DSpace at VNU: Design of non-traditional dynamic vibration absorber for damped linear structures

Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science http://pic.sagepub.com/ Design of non-traditional dynamic vibration absorber for damped linear structures ND Anh and NX Nguyen Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 2014 228: 45 originally published online 27 March 2013 DOI: 10.1177/0954406213481422 The online version of this article can be found at: http://pic.sagepub.com/content/228/1/45 Published by: http://www.sagepublications.com On behalf of: Institution of Mechanical Engineers Additional services and information for Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science can be found at: Email Alerts: http://pic.sagepub.com/cgi/alerts Subscriptions: http://pic.sagepub.com/subscriptions Reprints: http://www.sagepub.com/journalsReprints.nav Permissions: http://www.sagepub.com/journalsPermissions.nav Citations: http://pic.sagepub.com/content/228/1/45.refs.html >> Version of Record - Dec 26, 2013 OnlineFirst Version of Record - Mar 27, 2013 What is This? Downloaded from pic.sagepub.com at RICE UNIV on June 8, 2014 Original Article Design of non-traditional dynamic vibration absorber for damped linear structures Proc IMechE Part C: J Mechanical Engineering Science 2014, Vol 228(1) 45–55 ! IMechE 2013 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/0954406213481422 pic.sagepub.com ND Anh1,2 and NX Nguyen3 Abstract The Voigt-type of dynamic vibration absorber is a classical model and has attracted considerable attention in many years because of its simple design, high reliability and useful applications in the fields of civil and mechanical engineering Recently, a non-traditional type of dynamic vibration absorber was proposed Unlike the traditional damped absorber configuration, the non-traditional absorber has a linear viscous damper connecting the absorber mass directly to the ground instead of the main mass There have been some studies on the design of the non-traditional dynamic vibration absorber in the case of undamped primary structures Those studies have shown that the non-traditional dynamic vibration absorber has better performance than the traditional dynamic vibration absorber However, when damping is present at the primary system, there are very few studies on the design of non-traditional dynamic vibration absorber This article presents a simple approach to determine the approximate analytical solutions for the H1 optimization of the non-traditional dynamic vibration absorber attached to the damped primary structure subjected to force excitation The main idea of the study is based on the dual criterion suggested by Anh in order to replace approximately the original damped structure by an equivalent undamped structure Then the approximate analytical solution of dynamic vibration absorber’s parameters is given by using known results for undamped structure obtained The comparisons have been done to verify the effectiveness of the obtained results Keywords Dynamic vibration absorber, damped structure, dual criterion, analytical solutions, non-traditional, equivalent undamped structure Date received: September 2012; accepted: 14 February 2013 Introduction The dynamic vibration absorber (DVA) or tuned mass damper (TMD) is an auxiliary mass-spring system attached to a primary structure to reduce undesired vibration The DVA without damper was introduced by Frahm1 in 1909 but the first DVA1 had no damping element and it was only useful in a narrow range of frequencies very close to the natural frequency of the DVA In 1928, Ormondroyd and Den Hartog2 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 DVA, where a spring element and a viscous element are arranged in parallel, and has been considered as a standard model of the DVA Thenceforth, the DVA has been widely used in many fields of engineering and construction The reasons for those applications of the DVA were its efficient, reliable and low-cost characteristics 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 ratio of DVA’s frequency to the natural frequency of primary structure) and damping ratio 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 Corresponding author: Nguyen Xuan Nguyen, Hanoi University of Science, Vietnam National