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

Some approximate expressions of optimal tuning ratio of a TMD attached to a damped linear structure have been proposed.. This approximate analytical solution is obtained by using improve

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Design of TMD for damped linear structures using the dual criterion

of equivalent linearization method

N.D Anha,b, N.X Nguyenc,n

a

Institute of Mechanics, Vietnam Academy of Science and Technology, Hanoi, Vietnam

b University of Engineering and Technology, Vietnam National University, Hanoi, Vietnam

c

Hanoi University of Science, Vietnam National University, Hanoi, Vietnam

a r t i c l e i n f o

Article history:

Received 8 September 2012

Accepted 12 September 2013

Available online 16 October 2013

Keywords:

TMD

Improved equivalent linearization method

Damped structure

Dual criterion

Closed-form solution

Equivalent undamped structure

a b s t r a c t

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

1 Introduction

An auxiliary mass-spring-damper attached to a primary

struc-ture 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 TMDs for continuous structures and

multi-degrees-freedom structures have gained considerable attention of

many researchers, and TMD has been widely used in manyﬁelds 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, theﬁrst 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

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) H1 optimization (or ﬁxed-points theory), (2) H2 optimization, and (3) Stability max-imization The H1 optimization was ﬁrst proposed by Ormon-droyd and Den Hartog[2]when the primary structure is subjected

to harmonic excitation The purpose was to minimize the max-imum amplitude magniﬁcation factor of the primary structure The optimum tuning ratio of TMD wasﬁrst 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 H2optimization 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 accord-ing to H2optimization were presented by Iwata[7]and Asami[8] The stability maximization criterion and exact solutions of opti-mum parameters of TMD wereﬁrst 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 opti-mization criteria have been already solved analytically

Contents lists available atScienceDirect

journal homepage:www.elsevier.com/locate/ijmecsci

International Journal of Mechanical Sciences

n Corresponding author.

E-mail addresses: ndanh@imech.ac.vn (N.D Anh) ,

nguyennx12@gmail.com (N.X Nguyen)

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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

perturba-tion 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 H2optimization Based on the approximate

assump-tion of the existence of twoﬁx-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

lineariza-tion 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

2 Improved equivalent linearization method according

to dual criterion

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

dis-placement and velocity

€xþ2h_xþω2xþgðx; _xÞ ¼ f ðtÞ ð1Þ

where h andω0are constants, gðx; _xÞ 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,

RðτÞ ¼ E½f ðtÞf ðt þτÞ ð2Þ

SfðωÞ ¼21π

Z þ 1

1

RfðτÞeωτdτ ð3Þ

in which the notation E½U 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

€xþð2hþbÞ_xþðω2

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

e xð Þ ¼ g x; _xð Þb_xkx between Eq.(1)and its linearized Eq.(4)be minimum

E½e2ðxÞ ¼ E½ðgðx; _xÞb_xkxÞ2-min

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

E½ðgðx; _xÞb_xkxÞ2þE½ðb_xþkxλgðx; _xÞÞ2-min

where the ﬁrst term describes the conventional replacement and second term is its dual replacement

Using the dual criterion(6)and noting that E½x_x ¼ 0, we obtain

[30]

b¼ 1

2βEE½_xg½_x2

k¼ 1

2βEE½xg½x2

where it is denoted

β¼ ðE½_xgÞ2

E½_x2

E½g2þ

ðE½xgÞ2

E½x2E½g2 ð8Þ

To illustrate the above dual criterion, we now consider Van der Pol oscillator subjected to random excitation of white noise[30]

€xðαγx2Þ_xþω2x¼ sξðtÞ ð9Þ whereα;γ;ω0; s are positive real constants, the functionξðtÞ is a Gaussian white noise process of unit intensity with the Dirac delta

Table 1 The mean square response E½x 2 of Van der Pol oscillator versus the parameter s 2

α ¼ 0:2; ω ¼ 1; γ ¼ 2

s 2 E½x 2 simu E½x 2 conven Error (%) E½x 2 present Error (%)

