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ExtendingDenHartog’sVibrationAbsorberTechniquetoMulti-Degree-of-FreedomSystems

ARTICLEinJOURNALOFVIBRATIONANDACOUSTICS·AUGUST2005

ImpactFactor:0.71·DOI:10.1115/1.1924642

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ThomasJRoyston

UniversityofIllinoisatChicago

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Mehmet Bulent Ozer

Thomas J. Royston1

e-mail: troyston@uic.edu

Dept. of Mechanical-Engineering,University of Illinois at Chicago,

2039 Engineering Research Facility,Chicago, IL 60607

Extending Den Hartog’s VibrationAbsorber Technique toMulti-Degree-of-FreedomSystemsThe most common method to design tuned dynamic vibration absorbers is still that ofDen Hartog, based on the principle of invariant points. However, this method is optimalonly when attaching the absorber to a single-degree-of-freedom undamped main system.In the present paper, an extension of the classical Den Hartog approach to a multi-degree-of-freedom undamped main system is presented. The Sherman-Morrison matrixinversion theorem is used to obtain an expression that leads to invariant points for amulti-degree-of-freedom undamped main system. Using this expression, an analyticalsolution for the optimal damper value of the absorber is derived. Also, the effect oflocation of the absorber in the multi-degree-of-freedom system and the effect of theabsorber on neighboring modes are discussed. �DOI: 10.1115/1.1924642�

Keywords: Dynamic Vibration Absorber, Tuned Mass Damper, Sherman-MorrisonFormula, Den Hartog Method, Invariant Points, Multi Degree of Freedom Systems

1 IntroductionThe tuned mass vibration absorber is a commonly applied tech-

nique in the field of mechanical vibrations. The concept was firstdiscussed by Frahm �1�. His undamped mass-spring absorber wasable to set the vibration amplitude of the main system to zero at asingle frequency. An optimal broadband attenuation was thenachieved by a damped vibration-absorber design proposed by Or-mondroyd and Den Hartog �2�. The details of this approach aregiven in Den Hartog’s book �3�. The existence of invariant pointswas first introduced in the work of Ormondroyd and Den Hartog�2�, and is key to the existence of an analytical optimal solution.According to their work, with a single-degree-of-freedom �SDOF�undamped main system, there are two invariant points �frequen-cies� where the response is independent of the attached absorber’sdamping value. Following a clever approach they derived simpleexpressions for the optimal parameters.

Invariant points exist only when the main system is undamped,i.e., for a realistic system �damped� invariant points do not exist.However, for light damping levels the optimal parameters givenby Den Hartog’s method are “nearly optimal.” Much research hasbeen carried out to find the optimal absorber parameters when theSDOF main system has significant damping. Pennestri �4� applieda Chebysev’s min-max criterion for finding the optimal damperparameters. Thompson �5,6� approached the problem from a con-trols perspective and used a frequency locus approach to find theoptimal damper parameters that would minimize the main systemresponse as well as the motion of the absorber. In the work ofAsami et al. �7� it has been shown that an analytical series solu-tion exists for the optimal absorber parameters when the absorberis attached to a SDOF damped system. There are also numerousnumerical attempts to find the optimal absorber parameters whenthe main system is damped. Randall et al. introduced a numericalsearch method �8�; Soom and Lee �9� used a nonlinear programingmethod. Jordanov and Cheshankov �10,11� employed a numericalmethod, which optimizes one of several performance variables. In

1Corresponding author.Contributed by the Technical Committee on Vibration and Sound for publication

in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received January 16, 2004.

Final manuscript received October 26, 2004. Associate Editor: D. Dane Quinn.

Journal of Vibration and Acoustics Copyright © 20

their work these variables are relative and absolute displacementsof the masses, mass ratio, and broadness of the suppression band.

