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

Engineering Science Engineers, Part C: Journal of Mechanical http://pic.sagepub.com/content/228/1/45 The online version of this article can be found at: DOI: 10.1177/0954406213481422 or

Engineering Science Engineers, Part C: Journal of Mechanical

http://pic.sagepub.com/content/228/1/45

The online version of this article can be found at:

DOI: 10.1177/0954406213481422 originally published online 27 March 2013

2014 228: 45

Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science

ND Anh and NX Nguyen

Design of non-traditional dynamic vibration absorber for damped linear structures

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Design of non-traditional dynamic

vibration absorber for damped linear

structures

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 H1optimization 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: 5 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 Frahm1in 1909 but the first DVA1had

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

frequen-cies 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

paral-lel, 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

1

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

2

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

3

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

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

There have been many optimization criteria given

to design DVAs for undamped primary structures

Three typical optimization criteria are (a) H1

opti-mization, (b) H2 optimization and (c) stability

maxi-mization The H1optimization was first proposed by

Ormondroyd and Den Hartog2 when the primary

structure is subjected to harmonic excitation The

pur-pose was to minimize the maximum amplitude

mag-nification factor of the primary structure The

optimum tuning ratio of a DVA was first derived by

Hahnkamm3in 1932 and later in 1946, Brock4 gave

the optimum damping ratio The optimum

param-eters 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

pro-vided a very good approximation of the exact

solu-tions for the H1optimization in practice because the

exact solution was too complicated The H2

optimiza-tion criterion was suggested by Crandall and Mark7in

1963 for the case when the primary structure is

sub-jected to random excitation The purpose was to

min-imize 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

param-eters of a DVA according to the H2optimization were

presented by Iwata8 and Asami9 The stability

maxi-mization 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

cri-teria have already been solved analytically

When damping is present at the primary structure,

it is difficult to obtain analytical solutions for the

opti-mum parameters of a DVA Ioi and Ikeda11presented

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

proced-ures 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

struc-ture with a DVA subjected to both harmonic and

random excitation, and the optimal parameters of

the DVA for various values of mass ratio and

struc-tural 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 Matsuhisa16gave the exact solu-tion for the stability maximizasolu-tion criterion In 2002, Asami et al.17presented a series solution for the H1

optimization and an exact solution for the H2 opti-mization but their solution was extremely complicated

so they proposed an approximate solution for prac-tical use Based on the approximate assumption of the existence of two fix-points, Ghosh and Basu18gave 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, Tigli20proposed 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 accel-eration 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 struc-tures Liu and Liu22found 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 con-sidered the case when the primary structure is sub-jected 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 sug-gested 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, Anh31,32proposed an improved version of the equivalent linearization

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

struc-ture by replacing the damped primary strucstruc-ture with

an equivalent undamped structure in the case of

exci-tation 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

litera-ture for the case of an undamped primary struclitera-ture

Showing that there is an error in the expression of the

damping ratio of a DVA in Liu and Liu’s study,22we

try to recalculate and give the exact parameters of the

non-traditional DVA Section Equivalent undamped

structure introduces the dual criterion and the

equiva-lent undamped structure The main results of the

par-ameters 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 1 and 2 describe a traditional DVA and a

non-traditional DVA attached to an undamped

pri-mary structure, respectively, when a sinusoidal

excita-tion force f tð Þ ¼f0sin !t is applied immediately to the

main mass

In the model A with the traditional DVA, the

amp-litude magnification factor of the primary structure is

given by2

GA¼ xs

f0=ks



 

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

22

ð Þ2þð2dÞ2

2ð1 þ2þ2Þ2þ4

½ 2þ½2dð122Þ2

s

ð1Þ

where

 ¼md

ms

, !s¼

ffiffiffiffiffiffi

ks

ms

s , !d¼

ffiffiffiffiffiffi

kd

md

s ,

d¼ cd 2md!d

,  ¼!d

!s

,  ¼ !

!s

ð2Þ

Based on the fixed-points theory, Den Hartog derived the optimal parameters of the traditional DVA as follows5

 ¼ 1

d¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3

8 1 þ ð Þ

s

ð4Þ

In model B with the non-traditional DVA, the amplitude magnification factor of the primary struc-ture is25

GB¼ xs

f0=ks



 

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

22

ð Þ2þð2dÞ2

2ð1þ2þ2Þ2þ4

½ 2þ½2dð12þ2Þ2 s

ð5Þ Based on the fixed-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 non-traditional DVA are

 ¼ ffiffiffiffiffiffiffiffiffiffiffiffi1

1  

d¼1 2

ffiffiffiffiffiffiffiffiffiffiffiffi 3

2  

s

ð7Þ

Figure 1 Model A: a traditional DVA attached to an

undamped primary structure

DVA: dynamic vibration absorber

Figure 2 Model B: a non-traditional DVA attached to an undamped primary structure