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correlation function

RξðτÞ ¼ E½ξðtÞξðt þτÞ ¼δðτÞ ð10Þ

Using Eqs.(7)and(8)with gðx; _xÞ ¼γx2_x, we obtain

b¼35γE½x2

and the linearized equation of Eq.(9)now takes the following form

€xþ αþ35γE½x2

_xþω2x¼ sξðtÞ ð12Þ From Eq.(12)the mean square response E½x2 can be determined by

relation

E½x2 ¼ s2

2ω2αþ3γE½x2 ð13Þ

and then solving Eq.(13)with respect to unknown E½x2, we get

E½x2 ¼65γ αþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

α2þ6γs2

5ω2

ð14Þ 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 1 with given

parametersα¼ 0:2;ω¼ 1; γ¼ 2 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

struc-ture 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

3 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 kdand cd,

respectively Fig 1 shows two analytical models of the system

consisting of a damped primary structure and a TMD InFig 1(a), a sinusoidal excitation force fðtÞ ¼ f0 sinωt is applied directly to the main mass Whereas inFig 1(b), the sinusoidal vibratory motion

x0ð Þ ¼ xt 0 sinωt is transmitted to the main mass through the supporting spring and the damper

We introduce following parameters

μ¼md

ms; ωs¼

ffiffiffiffiffiffi

ks

ms

s

; ξs¼ cs

2msωs;

ωd¼

ffiffiffiffiffiffiffi

kd

md

s

; ξd¼ cd

2mdωd; α¼ωd

ωs; β¼ ω

where μ is the ratio of TMD's mass to the mass of primary structure,ωs;ξsandωd;ξdare natural frequencies and damping ratios of the structure and the TMD, respectively,αis the natural frequency ratio or tuning ratio, andβis 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]

In the case of undamped primary structure, it is clear that two models are the same Using H1optimization (i.e the minimization

of the maximum amplitude response), Den Hartog[5]introduced analytical expressions for the optimal tuning ratioαopt and the optimal damping ratioξdopt of the TMD as follows

ξdopt¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

3μ

8ð1þμÞ

s

ð19Þ

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 inFig 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

AM¼ xs

f0=ks

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðα2β2

Þ2þð2αβξdÞ2

α2ð1þ4αξsξdþα2þμα2Þβ2

þβ4

þ 2βξsðα2β2

Þþ2αβξdð1β2

μβ2

Þ

v

u

and in the modelFig 1(b), the steady-state response or the transmissibility of the primary structure is[15]

TR¼ xs

x0

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

½α2ð1þ4αξsξdÞβ2

2

þ½2βðα2ξsβ2ξsþαξdÞ2

½α2ð1þ4αξsξdþα2þμα2Þβ2

þβ4

2þ½2βξsðα2β2

Þþ2αβξdð1β2

μβ2

Þ2

v

u

ð17Þ

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InFig 2a with the original damped structure, the equation of

motion is

€xsþ2ξsωs_xsþω2

and in Fig 2b with the equivalent undamped structure, the

equation of motion has form as follows

€xsþðγþω2

whereγis an unknown constant that will be determined by using

the following dual criterion

A¼ ð2D ξsωs_xsγxsÞ2E

Tþ ðDγxs2λξsωs_xsÞ2E

T-min

in which we denote

ω2

e¼γþω2

and

:hiT¼1T

Z T

where T is an integral region and will be chosen later As observed,

theﬁrst term in the criterion(22)is the conventional replacement

while the second term describes its dual replacement The coef

ﬁ-cientsγandλare determined by the following set of equations

∂A

∂γ ¼ 0

∂A

Substituting the expression of function A in the criterion(22)into

the set of Eq.(25)leads to

2 x2

s

Tγ2ξsωs xs_xs

Tλ2ξsωs xs_xs

T¼ 0

2ξsωs _x2 s

D E

Tλ x s_xs

and then solving system of Eq.(26)in terms of unknown constants

γandλyields

γ¼ 2ξsωs

xs_xs

T

x2 s

T

" #

: 1

2 h ixs _x s 2

T

_x 2 s

T x 2 s

h iT

2 6 4

3 7 5

λ¼ xs_xs

2 T

2 x2 s

T _x2 s

D E

T xs_xs

2 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

:hiT¼ :hiΦ¼Φ1

ZΦ

0 ðUÞdφ; withΦ¼ωeT; ð28Þ hereby Eq.(27)can be rewritten in the form

γ¼ 2ξsωs

xs_xs

Φ

x2 s

Φ

" #

2 h ix s _x s 2

Φ

_x 2 s

Φh ix2s Φ

2 6 4

3 7 5

λ¼ xs_xs

2

Φ

2 x2 s

Φ _x2 s

D E

Φ x s_xs 2

Φ

ð29Þ

We also have from Eq.(21)

x ¼ a cosφ; φ¼ωtþφ ð30Þ

Fig 1 Two models of system consisting of TMD and damped primary structure.

