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doi:10.1016/j.ijmecsci.2006.06.008 International Journal of Mechanical Sciences 48 (2006) 1516–1524 Maximizing the fundamental frequency of laminated cylindrical panels using layerwise optimization Y. Naritaa,�, P. Robinsonb aDepartment of Mechanical Engineering, Hokkaido University, Sapporo 060-8628, Japan bAeronautics Department, Imperial College London, London SW7 2AZ, UK Received 10 March 2005; received in revised form 14 June 2005; accepted 21 June 2006 Available online 1 September 2006 Abstract A method of analysis is presented for determining the free vibration frequencies of cylindrically curved laminated panels under general edge conditions, and is implemented in a layerwise optimization (LO) scheme to determine the optimum fiber orientation angles for the maximum fundamental frequency. Based on the classical lamination theory applicable to thin panels, a method of Ritz is used to derive a frequency equation wherein the displacement functions are modified to accommodate arbitrary sets of edge conditions for both in-plane and out-of-plane motions. The LO approach, a recently developed scheme for laminated plates, is extended for the first time to the optimum design of curved panels. In a number of numerical examples, the accuracy of the analysis and the effectiveness of the LO approach are demonstrated. r 2006 Elsevier Ltd. All rights reserved. Keywords: Optimum design; Laminated composite; Cylindrical panel; Frequency design 1. Introduction Much progress has been made over the last two decades in the development of increasingly more efficient composite structures and industries continue to investigate strategies for fully exploiting the potential of composites for a variety of structural forms including the laminated cylindrical panel which is the subject of this paper. Given that a key advantage of composite laminates, in addition to their potential for high specific stiffness and strength, is the ability to tailor the properties of the laminate through lay- up design then it is clearly appropriate that efforts be applied to fully optimize this tailoring process. A curved panel is one form of general shells and falls into a category of shallow shell, and the vibration analysis of shallow shells has a long history of academic and practical interest, as summarized in a monograph [1] by Leissa and in review papers by Qatu [2] and Liew et al. [3]. A cylindrically curved panel can be regarded as a shallow shell that has small curvature (i.e., large curvature radius) in one direction. In Ref. [4], a complete set of equations for elastic deformation of laminated composite shallow shells are presented for static and vibration behaviors. For vibration of laminated composite curved panels, some relevant important discussions were found in the publications in the 1980s. Bert and Kumar [5] formulated vibration of the panel considering the bimodulus beha- viors. Soldatos compared some shell theories for the analysis of cross-ply laminated panels [6] and studied influence of thickness shear on the vibration of antisym- metric angle-ply panels [7]. Recently, Lam and Chun [8] dealt with dynamic analysis of clamped laminated curved panels and Berc- in [9] obtained natural frequencies of cross-ply laminated curved panels. In 1997, Bardell et al. [10] presented free and forced vibration analysis of cylindrically curved laminated panels and Selmane and Lakis [11] also analyzed anisotropic cylindrical shells. More recently, Soldatos and Messina [12] studied the influence of boundary conditions and trans- verse shear on the vibration of angle-ply laminated cylindrical panels. Kabir [13] applied a shallow shell theory of Reissner to determine the response of cylindrical panels with arbitrary lay-ups, and a mesh free approach was used ARTICLE IN PRESS www.elsevier.com/locate/ijmecsci 0020-7403/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmecsci.2006.06.008 �Corresponding author. Fax: +81 11 706 7889. E-mail address: ynarita@eng.hokudai.ac.jp (Y. Narita). by Zhao et al. [14] to analyze the vibration response of laminated cylindrical panels. As for tailoring, Raouf [15] considered the effect of tailoring on the dynamic characteristics of composite panels using fiber