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...

Design of three-element dynamic vibration absorber

for damped linear structures

N.D Anha,b, N.X Nguyenc,n, L.T Hoab

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 12 January 2013

Received in revised form

17 March 2013

Accepted 24 March 2013

Handling Editor: L.G Tham

Available online 9 May 2013

a b s t r a c t

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 element DVA for undamped primary structures Those studies have shown that the three-element 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

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1 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

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Journal of Sound and Vibration

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n Corresponding author Tel.: +84 989201949.

E-mail address: nguyennx12@gmail.com (N.X Nguyen).

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 H2optimization 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 H2optimization 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 2 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 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 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 H2criterion 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 three-element 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 H2 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 2presents the results in literature in the case of undamped primary structures Section 3 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 inSection 4.Section 5presents comparisons to validate the effectiveness of the obtained expressions Section 6 summarizes important findings of this study

2 Classical results for undamped structures

Figs 1 and 2describe 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]

GA¼

xs

f0=ks

¼

xs

u0

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðα2−β2Þ2þ ð2αβξdÞ2

½α2−ð1 þ α2þ μα2Þβ2þ β42þ ½2βαξdð1−β2−μβ2Þ2

s

where

μ ¼md

ms; ωs¼

ffiffiffiffiffiffi

ks

ms

s

; ωd¼

ffiffiffiffiffiffiffi

kd

md

s

; ξd¼ cd

2mdωd; α ¼ωd

ωs; β ¼ωω

s

(2) Based on the fixed-points theory, Den Hartog derived the optimal parameters of the standard DVA as follows[5]:

ξd¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

3μ

8ð1 þ μÞ

s

In model B with the three-element DVA, the amplitude magnification factor or the transmissibility of the primary structure

is[23]

GB¼

xs

f0=ks

¼

xs

u0

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

A2þ B2

C2þ D2

s

in which

A¼ καðα2−β2Þ;

B¼ 2ξdβðα2þ κα2−β2Þ;

C¼ κα3−β2καð1 þ α2þ μα2Þ þ β4κα;

D¼ 2ξdα2βð1 þ κÞ−2ξdβ3½1 þ α2ð1 þ κÞð1 þ μÞ þ 2ξdβ5; (6) and

μ ¼md

ms; ωs¼

ffiffiffiffiffiffi

ks

ms

s

; ωd¼

ffiffiffiffiffiffiffi

kd

md

s

; ξd¼ cd

2mdωd; α ¼ωd

ωs; β ¼ωω

s; κ ¼ka

Based on the fixed-points theory, Asami and Nishihara[23]introduced the parameters of the three-element DVA for the H∞ optimization as

α ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1

1þ μ 1−

ffiffiffiffiffiffiffiffiffiffiffiffiμ

1þ μ

r

s

κ ¼ 2ðμ þpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiμð1 þ μÞÞ; (9)

ξd¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ r r

−b−pffiffiffiffiffiffiffiffiffiffiffiffiffib2−ac a

s

where

r¼ ffiffiffiffiffiffiffi1 þμ μ

q

;

a¼ −2−2r þ 5r2þ 4r3−2r5þ r6;

b¼ 2−3r2−r4;

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 3shows the comparison of the amplitude magnification factors GA

where the parameters of the standard DVA are given by Eqs.(3) and (4)and GBwhere the parameters of the three-element DVA are given by Eqs.(8)–(10)for the case ofμ ¼ 0:2 We can see that the peaks of GBare 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

Fig 2 Model B: the three-element DVA attached to an undamped primary structure.

3 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

where h andω0are two constants, and gðx; _xÞ is a nonlinear function of two arguments x and _x Eq.(12)is linearized to become an equation in the following linear form:

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 eðxÞ ¼ gðx; _xÞ−b_x−kx between Eq.(12)and its linearized Eq.(13)is minimum

〈e2ðxÞ〉 ¼ 〈ðgðx; _xÞ−b_x−kxÞ2〉-min

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; _xÞ−b_x−kxÞ2

〉 þ 〈ðb_x þ kx−λgðx; _xÞÞ2

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:

ρ〈ðgðx; _xÞ−b_x−kxÞ2

〉 þ ð1−ρÞ〈ðb_x þ kx−λgðx; _xÞÞ2

〉-min

whereρ is an weighted parameter varying in the interval 0≤ρ≤1: When ρ ¼ 1=2 and ρ ¼ 1 we have the dual and conventional criteria, respectively Using criterion(16)we get the local linearization coefficients b, k as functions ofρ corresponding to a chosen local valueρ

The global values of the coefficients of the linearization are to be obtained as the averaging values of bðρÞ; kðρÞ over the interval 0≤ρ≤1:

b¼Z 1

0

bðρÞdρ; k ¼Z 1

0

Using the idea of the replacement of the equivalent linearization, the authors suggest the general replacement

Fig 3 Comparison of the amplitude magnification factors G A and G B

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

InSection 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 asFig 4a with an equivalent undamped-spring–mass structure as shown inFig 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

