Free Vibration of Antisymmetric Angle-Ply Composite Laminated Conical Shell under Classical Theory

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1 International Journal of Alie Engineering Research ISSN Volume, Number 5 (07) Research Inia Publications. htt:// Free Vibration of Antisymmetric Angle-Ply Comosite Laminate Conical Shell uner Classical Theory A.K. Nor Hafizah,, K.K. Viswanathan,,,*, Z.A. Aziz, an J.H. Lee 4 Deartment of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, 80 Skuai, Johor Bahru, Johor, Malaysia. UTM Centre for Inustrial an Alie Mathematics, Ibnu Sina Institute for Scientific & Inustrial research, Universiti Teknologi Malaysia, 80 Skuai, Johor Bahru, Johor, Malaysia. Kuwait College of Science an Technology, Doha District, Block 4, P.O. Box No. 75, Safat, Kuwait. 4 Det. of Naval Architecture an Ocean Engineering, Inha University, 00 Inharo, Nam-gu, Incheon, South Korea. *Corresoning author Orci: Abstract This aer focuses on free vibration behaviour of antisymmetric angle-ly laminate conical shell uner classical theory. The equations of motion are erive using Love s first aroximation thin shell theory. By alying oint collocation metho together with sline function aroximation, the free vibration equations of motion of the comosite laminate conical shell are transforme to a set of algebraic equations for the assume sline coefficients. The eigenvalue roblem is solve by emloying the eigensolution techniques with eigenvectors as sline coefficients to obtain the require frequencies. This stuy is conucte for Clame-Clame an Simly-Suorte bounary conitions. Detaile arametric investigation is carrie out to examine the influences of number of laminates, ly orientations, circumferential noe number an materials on the free vibration of conical shells. Keywors: free vibration, laminate conical shells, Love s first aroximation thin shell theory, sline function aroximation INTRODUCTION Thin shells are structural element in which the thickness is small comare to the length an with imensions. Shell structures mae an imortant contribution to the eveloment of several branches of engineering such as architecture an builing, chemical engineering, structural engineering an comosite construction. Other examles of the imact of shell structures inclue water cooling towers for ower stations, grain silos, armour, arch ams, tunnels an submarines. The alication of laminate comosite shell extens the characteristics of shells since it rovies higher strength-to-weight ratios, better corrosion resistance, longer fatigue life an also one can esign the irectional roerties. To obtain otimal esign of structures in engineering inustries, the vibrational behaviour analysis of shell structures is carrie out. Knowing more about vibration of structures hels us to control amage an take reventive measures from amages to occur. Conical shell is one of the revolution of shells that has great eman on inustries. Irie et. al [] an Sankaranarayanan et al. [] analyse the free vibration of a conical shell with variable thickness by means of transfer matrix aroach an Rayleigh-Ritz metho resectively. The stuy of axisymmetric free vibrations of conical shells ha been conucte by Tong [] where a simle an exact solution was obtaine irectly for the Donnell-tye governing equations. Ng et al. [4] use generalize ifferential quarature (GDQ) to erform free vibration analysis of rotating comosite laminate conical shells where the metho was eveloe to imrove the ifferential quarature (QD) metho. The ynamic analysis of the truncate conical shells subjecte to ressure ulse loaing was stuie by Jankowski an Kubiak [5] using Bubnov-Galerkin analytical-numerical metho. Flugge shell theory together with ower series metho was use by Xie et al. [6] to investigate free an force vibration of stee conical shells. Ghasemi et al. [7] alie Galerkin metho for stuying free vibration of truncate conical comosite shells uner various bounary conitions. Souza an Saraiva [8] use Rayleigh-Ritz metho to analyse free vibration of conical shells on rigily clame ege bounary conition. A moifie Fourier series solution was aote by Jin et al. [9] to investigate the vibration of truncate conical shells. Base on first orer shear eformation theory, free vibration of rotating grae comosite conical shells were stuie by Heyarour et al. 498

