DSpace at VNU: Design of three-element dynamic vibration absorber for damped linear structures

DSpace at VNU: Design of three-element dynamic vibration absorber for damped linear structures tài liệu, giáo án, bài gi...

Journal of Sound and Vibration 332 (2013) 4482–4495 Contents lists available at SciVerse ScienceDirect Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi Design of three-element dynamic vibration absorber for damped linear structures N.D Anh a,b, N.X Nguyen c,n, L.T Hoa b a b c Institute of Mechanics, Vietnam Academy of Science and Technology, Hanoi, Vietnam University of Engineering and Technology, Vietnam National University, Hanoi, Vietnam Hanoi University of Science, Vietnam National University, Hanoi, Vietnam a r t i c l e in f o abstract Article history: Received 12 January 2013 Received in revised form 17 March 2013 Accepted 24 March 2013 Handling Editor: L.G Tham Available online May 2013 The standard type of dynamic vibration absorber (DVA) called the Voigt DVA is a classical model and has long been investigated In the paper, we will consider an optimization problem of another model of DVA that is called three-element type DVA for damped primary structures Unlike the standard absorber configuration, the three-element DVA contains two spring elements in which one is connected to a dashpot in series and the other is placed in parallel There have been some studies on the design of the threeelement DVA for undamped primary structures Those studies have shown that the threeelement DVA produces better performance than the Voigt DVA does When damping is present at the primary system, to the best knowledge of the authors, there has been no study on the three-element dynamic vibration absorber This work presents a simple approach to determine the approximate analytical solutions for the H ∞ optimization of the three-element DVA attached to the damped primary structure The main idea of the study is based on the criteria of the equivalent linearization method in order to replace approximately the original damped structure by an equivalent undamped one Then the approximate analytical solution of the DVA's parameters is given by using known results for the undamped structure obtained The comparisons have been done to verify the effectiveness of the obtained results & 2013 Elsevier Ltd All rights reserved Introduction The dynamic vibration absorber is a passive vibration control device which is attached to a primary system to reduce undesired vibration The DVA without damper was introduced by Frahm [1] in 1909 but the first DVA [1] had no damping element and was only useful in a narrow range of frequencies very close to the natural frequency of the DVA In 1928, Ormondroyd and Den Hartog [2] found that the DVA with viscous damper was effective to an extended range of frequencies The damped DVA proposed by Den Hartog is now known as the Voigt type dynamic vibration absorber where a spring element and a viscous element are arranged in parallel, and it is has been considered as a standard model of the DVA In the design of the Voigt type DVA, the main objective is to give optimal parameters of the DVA so that its effect is maximal Because the mass ratio of the DVA to the primary structure is usually few percent, the principal parameters of the DVA are its tuning ratio (i.e the ratio of the DVA's frequency to the natural frequency of the primary structure) and damping ratio n Corresponding author Tel.: +84 989201949 E-mail address: nguyennx12@gmail.com (N.X Nguyen) 0022-460X/$ - see front matter & 2013 Elsevier Ltd All rights reserved http://dx.doi.org/10.1016/j.jsv.2013.03.032 N.D Anh et al / Journal of Sound and Vibration 332 (2013) 4482–4495 4483 There have been many optimization criteria given to design the DVA for undamped primary structures Three typical optimization criteria are H ∞ optimization, H optimization, and 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 the DVA was first derived