Acta Polytechnica CTU Proceedings doi:10.14311/APP.2020.26.0100 Acta Polytechnica CTU Proceedings 26:100–106, 2020 © Czech Technical University in Prague, 2020 available online at https://ojs.cvut.cz/ojs/index.php/app DESIGN OF TUNED MASS DAMPERS FOR LARGE STRUCTURES USING MODAL ANALYSIS 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. Tuned mass damper is a device, which can be highly useful when dealing with excessive vibration and is widely used in many engineering fields. However, its proper design and optimization is a complicated task. This study uses mode superposition method to speed up the evaluation of dynamic response. The speed of response calculation allows for a quick calculation of frequency response function and numerical optimization of tuned mass dampers. This optimization method is demonstrated on a numerical example of a cable stayed footbridge. The example compares a simplified and widely used design method of tuned mass damper with numerical optimization. Keywords: Dynamic response, modal coordinates, tuned mass damper, optimization. 1. Introduction The tuned mass damper (TMD) is a device composed of a spring, mass and a viscous damper, which is widely used in order to reduce mechanical vibration. However, its high efficiency is conditioned by appropriate design. Unfortunately, the closed form solution for its optimal parameters has been found only for special, simplified cases. Numerous studies dealing with optimal design of TMD have been published since the beginning of the 20th cetury, but most of them only consider a single dergee of freedom (SDOF) main structure. The first design method for stifness of TMD spring was pro- posed by Den Hartog [1]. It comes from observation of two invariant points on frequency response function (FRF), which are independent of TMD damping. The optimal damping was derived by Brock [2] for both motion and force excitation. These settings of param- eters are close to the optimal ones for undamped main structure. Thanks to simple application, it is still a recomended design method in several modern guide- lines [3–5]. The closed form solution of optimal tuning and damping for undamped main SDOF system was found by O. Nishihara and T. Asami [6]. Abubakar and Farid presented a numericaly derived formula for design of TMD on clasically damped SDOF struc- ture [7]. Simplification of main structure by SDOF model is sometimes insufficient and may lead to wrong tuning or unnecessary big mass of TMD. Especially, large structures such as bridges or high buildings, which may be sensitive to dynamic load, cannot be modeled as SDOF structures. Rana and Soong showed the importance of numerical tuning in case of multiple degrees of freedom (MDOF) structures and shows the example of multiple TMDs, each tuned for particular mode [8]. Ozer and Royston extended Hartog’s idea to MDOF systems [9], but the proposed method re- quires inversion of the dynamic stiffness matrix and numerical solution of system of equations, which may be impractical. This paper presents numerical optimization of sin- gle and multiple TMDs, on a MDOF damped stucture. The goal is to minimize the maximal amplitude (also called H∞ optimization) of the structure caused by simple harmonic load. In the case of H∞ numeri- cal optimization, the problem is that large amount of solutions needs to be evaluated for various TMD parameter values and forcing frequencies. To reduce the camputational complexity of the problem, method proposed in [10] was applied. This method operates with transformation to modal coordinates and allows us to find steady-state response of structures with TMD without solving a system of equations. This method is also suitable for design of TMDs for existing structures where excessive vibrations were measured and modal characteristics are known. 2. Response calculation The main structure is represented by classically damped MDOF system with n degrees of freedom and it is loaded by general dynamic load F(t). The system is described by equation of motion Kx + Cẋ + Mẍ = F(t) , (1) where K, C and M represent stiffness matrix, damp- ing matrix, and mass matrix, respectively, F (t) is the loading vector and x is vector of displacement. The vectors of velocity and acceleration are represented by symbols ẋ and ẍ. 2.1. single TMD TMD is attached to the m-th degree of freedom. The total system can be represented by system of n + 1 equations, but the equation which corresponds to the 100 https://doi.org/10.14311/APP.2020.26.0100 https://ojs.cvut.cz/ojs/index.php/app vol. 26/2020 Example of an Article with a Long Title mass of TMD can be separated [9], as can be seen in Eqs. (2) and (3). [K + Ka]x + [C + Ca]ẋ + Mẍ −Kamxa −C a mẋa = F(t) , (2) kaxa + caẋa + maẍa − [Cam] T ẋ− [Kam] T x = 0 , (3) where: Ka defined by Eq. (4), Ca defined by Eq. (5), Kam m-th column of matrix Ka, Cam m-th column of matrix Ca, ka stiffness of TMD, ca damping of TMD, ma mass of TMD, xa displacement of TMD, ẋa velocity of TMD, ẍa acceleration of TMD. Matrices Ka and Ca contain only a single non-zero element on the m×m position: Ka =   0 0 · · · 0 0 ... ... ... ... · · · ka(m×m) ... 