OPTIMUM TUNING OF MASS DAMPERS EMPLOYING FLOWER POLLINATION ALGORITHM IN FREQUENCY DOMAIN

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1 -3 Ekim 27 ANADOLU ÜNİVERSİTESİ ESKİŞEHİR OPTIMUM TUNING OF MASS DAMPERS EMPLOYING FLOWER POLLINATION ALGORITHM IN FREQUENCY DOMAIN S.M. Nigdeli and G. Bekdaş 2 Assoc. Prof. Dr., Department of Civil Engineering, Istanbul University, 3432, Avcılar, Istanbul, Turkey 2 Assoc. Prof. Dr., Department of Civil Engineering, Istanbul University, 3432, Avcılar, Istanbul, Turkey melihnig@istanbul.edu.tr, bekdas@istanbul.edu.tr ÖZET: Kütle sönümleyicileri doğru olarak ayarlandıklarında, yapılar, makinalar, gemiler veya başka mekanik sistemlerin titreşimlerinin azaltılmasında etkilidir. Ayarlanma işlemi ana yapının sönümü olmasından dolayı karışıktır ve optimum ayarlama için nümerik iterasyonlar gerekmektedir. Bu çalışmada yapının ivme transfer fonksiyonu değerini azaltan kütle sönümleyicisinin optimizasyonu için çiçek tozlaşma algoritması tabanlı metot sunulmuştur. Çiçek tozlaşma algoritması, çiçek açan bitkilerin tozlaşma işleminden ilham alan bir metazsezgisel algoritmadır. İki çeşit optimizasyon bir geçiş olasılığı ile kullanılmaktadır. Optimizasyon çeşitleri sırasıyla çapraz ve kendi kendini tozlaştırma işlemlerinden ilham alan genel ve yerel optimizasyondur. Sayısal örnekte, % sönüm oranlı ve çeşitli periyotlu tek serbestlik dereceli yapılar incelenmiştir. Sonuçlar klasik metotlar ve armoni araştırma yöntemi tabanlı metot ile karşılaştırılmıştır. Her iki metasezgisel tabanlı metot optimum sonuçların bulunmasında etkilidir ve klasik metotlara göre belirgin olarak üstündür. Amaç fonksiyonu yönünden çiçek tozlaşma algoritması tabanlı metot, armoni araştırma algoritması tabanlı metoda göre bir miktar üstündür. Ayrıca, çiçek tozlaşma algoritması daha az hesaplama eforu gerektirmektedir. ANAHTAR KELİMELER: Ayarlı kütle sönümleyicisi, optimizasyon, yapısal kontrol ABSTRACT: Tuned mass dampers (TMDs) are effective in the reduction of vibrations of structures, machines, ships or other mechanical systems if the parameter of TMDs is correctly tuned. The tuning process is complex because of inherent damping of the main structure and numerical iterations needed for an optimum tuning. In the present study, flower pollination algorithm (FPA) based methodology optimizing TMDs for reduction of the acceleration of transfer function value of structure is presented. FPA is a metaheuristic algorithm inspired from the pollination process of flowering plants. It uses two types of optimization with a switch probability. These types are global and local optimization inspired from the cross and self-pollinations, respectively. In the numerical example, single degree of freedom structures were investigated for different periods and % inherent damping ratio. The results were compared with the classical methods and harmony search (HS) based methodology. Both metaheuristic based methods are effective to find optimum TMD parameter and these methods are significantly effective comparing to the classical methods. The methodology employing FPA is slightly more effective than the HS based method in reduction of the objective function. Also, FPA needs less computational effect than HS. KEYWORDS: Tuned Mass Damper, Optimization, Structural Control.. INTROUCTION Tuned mass dampers (TMDs) are vibration absorbers used in mechanical systems including structures, machines, ship or several mechanical systems. The efficiency of TMDs are related to the tuning process in which the parameters such as mass (m d), stiffness coefficient (k d) (or period) and damping coefficient (c d) (or damping ratio) of TMD are found (Fig. ).