DVA: dynamic vibration absorber

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

mag-nification factor of the primary structure than the

traditional DVA does Figure 3 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

non-traditional DVA are given by equations (6) and (7),

for the case of  ¼ 0:2 We can see that the peaks of

GBare significantly lower than those of GA The

ana-lytical 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

criter-ion to study the damped primary structure by

con-sidering an equivalent undamped structure

Equivalent undamped structure using

dual technique

Dual equivalent linearization technique

Although the Caughey method of equivalent

linear-ization is normally used to linearise a non-linear

system, in this article, it is used to generate an

equiva-lent 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 þ2h _x þ !20x þ g x, _ð xÞ ¼f tð Þ ð8Þ

where h and !0 are constants, g x, _ð xÞis a non-linear

function of two arguments x and _x

Equation (8) is linearized to become an equation in the following linear form

€

x þð2h þ bÞx þ !_  20þk

x ¼ f tð Þ ð9Þ 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 e xð Þ ¼g x, _ð xÞ b _x  kx between equation (8) and its linearized equation (9) is minimum

e2ð Þx

 

¼ðg x, _ð xÞ b _x  kxÞ2

!min

b, k ð10Þ

where the operator :h iis the mean value on a period or

a part of the period in the case of deterministic sys-tems 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 equiva-lent 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

g x, _ð xÞ b _x  kx

þðb _x þ kx  g x, _ð xÞÞ2

!min

b, k, 

ð11Þ

Figure 3 Comparison of the amplitude magnification factors GAand GBfor the case of  ¼ 0:2

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

A  B

h i þB  A

!min

,  ð12Þ

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

equiva-lent undamped one Although the undamped

struc-ture 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

€

xsþ2s!sx_sþ!2sxs¼0 ð13Þ

In Figure 4(b) with the equivalent undamped

struc-ture, the motion equation has the form as follows

€

xsþ!2exs¼0 ð14Þ

where !eis an equivalent frequency and is denoted as

!2e¼ þ !2s ð15Þ Using the idea of the dual criterion, we replace 2s!sx_swith xs, where the term  will be determined

by using the following dual criterion

S ¼ð2s!sx_sxsÞ2

Dþðxs2s!sx_sÞ2

D!min

, 

ð16Þ

in which :

h iD¼ 1 D

ZD 0

:

ð Þdt ð17Þ

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

@¼0,

@S

@¼0:

ð18Þ

Substituting the expression of the function S in cri-terion (16) into the set of equation (18) leads to

2 x 2s

D 2s!shxsx_siD 2s!shxsx_siD¼0,

2s!s x_2s

D  xh sx_siD ¼0,

ð19Þ

and then solving the system of equation (19) in terms

of two unknown constants  and  yields

 ¼ 2s!shxsx_siD

x2 s

 

D

" #

: 1

2  hxs x _ s i2D _

x 2 s

h iD x 2 s

h iD

2 6

3 7 5,

 ¼ hxsx_si

2 D

2 x2 s

 

Dx_2 s

 

Dhxsx_si2D:

ð20Þ

Figure 4 The approximation of the primary structure

The first factor of the expression of  in equation

(20) is the result obtained via the conventional

criter-ion This result has been presented by Anh and

Nguyen.19We now use

:

h iD¼h i: ¼ 1



Z 0

:

ð Þd’, with  ¼ !eD ð21Þ Equation (20) can be rewritten in the form

 ¼ 2s!shxsx_si

x2 s

 



" #

: 1

2  hxs x _ s i2 _

x 2 s

h i x 2 s

h i

2 6

3 7 5,

 ¼ hxsx_si

2



2 x2

s

 

x_2

s

 

hxsx_si2:

ð22Þ

Using equation (14), we also have

xs¼acos ’, ’ ¼ !et þ ’0 ð23Þ

Therefore, combining equations (21) and (23) we

obtain

x2s

 

¼ a2

2  þ

1

2sin 2

,

xsx_s

h i¼a2!e

4 ðcos 2  1Þ,

_

x2s

 

¼a2!2e

2  

1

2sin 2

: ð24Þ

Substituting equation (24) into the first equation of

equation (22) and using equation (15), after some

cal-culations, we get

!2eþ 2 1  cos 2ð Þð2  sin 2Þ

822 sin22  1  cos 2ð Þ2s!s!e!2s¼0

ð25Þ Equation (25) is a quadratic equation in terms

of !e Solving this equation, we easily obtain

!e¼!s

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 þ ð1  cos 2Þð2  sin 2Þ

822 sin22  1  cos 2ð Þ2

2

s

0

@

2s  ð1  cos 2Þð2  sin 2Þ

822 sin22  1  cos 2ð Þ2s

ð26Þ

In this article, we get the mean value over a quarter

of the period of the primary system, i.e the value

 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

do not change their signs as well as directions.19 Putting the value  ¼ =2 into equation (26) yields