University, Hanoi, Vietnam Email: nguyennx12@gmail.com; nguyennx@vnu.edu.vn Downloaded from pic.sagepub.com at RICE UNIV on June 8, 2014 46 Proc IMechE Part C: J Mechanical Engineering Science 228(1) There have been many optimization criteria given to design DVAs for undamped primary structures Three typical optimization criteria are (a) H1 optimization, (b) H2 optimization and (c) stability maximization The H1 optimization was first proposed by Ormondroyd and Den Hartog2 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 a DVA was first derived by Hahnkamm3 in 1932 and later in 1946, Brock4 gave the optimum damping ratio The optimum parameters of a DVA 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 Asami6 proposed exact solutions and compared these with the results given by Den Hartog They found that both the 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 H1 optimization in practice because the exact solution was too complicated The H2 optimization criterion was suggested by Crandall and Mark7 in 1963 for the case 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 a DVA according to the H2 optimization were presented by Iwata8 and Asami9 The stability maximization criterion and exact solutions of optimum parameters of the DVA were first given by Yamaguchi10 in 1988 with the aim to improve the transient vibration of the structure In short, with undamped primary structures, all optimization criteria have already been solved analytically When damping is present at the primary structure, it is difficult to obtain analytical solutions for the optimum parameters of a DVA Ioi and Ikeda11 presented empirical formulae for the optimum parameters of the DVA attached to a damped primary structure based on the numerical method Randall et al.12 used numerical optimization procedures for evaluating the optimum DVA’s parameters, while considering damping in the structure Thompson13 gave procedures for a damped structure with a DVA, where the tuning ratio was optimized numerically and then using the optimum value obtained for the tuning ratio, the optimum damping ratio of the DVA was determined analytically Warburton14 carried out a detailed numerical study for a lightly damped structure with a DVA subjected to both harmonic and random excitation, and the optimal parameters of the DVA for various values of mass ratio and structural damping ratio were presented in the form of design tables Fujino and Abe15 employed a perturbation technique to derive formulae for the DVA’s optimal parameters, which may be used with good accuracy for mass ratio less than 2% and for very low values of the structural damping ratio In 1997, Nishihara and Matsuhisa16 gave the exact solution for the stability maximization criterion In 2002, Asami et al.17 presented a series solution for the H1 optimization and an exact solution for the H2 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 Basu18 gave a closed-form expression for optimal tuning ratio of DVA Anh and Nguyen19 suggested an approximate analytical solution of optimal tuning ratio of DVA by using the idea of the classical equivalent linearization method according to the conventional criterion Recently, Tigli20 proposed the exact optimum design parameters of the DVA for the H2 criterion in the case of minimizing the variance of the velocity and approximate solutions in the displacement and acceleration cases A non-traditional model of a DVA was proposed by Ren21 and Liu and Liu.22 Unlike the traditional damped absorber configuration, the non-traditional absorber has a linear viscous damper connecting the absorber mass directly to ground instead of the main mass Up to this time, there have been some studies on the design of the non-traditional DVA but these mainly focus on the case of undamped primary structures Liu and Liu22 found the optimum parameters of the non-traditional DVA for the case of a primary structure subjected to excitation force but their study has an error in the expression of the damping ratio of the DVA Cheung and Wong23 designed a non-traditional DVA by minimizing the maximum vibration velocity response Wong and Cheung24 considered the case when the primary structure is subjected to ground motion by minimizing the absolute displacement of the primary mass Cheung and Wong25,26 gave solutions of