STRUCTURAL CONTROL AND HEALTH MONITORING Struct Control Health Monit (2011) Published online in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/stc.446 Extension of equivalent linearization method to design of TMD for linear damped systems N D Anh1,Ã,y,z and N X Nguyen2 Institute of Mechanics, Vietnam Academy of Science and Technology, Hanoi, Vietnam Hanoi University of Science, Vietnam National University, Hanoi, Vietnam SUMMARY The vibration absorber has been used in many applications since invented In the case of vibration control by the tuned mass damper (TMD), the selection of optimum absorber parameters is extremely important This paper presents a closed-form expression for the optimum tuning ratio of a TMD attached to a damped primary system The result is obtained by using equivalent linearization method The values of the optimum tuning ratio derived from the expression proposed in this paper have been compared with those obtained numerically as well as the results obtained from other authors These values are reliable even the mass ratio of TMD to the primary structure and the structural damping ratio are quite high A simulation study has also been carried out to illustrate the obtained results Copyright r 2011 John Wiley & Sons, Ltd Received June 2010; Revised 14 November 2010; Accepted January 2011 KEY WORDS: TMD; equivalent linearization method; damped structure; fixed-point theory; closed-form expression INTRODUCTION The problem of undesired vibration reduction has been known many years and it has become more attractive nowadays Historically, an auxiliary mass–spring–damper control system attached to a primary structure was known as vibration absorber, tuned mass damper (TMD), or dynamic vibration absorber (DVA) The DVA without damper was proposed first in 1909 by Frahm [1], and later in 1956, Den Hartog [2] developed in the case of the absorber with viscous damper when primary structure modeled as a single-degree-of-freedom (SDOF) system After that, the designs of multi-TMDs for continuous structures and multi-degrees-freedom structures or mass–spring pendulum absorber for inverted pendulum-type structures have gained considerable attention of many researchers [3–5] Thenceforth, TMD has been widely used in many fields of engineering Perhaps the reasons for these applications were its efficient, reliable, and low-cost characteristics In the design of any control device for the reduction of undesired vibration, the aim would be to provide optimal parameters of the control device to maximize its effect Because the mass ratio of TMD to the primary structure is usually few percent, hence, the principal design parameters of the TMD are its tuning ratio (i.e ratio of the TMD’s frequency to the natural frequency of the primary structure) and its damping ratio The TMD can be used in two distinct ways to suppress the point *Correspondence to: N D Anh, Institute of Mechanics, Vietnam Academy of Science and Technology, Hanoi, Vietnam y E-mail: ndanh@imech.ac.vn z Full Professor Copyright r 2011 John Wiley & Sons, Ltd N D ANH AND N X NGUYEN vibration, collocated or non-collocated to the TMD, at the troublesome resonance frequency or at the troublesome frequency away from the resonance In the former case [1], the simplest method of tuning the neutralizer is by making its resonance frequency coincides with the resonance of the primary structure However, the TMD can become less effective, and may even increase the vibration of the primary structure when there is a change in the frequency of the excited force The increase can be clearly seen on each side of the operating frequency in the frequency response function graph In order to overcome this problem, an ingenious optimization method known as fixed-points theory was suggested by a proper selection of the neutralizer’s stiffness and damping In case of undamped primary structures, there exist two fixed-points at which the frequency response function is independent of the TMD’s damping The TMD’s stiffness is chosen so that the heights of the two fixed-points in the frequency