2 International Journal of Alie Engineering Research ISSN Volume, Number 5 (07) Research Inia Publications. htt:// [0] using ifferential quarature metho (DQM). Viswanathan et al. [,] ha conucte a few stuies on conical shells using sline metho. From the literature survey above, free vibration analysis of laminate comosite antisymmetric angle-ly conical shells using Bickley-tye sline still has not been investigate. The aim of this aer is to investigate this roblem. The avantage of this sline is that it has low-orer aroximation an yiel better accuracy comare to a global higher orer aroximation. Love s first aroximation thin shell theory is use to formulate the theoretical moels of the conical roblem. Bickley-tye sline together with collocation metho is use to aroximate the solution into a system of algebraic equation. The solution equations are then consiere as eigenvalue roblem which is solve using eigensolution technique. The results are obtaine for Clame-Clame an Simly- Suorte bounary conitions. The accuracy an reliability of the current solution are valiate by comaring with the results available in the literature. The effects of circumferential noe number, number of layers, length ratio, cone angle an material comosition on the frequency of conical shells are further investigate. MATHEMATICAL FORMULATION Figure. Geometry of layere conical shell A laminate comosite conical shell is consiere as eicte in Figure. The conical shell is referre to a coorinate system ( x,, z) which is fixe in the mile surface. The raius of the cone, the raius of the cone at the small en an raius of large en of the cone are enote by r xsin, r asin an r bsin resectively. is a the semi-vertical angle of the cone, is the length an h reresents the thickness of the conical shell. The equation of motion for conical shell in terms of stress an moment resultants are can be efine by [] b Nx Nx u Nx N h, x x xsin t Nx N Nx x x xsin Mx M v Mx h, x tan x x xsin t x x x M M M M x x x xsin x x x Mx M x sin x sin N x tan w h t where is the ensity. The stress resultants moment resultants M ij are written as,,,, N N N z x x x x,,,, z M M M zz x x x x z () N ij an The constitutive equations for stress an moment resultants are given in the following form N x A A A6 B B B6 x N A A A6 B B B6 N x A6 A6 A66 B6 B6 B 66 x M x B B B6 D D D6 x M B B B D D D 6 6 M x B6 B6 B66 D6 D6 D 66 x where the strain-islacements for conical shell are: u v x, u w, x x xsin x x tan u v w x v, x, xsin x x x v w x sin tan x x x w, x sin v v tan x tan sin w. x sin x x x x x x w The extensional stiffness, bening-extensional couling stiffness, an bening stiffness are reresente by Aij, Bij an Dij resectively which are obtaine by () () (4) 499

3 International Journal of Alie Engineering Research ISSN Volume, Number 5 (07) Research Inia Publications. htt:// A Q ( z z ); B Q ( z z ); ( k) ( k) ij ij k k ij ij k k k k D Q ( z z ). ( k ) ij ij k k k The stiffness comonents A 6, A 6, B, B, B, B, B 66, D 6 an D 6 are ientically zero since the antisymmetric anglely conical shells conition are consiere. The islacement comonents are assume to be in the following searable form: n u ( x,, t) U ( x) e 0 n v ( x,, t) V ( x) e 0 n w( x,, t) W ( x) e it it it in which, t an n reresent the angular frequency of vibration, time, an circumferential noe number resectively. The non-imensional arameters are introuce as follows: x a X, a x b an X [0,], frequency arameter a,the raii ratio b h, ratios of thickness to raius r a I A ' h, ratio of thickness to length a hk k, relative layer thickness of the k-th layer h By substituting Eq. (4) into Eq. () an then alying the resulting equations into the governing Eq. (), a set of secon orer artial ifferential equations in terms of islacement functions is obtaine. Substitution of Eq. (6) an Eq. (7) into the obtaine ifferential equations leas to the following matrix equation L L L U 0 L L L V 0 L L L W 0 (5) (6) (7) (8) The ifferential oerators, follow L ij of the matrix are given as L S S0n csc X X L S5 cot ( S ncsc X S0n csc S5 cot S6 cot ) X ( S n csc S n csc S cot 0 5 S cot S n csc cot ) 6 6 L S5n csc ( S cot X csc csc ) S5n S6n X ( S cot S n csc S n csc S n csc ) L S5 cot ( S n csc X S0n csc S5 cot S6 cot ) X ( S n csc S ncsc S cot S n csc cot ) L S5 (S 5 S6 S8n csccot X Sn csccot ) ( S 6 X S n csc S cot 6 6 S9n csc cot ) ( Sncsc cot X S n csc S cot S n csc cot ) (9) 490

