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Lei Zuoe-mail: [email protected]

Phone: 847-935-0086Abbott Laboratories,

Bldg. AP52S,200 Abbott Park Road,

Abbott Park, IL 60064-6212

Samir A. Nayfehe-mail: [email protected]

Phone: 617-253-2407Department of Mechanical Engineering,Massachusetts Institute of Technology,

Room 3-461A,77 Massachusetts Avenue,

Cambridge, MA 02139

The Two-Degree-of-FreedomTuned-Mass Damper forSuppression of Single-ModeVibration Under Random andHarmonic ExcitationWhenever a tuned-mass damper is attached to a primary system, motion of the absorberbody in more than one degree of freedom (DOF) relative to the primary system can beused to attenuate vibration of the primary system. In this paper, we propose that morethan one mode of vibration of an absorber body relative to a primary system be tuned tosuppress single-mode vibration of a primary system. We cast the problem of optimizationof the multi-degree-of-freedom connection between the absorber body and primary struc-ture as a decentralized control problem and develop optimization algorithms based on theH2 and H-infinity norms to minimize the response to random and harmonic excitations,respectively. We find that a two-DOF absorber can attain better performance than theoptimal SDOF absorber, even for the case where the rotary inertia of the absorber tendsto zero. With properly chosen connection locations, the two-DOF absorber achievesbetter vibration suppression than two separate absorbers of optimized mass distribution.A two-DOF absorber with a negative damper in one of its two connections to the primarysystem yields significantly better performance than absorbers with only positivedampers. �DOI: 10.1115/1.2128639�

1 IntroductionThe design of a single-degree-of-freedom �SDOF� tuned-mass

damper �TMD�, or dynamic vibration absorber �DVA�, to attenu-ate vibration of a single mode of a primary system under variousconditions has been studied extensively �e.g., �1–5��. To enhancethe effectiveness and robustness of TMD systems, multiple SDOFTMDs with frequencies tuned in the neighborhood of a mode of aprimary system have been proposed by Xu and Igusa �6� andoptimized by Zuo and Nayfeh �7�. Multiple SDOF TMDs havebeen used to damp more than one mode of a primary system bytuning each TMD to an individual mode of interest in the primarysystem �8,9�.

Whenever an absorber is attached to a primary system, there ispotential for utilization of motion in more than one degree offreedom of the absorber body relative to the primary system.Dahlbe �10� numerically optimized a continuous two-segmentcantilever beam for suppression of SDOF vibration and found thetwo-segment beam to be more effective than a SDOF TMD of thesame mass. Recently, Zuo and Nayfeh �11–13� and Verdirame andNayfeh �14� have optimized the stiffness and damping in multi-degree-of-freedom �MDOF� connections between a rigid bodyand primary structure to damp as many as six modes of a struc-ture. In these studies, one mode of vibration of the body relative tothe structure is tuned to each mode of interest in the primarysystem.

In this paper, we propose that more than one mode of vibrationof a body relative to a primary structure be tuned to one naturalfrequency of a primary system. Such an absorber is often easier toconstruct than an SDOF TMD or multiple SDOF TMDs becauseof the reduced need for guidance, and can achieve enhanced per-formance. We cast the design of MDOF TMD systems as decen-

Contributed by the Technical Committee on Vibration and Sound of ASME forpublication in the JOURNAL OF VIBRATIONS AND ACOUSTICS. Manuscript received June 3,
2003; final manuscript received April 1, 2005. Assoc. Editor: D. Dane Quinn.
56 / Vol. 128, FEBRUARY 2006 Copyright ©

aded 25 Jun 2009 to 129.49.32.179. Redistribution subject to ASM

tralized optimal control problems with static output feedback,where the feedback gain is a block diagonal matrix composed ofthe stiffness and damping parameters to be optimized.

Based on this formulation, we minimize the system response torandom excitation by adapting a gradient-based H2 optimizationtechnique based on Lyapunov equations. The optimal parametersare obtained and presented in dimensionless form to be useful fordesign. With properly chosen locations for the springs and damp-ers, the 2DOF TMD achieves better vibration suppression thantwo separate TMDs with optimized mass distribution. Next, tominimize the steady harmonic response under sinusoidal distur-bances, we propose an algorithm based on decentralized H� opti-mization, and find that the 2DOF TMD again offers performancebetter than the conventional SDOF TMD or two separate TMDs.

