design of multiple tuned mass dampers by using a numerical optimizer

EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICSEarthquake Engng Struct. Dyn. 2005; 34:125–144Published online 25 October 2004 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/eqe.413

Design of multiple tuned mass dampers byusing a numerical optimizer

Nam Hoang∗;† and Pennung Warnitchai

School of Civil Engineering; Asian Institute of Technology; P.O. Box 4 Klongluang;Pathumthani 12120; Thailand

SUMMARY

A new method to design multiple tuned mass dampers (multiple TMDs) for minimizing excessivevibration of structures has been developed using a numerical optimizer. It is a very powerful methodby which a large number of design variables can be e�ectively handled without imposing any restrictionbefore the analysis. Its framework is highly �exible and can be easily extended to general structureswith di�erent combinations of loading conditions and target controlled quantities. The method has beenused to design multiple TMDs for SDOF structures subjected to wide-band excitation. Some novelresults have been obtained. To reduce displacement response of the structure, the optimally designedmultiple TMDs have distributed natural frequencies and distinct damping ratios at low damping level.The obtained optimal con�guration of TMDs was di�erent from the earlier analytical solutions andwas proved to be the most e�ective. A robustness design of multiple TMDs has also been presented.Robustness is de�ned as the ability of TMDs to function properly despite the presence of uncertaintiesin the parameters of the system. Numerical examples of minimizing acceleration structural response havebeen given where the system parameters are uncertain and are modeled as independent normal variates.It was found that, in case of uncertainties in the structural properties, increasing the TMD dampingratios along with expanding the TMD frequency range make the system more robust. Meanwhile, ifTMD parameters themselves are uncertain, it is necessary to design TMDs for higher damping ratiosand a narrower frequency range. Copyright ? 2004 John Wiley & Sons, Ltd.

KEY WORDS: multiple tuned mass dampers; numerical optimization; optimal design; robustness design

1. INTRODUCTION

The tuned mass damper (TMD) is one of the most widely used control devices in civilstructures. When properly tuned, it can e�ectively suppress excessive vibrations of a structure[1–3]. However, the e�ectiveness of a single TMD may be impaired signi�cantly in realapplications because of its high sensitivity to frequency mistunings or to variation of the

∗Correspondence to: Nam Hoang, School of Civil Engineering, Asian Institute of Technology, P.O. Box 4Klongluang, Pathumthani 12120, Thailand.

†E-mail: [email protected]

Received 18 March 2004Revised 10 July 2004

Copyright ? 2004 John Wiley & Sons, Ltd. Accepted 10 July 2004

126 N. HOANG AND P. WARNITCHAI

damper damping ratio. Proposed by Igusa and Xu in 1990 [4], the use of multiple TMDs withdistributed natural frequencies o�ers a solution to such cases. It has been shown that multipleTMDs can be made more e�ective than a conventional optimal single TMD of the same totalmass and, furthermore, performance of multiple TMDs is less sensitive to uncertainties in theparameters of the system [5–7].In the early stage of designing multiple TMDs, certain design constraints were employed

in order to simplify the analysis with a large number of involved TMD parameters. TheTMDs were restricted to having identical masses and damping ratios, along with naturalfrequencies equally spaced over a frequency range. With these constraints, Yamaguchi andHarnpornchai [6] performed a parametric study on the e�ect of TMDs on a single degree-of-freedom (SDOF) structure under harmonic forces. They found that, for a given numberof TMDs, there exist an optimal frequency range and an optimal TMD damping ratio whichmake the structural response minimum. After that, Abe and Fujino [7] analytically studied theconstrained TMDs. A perturbation technique was used to derive explicitly modal propertiesof the combined system. Accordingly, a TMD damping ratio to maximize the lowest modaldamping and a critical TMD frequency range to make modal amplitudes equal were derivedin simple formulas. These formulas are relevant to the design of those simpli�ed TMDstoward the minimum response. Also shown in both the above studies, the sensitivity of themultiple TMDs against uncertainty in the structural frequency can be diminished signi�cantlyby widening the TMD frequency range.Igusa and Xu [8] analytically designed multiple TMDs for a SDOF structure subjected

to a wide-band force. In an attempt to generalize the optimal design problem, they relaxedthe previous restrictions on natural frequencies and damping ratio of TMDs. An asymptoticanalysis and an integral limit were used to obtain a simple approximation for the mean squaredisplacement response of the main structure. This approximate response is independent of theTMD damping ratios but these dampings must lie within certain bounds for its accuracy. Ananalytical optimization was then conducted for the approximate response using calculus ofvariations. Their optimal result shows that the natural frequencies of the TMDs are unequallydistributed over a frequency range centered at the controlled structural frequency and thewidth of this frequency range is proportional to the square root of the ratio of the total TMDmass and the main mass.The present paper proposes a new method of designing multiple TMDs to minimize unde-

sirable vibration of linear structures. It uses a numerical optimizer that follows the Davidon–Fletcher–Powell algorithm. In this method, parameters of TMDs are treated as unconstrainedoptimization variables. Analytical expressions of the quadratic performance function and thegradient are explicitly evaluated which help to avoid numerical errors and speed up the con-vergence. This is a very powerful method by which a large number of design variables canbe e�ectively handled without imposing any restriction before the analysis. Its framework ishighly �exible and can be easily applied to general structures with di�erent combinations ofloading conditions and target quantities to be controlled. Realistic control problems can thusbe considered.The developed method is used to design multiple TMDs for SDOF lumped-mass structures

subjected to wide-band excitation. For the optimal design problem, various con�gurations ofmultiple TMDs are numerically optimized to reduce the mean square displacement responseof the main structure. The optimally designed con�guration of TMDs is compared to theanalytical solution of Igusa and Xu [8] and is proved to be more e�ective. Toward practical