The problem of a tuned mass damper attached to a multi-degree-of-freedom �MDOF� main system has also been studiedextensively. A significant portion of these studies is in the struc-tural engineering field, with an interest in minimizing the responseof buildings due to seismic or wind loading. Different approacheshave been taken to find the optimal absorber parameters and theirlocations. Genetic algorithms have been used �12,13�. In the workof Rana and Soong �14� the effect of detuning is discussed and aneigenvector normalization technique is introduced to expand theresults of a numerical absorber optimization method for the SDOFsystem to the MDOF system. Sadek et al. �15� used a similareigenvector normalization method to find the optimal absorberparameters when the absorber is attached to a MDOF system.Their criterion was to find absorber parameters that result in equaldamping in the first two modes. Lin et al. �16� identified the modalproperties of a building through a random decrement method andIbrahim time-domain method. Using these modal values optimalabsorber parameters for a five-story-building case study were cal-culated. Xu and Kwok �17� used a semi-analytical method to findthe optimal absorber parameters that would minimize the windexcitation of a tall building.

In some studies, a controls theory approach has been taken tofind the optimal absorber parameters. In the studies of Haddad andRazavi �18� and Stech �19� a performance index is minimizedusing the linear quadratic regulator method to identify absorberparameters. The structure of the performance index determines ifthe absorber is H2 optimal or H� optimal. In the H2 optimal caseit is desired to minimize the area under the frequency response�energy minimization�. In the H� optimal case the maximum am-plitude of response is minimized �Den Hartog approach�. Eskinatand Ozturk �20� also drew attention to the similarities between anabsorber design problem and a controller design problem.

Vakakis and Paipetis �21� calculated the optimal absorber pa-rameters when the absorber system is attached to an undampedMDOF system using a polynomial series method. Kitis et al. �22�employed a numerical optimization method for finding the opti-mal absorber parameters of two damped vibration absorbers at-

tached to a MDOF system. In the recent work of Ozer and Roys-

AUGUST 2005, Vol. 127 / 34105 by ASME

ton �23�, the Sherman-Morrison matrix inversion theorem is usedto minimize the response of a particular mass or the weightedlinear summation of the responses of multiple masses when theabsorber is attached to a damped MDOF main system.

In the work of Lewis �24�, it has been shown that the invariantpoints exist when the absorber is attached to a 2-DOF undampedmain system. However, the method was not generalized. Plunkett�25� showed that invariant points exist when a damper �not atuned mass absorber� is attached from ground to a continuoussystem.

The objective of the present study is to directly extend Ormon-droyd and Den Hartog’s method to MDOF systems. To the best ofthe authors’ knowledge, this extension to MDOF systems has notbeen accomplished previously, aside from the work of Lewis,which is valid only for a 2-DOF main system. In order to obtainthe necessary expressions that would lead to determination of in-variant points of MDOF systems of any size, the Sherman-Morrison matrix inversion formula will be used. The Sherman-Morrison matrix inversion formula was first introduced bySherman and Morrison �26�. Hager �27� provides an extensivesurvey of science and engineering applications of this formula.This formula has been used by Ozer and Royston �28� to find theoptimal parameters of the electrical circuit, which is shuntedacross a piezoelectric patch to act as a dynamic vibration absorberon a mechanical structure. Also, the Sherman-Morrison methodhas been used in other vibrations applications, such as the work ofHong and Kim �29�, to show the effect of the absorber material onthe response of an acoustic-structural coupled system. Yang �30�employed the method to evaluate the exact receptances of nonpro-portionally damped systems.

2 Obtaining the Equations of MotionThe main system under consideration is an undamped multi-

degree-of-freedom �MDOF� vibrational system, which consists ofmasses and springs. A single-degree-of-freedom �SDOF� absorbersystem, consisting of a mass, spring, and a damper, is to be at-tached to the undamped MDOF main system �see Fig. 1�. Theobjective is to find the overall system response �after the attach-ment of the SDOF absorber� in the form of two terms. The firstterm gives the response of the system before the SDOF system isattached and the second term represents the effect of the SDOFabsorber on the main system.

It will be assumed that the force on the main system is appliedto the ith degree of freedom, the SDOF absorber is attached to themth degree of freedom, and the response expression for the kthdegree of freedom is desired. The equations of motion of theundamped main system are given as follows:

�M�x + �K�x = F�t� �1�

where �M�, �K�, and �F� are mass and stiffness matrices and theforcing vector, respectively.