2 Problem FormulationFigure 1 shows the configuration of a MDOF TMD attached to

a SDOF primary system. �We take the 2DOF TMD as an example;the general multi-DOF TMD can be handled similarly.� The pri-mary system has a natural frequency �s=�ks /ms and dampingratio �s=cs /2�ksms and is subject to a base excitation x0, an ex-ternal disturbance force f , or both. The absorber body has twoplanar degrees of freedom, translation xd and rotation �d. Its massis md and the rotational inertia about its center of mass is Id=md�2, where � is the radius of gyration. The absorber is con-nected to the primary system at distances d1 and d2 from its centerof mass via springs and dashpots. Our goal is to design the pa-rameters k1, c1, k2, and c2 as well as the locations of the connec-tions d1 and d2 in order to minimize the response of the primarysystem. We consider first the case where d1=d2=d, then in Sec.4.5 discuss the case where d1�d2. The effect of absorber mass mdand rotary inertia Id will also be explored in Sec. 4.

By taking the springs and dashpots as elements which feed backlocally the relative displacements and velocities, Zuo and Nayfeh
�15� have cast the optimization of a wide class of mechanical
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systems as zeroth-order decentralized control problems. In thisway, the role of the spring and dashpot pairs are replaced by thecontrol-force vector �u1 ,u2��, where the prime denotes the matrixtranspose, as shown in Fig. 2. The control forces in this case aregiven by

u1 = k1�x1 − xs� + c1�x1 − xs� �1�

u2 = k2�x2 − xs� + c2�x2 − xs� �2�

where x1 and x2 are the displacements of the absorber in the di-rection of xs at the connection locations.

The equations governing the vibration of the coupled systemwith small absorber rotation take the form

mdxd = − u1 − u2 �3�

Id�d = u1d − u2d �4�

msxs + csxs + ksxs = csx0 + ksx0 + f + u1 + u2 �5�

Noting that xd= �x1+x2� /2, �d= �x2−x1� /2d, and Id=md�2, wewrite the above governing equations in matrix form as

Fig. 1 2DOF TMD for one mode of a primary system

Fig. 2 Control formulation of a passive 2DOF TMD

Journal of Vibration and Acoustics


� md/2 md/2 0

− md/2 md/2 0

0 0 ms��x1

x2

xs + �0 0 0

0 0 0

0 0 cs��x1

x2

xs + �0 0 0

0 0 0

0 0 ks��x1

x2

xs

= � 0

0

cs�x0 + � 0

0

ks�x0 + �0

0

1� f + � − 1 − 1

d2/�2 − d2/�2

1 1�u1

u2� �6�

or

Mpp + Cpp + Kpp = Bpx0 + Bvx0 + Bdf + Buu �7�

where p= �x1 ,x2 ,xs��. From Eq. �6�, we see that the performanceof the TMD system does not depend on the rotational inertia Id.Rather, it depends on the ratio of the radius of gyration � to thedistance d from the mount points to the center of mass of theabsorber.

Defining the state variables of the system as

x = � p

p − Mp−1Bvx0

�8�

we can write the governing equations in first-order form as

x = Ax + B1w + B2u �9�

where w= �f ,x0�� and

A = � 0 I

− Mp−1Kp − Mp

−1Cp �10�

B1 = � 0 Mp−1Bv

Mp−1Bd Mp

−1�Bp − CpMp−1Bv�

�11�

B2 = � 0

Mp−1Bu

�12�

The cost output can be taken as the absolute or relative dis-placement, velocity, or acceleration of the primary system, whichcan be expressed in the form

z = C1x + D11w + D12u �13�

For example, if the cost output is the displacement response of theprimary system, we write the cost as z=xs=C1x, where C1= �0,0 ,1 ,0 ,0 ,0�.