InFig 4a with the original damped structure, the equation of motion is

€xsþ 2ξsωs_xsþ ω2

and inFig 4b with the equivalent undamped structure, the equation of motion has the form as follows:

€xsþ ω2

whereωeis an equivalent frequency and we denote

ω2

e¼ γ þ ω2

Using the idea of the weighted dual criterion(18), we replace 2ξsωs_xswithγxs, hence the termγ will be determined by the following criterion:

S¼ ρ〈ð2ξsωs_xs−γxsÞ2

〉Dþ ð1−ρÞ〈ðγxs−2λξsωs_xsÞ2

in which

〈:〉D¼D1Z D

with D is an integral region The coefficientsγ and λ are determined by the following set of equations:

∂S

∂γ¼ 0;

∂S

∂λ¼ 0:

(25)

Substituting the expression of the function S in the criterion(23)into the set of Eq.(25)leads to

〈x2

s〉Dγ−2ð1−ρÞξsωs〈xs_xs〉Dλ−2ρξsωs〈xs_xs〉D¼ 0;

2ξsωs〈_x2

and then solving system of Eq.(27)in terms of two unknown constantsγ and λ yields

γ ¼ 2ξsωs ρ〈_x2

s〉D〈xs_xs〉D

〈x2

s〉D〈_x2

s〉D−ð1−ρÞ〈xs_xs〉2

D

;

λ ¼ ρ〈xs_xs〉2

D

〈x2〉 〈_x2

〉 −ð1−ρÞ〈x _x〉2:

(27)

Fig 4 The approximation of the primary structure.

We now use

〈:〉D¼ 〈:〉Φ¼Φ1ZΦ

Therefore Eq.(27)can be rewritten in the form

γ ¼ 2ξsωs ρ〈_x 2

s 〉Φ〈xs_xs〉Φ

〈x 2 s〉Φ〈_x 2 s〉Φ−ð1−ρÞ〈xs_xs〉 2

Φ;

λ ¼ ρ〈xs_xs〉2

Φ

〈x2

s〉Φ〈_x2

s〉Φ−ð1−ρÞ〈xs_xs〉2

Φ

Using Eq.(21), we also have

Using Eqs.(28) and (30), we obtain

〈x2

s〉Φ¼a 2

2 ΦΦ þ1sin 2Φ;

〈xs_xs〉Φ¼a2ωe

4Φ ðcos 2Φ−1Þ;

〈_x2

s〉Φ¼a2ω2e

2Φ Φ−

1

2sin 2Φ

Then substituting Eq.(31)into the first equation of Eqs.(29)and using Eq.(22), after some calculations, we get

ω2

eþ 2ρð1−cos2ΦÞð2Φ−sin2ΦÞ

4Φ2−sin2

2Φ−ð1−ρÞð1−cos2ΦÞ2ξsωsωe−ω2

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

ωe¼ ωs

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ ρð1−cos2ΦÞð2Φ−sin2ΦÞ

4 Φ 2 −sin 2 2 Φ−ð1−ρÞð1−cos2ΦÞ 2

ξ2 s

r

− ρð1−cos2ΦÞð2Φ−sin2ΦÞ

4 Φ 2 −sin 2 2 Φ−ð1−ρÞð1−cos2ΦÞ 2ξs



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 do not change their signs as well as directions[21] Putting the valueΦ ¼ π=2 into

Eq.(33)yields

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ 2 ρπ

π 2 −4ð1−ρÞ

ξ2 s

r

þ 2 ρπ

π 2 −4ð1−ρÞξs

Whenρ ¼ 1=2 we have the result using the dual criterion

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ π

2 −2

2

ξ2 s

r

þ π

2 −2ξs

Using the weighted dual criterion, we obtain

ωe_wei¼

Z1 0

ωs

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ 2ρπ

π 2 −4ð1−ρÞ

ξ2 s

r

þ 2ρπ

π 2 −4ð1−ρÞξs

Integrating Eq.(36)yields

ωe_wei

ωs ¼ 1−πξs

2 þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiπξ2s

π2þ 4ξ2 s

q

þ πþ

πðπ2−4Þξs

8 ln

πð ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiπ2þ 4ξ2

s

q

þ 2ξsÞ

π2−4

−π2ðπ2−4Þξ2s

8 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

4þ π2ξ2 s

q ln2π2ð1 þ ξ2

sÞ þ π ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið4 þ π2ξ2

sÞðπ2þ 4ξ2

sÞ q

ðπ2−4Þð2 þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4þ π2ξ2

s

q

We have replaced the damped primary structure with an equivalent undamped structure where the approximate frequenciesωe_dua(using the dual criterion) andωe_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

4 Three-element DVA attached to damped linear structures

4.1 Force excitation

Fig 5describes a three-element DVA attached to the damped primary structure where the sinusoidal force excitation

fðtÞ ¼ f0sinωt applies directly to the main mass The amplitude magnification factor GBis