by Hahnkamm [3] in 1932 and later in 1946 Brock [4] gave the optimum damping ratio The optimum parameters of the DVA then were introduced by Den Hartog [5] The optimal tuning ratio and damping ratio of the Voigt type DVA derived by using the fixed-points theory are not exact because some approximations are taken when they are derived Nishihara and Asami [6] proposed the exact solutions and compared these with the results given by Den Hartog They found that both optimal tuning ratio and damping ratio presented by Den Hartog were very close to the exact solutions Therefore, the fixed-point theory provided a very good approximation of the exact solutions for the H ∞ optimization in practice because the exact solution was too complicated The H optimization criterion was suggested by Crandall and Mark [7] 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 optimal parameters of the DVA according to the H optimization were presented by Iwata [8] and Asami [9] The stability maximization criterion and the exact solutions of the optimum parameters of the DVA were first given by Yamaguchi [10] in 1988 with the aim to improve the transient vibration of the structure In short, all optimization criteria have been already solved analytically for undamped primary structures When damping is present at the primary structure, it is difficult to obtain analytical solutions for the optimum parameters of the DVA Ioi and Ikeda [11] presented the empirical formulae for the optimum parameters of the DVA attached to a damped primary structure based on the numerical method Randall et al [12] proposed numerical optimization procedures for evaluating the optimum DVA's parameters while considering damping in the structure Thompson [13] gave the procedures for a damped structure with a DVA, where the tuning ratio has been optimized numerically and then using the optimum value obtained for the tuning ratio, the optimum damping ratio of the DVA has been determined analytically Warburton [14] carried out a detailed numerical study for a lightly damped structure subjected to both harmonic and random excitation with DVA, and then the optimal parameters of the DVA for various values of the mass ratio and the structural damping ratio were presented in the form of design tables Fujino and Abe [15] employed a perturbation technique to derive formulae for the DVA's optimal parameters, which may be used with good accuracy for the mass ratio less than percent and for very low values of the structural damping ratio In 1997, Nishihara and Matsuhisa [16] gave the exact solution for the stability maximization criterion Pennestrì [17] proposed a min-max design of a DVA where a min–max objective function subject to six constraint equations with seven unknown variables was found In 2002, Asami et al [18] presented a series solution for the H ∞ optimization and an exact solution for the H optimization but their solution was extremely complicated so they proposed an approximate solution for practical use Based on the approximate assumption of the existence of two fix-points, Ghosh and Basu [19] gave a closed-form expression for the optimal tuning ratio of the DVA Brown and Singh [20] developed a min-max procedure to design a DVA in the presence of uncertainties in the forcing frequency range Anh and Nguyen [21] suggested an approximate analytical solution of the optimal tuning ratio of the DVA by using the idea of the classical equivalent linearization method according to the conventional criterion Recently, Tigli [22] proposed the exact optimum design parameters of the DVA for the H criterion in the case of minimizing the variance of the velocity and approximate solutions in the displacement and acceleration cases The three-element model of the dynamic vibration absorber was proposed by Asami and Nishihara [23] Unlike the standard absorber configuration, the three-element DVA contains two spring elements in which one is connected to a dashpot in series