MULTI-OBJECTIVE OPTIMIZATION OF VIBRATION CONTROL WITH VISCO-ELASTIC DAMPING
20 Tran Quang Hung MULTI-OBJECTIVE OPTIMIZATION OF VIBRATION CONTROL WITH VISCO-ELASTIC DAMPING Tran Quang Hung University of Science and Technology, The University of Danang; tqhung@dut.udn.vn Abstract - Visco-elastic damping is one of the passive control methods in structural vibration Multilayer visco-elastic patch is preferred due to high level of damping and easy implementation In order to obtain an economical design, many objectives must be considered and this concept leads to solving multi-objective optimization problem Genetic algorithm (GA) is an effective tool in such case This study shows how optimal solutions derived from NSGA algorithm can be A simple plate coupling with a cavity is considered Multi-objective optimization is simulated with many variables regarding geometry and material properties Key words - vibroacoustic; vibration control; visco-elastic patches; multilayer plate; multi-objective optimization; genetic algorithm K s −C Ms K f − f CT Where U and P - structural displacement and air pressure vectors respectively; Ks and Ms-structural stiffness and mass matrices; Kf and Mf – stiffness and mass matrices of fluid; C- fluid-structure coupled matrix; ρf=1,2kg/m3 – mass density of air; Za- acoustic impedance of cavity wall; - pulsation of periodic load Fs; j-complex number Table Material properties Layer Introduction In many cases such as vibration of industrial building floor, machinery vibration, etc., structures may be generally/ partly damaged by periodic load, resulting in high level of sound This problem can be solved by installing visco-elastic patches into structures Many authors have utilized such a method in literature [1, 2, 3, 8] Visco-elastic patches are bonded on the shell member of a structure to form a multilayer zone (Figure 1) in which the structure has the role of a basic layer; restrained layer is normally made of the same material as the structure Figure Visco-elastic treatment As visco-elastic materials possess high capacity of damping, vibration level can be reduced effectively The efficiency not only depends on the position of visco-elastic patches, but also on layer geometry and mechanical properties of materials Industrial problems need to be reduced simultaneously include vibration level and the weight of structure (i.e product cost) In reality, these goals are always in conflict and require solving a multi-objective optimization problem Description and formulation of vibroacoustic problem with visco-elastic damping Double cavity couple with an aluminum plate is considered and described in Figure Opening on the plate assures the interaction between two cavities 3M visco-elastic patches are chosen and their properties are shown in Table Finite element (FE) model of vibroacoustic problem lead to unsymmetrical system [5, 6]: j 0 U + = Fs (1) M f Z 0 A f P a Basic Restrained Viscoelastic Mass density (kg/m3) 2700 2700 1105 Young modulus (MPa) 7x104 7x104 Figure Poisson coefficient 0,33 0,33 0,49 Figure Vibroacoustic problem: FE model (left) and plate treated by visco-elastic patches (right) In the presence of a visco-elastic material, stiffness matrix Ks is complex, the function of and temperature T, which can be expressed as [1, 8, 9]: K s ( , T ) = K se + K sv = K se + G * ( , T )K sv (2) In which Kse is elastic part, Ksv is visco-elastic part, G* is the shear modulus of visco-elastic material (Fig 3) and K svis constant matrix It could be noted that the finite element model of multilayer visco-elastic patch was developed by several authors [1, 2, 3] In order to reduce numbers of DOF, component mode synthesis (CMS) method adapting to coupling system is chosen [4, 10, 11, 12] Let the reduced basis formed by m selected vectors, the full m DOF can be condensed to m DOF by projecting equation (1) on basis : K s −C − M s T Kf f C j 0 U F s = (3) + M f Z a 0 A f P Where X denotes reduced matrix of X, i.e: X = ΦT XΦ THE UNIVERSITY OF DANANG, JOURNAL OF SCIENCE AND TECHNOLOGY, NO 6(91).2015 In this study, full FE model contains 16342 DOF, there are 366 selected vectors introduced in basis The size of model is reduced to 2% 21 coded and represented as a string of the biologic gene Initial population must be generated, and then in order to obtain the next generation, genetic operations