4 International Journal of Alie Engineering Research ISSN Volume, Number 5 (07) Research Inia Publications. htt:// L S5 (S 5 S6 S8n csccot X Sn csc cot ) ( S 6 S6n csc X S6 cot S9n csc cot ) X ( Sn csc cot S6n csc S cot S n csc cot ) 6 9 L S5n csc ( S cot X S5n csc S6n csc ) X ( S cot S n csc S n csc ) L S5 ( S 5 S6 S8n csccot X Sn csc cot ) ( S 6 S6n csc X S cot S n csc cot S n csc cot 4Sn csc cot ) ( Sncsc cot X S S S cot S n csc S n csccot S n csccot 8 9 S n csc cot 6S n csc cot ) 9 4 L S7 S 4 7 (S 8n csc X X S9 4Sn csc ) (4S 6n csc cot X S8n csc S9 4Sn csc ) X ( S cot S n csc cot S n csc 6 8 S n csc S n csc ) S n csc 4 N i i j j j U X a X b X X H X X i0 j0 N i i j j j V X c X X X H X X i0 j0 4 N i i j j j W X e X f X X H X X i0 j0 5 (0) Here H X X j is the Heavisie ste function an ai, ci, ei, bj, j an f j are unknown coefficients. The number of sub-intervals are reresente by N of X ivie within the range [0,]. The knots of the slines or the collocation oints are assume at X = X s= s / N, where s = 0,,..., N. A set of N + homogenous equations with unknown sline coefficients is obtaine by satisfying Eq. (8) with the slines at the knots. The bounary conitions are alie at the two ens of the conical shell. Consiering Clame-Clame an Simly- Suorte bounary conitions in this work, the conition are as follow: (i) Clame-Clame (C-C) (both the ens are clame) U V W 0 at X 0 an X. (ii) X Simly-Suorte (S-S) (both ens are simly suorte) V W N M 0 at X 0 an X X X After alying bounary conitions, 8 equations on sline coefficients are obtaine. Aing them with those obtaine earlier, a set of N + homogenous equations, with the same number unknown are accumulate. Then, the system of equations are written in the form of eigenvalue roblem as in (0) Pq Qq In this relation, [P] an [Q] are square matrices, [q] is a column matrix, is the frequency arameter an [q] is the eigenvector. SOLUTION PROCEDURE The islacement functions U(X), V(X) an W(X) are aroximate using cubic sline functions an can be exresse as RESULT AND DISCUSSION The reliability of the resent numerical formulation for conical shells is investigate by firstly, convergence tests, an seconly, comarisons with the existing results in the literature. Table shows the convergence frequency arameter for a conical shell with 0, 0.5 an n=. The comutation is conucte by choosing N= as 49

5 International Journal of Alie Engineering Research ISSN Volume, Number 5 (07) Research Inia Publications. htt:// the initial value an the comutation is carrie on until it is foun that N=6 is sufficient to achieve low value of ercentage of change. The resent results are comare with Irie et al. [] for isotroic conical shell with arameters: 0, 45,60, 0.5,0.5,0.75 an 0. which are eicte in Table an 4. It can be seen from the table that the values of the funamental frequency of resent stuy are in goo agreement with the results of Irie et al. []. Table : Convergence stuy on frequency arameter for antisymmetric angle ly laminate conical shells with C-C bounary conitions N % change Table : Comarison of frequency arameter for isotroic conical shells uner C-C bounary conitions: Present Irie [] Table : Comarison of frequency arameter for isotroic conical shells uner C-C bounary conitions: Present Irie [] Table 4: Comarison of frequency arameter for isotroic conical shells uner C-C bounary conitions: Present Irie [] Table 5: The effect of circumferential noe number n on the frequency arameter of C-C conical shells n C-C Table 6: The effect of circumferential noe number n on the frequency arameter of S-S conical shells n S-S The effect of the variation of the circumferential noe number, n on the values of frequency arameter for antisymmetric angle-ly comosite laminate conical shells uner C-C an S-S bounary conitions are emonstrate in Table 5 an 6. The arameters of the shells are fixe as: length ratio 0.5, ratio of thickness to a raius 0.05 an cone angle 0. The material HSG an SGE are use which are arrange in the orer of HSG-SGE. From the tables, it is foun that the variation of circumferential noe number leas to the u an own of the frequency arameter of conical shells. For the case of C-C conical shell, it is observe that the value increases u to n= an then ecreases afterwars. The value of for m= are seen to ecrease graually from n=0 to n= an then increases afterwars. For frequency arameter, it can be seen that the value ecreases an increases over the range 0n 5. The vibrational behaviour for S-S conical shell case gives similar attern to the case before. As can be seen in the table, it is clear that the value of is higher for higher moe. Also, it is evient that the frequency arameter of C-C conical shells is greater than those of the S-S conical shells. 49

