DSpace at VNU: Design of TMD for damped linear structures using the dual criterion of equivalent linearization method

International Journal of Mechanical Sciences 77 (2013) 164–170 Contents lists available at ScienceDirect International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci Design of TMD for damped linear structures using the dual criterion of equivalent linearization method N.D Anh a,b, N.X Nguyen c,n a Institute of Mechanics, Vietnam Academy of Science and Technology, Hanoi, Vietnam University of Engineering and Technology, Vietnam National University, Hanoi, Vietnam c Hanoi University of Science, Vietnam National University, Hanoi, Vietnam b art ic l e i nf o a b s t r a c t Article history: Received September 2012 Accepted 12 September 2013 Available online 16 October 2013 Design of tuned mass damper (TMD) for damped linear structures has attracted considerable attention in recent years Some approximate expressions of optimal tuning ratio of a TMD attached to a damped linear structure have been proposed In the paper, another closed-form expression for the optimum tuning ratio is presented for two models, excitation force and ground motion This approximate analytical solution is obtained by using improved equivalent linearization method according to dual criterion The values of optimum tuning ratio derived from the expression proposed in the present study have been compared with those obtained numerically and from results investigated by other authors The comparisons have verified the accuracy of the suggested expression for both small and large structural damping & 2013 Elsevier Ltd All rights reserved Keywords: TMD Improved equivalent linearization method Damped structure Dual criterion Closed-form solution Equivalent undamped structure Introduction An auxiliary mass-spring-damper attached to a primary structure was known as vibration absorber, tuned mass damper (TMD), or dynamic vibration absorber (DVA) The TMD without damper was first introduced by Frahm [1] in 1909, and then in 1928 Ormondroyd and Den Hartog [2] developed to the case of TMD with viscous damper when primary structure modeled as undamped single-degree-of-freedom (SDOF) system Thenceforth, the designs of multi-TMDs for continuous structures and multidegrees-freedom structures have gained considerable attention of many researchers, and TMD has been widely used in many fields of engineering and construction The reasons for those applications of TMD were its efficient, reliable, and low-cost characteristics In the design of TMD for reduction of undesired vibration, the main aim is to give optimal parameters of the TMD so that its effect is maximum Because the mass ratio of TMD to primary structure is usually few percent, the principal design parameters of the TMD are its tuning ratio (i.e ratio of TMD's frequency to the natural frequency of primary structure) and its damping ratio In case of undamped primary structures, the first invented TMD [1] had no damping element and it was only useful in a narrow range of frequencies very close to the natural frequency of TMD n Corresponding author E-mail addresses: ndanh@imech.ac.vn (N.D Anh), nguyennx12@gmail.com (N.X Nguyen) 0020-7403/$ - see front matter & 2013 Elsevier Ltd All rights reserved http://dx.doi.org/10.1016/j.ijmecsci.2013.09.014 Ormondroyd and Den Hartog [2] found that the TMD with viscous damper was effective to an extended range of frequencies The damped TMD proposed by Den Hartog is now known as the Voigt type TMD where a spring element and a viscous element are arranged in parallel, and is has been considered as a standard model of TMD Since then, there have been many optimization criteria given to design of TMD for undamped primary structures Three typical optimization criteria are (1) H optimization (or fixed-points theory), (2) H2 optimization, and (3) Stability maximization The H optimization