0 · · · · · · 0   , (4) Ca =   0 0 · · · 0 0 ... ... ... ... · · · ca(m×m) ... 0 · · · · · · 0   . (5) Assuming the loading force is harmonic with ampli- tude Fa and circular frequency ω, we only need to find a solution in the form of steady-state response. The equations (2) and (3) can be transformed to modal coordinates according to Eq. (7). Substituting xa expressed from Eq. (3) to Eq. (2), the following result is obtained: [ Ω + iωC′ −ω2I+ + (iωca + ka)(−ω2ma) −ω2ma + iωca + ka ΦTm,∗Φm,∗ ] q = ΦT Fa , (6) x = Φq , (7) where Ω, Φ, and C′ are spectral matrix, modal ma- trix, and modal damping matrix, respectively. Φm,∗ denotes the m-th row of modal matrix and q is the vector of modal coordinates. The modal matrix must be mass orthonormal with mode shapes φ arranged as its columns. The spectral matrix contains squares of natural circular frequencies ωj on its diagonal. For a classically damped structure, C′ is also diagonal containing members 2ξjωj where ξj is the damping ratio of j-th natural frequency. Eq. (6) is a system of n linear equations. Thanks to its structure, Sherman–Morrison formula can be used to find the inversion of the matrix on the left-hand side. The solution of separate modal coordinate qj with used substitutions is given by following equations: [ ω2ma(iωca + ka)Φm,jST2 (−ω2ma + iωca + ka) −ω2ma(iωca + ka)s1 + + φTj ] FA ω2j −ω2 + 2iξjωωj = qj , (8) s1 = n∑ j=1 Φ2m,j ω2j −ω2 + 2iξjωωj , (9) ST2 = n∑ j=1 Φm,jφTj ω2j −ω2 + 2iξjωωj . (10) As can be seen in Eq. (8-10), all mode shapes must be known to find an exact contribution of one par- ticular mode shape to the response. However, mode shapes with frequencies, which are far from forcing frequency have negligible influence on the response. Therefore using only several selected mode shapes in Eqs. (7-10) can lead to very precise results. The reduction of number of mode shapes is necessary for fast response evaluation. 2.2. multiple TMDs In the case of multiple TMDs, the evaluation of sep- arate modal coordinates using Eq. (8) is no longer possible, but Sherman–Morrison formula A−1i+1 = [Ai + uv T ]−1 = A−1i − A−1i uv T A−1i 1 + vT A−1i u (11) can be repeatedly applied to get a vector of modal coordinates q. Each addition of TMD requires one evaluation of Eq. (11) with substitutions given by Eqs. (12- 14). A0 = Ω + iωC′ −ω2I , (12) u = (iωca + ka)(−ω2ma) −ω2ma + iωca + ka ΦTm,∗ . (13) vT = Φm,∗ (14) For j TMDs, the vector of modal coordinates is q = A−1j Φ T Fa . (15) This method may be inappropriate for large number of TMDs because of its high numerical complexity caused by multiplication of full matrices. However, its usage still avoids the solution of the system of 101 Jan Štěpánek, Jiří Máca Acta Polytechnica CTU Proceedings equations. In section 4.5, this method is successfully used for the evaluation of structural response with three TMDs. As can be seen in Eqs. (11-14), the system recep- tance matrix A−1j is found without necessity of matrix inversion. The only matrix which must be inverted is A0, but it is a simple procedure, because A0 is a diagonal matrix. The receptance matrix can be used for fast evaluation of response to various load vectors. 3. Simplified TMD design Well known simplified design of TMD for minimiza- tion of maximal amplitude, first introduced in [1, 2] by Hartog and Brock, is used in this study as reference method and is compared to numerical optimization of TMDs. This method was chosen for its easy applica- tion and the fact it is widely used and recommended. Using following notation: Mef f effective mass of the main struc- ture, Ωj n-th natural frequency of the main structure, ωa = √ ka/ma natural frequency of TMD, µ = ma/Mef f mass ratio, β = ωa/Ωj frequency ratio, ξa = ca/ca,cr damping ratio of TMD, ca,cr = 2 √ kama critical damping of TMD. the optimal values of parameters β and ξa for force excitation are βopt = 1 1 + µ , (16) ξa,opt = √ 3µ 8(1 + µ) . (17) This method was firstly proposed for SDOF main structure, but it can be used for design of TMD which is supposed to reduce vibration of j-th mode shape. The effective mass of the main structure Mef f can be found according to the following equation: Mef f = 1 Φ2m,j = φ′Tj Mφ ′ j , (18) where φ′j denotes the j-th mode shape normalized with respect to the m-th ordinate where TMD is attached hence the m-th ordinate is equal to one. 4. Numerical example This chapter analysis a cable stayed footbridge. The structure is suspected of being sensitive to dynamic pedestrian load. A Numerical model of the struc- ture was created in MATLAB. Design of TMDs was performed both using simplified method indicated in section 3 and numerical optimization. mode frequency [Hz] damping ratio [-] 1 0.733 0.0190 2 0.923 0.0148 3 1.300 0.0106 4 1.595 0.0087 5 1.818 0.0078 6 1.924 0.0075 7 2.002 0.0071 8 2.218 0.0063 Table 1. Natural frequencies and damping ratios. 