DSpace at VNU: Extension of equivalent linearization method to design of TMD for linear damped systems

This paper presents a closed-form expression for the optimum tuning ratio of a TMD attached to a damped primary system.. The comparison has shown that the values of optimal tuning ratio

Published online in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/stc.446

Extension of equivalent linearization method to design of TMD

for linear damped systems

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

2

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

SUMMARY

The vibration absorber has been used in many applications since invented In the case of vibration control

by the tuned mass damper (TMD), the selection of optimum absorber parameters is extremely important This paper presents a closed-form expression for the optimum tuning ratio of a TMD attached to a damped primary system The result is obtained by using equivalent linearization method The values of the optimum tuning ratio derived from the expression proposed in this paper have been compared with those obtained numerically as well as the results obtained from other authors These values are reliable even the mass ratio of TMD to the primary structure and the structural damping ratio are quite high A simulation study has also been carried out to illustrate the obtained results Copyright r 2011 John Wiley & Sons, Ltd

Received 7 June 2010; Revised 14 November 2010; Accepted 8 January 2011

KEY WORDS: TMD; equivalent linearization method; damped structure; fixed-point theory; closed-form

expression

1 INTRODUCTION The problem of undesired vibration reduction has been known many years and it has become more attractive nowadays Historically, an auxiliary mass–spring–damper control system attached to a primary structure was known as vibration absorber, tuned mass damper (TMD),

or dynamic vibration absorber (DVA) The DVA without damper was proposed first in 1909 by Frahm [1], and later in 1956, Den Hartog [2] developed in the case of the absorber with viscous damper when primary structure modeled as a single-degree-of-freedom (SDOF) system After that, the designs of multi-TMDs for continuous structures and multi-degrees-freedom structures

or mass–spring pendulum absorber for inverted pendulum-type structures have gained considerable attention of many researchers [3–5] Thenceforth, TMD has been widely used in many fields of engineering Perhaps the reasons for these applications were its efficient, reliable, and low-cost characteristics

In the design of any control device for the reduction of undesired vibration, the aim would be to provide optimal parameters of the control device to maximize its effect Because the mass ratio of TMD to the primary structure is usually few percent, hence, the principal design parameters of the TMD are its tuning ratio (i.e ratio of the TMD’s frequency to the natural frequency of the primary structure) and its damping ratio The TMD can be used in two distinct ways to suppress the point

*Correspondence to: N D Anh, Institute of Mechanics, Vietnam Academy of Science and Technology, Hanoi, Vietnam.

y

E-mail: ndanh@imech.ac.vn

z

Full Professor.

vibration, collocated or non-collocated to the TMD, at the troublesome resonance frequency or at the troublesome frequency away from the resonance In the former case [1], the simplest method of tuning the neutralizer is by making its resonance frequency coincides with the resonance of the primary structure However, the TMD can become less effective, and may even increase the vibration of the primary structure when there is a change in the frequency of the excited force The increase can be clearly seen on each side of the operating frequency in the frequency response function graph In order to overcome this problem, an ingenious optimization method known as fixed-points theory was suggested by a proper selection of the neutralizer’s stiffness and damping

In case of undamped primary structures, there exist two fixed-points at which the frequency response function is independent of the TMD’s damping The TMD’s stiffness is chosen so that the heights of the two fixed-points in the frequency response function become equal and then the TMD’s damping is determined when allowing these fixed-points to be peaks of the frequency response function This optimization technique is described in detail by Den Hartog [2] for undamped primary system that is subjected to harmonic excitation Since then, the fixed-points theory has become one of the design laws used in fabricating a TMD for the control of vibration of

a relatively simple undamped system In the case of damped primary structure, it is difficult to obtain analytical solutions for the optimum parameters of the TMD Ioi and Ikeda [6], based on the numerical method, have presented the empirical formulae for the optimum parameters of the TMD attached to damped primary structure Randall et al [7] have used numerical optimization procedures for evaluating the optimum TMD’s parameters while considering damping in the structure Thomson [8] has proposed the procedures for a damped structure with TMD, where the tuning ratio has been optimized numerically and then using the optimum value of the tuning ratio, the optimum damping ratio of the TMD has been obtained analytically Warburton [9] 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 the TMD (i.e tuning ratio and damping ratio) for various values of mass ratio and structural damping ratio have been presented

