DOI:10.14311/APP.2021.30.0098 Acta Polytechnica CTU Proceedings 30:98–103, 2021 © Czech Technical University in Prague, 2021 available online at http://ojs.cvut.cz/ojs/index.php/app OPTIMIZATION OF TUNED MASS DAMPERS ATTACHED TO DAMPED STRUCTURES - MINIMIZATION OF MAXIMUM DISPLACEMENT AND ACCELERATION Jan Štěpánek∗, Jiří Máca Czech Technical University in Prague, Faculty of Civil Engineering, Department of Mechanics, Thákurova 7, 166 29 Prague 6, Czech Republic ∗ corresponding author: jan.stepanek@fsv.cvut.cz Abstract. A tuned mass damper is a device, which can be highly helpful while dealing with dynamic behaviour of structures. Its proper design is conditioned by knowledge of both loading and the structure properties. In many cases, the structure can be represented by single degree of freedom model, which simplifies the design and optimization of tuned mass dampers. Most of studies focus only on minimization of displacement of the main structure under harmonic force load, however, in many cases, different frequency response function would be more appropriate. This paper presents an extension of design formulas for the H∞ optimization of tuned mass dampers for damped structures and various frequency response functions. Keywords: Acceleration, damped structure, displacement, dynamic response, optimization, tuned mass damper. 1. Introduction Optimal parameters of a tuned mass damper (TMD) have been a subject of many researches since the first half of the 20th century. Since then, several differ- ent optimization criteria have been proposed. The most common ones are called H2, which minimizes the area under a frequency response function (FRF) and H∞, which minimizes the maximum amplitude magnification factor. A first set of nearly optimal TMD parameters for undamped structure came from the existence of invariant points, where the amplitude is independent of TMD damping [1, 2]. A closed form solution of H∞ optimum parame- ters of TMD attached to a damped structure have not been found. In 1978, Ioi and Ikeda presented em- piric design formulas [3], however, they were derived only for an insufficiently large mass ratio µ > 0.03, which makes them inappropriate for certain civil en- gineering structures. It is interesting that despite the formulas being derived for a large mass ratio, they are recommended in footbridge design guidebook SE- TRA [4]. Design charts allowing use of lower mass ra- tio µ > 0.01 were published in 1981 [5]. Asami et al. used a perturbation method to modify the fixed-point solution regarding a main system damping ratio as a perturbation [6]. They obtained a series solution, which is, however, too complicated for engineering application. Therefore, they have proposed a sim- plified version. Abubakar presented a simple modifi- cation of the fixed-point TMD optimum for damped structure. This modification provides very accurate solution for minimization of displacement [7]. Despite many useful formulas and design methods being pro- posed, some modern guidebooks for civil engineering, for example [4, 8], keeps using the original approxi- mate solutions thanks to their easy application. Most of studies focus on a reduction in a maxi- mum amplitude of displacement. This is a suitable criterion for the reduction of structural stress due to a dynamic load. However, in many cases, the main purpose of TMD is to reduce the maximum ampli- tude of acceleration of the main structure. These cases contain machines and sensors which are sensi- tive to vibration, because the vibration tolerance is usually defined in the units of acceleration. Further- more, human comfort criteria inside buildings or on footbridges are usually limited by acceleration. This paper points out the importance of distinction between displacement and acceleration criteria of the H∞ optimization. The approximate optimal stiffness and damping ratio for both criteria are proposed in this paper covering both ground and force excitation. Considering the importance of easy practical applica- tion of the result, the formulas for optimal parame- ters are composed of standard solutions given by the fixed-point method and a correction, which takes non- zero damping into account. Therefore, in the case of zero structural damping, we obtain standard result given by the fixed-point method. In the case of non- zero damping, we obtain results given by the modified formulas. 