response function become equal and then the TMD’s damping is determined when allowing these fixed-points to be peaks of the frequency response function This optimization technique is described in detail by Den Hartog [2] for undamped primary system that is subjected to harmonic excitation Since then, the fixed-points theory has become one of the design laws used in fabricating a TMD for the control of vibration of a relatively simple undamped system In the case of damped primary structure, it is difficult to obtain analytical solutions for the optimum parameters of the TMD Ioi and Ikeda [6], based on the numerical method, have presented the empirical formulae for the optimum parameters of the TMD attached to damped primary structure Randall et al [7] have used numerical optimization procedures for evaluating the optimum TMD’s parameters while considering damping in the structure Thomson [8] has proposed the procedures for a damped structure with TMD, where the tuning ratio has been optimized numerically and then using the optimum value of the tuning ratio, the optimum damping ratio of the TMD has been obtained analytically Warburton [9] has carried out a detailed numerical study for a lightly damped structure subjected to both harmonic and random excitation with TMD, and then the optimal parameters of the TMD (i.e tuning ratio and damping ratio) for various values of mass ratio and structural damping ratio have been presented in the form of design tables Fujino and Abe [10] have employed a perturbation technique to derive formulae for optimal TMD parameters, which may be used with good accuracy for mass ratios less than 2% and for very low values of structural damping Thus, in the general case of damping in the primary system, the optimal TMD’s parameters have to be evaluated either numerically or from empirical expressions The reason for this is that when the primary system takes account of damping, a very useful feature of the classical structure–damper system is lost This is the existence of fixed-point frequencies, i.e frequencies at which the transmissibility of vibration is independent of the damping in the attached control device Recently, Ghosh and Basu [11] have presented a closed-form expression for optimal tuning ratio of TMD based on the approximate assumption about the existence of two fixed-points In this paper, another closed-form expression for optimal tuning ratio of TMD is proposed This result is obtained based on the equivalent linearization method, where the damped primary system is replaced equivalently by an undamped system and then using the known result for undamped primary systems to give the expression of optimal tuning ratio The equivalent linearization has been used widely in many applications since invented [12–16] The result in this paper is compared with the result obtained from Ghosh and Basu’s expression as well as the result obtained numerically from Ioi and Ikeda [6] The comparison has shown that the values of optimal tuning ratio derived from the expression in this paper are closer to the values from the result given by Ioi and Ikeda than those from the expression proposed by Ghosh and Basu Finally, a simulation example is carried out to illustrate the obtained results THE TMD–STRUCTURE SYSTEM AND DEN HARTOG’S CLASSICAL RESULTS IN THE CASE OF UNDAMPED STRUCTURE The structure is modelled as an SDOF system by considering only the predominant mode in energy dissipation As shown in Figure 1, the SDOF system consists of the mass ms, the spring ks and the damping coefficient cs The mass of the TMD is md and its stiffness and damping coefficients are kd and cd, respectively Copyright r 2011 John Wiley & Sons, Ltd Struct Control Health Monit (2011) DOI: 10.1002/stc EXTENSION OF EQUIVALENT LINEARIZATION METHOD TMD md xd + Δ kd SDOF system cd ms xs f (t ) cs ks Figure Damped vibration absorber applied to a force-excited system with damping Let xs denote the vertical displacement of SDOF system relatively to its base from the equilibrium position, and let xd denote the vertical displacement of TMD