was first proposed by Ormondroyd and Den Hartog [2] when the primary structure is subjected to harmonic excitation The purpose was to minimize the maximum amplitude magnification factor of the primary structure The optimum tuning ratio of TMD was first derived by Hahnkamm [3] in 1932 and later in 1946 Brock [4] given the optimum damping ratio The optimum parameters of TMD then were introduced by Den Hartog [5] The H2 optimization criterion was suggested by Crandall and Mark [6] in 1963 when the primary structure is subjected to random excitation The purpose was to minimize the area under the frequency response curve of the system (i.e total vibration energy of the structure over all frequencies) After that, the optimum parameters of TMD according to H2 optimization were presented by Iwata [7] and Asami [8] The stability maximization criterion and exact solutions of optimum parameters of TMD were first given by Yamaguchi [9] in 1988 with the aim was to improve the transient vibration of the structure In short, with undamped primary structures, all optimization criteria have been already solved analytically N.D Anh, N.X Nguyen / International Journal of Mechanical Sciences 77 (2013) 164–170 When primary structures take into account damping, however, it is difficult to obtained analytical solutions for the optimum parameters of TMD Ioi and Ikeda [10] have presented the empirical formulae for the optimum parameters of the TMD attached to a damped primary structure based on the numerical method Randall et al [11] have used numerical optimization procedures for evaluating the optimum TMD's parameters while considering damping in the structure Thompson [12] has given the procedures for a damped structure with TMD, where the tuning ratio has been optimized numerically and then using the optimum value obtained for the tuning ratio, the optimum damping ratio of TMD has been determined analytically Warburton [13] 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 TMD for various values of mass ratio and structural damping ratio have been presented on the form of design tables Fujino and Abe [14] have employed a perturbation technique to derive formulae for TMD’s optimal parameters, which may be used with good accuracy for mass ratio less than 2% and for very low values of structural damping ratio Asami et al [15] have presented a series solution for the H1 optimization and a closed-form solution for the H2 optimization Based on the approximate assumption of the existence of two fix-points, Ghosh and Basu [16] have given a closed-form expression for optimal tuning ratio of TMD Thus, in the general case of damping in the primary structures, the optimal TMD's parameters have to be evaluated either numerically or from approximate solutions Equivalent linearization method is one of the common approaches to approximate analysis of dynamical systems The original stochastic version of this method was proposed by Caughey [17,18] which is based on the replacement of a nonlinear oscillator under Gaussian excitation with a linear one under the same excitation and the coefficients of linearization can be found from the conventional mean-square criterion Thenceforward, there have been some extended versions of equivalent linearization method [19–27] Recently, Anh and Nguyen [28] suggested an approximate analytical solution of optimal tuning ratio of TMD by using classical equivalent linearization method according to the conventional criterion The main objective of this article is to give another closed-form expression for optimal tuning ratio of the TMD attached to a damped structure This result is obtained by utilizing an improved equivalent linearization method based on the dual criterion proposed by Anh [29] The solution derived from the present paper is validated by comparing with the results given by numerical method, by original version of classical equivalent