Copyright ? 2004 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2005; 34:125–144

DESIGN OF MULTIPLE TUNED MASS DAMPERS 127

applications of multiple TMDs, the remainder of the paper addresses the robustness designof TMDs. Robustness is de�ned as the ability of TMDs to function properly despite thepresence of uncertainties in the parameters of the system. Numerical examples of minimiz-ing acceleration structural response using �ve TMDs are given where the system parametersare uncertain and are modeled as independent normal variates. It is found that, in general,increasing damping ratios along with adjusting the frequency range of TMDs can make thesystem less sensitive with uncertainties.

2. FORMULATION OF OPTIMAL DESIGN PROBLEM

2.1. Equations of motion

Consider a combined system of a main linear structure and mass dampers. The main structureis described by m vibration modes with shape function i(x), modal mass msi, modal frequency!si and modal damping ratio �si, where i=1; : : : ; m. A stationary zero-mean random force u(t)is applied to the main structure. The structural displacement response w(x; t) is presented interms of modal coordinates zsi(t) as

w(x; t)=m∑i=1

i(x)zsi(t) (1)

Attached to the structure is a set of n TMDs acting along the structural response direction.The j-th TMD ( j=1; : : : ; n) has mass mj, natural frequency !j, damping ratio �j and is locatedat xj. The combined system has nc=m + n degrees of freedom and its matrix equation ofmotion can be derived, using Lagrange’s equation, as

M�z(t) +Cz(t) +Kz(t)= fu(t) (2)

Here, z(t)= {zs1 zs2 : : : zsm z1 z2 : : : zn}T in which the last n components correspond todisplacements of TMDs; M, K and C are symmetric nc × nc-matrices given by

M=diag{ms1; ms2; : : : ; msm; m1; m2; : : : ; mn} (3)

K=

[KSS KST

KTST KTT

]and C=

[CSS CST

CTST CTT

](4)

where sub-matrices of K and C are

KST =−�TKTT; CST =−�TCTT (5)

KSS =KS +�TKTT�; CSS =CS +�

TCTT� (6)

KS = diag{ks1; ks2; : : : ; ksm}; CS =diag{cs1; cs2; : : : ; csm} (7)

KTT = diag{k1; k2; : : : ; kn}; CTT =diag{c1; c2; : : : ; cn} (8)



in which ksi=msi!2si, csi=2msi!si�si (i=1; : : : ; m), and kj=mj!2j , cj=2mj!j�j ( j=1; : : : ; n); � is an n×m-matrix of mode-shape values evaluated at attachment points of TMDs:

�=

1(x1) · · · m(x1)

.... . .

...

1(xn) · · · m(xn)

(9)

Vector f on the right hand side of Equation (2) depends on the location of the excitationu(t). Note that if the main structure is a SDOF lumped-mass system, m=1 and the matrix� reduces to an n-vector of ones.To formulate the optimal design problem, the equation of motion (2) needs to be converted

into the state-space form. Among several ways to describe a state-space equation, here anapproach without rendering inversion of the mass matrix is taken. If � denotes a matrix oforthonormal eigenvectors obtained from an eigen-analysis of the undamped form of Equation(2), that is

�TM�= I and �TK�=� (10)

where � is a diagonal matrix of eigenvalues, then by applying transformation z(t)=�q(t),Equation (2) can be rewritten in terms of a set of generalized coordinates q(t) as

�q(t) +Dq(t) +�q(t)=�Tfu(t) (11)

The damping matrix D=�TC� is usually not diagonal because the matrix C is non-proportional. A �rst-order state equation is then expressed by

x(t)=Ax(t) + bu(t) (12)

where

x(t)=

{q(t)

q(t)

}; A=

[0 I

−� −D

]and b=

{0

�Tf

}(13)

Considering Equation (12), with a random excitation input u(t), the random response inthe state vector terms can be described by a covariance matrix X(t)=E[x(t)xT(t)], where theoperator E denotes the expected value. In this case since u(t) is a stationary process, the statecovariance matrix is time-invariant and can be obtained by solving a Lyapunov equation. Ifu(t) is a white-noise force with spectral intensity S0, X is the solution of [9]:

AX+XAT + 2�S0bbT = 0 (14)

The above associated Lyapunov equation can be e�ciently solved by the algorithm ofBartels and Stewart [10] using triangular (Schur) matrix decomposition.