2. Equations of Motion The main structure is represented by a single degree of freedom (SDOF) system and it consists of a mass m1, stiffness k1, and damping c1. TMD is composed of a mass m2, an elastic member with stiffness k2 and a viscous member with damping c2. The struc- 98 http://dx.doi.org/10.14311/APP.2021.30.0098 http://ojs.cvut.cz/ojs/index.php/app vol. 30/2021 Optimization of tuned mass dampers in damped structures Figure 1. a) Force excitation, b) Support excitation. ! k1 + k2 −k2 −k2 k2 " # x1 x2 $ + ! c1 + c2 −c2 −c2 c2 " # ẋ1 ẋ2 $ + ! m1 0 0 m2 " # ẍ1 ẍ2 $ = F (t) (1) ture can be loaded by a harmonic force (Figure 1a) or excited by support motion (Figure 1b). This pa- per covers both loading cases, because the optimal parameters of TMD are not identical for force and support excitation. The equations of motion are de- scribed by Eq. 1, where F (t) = # f (t) 0 $ in the case of force excitation, and F (t) = # −ẍ0(t)m1 −ẍ0(t)m2 $ in the case of support excitation. x, ẋ and ẍ denote the displace- ment, velocity and acceleration, respectively. It is im- portant to note that x induced by support excitation is a relative displacement between the support and the mass. The absolute displacement is xa = x + x0. 3. Optimal parameters In order to solve Equation 1, it is advantageous to use following dimensionless substitutions and notation for further operations [9]: ω1 = % k1/m1 natural frequency of the main structure, ω2 = % k2/m2 natural frequency of TMD, µ = m2/m1 mass ratio, β = ω2/ω1 frequency ratio, Ω = ω/ω1 forcing frequency ratio, c1,cr = 2 √ k1m1 critical damping of the main structure, c2,cr = 2 √ k2m2 critical damping of TMD, ξ1 = c1/c1,cr damping ratio of the main struc- ture, ξ2 = c2/c2,cr damping ratio of TMD, Nf (Ω)= (2ξ2βΩ)2 + (β2 − Ω2)2, Na(Ω)= [1 + (2ξ1Ω)2][(β2 − Ω2)2 + (2ξ2βΩ)2], Nr (Ω) = [(1 + µ)β2 − Ω2] 2Ω4 + (1 + µ)2(2ξ2βΩ)2Ω4, D(Ω) = [(1 − Ω2)(β2 − Ω2) − µβ2Ω2 − 4ξ1ξ2βΩ2] 2 + +4Ω2[(β2 − Ω2)ξ1 + (1 − Ω2 − µΩ2)βξ2] 2. Assuming a harmonic load according to Figure 1, a steady state part of response can be expressed from Eq. 1. The frequency response functions which de- scribe the normalized response of the structure are summarized in Table 1. 3.1. fixed-point approach The approximate H∞ optimum for main structure with no or negligible damping can be reached by the fixed-point method. This method was firstly de- scribed by Den Hartog [1]. He postulated that the approximate optimum frequency ratio βuopt is reached if the value of FRF is equal in invariant points P and Q, where the value of FRF is not affected by TMD damping c2. Brock lately stated that the optimum damping ratio ξu2,opt can be taken as the average of two optima, each calculated separately for one of two points P and Q and he derived the well known for- mula of optimum damping for FRF No. 1 in Table 1 [2]. Using this approach, we can derive the ap- proximate optimum solutions for various FRFs. The results are summarized in Table1. It should be noted that the optimum parameters are identical for trans- fer functions 1 and 3, as well as for 2 and 4. This is can be seen in Na(Ω, ξ1 = 0) = Nf (Ω). However, this is true only if ξ1 = 0. 