and the other is placed in parallel Up to this time, there have been some studies on the design of the threeelement DVA but most of them have been investigated for undamped primary structures Asami and Nishihara [23–25] found the optimal parameters of the three-element DVA for the H ∞ optimization, the H optimization and the stability maximization In the case of damped primary structures, to the best knowledge of the authors, there has been no study on the three-element dynamic vibration absorber Perhaps the reason is that the calculation of the three-element DVA's parameters for a damped structure is too complicated The equivalent linearization method is one of the common approaches to approximate analysis of dynamical systems The original linearization for deterministic systems was proposed by Krylov and Bogoliubov [26] Then Caughey [27,28] expanded the method for stochastic systems Thenceforward, there have been some extended versions of the equivalent linearization method Recently, using the dual approach Anh [29] proposed a so-called dual criterion for the equivalent linearization method The effectiveness of that dual criterion has been shown for Duffing, Van der Pol and Lutes-Sakani oscillators with the white noise excitation by Anh et al [30] In this paper, the authors suggest a new criterion called the weighted dual criterion The conventional criterion and the dual criterion can be obtained from the weighted dual criterion as special cases Further, based on the idea of “equivalent replacement” of the equivalent linearization method, the article deals with the design of the three-element DVA attached to a damped structure by replacing the damped primary structure with an equivalent undamped structure Comparisons have been done to verify the accuracy of the obtained results The rest of the paper is organized as follows: Section presents the results in literature in the case of undamped primary structures Section introduces the weighted dual criterion and the equivalent undamped structure The main results of the parameters of the three-element DVA attached to a damped primary structure are given in Section Section presents comparisons to validate the effectiveness of the obtained expressions Section summarizes important findings of this study 4484 N.D Anh et al / Journal of Sound and Vibration 332 (2013) 4482–4495 Classical results for undamped structures Figs and describe a standard DVA and a three-element DVA attached to an undamped primary structure, respectively It is obvious that the force excitation case and the ground motion case are similar when the primary structure is undamped In model A with the standard DVA, the amplitude magnification factor or the transmissibility of the primary structure is given by [2]    sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   x  x  2 ị2 ỵ 2d ị2  s   s GA ¼  ; (1) ¼ ¼ 2 f =ks  u0  ½α −ð1 ỵ ỵ ị2 ỵ ỵ ẵ2d 12 ị2 where m ẳ d ; s ¼ ms sffiffiffiffiffiffi sffiffiffiffiffiffiffi ks kd cd ω ω ; ωd ¼ ; α¼ d; β¼ ; ξ ¼ ωs ms md d 2md ωd ωs (2) Based on the fixed-points theory, Den Hartog derived the optimal parameters of the standard DVA as follows [5]: ẳ ; 1ỵ s : d ẳ 81 ỵ ị (3) (4) In model B with the three-element DVA, the amplitude magnification factor or the transmissibility of the primary structure is [23]    sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   x  x  A2 þ B2  s   s GB ¼  ; (5) ¼ ¼ f =ks  u0  C ỵ D2 in which A ẳ 2 ị; B ẳ 2d ỵ 2 ị; C ẳ ỵ ỵ ị ỵ β4 κα; Fig Model A: the standard DVA attached to an undamped primary structure N.D Anh et al / Journal of Sound and Vibration 332 (2013) 4482–4495 4485 Fig Model B: the three-element DVA attached to an undamped primary structure D ẳ 2d ỵ ị2d ẵ1 ỵ ỵ ị1 ỵ ị ỵ 2d β5 ; (6) and m μ ¼ d ; ωs ¼ ms sffiffiffiffiffiffi sffiffiffiffiffiffiffi ks kd cd ω ω ka ; ωd ¼ ; α¼ d; β¼ ; κ¼ : ; ξ ¼ ωs ms md d 2md ωd ωs kd (7) Based on the fixed-points theory, Asami and Nishihara [23] introduced the parameters of the