2.2. Performance function

To mathematically de�ne an optimal design problem, a performance index J is �rst needed. Fora stationary random excitation, the mean square of structural response is the most fundamental



description. The response which is of interest in practice is often a physical quantity suchas displacement, acceleration, stress, etc., at a selected point of the main structure. If y(t) isdesignated for such a physical response, J can be expressed as

J =E[y2(t)] (15)

Obviously, the above expression can be easily extended to combine structural responsesat various points or to investigate total vibration energy of the structure, or furthermore totake the TMD motions into account. The optimal design problem is then to search for a setof parameters of TMDs that minimizes J, given the structure and the excitation. The TMDparameters to be optimized are viewed as design variables and are collectively representedby a design np-vector p. The optimal search of a large-dimension vector p is feasible onlyby means of a computer-based numerical optimizer provided that at least the performancefunction J(p), preferably along with its gradient, is evaluated in an e�cient manner.To e�ciently evaluate J, the physical structural response y(t) is represented by the state

coordinates x(t) so that J can be expressed in terms of the state covariance matrix X whichis obtainable by solving the Lyapunov equation. For example, if the structural displacementat a point xs is of interest, from Equation (1), y(t) may be written in matrix notation as

y(t)=w(xs; t)= sTz(t) (16)

in which s is a constant nc-vector: s= { 1(xs) : : : m(xs) 0 : : : 0}T. Using the transformationz(t)=�q(t) and the de�nition of the state vector x(t) leads to

y(t)= rTx(t) (17)

where r= {sT� 0}T. In this case the expression of J in Equation (15) becomesJ =E[xT(t)Rx(t)] (18)

from which introducing the trace operation and re-arranging terms give the result

J = tr[RX] (19)

where R= rrT and tr stands for the trace operator. Similarly, the structural velocity at a pointxs is written as

w(xs; t)= {0 sT�}x(t) (20)

and it is clear that the mean square of velocity response is evaluated by the same formula (19),except r= {0 sT�}T. Also by properly reformulating vector r, Equation (19) can representperformance functions of other response quantities such as stresses, strains, vibration energy.In studying human tolerance to mechanical vibrations, measurement of acceleration is

usually preferable to that of velocity or displacement. For such cases, di�erentiatingEquation (20)

y(t)= �w(xs; t)= rTx(t) (21)

where r= {0 sT�}T and substituting Equation (12) yieldsy(t)= rT[Ax(t) + bu(t)] (22)



Formulate M, K, C

from eigenanalysisNumerical

optimization:

Iterativesearch for

pwhich yields

minimum J(p)

Formulate A, b, r

Evaluate J

p

J(p)

J

∆

Solve for Xfrom Lyapunov Eq.

Evaluate gradients

ΦDΛ ,,Obtain

Figure 1. Flow diagram of optimal design of multiple TMDs by numerical optimization.

i.e., y(t) is now a combination of both the state and the excitation input, leading to a longerexpression of J. For a white-noise excitation of intensity S0, it can be shown that

J = tr[R[AXAT + 2�S0(A+ I)bbT]] (23)

in which R= rrT and X is the solution of Equation (14).

2.3. Gradient of performance function

To accelerate the optimal searching, the gradient of the performance function ∇J≡{@J=@p1 : : : @J=@pnp}T is needed because it provides information about the best searchingdirection in np variable space. The evaluation of the gradient requires the computation of thepartial derivatives @J=@pr , r=1; : : : ; np. With the formulation of J described in the previoussection, explicit expressions for @J=@pr can also be derived, from which an accurate evaluationof the gradient is achieved with least computational e�ort. This speeds up the convergenceand makes the proposed approach attractive for practical use.For convenience, a �ow diagram of optimal design of multiple TMDs by numerical

opimization is shown in Figure 1. From the diagram, it is seen that J is determined viaeigenmatrices �, � and D which are all in turn functions of TMD parameters in the de-sign vector p. On that account, @J=@pr can be accomplished using the chain rule of partialderivative as follows.Firstly, the partial derivatives of J with respect to matrix A, vectors b and r are derived.

The derivation in the case of a white-noise input is given in Appendix A. Let aij, bi and rj



denote components of A, b and r, respectively, where i; j=1; : : : ; N =2nc, the results for theperformance function formulated by Equation (19) are

@J=@A≡ [@J=@aij]= 2VX (24)

where V is the solution of

VA+ATV+R= 0 (25)

@J=@b≡ {@J=@b1 : : : @J=@bN}T =4�S0Vb (26)

and

@J=@r≡ {@J=@r1 : : : @J=@rN}T =2Xr (27)

For the performance function formulated by Equation (23), the results are given by

@J=@A=2(VX+RAX+ �S0RbbT) (28)

where V is the solution of

VA+ATV+ATRA= 0 (29)

@J=@b=2�S0[2V+R(A+AT + 2I)]b (30)

and

@J=@r=2[AXAT + �S0(A+AT + 2I)bbT]r (31)

By de�nition, matrix A is composed of matrices � and D. Thus, the partial derivativesof J with respect to � and D are directly obtained from @J=@A. Meanwhile, both vectors b andr depend on matrix � through sub-vectors �Tf and �Ts, respectively. Extracting @J=@(�Tf)and @J=@(�Ts) from the corresponding @J=@b and @J=@r, the partial derivative of J with respectto � is then found by

@J=@�= f[@J=@(�Tf)]T + s[@J=@(�Ts)]T (32)