4.1. Model of the structure The structure is 243 m long, symmetric, cable- stayed bridge with prestressed concrete bridge deck. A scheme of the structure and numbering of important nodes can be seen in Figure 1. Two-dimensional finite element (FEM) model was created in order to exam- ine vertical vibrations. The model contains nodes which define geometry of the structure and also uses nodes added by FEM mesh. The final model contains 810 degrees of freedom. Eight lowest natural frequencies, mode shapes and associated damping ratios were measured on the real structure. Only vertical bending mode shapes are taken into account in this paper. The numerical model was identified to be in accordance with measurement. Natural frequencies and damping ratios are summa- rized in Table 1. 4.2. load As can be seen in Table 1, several frequencies are close to 2 Hz, which is typical frequency of human walk and thus resonance effect may occure. According to guidebook [5], a stationary load model which repre- sents a group of 8-15 walkers can be used to evaluate the response and to decide whether the vibration is excessive. The group of walkers is simplified to one harmonic force in resonance with mode, which has the frequency closest to 2 Hz. The force should be applied to the most adverse position, which may be under- stood as a place with the highest vertical ordinate of the forced mode shape. In the case given, it was decided to find maximal displacement over the bridge span for three positions, which corresponds to the maximal ordinate of mode shapes. They are node 19 for the 5th mode shape, node 5 for the 6th, and node 27 for the 7th one. The results can be seen in Figure 1. The highest peak in range between 1.5-2.5 Hz is caused by force situated in node 5 in resonance with 6th natural frequency. Therefore the harmonic force with amplitude fv is placed in node 5. fv = kv × 180 = 3 × 180 = 540 N . (19) However, the amplitude fv is important only for taking the absolute value of the response, but it has no impact on the design of TMD. 102 vol. 26/2020 Example of an Article with a Long Title Figure 1. Static scheme of the bridge. Figure 2. FRFs of force in nodes 5,19, and 27. Figure 3. Mode shape No. 5. Figure 4. Mode shape No. 6. Figure 5. Mode shape No. 7. Node 5 is appropriate place for simple harmonic load, because mode shapes 5,6 and 8 also have high ordinate in this node, therefore multiple modes are forced. For simplified TMD design from section 3, the load does not play a role, but it is important for numerical optimization and calculation of FRF. 4.3. objective function An objective function must be defined for the opti- mization of TMD. In this paper, it was decided to use H∞ optimization - maximal value of displacement am- plitude along the bridge deck between 1.7 and 2.2 Hz, where the load causes highest response. That means 103 Jan Štěpánek, Jiří Máca Acta Polytechnica CTU Proceedings maximal values of vector of displacement x create FRF, which creates a continuous FRF with discon- tinuous derivative. Highest peak of FRF is then the value of the objective function which is minimized. The step of forcing frequency for evaluating the FRF was set to 0.001 Hz to catch the sharp peaks but with respect to the speed of calculation. The transformation to modal coordinates allows us to use reduced number of mode shapes to speed up the evaluation of response. Observation of FRF showed that using more than first 12 mode shapes and natural frequencies does not affect the values of peak response in chosen frequency range. Therefore it was decided to use only first 12 mode shapes. The reduction of number of mode shapes reduced the size of the problem dramatically (from 810 equations to 12 equations) without noticeable decrease of accuracy. 4.4. single TMD 4.4.1. simplified design The simplified design method presented in section 3 was performed in order to reduce vibration of the sixth mode shape, because the pacing force causes the highest peak on FRF in resonance with the sixth natural frequency, as can be seen in Figure 2. TMD with one vertical degree of freedom and mass ma = 600 kg is attached to node 5, and its parameters were designed according to section 3, which resulted into values ka = 86300 Nm−1, and ca = 772 Nsm −1. The mass of 600 kg is approximately only 0.15 % of the total mass of the structure, but it can reduce the vibration significantly. 