are applied (selection, mutation, and crossover) Ranking process assures the rate of convergence Optimal Results Now these variables are considered: - Basic layer (structure): plate thickness h1; Young modulus E1; mass density 1 Real problem does not allow modify Young modulus and mass density of structure, but this study considers these two parameters to get a general case - Visco-elastic layer: thickness h2 - Restrained layer: thickness h3 Initial Population Selection Figure Shear modulus and lost factor of 112 3M visco-elastic material Ranking Based on maximum strain energy law, initial position of visco-elastic patches is detected as shown in Figure (i.e they are installed in zones where shear strain is maximal) This treatment helps to control the first three vibration modes of the plate Mutation Crossover Condensation of model In the next section, one will find optimal parameters of theproblem in condition of many cost functions Evaluation Multiobjective optimization using NSGA algorithm No Convergence In this study, cost functions are the following: - V = S n - S v dS n 2 p dV + c V p dV 4 f V f Figure NSGA algorithm (5) Constrained conditions g(x) of the optimization problem are defined by variation ranges of variables as shown in Table Equal condition h(x)=0 is not considered here Table Initial value and variation range of variables Total mass Mp of the whole structure In the equation (5), V is air volume, p is air pressure and pis gradient of p Multi-objective optimization problem can be written as: 𝑚𝑖𝑛 𝑓(𝑥), 𝑥 ∈ 𝐷𝑛 𝑤𝑖𝑡ℎ: { 𝑔(𝑥) ≤ ℎ(𝑥) = Stop (4) Mean value of sound level [6]: a = - Yes Mean value of square normal velocity of the plate surface S: (6) In which x denotes vector of n variables, vector f(x) collects all cost functions above (i.e Vn2, Pa and M p ), g(x) and h(x) are constrained conditions which will be described in the next section, Dn is feasible space of the problem Because of the conflict between cost functions, this problem has not only one solution, it has in fact an infinite number of solutions Therefore, designer must find optimum surface containing these solutions – that is so called Pareto-optimum One of the methods to solve effectively multi-objective optimization problem is the use of genetic algorithms [15], in which NSGA algorithm adapted to vibroacoustic problem is described in Figure [13, 14] Variable x is Variable h1 E1 ρ1 h2 h3 Initial value 3.10-3m 7.103MPa 2700kg/m3 2.10-4m 5.10-4m Variation (%) ±20 ±20 ±30 ±90 ±50 The objectives are: sound level Pa at frequency f=35.4Hz (first mode); the mean value Vn at frequency f=168.3Hz (sixth mode); the total mass of treated plate Mp The values of Pa and Vn are presented as dB with the reference of 106 NSGA parameters are: number of initial population P=50; probability of crossover pc=0.5; probability of mutation pm=0.05 A good convergence is obtained after 26 generations If we increase the values of P, pc and pm the rate of convergence is better but the number of population in each generation is larger, consequently the simulation time is more important Figure shows 3D Pareto-optimum surface versus the initial design Plots in 2D are also observed in figures 68 22 Tran Quang Hung Figure Pareto-optimum surface; +: all solutions; o:optimal surface It is important to note that all points located at Paretooptimum surface can be the solution of the problem The final solution is decided by the designer For example: Figure Pareto-optimum, Mp versus Pa - If minimizing of sound level Pa and structural weight Mp is important, one can choose solution 1, and associated value of objectives functions are: Pa=77.1dB; Vn=76.55dB and Mp=2.201kg - If minimizing of sound level Pa and structural vibration Vn is important, one can choose solution and associated value of objectives functions are: Pa=74.29dB; Vn=67.62dB and Mp=3.676kg Values of solution and solution are shown in Table Table Optimal solution and Variable Initial value Structure h1 3.10-3 m E1 7.104MPa ρ1 Solution Solution2 3.26.10-3 m 3.60.10-3 m 5.6.104MPa 8.4.104MPa 2700 kg/m3 2227 kg/m3 3380 kg/m3 h2 2.10-4m 0.20.10-4 m 0.20.10-5 m h3 5.10-4 m 7.00.10-4 m 5.38.10-4 m Visco-elastic patch h2 2.10-4 m 0.20.10-4 m 0.20.10-4 m h3 5.10-4 m 6.34.10-4 m 6.81.10-4 m Visco-elastic patch h2 2.10-4 m 0.20.10-4 m 0.20.10-4 m h3 5.10-4 m 7.50.10-4 m 2.50.10-4 m Visco-elastic patch