The next step is to evaluate the partial derivatives of matrices �, � and D with respect todesign variables. Di�erentiation of a matrix is here de�ned by di�erentiating each element. Itis noted that partial derivatives of eigenvalues and eigenvectors were completely derived byFox and Kapoor [11] in 1968 and are presented here for convenience. If �i denotes the i-thdiagonal element of matrix � and Mi is the corresponding eigenvector (i=1; : : : ; nc) then

@�i=@pr =MTi (@K=@pr − �i@M=@pr)Mi and @Mi=@pr =nc∑j=1

�ijMj (33)

where

�ij=

{− 12 M

Ti (@M=@pr)Mi for i= j

MTj (@K=@pr − �i@M=@pr)Mi=(�i − �j) for i �= j(34)



from which, by introducing Q=[�ij], Equation (33) is written in matrix form,

@�=@pr =diag{@�1=@pr; : : : ; @�nc =@pr} and @�=@pr =�QT (35)

As for @D=@pr , di�erentiating its de�nition expression D=�TC� with respect to pr yields

@D=@pr =�[@C=@pr]�T +DQT + QD (36)

Finally, @J=@pr is calculated using the chain rule of partial derivative for matrix functions as

@J@pr

= tr

[@J@�

(@�@pr

)T]+ tr

[@J@D

(@D@pr

)T]+ tr

[@J@�

(@�@pr

)T](37)

Therefore, to evaluate the gradient ∇J, only one additional Lyapunov equation has to besolved, Equations (25) or (29), and the remaining calculations are merely matrix manipula-tions. A highly e�cient computer program accordingly can be made.

2.4. Numerical optimization

Once the calculation program to compute J(p) and ∇J is established, various types of numer-ical optimizer may be incorporated. One of the most robust gradient-based unconstrained op-timization techniques is employed herein. It is the technique following the Davidon–Fletcher–Powell algorithm [12]. To implement this technique, TMD parameters need to be converted tounconstrained design variables by making a change of variables. Some typical transformations[12] using square, sine or cosine functions can be applied for this purpose.

3. OPTIMAL DESIGN OF MULTIPLE TMDS

The optimal design problem of multiple TMDs is very complicated because a system of astructure with attached TMDs has many closely-spaced natural frequencies and non-proportionaldamping. Even when the main structure is modeled as a SDOF structure, many approximationsand assumptions are needed to simplify the problem in the earlier analytical [5, 7] and para-metric [6] studies. Although much insight into mechanism of vibration control using multipleTMDs was provided from those studies, the question of whether the simpli�cations are justi-�ed or whether the obtained multiple TMDs are really optimal still remains. From this viewpoint, the numerical optimization approach, which is developed for general multi-degree-of-freedom structures without imposing any restriction on TMDs before the analysis, is employedto design multiple TMDs for SDOF lumped-mass structures (m=1) in the remainder of thepaper.In this section, an optimal design of n TMDs for a SDOF structure in Figure 2 is numerically

performed. The structural properties, with the subscript i=1 in the notation hereafter droppedfor convenience, are given as ms=1000kg, !s=(2�)=1Hz and �s=0:5%. A wide-band forcemodeled by white noise is applied to the main mass, and the mean square displacementresponse of that mass is to be minimized. A set of 20 TMDs is used where all TMDs, forsimplicity, are assigned to have equal masses. The total mass of the TMDs is 10 kg, that is1% of the main mass. This is the same example investigated by Igusa and Xu [8].



Structure

�s �s

ms

(n) (1) (2)

TMDs

Figure 2. A SDOF structure with n TMDs.

To gain more information on the dynamic characteristics of multiple TMDs, two di�erentoptimization problems are considered: (I) the damping ratios of TMDs �j are identically setto a given value � so that their natural frequencies are design variables, p= {!1; : : : ; !20}Tand (II) both the natural frequencies and damping ratios of TMDs are to be optimized,p= {!1; : : : ; !20; �1; : : : ; �20}T. The optimal search with the performance function given byEquation (19) then converges rapidly to a unique global minimum from random initial valuesof variables. Note that the optimal solution in problem (I), Con�guration I, is a function of �which is varied from 0.1% to 9%, whereas there is only a single optimal solution in problem(II), Con�guration II. These optimal con�gurations of TMDs are depicted in Figure 3 wherethe abscissa is the TMD damping ratio �j and the ordinate is the tuning frequency, i.e. ratioof the damper frequency to the structural frequency !j=!s.In Figure 3, the optimal natural frequencies of TMDs for problem (I) greatly varies for

di�erent values of �. When �¡1:5%, the optimal TMD frequencies are well distributed over awide frequency range around the structural frequency !s. As � increases, some TMD frequen-cies start to group together in a reducing frequency range to produce the minimum response.As � continues to increase, more and more optimal frequencies of TMDs are found in groupstoward two edges of an even narrower frequency range. When TMDs are highly damped(�¿6:1%), optimal tuning frequencies totally coincide, implying a set of identical TMDs ora single TMD of equal total mass should be used.For problem (II), the condition of equal dampings is removed and consequently, the best

performance of TMDs is anticipated. The obtained optimal con�guration of multiple TMDs(Con�guration II) has a nearly uniform frequency distribution and low-and-nearly-equaldamping ratios. This con�guration is very similar to the Con�guration I at �=0:5%. Theresult implies that the assumption of uniform frequency distribution and equal damping ratiosof TMDs, which has been employed in most of the past studies on optimal design of multipleTMDs [5–7], is quite reasonable for practical design purposes. Note that some investigators[13, 14] have tried to examine the assumption by comparing performances of multiple TMDswith several di�erent frequency distributions, and found that ‘uniform frequency distribution’is superior to other tested non-uniform frequency distributions. But never before has this