three-element DVA for the H ∞ optimization as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  rffiffiffiffiffiffiffiffiffiffiffiffi μ 1− ; (8) ẳ 1ỵ 1ỵ ẳ ỵ p ỵ ịị; s p ỵ r b b −ac ξd ¼ ; r a (9) (10) where rẳ q 1ỵ ; a ẳ 22r ỵ 5r ỵ 4r 2r ỵ r ; b ¼ 2−3r −r ; c ¼ −2 þ 2r þ r : (11) Asami and Nishihara [23] have shown that the three-element DVA provides a greater reduction in the maximum amplitude magnification factor than the standard DVA does Fig shows the comparison of the amplitude magnification factors GA where the parameters of the standard DVA are given by Eqs (3) and (4) and GB where the parameters of the three-element DVA are given by Eqs (8)–(10) for the case of μ ¼ 0:2 We can see that the peaks of GB are significantly lower than those of GA The analytical solutions in Eqs (3), (4), (8)–(10) obtained for undamped structures have required an extension to damped structures because the damping always exists in real structures In next section, we will use the idea of the equivalent linearization method to study the damped primary structure by considering an equivalent undamped structure 4486 N.D Anh et al / Journal of Sound and Vibration 332 (2013) 4482–4495 Fig Comparison of the amplitude magnification factors GA and GB Equivalent undamped structure 3.1 Dual equivalent linearization technique The conventional linearization for deterministic systems was proposed by Krylov and Bogoliubov [26] Then Caughey [27,28] expanded the method for stochastic systems We here consider a single degree of freedom system with the nonlinear function depending on displacement and velocity x ỵ 2hx_ ỵ 20 x ỵ gx; x_ Þ ¼ f ðtÞ; (12) where h and ω0 are two constants, and gðx; x_ Þ is a nonlinear function of two arguments x and x_ Eq (12) is linearized to become an equation in the following linear form: x ỵ 2h ỵ bịx_ ỵ 20 ỵ kịx ẳ f ðtÞ; (13) where the linearization coefficients b; k are found by an optimal criterion There are some criteria for determining these coefficients but the most extensively used criterion is the mean square error criterion which requires that the mean square of the error exị ẳ gx; x_ ịbx_ kx between Eq (12) and its linearized Eq (13) is minimum e2 xị ẳ gx; x_ ịbx_ kxị2 -min; (14) b;k where the operator 〈.〉 is the mean value on a period or a part of the period in the case of deterministic systems, and is the expectation operator in the case of stochastic systems Although the mean square criterion (14) gives a quite good prediction, the solution error according to criterion (14) may be unacceptable for the case of the major nonlinearity 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 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: 〈ðgðx; x_ Þ−bx_ −kxÞ2 ỵ bx_ ỵ kxgx; x_ ịị2 -min; b;k; (15) where the first term describes the conventional replacement and the second term is its dual replacement In the paper, we suggest a new criterion called weighted dual criterion as follows: gx; x_ ịbx_ kxị2 ỵ 1ịbx_ ỵ kxgx; x_ ÞÞ2 〉-min b;k;λ (16) where ρ is an weighted parameter varying in the interval 0≤ρ≤1: When ρ ¼ 1=2 and ρ ¼ we have the dual and conventional criteria, respectively Using criterion (16) we get the local linearization coefficients b, k as functions of ρ corresponding to a chosen local value b ẳ bị; k ẳ kị: (17) The global values of the coefficients of the linearization are to be obtained as the averaging values of bðρÞ; kðρÞ over the interval 01: Z Z bẳ bịd; k ẳ kịd: (18) 0 Using the idea of the replacement of the equivalent linearization, the authors suggest the general replacement AB ỵ 1ịBA-min: ; (19) N.D Anh et al / Journal of Sound and Vibration 332 (2013) 4482–4495 4487 Fig The approximation of the primary structure When A is a nonlinear system and B is a linear system, we have the equivalent linearization method When A is a damped structure and B is an undamped structure, we have the problem considered in the paper In Section 3.2, we are going to use the idea of the above criteria in the