6 International Journal of Alie Engineering Research ISSN Volume, Number 5 (07) Research Inia Publications. htt:// Figure shows the effect of cone angle (0 90) on the frequency arameter of two-layere [0 0 /-0 0 ] conical shells. The stuy is conucte for conical shells with C-C an S-S bounary conitions for frequency arameter with moe, an reresente by Figs. (a), (b) an (c) resectively. The arameters such as n=, 0.5, ' 0.0 an material HSG-SGE are fixe for both conical shells uner C-C an S-S bounary conitions. From Fig., it is observe that by increasing the cone angle of shells, the frequency arameter ecreases. As is shown in Fig (a), (b) an (c), the frequency arameter is seen to ro significantly from 0 until 0 an then ecreases steaily for For higher moe, the frequency arameter is higher. Also, the S-S conical shell has lower frequency arameter comare to C-C conical shell. Figure eicts the frequency arameter of C-C an S-S bounary conitions conical shells with arameters n=, 0.5, ' 0.0 an material HSG-SGE. The influence of cone angle, on the values of for two-layere [45 0 /-45 0 ] conical shells is islaye. As one can see in Fig., the frequency arameter shows a eclining tren with the increase in cone angle which is similar as shown in Fig.. Observing frequency arameter for two-layere [0 0 /-0 0 ] conical shells in Fig. with frequency arameter for twolayere [45 0 /-45 0 ] conical shells in Fig., the ifference in the values of are seen to become smaller with the increase of cone angle. The frequency arameter of antisymmetric angle-ly laminate conical shells uner C-C an S-S bounary conitions for four-layere conical shells with lyangle [0 0 /-0 0 / 0 0 /-0] an [45 0 /-45 0 / 45 0 /-45] are illustrate in Figs. 4 an 5 resectively. The materials are arrange in the form of HSG-SGE-SGE-HSG. Figure. Variation of frequency arameter with resect to cone angle for two-layere [0 0 /-0 0 ] conical shells Figure. Variation of frequency arameter with resect to cone angle for two-layere [45 0 /45 0 ] conical shells 49

7 International Journal of Alie Engineering Research ISSN Volume, Number 5 (07) Research Inia Publications. htt:// Figure 4. Variation of frequency arameter with resect to cone angle for four-layere [0 0 /-0 0 / 0 0 /-0] conical shells Figure 5. Variation of frequency arameter with resect to cone angle for four-layere [45 0 /-45 0 / 45 0 /-45] conical shells It is seen that the values of for conical shells with lyangle [45 0 /-45 0 / 45 0 /-45] are higher for smaller cone angle when comare with ly-angle [0 0 /-0 0 / 0 0 /-0] but the values o not have much ifference as cone angle increases. For both figures, the frequency arameter for C-C conical shells is greater than S-S conical shells case. Furthermore, the results from Figs.,, 4 an 5 shows that the number of layers an ifferent ly-angles have consierable effect on the frequency arameter of conical shells. In Fig. 6, the effects of length ratio on the angular frequency of C- C conical shells are investigate. The length ratio, is varie from 0. to 0.9. The conical shells arameters are taken to be n=, 0an 0.05 with material orientation HSG-SGE. The stuy is conucte for three ifferent ly-angles: [0 0 /-0 0 ], [45 0 /-45 0 ] an [60 0 /-60 0 ] eicte by Figs. 6(a), 6(b) an 6(c) resectively. From the figures, it is foun that as increases, the values of angular frequency increase. It can be seen the angular frequency increases slowly u to certain value of an then climbs raily afterwars. Also, the values of are slightly ifference for ifferent ly angles for but the change in the values of are more ronounce for 0.5 onwars. The maximum value of of conical shells occurs for ly-angle [0 0 /-0 0 ] at 0.9. Figure 7 resents the variation of angular frequency for S-S conical shells against length ratio with arameters n=, 0an The results are resente for three ifferent ly-angles. It can be observe that the vibrational behaviour of conical shells in Fig. 7 has the same attern as shown in Fig. 6. Observing the results in Figs. 6 an 7, it is note that the values of are lower for S-S conical shells than the values of for C-C conical shells. Also, the frequencies are lower for higher ly-angle as the values of increases. Figures 8 shows the angular frequency for C- C an S-S conical shells with resect to length ratio for two ifferent cone angle, 45an 60. It can be seen that the effect of length ratio on the frequencies of conical shells are more ronounce for higher cone angle ( 60 ). Figures 9 eicts the angular frequency values of C-C an S-S four- layere conical shells with two ifferent ly orientation ; [0 0 /-0 0 / 0 0 /-0] an [45 0 /-45 0 / 45 0 /-45]. It is conclue that the ly-angles gives influence 494