3.2. Non-zero damping The fixed-point approach is not available if we intro- duce damping of the main structure. An analytical solution of optimum TMD parameters attached to a damped main structure have not been found and according to Asami et al., the solution is probably impossible [6]. Therefore, we need to rely on empiric and approximate solutions. In order to define an empiric formulas for optimum frequency ratio βopt and optimum damping ratio ξopt,2, the values of the parameters were found numer- ically on a surface of 44x41 points for 0.001 ≤ µ ≤ 0.2 and 0 ≤ ξ1 ≤ 0.2. The density of mesh was increased in the area of low mass ratio 0.001 ≤ µ ≤ 0.005, be- cause in this range, the optimum values of βopt and 99 Jan Štěpánek, Jiří Máca Acta Polytechnica CTU Proceedings No. excitation frequency responsefunction type frequency response function β u opt ξ u 2,opt 1 force displacement &&&& x1 f0/k1 &&&& = ' Nf (Ω) D(Ω) 1 1 + µ ' 3µ 8(1 + µ) 2 force acceleration &&&& ẍ1 ω21 f0/k1 &&&& = Ω 2 ' Nf (Ω) D(Ω) ( 1 1 + µ ' 3µ 4(2 + µ) 3 support abs. displacement &&&& xa1 x0 &&&& = ' Na(Ω) D(Ω) 1 1 + µ ' 3µ 8(1 + µ) 4 support abs. acceleration &&&& ẍ1 a ω21 x0 &&&& = Ω 2 ' Na(Ω) D(Ω) ( 1 1 + µ ' 3µ 4(2 + µ) 5 support rel. displacement &&&& ω21 x1 ẍ0 &&&& = 1 Ω2 ' Nr (Ω) D(Ω) ' 2 − µ 2(1 + µ)2 ' 3µ 4(1 + µ)(2 − µ) 6 support rel. acceleration &&&& x1 x0 &&&& = ' Nr (Ω) D(Ω) ' 2 + µ 2(1 + µ)2 ' 3µ 8(1 + µ) Table 1. Optimum parameters of TMD based on the fixed-point method. ξopt,2 change more rapidly then in the rest of the func- tion. A convergence problem may occur while find- ing ξopt,2 numerically, because a structural response is much more sensitive to TMD tuning than to its damping. The empiric formulas are combination of the fixed- point solution and a correction function, which takes the structural damping into account. The original ap- proach [3] uses a correction function composed of 2nd order polynomial of µ and ξ1, but we found it neither simplest nor best-fitting. Numerous types of func- tions, including polynomial, exponential, logarithmic and hyperbolic ones, were tested using a non-linear regression to find the best approximation of the op- timum TMD setting regarding a simple application. Thanks to similar characteristics of the optimum pa- rameters for all frequency response functions, two common function were defined: βopt = βuopt + ξ1 ) a1ξ1 + a2 * µ 1 + µ + 1 3 , (2) ξ2,opt = ξu2,opt + ξ1[b1 + b2 ln(µ) + b3µ + b4µξ 2 1 ] (3) As we can see in the shape of Eqs. (2) and (3), the optimum values of TMD parameters βopt = βuopt and ξ2,opt = ξu2,opt if ξ1 = 0. In other words, the for- mulas preserve the fixed-point approach if the main structure remains undamped. The coefficient of determination was chosen as the comparative criterion for function fitting and it reached higher than 0.997 in all cases of FRFs for our selected function types. The maximum error of βopt was lower than 1%. The maximum error of ξ2,opt reached up 5%, but it occurred only for extremely small mass ratio, where the optimum damping is close to zero and such a high error may be produced rather by a precisions of the numerical solution than by an inaccurate approximation. The average relative er- ror of ξ2,opt was lower than 1% for all transfer func- tions. The coefficients ai and bi were found using linear regression and they are summarized in Table 2. Figure 2 presents the improvement of FRF using desing formulas (2) and (3) for relative displacement of structure under support excitation. Figure 2. Improvement of TF 5 (µ = 0.05, ξ1 = 0.05) using desing formulas (2) and (3). Figures 3-8 compare the numerical results with the approximations given by design formulas. We can see that in most cases, the empiric formulas give very pre- cise results. It is also visible that ξ2,opt rises as the structural damping increases. However, The effect of ξ1 to the optimum frequency ratio is slightly more 100 vol. 30/2021 Optimization of tuned mass dampers in damped structures FRF No. a1 a2 b1 b2 b3 b4 1 -0.7636 -0.8748 0.1801 0.0140 0 -4.6350 2 1.2470 0.2644 0.2101 0.0178 1.264 0 3 -0.5561 -0.8466 0.1782 0.01367 0.01865 -4.3310 4 1.5600 0.2778 0.1985 0.01552 1.1420 29.0900 5 -0.7639 -1.5300 0.2005 0.01650 0.4629 0 6 0.9731 -0.2176 0.1783 0.0145 0 0 Table 2. Coefficients ai and bi for design equations (2) and (3). Figure 3. Optimum TMD parameters: TF No. 1. Figure 4. Optimum TMD parameters: TF No. 2. Figure 5. Optimum TMD parameters: TF No. 3. 