When the damped SDOF absorber is attached to the main sys-tem, the equation of the total system can be written as follows:

¨ ˙ ˙

Fig. 1 MDOF main syste

�M�x + �Ca�x + ��K� + �Ka��x + Caddxa + Kaddxa = F�t� �2�

342 / Vol. 127, AUGUST 2005

maxa + caxa + kaxa + CaddT x + Kadd

T x = 0 �3�

where ma, ka, and ca are the absorber mass, stiffness, and dampingvalues, respectively. The term xa represents the displacement ofthe absorber mass. Superscript T represents the transpose of amatrix. Here, the �Ka� and �Ca� expressions contain matrices withall of their elements being zero except the �m�m�th element. The�m�m�th elements are ka and ca in the expressions for �Ka� and�Ca�, respectively. Similarly, Kadd and Cadd are two vectorswhose �1�m�th elements are −ka and −ca, respectively. All otherelements of Kadd and Cadd are zero

�K� = �0�1,1� … 0�1,m� … 0�n,1�

… … … … …… … ka�m,m� … …… … 0 … …

0�n,1� … 0�n,m� … 0�n,n�

� �4�

�C� = �0�1,1� … 0�1,m� … 0�n,1�

… … … … …… … ca�m,m� … …… … … … …

0�n,1� … 0�n,m� … 0�n,n�

� �5�

KaddT = �0�1,1� … − ka�1,m� … 0�1,n� � �6�

CaddT = �0�1,1� ¯ − ca�1,m� ¯ 0�1,n�� �7�

Let us assume that the applied force is sinusoidal. The trans-formed equations of motion ��2� and �3��, which represent thesteady-state behavior of the system, are given as follows:

�− �2�M� + �K� + i��Ca� + �Ka��x + �i�Cadd + Kadd�xa = F

�8�

�− �2ma + i�ca + ka�xa + �i�CaddT + Kadd

T �x = 0 �9�

where � is the excitation frequency.The expression for displacement of absorber mass xa can be

solved for by using Eq. �9�. Inserting the expression for xa intoEq. �8� results in the following equation:

�− �2�M� + �K� + i��Ca� + �Ka� − �i�Cadd + Kadd�

��i�Cadd

T + KaddT �

�− �2ma + i�ca + ka��x = F �10�

Solving for response vector x results in the following expression:

x = ��Zmain� + �Za� − Za1Za2

T �−1F �11�

where

2

and the absorber system

m

�Zmain� = − � �M� + �K� �12�

Transactions of the ASME

�Za� = �Ka� + i��Ca� �13�

Za1= �i�Cadd + Kadd� �14�

Za2

T =�i�Cadd

T + KaddT �

ka − �2ma + i�ca�15�

In order to evaluate the inversion in the response expression, theSherman-Morrison matrix inversion formula will be used. TheSherman-Morrison matrix inversion formula is given as follows:

��A� + uvT�−1 = �A�−1 −�A�−1uvT�A�−1

1 + vT�A�−1u�16�

In Eq. �16�, �A� is a n�n full-rank matrix and u and v are n�1 vectors. So if one applies this method in the response expres-sion given in Eq. �11�, the following expression can be obtained:

system is attached and xmainkrepresents the coordinate response of

Journal of Vibration and Acoustics

��� = Zmain� + �Za−1 +Zmain� + �Za−1Za1

Za2

T Zmain� + �Za−1

1 − Za2

T Zmain� + �Za−1Za1

�17�where

��� = ��Zmain� + �Za� − Za1Za2

T �−1 �18�

The first term of the left-hand side in Eq. �17� is the inversion ofthe two summed matrices, and it is in the Sherman-Morrison formif �Za� can be expressed as the multiplication of two vectors

�Za� = �0

…�ka + i�ca

…0

��0 … �ka + i�ca … 0 � �19�

Using the Sherman-Morrison expression one can obtain the

following:

��Zmain� + �Za��−1 = ��main� −

��main��0

…�ka + i�ca

…0

��0 … �ka + i�ca … 0 ���main�

1 + �0 … �ka + i�ca … 0 ���main��0

…�ka + i�ca

…0

��20�

Simplifying the above equation results in the following expres-sion:

��Zmain� + �Za��−1 = ��main� − �main�m

���main�m �T ka + i�ca

1 + ��main�mm�ka + i�ca��21�

where �main�m represents the mth column of the main system

receptance matrix and ��main�mm is the m�m element of the mainsystem receptance matrix before the inclusion of the added sys-tem.