0 1 2 3 4 5 6 7 8 9 100.9

0.95

1

1.05

1.1ω

j /ω

sConfiguration II

1

2

3

4

5

6

7

8

9

10

11

12

20

Configuration I

No. of TMDs–

ξj ,%

Figure 3. Optimal con�gurations of 20 TMDs (�=1%) for minimizing displacement of the SDOFstructure subject to a white-noise excitation.

assumption been veri�ed by the ‘true’ optimal con�guration of multiple TMDs as presentedhere.As stated before, Igusa and Xu [8] analytically derived an optimal con�guration of multiple

TMDs with non-uniform frequency distribution. It is now of interest to compare their analyticalsolution with the numerical solution of the present study. In the paper of Igusa and Xu, amass density function of TMDs, p(!j), was de�ned as

p(!j)=mj=�j (38)

where �j=(!j+1 − !j)=!j, for j=1; : : : ; n, is a non-dimensional frequency spacing para-meter. The optimal design of the multiple TMDs was then speci�ed by the mass densitywhich minimizes the mean square response of the main structure. A closed-form optimalsolution for p(!) was derived using calculus of variations. It is a half ellipse with majorand minor axis points �xed for a given total mass ratio of TMDs. Using parameters of theexample in this section, the optimal mass density of Igusa and Xu is plotted in Figure 4.In comparison with the optimal mass density of Igusa and Xu, the mass density values

of the numerically optimized solution, Con�guration II, are evaluated and presented inFigure 4. It is seen that the numerical solution forms a curved shape of wider bandwidth,smaller amplitude and biased to the lower frequency. Furthermore, it should be pointed outthat the optimal frequency range in Igusa and Xu’s solution is a function of total mass ratioonly, and is independent from the number of TMDs. This is di�erent from what was reportedin several studies [6, 7, 15] that for a given total mass ratio the optimal frequency range ex-pands with the increase in the number of TMDs. To elaborate this point, di�erent numbersof TMDs, e.g. n=10 and 40, for the same total mass (10 kg) are numerically optimized. In



0.9 0.95 1 1.05 1.1

j / ω ω s

0

40

80

120

p(ω

j),kg

Igusa and Xu

Configuration IIn = 10

n = 20

n = 40

n = 20

Figure 4. Optimal distributions for mass density of multiple TMDs (�=1%).

each optimization case, TMDs are still assumed to have equal masses for simplicity. From theoptimization results, the optimal mass densities are calculated by Equation (38) and shownin Figure 4. The expanding of the optimal frequency range with the increase in the numberof TMDs is then noticed.The di�erence between the numerical and analytical optimal solutions lies in the approxi-

mations used in the latter approach. To illustrate this point, the performance of the obtainedmultiple TMDs is to be re-examined. In this case, the performance index is the mean squaredisplacement response of the main structure, which, instead of using Equation (19), can bedirectly evaluated by the integral [16]:

J = S0∫ ∞

−∞|Hs(!)|2 d! (39)

where S0 is the spectral intensity of the white-noise force and Hs(!) is the frequency responseof the main structure. Hs(!) can be formulated exactly as [8]:

Hs(!)= [ms(!2s − !2 − 2i�s!s!)− i!Z(!)]−1 (40)

where i=√−1 and Z(!) is the total impedance of n TMDs given by

Z(!)=−i!n∑

j=1mj(!2j − 2i�j!j!)=(!2j − !2 − 2i�j!j!) (41)

It is this impedance of TMDs for which Igusa and Xu derived an approximate expressionto make their optimization problem analytically tractable. By means of an asymptotic analysis



Configuration I, Configuration II, Igusa and Xu's

0 1 2 3 4 5 6 7 8 9 10

ξj ,%

0.05

0.1

0.15

0.2

0.25

J/J 0

Exact

Approximate

Figure 5. Normalized mean square of displacement response of the SDOF structurewith optimal con�gurations of 20 TMDs.

and an integral limit, a simple approximation for Z(!) was given as

Z(!)≈ (�=2)!p(!) (42)

Note that Equation (42) excludes the dependence of Z(!) on dampings of TMDs but thereare conditions on TMD damping ratios instead. For the optimal mass density of Igusa andXu in Figure 4, the TMD damping ratios must lie within the bounds 0:94%¡�j¡5:6%.Therefore, to each con�guration of multiple TMDs, the performance index J can be eval-

uated by formulas (39)–(40) in which either the exact expression of TMD impedance inEquation (41) or the simple approximation in Equation (42) is employed. The performancesof the optimal con�gurations of 20 TMDs achieved by numerical optimization and by Igusaand Xu are plotted against TMD damping ratio in Figure 5. In this �gure, J is normalizedby the mean square response of the structure without TMDs J0. The dashed lines indicate theperformance of the analytical solution with the TMD damping ratios out of the mentionedbounds. The approximate J of the analytical solution is a constant line since Equation (42) isindependent of TMD dampings. As for numerical solutions, the approximate J can be calcu-lated and plotted only for TMD con�gurations from which the mass density can be evaluated,after the de�nition in Equation (38). They are con�gurations having TMD damping ratios lessthan 3.9% in Figure 3.From Figure 5, by the exact calculation, the e�ectiveness of multiple TMDs provided low