To complete the state-space description, we rewrite the “controlforce” given by Eqs. �1� and �2� as a static feedback gain F mul-tiplied by the “measurement output” y. That is,

u = �k1 c1 0 0

0 0 k2 c2 y =

def

Fy �14�

where y is given by

y = �x1 − xs, x1 − xs,x2 − xs, x2 − xs�� = C2x + D21w + D22u

�15�

The form of C2 follows from the definition of the state given byEq. �8� and the matrices D22=0 and

D21 = C2� 0

Mp−1Bv

�16�

Equations �9�, �13�, and �15� cast the design of the MDOFTMD system as a decentralized control problem, as shown by theblock diagram of Fig. 3.

Based on this formulation, we use decentralized control tech-niques to directly optimize the stiffness and damping coefficientsof the springs and dampers to achieve performance �measured byz� under the disturbance w. Decentralized H2 optimization mini-mizes the output variance under random excitation, and decentral-
ized H� optimization minimizes the worst-case response magni-
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tude under harmonic excitation �15�. In the following section, wewill briefly review our method for decentralized H2 optimization,and then present the results of optimization of a 2DOF TMDunder random excitation in Sec. 4. In Sec. 5, an algorithm fordecentralized H� optimization is proposed along with some re-sults for the 2DOF TMD under harmonic excitation.

3 Review of Decentralized H2 OptimizationThe system H2 norm is defined as the energy of the system

impulse response:

�H�22 =�

0

�

trace�hzw� �t�hzw�t��dt

=1

2��

−�

�

trace�Hzw� �j��Hzw�j��d� �17�

where hzw is the system impulse-response matrix and Hzw is thesystem transfer matrix. Another interpretation of system H2 normis the asymptotic value of output variance under unit white noiseinput. That is, if E�w�t��=0 and E�w��t�w�t+��= I��, then

zz2 = lim

T→�E� 1

T�0

T

z�t��z�t�dt = �H�22 �18�

where zz is the RMS value of z. Therefore, minimization of theH2 norm minimizes the RMS response of the output z�t� underwide-band random excitation w�t�.

Based on the definition given by Eq. �17�, it can be shown thatthe H2 norm of an LTI system can be evaluated by solving aLyapunov equation �16�, as summarized in the following: The H2norm of the LTI system from w→z given by

x = Acx + Bcw

z = Ccx + Dcw �19�

is infinite if Ac is unstable or Dc is nonzero. Otherwise

�H�22 = trace�Bc�KBc� �20�

where K is a symmetric matrix �known as the observability Gram-mian� which satisfies the Lyapunov equation

Ac�K + KAc + Cc�Cc = 0

Under the decentralized feedback u=Fy, the closed-loop sys-tem �w→z� shown in Fig. 3 is given by

� Ac Bc

Cc Dc = � A + B2FC2 B1 + B2FD21

C1 + D12FC2 D12FD21 �21�

To obtain a finite H2 norm, D11+D12FD21 must be zero. Then the

Fig. 3 Block diagram of the primary system and MDOF TMDviewed as a system with decentralized control

decentralized H2 optimal control problem becomes

58 / Vol. 128, FEBRUARY 2006


minF

�H�22 = trace��B1 + B2FD21��K�B1 + B2FD21�� 22�

subject to

K�A + B2FC2� + �A + B2FC2��K + �C1 + D12FC2��C1 + D12FC2�

= 0 �23�

where F is in the given block-diagonal form. In practice, we re-quire that the parameters be nonnegative. So we replace F byFd�Fd, where the symbol “�” denotes multiplication entry byentry.

Employing the Lagrange multiplier method �17� and matrix cal-culus, we obtain the gradient of the square of the H2 norm withrespect to Fd �where Fd�Fd is the controller gain� in the form �11�

��H�22

�Fd= 4��D12� D12�Fd�Fd�C2 + D12� C1 + B2�K�LC2�� 24�

+ B2�K��B1 + B2�Fd�Fd�D21�D21� ��Fd �25�

where the observability Grammian K and Lagrange multiplier ma-trix L are obtained by solving the two decoupled Lyapunov matrixequations with a given matrix Fd:

K�A + B2�Fd�Fd�C2� + �A + B2�Fd�Fd�C2��K + �C1

+ D12�Fd�Fd�C2��C1 + D12�Fd�Fd�C2� = 0 �26�

L�A + B2�Fd�Fd�C2�� + �A + B2�Fd�Fd�C2�L + �B1

+ B2�Fd�Fd�D21��B1 + B2�Fd�Fd�D21�� = 0 �27�We can therefore adapt a gradient-based optimization method,

such as the FBGS quasi-Newton method �18�, to solve for thematrix Fd efficiently and then obtain the stiffness and dampingparameters from controller gain Fd�Fd. A good initial guess forFd is

Fd = ��k/2 �c/2 0 0

0 0 �k/2 �c/2 �28�

where k and c are the optimal stiffness and damping of an SDOFTMD �of the same mass ratio md /ms� obtained from the analyticaltuning formulas developed by Den Hartog �1� or Asani et al. �5�.

4 The H2 Optimal 2DOF TMDUsing gradient-based decentralized H2 optimization, we obtain

the optimal parameters of k1, k2, c1, and c2 as a function of theratio � /d. �We consider initially the case where d1=d2=d, andlater the case where d1�d2.� As for the SDOF absorber �5�, theoptimal H2 norm is proportional to the square-root of the naturalfrequency �s of the primary system, and we therefore normalize

the H2 norm by ��s in the following. In this section, we give acomprehensive study of the 2DOF TMD, taking the excitation asthe base motion x0�w=x0� and the performance index as the RMSvalue of the displacement xs.

4.1 Effect of � /d. As pointed out following Eq. �6�, the per-formance of the TMD system depends on the ratio of the absorb-er’s radius of gyration � to the distance d from the mount points tothe center of mass of the absorber. The solid line in Fig. 4 showsthe normalized optimal H2 norm as a function of � /d for =md /ms=5% and �s=0.

On this plot, � /d=� �or d=0� corresponds to the optimal

SDOF TMD, which attains �H�2=2.108��s. If � /d=1, we obtaina system equivalent to two separate SDOF TMDs with m1=m2

=md /2, as shown in Fig. 5. And � /d=1/�3 is the case of auniform-density bar supported at its two ends. But the optimal � /dfor =5% and �s=0 is 0.780, which yields �H�2=2.0189��s. The
trend is similar if the primary system is lightly damped, as indi-
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cated by the dashed line in Fig. 4, where the optimal � /d for =5% and �s=1% is 0.777. The frequency responses are comparedin Fig. 6.

Figure 4 also shows that the optimal H2 norm at � /d=0 issmaller than that of a SDOF TMD �� /d=��, indicating that a2DOF TMD without rotary inertia can achieve better vibrationsuppression than the SDOF TMD. This can be explained by com-parison of the impedance of the 2DOF TMD with no rotary inertiaas sketched in Fig. 7�a� to that of the SDOF TMD sketched in Fig.7�b�. The latter is given by a second-order transfer function,whereas the former is given by a third-order transfer function,which �when optimized� accounts for the improvement inperformance.

4.2 Comparison to Two TMDs With Optimal MassDistribution. As mentioned previously, the performance of theoptimal 2DOF TMD exceeds that of two separate TMDs withequal masses and optimal springs and dampers. The authors �7�have also obtained the optimal H2 norm of two SDOF TMDs withunequal mass distribution as plotted in Fig. 8 for =5% and �s=0.

In this curve, m1 / �m1+m2�=0.5 corresponds to the case ofequal masses, and m1 / �m1+m2�=1 corresponds to the case of asingle SDOF TMD. The optimal mass distribution is m1 / �m1

Fig. 4 The normalized system H2 norm as a function of � /dwith �=5% for �s=0 „solid… and for �s=1% „dashed…

Fig. 5 Two separate SDOF TMDs for one mode of a primary
system


+m2�=53.35% with the corresponding �H�2=2.0426��s, which isonly 0.02% less than that of two SDOF TMDs with equal masses,and is larger than the H2 norm of the 2DOF TMD with optimal� /d. Thus the performance of the 2DOF TMD generally exceedsthat of two separate SDOF TMDs, even if the mass is distributedoptimally among them.