the optimal parameters of the non-traditional DVA for the H1 optimization and H2 optimization In the case of damped primary structures, there are very few studies on the design of the non-traditional DVA Liu and Coppola27 suggested an approximate solution based on Ghosh and Basu’s method18 and employed two numerical approaches to obtain the optimum parameters of the non-traditional DVA attached to damped primary structures 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.28 Then Caughey29,30 expanded the method for stochastic systems Going forward, there have been some extended versions of the equivalent linearization method Recently, Anh31,32 proposed an improved version of the equivalent linearization Downloaded from pic.sagepub.com at RICE UNIV on June 8, 2014 Anh and Nguyen 47 method with the dual criterion Based on the idea of the dual criterion, this article deals with the design of a non-traditional DVA attached to a damped structure by replacing the damped primary structure with an equivalent undamped structure in the case of excitation force Comparisons have been undertaken to verify the accuracy of the obtained results The article is organized as follows: section Classical results for undamped structures presents the results in the literature for the case of an undamped primary structure Showing that there is an error in the expression of the damping ratio of a DVA in Liu and Liu’s study,22 we try to recalculate and give the exact parameters of the non-traditional DVA Section Equivalent undamped structure introduces the dual criterion and the equivalent undamped structure The main results of the parameters of the non-traditional DVA attached to the damped primary structure are given in the subsequent section Comparisons to validate the effectiveness of the obtained expressions are discussed next The last section summarizes important findings of this study Classical results for undamped structures Figures and describe a traditional DVA and a non-traditional DVA attached to an undamped primary structure, respectively, when a sinusoidal excitation force ftị ẳ f0 sin !t is applied immediately to the main mass In the model A with the traditional DVA, the amplitude magnification factor of the primary structure is given by2    xs   GA ẳ  f0 =ks  s 2 2 ị2 ỵ2 d ị2 ẳ 2 ẵ2 1ỵ2 ỵ2 ị2 ỵ4 ỵẵ2 d 12 2 ị 1ị Figure Model A: a traditional DVA attached to an undamped primary structure DVA: dynamic vibration absorber where sffiffiffiffiffiffi sffiffiffiffiffiffi md ks kd , !s ¼ , !d ¼ , ¼ ms ms md cd !d ! , ¼ , ¼ d ¼ !s 2md !d !s ð2Þ Based on the fixed-points theory, Den Hartog derived the optimal parameters of the traditional DVA as follows5 1ỵ s 3 d ẳ 81 ỵ ị ẳ 3ị 4ị In model B with the non-traditional DVA, the amplitude magnification factor of the primary structure is25    xs   GB ¼  f0 =ks  s 2 2 ị2 ỵ2 d ị2 ẳ ẵ2 1ỵ2 ỵ2 ị2 ỵ4 ỵẵ2 d 12 ỵ2 ị2 5ị Based on the xed-points theory, Liu and Liu22 proposed the parameters of the non-traditional DVA but their study has an error in the expression of the damping ratio of the DVA Therefore, we try to recalculate The exact parameters of the nontraditional DVA are ¼ pffiffiffiffiffiffiffiffiffiffiffiffi 1À sffiffiffiffiffiffiffiffiffiffiffiffi 3 d ¼ 2À Figure Model B: a non-traditional DVA attached to an undamped primary structure DVA: dynamic vibration absorber Downloaded from pic.sagepub.com at RICE UNIV on June 8, 2014 ð6Þ ð7Þ 48 Proc IMechE Part C: J Mechanical Engineering Science 228(1) Liu and Liu22 showed that the non-traditional DVA with the parameters in equations (6) and (7) provides a greater reduction in the maximum amplitude magnification factor of the primary structure than the traditional DVA does Figure shows the comparison of the amplitude magnification factors GA , where the parameters of the Voigt DVA are given by equations (3) and (4), and GB , where the parameters of the nontraditional DVA are given by equations (6) and (7), for the case of  ¼ 0:2 We can see that the peaks of GB are significantly lower than those of GA The analytical solutions in equations (3), (4), (6) and (7) obtained for undamped linear structures have required an extension to damped linear structures because the damping always exists in real structures In the next section, we use the idea of the dual criterion to study the