linearization method and by other authors The comparisons have justified the significant accuracy of the proposed expression for both small and large structural damping Z ỵ1 Rf ịe d 3ị in which the notation Eẵ U denotes the mathematical expectation operator We restrict to the case of stationary response of Eq (1) if it exists Eq (1) is linearized to become an equation in the following linear form x€ ỵ 2h ỵ bịx_ ỵ 20 ỵkịx ẳ f tị ð4Þ where the coefficients of linearization b; k are found by an optimal criterion There are some criteria for determining this coefficients b; k but the most extensively used criterion is the mean square error criterion which requires that the mean square of error exị ẳ g x; x_ ị À bx_ À kx between Eq (1) and its linearized Eq (4) be minimum Eẵe2 xị ẳ Eẵgx; x_ ị Àbx_ À kxÞ2 Š-min ð5Þ b;k In general, although the mean square criterion (5) gives a quite good prediction as has been shown by many authors, however, in the case of major nonlinearity, the solution error according to criterion (5) may be unacceptable [22,25] In order to reduce the solution error we may use the dual approach to the equivalent linearization method as proposed by Anh [29] and investigated in detail by Anh et al [30] The classical linearization method is based on replacing the original nonlinear system by a linear system that is equivalent to the original one in some probabilistic sense Using the dual conception, we also can replace the obtained equivalent linear system by a nonlinear one that belongs to the same class of the original nonlinear system Combining those two steps we may consider a following dual criterion Eẵgx; x_ ị bx_ kxị2 ỵ Eẵbx_ ỵ kx gx; x_ ịị2 -min 6ị b;k; where the first term describes the conventional replacement and second term is its dual replacement Using the dual criterion (6) and noting that Eẵxx_ ẳ 0, we obtain [30] E½x_ gŠ À β E½x_ Š Eẵxg kẳ Eẵx2 bẳ ẳ 7ị where it is denoted ẳ Eẵx_ gị2 Eẵxgị2 þ E½x_ ŠE½g Š E½x2 ŠE½g Š 8ị x x2 ịx_ ỵ 20 x ẳ stị Content of the conventional linearization was described in the works of Caughey [17,18] Here, we consider a single degree of freedom system with the nonlinear function depending on displacement and velocity ð1Þ where h and ω0 are constants, gðx; x_ Þ is a nonlinear function of two arguments x and x_ The function f ðtÞ is a zero mean Gaussian stationary process with the correlation function and spectral density given by, respectively, Rf ị ẳ Eẵf tịf t ỵ ị To illustrate the above dual criterion, we now consider Van der Pol oscillator subjected to random excitation of white noise [30] Improved equivalent linearization method according to dual criterion x ỵ 2hx_ ỵ 20 x ỵgx; x_ ị ẳ f tị Sf ị ¼ 165 ð2Þ ð9Þ where α; γ ; ω0 ; s are positive real constants, the function ξðtÞ is a Gaussian white noise process of unit intensity with the Dirac delta Table The mean square response E½x2 Š of Van der Pol oscillator versus the parameter s2 ðα ¼ 0:2; ẳ 1; ẳ 2ị [30] s2 Eẵx2 Šsimu E½x2 Šconven Error (%) E½x2 Špresent Error (%) 0.02 0.20 1.00 2.00 4.00 0.2081 0.3638 0.7325 1.0255 1.4525 0.1366 0.2791 0.5525 0.7589 1.0512 34.3573 23.2741 24.5742 25.9998 27.6248 0.2069 0.3838 0.7342 1.0000 1.3770 0.5590 5.4964 0.2304 2.4866 5.1969 166 N.D Anh, N.X Nguyen / International Journal of Mechanical Sciences 77 (2013) 164170 correlation function R ị ẳ Eẵtịt ỵ ị ẳ ị 10ị Using Eqs (7) and (8) with gx; x_ ị ẳ x2 x_ , we obtain bẳ Eẵx2 kẳ0 11ị and the linearized equation of Eq (9) now takes the following form   x ỵ ỵ Eẵx2 x_ ỵ 20 x ẳ stị 12ị From Eq (12) the mean square response E½x2 Š can be determined by relation Eẵx2 ẳ 2 s2 ỵ 35 Eẵx2 13ị consisting of a damped primary structure and a TMD In Fig 1(a), a sinusoidal excitation force f tị ẳ f sin ωt is applied directly to the main mass Whereas in Fig 1(b), the sinusoidal vibratory motion x0 ðt Þ ¼ x0 sin ωt is transmitted to the main mass through the supporting spring