Global Optimization of Stacking Sequence in a Laminated Cylindrical Shell Using Differential Quadrature Method
الموضوعات :
M.R Saviz
1
(Mechanical Engineering Department, Azarbaijan Shahid Madani University, Tabriz, Iran)
A Ziaei Asl
2
(Mechanical Engineering Department, Azarbaijan Shahid Madani University, Tabriz, Iran)
الکلمات المفتاحية: Vibration analysis, Stacking sequence optimization, Globalized Nelder–Mead, Laminated cylinder, Differential quadrature method,
ملخص المقالة :
Based on 3-D elasticity approach, differential quadrature method (DQM) in axial direction is adopted along with Globalized Nelder–Mead (GNM) algorithm to optimize the stacking sequence of a laminated cylindrical shell. The anisotropic cylindrical shell has finite length with simply supported boundary conditions. The elasticity approach, combining the state space method and DQM is used to obtain a relatively accurate objective function. Shell thickness is fixed and orientations of layers change in a set of angles. The partial differential equations are reduced to ordinary differential equations with variable coefficients by applying DQM to the equations, then, the equations with variables at discrete points are obtained. Natural frequencies are attained by solving the Eigen-frequency equation, which appears by incorporating boundary conditions into the state equation. A GNM algorithm is devised for optimizing composite lamination. This algorithm is implemented for maximizing the lowest natural frequency of cylindrical shell. The results are presented for stacking sequence optimization of two to five-layered cylindrical shells. Accuracy and convergence of developed formulation is verified by comparing the natural frequencies with the results obtained in the literature. Finally, the effects of mid-radius to thickness ratio, length to mid-radius ratio and number of layers on vibration behavior of optimized shell are investigated. Results are compared with those of Genetic Algorithm (GA) method, showing faster and more accurate convergence.
[1] Schmit L.A.,1979, Optimum design of laminated fiber composite plates, International Journal for Numerical Methods in Engineering 11: 623-640.
[2] Haftka R.T., Walsh J.L., 1992, Stacking sequence optimization for buckling of laminated plates by integer programming, AIAA Journal 30: 814-819.
[3] Nagendra S., Haftka R.T., Gurdal Z., 1992, Stacking sequence of simply supported laminated with stability and strain constraints, AIAA Journal 30: 2132-2137.
[4] Kam T.Y., Lai F.M., 1995, Design of laminated composite plates for optimal dynamic characteristics using a constrained global optimization technique, Computer Methods in Applied Mechanics and Engineering 20(3-4): 384-402.
[5] Narita Y., Zhao X., 1998, An optimal design for the maximum fundamental frequency of laminated shallow shells, International Journal of Solids and Structures 35(20): 2571-2583.
[6] Narita Y., Zhao X., 1997, Maximization of fundamental frequency for generally laminated rectangular plates by the complex method, Transaction of the Japan Society of Mechanical Engineers Part C 63: 364-370.
[7] Tsau L.R., Chang Y.H., Tsao F.L., 1995, The design of optimal stacking sequence for laminated FRP plates with inplane loading, Computers & Structures 55(4): 565-580.
[8] Spendley W., Hext G.R., Himsworth F.R., 1962, Sequential application of simplex designs in optimization and evolutionary operation, Techno Metrics 4: 441-461.
[9] Nelder J.A., Mead R., 1965, A simplex for function minimization, The Computer Journal 7: 308-313.
[10] Lagarias J.C., Reeds J.A., Wright M.H., Wright P.E., 1999, Convergence behavior of the Nelder–Mead simplex algorithm in low dimensions, The SIAM Journal on Optimization 9: 112-147.
[11] Abouhamze M., Shakeri M., 2007, Multi-objective stacking sequence optimization of laminated cylindrical panels using a genetic algorithm and neural networks, Composite Structures 81: 253-263.
[12] Margarida F.C., Salcedo R.L., 1996, The simplex-simulated annealing approach to continuous non-linear optimization, Computers & Chemical Engineering 20(9): 1065-1080.