4.3 Negative Damping. Figure 4 shows two sharp corners inthe dependence of the optimal H2 norm on � /d. This is because atthese values of � /d the constraint ci�0 become active. That is,

Fig. 6 Frequency responses of the H2 optimal TMD system for�=5% and �s=0: � /d=0.780 „solid…, two separate TMDs „dash…,uniform bar supported at two ends „dash-dot…, and SDOF TMD„dot…

Fig. 7 „a… Two-DOF TMD with � /d=0 and „b… SDOF TMD

Fig. 8 Normalized system H2 norm versus mass distribution
among two SDOF TMDs for �=5% and �s=0
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the constraint is active for � /d greater than about 1.25 or less than0.751. An interesting set of designs are found beyond these twovalues if we do not constrain the parameters to be positive.

Take as an example the case of � /d=0.2, =5%, and �s=0with the stiffness ks and mass ms of the primary system normal-ized to one. We optimize the parameters of stiffness and dampingusing the algorithm of Sec. 3 �which guarantees that ki�0 andci�0� and list as the first entry in Table 1 the optimal parameters,the poles of the coupled absorber and primary system, and thepoles of the absorber mounted to ground. With the parametersconstrained to be nonnegative, we see that for this value of � /d,one of the modes of the absorber is tuned close to �s, but the otheris over damped. The Bode plot of xs�s� /x0�s� is shown as a dashedline in Fig. 9.

If we modify the algorithm to allow negative stiffness anddamping, we obtain the results shown as the second entry of Table1 and indicated by the solid line in Fig. 9. From Table 1, we seethat although there is a negative damper in the system, the coupledabsorber and primary system are stable. Moreover, the perfor-mance of this system is significantly better than that of the systemwith the dampers constrained to be non-negative. Such a designcould readily be implemented in an active absorber, though therewould be potential for instabilities in the presence of modelinguncertainties.

It may be suggested that a different configuration of springs anddampers could attain the same performance without resort tonegative dampers. For this particular example, it can be shownalgebraically that it is not possible to find positive values of k1, k2,c1, and c2 that yield the same dynamics, even if their locations areallowed to vary independently. However, it may be possible toobtain improved performance in passive systems by considering asomewhat more general configuration �e.g., including torsionalsprings and dampers�.

Table 1 Comparison of designs obtained withremain nonnegative for � /d=0.2, �=5%, cs=0

Fig. 9 Bode plots of xs„s… /x0„s… for � /„dot…, optimized with nonnegative con
negative constraint „solid…
60 / Vol. 128, FEBRUARY 2006


4.4 Optimal Parameters. The optimal normalized modal fre-quencies �undamped� �1 /�s and �2 /�s, and modal damping ra-tios �1 and �2 of the TMD subsystem �attached to ground� for =5% and �s=0 are shown in Fig. 10 as functions of � /d. As wehave observed in the foregoing, for small � /d, we can only tuneone of the natural frequencies of the TMD close to �s and theother mode is over damped.

To provide results more readily used for design, in Fig. 11 wegive the optimal parameters in the form of dimensionless “fre-

quencies” �1 /�s and �2 /�s and “damping factors” �1 and �2

�1 = �k1/md, �2�k2/md

�1 = c1/�2�mdk1�, �2 = c1/�2�mdk1�

from which the optimal stiffness and damping can be constructed.These are convenient dimensionless parameters, but they do notcorrespond to the resonant frequencies or damping ratios of themodes of the TMD subsystem.