!e¼ !s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 þ  2

 2 2

ð Þ22 s

q

þ22s

ð27Þ

We have replaced the damped primary structure by

an equivalent undamped structure with the approxi-mate frequency !egiven in equation (27) In the next section, we use this result to give parameters of non-traditional DVA attached to damped linear structures

Parameters of non-traditional DVA attached to damped linear structures by using the equivalent undamped structure

Figure 5 depicts a non-traditional DVA attached to the damped primary structure with the sinusoidal excitation force f tð Þ ¼f0sin !t The amplitude magni-fication factor GB is

GB¼ xs

f0=ks



 

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

22

ð Þ2þð2dÞ2

2 1 þ 2þ2þ4sd

2þ4

þ2d1  2þ2

þ2s  22

v u u t

ð28Þ where s¼cs=2ms!s is the structural damping ratio Using equation (6) for the equivalent undamped structure, i.e e¼1= ffiffiffiffiffiffiffiffiffiffiffiffi

1  

p

and equation (27) for the equivalent frequency and noting that

e¼!d

!e

,

 ¼!d

!s

,

ð29Þ

we obtain

 ¼ 1 ffiffiffiffiffiffiffiffiffiffiffiffi

1  

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 þ  2

 2 2

ð Þ22 s

q

þ22s

When s¼0, the analytical solution (30) will reduce to equation (6) for undamped primary struc-tures The expression of the optimal tuning ratio in equation (30) is independent of the DVA’s damping This optimal tuning ratio together with the appropri-ate DVA’s damping will minimize the maximum dis-placement 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

little even when the damping ratio changes

consider-ably To express this comment, we consider an

exam-ple, say, a undamped primary structure (i.e s¼0)

with the mass ratio  ¼ 0:05 Figure 6 describes the

steady-state response curves in two cases: in the first

case, the tuning ratio  ¼ 1:0260 and damping ratio

d¼0:1387, obtained from equations (6) and (7), and

in the second case, the tuning ratio changes by 1%

Figure 7 depicts the case in which the damping ratio

changes by 5% We can see that in Figure 6 the

max-imum amplitude magnification factor GB increases

significantly but in Figure 7, it changes very little

Therefore, we use the damping ratio of the

non-tradi-tional DVA as equation (7) and we obtain the

param-eters of the non-traditional DVA as follows

 ¼ 1

ffiffiffiffiffiffiffiffiffiffiffiffi

1  

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 þ  2

 2 2

ð Þ22 s

q

þ22s

d¼1

2

ffiffiffiffiffiffiffiffiffiffiffiffi

3

2  

s

:

ð31Þ

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 (equa-tion 31) proposed in this study, these solu(equa-tions will be compared with those of other authors in the literature Figure 8 shows the amplitude magnification factor GB

in the case of a non-traditional DVA with  ¼ 0:1 and

s¼0:08 The dashed line is Liu and Liu’s result22 (equations (6) and (7)), with  ¼1:0541 and

d ¼0:1987 The solid line is the results of this study (equation (40)), with  ¼ 1:0210 and d¼0:1987, and the dot-dashed line is Liu and Coppola’s result,27with

 ¼1:0405 and d¼0:1987 Figure 9 is the case where

 ¼0:1 and s¼0:15 Figures 10 and 11 present com-parisons when  ¼ 0:15, s¼0:08 and s¼0:15, 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 ampli-tude magnification factor GB is the smallest when

Figure 5 The non-traditional DVA-damped primary structure system and the approximate non-traditional DVA-undamped structure system

DVA: dynamic vibration absorber

Figure 6 Graph of the amplitude magnification factor GBversus  with  ¼ 0:05, s¼0 when the tuning ratio change by 1%

Figure 8 Comparison of the amplitude magnification factors GBin the case of a non-traditional DVA with  ¼ 0:1 and s¼0:08 DVA: dynamic vibration absorber

Figure 9 Comparison of the amplitude magnification factors GBin the case of a non-traditional DVA with  ¼ 0:1 and s¼0:15 DVA: dynamic vibration absorber

Figure 7 Graph of the amplitude magnification factor GBversus  with  ¼ 0:05, s¼0 when the damping ratio change by 5%

Figure 10 Comparison of the amplitude magnification factors GBin the case of a non-traditional DVA with  ¼ 0:15 and s¼0:08 DVA: dynamic vibration absorber

Figure 11 Comparison of the amplitude magnification factors GBin the case of a non-traditional DVA with  ¼ 0:15 and s¼0:15 DVA: dynamic vibration absorber

Figure 12 Comparison of the maximum amplitude magnification factor GBwith  ¼ 0:1