Fig 2 The approximation of the primary structure.

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Therefore, using Eqs.(28)and(30)we obtain

x2

s

Φ¼ a2

2Φ Φ þ12 sin 2Φ

xs_xs

Φ¼a2ωe

4Φ ðcos 2Φ1Þ

_x2

s

D E

Φ¼a2ω2

e

2Φ Φ 12 sin 2Φ ð31Þ

then substituting Eq (30)into the ﬁrst equation of Eq.(27) and

using Eq.(23), after some calculations, we get

ω2

eþ 2ð1 cos 2ΦÞð2Φ sin 2ΦÞ

8Φ2

2 sin2

2Φð1 cos 2ΦÞ2ξsωsωeω2

s¼ 0 ð32Þ

Eq.(32)is a quadratic equation in terms ofωe Solving this equation,

we easily obtain

ωe¼ωs

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ ð1 cos 2ΦÞð2Φ sin 2ΦÞ

8Φ2

2 sin2

2Φð1 cos 2ΦÞ2

ξ2 s

v

u

0

B

ð1 cos 2ΦÞð2Φ sin 2ΦÞ

8Φ2

2 sin2

2Φð1 cos 2ΦÞ2ξs

!

ð33Þ Now, using the result in Eq (18) for the obtained undamped

structure as shown inFig 2b, we have

and noting that

αeopt¼ωd

ωe

αopt¼ωd

then combining with Eq (33), weﬁnd an approximate analytical

solution for the optimal tuning ratio of TMD in the case of damped

primary structures for both two models inFig 1as follows

αopt¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ ð1 cos 2 Φ Þð2 Φ sin 2 Φ Þ

8 Φ 2 2 sin 2 2 Φ ð1 cos 2 Φ Þ 2

ξ2 s

r

ð1 cos 2 Φ Þð2 Φ sin 2 Φ Þ

8 Φ 2 2 sin 2 2 Φ ð1 cos 2 Φ Þ 2ξs

1þμ

ð36Þ The choice of the constantΦ, 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.Φ¼π=2 Putting this valueΦ¼π=2 into

Eq.(36)yields

αopt¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ π2

ðπ2 2Þ 2ξ2

s

q

π

2 2ξs

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

appro-priate TMD’s damping will minimize the maximum of the

displace-ment 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 ξs¼ 0) with the mass ratio

μ¼ 0:05 In this case, it is clear that the expressions for AM and

TR in Eqs.(16)and(17)are the same.Fig 3describes the

steady-state response curves in two cases: theﬁrst case, the tuning ratio

α¼ 0:9524 and the damping ratioξ ¼ 0:1336 are obtained from

Eqs (18) and (19), and the second case when the tuning ratio changes 1% Fig 4 depicts the case in which the damping ratio changes 10% We can see that inFig 3the amplitude magniﬁcation factor AMchanges considerably, but inFig 4it is nearly unchanged

4 Comparisons

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]

ð1þμÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 4

π2ξ2 s

q

þ2

πξs

and the closed-form expression proposed by Ghosh and Basu[16]

αopt¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

14ξ2

sμð2ξ2

s1Þ ð1þμÞ3

v u

ð39Þ for excitation force model inFig 1(a)

These comparisons have done inTables 2–4, in which the values

of mass ratioμ are 0.01, 0.03, 0.05, respectively, and the different values of structural damping ratioξsare 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 2–4proclaim 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

Fig 3 Graph of the amplitude magniﬁcation factor AM versus β with

μ ¼ 0:05; ξ s ¼ 0 when the tuning ratio change 1%.

Fig 4 Graph of the amplitude magniﬁcation factor AM versus β with

μ ¼ 0:05; ξ s ¼ 0 when the damping ratio change 10%.

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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

factors for the excitation force model inFig 1a where the mass

ratioμ¼ 0:05 and the structural dampingξs¼ 0:15 for three cases:

the ﬁrst case is Den Hartog's results (18) and (19) with

α¼ 0:9524; ξd¼ 0:1336, the second case is Ghosh and Basu's

expression(39) with α¼ 0:9096;ξd¼ 0:1336 and the third case

is the present expression(37)withα¼ 0:8971;ξd¼ 0:1336

motion model inFig 1b withμ¼ 0:05 andξs¼ 0:15 in two cases,

Den Hartog's results (18) and (19) with α¼ 0:9524; ξd¼ 0:1336

and the present expression(37)withα¼ 0:8971;ξ ¼ 0:1336

5 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 structural damping

Acknowledgments

This research is funded by Vietnam National Foundation for Science and Technology Development (Nafosted)

References

[1] Frahm H Device for damped vibration of bodies U.S Patent no 989958, 30 October 1909.