relative to the primary structure from its equilibrium position The symbol D in Figure indicates the length of spring kd at equilibrium position of TMD–structure system, f(t) is the external force Kinetic energy T, potential energy P, and dissipative energy F are expressed as follows T ¼ 12 ms x_ 2s 12 md _xd 1_xs ị2 1ị ẳ 12 ks x2s 12 kd x2d 2ị F ẳ 12 cs x_ 2s 12 cd x_ 2d ð3Þ The equations of motion of the TMD–structure system are derived by using Lagrange’s equations d @T @T @Å @F ¼À À 1f ðtÞ ð4Þ À dt @_xs @xs @xs @_xs d @T @T @Å @F ¼À À À dt @_xd @xd @xd @_xd ð5Þ Substituting Equations (1), (2) and (3) into Equations (4) and (5) yields ms x€ s 1cs x_ s 1ks xs ¼ md x€ d 1md x€ s 1f ðtÞ ð6Þ md x€ d 1cd x_ d 1kd xd ẳ md x s 7ị Equations (6) and (7) can be rewritten as follows ms x€ s 1cs x_ s 1ks xs ¼ cd x_ d 1kd xd 1f ðtÞ ð8Þ md x€ d 1cd x_ d 1kd xd ẳ md x s 9ị Introduce the parameters sffiffiffiffiffi md ks ; os ¼ ; m¼ ms ms cs ; xs ¼ 2ms os sffiffiffiffiffiffi kd od ¼ ; md xd ¼ cd ; 2md od a¼ od os where m is the ratio of the mass of the TMD to the mass of the primary structure, os, xs and od, xd are natural frequencies and damping ratios of the structure and the TMD, respectively, a is the natural frequency ratio or tuning ratio Copyright r 2011 John Wiley & Sons, Ltd Struct Control Health Monit (2011) DOI: 10.1002/stc N D ANH AND N X NGUYEN Equations (8) and (9) can be rewritten in the dimensionless form as f ðtÞ x€ s 12os xs x_ s 1o2s xs ¼ 2maxd os x_ d 1ma2 o2s xd ms x€ d 12axd os x_ d 1a2 o2s xd ¼ À€xs ð10Þ ð11Þ In the case of undamped primary structure, i.e xs is equal to zero, Equations (10) and (11) have the simple form f ðtÞ x€ s 1o2s xs ¼ 2maxd os x_ d 1ma2 o2s xd ð12Þ ms x€ d 12axd os x_ d 1a2 o2s xd ẳ xs 13ị By using xed-point method, Den Hartog [2] has given the expression for the optimum tuning ratio aopt and the damping ratio xdopt of TMD in case of an undamped primary system subjected to sinusoidal excitation as follows 14ị aopt ẳ 11m s 3m xdopt ẳ ð15Þ 8ð11mÞ The results (14)–(15) obtained for undamped linear systems have required an extension to damped linear systems since the damping does always exist for real structures USING EQUIVALENT LINEARIZATION METHOD TO OBTAIN THE CLOSEDFORM EXPRESSION FOR OPTIMAL TUNING RATIO The aim in present paper is using equivalent linearization method in order to replace approximately the damped–spring–mass primary structure (Figure 2(a)) by a spring–mass structure (Figure 2(b)) and then using Den Hartog’s results to obtain the closed-form expression for optimal tuning ratio In the case (a) of Figure with damped structure, the equation of motion has form x€ s 12xs os x_ s 1o2s xs ẳ 16ị And in the case (b) of Figure with undamped structure, the equation of motion is x€ s 1o2e xs ¼ ð17Þ where oe is unknown constant and will be determined by minimizing the following function A ¼ hð2xs os x_ s 1o2s xs À o2e xs Þ2 iTt with respect to o2e , where Z T ð:Þdt 18ị h:iTt ẳ T where T is constant and will be chosen later This leads to dA=do2e ¼ 0, and so we have hððo2e À o2s Þxs À 2xs os x_ s ịxs iTt ẳ 19ị ms ms approximately ks cs (a) ke (b) Figure The approximation of the primary structure Copyright r 2011 John Wiley & Sons, Ltd Struct Control Health Monit (2011) DOI: 10.1002/stc EXTENSION OF EQUIVALENT LINEARIZATION METHOD or ðo2e À o2s Þhx2s ij" À 2xs os h_xs xs ij" ¼ where h:ij" ¼ " j Z ð20Þ j" ð:Þ dj ð21Þ " ¼ oe T For further analysis, one gets from Equation (17) with j xs ¼ a cos j; j ẳ oe t1j0 22ị Using denitions (19), (21) and expression (22), one has a2 " T j " sin2j" hxs it ¼ hxs i ¼ j1 " 2j h_xs xs iTt ¼ h_xs xs ij" ẳ 23ị o e a2 " 1Þ ðcos 2j " 4j Substituting Equations (23) and (24) into Equation (20) we obtain " À cos 2j xs os oe À o2s ¼ o2e " " 12 sin 2j j1 Solving Equation (25), we easily obtain solution os oe ¼ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2 u u " " 1 À cos 2j t11 1 À cos 2j xs x2s 1 j1 " " " sin 2j " sin 2j j1 ð24Þ ð25Þ ð26Þ Using Den Hartog’s result (14) for undamped primary structure as shown in Figure 2(b), we have aeopt ¼ 11m Note that aeopt ¼ aopt ¼ od oe od os and using Equation (26), finally we obtain a closed-form expression for optimal tuning ratio as follows 0v aopt ẳ 27ị !2 u u 1 À cos 2j " " C 1 À cos 2j B ð11mÞ@t11 xs A x2s j1 j1 " 12 sin 2j" " 12 sin 2j" " so that the values of the optimal tuning ratio aopt The problem of choice is the constant j derived from expression (27) are closest to those obtained numerically need considering in next " ¼ p=2, i.e we get the mean value over a quarter of period of study In this paper, the value j primary system, is proposed The reason for this choice is that in the first quarter of vibration period, the displacement and the velocity of primary system not change their signs as well as directions The integration over a quarter of period has been used in some previous studies by " ¼ p=2 into expression (27) leads to some authors [17,18] Putting this value j ! aopt ẳ 28ị r 2 11mị 11 xs xs p p Copyright r 2011 John Wiley & Sons, Ltd Struct Control Health Monit (2011) DOI: 10.1002/stc N D ANH AND N X NGUYEN Expression (27) in general and expression (28) in particular would reduce to Den Hartog’s expression (14) for the optimum tuning ratio of a TMD for an undamped SDOF system The expression for aopt in Equation (28) is independent of the damping in the TMD This optimum tuning ratio together with appropriate damping in the TMD will minimize the maxima of the displacement of the primary structure Thus, it will be possible to optimize the response reduction of the structure subjected to an external force which has a wide banded energy content or which has dominant energy at the natural period of the structure COMPARISONS To compare the values of the optimum tuning ratio from the expression proposed in this paper with those available otherwise, the values of the optimum tuning ratio corresponding to typical values of the structure–TMD parameter are presented in a tabular form For the design of the TMD it would be convenient to have the values of the optimum tuning ratio corresponding to typical values of the structure–TMD parameter in tabular form The values of the optimum tuning ratio from the proposed expression (28) would be compared with the values calculated from the empirical expression given by Ioi and Ikeda [6] and those from the expression proposed by Ghosh and Basu [11] The tolerance or permissible error ranges attached to equations of Ioi and Ikeda are given to be less than 1% for 0.03o mo0.4 and xso0.15 In Ghosh and Basu’s paper, they have given expression sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4x2s m2x2s 1ị 29ị aopt ẳ ð11mÞ3 These comparisons have been done in Tables I–IV, where the mass ratio m has been considered as 0.03; 0.05; 0.1; 0.35, respectively, and the different values of the structural Table I Optimum tuning ratio of TMD for different structural damping ratios and the mass ratio m 0.03 Structural Optimum tuning ratio aopt Optimum tuning ratio aopt Optimum tuning ratio aopt damping ratio xs given by Ghosh and Basu [11] proposed in this paper given by Ioi and Ikeda [6] 0.005 0.01 0.02 0.03 0.05 0.07 0.10 0.12 0.14 0.15 0.9708 0.9707 0.9701 0.9692 0.9661 0.9615 0.9515 0.9429 0.9326 0.9268 (0.14)à (0.29) (0.56) (0.82) (1.27) (1.64) (2.04) (2.17) (2.28) (2.27) 0.9678 0.9647 0.9586 0.9525 0.9405 0.9286 0.9110 0.8995 0.8882 0.8826 (0.17) (0.33) (0.63) (0.92) (1.42) (1.84) (2.31) (2.49) (2.59) (2.60) 0.9694 0.9679 0.9647 0.9613 0.9540 0.9460 0.9325 0.9225 0.9118 0.9062 ÃDifferent from Ioi and Ikeda’s results in percentage terms Table II Optimum tuning ratio of TMD for different structural damping ratios and the mass ratio m 0.05 Structural Optimum tuning ratio aopt Optimum tuning ratio aopt Optimum tuning ratio aopt damping ratio xs given by Ghosh and Basu [11] proposed in this paper given by Ioi and Ikeda [6] 0.005 0.01 0.02 0.03 0.05 0.07 0.10 0.12 0.14 0.15 0.9523 0.9522 0.9516 0.9507 0.9477 0.9432 0.9336 0.9252 0.9152 0.9096 (0.16) (0.33) (0.63) (0.92) (1.46) (1.90) (2.44) (2.69) (2.84) (2.90) Copyright r 2011 John Wiley & Sons, Ltd 0.9494 0.9463 0.9403 0.9344 0.9225 0.9109 0.8937 0.8824 