4.4.2. numerical optimization To optimize TMD numerically, function fmincon im- plemented in MATLAB was used. The function uses interior-point algorithm and is designed to solve con- strained multivariable optimization problems. The only implemented constraints were ka > 0 and ca > 0. The function was used to find optimal values of ka and ca which minimize the objective function. Gener- ally, in the case of single TMD, the objective function can have at most one local minimum for each natural frequency. In our case, the function has only one local minimum for the sixth frequency, because the peak of FRF reaches the highest level in resonance with the sixth natural frequency. The initial values of ka and ca define the local min- imum to which interior-point algorithm converges, therefore it is advantageous to begin with TMD tuned close to the natural frequency which response is mini- mized. For parameters of TMD, which are "far from optimal", a gradient of objective function may be very low, therefore convergence problems may occur, or it can take a large number of iterations to find the local minimum. Parameters designed according to section 3 are quality starting point for quick convergence. The parameters, which optimize the objective func- tion ka = 84100 Nm−1 and ca = 1259 Nsm −1 were mode shape node ka [Nm−1] ca [Nsm−1] ma [kg] 5 19 26000 98 200 6 5 29100 150 200 7 27 31500 129 200 Table 2. TMDs designed to reduce vibration of the 5th, 6th, and 7th mode shape. found numerically. Corresponding FRFs for both methods of design can be seen in Figure 6. 4.5. Multiple TMDs As we can see in Figure 2, there are three dominant natural frequencies between range 1.7-2.2 Hz. There- fore three TMDs are designed in this section to show that a higher number of TMDs can reduce the ob- jective function more effectively than one. It was decided to keep the sum of TMDs mass constantly at 600 kg. The constant sum of weight allows for a relevant comparison of one and multiple TMDs. 4.5.1. simplified design In order to reduce FRF, three tuned mass dampers were designed according to section 3 to reduce vibra- tion in resonance with the 5th, 6th, and 7th natural frequencies. All of them were positioned to the node where the highest ordinate of damped mode is located. Their properties are summarized in Tab. 2. The mass was equally distributed among TMDs because its op- timal distribution remains unknown. 4.5.2. Numerical optimization The numerical optimization was performed with two basic settings. The first one uses position of TMDs used by previous section (nodes 19, 5, and 27) and was expected to provide the best results. The second one places all TMDs in node 5. This calculation was proceeded in order to show that dividing the mass among more TMDs can perform better than one TMD, if more than one mode participate on the response. Theoretically, one TMD is only a subset of multiple TMDs with the same sum of mass. Therefore, dividing the mass provided additional room for a performance improvement. The objective function was minimized finding ca,i, ka,i, and ma,i for i =1. . . 3. The following constraints were implemented: • ca,i ≥ 0, • ka,i ≥ 0, • ma,i ≥ 0, • ∑3 i=1 ma,i = 600 kg. From the numerical point of view, the problem is much more complicated than the one with single TMD because 9 variables were optimized. Moreover, the function contains multiple local minima. Therefore, 104 vol. 26/2020 Example of an Article with a Long Title settings No. design method node ka[Nm−1] ca [Nsm−1] ma [kg] maximal displacement [mm] - no TMD - - - - 3.447 1 simplified 5 86300 772 600 1.349 2 numerical optimization 5 84100 1259 600 1.103 3 simplified 19 26000 98 200 1.5435 29100 150 200 27 31500 129 200 4 numerical optimization 19 26800 213 208 1.1005 24500 198 157 27 33500 312 236 5 numerical optimization 5 29800 191 189 0.9205 21600 147 176 5 34000 391 235 Table 3. Comparison of design methods and peak response between 1.7-2.2 Hz. it was decided to run the optimization multiple times from one hundred randomly generated initial points, which fulfilled the constraints. This procedure allowed us to find good results, but it is not possible to say that the global minimum was found. 