damped primary structure by considering an equivalent undamped structure Equivalent undamped structure using dual technique Dual equivalent linearization technique Although the Caughey method of equivalent linearization is normally used to linearise a non-linear system, in this article, it is used to generate an equivalent undamped model of the DVA, which is more amenable to solution The conventional linearization for deterministic systems was proposed by Krylov and Bogoliubov.28 Then Caughey29,30 expanded the method for stochastic systems Here, we consider a single degree of freedom system with the non-linear function depending on displacement and velocity x ỵ 2hx_ ỵ !20 x ỵ gx, x_ ị ẳ ftị ð8Þ Equation (8) is linearized to become an equation in the following linear form x ỵ 2h ỵ bịx_ ỵ !20 ỵ k x ẳ ftị 9ị where the coefficients of the linearization b, k are found by an optimal criterion There are some criteria for determining these coefficients b, k 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_ ị bx_ À kx between equation (8) and its linearized equation (9) is minimum     e2 xị ẳ ðgðx, x_ Þ À bx_ À kxÞ2 ! b, k ð10Þ where the operator h:i 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 (10) gives a quite good prediction, the solution error according to the criterion (10) may be unacceptable for the case of the major non-linearity In order to reduce the solution error, we may use the dual approach to the equivalent linearization method, as proposed by Anh31 and investigated in detail by Anh et al.32 The classical linearization method is based on replacing the original non-linear 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 non-linear one that belongs to the same class of the original non-linear system Combining those two steps, we may consider the following dual criterion     gx, x_ ị bx_ kxị2 ỵ bx_ þ kx À gðx, x_ ÞÞ2 ! where h and !0 are constants, gðx, x_ Þ is a non-linear _ function of two arguments x and x Figure Comparison of the amplitude magnification factors GA and GB for the case of  ¼ 0:2 Downloaded from pic.sagepub.com at RICE UNIV on June 8, 2014 b, k,  ð11Þ Anh and Nguyen 49 where the first term describes the conventional replacement and the second term is its dual replacement Using the idea of the replacement of the equivalent linearization, the authors suggest the general replacement   hA À Bi ỵ B A ! 12ị , where !e is an equivalent frequency and is denoted as !2e ẳ ỵ !2s 15ị Using the idea of the dual criterion, we replace 2s !s x_ s with xs , where the term will be determined by using the following dual criterion     S ¼ ð2s !s x_ s xs ị2 D ỵ xs 2s !s x_ s Þ2 D ! ,  When A is a non-linear 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 this article In the next section, we are going to use the idea of the above dual criterion in the problem of the design a DVA for damped linear structures, namely, we will replace the damped primary structure with an equivalent undamped one Although the undamped structure standing alone will produce infinite response in the resonant range, it is emphasized that our system consists of the DVA and the primary structure; so, in total, this system includes the damping Equivalent undamped structure The main idea of this study is using the dual criterion in order to replace approximately the original damped-spring-mass structure, shown Figure 4(a), with an equivalent undamped-spring-mass structure (see Figure 4(b)) As mentioned above, it is noted that although the undamped structure standing alone will produce 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 Figure 4(a) with the original damped structure, the motion equation is 16ị in which h: iD ẳ D Z :ịdt ð13Þ In Figure 4(b) with the equivalent undamped structure, the motion equation has the form as follows x s ỵ !2e xs ẳ 14ị 17ị where D is an integral region and will be chosen later The first term in criterion (16) is the conventional replacement, while the second term describes its dual replacement The coefficients and  are determined by the following set of equations @S ¼ 0, @ @S ẳ 0: @ 18ị Substituting the expression of the function S in criterion (16) into the set of equation (18) leads to   x2s D À 2s !s hxs x_ s iD  À 2s !s hxs x_ s iD ¼ 0,   2s !s x_ 2s D  À hxs x_ s iD ¼ 0, ð19Þ and then solving the system of