0.8713 0.8658 (0.15) (0.30) (0.56) (0.81) (1.24) (1.59) (1.94) (2.06) (2.09) (2.06) 0.9508 0.9491 0.9456 0.9420 0.9341 0.9256 0.9114 0.9010 0.8899 0.8840 Struct Control Health Monit (2011) DOI: 10.1002/stc EXTENSION OF EQUIVALENT LINEARIZATION METHOD Table III Optimum tuning ratio of TMD for different structural damping ratios and the mass ratio m 0.1 Structural Optimum tuning ratio aopt Optimum tuning ratio aopt Optimum tuning ratio aopt damping ratio xs given by Ghosh and Basu [11] proposed in this paper given by Ioi and Ikeda [6] 0.005 0.01 0.02 0.03 0.05 0.07 0.10 0.12 0.14 0.15 0.9090 0.9089 0.9084 0.9075 0.9047 0.9005 0.8916 0.8837 0.8744 0.8692 (0.21) (0.41) (0.81) (1.19) (1.90) (2.55) (3.39) (3.83) (4.21) (4.36) 0.9062 0.9033 0.8976 0.8919 0.8806 0.8695 0.8531 0.8423 0.8317 0.8264 (0.10) (0.21) (0.39) (0.55) (0.81) (0.98) (1.08) (1.03) (0.88) (0.78) 0.9071 0.9052 0.9011 0.8968 0.8878 0.8781 0.8624 0.8511 0.8391 0.8329 Table IV Optimum tuning ratio of TMD for different structural damping ratios and the mass ratio m 0.35 Structural Optimum tuning ratio aopt Optimum tuning ratio aopt Optimum tuning ratio aopt damping ratio xs given by Ghosh and Basu [11] proposed in this paper given by Ioi and Ikeda [6] 0.005 0.01 0.02 0.03 0.05 0.07 0.10 0.12 0.14 0.15 0.7407 0.7406 0.7402 0.7396 0.7375 0.7344 0.7277 0.7219 0.7150 0.7111 (0.35) (0.84) (1.37) (2.04) (3.33) (4.57) (6.33) (7.41) (8.45) (8.93) 0.7384 0.7360 0.7314 0.7267 0.7175 0.7085 0.6951 0.6863 0.6777 0.6734 (0.04) (0.22) (0.16) (0.26) (0.53) (0.88) (1.56) (2.11) (2.79) (3.16) 0.7381 0.7344 0.7302 0.7248 0.7137 0.7023 0.6844 0.6721 0.6593 0.6528 damping ratio xs as 0.005; 0.01; 0.02; 0.03; 0.05; 0.07; 0.1; 0.12; 0.14; 0.15 have been considered Tables II–IV show that the values of optimum tuning ratio from the expression (28) proposed in this paper are closer to the values from the empirical expression given by Ioi and Ikeda than those derived from the expression (29) given by Ghosh and Basu Moreover, the expression for optimum tuning ratio presented in this study is significant even when the mass ratio m and the structural damping ratio xs are quite high As can be seen from above tables, the values of tuning ratio decrease when increasing the structural ratio as well as when the mass ratio increases SIMULATION EXAMPLE A simulation example has been carried out to illustrate the performance of the TMD design with a tuning ratio that has been evaluated from the expression proposed in this paper The example primary structure SDOF is considered with the structural damping ratio xs is equal to 0.15, the mass ratio of TMD to the structural m is equal to 0.1, the tuning ratio is equal to that obtained from the proposed expression (28) (i.e aopt 0.8264) The damping ratio of the TMD is assumed to be equal to the optimum value obtained from expression (15): sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3m xdopt ¼ 8ð11mÞ i.e xd is equal to 0.1864 Solving Equations (10) and (11) by using the fourth-order Runge–Kutta method, the time history integrations for the displacement responses of the primary structure subjected to random excitation with the TMD from expression proposed in Copyright r 2011 John Wiley & Sons, Ltd Struct Control Health Monit (2011) DOI: 10.1002/stc N D ANH AND N X NGUYEN Figure Displacement time history of the SDOF system with TMD this paper and the TMD from Ghosh and Basu’s expression have been performed The results presented in Figure have examined the accuracy of result presented in this study CONCLUSIONS By using the equivalent linearization method, a closed-form expression for the optimum tuning ratio of a TMD attached to a damped primary structure, modelled as an SDOF system, has been presented in this paper This is the first time the damped primary structure is replaced equivalently by an undamped system and then using known results in case of an undamped structure to obtain the analytical expression of optimal tuning ratio Furthermore, this result is received by taking mean value over a quarter of vibration period but is not over the whole