101 Jan Štěpánek, Jiří Máca Acta Polytechnica CTU Proceedings Figure 6. Optimum TMD parameters: TF No. 4. Figure 7. Optimum TMD parameters: TF No. 5. Figure 8. Optimum TMD parameters: TF No. 6. complicated. As we can see in Figures 3-8, increase of ξ1 leads to decrase of βopt in the case of displacement FRFs (No. 1,3,5). In the case acceleration FRFs (No. 2,4,6), the increase of ξ1 has an opposite effect. Sen- sitivity of optimum parameters to the damping ratio ξ1 varies among FRFs. It is interesting to note that in the case of acceleration FRFs, the optimum tuning βopt may slightly rise above one, which is an unusual situation in TMD design. 4. Conclusions The design formulas for H∞ optimization of TMD attached to a damped structure are presented in this paper. The formulas are appropriate for TMDs with the mass ratio µ between 0.001 and 0.2, and for the damping ratio of the main structure between 0 and 102 vol. 30/2021 Optimization of tuned mass dampers in damped structures 0.2. This is a sufficient range for most of TMDs lo- calized in civil engineering structures and we consider the error of design parameters negligible in this range. The formulas were optimized for six frequency re- sponse functions, which describe relative and abso- lute displacement or acceleration for both force and support excitation. It is necessary to chose a suit- able FRF for TMD optimization regarding the pur- pose of TMD. For the purpose of a stress reduction of the main structure, FRFs which describe displace- ment should be used. On the other hand, in order to increase a comfort of people in high-rise buildings and pedestrian on footbridges, or to reduce the vi- bration of equipment localized in the main structure, the acceleration FRFs are an appropriate choice. We also find out that reduction of acceleration may lead to TMD tuned to a resonant or even slightly higher frequency. An easy application is preserved. Only two simple formulas are presented, however, they can be used for all considered transfer functions with different param- eters ai and bi. Acknowledgements The authors gratefully acknowledge support from the Czech Technical University in Prague, project SGS20/038/OHK1/1T/11 Development and application of numerical algorithms for analysis and modelling in mechanics of structures and materials. References [1] J. P. Den Hartog. Mechanical Vibrations. McGraw Hill, New York, 1934. [2] J. Brock. A note on the damped vibration absorber. Journal of Applied Mechanics 68(A):284, 1946. [3] T. Ioi, K. Ikeda. On the dynamic vibration damped absorber of the vibration systems. Bulletion of the JSME 21(151):64–71, 1978. [4] SETRA. Footbridges: Assessment of vibrational behaviour of footbridges under pedestrian loading, 2006. [5] S. E. Randall, D. M. Halsted, D. L. Taylor. Optimum vibration absorbers for linear damped systems. Journal of Mechanical Design 103:908–913, 1981. [6] T. Asami, O. Nishihara, A. M. Baz. Analytical solutions to H∞ and H2 optimization of dynamic vibration absorbers attached to damped linear systems. Journal of Vibration and Acoustics 124:284–295, 2002. [7] I. M. Abubakar, B. J. M. Farid. Generalized den hartog tuned mass damper system for control of vibrations in structures. Earthquake Resistant Engineering Structures 104:185–193, 2009. [8] FIB Bulletin 32: Guidelines for the design of footbridges, 2005. [9] O. Nishihara, T. Asami. 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