8 International Journal of Alie Engineering Research ISSN Volume, Number 5 (07) Research Inia Publications. htt:// on the frequency arameter of four-layere conical shells. With the increase in the length ratio, the ifference in the value of for [0 0 /-0 0 / 0 0 /-0] conical shells are higher than conical shells with ly-angle [45 0 /-45 0 / 45 0 /-45]. Figure 6. Variation of angular frequency with resect to length of cone for ifferent lamination angles uner C-C bounary conitions Figure 7. Variation of angular frequency with resect to length of cone for ifferent lamination angles uner S-S bounary conitions Figure 8. Variation of angular frequency with resect to length of cone for 45 an 60 uner C-C an S-S bounary conitions 495

9 International Journal of Alie Engineering Research ISSN Volume, Number 5 (07) Research Inia Publications. htt:// Figure 9. Variation of angular frequency with resect to length of cone for four-layere conical shells uner C-C an S-S bounary conitions CONCLUSION In this work, free vibration of antisymmetric angle-ly comosite laminate conical shells uner C-C an S-S bounary conitions are investigate. Sline metho is use to aroximate the solution into a set of algebraic equations which is then solve using eigensolution technique. From the results of the stuy, it is conclue that among two consiere bounary conitions for conical shell, natural frequencies in C-C bounary conitions have higher value comare to S-S bounary conitions. In aition, geometric roerties such as circumferential noe number, cone angle, length ratio, ly orientation, number of layers an materials give areciable effect to the frequency values of the conical shells. The frequencies are increase with the ecrease of cone angle. Also, by increasing the length ratio, the frequencies are seen to ecrease. The obtaine results coul serve as reference solutions for investigators an engineers. ACKNOWLEDGEMENTS This work was suorte by a Secial Eucation Program for Offshore Plant by the Ministry of Trae, Inustry an Energy Affairs (MOTIE), South Korea an artly suorte from Ministry of Higher Eucation(MOHE), GUP roject Vote No. Q.J H90 uner Research Management Centre(RMC), Universiti Teknologi Malaysia, Malaysia. REFERENCES [] Irie, T., Yamaa, G., an Kaneko, Y., 98, Free Vibration of Conical Shells with Variable Thickness, J. Soun Vib., 8(), [] Irie, T., Yamaa, G., an Tanaka, K., 984, Natural Frequencies of Truncate Conical Shells,, J. Soun Vib., 9(), [] Sankaranarayanan, N., Chanrasekaran, K., an Ramaiyan, G., 988, Free Vibrations of Laminate Conical Shells of Variable Thickness, J. Soun Vib., (), [4] Tong, L., 99, Free Vibration of Orthotroic Conical Shells, Int. J. Engng Sci, (5), [5] Ng, T. Y., Li, H., an Lam, K. Y., 00, Generalize Differential Quarature for Free Vibration of Rotating Comosite Laminate Conical Shells with Various Bounary Conitions, Int. J. Mech. Sci, 4, [6] Jankowski, J., an Kubiak, T., 00, Dynamic Resonse of the Truncate Conical Shell Subjecte to Pressure Pulse Loaing, Mech. Mech. Engng., 4(),. 5-. [7] Ghasemi, F. A., Ansari, R., an Paskiaby, R. B., 0, Free Vibration Analysis of Truncate Conical Comosite Shells using The Galerkin Metho, J. Al. Sci., (7),

10 International Journal of Alie Engineering Research ISSN Volume, Number 5 (07) Research Inia Publications. htt:// [8] Xie, K., Chen, M., an Liu, Z., 07, An Analytic Metho For Free An Force Vibration Analysis of Stee Conical Shells with Arbitrary Bounary Conitions, Thin-Walle Structures.,, [9] Souza, V. D., an Saraiva, J., 995, Analysis of Free Vibrations Of Conical Shells using Donnell s Aroximations, Al. Mech. Rev., 48(), [0] Jin, G., Ma, X., Shi, S., an Liu, Z., 04, A Moifie Fourier Series Solution for Vibration Analysis of Truncate Conical Shells with General Bounary Conitions, Al. Acoust., 85, [] Heyarour, Y., Agham, M. M., an Malekzaech, P., 04, Free Vibration Analysis of Rotating Functionally Grae Carbon Nanotube-Reinforce Comosite Truncate Conical Shells, Comosite Structures., 7, [] Viswanathan, K. K., Navaneethakrishnan, P. V., 04, Free Vibration of Layere Truncate Conical Shell Frusta of Differently Varying Thickness by the Metho of Collocation with Cubic an Quintic Slines, Int. J. Solis Struct., 4,. 9-50, 005. [] Soeel, W., 004, Vibration of Shells an Plates, West Lafayette, Iniana: Purue University. 497