period as classical method This taking mean value over a quarter of period has been used by some other authors The obtained expression of tuning ratio is compared with known results in the literature and the comparison shows that the expression of tuning ratio proposed in this paper is significant even when the mass ratio and structural damping are high The next study in the future may be to replace simultaneously the TMD–damped primary structure system by a TMD–undamped structure system and then determining not only the optimal tuning ratio but also the damping ratio of TMD ACKNOWLEDGEMENTS This paper is supported by Vietnam’s National Foundation for Science and Technology Development (NAFOSTED) REFERENCES Frahm H Device for Damped Vibration of Bodies U.S Patent No 989958, 30 October 1909 Den Hartog JP Mechanical Vibrations McGraw-Hill: New York, 1956 Casciati F, Giuliano F Performance of multi-TMD in the towers of suspension bridges Journal of Vibration and Control 2009; 15(6):821–847 Anh ND, Matsuhisa H, Viet LD, Yasuda M Vibration control of an inverted pendulum type structure by passive mass–spring-pendulum dynamic vibration absorber Journal of Sound and Vibration 2007; 307:187–201 Hoang N, Warnitchai P Optimal placement and tuning of multiple tuned mass dampers for suppressing multimode structural response Smart Structures and Systems 2006; 2(1):1–24 Ioi T, Ikeda K On the dynamic vibration damped absorber of the vibration system Bulletin of the Japanese Society of Mechanical Engineering 1978; 21(151):64–71 Copyright r 2011 John Wiley & Sons, Ltd Struct Control Health Monit (2011) DOI: 10.1002/stc EXTENSION OF EQUIVALENT LINEARIZATION METHOD Randall SE, Halsted DM, Taylor DL Optimum vibration absorbers for linear damped systems Journal of Mechanical Design (ASME) 1981; 103:908–913 Thompson AG Optimum tuning and damping of a dynamic vibration absorber applied to a force excited and damped primary system Journal of Sound and Vibration 1981; 77:403–415 Warburton GB Optimal absorber parameters for various combinations of response and excitation parameters Earthquake Engineering and Structural Dynamics 1982; 10:381–401 10 Fujino Y, Abe M Design formulas for tuned mass dampers based on a perturbation technique Earthquake Engineering and Structural Dynamics 1993; 22:833–854 11 Ghosh A, Basu B A closed-form optimal tuning criterion for TMD in damped structures Structural Control and Health Monitoring 2007; 14:681–692 12 Roberts JB, Spanos PD Random Vibration and Stochastic Linearization Wiley: New York, 1990 13 Socha L, Soong TT Linearization in analysis of non-linear stochastic systems Applied Mechanics Reviews 1991; 44(10):399–422 14 Casciati F, Faravelli L, Hasofer AM A new philosophy for stochastic equivalent linearization Probabilistic Engineering Mechanics 1993; 8:179–185 15 Anh ND, Di Paola M Some extensions of Gaussian equivalent linearization International Conference on Nonlinear Stochastic Dynamics, Hanoi, Vietnam, 7–10 December 1995; 5–16 16 Anh ND, Schiehlen W New criterion for Gaussian equivalent linearization European Journal of Mechanics A/Solids 1997; 16(6):1025–1039 17 Aderson D, Desaix M, Lisak M, Rasch J Galerkin approach to approximate solutions of some nonlinear oscillator equations American Journal of Physics 2010; 78(9):920–924 18 He J-H Variational approach for nonlinear oscillators Chaos, Solitons and Fractals 2007; 34:1430–1439 Copyright r 2011 John Wiley & Sons, Ltd Struct Control Health Monit (2011) DOI: 10.1002/stc ... obtained for undamped linear systems have required an extension to damped linear systems since the damping does always exist for real structures USING EQUIVALENT LINEARIZATION METHOD TO OBTAIN... 10.1002/stc EXTENSION OF EQUIVALENT LINEARIZATION METHOD Table III Optimum tuning ratio of TMD for different structural damping ratios and the mass ratio m 0.1 Structural Optimum tuning ratio aopt... CONCLUSIONS By using the equivalent linearization method, a closed-form expression for the optimum tuning ratio of a TMD attached to a damped primary structure, modelled as an SDOF system, has been