design of three-element DVA for damped primary structures, namely, we will replace the damped primary structure with an equivalent undamped one Although the undamped primary structure standing alone will produce an infinite response in the resonant range, it is emphasized that our system consists of the DVA and the primary structure, therefore in total this system includes the damping and will not have infinite responses 3.2 Equivalent undamped primary structure The main idea of the present work is using the above criteria in order to replace approximately the original damped-spring– mass structure as Fig 4a with an equivalent undamped-spring–mass structure as shown in Fig 4b As above mentioned, it is noted that although the undamped primary structure standing alone will produce an infinite response in the resonant range, our system consists of the damped DVA and the primary structure, so in total this system includes the damping In Fig 4a with the original damped structure, the equation of motion is x s ỵ 2s s x_ s þ ω2s xs ¼ 0; (20) and in Fig 4b with the equivalent undamped structure, the equation of motion has the form as follows: x s ỵ 2e xs ẳ 0; (21) 2e ẳ ỵ 2s : (22) where ωe is an equivalent frequency and we denote Using the idea of the weighted dual criterion (18), we replace 2ξs ωs x_ s with γxs , hence the term γ will be determined by the following criterion: S ¼ ρ〈ð2ξs s x_ s xs ị2 D ỵ 1ịxs 2s s x_ s Þ2 〉D -min; γ;λ (23) in which 〈:〉D ¼ D Z D ð:Þdt; (24) with D is an integral region The coefficients γ and λ are determined by the following set of equations: ∂S ¼ 0; ∂γ ∂S ¼ 0: ∂λ (25) Substituting the expression of the function S in the criterion (23) into the set of Eq (25) leads to 〈x2s 〉D γ−2ð1−ρÞξs ωs 〈xs x_ s 〉D λ−2ρξs ωs 〈xs x_ s 〉D ¼ 0; 2ξs ωs 〈x_ 2s 〉D λ−〈xs x_ s 〉D γ ¼ 0; (26) and then solving system of Eq (27) in terms of two unknown constants γ and λ yields ρ〈x_ 2s 〉D 〈xs x_ s 〉D ; 〈xs 〉D 〈x_ 2s 〉D −ð1−ρÞ〈xs x_ s 〉2D ρ〈xs x_ s 〉2D : ¼ 2 〈xs 〉D 〈x_ s D 1ịxs x_ s 2D ẳ 2s s λ (27) 4488 N.D Anh et al / Journal of Sound and Vibration 332 (2013) 4482–4495 We now use 〈:〉D ¼ 〈:〉Φ ¼ Φ Z Φ ð:Þdφ; with Φ ¼ ωe D: (28) Therefore Eq (27) can be rewritten in the form γ ¼ 2ξs ωs λ¼ ρ〈x_ 2s 〉Φ 〈xs x_ s 〉Φ 〈x2s 〉Φ 〈x_ 2s 〉Φ −ð1−ρÞ〈xs x_ s 〉2Φ ; ρ〈xs x_ s 〉2Φ 2 〈xs 〉Φ 〈x_ s 〉Φ −ð1−ρÞ〈xs x_ s 〉2Φ : (29) Using Eq (21), we also have xs ẳ a cos ; ẳ e t ỵ : (30) Using Eqs (28) and (30), we obtain x2s ẳ a2 ỵ 12 sin 2Φ ; a ωe ðcos 2Φ−1Þ; 4Φ   2 a ωe Φ− sin 2Φ : 〈x_ 2s 〉Φ ¼ 2Φ 〈xs x_ s 〉Φ ¼ (31) Then substituting Eq (31) into the first equation of Eqs (29) and using Eq (22), after some calculations, we get 2e ỵ 21cos2ị2sin2ị 42 sin2 21ị1cos2ị2 s s ωe −ω2s ¼ 0: Eq (32) is a quadratic equation in terms of ωe Solving this equation, we easily obtain r h i e ẳ s 1ỵ ρð1−cos2ΦÞð2Φ−sin2ΦÞ 4Φ2 −sin2 2Φ−ð1−ρÞð1−cos2ΦÞ2 ξ2s − ρð1−cos2ΦÞð2Φ−sin2ΦÞ ξ 4Φ2 −sin2 2Φ−ð1−ρÞð1−cos2ΦÞ2 s (32)  : (33) In this paper, we get the mean value over a quarter of the period of the primary system, i.e the following integration domain,Φ ¼ π=2, is proposed The reason for this choice is that in the first quarter of the vibration period, the displacement and the velocity of the primary system not change their signs as well as directions [21] Putting the value Φ ¼ π=2 into Eq (33) yields ωs ωe ẳ r h i 1ỵ 2 41ị : (34) 2s ỵ 41ị s When ¼ 1=2 we have the result using the dual criterion s e_dua ẳ r   1ỵ 2 : (35) 2s ỵ 22 s Using the weighted dual criterion, we obtain Z ωe_wei ¼ s r h i 1ỵ 41ị d: (36) 2s ỵ 41ị s Integrating Eq (36) yields qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ωe_wei πξs πξ2s πðπ 4ịs ỵ 4s ỵ 2s ị ỵ q ln ỵ ẳ s ỵ 42s ỵ q 2 þ ξ2s Þ þ π ð4 þ π ξ2s ị ỵ 42s ị 4ị2s q ln q 4ị2 ỵ ỵ 2s ị ỵ 2s (37) We have replaced the damped primary structure with an equivalent undamped structure where the approximate frequencies ωe_dua (using the dual criterion) and ωe_wei (using the weighted dual criterion) are given in Eqs (35) and (37), respectively