orientation. Narita et al. [16] applied a genetic algorithm to determine the maximum fundamental frequency of laminated shallow shells that are supported by shear diaphragms and for the same problem they used Kuhn–Tucker condition to derive the maximum funda- mental frequency of laminated shallow shells [17]. The effect of using various solutions upon optimizing vibration characteristics of laminated shallow shells are also studied [18]. These papers [15–18] are however limited to a simple case with the edges fully supported by shear diaphragms where a simplified frequency formula is derivable by neglecting the cross-elasticity terms. In the present paper, a semi-analytical solution is presented for the free vibration of cylindrically curved laminated panels with arbitrary boundary conditions with respect to in-plane and out-of-plane displacements. The solution is based on a method of Ritz wherein the displacement functions are modified by boundary indices [19] to satisfy the required kinematical boundary conditions. Furthermore the analytical method is combined with the layerwise optimization (LO) scheme, recently developed for laminated flat plates [20,21]. Numerical examples demon- strate the accuracy of the present Ritz solution to determine natural frequencies of cylindrically curved panels with various edge conditions, and also the extension of the LO approach to the curved panels is shown to be quite effective in obtaining the optimum fiber orientation angles which maximize the fundamental frequencies of the laminated panels. 2. Analysis The quadratic mid-surface of a shallow shell (panel) may be expressed in a rectangular coordinate system as fðx; yÞ ¼ � 1 2 x2 Rx þ 2 xy Rxy þ y 2 Ry � � , (1) where Rx and Ry are the radii of curvature in the x and y directions, respectively, and Rxy is the radius of twist. For a cylindrically curved panel, the orientation of the xy coordinates may be chosen so that R ¼ Rx is a principal constant curvature radius and Ry and Rxy are infinite, as shown in Fig. 1. The dimension of its planform is given by a� b and the panel thickness is h. The four sides are subjected to uniform in-plane (i.e., stretching) and out-of- plane (bending) boundary conditions. Using the Kirchhoff hypothesis, the displacements u*(x, y, z, t), v*(x, y, z, t) and w*(x, y, z, t) of an arbitrary point in a panel are written as u� ¼ u� z qw qx ; v� ¼ v� z qw qy ; w� ¼ w, (2) where z is the coordinate measured from the mid-surface in the direction of outer normal. The u(x, y, t) and v(x, y, t) are displacement components, tangent to the mid-surface and parallel to the xz and yz planes, respectively, and w(x, y, t) is a displacement component normal to the mid-surface at a typical point on the mid-surface. In the linear theory, the strain components at an arbitrary point (x, y, z) are ��x ¼ �x þ zkx; � � y ¼ �y þ zky; g � xy ¼ gxy þ zkxy, (3) assuming that z is negligible in comparison with R, where the membrane strains are given by �x ¼ qu qx þ w R ; �y ¼ qv qy ; gxy ¼ qv qx þ qu qy , (4) and the curvature changes due to the vibratory displace- ments are kx ¼ � q2w qx2 ; ky ¼ � q2w qy2 ; kxy ¼ �2 q2w qx qy . (5) The kinetic approximations (2)–(5) used in the present shallow shell analysis are those of the Donnell type [4]. For a laminated panel composed of fibrous composite thin layers, each layer may be regarded as a macroscopi- cally orthotropic but the fiber orientation may not be parallel to the coordinate axes. The stress–strain equations for an element of material in the kth layer can be written as sx sy txy 8><>: 9>= >; ðkÞ ¼ Q̄11 Q̄12 Q̄16 Q̄12 Q̄22 Q̄26 Q̄16 Q̄26 Q̄66 2 64 3 75 ðkÞ �x �y gxy 8><>: 9>= >;, (6) ARTICLE IN PRESS x y v u R a/2 b/2 w b/2 a/2 Edge (4) Edge (1) Edge (2) Edge (3) O Fig. 1. Mid-surface of cylindrically curved panel. Y. Narita, P. Robinson / International Journal of Mechanical Sciences 48 (2006) 1516–1524 1517 where the constants Q̄ ðkÞ ij are the elastic constants of the kth layer. The Q̄ ðkÞ ij are determined from the transformation relationships using the fiber orientation angle y and the stiffness Q11 ¼ EL 1� nLTnTL ; Q12 ¼ nTLQ11, Q22 ¼ ET 1� nLTnTL ; Q66 ¼ GLT , ð7Þ where EL and ET are the moduli of elasticity in the L and T directions, respectively, GLT is the