2 -3 Ekim 27 ANADOLU ÜNİVERSİTESİ ESKİŞEHİR x d k d x N cd m d m N c N k N x i m i c i k i x m c k Figure. A shear building model with a TMD on the top There are several well-known expressions about the tuning of dampers and some of these equations are given in Table. Table. The closed form expressions for TMD tuning ωd, opt cd, Method f opt = ξ d, opt = ω 2m ω (947) (982) (997).. x g + µ + µ s ( µ 2) ξ + µ µ + µ d 3µ 8( + µ ) opt d, opt µ ( µ 4) 4( + µ )( µ 2) In the equations, the optimum frequency (f opt) and damping ratio (ξ d,opt) of TMD is calculated according to the mass ratio of TMD and structure (μ) and inherent damping (ξ). The frequency of the structure is ω s, while ω d,opt, c d,opt and m d are optimum frequency of TMD, optimum damping of coefficient of TMD and mass of TMD, respectively. The best and optimum tuning option for TMD are the usage of metaheuristic algorithms such as genetic algorithm (Hadi and Arfiadi 998; Marano et al. 2; Marano and Greco 24), particle swarm optimization (Leung and Zhang 29; Leung et al. 28), bionic algorithm (Steinbuch 2), harmony search (Bekdaş and Nigdeli 2; 27; Bekdaş and Nigdeli 27; Nigdeli et al. 26), ant colony optimization (Farshidianfar and Soheili 23a), artificial bee optimization (Farshidianfar and Soheili 23b), shuffled complex evolution (Farshidianfar and Soheili 23c), bat algorithm (Bekdaş and Nigdeli 27), teaching learning based optimization (Nigdeli and Bekdaş 2) and flower pollination algorithm (Nigdeli et al. 26). In this paper, FPA based methodology using frequency domain analyses of mechanical system is presented. The investigations were done for the structural system with.s, s,.s and 2s period and % inherent damping ratio. The compared methods are HS based frequency domain approach (Nigdeli and Bekdaş 27, Nigdeli et al. 26) and closed form expressions given in Table. ξ + µ + µ + µ

3 -3 Ekim 27 ANADOLU ÜNİVERSİTESİ ESKİŞEHİR 2. METHODOLODGY In nature, every process has a final goal. To reach to the final goal, a special procedure is in progress. Metaheuristic algorithms are the mathematical form of these algorithms developed for reaching to an objective in an engineering design. The objectives may be related to cost, performance and safety. Thus, the problem is mentioned as an optimization process. Flower pollination algorithm (FPA) is also a metaheuristic algorithm inspired from the two types of pollination process which are biotic and abiotic pollination. In biotic pollination, pollens are carried by the pollinators such as birds, insects or bees. These flying living beings obey the rules of Lévy flight. For that reason, biotic or cross pollination is formulated as global optimization part of the FPA and formulated as Eq. () by using a Lévy distribution (LD). X = X + LD X X ) () new, i old, i ( best old, i i is the modified vector and the total number of vectors are equal to the pollination number (p). X is the set of modified design variables including mass (m d), period (T d) and damping ratio (ξ d) of the TMD. The subscripts new and old represent modified and existing design variable with a best value (X best) in means of the objective function (f) which is the maximum acceleration transfer function in decibel as seen in Eq. (2). f = 2Log max(tfn (ω)) (2) The transfer function is the ratio of the Laplace transforms of the accelerations of the top story of the structure and ground as symbolized as TF N (ω) which is a function of frequency (ω). Abiotic pollination is the reproduction process in which the fertilization occurs from the pollens of the same flower of plant. Abiotic or self-pollination is used as the local optimization. In this part, the convergence is essential while the trapping of a local optimum is prevented by global optimization. Local optimization uses a linear distribution and a random number between and (rand(,)) is used. It is formulated as Eq. (3). Two randomly chosen existing solutions (m and k) are used in local optimization. X = X + rand (,)( X X ) (3) new, i old, i m k By combining these two types of pollination with the flower constancy which is the probability of reproduction considering the similarity of two flowers, FPA is developed by Yang (22) and a switch probability is used to control the type of the optimization. 3. NUMERICAL EXAMPLES Four single degree of freedom structures were investigated. The natural periods of the structures are.s, s,.s and 2s. The inherent damping of structures is equal to %. The mass ratio was searched between % and %, while the period of TMD (T d) was taken between.8 and.2 times of the period of the structure. The cases of the ranges of damping ratio of TMD (ξ d) were investigated. In case, ξ d was search between % and %, while 3% is the upper bound for case 2. The results are presented in Table 2 with the comparisons of classical methods and harmony search (HS) based methodology []. The objective function is f which the maximum acceleration transfer function value in db.