and the damper We introduce following parameters sffiffiffiffiffiffi m ks cs μ ¼ d ; ωs ¼ ; ; ξs ¼ ms ms 2ms ωs sffiffiffiffiffiffiffi kd cd ω ω ; ξd ¼ ωd ¼ ; ẳ d; ẳ 15ị md 2md d s s where μ is the ratio of TMD's mass to the mass of primary structure, ωs ; ξs and ωd ; ξd are natural frequencies and damping ratios of the structure and the TMD, respectively, α is the natural frequency ratio or tuning ratio, and β is the force frequency ratio In the model Fig 1(a), the steady-state response or the amplitude magnification of the primary structure is given by [15] vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   u  xs  u ðα2 ị2 ỵ 2d ị2  ẳ uh AM ¼  i2 i2 h t  f =ks ỵ 4s d ỵ ỵ ị2 ỵ ỵ 2s ị ỵ 2αβξd ð1 À β2 À μβ2 Þ and in the model Fig 1(b), the steady-state response or the transmissibility of the primary structure is [15] vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   u 2  xs  u ẵ2 ỵ 4s d ị ỵ ẵ22 s s ỵ d ފ2   TR ¼   ¼ t 2 2 2 2 x0 ẵ ỵ 4s d ỵ ỵ ị ỵ ỵ ẵ2s ị þ 2αβξd ð1 À β À μβ ފ2 and then solving Eq (13) with respect to unknown E½x2 Š, we get s! s2 ỵ ỵ Eẵx2 ẳ 14ị 520 The results of mean square response of Van der Pol oscillator (9) obtained from Eq (14) and the conventional linearization are compared with Monte-Carlo simulation in Table with given parameters α ¼ 0:2; ω ¼ 1; γ ¼ and various values of s2 [22] It is seen that the errors of the present method are considerably smaller than those of the conventional method, namely the greatest error of present method is only 5.4964% whereas the smallest error of the conventional method is 23.2741% In next section, we are going to use the idea of the above dual criterion in the problem of design TMD for damped linear structures, namely, we will replace the damped primary structure by an equivalent undamped one Although the undamped structure standing alone would procedure infinite response in resonant range, it is emphasized that our system consists of TMD and primary structure so in total this system includes the damping Using improved equivalent linearization method to obtain the approximate analytical solution for optimal tuning ratio 3.1 Formulation of problem and classical results for undamped structures The primary structure is modeled as a single degree of freedom system (SDOF) by considering only the predominant mode in energy dissipation The SDOF system consists of a mass ms, a spring with spring constant ks, and a viscous damper with damping coefficient cs We assume that the mass of TMD is md, its spring stiffness constant and damping coefficient are kd and cd, respectively Fig shows two analytical models of the system ð16Þ ð17Þ In the case of undamped primary structure, it is clear that two models are the same Using H optimization (i.e the minimization of the maximum amplitude response), Den Hartog [5] introduced analytical expressions for the optimal tuning ratio αopt and the optimal damping ratio ξdopt of the TMD as follows opt ẳ dopt 1ỵ s ẳ 81 ỵ ị 18ị 19ị The analytical solutions in Eqs (18) and (19) obtained for undamped linear structures have required an extension to damped linear structures because the damping always exists in real structures In next part, an approximate expression for the optimal tuning ratio of TMD is presented by using the improved version of equivalent linearization method according to dual criterion 3.2 The optimal tuning ratio of TMD based on the improved linearization method The main idea of the present study is using the improved equivalent linearization method with the dual criterion in order to replace approximately the original damped-spring-mass structure as in Fig 2a with an equivalent undamped-spring-mass structure as shown in Fig 2b, then we use the result in Eq (18) for the obtained undamped-spring-mass structure to get an approximate analytical solution for optimal tuning ratio of TMD As above mentioned, it is emphasized that we only use the