in the form of design tables Fujino and Abe [10] have employed a perturbation technique to derive formulae for optimal TMD parameters, which may be used with good accuracy for mass ratios less than 2% and for very low values of structural damping Thus, in the general case of damping in the primary system, the optimal TMD’s parameters have to be evaluated either numerically or from empirical expressions The reason for this is that when the primary system takes account of damping, a very useful feature of the classical structure–damper system is lost This is the existence

of fixed-point frequencies, i.e frequencies at which the transmissibility of vibration is independent

of the damping in the attached control device

Recently, Ghosh and Basu [11] have presented a closed-form expression for optimal tuning ratio of TMD based on the approximate assumption about the existence of two fixed-points

In this paper, another closed-form expression for optimal tuning ratio of TMD is proposed This result is obtained based on the equivalent linearization method, where the damped primary system is replaced equivalently by an undamped system and then using the known result for undamped primary systems to give the expression of optimal tuning ratio The equivalent linearization has been used widely in many applications since invented [12–16] The result in this paper is compared with the result obtained from Ghosh and Basu’s expression as well as the result obtained numerically from Ioi and Ikeda [6] The comparison has shown that the values of optimal tuning ratio derived from the expression in this paper are closer to the values from the result given by Ioi and Ikeda than those from the expression proposed by Ghosh and Basu Finally, a simulation example is carried out to illustrate the obtained results

2 THE TMD–STRUCTURE SYSTEM AND DEN HARTOG’S CLASSICAL RESULTS IN

THE CASE OF UNDAMPED STRUCTURE The structure is modelled as an SDOF system by considering only the predominant mode in energy dissipation As shown in Figure 1, the SDOF system consists of the mass ms, the spring

ksand the damping coefficient cs The mass of the TMD is mdand its stiffness and damping coefficients are kdand cd, respectively

Let xs denote the vertical displacement of SDOF system relatively to its base from the equilibrium position, and let xddenote the vertical displacement of TMD relative to the primary structure from its equilibrium position The symbol D in Figure 1 indicates the length of spring

kdat equilibrium position of TMD–structure system, f(t) is the external force

Kinetic energy T, potential energy P, and dissipative energy F are expressed as follows

T ¼12m s _x2

 ¼1

F ¼1

2c s _x2

s112c d _x2

The equations of motion of the TMD–structure system are derived by using Lagrange’s equations

d

dt

@T

@_xs

@T

@x s

@x s

@_xs

d

dt

@T

@_xd

@T

@x d

@x d

@_xd

ð5Þ Substituting Equations (1), (2) and (3) into Equations (4) and (5) yields

m s €x s 1c s _x s 1k s x s ¼ m d €x d 1m d €x s 1f ðtÞ ð6Þ

m d €x d 1c d _x d 1k d x d ¼ m d €x s ð7Þ Equations (6) and (7) can be rewritten as follows

m s €x s 1c s _x s 1k s x s ¼ c d _x d 1k d x d 1f ðtÞ ð8Þ

m d €x d 1c d _x d 1k d x d ¼ m d €x s ð9Þ Introduce the parameters

m ¼m d

m s

ffiffiffiffiffi

k s

m s

s

; xs¼ c s

2m sos

ffiffiffiffiffiffi

k d

m d

s

2m dod

; a ¼od

os

where m is the ratio of the mass of the TMD to the mass of the primary structure, os, xsand od,

xdare natural frequencies and damping ratios of the structure and the TMD, respectively, a is the natural frequency ratio or tuning ratio

d m

d

s m

s

s x

d

x + Δ TMD

SDOF system

( )

f t

Figure 1 Damped vibration absorber applied to a force-excited system with damping

Equations (8) and (9) can be rewritten in the dimensionless form as

€x s12osxs _x s1o2s x s¼ 2maxdos _x d1ma2o2s x d1f ðtÞ

In the case of undamped primary structure, i.e xsis equal to zero, Equations (10) and (11) have the simple form

€x s1o2s x s¼ 2maxdos _x d1ma2o2s x d1f ðtÞ

m s

ð12Þ

By using fixed-point method, Den Hartog [2] has given the expression for the optimum tuning ratio aoptand the damping ratio xdoptof TMD in case of an undamped primary system subjected to sinusoidal excitation as follows

aopt¼ 1

xdopt¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3m 8ð11mÞ

s

ð15Þ

The results (14)–(15) obtained for undamped linear systems have required an extension to damped linear systems since the damping does always exist for real structures

