpressed Laminated Curved Panels. M.S. Thesis, Department f Civil Engineering, Natinal Cheng Kung University, Tainan, Taiwan (in Chinese). Chen, C.-S.; Cheng, W.-S.; Chien, R.-D.; Dng, J.-L. (2002): Large Amplitude Vibratin f an Initially Stressed Crss Ply Laminated Plates. Applied Acustics, Vl. 63, N. 9, pp Chun, L.; Lam, K. Y. (1995): Dynamic Analysis f Clamped Laminated Curved Panels. Cmpsite Structures, Vl. 30, N. 4, pp Ck, R. D.; Malkus, D. S.; Plesha, M. E.; Witt, R. J. (2002): Cncepts and Applicatins f Finite Element Analysis, Furth Editin. Jhn Wiley & Sns Inc. Crawley, E. F. (1979): The Natural Mdes f Graphite/Epxy Cantilever Plates and Shells. Jurnal f Cmpsite Materials, Vl. 13, N. 3, pp

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31 Maximizatin f Fundamental Frequencies 211 Tpal, U.; Uzman, U. (2006): Optimal Design f Laminated Cmpsite Plates t Maximize Fundamental Frequency Using MFD Methd. Structural Engineering and Mechanics, Vl. 24, N. 4, pp Vanderplaats, G. N. (1984): Numerical Optimizatin Techniques fr Engineering Design with Applicatins, Chapter 2. McGraw-Hill. Whitney, J. M. (1973): Shear Crrectin Factrs fr Orthtrpic Laminates Under Static Lad. Jurnal f Applied Mechanics, Vl. 40, N. 1, pp

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