shear modulus and nLT and nTL are the major and minor Poisson’s ratios in an orthotropic layer. The transformation relationship (6) is found in well-known textbooks [22,23]. The force resultants and the moment resultants are obtained by integrating the stresses and the stresses multiplied by z, respectively, over the panel thickness h, and are written in matrix form as N M � � ¼ A B B D � � � k � � , (8) where {N}, {M}, {e} and {k} are the vectors of force resultants, moment resultants, mid-surface strains and curvatures, respectively, given by fNg ¼ Nx Ny Nxy 8>><>>: 9>>= >>;; fMg ¼ Mx My Mxy 8>><>>: 9>>= >>;, f�g ¼ �x �y gxy 8>><>>: 9>>= >>;; fkg ¼ kx ky kxy 8>><>>: 9>>= >>;, ð9a2dÞ and [A], [B] and [D] are the matrices of stiffness coefficients defined by ½A� ¼ A11 A12 A16 A12 A22 A26 A16 A26 A66 2 664 3 775; ½B� ¼ B11 B12 B16 B12 B22 B26 B16 B26 B66 2 664 3 775, ½D� ¼ D11 D12 D16 D12 D22 D26 D16 D26 D66 2 664 3 775. ð10a2cÞ The stiffness coefficients in Eqs. (10a–c) are determined by Aij ¼ XK k¼1 Q̄ ðkÞ ij zk � zk�1ð Þ; Bij ¼ 1 2 XK k¼1 Q̄ ðkÞ ij z 2 k � z 2 k�1 � � , Bij ¼ 1 3 XK k¼1 Q̄ ðkÞ ij z 3 k � z 3 k�1 � � , ð11a2cÞ (i, j ¼ 1, 2, 6) where zk is the distance from the mid-surface to the upper surface of the kth layer and K is the total number of layers [22,23]. In the present study, the free vibration problem can be solved by means of the Ritz method. This requires the evaluation of energy functional. The strain energy stored in a panel during elastic deformation is written in the classical (thin) shallow shell theory by V ¼ V s þ Vbs þ V b, (12) where Vs, Vbs and Vb are the parts of the total strain energy due to stretching, bending–stretching coupling and bend- ing, respectively, Vs ¼ 1 2 ZZ f�gT½A�f�gdArea, (13a) Vbs ¼ 1 2 ZZ fkgT½B�f�g þ f�gT½B�fkg � � dArea, (13b) Vb ¼ 1 2 ZZ fkgT½D�fkgdArea. (13c) The kinetic energy of the panel due to translational motion only is given by T ¼ 1 2 r ZZ qu qt � �2 þ qv qt � �2 þ qw qt � �2" # dArea, (14) where r is the average mass density of the panel per unit area of the mid-surface. For clarity in the formulation, the following dimension- less quantities are introduced: x ¼ 2x a ; Z ¼ 2y b ðdimensionless coordinatesÞ, (15a) O ¼ oa2 ffiffiffiffiffiffi r D0 r ðdimensionless frequency parameterÞ, (15b) with D0 ¼ ET h 3 12 1� nLTnTLð Þ ðreference plate stiffnessÞ. (15c) In the Ritz method the displacements may be assumed in the form uðx; Z; tÞ ¼ XM�1 i¼0 XN�1 j¼0 PijX iðxÞY jðZÞsinot, (16a) vðx; Z; tÞ ¼ XM�1 k¼0 XN�1 l¼0 QklX kðxÞY lðZÞsinot, (16b) wðx; Z; tÞ ¼ XM�1 m¼0 XN�1 n¼0 RmnX mðxÞY nðZÞsinot, (16c) where Pij, Qkl and Rmm are unknown coefficients and Xi(x), Yj(Z),y and Yn(Z) are the functions that satisfy at least the kinematical boundary conditions at the edges. The upper limit in each of the summations (16) is arbitrary but is unified here for simplicity in the convergence test. ARTICLE IN PRESS Y. Narita, P. Robinson / International Journal of Mechanical Sciences 48 (2006) 1516–15241518 After substituting Eqs. (16) into the functional L L ¼ Tmax � Vmax (17) composed of the maximum strain and kinetic energies obtained from Eqs. (12) and (14), the stationary value is obtained by qL qPij ¼ 0; qL qQkl ¼ 0; qL qRmn ¼ 0 i; k;m ¼ 0; 1; 2; . . . ; ðM � 1Þ; j; l; n ¼ 0; 1; 2; . . . ; ðN � 1Þð Þ. ð18a2cÞ The result of the minimization process (18) yields a set of homogeneous, linear simultaneous equations in the un- knowns fPij ;Qkl ;Rmng. For non-trivial solutions the determinant of the coefficient matrix is set to zero. The (M�N)� 3 eigenvalues may be extracted and the lowest several eigenvalues (natural frequencies) are important from a practical viewpoint. The above procedure is a standard routine of the Ritz method, and is modified to incorporate arbitrary edge conditions [19]. This approach introduces the following polynomials: X i xð Þ ¼ xi 1þ xð ÞBu1 1� xð ÞBu3, Y j Zð Þ ¼ Zj 1þ Zð ÞBu2 1� Zð ÞBu4, ð19a;bÞ X k xð Þ ¼ xk 1þ xð ÞBv1 1� xð ÞBv3, Y l Zð Þ ¼ Zl 1þ Zð ÞBv2 1� Zð ÞBv4, ð19c;dÞ X m xð Þ ¼ xm 1þ xð ÞBw1 1� xð ÞBw3, Y n Zð Þ ¼ Zn 1þ Zð ÞBw2 1� Zð ÞBw4, ð19e; fÞ where Brs (r ¼ u, v,w; s ¼ 1, 2, 3, 4) is the boundary index [19] which is used to satisfy the kinematic boundary conditions. The capital letter B stands for Boundary. The first subscript letter