Abstract
In this study, the multi-objective optimal design of hybrid viscoelastic/composite sandwich beams for minimum weight and minimum vibration response is aimed. The equation of motion for linear vibrations of a multi-layer beam is derived by using the principle of virtual work in the most general form. These governing equations together with the boundary conditions are discretized by the generalized differential quadrature method (GDQM) in the frequency domain for the first time. Also, the time and temperature dependent properties of the viscoelastic materials are taken into consideration by a novel ten-parameter fractional derivative model that can realistically capture the response of these materials. The material variability is accounted for by letting an optimization algorithm choose a material freely out of four fiber-reinforced composite materials and five viscoelastic damping polymers for each layer. The design parameters, i.e., the orientation angles of the composites, layer thicknesses and the layer materials that give the set of optimal solutions, namely the Pareto frontier, is obtained for the three and nine-layered clamped-free sandwich beams by using a variant of the non-dominated sorting genetic algorithms (NSGA II).
Similar content being viewed by others
References
Araujo AL, Martins P, Soares CMM, Soares CAM, Herskovits J (2009) Damping optimization of viscoelastic laminated sandwich composite structures. Struct Multidiscip Optim 39:569–579. doi:10.1007/s00158-009-0390-4
Araujo AL, Soares CMM, Soares CAM (2008) Optimal design of active, passive, and hybrid sandwich structures. Proc SPIE 6926. doi:10.1117/12.758224
Araujo AL, Soares CMM, Soares CAM, Herskovits J (2010) Optimal design and parameter estimation of frequency dependent viscoelastic laminated sandwich composite plates. Compos Struct 92:2321–2327. doi:10.1016/j.compstruct.2009.07.006
Arikoglu A (2014) A new fractional derivative model for linearly viscoelastic materials and parameter identification via genetic algorithms. Rheol Acta 53:219–233
Arikoglu A, Ozkol I (2010) Vibration analysis of composite sandwich beams with viscoelastic core by using differential transform method. Compos Struct 92:3031–3039. doi:10.1016/j.compstruct.2010.05.022
Arikoglu A, Ozkol I (2012) Vibration analysis of composite sandwich plates by the generalized differential quadrature method. AIAA J 50:620–630. doi:10.2514/1.j051287
Chen WJ, Liu ST (2016) Microstructural topology optimization of viscoelastic materials for maximum modal loss factor of macrostructures. Struct Multidiscip Optim 53:1–14. doi:10.1007/s00158-015-1305-1
Deb K, Pratap A, Agarwal S, Meyarivan T (2002) A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans Evol Comput 6:182–197. doi:10.1109/4235.996017
Ditaranto RA (1965) Theory of vibratory bending for elastic and viscoelastic layered finite-length beams. J Appl Mech 32:881–886
Fasana A, Marchesiello S (2001) Rayleigh-Ritz analysis of sandwich beams. J Sound Vibr 241:643–652. doi:10.1006/jsvi.2000.3311
Gray F (1953) Pulse code communication. United States Patent Number 2632058
Hamdaoui M, Robin G, Jrad M, Daya EM (2015) Optimal design of frequency dependent three-layered rectangular composite beams for low mass and high damping. Compos Struct 120:174–182. doi:10.1016/j.compstruct.2014.09.062
He J, Fu Z-F (2001) Modal analysis. Butterworth-Heinemann, Oxford
Johnson C, Kienholz D, Rogers L (1981) Finite element prediction of damping in beams with constrained viscoelastic layers. Shock and Vibration Bulletin 51:71–81
Kerwin EM (1959) Damping of flexural waves by a constrained viscoelastic layer. The Journal of the Acoustical Society of America 31:952–962
Lall AK, Nakra BC, Asnani NT (1983) Optimum Design of Viscoelastically Damped Sandwich Panels. Eng Optimiz 6:197–205. doi:10.1080/03052158308902470