3 USING EQUIVALENT LINEARIZATION METHOD TO OBTAIN THE

CLOSED-FORM EXPRESSION FOR OPTIMAL TUNING RATIO The aim in present paper is using equivalent linearization method in order to replace approximately the damped–spring–mass primary structure (Figure 2(a)) by a spring–mass structure (Figure 2(b)) and then using Den Hartog’s results to obtain the closed-form expression for optimal tuning ratio

In the case (a) of Figure 2 with damped structure, the equation of motion has form

And in the case (b) of Figure 2 with undamped structure, the equation of motion is

where oe is unknown constant and will be determined by minimizing the following function

A ¼ hð2xsos _x s1o2

s x s o2

e x sÞ2iT

t with respect to o2

e, where :

h iT

t¼1

T

Z T

0

where T is constant and will be chosen later

This leads to dA=do2

e¼ 0, and so we have hððo2e o2s Þx s 2xsos _x s Þx siT t ¼ 0 ð19Þ

approximately

s

m

s

k

s

c

s

m

e

k

Figure 2 The approximation of the primary structure

ðo2e o2s Þhx2sij  2xsos h_x s x sij ¼ 0 ð20Þ where

h:ij ¼1

 j

Z j  0

with j ¼ oe T For further analysis, one gets from Equation (17)

Using definitions (19), (21) and expression (22), one has

hx2

siT

t ¼ hx2

sij  ¼ a

2

2 j j1

1

2sin2 j

ð23Þ

h_x s x siT t ¼ h_x s x sij ¼oe a

2

Substituting Equations (23) and (24) into Equation (20) we obtain

o2e1 1  cos 2 j

 j11

2sin 2 jxsosoe o

2

Solving Equation (25), we easily obtain solution

oe¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffios

111 4

1  cos 2 j

 j11

2sin 2 j

!2

x2s

v u

11 2

1  cos 2 j

 j11

2 sin 2 jxs

ð26Þ

Using Den Hartog’s result (14) for undamped primary structure as shown in Figure 2(b), we have

aeopt¼ 1 11m Note that

aeopt¼od

oe

aopt¼od

os

and using Equation (26), finally we obtain a closed-form expression for optimal tuning ratio as follows

ð11mÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

111 4

1  cos 2 j

 j11

2sin 2 j

!2

x2s

v u

11 2

1  cos 2 j

 j11

2sin 2 jxs

0 B

1 C

ð27Þ

The problem of choice is the constant j so that the values of the optimal tuning ratio aopt derived from expression (27) are closest to those obtained numerically need considering in next study In this paper, the value j ¼ p=2, i.e we get the mean value over a quarter of period of primary system, is proposed The reason for this choice is that in the first quarter of vibration period, the displacement and the velocity of primary system do not change their signs as well as directions The integration over a quarter of period has been used in some previous studies by some authors [17,18] Putting this value j ¼ p=2 into expression (27) leads to

ð11mÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

p2x2s

r

12

pxs

Expression (27) in general and expression (28) in particular would reduce to Den Hartog’s expression (14) for the optimum tuning ratio of a TMD for an undamped SDOF system The expression for aoptin Equation (28) is independent of the damping in the TMD This optimum tuning ratio together with appropriate damping in the TMD will minimize the maxima of the displacement of the primary structure Thus, it will be possible to optimize the response reduction of the structure subjected to an external force which has a wide banded energy content

or which has dominant energy at the natural period of the structure

4 COMPARISONS

To compare the values of the optimum tuning ratio from the expression proposed in this paper with those available otherwise, the values of the optimum tuning ratio corresponding to typical values of the structure–TMD parameter are presented in a tabular form For the design of the TMD it would be convenient to have the values of the optimum tuning ratio corresponding to typical values of the structure–TMD parameter in tabular form