[2] Ormondroyd J, Den Hartog JP The theory of the dynamic vibration absorber Trans ASME, J Appl Mech 1928;50(7):9–22

[3] Hahnkamm E The damping of the foundation vibrations at varying excitation frequency Master of Archit 1932;4:192–201 (in German)

Table 2

Optimal tuning ratio of TMD for different structural damping ratios and the mass

ratio μ ¼ 0:01.

ξ s Ioi and Ikeda

[10]

Ghosh and Basu [16]

The conventional criterion [28]

The present paper 0.005 0.9888 0.9900 (0.12) 0.9870 (0.18) 0.9881 (0.07)

0.01 0.9874 0.9899 (0.25) 0.9838 (0.36) 0.9862 (0.12)

0.02 0.9846 0.9893 (0.48) 0.9776 (0.71) 0.9822 (0.24)

0.03 0.9815 0.9883 (0.69) 0.9714 (1.03) 0.9783 (0.33)

0.05 0.9748 0.9852 (1.07) 0.9591 (1.61) 0.9705 (0.44)

0.07 0.9672 0.9804 (1.36) 0.9470 (2.09) 0.9628 (0.45)

0.10 0.9545 0.9702 (1.64) 0.9291 (2.66) 0.9514 (0.31)

0.12 0.9450 0.9613 (1.72) 0.9173 (2.93) 0.9438 (0.13)

0.14 0.9348 0.9507 (1.70) 0.9058 (3.10) 0.9363 (0.16)

0.15 0.9294 0.9447 (1.65) 0.9001 (3.15) 0.9326 (0.34)

Table 3

ratio μ ¼ 0:03.

ξ s Ioi and Ikeda

[10]

Ghosh and Basu [16]

0.01 0.9679 0.9707 (0.29) 0.9647 (0.33) 0.9670 (0.09)

0.02 0.9647 0.9701 (0.56) 0.9586 (0.63) 0.9632 (0.16)

0.03 0.9613 0.9692 (0.82) 0.9525 (0.92) 0.9593 (0.21)

0.05 0.9540 0.9661 (1.27) 0.9405 (1.42) 0.9517 (0.24)

0.07 0.9460 0.9615 (1.64) 0.9286 (1.84) 0.9441 (0.20)

0.10 0.9325 0.9515 (2.04) 0.9110 (2.31) 0.9329 (0.04)

0.12 0.9225 0.9429 (2.17) 0.8995 (2.49) 0.9255 (0.32)

0.14 0.9118 0.9326 (2.28) 0.8882 (2.59) 0.9181 (0.69)

0.15 0.9062 0.9268 (2.27) 0.8826 (2.60) 0.9145 (0.92)

Table 4

ratio μ ¼ 0:05.

ξ s Ioi and Ikeda

[10]

Ghosh and Basu [16]

0.01 0.9491 0.9522 (0.33) 0.9463 (0.30) 0.9486 (0.05)

0.02 0.9456 0.9516 (0.63) 0.9403 (0.56) 0.9448 (0.08)

0.03 0.9420 0.9507 (0.92) 0.9344 (0.81) 0.9410 (0.11)

0.05 0.9341 0.9477 (1.46) 0.9225 (1.24) 0.9336 (0.05)

0.07 0.9256 0.9432 (1.90) 0.9109 (1.59) 0.9261 (0.05)

0.10 0.9114 0.9336 (2.44) 0.8937 (1.94) 0.9151 (0.41)

0.12 0.9010 0.9252 (2.69) 0.8824 (2.06) 0.9078 (0.75)

0.14 0.8899 0.9152 (2.84) 0.8713 (2.09) 0.9006 (1.20)

0.15 0.8840 0.9096 (2.90) 0.8658 (2.06) 0.8971 (1.48)

Fig 5 Comparison of the amplitude magniﬁcation factors AM ¼ x s

f 0 =k s

with μ ¼ 0:05 and ξ s ¼ 0:15.

Fig 6 Comparison of the transmissibilities TR ¼ x s

x 0

with μ ¼ 0:05 and ξ

s ¼ 0:15.

Trang 7

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