In Section 4, we will use the above results to give the parameters of the three-element DVA attached to a damped linear structure N.D Anh et al / Journal of Sound and Vibration 332 (2013) 4482–4495 4489 Fig The three-element DVA attached to the damped primary structure in the case of force excitation and the equivalent system Three-element DVA attached to damped linear structures 4.1 Force excitation Fig describes a three-element DVA attached to the damped primary structure where the sinusoidal force excitation f tị ẳ f sin t applies directly to the main mass The amplitude magnification factor GB is   s  x  A2 ỵ B2  s  GB ¼  ; (38) ¼ f =ks  C ỵ D2 where A ẳ 2 ị; B ẳ 2d ỵ 2 ị; C ẳ ẵ1 ỵ þ μα2 Þ þ 4αξs ξd ðκ þ 1ފαβ2 þ ỵ 4s d ị4 ; D ẳ ẵ2d ỵ ị ỵ 2s 2ẵd ỵ s ỵ d ỵ ị1 ỵ ị3 ỵ 2d ; (39) and ξs ¼ cs =2ms ωs is the structural damping ratio Using Eqs (8)–(10) for the equivalent undamped structure, we have sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffi μ αe ¼ 1ỵ 1ỵ ; p e ẳ ỵ þ μÞÞ; sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ r −b− b −ac ξde ¼ : r a (40) It is noted that αe ¼ α¼ ωd ωe ; ωd : ωs (41) Using the dual criterion with ωe_dua in Eq (35), we obtain the parameters of the three-element DVA as follows: !sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffi  2 μ π π dua ẳ ỵ 2 s 2 s 1ỵ 1ỵ ; p dua ẳ ỵ ỵ ịị; s p ỵ r −b− b −ac ξddua ¼ : r a (42) And using the weighted dual criterion Eq (37) yields   q p ps ỵ42s ỵ2s ị 2 1ỵ2s ịỵ 4ỵ 2s ị ỵ42s Þ πξ2s π ðπ −4Þξ2s πξs πðπ −4Þξs μ p ffiffiffiffiffiffiffiffiffiffiffi ffi p ffiffiffiffiffiffiffiffiffiffiffi ffi p ln ln wei ẳ ỵ 2 ỵ 1ỵ 1ỵ ; 2 2 ỵ4s ỵ p wei ẳ ỵ ỵ ịị; q p b b ac dwei ẳ 1ỵr : r a 4ỵ s 4ị2ỵ 4ỵ s ị (43) Eqs (42) and (43) are the approximate analytical solutions for the parameters of the three-element DVA attached to the primary damped structure 4490 N.D Anh et al / Journal of Sound and Vibration 332 (2013) 4482–4495 4.2 Ground motion Fig shows a three-element DVA attached to the damped structure when the primary structure is subjected to ground motion The transmissibility T B is   s x  A2t ỵ B2t  s (44) TB ¼   ¼ u0  C ỵ D2 in which At ẳ ẵ þ 4ξs ξd αð1 þ κފαβ2 þ 4ξs ξd β4 ; Bt ẳ 2s 2 ị ỵ 2d þ κα2 −β2 Þβ; (45) and C, D are determined in Eq (39) The approximate expressions of the three-element DVA's parameters are similar to Eqs (42) and (43) The effectiveness of the above results will be verified in Section 5 Comparisons 5.1 Force excitation To validate the effectiveness of results (42) and (43) proposed in the present study, these solutions will be compared with Asami and Nishihara's result [23] Fig shows the graph of the amplitude magnification factors GB where μ ¼ 0:03 and ξs ¼ 0:1 Fig is the case where μ ¼ 0:03 and ξs ¼ 0:15 Figs and 10 present the comparisons when μ ¼ 0:05, ξs ¼ 0:1 and μ ¼ 0:05, ξs ¼ 0:15, respectively Figs 11–14 show the comparisons of the maximum of the amplitude magnification factor GB when the damping ratio ξs of the primary structure is changed 5.2 Ground motion Figs 15–18 present the graph of the transmissibility T B versus the force frequency ratio β Figs 19–22 describe the comparisons the graph of the maximum of the transmissibility versus the damping structural ratio ξs Some remarks can be drawn from the comparisons in Sections 5.1 and 5.2 as follows: Fig The three-element DVA attached to the damped primary structure in the case of ground motion and the equivalent system Fig Comparison of the amplitude magnification factors GB where μ ¼ 0:03 and ξs ¼ 0:1 N.D Anh et al / Journal of Sound and Vibration 332 (2013) 4482–4495 Fig Comparison of the amplitude magnification factors GB where μ ¼ 0:03 and ξs ¼ 0:15 Fig Comparison of the amplitude magnification factors GB where μ ¼ 0:05 and ξs ¼ 0:1 Fig 10 Comparison of the amplitude magnification factors GB where μ ¼ 0:05 and ξs ¼ 0:15 Fig 11 Comparison of the maximum of the amplitude magnification factors GB where μ ¼ 0:03 4491 4492 N.D Anh et al / Journal of Sound and Vibration 