Next, we minimize the H2 norm of xs /x0 as the mass ratio isvaried. Figure 12 shows the optimal � /d for different values of .The achieved minimal H2 norms of 2DOF TMD system withoptimal � /d are shown in Fig. 13 and compared with those of theSDOF TMD and two separate TMDs. The corresponding optimaltuning of stiffness and damping for the optimal � /d are given inFig. 14 in terms of the dimensionless parameters �1 /�s, �2 /�s,

�1, and �2. Figures 13 and 14 can be used to design the optimal2DOF TMD for a given mass ratio. Comparing the curves in Fig.13, we see that the optimal 2DOF TMD whose mass is 5% of thatof the primary system provides roughly the same performance asthe optimal SDOF absorber whose mass is 6% of that of theprimary system.

nd without the constraint that the parameterss=1, and ks=1

0.2, �=5%, and �s=0: original systemaint „dashed…, optimized without non-

a, m

d=str



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4.5 Asymmetric Connection Locations. Thus far, we havemaintained the symmetry of the locations of the connections be-tween the TMD and primary system by holding d1=d2=d. In thissubsection, we relax this constraint and examine the H2 optimal2DOF TMD allowing d1 and d2 to vary independently. The designmethod is the same as before: For a given d1 and d2, we usedecentralized control techniques to optimize the stiffness anddamping values.

The achievable performance is a function of � /d1 and � /d2.Figure 15 shows the minimal normalized H2 norm for =5% and�s=0 attained by optimizing k1, k2, c1, and c2 for a range of � /d1and � /d2. From this figure, we see that the optimal � /d1=� /d2=0.780 for symmetric connections is a saddle point and that wecan attain better performance by allowing d1 and d2 vary indepen-dently. The contour plot indicates that the global minimum for amass ratio of 5% is attained when � /d1 tends to zero and � /d2is approximately 3. To avoid a singular computation, we take� /d1=0.01 and find the corresponding best � /d2 to be 3.055. Thecorresponding optimal parameters and closed-loop poles areshown in Table 2 and compared with those of the optimal sym-metric case. The corresponding frequency responses are comparedin Fig. 16.

5 Decentralized H� OptimizationWe now turn to the problem of minimizing the worst-case re-

sponse to sinusoidal inputs of the primary system with a 2DOFTMD. For single-input-single-output LTI systems, the H� norm is

Fig. 10 Optimal modal frequencies ansystem as a function for � /d for �=5%

Fig. 11 Optimal parameters of the 2DOF



the peak of the magnitude of the frequency-response. For multi-input-multi-output LTI systems, it is the supremum of the largestsingular value over all frequencies:

�H��2 = sup

��Rmax

2 �Hzw�j�� = sup��R

�max�Hzw� �j��Hzw�j�� 29�

where max�·� and �max�·� respectively, denote the largest singularvalue and eigenvalue of their arguments. We see that the H� normis the steady magnitude response under worst-case sinusoidal ex-citation. Note that Den Hartog’s design formulas for an SDOFTMD mounted to a SDOF primary system yield a close approxi-mation to the H� optimal design.

For an SDOF TMD attached to multi-DOF primary system, theefficient algorithms for optimization of centralized static outputfeedback proposed by El Ghaoui et al. �19� or by Geromel et al.�20� can be used. However, as we have seen in Sec. 2, the opti-mization of an MDOF TMD is equivalent to decentralized optimalcontrol with static output feedback.

Decentralized H� optimization has been investigated by manyresearchers in the controls community, and various techniquessuch as alternative linear matrix inequalities �LMIs� �21�, homo-topy �22,15�, and LMI iteration �23� have been proposed. Butnone of these algorithms can be guaranteed to converge to a localoptimum or a stationary point. Our experience is that these meth-ods generate sequences that decrease quickly when the controllergain is far from the optimum, but become very inefficient whenthe sequences come close the optimum. In the engineering appli-cation of structural optimization, a commonly used frequency-

amping ratios of the 2DOF TMD sub-nd �s=0

d da

TMD versus � /d for �=5% and �s=0

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nection spacing d versus mass ratio � for �s=0

for �s=0

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domain optimization method is based on the evaluation of a trans-fer function at discrete frequencies �e.g., �24,10��. To capture themaximal magnitude response, the frequencies have to be closelyspaced and hence the method is also computationally inefficient.