damping level is con�rmed. The numerically optimized TMDs yield the minimum responsewhile the analytically designed TMDs of Igusa and Xu lead to a higher response, i.e. a lowercontrol e�ect. In contrast, the approximate curves suggest that the analytical solution would



provide the lowest response. Hence, by using the approximation in Equation (42), the controle�ect of the analytical solution has been over-estimated.In summary, this section deals with the optimal design problem of multiple TMDs using

numerical optimization. The method is e�cient and applicable for any large number of TMDswithout recourse to simplifying assumptions. It is found that the optimally designed multipleTMDs have distributed natural frequencies and distinct damping ratios at low damping level.The obtained optimal con�guration of TMDs is di�erent from the earlier analytical solutionand is proved to be more e�ective.

4. ROBUSTNESS DESIGN OF MULTIPLE TMDS

The optimal design of multiple TMDs in the previous section is based on the assumptionthat all the parameters of the system are accurately estimated. In real applications, however,structural properties are possibly in error due to uncertainty in modeling or measurement.Likewise, TMD parameters may shift away from their design values due to uncertainty infabrication. These uncertainties are inevitable to some degree and since the system responseis sensitive to variations in the system parameters [17], the control e�ect of multiple TMDswould be then degraded. It is thus essential to design TMDs that function properly despite thepresence of uncertainties in the parameters of the system, i.e. robust TMDs. The numericaloptimization program can be modi�ed to accommodate this problem.The example structure for this section is a mass attached by �ve TMDs (n=5) as in Figure

2. The main mass is subjected to a white-noise excitation and its mean square accelerationresponse is expressed by the performance index J in Equation (23). The robustness design ofTMDs is here demonstrated in two problems: the �rst concerns uncertainties in the propertiesof the main structure and the second deals with those in the parameters of the TMDs. Amongmany possible sources of uncertainty, the uncertainty in the natural frequency, either of themain structure or of the TMDs, would be a critical factor due to the tuning requirements.In the �rst design problem, the frequency mistuning is solely caused by uncertainty in the

spring sti�ness of the main structure ks. Uncertainties in the other structural properties as wellas in the TMD parameters are assumed to be negligible. The structural sti�ness is modeledas a random variable, and as a result the response of the main structure, measured by theperformance index J, is a function of the variate ks. The problem is then how to searchfor a vector of natural frequencies and damping ratios of TMDs, p= {!1; : : : ; !5; �1; : : : ; �5}T,that suppresses e�ectively the random J. A new performance function, JR, is de�ned for thispurpose as

JR(p)=1ns

ns∑i=1J(p)|ks = k(i)s

(43)

where k(i)s is the i-th sample value of the sti�ness variate ks and ns is the total number ofsamples. The value of ns should be su�ciently high to represent the sample space of thevariate. The index JR expresses an average performance of ns distinct systems whose struc-tural sti�nesses follow a speci�ed probability distribution. Accordingly, the multiple TMDsoptimized by this index a�ect a broad range of erroneous systems.In this example, the mass and damping ratio of the main structure are given as ms=1000kg

and �s=0:5%, respectively. The TMDs are of equal masses with the total weight of 10kg. The



1 2 3 4 5

ξ j ,%

0.80.8

0.90.9

1

1.11.1

1.21.2

j/ ω

ω s

S1S0

S2

S3

Figure 6. Robust con�gurations of TMDs for di�erent levels of uncertainty in structural sti�ness(S0, S1, S2, and S3 are for �s=0, 5%, 10% and 20%, respectively).

random structural sti�ness is assumed to have normal distribution with mean value chosenas 1000 N=m to make the mean value of the structural frequency equal to 1 rad=s. Threedi�erent coe�cients of variation of the sti�ness variable, �s=5, 10 and 20%, which indicatelow, moderate and high variations, are considered. For each value of �s, ns samples of therandom sti�ness are generated. Optimization of the performance function JR in Equation (43)then produces a set of TMDs e�ective against those ns sampled systems. With a su�cientlyhigh ns number, for example ns=50, 200 and 800 for �s=5, 10 and 20%, respectively,optimization for di�erent trials of ns samples almost converge to a unique solution. This isthe robust con�guration of TMDs whose e�ectiveness is statistically insensitive to uncertaintyin the structural sti�ness. The numerical optimization typically took 30 minutes of runningtime on a Pentium III personal computer if ns=50 and as much as 2 hours if ns=800.Figure 6 reveals the robust con�gurations S1, S2 and S3 of multiple TMDs for �S =5, 10

and 20%, respectively. The optimally designed TMDs for the deterministic system (�s=0),con�guration S0, is also plotted for comparison. It can be noticed from the �gure that thehigher the coe�cient of variation of the structural sti�ness, the larger the TMD damping ratiosand the wider the TMD frequency range. The spread of TMD frequency range is probably tocover the mistuned structural frequency due to sti�ness uncertainty, and the increase in TMDdampings is to �atten the structural response curve in that range.To illustrate how robustness is improved, performance of the given structure with uncertain

sti�ness using the robust TMDs is re-examined. For each level of uncertainty, measuredby �s, 1000 samples of the random structural sti�ness are generated. Data of performanceindex J, that is mean square acceleration response of sampled structure-TMDs systems to theconsidered white-noise excitation are then computed, and probabilistic information of J can