Figure Pareto-optimum, Mp versus Vn Figure Pareto-optimum, Vn versus Pa Figure Response corresponding to solution and THE UNIVERSITY OF DANANG, JOURNAL OF SCIENCE AND TECHNOLOGY, NO 6(91).2015 Finally, Figure shows the responses of system corresponding to solution and solution versus initial design over a band of f = [0 300]Hz Conclusion and remarks This study has solved the multi-objective optimization of vibroacoustic problem in which visco-elastic damping is introduced The set of optimal solutions is found and presented as Pareto-optimum surface by exploring NSGA algorithm Some solutions are shown to enhance their optimal effect in detail It is clear that the result of NSGA algorithm depends on input parameters: number of the initial population, probability of mutation or crossover, etc Therefore, if we would like to obtain more exact solutions, other simulations with new input data could be run REFERENCES [1] M.L Drake and J Soovere, A design guide for damping of aerospace structures, In AFWAL Vibration Damping Workshop Proceedings 3, 1984 [2] T.P Khatua and Y.K Cheung, Bending and vibration of multilayer sandwich beams and plates, International Journal for Numerical Methods in Engineering, Vol 6:11–24, 1973 [3] A.M.G Lima and D.A Rade, Modelling of structures supported on viscoelastic mounts using frf substructuring, In Proceedings of the Twelfth Int Congress on Sound and Vibration, ICSV12, Lisbon, Portugal, 2005 [4] G Masson, B Ait-Brik, S Cogan and N Bouhaddi, Component mode synthesis (cms) based on an enriched ritz approach for efficient structural optimization, Journal of Sound and Vibration, 296:845–860, 2006 23 [5] H.J-P Morand and R Ohayon, Variational formulations for elastoacoustic vibration problem: finite element results, In Second Int Symp on finite element method applied to flow problems, Rappalo (Italy), 14-18 June 1976 [6] H.J-P Morand and R Ohayon, Interactions fluide-structures, Masson, Paris, 1992 [7] C.H Park and A Baz, Vibration control of bending modes of plates using active constrained layer damping, Journal of Sound and Vibration, 227(4):711–737, 1999 [8] M.D Rao, Recent applications of viscoelastic damping for noise control in automobiles and commercial airplanes, In USA Symposium on Emerging Trends in Vibration and Noise Engineering, 2001, India [9] L.C Rogers, Operators and fractional derivatives for viscoelastic constitutive equations, J Rheology, 27(4):351–372, 1983 [10] Q.H Tran, M Ouisse and N Bouhaddi, A robust CMS method for stochastic vibroacoustic problem, In The Ninth International Conference on Computational Structures Technology, Athens, Greece, Seprember 2-5, 2008 [11] Q.H Tran, M Ouisse and N Bouhaddi, A Robust Component Mode Synthesis Method for Stochastic Damped Vibroacoustics, Mechanical Systems and Signal Processing, Volume 24, Issue 1, January 2010, Pages 164-181 (DOI: 10.1016/j.ymssp.2009.06.016) [12] Tran Quang Hung, Model reduction of vibroacoustic problem with viscoelastic and poro-elastic dampings, Journal of Science and Technology, The University of Danang, Volume 1(50), Pages 19-25, 2012 [13] K Deb, Multi-Objective Optimization using Evolutionary Algorithms, John Wiley & Sons, Chichester, UK, 2001 [14] K Deb, S Agrawal, A Pratab and T Meyarivan, A fast elitist nondominated sorting genetic algorithm for multi-objective optimization: Nsga-ii, In KanGAL report 200001, Indian Institute of Technology, Kanpur, India, 2000 [15] C.M Fonseca and P.J Fleming, Genetic algorithms for multiobjective optimization: formulation, discussion and generalization, Proceedings of the Fifth International Conference on Genetic Algorithms, pages 416–423, San Mateo, CA, 1993 (The Board of Editors received the paper on 10/26/2014, its review was completed on 11/10/2014) ... Baz, Vibration control of bending modes of plates using active constrained layer damping, Journal of Sound and Vibration, 227(4):711–737, 1999 [8] M.D Rao, Recent applications of viscoelastic damping. .. mode); the total mass of treated plate Mp The values of Pa and Vn are presented as dB with the reference of 106 NSGA parameters are: number of initial population P=50; probability of crossover pc=0.5;... Conclusion and remarks This study has solved the multi-objective optimization of vibroacoustic problem in which visco-elastic damping is introduced The set of optimal solutions is found and presented