We therefore develop a method with better efficiency for mini-mization of the peak of the frequency response. It is based onthese two points: �1� The peak of the frequency response, or theH� norm of a LTI system, can be computed very efficiently using -iteration. �2� Finite differences can be used to approximate thegradient, and the computational efficiency should be much betterthan direct search if the objective function is easy to evaluate. The -iteration algorithm is based on this fact: For the stable LTIsystem of Eq. �19� �assuming that Dc=0�, �H�� if and only ifthe Hamiltonian matrix

� Ac1

BcBc�

−1

Cc�Cc − Ac�

�has no eigenvalue on the imaginary axis. So starting with an upperand a lower bound we can use a bisection algorithm to calculatethe system H� norm �i.e., the value of that makes the aboveHamiltonian matrix have an imaginary eigenvalue�. A similarHamiltonian matrix exists for the case Dc�0; details can befound in the text �16�. Standard routines for -iteration, such asthe Matlab function normhinf in the Robust Control Toolbox, areavailable.

The algorithm for minimization of the worst-case response toharmonic excitation is summarized as follows:

TMD as a function of the mass ratio �

Fig. 15 Contour map of normalized minimal H2 norm for vari-ous values of � /d1 and � /d2 for �=5% and �s=0

Fig. 13 Optimal H2 norm of xs /x0 versus the mass ratio for�s=0: 2DOF TMD with optimal � /d „solid…, two separate TMDs„dashed…, and SDOF TMD „dotted…

Fig. 12 Optimal ratio � /d of the radius of gyration to the con-

Fig. 14 Parameters of the optimal 2DOF



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Step 1: Choose an initial value of Fd composed of �k1, �k2,�c1, and �c2. �To ensure that we obtain nonnegative param-eters, we take controller gain as Fd�Fd.�Step 2: Evaluate the H� norm J� using the Matlab functionnormhinf �or any other -iteration routine �16�� for a givenmatrix Fd and Fd+�Fd. Then approximate the gradient�J� /�Fd using the finite difference �J� /�Fd. If the decrease of�J� is small enough, stop; otherwise go to Step 3.Step 3: Based on the gradient �J� /�Fd, calculate a searchdirection DF �e.g., the FBGS quasi-Newton direction �18��.Choose a proper step size � using the Armijo rule, or any otherrule �18�. Update Fd with Fd+�DF. Go to Step 2.

Note that the system H� norm is a nonsmooth function of thematrix Fd and therefore, at some points in the design space,�J� /�Fd may approximate a mixture of subgradients. Such amixture might not yield a true descent direction, in which case, wecan either provide a slight perturbation to the current iterate, orchoose a descent direction directly from the finite difference.

Using this algorithm, we minimize the H� norm of the 2DOFTMD system to obtain the optimal stiffness and damping param-eters for harmonic excitation. Figure 17 shows the minimal H�

norms achieved using a 2DOF absorber of mass ratio =5% at-tached to a primary system with initial damping �s=0 and �s=1% as a function of � /d. The trend is similar to that obtained forH2 optimal design �as shown in Fig. 4�. The optimal values of � /dfor =5% are, respectively, 0.751 and 0.747 with �s=0 and �s=1%, which are close to the optimal values of 0.780 and 0.777 forthe H2 optimal designs with =5%. As in the case of H2 optimaldesign, the frequency peak of the optimal 2DOF TMD withoutrotary inertia �� /d=0� is smaller than that of the optimal SDOF

Table 2 Comparison of symmetric and asymms=1, and ks=1

Fig. 16 Frequency responses of the H2 optimal 2DOF TMDwith symmetric and asymmetric connection locations for �=5% and �s=0: � /d1=� /d2=0.780 „solid…, � /d1=0.010 and � /d2
=3.055 „dashed…, SDOF TMD „dotted…


TMD �corresponding to � /d=��. The frequency responses of theoptimal 2DOF TMD, SDOF TMD, and two SDOF TMDs arecompared in Fig. 18. It is seen that the 2DOF TMD attains muchbetter performance than either the SDOF TMD or two SDOFTMDs.