idea of dual criterion, i.e it is not replacing a nonlinear system with an equivalent linear one but replacing a damped structure with an equivalent undamped one N.D Anh, N.X Nguyen / International Journal of Mechanical Sciences 77 (2013) 164–170 167 Fig Two models of system consisting of TMD and damped primary structure Fig The approximation of the primary structure In Fig 2a with the original damped structure, the equation of motion is x€ s ỵ2s s x_ s ỵ 2s xs ẳ ð20Þ and in Fig 2b with the equivalent undamped structure, the equation of motion has form as follows x€ s ỵ ỵ 2s ịxs ẳ 21ị where is an unknown constant that will be determined by using the following dual criterion D E D E 22ị A ẳ 2s s x_ s xs ị2 ỵ xs À 2λξs ωs x_ s Þ2 -min T T ; in which we denote 2e ẳ ỵ ω2s T Z T ð U Þdt ð26Þ and then solving system of Eq (26) in terms of unknown constants γ and λ yields "   # xs x_ s γ ¼ 2ξs ωs   T :6 xs x_ s iT xs T h À  2 x_ s hxs iT T  x x_ 2 s s λ ¼   D 2E 2 xs T x_ s T T  2 À xs x_ s T ð27Þ The first factor in the expression of γ in Eq (27) is the result obtained via the conventional criterion This result has been presented by Anh and Nguyen [28] We now use ð24Þ h:iT ¼ h:iΦ ¼ where T is an integral region and will be chosen later As observed, the first term in the criterion (22) is the conventional replacement while the second term describes its dual replacement The coefficients γ and λ are determined by the following set of equations ∂A ¼0 A ẳ0 T 23ị and h:iT ẳ D E   2ξs ωs x_ 2s λ À xs x_ s T γ ¼ Z Φ Φ U ịd; with ẳ e T; 28ị hereby Eq (27) can be rewritten in the form "   # _ xs xs γ ¼ 2ξs ωs   Φ :6 _ x x xs Φ h i À  s s Φ2 x x_ s Φ h s iΦ  ð25Þ Substituting the expression of function A in the criterion (22) into the set of Eq (25) leads to       x2s T γ À 2ξs ωs xs x_ s T λ À2ξs ωs xs x_ s T ¼ 2 x x_ Φ  2 xs Φ x_ s À xs x_ s Φ Φ s s ẳ   D 2E 29ị We also have from Eq (21) xs ¼ a cos φ; φ ẳ e t ỵ 30ị 168 N.D Anh, N.X Nguyen / International Journal of Mechanical Sciences 77 (2013) 164–170 Therefore, using Eqs (28) and (30) we obtain    2 a2 ỵ sin xs ¼ 2Φ   a2 ωe xs x_ s ẳ cos 1ị   D E 2 a ωe Φ À sin 2Φ ¼ x_ 2s Φ 2Φ ð31Þ then substituting Eq (30) into the first equation of Eq (27) and using Eq (23), after some calculations, we get 2e ỵ 21 À cos 2ΦÞð2Φ À sin 2ΦÞ 8Φ À2 sin 2Φ À ð1 À cos 2ΦÞ2 2 ξs ωs ωe 2s ẳ 32ị Eq (32) is a quadratic equation in terms of ωe Solving this equation, we easily obtain 0vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " #2 u u ð1 À cos 2ị2 sin 2ị t e ẳ s B 2s @ 1ỵ 2 sin À cos 2ΦÞ2 ! ð1 À cos 2ΦÞð2Φ À sin 2ΦÞ À ξ ð33Þ s 8Φ2 À sin 2Φ À ð1 À cos 2ΦÞ2 Fig Graph of the amplitude magnification factor AM versus β with μ ¼ 0:05; ξs ¼ when the tuning ratio change 1% Now, using the result in Eq (18) for the obtained undamped structure as shown in Fig 2b, we have αeopt ẳ 1ỵ 34ị and noting that d e d αopt ¼ ωs Fig Graph of the amplitude magnification factor AM versus β with μ ¼ 0:05; ξs ¼ when the damping ratio change 10% eopt ẳ 35ị then combining with Eq (33), we find an approximate analytical solution for the optimal tuning ratio of TMD in the case of damped primary structures for both two models in Fig as follows rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i ffi αopt ¼ 1ỵ cos 2ị2 sin 2ị 82 sin 2Φ À ð1 À cos 2ΦÞ2 2 ð1 À cos 2ΦÞð2Φ À sin 2ΦÞ s À 8Φ2 À sin 2Φ À ð1 À cos 2ΦÞ2 ξ Eqs (18) and (19), and the second case when the tuning ratio changes 1% Fig depicts the case in which the damping ratio changes 10% We can see that in Fig the amplitude magnification factor AMchanges considerably, but in Fig it is nearly unchanged ξs 1ỵ Comparisons 36ị The choice of the constant , that is the integral region, so that the values of optimal tuning ratio αopt from the expression (36) are closest to those obtained by numerical method needs a further investigation In the present study, we get the mean value over a quarter of period [28], i.e Φ ¼ π =2 Putting this value Φ ¼ π =2 into Eq (36) yields q 2 ỵ 2ị2 s 2 s opt ẳ 37ị 1ỵ To validate the results proposed in this paper, the values of optimal tuning ratio obtained from the expression (37) are compared with the values calculated via numerical method given by Ioi and Ikeda [10] and the values from approximate analytical expression obtained by using the original version of classical equivalent linearization method with conventional criterion [28] The analytical solution (36) in general and the solution (37) in particular will reduce to Eq (18) for undamped primary structures The expression of optimal tuning ratio in Eq (37) is independent of TMD’s damping This optimal tuning ratio together with appropriate TMD’s damping will minimize the maximum of the displacement of the primary structures However, it is noted that the amplitude magnification factor AM and the transmissibility TR will change considerably (namely the change of peak) when the tuning ratio has slight change Whereas AM and TR will be nearly unchanged even when the damping ratio change considerably To express this comment, we consider an example, say, an undamped primary structure (i.e ξs ¼ 0) with the mass ratio μ ¼ 0:05 In this case, it is clear that the expressions for AM and TR in Eqs (16) and (17) are the same Fig describes the steadystate response curves in two cases: the first case, the tuning ratio α ¼ 0:9524 and the damping ratio ξd ¼ 0:1336 are obtained from and the closed-form expression proposed by Ghosh and Basu [16] vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u1 42 22 1ị s s opt ẳ t 39ị ỵ ị3 opt ẳ q  ỵ ị ỵ 42 s ỵ ξs ð38Þ for excitation force model in Fig 1(a) These comparisons have done in Tables 2–4, in which the values of mass ratio μ are 0.01, 0.03, 0.05, respectively, and the different values of structural damping ratio ξs are 0.005, 0.01, 0.02, 0.03, 0.05, 0.07, 0.1, 0.12, 0.14, and 0.15 The number in brackets indicates the difference from Ioi and Ikeda’s results in percentage term As we can seen, Tables 2–4 proclaim that the values of optimal tuning ratio from the expression (37) presented in this study for excitation force model are closer to the values from numerical method given by Ioi and Ikeda [10] than those derived from the expression (39) proposed by Ghosh and Basu [16] and from the N.D Anh, N.X Nguyen / International Journal of Mechanical Sciences 77 (2013) 164–170 169 Table Optimal tuning ratio of TMD for different structural damping ratios and the mass ratio μ ¼ 0:01 ξs Ioi and Ikeda [10] Ghosh and Basu [16] The conventional criterion [28] The present paper 0.005 0.01 0.02 0.03 0.05 0.07 0.10 0.12 0.14 0.15 0.9888 0.9874 0.9846 0.9815 0.9748 0.9672 0.9545 0.9450 0.9348 0.9294 0.9900 0.9899 0.9893 0.9883 0.9852 0.9804 0.9702 0.9613 0.9507 0.9447 0.9870 0.9838 0.9776 0.9714 0.9591 0.9470 0.9291 0.9173 0.9058 0.9001 0.9881 0.9862 0.9822 0.9783 0.9705 0.9628 0.9514 0.9438 0.9363 0.9326 (0.12) (0.25) (0.48) (0.69) (1.07) (1.36) (1.64) (1.72) (1.70) (1.65) (0.18) (0.36) (0.71) (1.03) (1.61) (2.09) (2.66) (2.93) (3.10) (3.15) (0.07) (0.12) (0.24) (0.33) (0.44) (0.45) (0.31) (0.13) (0.16) (0.34)    s  Fig Comparison of the amplitude magnification factors AM ¼ f x=k  with μ ¼ 0:05 s and ξs ¼ 0:15 Table Optimal tuning ratio of TMD for different structural damping ratios and the mass ratio μ ¼ 0:03 ξs Ioi and Ikeda [10] Ghosh and Basu [16] The conventional criterion [28] The present paper 0.005 0.01 0.02 0.03 0.05 0.07 0.10 0.12 0.14 0.15 0.9694 0.9679 0.9647 0.9613 0.9540 0.9460 0.9325 0.9225 0.9118 0.9062 0.9708 0.9707 0.9701 0.9692 0.9661 0.9615 0.9515 0.9429 0.9326 0.9268 0.9678 0.9647 0.9586 0.9525 0.9405 0.9286 0.9110 0.8995 0.8882 0.8826 0.9689 0.9670 0.9632 0.9593 0.9517 0.9441 0.9329 0.9255 0.9181 0.9145 (0.14) (0.29) (0.56) (0.82) (1.27) (1.64) (2.04) (2.17) (2.28) (2.27) (0.17) (0.33) (0.63) (0.92) (1.42) (1.84) (2.31) (2.49) (2.59) (2.60) (0.05) (0.09) (0.16) (0.21) (0.24) (0.20) (0.04) (0.32) (0.69) (0.92) Table