equation (19) in terms of two unknown constants and  yields " x s ỵ 2s !s x_ s ỵ !2s xs ¼ D # hxs x_ s i ¼ 2s !s   D :4 5, hxs x_ s i2D xs D À hx_ i hx2 i s D s D ¼ hxs x_ s i2     D x2s D x_ 2s D Àhxs x_ s i2D Figure The approximation of the primary structure Downloaded from pic.sagepub.com at RICE UNIV on June 8, 2014 : ð20Þ 50 Proc IMechE Part C: J Mechanical Engineering Science 228(1) The first factor of the expression of in equation (20) is the result obtained via the conventional criterion This result has been presented by Anh and Nguyen.19 We now use h:iD ¼ h:iÈ ¼ È with È ¼ !e D ð:Þd’, ð21Þ We have replaced the damped primary structure by an equivalent undamped structure with the approximate frequency !e given in equation (27) In the next section, we use this result to give parameters of nontraditional DVA attached to damped linear structures # hxs x_ s i2     È x2s È x_ 2s È Àhxs x_ s i2È ð22Þ : Using equation (14), we also have xs ¼ a cos ’, ’ ẳ !e t ỵ 23ị Therefore, combining equations (21) and (23) we obtain    2 a2 xs ẩ ẳ ẩ ỵ sin 2ẩ , 2ẩ a !e hxs x_ s iÈ ¼ ðcos 2È À 1Þ, 4È    2 a2 !2e È À sin 2È : x_ s È ¼ 2È Parameters of non-traditional DVA attached to damped linear structures by using the equivalent undamped structure Figure depicts a non-traditional DVA attached to the damped primary structure with the sinusoidal excitation force ftị ẳ f0 sin !t The amplitude magnification factor GB is    xs    GB ¼  f0 =ks  vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u 2 ị ỵ2 d ị2 ẳu u u ỵ þ 2 þ 4s d þ t  ỵ d þ 2 þ 2s À ð28Þ ð24Þ Substituting equation (24) into the first equation of equation (22) and using equation (15), after some calculations, we get !2e ỵ 27ị ẩ hxs x_ s i ẳ 2s !s   È :4 5, hxs x_ s i2È xs È À hx_ i hx2 i s È s È ¼ !s ffi !e ¼ q 2 ỵ 2 s2 ỵ 22 s ð À2Þ Z Equation (20) can be rewritten in the form " not change their signs as well as directions.19 Putting the value È ¼ =2 into equation (26) yields 2ð1 À cos 2ÈÞð2È À sin 2ÈÞ s !s !e À !2s ¼ 8È2 À sin2 2È À ð1 À cos 2ÈÞ2 ð25Þ Equation (25) is a quadratic equation in terms of !e Solving this equation, we easily obtain 0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2 ð1 À cos 2ÈÞð2È À sin 2ẩị @ 1ỵ !e ẳ !s 8ẩ2 sin2 2È À ð1 À cos 2ÈÞ2  ð1 À cos 2ÈÞð2È À sin 2ÈÞ s2 À  s 8È2 À sin2 2È À ð1 À cos 2ÈÞ2 ð26Þ In this article, we get the mean value over a quarter of the period of the primary system, i.e the value È ¼ =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 where s ¼ cs =2ms !s is the structural damping ratio Using equation (6) p for the equivalent undamped ffiffiffiffiffiffiffiffiffiffiffiffi structure, i.e e ¼ 1= À  and equation (27) for the equivalent frequency and noting that !d , !e !d ẳ , !s e ẳ 29ị we obtain ¼   ffi pffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2   ỵ 2 s ỵ 2 s 30ị  2ị When s ẳ 0, the analytical solution (30) will reduce to equation (6) for undamped primary structures The expression of the optimal tuning ratio in equation (30) is independent of the DVA’s damping This optimal tuning ratio together with the appropriate DVA’s damping will minimize the maximum displacement of the primary structure However, it is noted that the maximum amplitude magnification factor GB will change significantly when the tuning ratio changes slightly Whereas GB changes very Downloaded from pic.sagepub.com at RICE UNIV on June 8, 2014 Anh and Nguyen 51 Figure The non-traditional DVA-damped primary structure system and the approximate non-traditional DVA-undamped structure system DVA: dynamic vibration absorber Figure Graph of the amplitude magnification factor GB versus with  ¼ 0:05, s ¼ when the tuning ratio change by 1% little even when the damping ratio changes considerably To express this comment, we consider an example, say, a undamped primary structure (i.e s ¼ 0) with the mass ratio  ¼ 0:05 Figure describes the steady-state response curves in two cases: in the first case, the tuning ratio ¼ 1:0260 and damping ratio d ¼ 0:1387, obtained from equations (6) and (7), and in the second case, the tuning ratio changes by 1% Figure depicts the case in which the damping ratio changes by 5% We can see that in Figure the maximum amplitude magnification factor GB increases significantly but in Figure 7, it changes very little Therefore, we use the damping ratio