In Fig. 17 the two sharp corners of the curve are again the resultof the constraint that the parameters of stiffness and dampingremain non-negative. Without taking the feedback gain F to beFd�Fd, the forgoing algorithm can also be used to minimize the

etric connection locations for �=5%, cs=0,

Fig. 17 The effect of � /d for H� optimal design with �=5% for�s=0 „solid… and for �s=1% „dashed…

Fig. 18 Frequency responses of the H� optimal 2DOF TMDdesign for �=5% and �s=0: optimal � /d=0.751 „solid…, twoseparate TMDs „dashed…, uniform bar supported at two ends

m

„dashed-dotted…, and SDOF TMD „dotted…

FEBRUARY 2006, Vol. 128 / 63


.38

Downlo

H� norm while allowing the parameters to be negative. �Specialattention must be paid to the step size to avoid a destabilizingdesign.� The frequency responses obtained with and without theconstraint that the parameters remain nonnegative are compared inFig. 19. As in H2 optimization, decentralized H� optimizationyields significant improvement with one damper allowed to benegative, and the total system is still stable.

To better understand the physics of the 2DOF TMD, it is usefulto examine the apparent mass of the TMD, i.e., the transfer func-

Fig. 19 Bode plots of xs„s… /x0„s… for �optimization: original system „dotted…„dashed line, peak magnitude of 6.07straint „solid line, peak magnitude of 3

Fig. 20 Normalized apparent mass F / „s2mdxtained without parameters constraints: „a… � /lines denote the total apparent mass and the
absorber mode
64 / Vol. 128, FEBRUARY 2006


tion from the acceleration s2xs of the primary system to the forceF exerted upon it by the TMD. Figure 20 gives the normalized

apparent mass F / �s2mdxs� for three designs of the 2DOF TMD.The total apparent mass �denoted by the solid lines� is the sum ofcontributions from the two modes of the TMD �denoted by dashedlines�. The total normalized apparent mass at zero frequency mustbe unity and is equal to the sum of the residues of the two modes.

From Fig. 20�a�, we see that the two modes of the optimum

=0.2, �=5%, and �s=0 obtained by H�

timized with non-negative constraintoptimized without nonnegative con-4…

of various H� optimal 2DOF absorbers ob-1, „b… � /d=0.751, and „c… � /d=0.2. The solidashed lines denote the contribution of each

/d, op1…,

s…

d=d



Downlo

design with � /d=1 have equal residues and are tuned to frequen-cies just below and just above the resonance frequency of theprimary system. This is in contrast to the absorber obtained with� /d=0.751, shown in Fig. 20�b�, in which the residue of the firstmode is slightly larger than unity and that of the second mode isnegative. In this case, most of the vibration suppression arisesfrom the first absorber mode. The difference in the contribution ofthe absorber modes is even more pronounced when � /d=0.2 anda parameter is allowed to be negative, as shown in Fig. 20�c�.Here the residue of the first mode is approximately three and thatof the second mode is approximately negative two. The first modeacts to suppress vibration and is tuned close to the resonance ofthe primary system, whereas the second mode magnifies vibrationand is tuned to a significantly higher frequency.

6 ConclusionsIn this paper, we propose the use of a MDOF TMD for one

mode of primary system and show that, for a given mass, anoptimal 2DOF TMD performs better than a traditional SDOFTMD or two separate TMDs with optimal mass distribution. Wecast the parameter optimization of MDOF TMD systems as adecentralized control problem, where the block-diagonal control-ler gain is directly composed of the stiffness and damping param-eters of the connections between the absorber and primary system.Based on this formulation, we adapt decentralized H2 and H�

optimization techniques to optimize the system response underrandom and harmonic excitation, respectively.

First, we employ gradient-based decentralized H2 optimizationto minimize the RMS response under random excitation and pro-vide a comprehensive study of the performance of a 2DOF TMDattached to a SDOF primary system. Design charts for passiveTMD implementation �in which all of the springs and dampers arerequired to be positive� are provided. We then discuss the casewhere the dampers are allowed to be negative, and find that theperformance is considerably improved. This suggests that an ef-fective reaction-mass actuator can be constructed with a 2DOFreaction mass.

We propose an algorithm for decentralized H� optimization tominimize the peak of the frequency response under harmonic ex-citation. The maximal response is obtained efficiently using -iteration and finite differences are used to approximate its gra-dient with respect to the design parameters. We then optimize the2DOF absorber and find that its frequency-domain performanceis again better than that of the SDOF absorber or two SDOFabsorbers.

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