4 -3 Ekim 27 ANADOLU ÜNİVERSİTESİ ESKİŞEHİR Period (s) FPA Case FPA Table 2. The optimum results HS (Nigdeli et HS (Nigdeli et al. 26) al. 26) Case Den Hartog (947) (982) (997) μ T d (s) ξ d f μ T d (s) ξ d f μ T d (s) ξ d f μ T d (s) ξ d f CONCLUSION The both metaheuristic algorithms are effective to find the optimum values. The optimum results are the same with minor differences. The top story acceleration transfer function plots were given in Figure Structure with. s period Case Figure 2. Frequency domain plots for structure with. s period

5 -3 Ekim 27 ANADOLU ÜNİVERSİTESİ ESKİŞEHİR 2 2 Structure with s period Case Figure 3. Frequency domain plots for structure with s period 2 2 Structure with. s period Case Figure 4. Frequency domain plots for structure with. s period 2 Structure with 2 s period Case Figure. Frequency domain plots for structure with 2 s period

6 -3 Ekim 27 ANADOLU ÜNİVERSİTESİ ESKİŞEHİR In these plots, both metaheuristic algorithm are shown in a single representation since the results have only minor differences. These minor differences are resulted from the effectiveness of FPA algorithm on finding precise solutions. This difference between the algorithms are not counted as an advantage, but the computational effort needed for FPA algorithms is less than HS in optimization of TMD. REFERENCES Bekdaş, G. and Nigdeli, S.M. (2). Estimating Optimum Parameters of Tuned Mass Dampers using Harmony Search. Engineering Structures, 33, Bekdaş, G., & Nigdeli, S. M. (27). Metaheuristic based optimization of tuned mass dampers under earthquake excitation by considering soil-structure interaction. Soil Dynamics and Earthquake Engineering 92, J.P. (99). Mechanical Vibrations, McGraw-Hill, New York. Farshidianfar, A., Soheili, S. (23a). Ant colony optimization of tuned mass dampers for earthquake oscillations of high-rise structures including soil structure interaction. Soil Dyn. Earthquake Eng.,, Farshidianfar, A., Soheili, S. (23b). ABC optimization of TMD parameters for tall buildings with soil structure interaction. Interact. Multiscale Mech., 6, Farshidianfar, A., Soheili, S. (23c). Optimization of TMD parameters for Earthquake Vibrations of Tall Buildings Including Soil Structure Interaction. Int. J. Optim. Civ. Eng., 3, Hadi, M.N.S. and Arfiadi, Y. (998). Optimum design of absorber for MDOF structures. Journal of Structural Engineering-ASCE, 24, Leung, A.Y.T. and Zhang, H. (29). Particle swarm optimization of tuned mass dampers. Engineering Structures, 3, Leung, A.Y.T., Zhang, H., Cheng, C.C. and Lee, Y.Y. (28). Particle swarm optimization of TMD by nonstationary base excitation during earthquake. Earthquake Engineering and Structural Dynamics, 37, Marano, G.C., Greco, R. and Chiaia, B. (2). A comparison between different optimization criteria for tuned mass dampers design. Journal of Sound and Vibration, 329, Marano, G.C. and Greco, R. (2). Optimization criteria for tuned mass dampers for structural vibration control under stochastic excitation. Journal of vibration and control, 7(), Nigdeli, S. M., & Bekdaş, G. (27). Optimum tuned mass damper design in frequency domain for structures. KSCE Journal of Civil Engineering, 2(3), Nigdeli SM, Bekdaş G (2). Teaching-Learning-Based Optimization for Estimating Tuned Mass Damper Parameters. 3rd International Conference on Optimization Techniques in Engineering (OTENG '), 7-9 November 2, Rome, Italy. Nigdeli SM, Bekdas G, Yang XS (26). Optimum Tuning of Mass Dampers for Seismic Structures Using Flower Pollination Algorithm. 7th European Conference of Civil Engineering (ECCIE '6), 7-9 December 26, Bern, Switzerland. Nigdeli SM, Bekdas G, Sayin B (26). Optimum Tuned Mass Damper Design using Harmony Search with Comparison of Classical Methods. 4th International Conference of Numerical Analysis and Applied Mathematics, 9-2 September 26, Rhodes, Greece. Sadek F., Mohraz B., Taylor A.W., Chung R.M. (997). A method of estimating the parameters of tuned mass dampers for seismic applications. Earthq. Eng. Struct. D. 26, Steinbuch, R. (2). Bionic optimisation of the earthquake resistance of high buildings by tuned mass dampers. Journal of Bionic Engineering, 8, Yang X.-S (22). Flower pollination algorithm for global optimization. International Conference on Unconventional Computing and Natural Computation. Springer Berlin Heidelberg. G.B. (982). Optimum absorber parameters for various combinations of response and excitation parameters. Earthq. Eng. Struct. D., 38 4.