Lifshitz JM, Leibowitz M (1987) Optimal sandwich beam Design for Maximum Viscoelastic Damping Int. J Solids Struct 23:1027–1034. doi:10.1016/0020-7683(87)90094-1
Loja MAR, Barbosa JI, Soares CMM (2015) Dynamic behaviour of soft core sandwich beam structures using kriging-based layerwise models. Compos Struct 134:883–894. doi:10.1016/j.compstruct.2015.08.096
Mace M (1994) Damping of beam vibrations by means of a thin constrained viscoelastic layer - evaluation of a new theory. J Sound Vibr 172:577–591. doi:10.1006/jsvi.1994.1200
Madeira JFA, Araujo AL, Soares CMM, Soares CAM (2015) Multiobjective optimization of viscoelastic laminated sandwich structures using the direct MultiSearch method. Comput Struct 147:229–235. doi:10.1016/j.compstruc.2014.09.009
Marcelin JL, Trompette P, Dornberger R (1995) Optimization of composite beam structures using a genetic algorithm. Struct Optimization 9:236–244. doi:10.1007/Bf01743976
Mead DJ, Markus S (1969) FORCED VIBRATION OF A 3-LAYER, DAMPED SANDWICH BEAM WITH ARBITRARY BOUNDARY CONDITIONS. J Sound Vibr 10:163–175. doi:10.1016/0022-460x(69)90193-x
Mead DJ, Markus S (1970) Loss factors and resonant frequencies of encastre damped sandwich beams. J Sound Vibr 12:99–112. doi:10.1016/0022-460x(70)90050-7
Nashif AD, Jones DIG, Henderson JP (1985) Vibration damping. Wiley, New York
Pradhan SC, Ng TY, Lam KY, Reddy JN (2001) Control of laminated composite plates using magnetostrictive layers. Smart Mater Struct 10:657–667. doi:10.1088/0964-1726/10/4/309
Rikards R, Chate A (1995) Optimal-Design of Sandwich and Laminated Composite Plates Based on the planning of experiments. Struct Optimization 10:46–53. doi:10.1007/Bf01743694
Shi YM, Sol H, Hua HX (2006) Material parameter identification of sandwich beams by an inverse method. J Sound Vibr 290:1234–1255. doi:10.1016/j.jsv.2005.05.026
Shu C, Du H (1997) Implementation of clamped and simply supported boundary conditions in the GDQ free vibration analysis of beams and plates. Int J Solids Struct 34:819–835. doi:10.1016/s0020-7683(96)00057-1
Tang SJ, Lumsdaine A (2008) Analysis of constrained damping layers, including normal-strain effects. AIAA J 46:2998–3011. doi:10.2514/1.33068
Tornabene F, Viola E (2007) Vibration analysis of spherical structural elements using the GDQ method. Comput Math Appl 53:1538–1560. doi:10.1016/j.camwa.2006.03.039
Tornabene F, Viola E (2008) 2-D solution for free vibrations of parabolic shells using generalized differential quadrature method. Eur J Mech a-Solid 27:1001–1025. doi:10.1016/j.euromechsol.2007.12.007
Xu C, Lin S, Yang YZ (2015) Optimal design of viscoelastic damping structures using layerwise finite element analysis and multi-objective genetic algorithm. Comput Struct 157:1–8
Zheng H, Cai C (2004) Minimization of sound radiation from baffled beams through optimization of partial constrained layer damping treatment. Appl Acoust 65:501–520. doi:10.1016/j.apacoust.2003.11.008
Zheng H, Cai C, Pau GSH, Liu GR (2005) Minimizing vibration response of cylindrical shells through layout optimization of passive constrained layer damping treatments. J Sound Vibr 279:739–756. doi:10.1016/j.jsv.2003.11.020
Zheng H, Cai C, Tan XM (2004) Optimization of partial constrained layer damping treatment for vibrational energy minimization of vibrating beams. Comput Struct 82:2493–2507. doi:10.1016/j.compstruc.2004.07.002
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Arikoglu, A. Multi-objective optimal design of hybrid viscoelastic/composite sandwich beams by using the generalized differential quadrature method and the non-dominated sorting genetic algorithm II. Struct Multidisc Optim 56, 885–901 (2017). https://doi.org/10.1007/s00158-017-1695-3
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00158-017-1695-3