4.6. results The results are summarized in Table 3. A total of 5 different TMD settings were found using simplified method and numerical optimization. As expected, TMDs designed using Den Hartog’s criteria were able to reduce the vibration though they were originally proposed for structure with one degree of freedom. It also can be noticed in Figure 6 that single TMD can positively affect vibration of adjacent mode shapes if their frequencies are close enough, and if the adjacent mode shapes have sufficiently high ordinate in the position of TMD. Figure 6. FRF - settings No. 1-2. As is shown in Figures 6 and 7, the numerical op- timization was able to improve the performance of both single and multiple TMDs. In the case when starting parameters of TMDs were randomly chosen, we expected that each tuned mass damper would con- verge to a point where it is able to damp one of the Figure 7. FRF - settings No. 3-5. dominant frequencies. This assumption was confirmed and can be seen in Figure 8, which shows the FRFs of tuned mass dampers. Figure 8. displacement of TMDs - settings No. 5. Unexpectedly, the best results were provided by settings where all three TMDs were placed in node 5, despite the fact that this is not the place where the highest ordinates of some mode shapes are lo- cated. This phenomenon can be explained by closer observation of mode shapes in Figures 3, 4, and 5. The force is placed in node 5, therefore all the active mode shapes oscillate with the same phase in this 105 Jan Štěpánek, Jiří Máca Acta Polytechnica CTU Proceedings node. However, in nodes 19 and 27, the sixth mode shape oscillates in opposite phase than mode shapes 5 and 7. Therefore neither node 19 nor node 27 are the best place for TMD. However, this situation is closely connected to the fact that the optimization was performed to reduce vibration caused by only one specific force. For design of this footbridge, symmetric placement of smaller TMDs to both sides of the bridge would solve the problem. 5. conclusions Modified mode superposition method was used to eval- uate response of structure with tuned mass dampers to speed up the response calculation. Utilization of this method provided a powerful tool for numerical optimization, which requires numerous evaluations of steady state response. This response evaluation method provides several important advantages in com- parison to standard response calculation: • Reduction of numerical complexity without notice- able error thanks to reduced number of mode shapes used. • Allows for a fast evaluation of various load vectors, because the system receptance matrix is known. • Provides possibility to use experimentally measured mode shapes and natural frequencies for more pre- cise response estimate. • Can be used for both single and multiple TMDs. Simplified TMD design method proposed by Den Hartog was compared to numerical optimization. The results demonstrate that numerical design is more appropriate for MDOF structures and can improve the overall performance of TMDs. In all cases of com- parison, the analysis showed that optimal damping is higher than the one designed by simplified criterion. Further it was shown that multiple TMDs with the same sum of mass as a single TMD can provide better results. The best settings of TMDs was able to reduce the peak response between 1.7-2.2 Hz by 73 %, with TMDs total mass of only 0.15 % of the structural mass. This result shows that the common recommendations for TMDs to have 1-3 % of mass of the structure may lead to uneconomical design. Acknowledgements The authors gratefully acknowledge support from the Czech Technical University in Prague, project SGS19/032/OHK1/1T/11 Development and application of numerical algorithms for analysis and modeling in me- chanics 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] SETRA. Footbridges: Assessment of vibrational behaviour of footbridges under pedestrian loading, 2006. [4] E. Caetano, A. Cunha, W. Hoorpah, J. Raoul. Footbridge Vibration Design. CRC Press, 2009. [5] FIB Bulletin 32: Guidelines for the design of footbridges, 2005. [6] O. Nishihara, T. Asami. Closed-form exact solution to h infinity optimization of dynamic vibration absorber: Ii. development of an algebraic approach and its application to a standard problem. Proceedings of the SPIE, 3989, 2000. doi:10.1117/12.384589. [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] R. Rana, T. Soong. Parametric study and simplified design of tuned mass dampers. Engineering Structures 20(3):193–204, 1998. [9] M. B. Ozer, T. J. Royston. Extending den hartog’s vibration absorber technique to multi-degree-of-freedom systems. Journal of Vibration and Acoustics 127(4):341–350, 2005. doi:10.1115/1.1924642. [10] J. Štěpánek, J. Máca. Dynamic response of structures with tuned mass dampers in modal coordinates. Vibroengineering PROCEDIA 23:13–17, 2019. doi:10.21595/vp.2019.20672. 106 https://doi.org/10.1117/12.384589 https://doi.org/10.1115/1.1924642 https://doi.org/10.21595/vp.2019.20672 Acta Polytechnica CTU Proceedings 26:100–106, 2020 1 Introduction 2 Response calculation 2.1 single TMD 2.2 multiple TMDs 3 Simplified TMD design 4 Numerical example 4.1 Model of the structure 4.2 load 4.3 objective function 4.4 single TMD 4.4.1 simplified design 4.4.2 numerical optimization 4.5 Multiple TMDs 4.5.1 simplified design 4.5.2 Numerical optimization 4.6 results 5 conclusions Acknowledgements References