GB¼

xs

f0=ks

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

A2þ B2

C2þ D2

s

where

A¼ καðα2−β2Þ;

B¼ 2βξdðα2þ κα2−β2Þ;

C¼ κα3−½κð1 þ α2þ μα2Þ þ 4αξsξdðκ þ 1Þαβ2þ ðκα þ 4ξsξdÞβ4;

D¼ ½2ξdð1 þ κÞ þ 2ξsκαα2

β−2½ξdþ ξsκα þ ξdα2

ð1 þ κÞð1 þ μÞβ3

þ 2ξdβ5

andξs¼ cs=2msωsis the structural damping ratio

Using Eqs.(8)–(10)for the equivalent undamped structure, we have

αe¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1

þμ 1− ffiffiffiffiffiffiffiμ

1 þμ

q

s

;

κe¼ 2ðμ þpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiμð1 þ μÞÞ;

ξde¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ r r

−b−pffiffiffiffiffiffiffiffiffiffiffiffiffib2−ac a

s

It is noted that

αe¼ωd

ωe;

α ¼ωd

Using the dual criterion withωe_duain Eq.(35), we obtain the parameters of the three-element DVA as follows:

αdua¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ π

2 −2

2

ξ2 s

r

− π

2 −2ξs

! ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1

þμ 1− ffiffiffiffiffiffiffiμ

1 þμ

q

s

;

κdua¼ 2ðμ þpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiμð1 þ μÞÞ;

ξddua¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ r r

−b−pffiffiffiffiffiffiffiffiffiffiffiffiffib2−ac a

s

And using the weighted dual criterion Eq.(37)yields

αwei¼ 1−πξs

2 þ ffiffiffiffiffiffiffiffiffiffiffiffiπξ2s

π 2 þ4ξ 2 s

p

þπþπðπ 2 −4Þξs

8 lnπð

ffiffiffiffiffiffiffiffiffiffiffiffi

π 2 þ4ξ 2 s

p

þ2ξsÞ

π 2 −4 −π2ðπ2−4Þξ2s

8 ffiffiffiffiffiffiffiffiffiffiffiffi

4 þπ 2 ξ 2 s

p ln2π2ð1þξ2sÞþπ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð4þπ 2 ξ 2 sÞðπ 2 þ4ξ 2 sÞ

p

ðπ 2 −4Þð2þ ffiffiffiffiffiffiffiffiffiffiffiffi

4 þπ 2 ξ 2 s

p

Þ

1

þμ 1− ffiffiffiffiffiffiffiμ 1þμ

q

s

;

κwei¼ 2ðμ þpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiμð1 þ μÞÞ;

ξdwei¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þr

r −b− ffiffiffiffiffiffiffiffiffiffi

b 2 −ac

p

a

q

:

(43)

Eqs.(42) and (43)are the approximate analytical solutions for the parameters of the three-element DVA attached to the primary damped structure

Fig 5 The three-element DVA attached to the damped primary structure in the case of force excitation and the equivalent system.

4.2 Ground motion

Fig 6shows a three-element DVA attached to the damped structure when the primary structure is subjected to ground motion The transmissibility TBis

TB¼

xs

u0

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

A2tþ B2 t

C2

þ D2

s

(44)

in which

At¼ κα3−½κ þ 4ξsξdαð1 þ κÞαβ2þ 4ξsξdβ4;

Bt¼ 2ξsκαðα2

−β2

Þβ þ 2ξdðα2

þ κα2

−β2

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 inSection 5

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 7 shows the graph of the amplitude magnification factors GB whereμ ¼ 0:03 and

ξs¼ 0:1.Fig 8is the case whereμ ¼ 0:03 and ξs¼ 0:15.Figs 9 and 10present the comparisons whenμ ¼ 0:05, ξs¼ 0:1 and

μ ¼ 0:05, ξs¼ 0:15, respectively

Figs 11–14show the comparisons of the maximum of the amplitude magnification factor GBwhen the damping ratioξs

of the primary structure is changed

5.2 Ground motion

Figs 15–18 present the graph of the transmissibility TB versus the force frequency ratio β.Figs 19–22 describe the comparisons the graph of the maximum of the transmissibility versus the damping structural ratioξs

Some remarks can be drawn from the comparisons inSections 5.1 and 5.2as follows:

Fig 6 The three-element DVA attached to the damped primary structure in the case of ground motion and the equivalent system.

Fig 7 Comparison of the amplitude magnification factors G where μ ¼ 0:03 and ξ ¼ 0:1.

Fig 9 Comparison of the amplitude magnification factors G B where μ ¼ 0:05 and ξ s ¼ 0:1.

Fig 10 Comparison of the amplitude magnification factors G B where μ ¼ 0:05 and ξ s ¼ 0:15.

Fig 8 Comparison of the amplitude magnification factors G B where μ ¼ 0:03 and ξ s ¼ 0:15.

Fig 11 Comparison of the maximum of the amplitude magnification factors G where μ ¼ 0:03.