Optimal tuning ratio of TMD for different structural damping ratios and the mass ratio μ ¼ 0:05 ξs Ioi and Ikeda [10] Ghosh and Basu [16] The conventional criterion [28] The present paper 0.005 0.01 0.02 0.03 0.05 0.07 0.10 0.12 0.14 0.15 0.9508 0.9491 0.9456 0.9420 0.9341 0.9256 0.9114 0.9010 0.8899 0.8840 0.9523 0.9522 0.9516 0.9507 0.9477 0.9432 0.9336 0.9252 0.9152 0.9096 0.9494 0.9463 0.9403 0.9344 0.9225 0.9109 0.8937 0.8824 0.8713 0.8658 0.9505 0.9486 0.9448 0.9410 0.9336 0.9261 0.9151 0.9078 0.9006 0.8971 (0.16) (0.33) (0.63) (0.92) (1.46) (1.90) (2.44) (2.69) (2.84) (2.90) (0.15) (0.30) (0.56) (0.81) (1.24) (1.59) (1.94) (2.06) (2.09) (2.06) (0.03) (0.05) (0.08) (0.11) (0.05) (0.05) (0.41) (0.75) (1.20) (1.48) result (38) obtained by using classical equivalent linearization method with conventional criterion Furthermore, the comparison of the proposed expression in this paper with the numerical results also shows that our solution has virtually no error up to the structural damping is equal 0.15 Fig describes the comparison of the amplitude magnification factors for the excitation force model in Fig 1a where the mass ratio μ ¼ 0:05 and the structural damping ξs ¼ 0:15 for three cases: the first case is Den Hartog's results (18) and (19) with α ¼ 0:9524; ξd ¼ 0:1336, the second case is Ghosh and Basu's expression (39) with α ¼ 0:9096; ξd ¼ 0:1336 and the third case is the present expression (37) with α ¼ 0:8971; ξd ¼ 0:1336 Fig shows the curves of the transmissibility for the ground motion model in Fig 1b with μ ¼ 0:05 and ξs ¼ 0:15 in two cases, Den Hartog's results (18) and (19) with α ¼ 0:9524; ξd ¼ 0:1336 and the present expression (37) with α ¼ 0:8971; ξd ¼ 0:1336     Fig Comparison of the transmissibilities TR ¼ xx0s  with μ ¼ 0:05 and ξs ¼ 0:15 Concluding remarks Although the design of TMD for a linear system is classical problem and there have been many works on this problem, in the case of damped linear structures there have been only either numerical methods or approximate analytical solutions for optimal parameters of TMD so far This paper has presented a closed-form expression for the optimal tuning ratio of TMD attached to a damped primary structure modeled as a single-degree-of-freedom system for two cases: excitation force and ground motion The main idea of this study is based on the improved version of equivalent linearization method with dual criterion suggested by Anh in order to replace approximately the original damped structure by an equivalent undamped structure, then using known results for obtained undamped structure to get an approximate analytical solution for the optimal tuning ratio of TMD The solution derived from the present paper is validated by comparing with the results given by numerical method, by the original version of classical equivalent linearization method and by other authors The comparisons have justified the significant accuracy of the proposed expression for both small and large 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2010;32(4):263–6 [30] Anh ND, Hieu NN, Linh NN A dual criterion of equivalent linearization method for nonlinear systems subjected to random excitation Acta Mech 2012;223(3):645–54  ... version of equivalent linearization method according to dual criterion 3.2 The optimal tuning ratio of TMD based on the improved linearization method The main idea of the present study is using the. .. use the idea of the above dual criterion in the problem of design TMD for damped linear structures, namely, we will replace the damped primary structure by an equivalent undamped one Although the. .. excitation and the coefficients of linearization can be found from the conventional mean-square criterion Thenceforward, there have been some extended versions of equivalent linearization method