of the non-traditional DVA as equation (7) and we obtain the parameters of the non-traditional DVA as follows qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , pffiffiffiffiffiffiffiffiffiffiffiffi À  ỵ 2 s2 ỵ 22 s  2ị s 3 : d ẳ 2 ẳ 31ị Equation (31) shows the main results proposed in this article The effectiveness of the results are verified in the following part To validate the effectiveness of the results (equation 31) proposed in this study, these solutions will be compared with those of other authors in the literature Figure shows the amplitude magnification factor GB in the case of a non-traditional DVA with  ¼ 0:1 and s ¼ 0:08 The dashed line is Liu and Liu’s result22 (equations (6) and (7)), with ¼ 1:0541 and d ¼ 0:1987 The solid line is the results of this study (equation (40)), with ¼ 1:0210 and d ¼ 0:1987, and the dot-dashed line is Liu and Coppola’s result,27 with ¼ 1:0405 and d ¼ 0:1987 Figure is the case where  ¼ 0:1 and s ¼ 0:15 Figures 10 and 11 present comparisons when  ¼ 0:15, s ¼ 0:08 and s ¼ 0:15, respectively Figures 12 and 13 depict comparisons of the maximum amplitude magnification factor GB when the damping ratio s of the primary structure is changed We observe that the maximum amplitude magnification factor GB is the smallest when Downloaded from pic.sagepub.com at RICE UNIV on June 8, 2014 52 Proc IMechE Part C: J Mechanical Engineering Science 228(1) Figure Graph of the amplitude magnification factor GB versus with  ¼ 0:05, s ¼ when the damping ratio change by 5% Figure Comparison of the amplitude magnification factors GB in the case of a non-traditional DVA with  ¼ 0:1 and s ¼ 0:08 DVA: dynamic vibration absorber Figure Comparison of the amplitude magnification factors GB in the case of a non-traditional DVA with  ¼ 0:1 and s ¼ 0:15 DVA: dynamic vibration absorber Downloaded from pic.sagepub.com at RICE UNIV on June 8, 2014 Anh and Nguyen 53 Figure 10 Comparison of the amplitude magnification factors GB in the case of a non-traditional DVA with  ¼ 0:15 and s ¼ 0:08 DVA: dynamic vibration absorber Figure 11 Comparison of the amplitude magnification factors GB in the case of a non-traditional DVA with  ¼ 0:15 and s ¼ 0:15 DVA: dynamic vibration absorber Figure 12 Comparison of the maximum amplitude magnification factor GB with  ¼ 0:1 Downloaded from pic.sagepub.com at RICE UNIV on June 8, 2014 54 Proc IMechE Part C: J Mechanical Engineering Science 228(1) Figure 13 Comparison of the maximum amplitude magnification factor GB with  ¼ 0:15 we use the parameters of the non-traditional DVA given by equation (31) Therefore, the results (equation 31) obtained in this article are useful in practice Concluding remarks This article proposes a simple approach to design a non-traditional DVA attached to the damped primary structure In the case of undamped primary structures, we find that there is an error in the expression of the DVA’s damping ratio in Liu and Liu’s article.22 We try to recalculate and give the exact parameters of the non-traditional DVA When damping is present at the primary system, the main objective of the study is to replace approximately the original damped structure by an equivalent undamped structure based on the idea of the dual criterion Then we have obtained the approximate analytical solutions for the H1 optimization by using the known results for the undamped structure obtained The results are validated and the effectiveness is compared with other such studies in the literature The comparisons show that the expressions proposed in this study are useful in practice Funding This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) References Frahm H Device for damped vibration of bodies US Patent 989958, 30 October 1909 Ormondroyd J and Den Hartog JP The theory of the dynamic vibration absorber ASME J Appl Mech 1928; 50(7): 9–22 Hahnkamm E The damping of the foundation vibrations at varying excitation frequency (in German) Ing Arch 1932; 4: 192–201 Brock JE A note on the damped vibration absorber ASME J Appl Mech 1946; 13(4): A–A284 Den Hartog JP Mechanical vibrations New York: McGraw-Hill, 1956 Nishihara O and Asami T Close-form solutions to the exact optimizations of dynamic vibration absorber (minimizations of the maximum amplitude magnification factors) J Vib Acoust 2002; 124: 576–582 Crandall