If Eq. �21� is substituted into Eq. �17�, the receptance expres-sion can be found as the summation of two terms. The first termrepresents the response of the main systems before the absorbersystem is attached, and the second term represents the effect of theabsorber system on the main system response. If the overall sys-tem receptance expression is multiplied by the forcing term, theexpression for the kth degree of freedom can be obtained as fol-lows:

xk = xmaink−

xmainm��main�km

1

− �2ma+

1

ka + i�ca+ ��main�mm

�22�

where xk represents the kth coordinate response after the absorber

the kth coordinate before the absorber system is attached.Equation �22� gives the response of the main system after the

introduction of the attached system. The response is composed ofthe first term, which is the response of the main system before theSDOF absorber is added, and the second term, which is the effectof the added system on the main system.

3 Den Hartog’s Method for Finding the OptimalDamped Vibration-Absorber Parameters

Den Hartog’s vibration-absorber parameter optimizationmethod was introduced in 1928. The optimization of vibration-absorber parameters can be a complicated problem if the optimi-zation is to be done through combined partial differentiation withrespect to stiffness, damping, and frequency. However, Den Har-tog and Ormondroyd employed a very different approach to thisproblem. First, the existence of invariant points in the frequencyresponse curve was shown �see Fig. 2�. Their argument states thatthere are two invariant points in the frequency response curve forthe main system mass, where its response amplitude is indepen-dent of the damping of the absorber system. The stiffness valuethat results in equal amplitudes at the invariant points is taken tobe optimal.

In order to find the appropriate damping value one needs toevaluate the derivative of the amplitude of the response at aninvariant point, set it equal to zero, and solve for the correspond-

ing damping value. This procedure is carried out for both of the

AUGUST 2005, Vol. 127 / 343

invariant points, and the mean of the two damping values is takenas the optimal damping value. This derivation is full of elegantsimplifications, which result in simple expressions for optimalstiffness and damping values. Unfortunately, when this method isextended to MDOF main systems some of the simplifications areno longer applicable, resulting in a more complex form of optimalparameter expressions.

4 Calculation of Optimal Stiffness Value

4.1 Calculation of Optimal Stiffness Value in the GeneralCase. The derivation of optimal stiffness value starts with thecalculation of the invariant points in the frequency response curve.Den Hartog’s derivation shows that invariant points exist in thefrequency response curve for an undamped SDOF main system.The first task here is to show that invariant points also exist whenthe main system is an undamped MDOF system.

In order to show the existence of invariant points, one can usethe response expression that is derived in Sec. 2. Using Eq. �22�,it can be shown that the amplitude of the kth coordinate responseis as follows:

�xk� = �xmaink��

1

− �2ma+

1

ka + i�ca+ ��main�mm −

xmainm

xmaink

��main�km

1

− �2ma+

1

ka + i�ca+ ��main�mm

��23�

Equation �23� can be written as follows after renaming some ofthe terms:

�xk� = �xmaink�

���� 1

− �2ma+

ka

ka2 + �2ca

2 + �2�2

+ � �ca

ka2 + �2ca

2�2

� 1

− �2ma+

ka

ka2 + �2ca

2 + �1�2

+ � �ca

ka2 + �2ca

2�2��24�

where

Fig. 2 The two invariant point locations

�1 = ��main�mm �25�

344 / Vol. 127, AUGUST 2005

�2 = ��main�mm −xmainm

xmaink

��main�km �26�

After several steps of simplifications the response amplitude ex-pression for the kth-degree-of-freedom system can be found asfollows:

�xk� = �xmaink����ka − �2ma − �2maka�2�2 + ca

2���1 − �2ma�2��2

�ka − �2ma − �2maka�1�2 + ca2���1 − �2ma�1��2�

�27�

In Eq. �27�, the �xmaink� term is not a function of the damping value

ca. The expression inside the square root is in the following form:

�xk� = �xmaink���A2 + ca

2B2

C2 + ca2D2� �28�

As explained in the original work of Den Hartog, the value of theterm in the square root will be independent of ca if the followingrelation holds:

A2

C2 =B2

D2 �29�

The terms A, B, C, and D can be determined by observing Eqs.�27� and �28�. If one expands Eq. �29� and then simplifies theresulting expression, the following equation containing the poly-nomial in terms of � can be obtained:

�ma2��1 + 2ka�1�2 + �2���4 − �2ma�1 + ka��1 + �2����2 + 2ka = 0

�30�If Eq. �30� is solved, the following expressions for the two

invariant points can be obtained:

�12 =

1 + ka��1 + �2� + �1 + ka2��1 − �2�2

ma��1 + 2ka�1�2 + �2��31�

�22 =

1 + ka��1 + �2� − �1 + ka2��1 − �2�2

ma��1 + 2ka�1�2 + �2��32�

In the above expressions �1 and �2 are the two invariant pointlocations in the frequency response curve. However, it is impor-tant to note that they are not explicit expressions in terms of �.There are �1 and �2 terms on the right-hand sides of the equa-tions, which themselves are functions of �. Therefore, a root-searching numerical method should be applied to find the invari-ant points.

Also, it should be noted that in Eqs. �31� and �32� the unknownsare �1, �2, and ka. Therefore Eqs. �31� and �32� are not enough todetermine the three unknowns. It was mentioned in Sec. 3 that theka value that results in the equal amplitude at the invariant pointsis needed. The third equation will be obtained using the condition,as shown in Eq. �33�

�xk��1�� = �xk��2�� �33�

One can evaluate the above expression using Eq. �27�. It is knownthat the response is independent of the damping value ca at theinvariant points. So as Den Hartog suggests one can take the cavalue to be infinite and Eq. �33� can be written as follows:

�xmaink��1��� 1 − �1

2ma�2��1�1 − �1

2ma�1��1�� − �xmaink

��2��� 1 − �22ma�2��2�

1 − �22ma�1��2�

�= 0 �34�

Using Eqs. �31�, �32�, and �34� one can numerically solve for theunknown values of ka, �1, and �2. In the following two sectionstwo special conditions will be considered where Eqs. �31�, �32�,

and �34� take simpler forms.

Transactions of the ASME

4.2 Calculation of the Optimal Stiffness Value When theAbsorber is Attached to the Target Mass. There are simplifica-tions to the optimal stiffness and invariant point expressions whenthe absorber mass is attached to the target mass. Recall that thetarget mass �mass whose response is to be minimized� is coordi-nate k and the mass that the absorber is attached to is coordinatem. So in this case subscript k is equal to subscript m. Setting k andm equal Eq. �26� becomes

�2 = 0 �35�

Setting �2 equal to zero in Eq. �31�, �32�, and �34� results in

�12 =

1 + ka�1 + �1 + ka2�1

2

ma�1�36�

�22 =

1 + ka�1 − �1 + ka2�1

2

ma�1�37�

�xmaink��1��� 1

1 − �12ma�1��1�

� − �xmaink��2��� 1

1 − �22ma�1��2�

� = 0

�38�

It can be observed from Eq. �36�–�38� that the expressions for thecalculation of optimal stiffness and the invariant points are sim-pler than the general case given in Eq. �31�, �32�, and �34�.

4.3 Calculation of the Optimal Stiffness Value When theMain System is a Single Degree of Freedom System. In thissection the expressions that gives the invariant points and optimalstiffness will be derived using the equations given in the generalcase section. It will be shown that the general case simplifies toDen Hartog’s optimal vibration-absorber expression for a SDOFmain system.