332 (2013) 4482–4495 Fig 12 Comparison of the maximum of the amplitude magnification factors GB where μ ¼ 0:05 Fig 13 Comparison of the maximum of the amplitude magnification factors GB where μ ¼ 0:1 Fig 14 Comparison of the maximum of the amplitude magnification factors GB where μ ¼ 0:2  The peaks of the amplitude magnification factor GB and the transmissibility T B using results (43) and (44) are  significantly lower than those using Asami and Nishihara [23] Hence, results (43) and (44) proposed in the paper are useful in practice for both force excitation and ground motion Result (44) using the weighted dual criterion is better than result (43) using the dual criterion when the mass ratio μ and the damping structural ratio ξs are small This is significant because the ratios μ and ξs are usually small in practice The difference between results (43) and (44), however, is quite small Conclusions The three-element model of dynamic vibration absorber has better performance than the standard model DVA does To the best knowledge of the authors, however, there has been no study on the three-element dynamic vibration absorber when the primary structure is damped The present paper proposes a simple approach to design the three-element DVA attached to the damped primary structure Based on the idea of the equivalent linearization method, the work suggests a N.D Anh et al / Journal of Sound and Vibration 332 (2013) 4482–4495 Fig 15 Comparison of the transmissibility T B where μ ¼ 0:03 and ξs ¼ 0:1 Fig 16 Comparison of the transmissibility T B where μ ¼ 0:03 and ξs ¼ 0:15 Fig 17 Comparison of the transmissibility T B where μ ¼ 0:05 and ξs ¼ 0:1 Fig 18 Comparison of the transmissibility T B where μ ¼ 0:05 and ξs ¼ 0:15 4493 4494 N.D Anh et al / Journal of Sound and Vibration 332 (2013) 4482–4495 Fig 19 Comparison of the maximum of the transmissibility T B where μ ¼ 0:03 Fig 20 Comparison of the maximum of the transmissibility T B where μ ¼ 0:05 Fig 21 Comparison of the maximum of the transmissibility T B where μ ¼ 0:1 Fig 22 Comparison of the maximum of the transmissibility T B where μ ¼ 0:2 N.D Anh et al / Journal of Sound and Vibration 332 (2013) 4482–4495 4495 new criterion called the weighted dual criterion to replace approximately the damped primary structure with an equivalent undamped structure After that, the parameters of the three-element DVA are obtained by using the known results for undamped structure This paper has given the approximate analytical solutions of the H ∞ optimization for both force excitation and ground motion The effectiveness of the results is validated by numerical simulations The comparisons show that the expressions for the three-element DVA's parameters proposed in the present study are useful in practice Acknowledgments This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] H Frahm, Device for damped vibration of bodies, U.S Patent No 989958, 30 October 1909 J Ormondroyd, J.P Den Hartog, The theory of the dynamic vibration absorber, Transactions of ASME, Journal of Applied Mechanics 50 (7) (1928) 9–22 E Hahnkamm, The damping of the foundation vibrations at varying excitation frequency, Master of Architecture (1932) 192–201 (in German) J.E Brock, A note on the damped vibration absorber, Transactions of ASME, Journal of Applied Mechanics 13 (4) (1946) A-284 J.P Den Hartog, Mechanical Vibrations, McGraw-Hill, New York, 1956 O Nishihara, T Asami, Close-form solutions to the exact optimizations of dynamic vibration absorber (minimizations of the maximum amplitude magnification factors), Journal of Vibration and Acoustics 124 (2002) 576–582 S.H Crandall, W.D Mark, Random Vibration in Mechanical Systems, Academic Press, New York, 1963 Y Iwata, On the construction of the dynamic vibration absorbers, Preparation of the Japan Society of Mechanical Engineering 820 (8) (1982) 150–152 (in Japanese) T Asami, et al., Optimum design of dynamic absorbers for a system subjected to random