[13] Chelouah R., Siarry P., 2003, Genetic and Nelder–Mead algorithms hybridized for a more accurate global optimization of continuous multiminima functions, The European Journal of Operational Research 148: 335-348.
[14] Chelouah R., Siarry P., 2005, A hybrid method combining continuous tabu search and Nelder–Mead simplex algorithms for the global optimization of multiminima functions, The European Journal of Operational Research 161: 636-654.
[15] Luersen M.A., Riche R.L., 2004, Globalized Nelder–Mead method for engineering optimization, Computers & Structures 82: 2251-2260.
[16] Siarry P., Chelouah R., 2005, A hybrid method combining continuous Tabu search and Nelder-Mead simplex algorithms for the global optimization of multiminima functions, The European Journal of Operational Research 161: 634-654.
[17] Bert C.W., Malik M., 1997, Differential quadrature method: a powerful new technique for analysis of composite structures, Composite Structures 39: 179-189.
[18] Malekzadeh P., Farid M., Zahedinejad P., 2008, A three-dimensional layer wise differential quadrature free vibration analysis of laminated cylindrical shells, International Journal of Pressure Vessels and Piping 85: 450-458.
[19] Malekzadeh P., Fiouz A.R., Razi H., 2009, Three-dimensional dynamic analysis of laminated composite plates subjected to moving load, Composite Structures 90: 105-114.
[20] Li H., Lam K.Y., 1998, Frequency characteristics of a thin rotating cylindrical shell using the generalized differential quadrature method, International Journal of Mechanical Sciences 40(5): 443-459.
[21] Li H., Lam K.Y., 2001, Orthotropic influence on frequency characteristics of a rotating composite laminated conical shell by the generalized differential quadrature method, International Journal of Solids and Structures 38: 3995-4015.
[22] Wu C.P., Lee C.Y., 2001, Differential quadrature solution for the free vibration analysis of laminated conical shells with variable stiffness, International Journal of Mechanical Sciences 43(8): 1853-1869.
[23] Chen W.Q., Lv C.F., Bian Z.G., 2003, Elasticity solution for free vibration of laminated beams, Composite Structures 62: 75-82.
[24] Soong T.V., 1970, A subdivisional method for linear system, AIAA/ASME Structures 1970: 211-223.
[25] Duda O.R., Hart P.E., Strol D.G., 2001, Pattern Classification, New York, John Wiley & Sons.
[26] Gilli M., Winker P., 2003, A global optimization heuristic for estimating agent based models, Computational Statistics and Data Analysis 42: 299-312.
[27] Luersen M.A., Riche R.L., 2004, Globalized Nelder–Mead method for engineering optimization, Computers & Structures 82: 2251-2260.
[28] Shu C., Richards B.E., 1992, Application of generalized differential quadrature to solve two-dimensional incompressible Navier–Stoaks equations, International Journal for Numerical Methods in Fluids 15: 791-798.
[29] Chen W.Q., Lu C.F., 2005, 3D free vibration analysis of cross-ply laminated plates with one pair of opposite edges simply supported, Composite Structures 69: 77-87.
[30] Lam K.Y., Loy C.T., 1995, Analysis of rotating laminated cylindrical shells by different thin shell theories, Journal of Sound and Vibration 1995: 23-35.
[31] Zhang X.M., 2001, Vibration analysis of cross-ply laminated composite cylindrical shells using the wave propagation approach, Applied Acoustics 62: 1221-1228.
[32] Khalili S.M.R., Davar A., Malekzadeh Fard K., 2012, Free vibration analysis of homogeneous isotropic circular cylindrical shells based on a new three-dimensional refined higher-order theory, International Journal of Mechanical Sciences 56: 1-25.
[33] Shakeri M., Yas M.H., Ghasemi G.M., 2005, Optimal stacking sequence of laminated cylindrical shells using genetic algorithm, Mechanics of Advanced Materials and Structures 12: 305-312.