The values of the optimum tuning ratio from the proposed expression (28) would be compared with the values calculated from the empirical expression given by Ioi and Ikeda [6] and those from the expression proposed by Ghosh and Basu [11] The tolerance or permissible error ranges attached to equations of Ioi and Ikeda are given to be less than 1% for 0.03o mo0.4 and xso0.15 In Ghosh and Basu’s paper, they have given expression

aopt¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1  4x2s  mð2x2

s 1Þ ð11mÞ3

s

ð29Þ

These comparisons have been done in Tables I–IV, where the mass ratio m has been considered as 0.03; 0.05; 0.1; 0.35, respectively, and the different values of the structural

Table I Optimum tuning ratio of TMD for different structural damping ratios and the mass ratio m 5 0.03 Structural

damping ratio xs

Optimum tuning ratio aopt given by Ghosh and Basu [11]

Optimum tuning ratio aopt proposed in this paper

Optimum tuning ratio aopt given by Ioi and Ikeda [6]

Different from Ioi and Ikeda’s results in percentage terms.

Table II Optimum tuning ratio of TMD for different structural damping ratios and the mass ratio m 5 0.05 Structural

damping ratio xs

Optimum tuning ratio aopt given by Ghosh and Basu [11]

Optimum tuning ratio aopt proposed in this paper

Optimum tuning ratio aopt given by Ioi and Ikeda [6]

damping ratio xs as 0.005; 0.01; 0.02; 0.03; 0.05; 0.07; 0.1; 0.12; 0.14; 0.15 have been considered

Tables II–IV show that the values of optimum tuning ratio from the expression (28) proposed

in this paper are closer to the values from the empirical expression given by Ioi and Ikeda than those derived from the expression (29) given by Ghosh and Basu Moreover, the expression for optimum tuning ratio presented in this study is significant even when the mass ratio m and the structural damping ratio xs are quite high As can be seen from above tables, the values of tuning ratio decrease when increasing the structural ratio as well as when the mass ratio increases

5 SIMULATION EXAMPLE

A simulation example has been carried out to illustrate the performance of the TMD design with a tuning ratio that has been evaluated from the expression proposed in this paper The example primary structure SDOF is considered with the structural damping ratio xsis equal

to 0.15, the mass ratio of TMD to the structural m is equal to 0.1, the tuning ratio is equal to that obtained from the proposed expression (28) (i.e aopt50.8264) The damping ratio of the TMD

is assumed to be equal to the optimum value obtained from expression (15):

xdopt¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3m 8ð11mÞ s

i.e xd is equal to 0.1864 Solving Equations (10) and (11) by using the fourth-order Runge–Kutta method, the time history integrations for the displacement responses of the primary structure subjected to random excitation with the TMD from expression proposed in

Table III Optimum tuning ratio of TMD for different structural damping ratios and the mass ratio m 5 0.1 Structural

damping ratio xs

Optimum tuning ratio aopt given by Ghosh and Basu [11]

Optimum tuning ratio aopt proposed in this paper

Optimum tuning ratio aopt given by Ioi and Ikeda [6]

Table IV Optimum tuning ratio of TMD for different structural damping ratios and the mass ratio m 5 0.35 Structural

damping ratio xs

Optimum tuning ratio aopt given by Ghosh and Basu [11]

Optimum tuning ratio aopt proposed in this paper

Optimum tuning ratio aopt given by Ioi and Ikeda [6]

this paper and the TMD from Ghosh and Basu’s expression have been performed The results presented in Figure 3 have examined the accuracy of result presented in this study

6 CONCLUSIONS

By using the equivalent linearization method, a closed-form expression for the optimum tuning ratio of a TMD attached to a damped primary structure, modelled as an SDOF system, has been presented in this paper This is the first time the damped primary structure is replaced equivalently by an undamped system and then using known results in case of an undamped structure to obtain the analytical expression of optimal tuning ratio Furthermore, this result is received by taking mean value over a quarter of vibration period but is not over the whole period as classical method This taking mean value over a quarter of period has been used by some other authors

The obtained expression of tuning ratio is compared with known results in the literature and the comparison shows that the expression of tuning ratio proposed in this paper is significant even when the mass ratio and structural damping are high

The next study in the future may be to replace simultaneously the TMD–damped primary structure system by a TMD–undamped structure system and then determining not only the optimal tuning ratio but also the damping ratio of TMD

ACKNOWLEDGEMENTS

This paper is supported by Vietnam’s National Foundation for Science and Technology Development (NAFOSTED)

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