excitation, JSME International Journal, Series 3, Vibration, Control Engineering, Engineering for Industry 34 (2) (1991) 218–226 H Yamaguchi, Damping of transient vibration by a dynamic absorber, Transactions of the Japan Society of Mechanical Engineering, Series C 54 (1988) 561–568 (in Japanese) T Ioi, K Ikeda, On the dynamic vibration damped absorber of the vibration system, Bulletin of the Japanese Society of Mechanical Engineering 21 (1978) 64–71 S.E Randall, D.M Halsted, D.L Taylor, Optimum vibration absorbers for linear damped systems, ASME Journal of Mechanical Design 103 (1981) 908–913 A.G Thompson, Optimum tuning and damping of a dynamic vibration absorber applied to a force excited and damped primary system, Journal of Sound and Vibration 77 (1981) 403–415 G.B Warburton, Optimal absorber parameters for various combinations of response and excitation parameters, Earthquake Engineering and Structural Dynamics 10 (1982) 381–401 Y Fujino, M Abe, Design formulas for tuned mass dampers based on a perturbation technique, Earthquake Engineering and Structural Dynamics 22 (1993) 833–854 O Nishihara, H Matsuhisa, Design and tuning of vibration control devices via stability criterion, Preparation of the Japan Society of Mechanical Engineering 97-10-1 (1997) 165–168 E Pennestrì, An application of Chebyshev's min–max criterion to the optimal design of a damped dynamic vibration absorber, Journal of Sound and Vibration 217 (4) (1998) 757–765 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 (2002) 284–295 A Ghosh, B Basu, A closed-form optimal tuning criterion for TMD in damped structures, Structural Control and Health Monitoring 14 (2007) 681–692 B Brown, T Singh, Minimax design of vibration absorbers for linear damped systems, Journal of Sound and Vibration 330 (11) (2010) 2437–2448 N.D Anh, N.X Nguyen, Extension of equivalent linearization method to design of TMD for linear damped systems, Structural Control and Health Monitoring 19 (6) (2012) 565–573 O.F Tigli, Optimum vibration absorber (tuned mass damper) design for linear damped systems subjected to random loads, Journal of Sound and Vibration 331 (13) (2012) 3035–3049 T Asami, O Nishihara, Analytical and experimental evaluation of an air damped dynamic vibration absorber: design optimizations of the threeelement type model, Journal of Vibration and Acoustics 121 (1999) 334–342 T Asami, O Nishihara, H2 optimization of the three-element type dynamic vibration absorbers, Journal of Vibration and Acoustics 124 (2002) 583–592 T Asami, O Nishihara, Design optimization of the three-element type dynamic vibration absorbers based on the stability maximization, Dynamics & Design Conference 19 (2002) 1463–1468 N Krylov, N Bogoliubov, Introduction to Nonlinear Mechanics, Princeton University Press, Princeton, 1943 T.K Caughey, Response of Van der Pols oscillator to random excitations, Transactions of ASME, Journal of Applied Mechanics 26 (1) (1956) 345–348 T.K Caughey, Random excitation of a system with bilinear hysteresis, Transactions of ASME, Journal of Applied Mechanics 27 (1) (1960) 649–652 N.D Anh, Duality in the analysis of responses to nonlinear systems, Vietnam Journal of Mechanics, VAST 32 (4) (2010) 263–266 N.D Anh, N.N Hieu, N.N Linh, A dual criterion of equivalent linearization method for nonlinear systems subjected to random excitation, Acta Mechanica 223 (3) (2012) 645–654  ... design of vibration absorbers for linear damped systems, Journal of Sound and Vibration 330 (11) (2010) 2437–2448 N.D Anh, N.X Nguyen, Extension of equivalent linearization method to design of. .. Device for damped vibration of bodies, U.S Patent No 989958, 30 October 1909 J Ormondroyd, J.P Den Hartog, The theory of the dynamic vibration absorber, Transactions of ASME, Journal of Applied... the case of force excitation and the equivalent system Three-element DVA attached to damped linear structures 4.1 Force excitation Fig describes a three-element DVA attached to the damped primary