0 5 10 15 20

δs ,%

0 5 10 15 20

δs ,%0 5 10 15 20

δs ,%

0 5 10 15 20

δs ,%

0

0.2

0.4

0.6

0.8

J/J 0

0

0.2

0.4

0.6

0.8

J/J 0

J/J 0

0

0.2

0.4

0.6

0.8

J/J 0

0

0.2

0.4

0.6

0.8

Configuration S1

Configuration S2 Configuration S3

90-percentile

10-percentile

50-percentile

Configuration S0

Figure 7. Percentiles of normalized mean square acceleration versus levelof uncertainty in structural sti�ness.

be estimated. The median and bounds of J are represented by 50-percentile, 10-percentile and90-percentile values, respectively. Basically the 90-percentile says that 90% of the time, J isbelow this amount [18]. The performance index J is normalized by that of the correspondingmain structure without TMDs J0 and is plotted in Figure 7 for all con�gurations S0–S3.As in Figure 7, performance of the TMDs designed without considering uncertainty (con-

�guration S0) is stable only if the variation of the structural sti�ness is small (�s¡5%). Forlarger �s, its median grows and its bounds of scatterness expand signi�cantly i.e. not robust atall. In contrast, J=J0 for con�guration S3 is almost insensitive with �s, which is seen from itsstable median between very close upper and lower bounds, over a broad range of �s. Similaris the performance of con�gurations S1 and S2 within their corresponding design ranges of�s. It shows that the degree of robustness of multiple TMDs can be designed by the proposednumerical optimization approach. The result in Figures 6 and 7 clearly indicates that wideningTMD frequency range together with increasing TMD damping ratios would make TMDs morerobust against uncertainty in the structural frequency.The second design problem examines uncertainty in the TMD parameters. All the struc-

tural properties are now assumed to be accurately known, ms=1000 kg, ks=1000 N=m and�s=0:5%. The uncertainty is associated with the sti�nesses and damping ratios of TMDs,while their masses are deterministic, mi=2kg, for i=1; : : : ; 5, as before. Those uncertain TMDparameters are modeled by a vector of independent, normal variates p= {k1; : : : ; k5; �1; : : : ; �5}T



1 2 3 4 5

ξ j ,%

0.8

0.9

1

1.1

1.2

j/

ω

ω s

T1S0

T2

T3

Figure 8. Robust con�gurations of TMDs for di�erent levels of uncertainty in their parameters (S0, T1,T2, and T3 are for �T =0, 3%, 5% and 10%, respectively).

and the index J is accordingly a random function. In this case, the performance function JRis modi�ed from Equation (43) as

JR =1ns

ns∑i=1J∣∣∣∣p=pi

(44)

where pi is the i-th sample of the random vector p; and the target is to �nd the mean valuesof p that minimize JR. By doing so, the obtained solution will be a set of TMDs that remainse�ective even if their parameters are varied from the designed values. The same coe�cientof variation �T is here applied to all TMD variables for simplicity.The robust con�gurations of TMDs T1, T2 and T3 for �T =3, 5 and 10%, respectively,

are displayed by comparison with con�guration S0 (�T =0) in Figure 8. From the �gure, thelarger variation in the TMD parameters demands also the higher TMD damping ratios but, incontrast to the �rst problem, the narrower TMD frequency range. When �T =10%, the robustsolution is a set of nearly identical TMDs. It is noticed that these robust solutions are in termsof mean values of the TMD sti�nesses and damping ratios which are possibly dispersed byuncertainties in real cases.Percentile plots of the normalized mean square acceleration response (J=J0) against �T are

given in Figure 9. From the �gure, the performance of con�guration S0 also deteriorates andscatters with increasing �T , but less than with �S in the �rst problem. In other words, thestructural response is less sensitive to the uncertainty in the TMD parameters than that in thestructural properties. The �gure also exhibits stable performance of the obtained robust TMDcon�gurations T1–T3 as expected. The result suggests that, in the case of highly uncertainfabrication, TMDs should be designed for identical properties.



0 5 10 15 20

δT ,%0 5 10 15 20

δT ,%

0 5 10 15 20

δT ,%0 5 10 15 20

δT ,%

0

0.2

0.4

0.6

0.8J/

J 0

0

0.2

0.4

0.6

0.8

J/J 0

J/J 0

0

0.2

0.4

0.6

0.8

J/J 0

0

0.2

0.4

0.6

0.8

Configuration T1

Configuration T2 Configuration T3

90-percentile

10-percentile

50-percentile

Configuration S0

Figure 9. Percentiles of normalized mean square acceleration versuslevel of uncertainty in TMD parameters.