SH and Mark WD Random vibration in mechanical systems New York: Academic Press, 1963 Iwata Y On the construction of the dynamic vibration absorbers (in Japanese) Prepr Jpn Soc Mech Eng 1982; 820(8): 150–152 Asami T Optimum design of dynamic absorbers for a system subjected to random excitation JSME Int J Ser III 1991; 34(2): 218–226 10 Yamaguchi H Damping of transient vibration by a dynamic absorber (in Japanese) Trans Jpn Soc Mech Eng Ser C 1988; 54(499): 561–568 11 Ioi T and Ikeda K On the dynamic vibration damped absorber of the vibration system Bull Jpn Soc Mech Eng 1978; 21(151): 64–71 12 Randall SE, Halsted DM and Taylor DL Optimum vibration absorbers for linear damped systems J Mech Des (ASME) 1981; 103: 908–913 13 Thompson AG Optimum tuning and damping of a dynamic vibration absorber applied to a force excited and damped primary system J Sound Vib 198; 77: 403–415 14 Warburton GB Optimal absorber parameters for various combinations of response and excitation parameters Earthquake Eng Struct Dynam 1982; 10: 381–401 15 Fujino Y and Abe M Design formulas for tuned mass dampers based on a perturbation technique Earthquake Eng Struct Dynam 1993; 22: 833–854 16 Nishihara O and Matsuhisa H Design and tuning of vibration control devices via stability criterion Prepr Jpn Soc Mech Eng 1997; 97-10-1: 165–168 17 Asami T, Nishihara O and Baz AM Analytical solutions to H1 and H2 optimization of dynamic vibration absorbers attached to damped linear systems J Vib Acoust 2002; 124: 284–295 Downloaded from pic.sagepub.com at RICE UNIV on June 8, 2014 Anh and Nguyen 55 18 Ghosh A and Basu B A closed-form optimal tuning criterion for TMD in damped structures Struct Control Health Monit 2007; 14: 681–692 19 Anh ND and Nguyen NX Extension of equivalent linearization method to design of TMD for linear damped systems Struct Control Health Monit 2012; 19(6): 565–573 20 Tigli OF Optimum vibration absorber (tuned mass damper) design for linear damped systems subjected to random loads J Sound Vib 2012; 331(13): 3035–3049 21 Ren MZ A variant design of the dynamic vibration absorber J Sound Vib 2001; 245(4): 762–770 22 Liu K and Liu J The damped dynamic vibration absorbers: revisited and new result J Sound Vib 2005; 284: 1181–1189 23 Cheung YL and Wong WO Design of a non-traditional dynamic vibration absorber (L) J Acoust Soc Am 2009; 126(2): 564–567 24 Wong WO and Cheung YL Optimal design of a damped dynamic vibration absorber for vibration control of structure excited by ground motion Eng Struct 2008; 30: 282–286 25 Wong WO and Cheung YL H-infinity optimization of a variant design of the dynamic vibration absorberrevisited and new results J Sound Vib 2011; 330: 3901–3912 26 Wong WO and Cheung YL H2 optimization of a nontraditional dynamic vibration absorber for vibration control of structures under random force excitation J Sound Vib 2011; 330: 1039–1044 27 Liu K and Coppola G Optimal design of damped dynamic vibration absorber for damped primary systems Trans Canadian Soc Mech Eng 2010; 34(1): 119–135 28 Krylov N and Bogoliubov N Introduction to nonlinear mechanics Princeton, NJ: Princeton University Press, 1943 29 Caughey TK Response of Van der Pols oscillator to random excitations Trans ASME J App Mech 1956; 26(1): 345–348 30 Caughey TK Random excitation of a system with bilinear hysteresis Trans ASME J App Mech 1960; 27(1): 649–652 31 Anh ND Duality in the analysis of responses to nonlinear systems Vietnam J Mech VAST 2010; 32(4): 263–266 32 Anh ND, Hieu NN and Linh NN A dual criterion of equivalent linearization method for nonlinear systems subjected to random excitation Acta Mechanica 2012; 223(3): 645–654 Downloaded from pic.sagepub.com at RICE UNIV on June 8, 2014 ... WO Design of a non-traditional dynamic vibration absorber (L) J Acoust Soc Am 2009; 126(2): 564–567 24 Wong WO and Cheung YL Optimal design of a damped dynamic vibration absorber for vibration. .. optimization of a nontraditional dynamic vibration absorber for vibration control of structures under random force excitation J Sound Vib 2011; 330: 1039–1044 27 Liu K and Coppola G Optimal design of. .. in the case of undamped primary structures Those studies have shown that the non-traditional dynamic vibration absorber has better performance than the traditional dynamic vibration absorber However,