5. CONCLUSIONS

This paper has presented a new method to design multiple TMDs for minimizing excessivevibration of linear structures. The method uses a numerical optimizer following a gradient-based non-linear programming algorithm to search for optimal parameters of TMDs. The targetresponse is formulated as a quadratic performance index which can be e�ciently computed bysolving a Lyapunov equation. By that formulation, analytical expressions for the gradient ofthe performance index are also explicitly evaluated which help to avoid numerical errors andspeed up the convergence. This is a very powerful method by which a large number of designvariables can be e�ectively handled without imposing any restriction before the analysis. Itsframework is highly �exible and can be easily extended to general structures with di�erentcombinations of loading conditions and target controlled quantities.The developed method is applied to optimize TMDs for a SDOF structure subjected to

wide-band excitation. The result shows that the most e�ective con�guration of TMDs hasdistributed natural frequencies and distinct damping ratios at low damping level. This opti-mal con�guration, for practical design purposes, can be supposed to have uniform frequencydistribution and equal damping ratios.



In addition, it is demonstrated that the method can be modi�ed to accommodate other real-istic problems where the system parameters are possibly in error due to random uncertainties.Numerical results indicate that, in the presence of uncertainties in the structural properties,increasing the TMD damping ratios along with expanding the TMD frequency range make thesystem more robust. Meanwhile, if TMD parameters themselves are uncertain, it is necessaryto design TMDs for higher damping ratios and narrower frequency range.

APPENDIX A

In this Appendix, partial derivatives of J with respect to matrix A, and vectors b and rare derived. To facilitate the derivation, the covariance matrix X that formulates J needs tobe explicitly expressed in terms of A and b. From the well-known general solution of thedi�erential state equation (12):

x(t)= eAtx(0) +∫ t

0eA(t−�)bu(�) d� (A1)

Assuming that the system starts from rest, i.e. x(0)= 0, post-multiplying Equation (A1) withxT(t) and taking the expected value lead to

X(t)=E[x(t)xT(t)]=∫ t

0

∫ t

0eA(t−�)bE[u(�)u()]bTeA

T(t−) d� d (A2)

As u(t) is a stationary zero-mean white-noise excitation of intensity S0, E[u(�)u()]=2�S0�(� − ), in which �(t) is the Dirac delta function; the covariance matrix in Equation (A2)then becomes

X=2�S0∫ ∞

0eAtbbTeA

Tt dt (A3)

For the performance function formulated by Equation (19), substituting Equation (A3) yields

J = tr[2�S0R

∫ ∞

0eAtbbTeA

Tt dt]

(A4)

Partial derivative of J with respect to matrix A

The following derivation is a modi�cation of that initially given by Levine and Athan [19]and employed later in the paper of Xu et al. [20]. For an in�nitesimal change in A, theperformance function is

J(A+ �A)= tr[2�S0R

∫ ∞

0e(A+�A)tbbTe(A+�A)Tt dt

](A5)

where is small relative to unity. The Bellman’s �rst-order expansion of the exponentialmatrix function is

e(A+�A)t =eAt + ∫ t

0eA(t−s)�AeAs ds (A6)



The increment of the performance function �J is obtained by combining Equations (A5) and(A6) and subtracting Equation (A4):

�J= J(A+ �A)− J (A)=2 tr[2�S0R

∫ ∞

0

∫ t

0eA(t−s)�AeAsbbTeA

Tt ds dt]

(A7)

Using the commutative property of matrix multiplication within the trace operator yields

�J= tr[4�S0

∫ ∞

0

∫ t

0eAsbbTeA

TtReA(t−s)�A ds dt]

(A8)

Changing the order of integration and using the transformation of variables �= t − s gives

�J= tr[4�S0

∫ ∞

0

∫ ∞

0eAsbbTeA

TseAT�ReA� d� ds�A

](A9)

from which, according to Kleinman’s lemma, the derivative of J with respect to A is

@J@A

=4�S0∫ ∞

0

∫ ∞

0eA

T�ReA�eAsbbTeATs d� ds (A10)

This result is further simpli�ed by noting that, similarly to X given in Equation (A3), thematrix

V=∫ ∞

0eA

TtReAt dt (A11)

can be evaluated without numerical integration by solving the associated Lyapunov equation.The �nal formulas for calculating @J=@A are given in Equations (24)–(25) of the main text.

Partial derivative of J with respect to vector b

The vector @J=@b is to be derived from the partial derivative of J with respect to matrixB= bbT. The performance index J in terms of an in�nitesimal change in B is

J(B+ �B)= tr[2�S0R

∫ ∞

0eAt(B+ �B)eA

Tt dt]

(A12)

The increment of the performance function, �J, is then computed by subtracting Equation(A4) from Equation (A12) and next employing the commutative property inside the traceoperator:

�J= tr[2�S0

∫ ∞

0eA

TtReAt dt�B]

(A13)

Again Kleinman’s lemma is applied to obtain @J=@B as

@J@B=2�S0V (A14)



Finally, @J=@b is obtained by the chain rule of partial derivative

@J@b=

[@J@B+

(@J@B

)T]b (A15)

and the result is given in Equation (26).

Partial derivative of J with respect to vector r

By de�nition, R= rrT and the derivation for @J=@r becomes quite similar to that for vector b.The result is in Equation (27).For the performance function given by Equation (23), applying similar mathematical ma-

nipulations to the X-related term and taking common derivative of trace for the other termsleads to the formulas (28)–(31).

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design of multiple tuned mass dampers by using a numerical optimizer

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