In the first section the expression that gives the location of theinvariant points is provided in Eq. �30�. Since the main system isa SDOF system, the following relations hold:

�1 =1

K − �2M�39�

�2 = 0 �40�

where K and M are main system �SDOF� stiffness and mass val-ues. Substituting Eqs. �39� and �40� into Eq. �30� results in thefollowing:

�ma2 1

K − �2M��4 − �2ma�1 + ka

1

K − �2M���2 + 2ka = 0

�41�Rearranging the above equation into the form of a polynomial

results in

�4 − 2�ka�ma + M� + maK�

ma�ma + 2M��2 + 2

kaK

ma�ma + 2M�= 0 �42�

Using a simple change of variables procedure the above polyno-mial can become a second-order polynomial. It is known that thevalue of the summation of the roots of a second-order polynomialis the negative of the first-order term. Therefore, the following canbe obtained:

�12 + �2

2 =2�ka�ma + M� + maK�

ma�ma + 2M��43�

At this point recall Eq. �34�, which enforces the condition thatthe amplitude of the invariant points should be the same. InsertingEqs. �39� and �40� into Eq. �34� and taking the expressions out of

the absolute values result in

Journal of Vibration and Acoustics

F

K − �12M

1

1 −�1

2ma

K − �12M

+F

K − �22M

1

1 −�2

2ma

K − �22M

= 0 �44�

Rearranging and simplifying the above equation results in the fol-lowing expression:

�12 + �2

2 =2K

M + ma�45�

Now, equating Eqs. �43� and �45� and then solving for ka resultsin the following expression:

ka =maMK

�M + ma�2 �46�

If both the numerator and the denominator of Eq. �46� is dividedby M2, the following expression is obtained:

ka

ma=

KM

�1 +ma

M �2 �47�

Now, by renaming some of the terms as follows �to match DenHartog’s notation�:

�a2 =

ka

ma, �48a�

�n2 =

K

M, �48b�

� =ma

M, �48c�

f =�a

�n�48d�

the following expression can be obtained:

f =1

1 + ��49�

This is the same expression as in Den Hartog’s derivation for theoptimal absorber stiffness. So, it has been shown that Eqs. �31�,�32�, and �34� are general equations for determining the invariantpoints and the optimal absorber stiffness. The SDOF main system,which is used in Den Hartog’s derivation, is a simplified case ofthis general method.

5 Calculation of the Optimal Damping Values

5.1 Applying Den Hartog’s Method. After finding the opti-mal value of the absorber spring, the next step of the absorberdesign is the calculation of the value of the viscous damper ele-ment. It is discussed in Sec. 3, how to find the optimal dampervalue for a SDOF main system. The algorithm is almost the same.One needs to take the derivative of the response expression givenin Eq. �27� and find the damper value that makes the derivativezero at the invariant point. If the derivative of the response am-plitude is evaluated and the derivative of the expression with re-spect to � is set to zero, the following expression can be obtained:

���xk�2���

= 0 �50�

Evaluation of Eq. �50� results in a complicated expression, whichcan be solved for the damping value ca. The expression is afourth-order polynomial in terms of the damping value. It can bereduced to a second-order polynomial by a simple change-of-variables procedure. The coefficients of the second-order polyno-

mial are calculated as follows:

AUGUST 2005, Vol. 127 / 345

a1 =� �xmaink

�

���4F1

2��1�F12��2� + 2�xmaink

��4�F1��1�F1��2�

��F1��1�F2��1� − F1��1�F2��2��� �51�

a2 =� �xmaink

�

���2��− �2ma + kaF1��2��2F1

2��1� + �− �2ma

+ kaF1��1��2F12��2�� + 2�xmaink

��2��− �2ma + kaF1��1���2�ma

+ kaF2��1��F12��2� − �− �2ma + kaF1��2���2�ma

+ kaF2��2��F12��1�� + 2�xmaink

��2�F1��1�F2��1��− �2ma

+ kaF1��2��2 − F1��2�F2��2��− �2ma + kaF1��1��2�

+ 2�xmaink���F1

2��1��− �2ma + kaF1��2��2

− F12��2��− �2ma + kaF1��1��2� �52�

a3 =� �xmaink

�

���− �2ma + kaF1��2��2�− �2ma + kaF1��1��2 + 2�xmaink

�

��− �2ma + kaF1��1���− �2ma + kaF1��2����2�ma

+ kaF2��1���− �2ma + kaF1��2��

− �2�ma + kaF2��2���− �2ma + kaF1��1��� �53�

where a1 is the coefficient of the second-order term, a2 is thecoefficient of the first-order term, and a3 is the constant term, andthe expressions for F1��� and F2��� are given as follows:

F1��� = 1 − �2ma� �54�

F2��� = 2�ma� + �2ma

��

���55�

Using Eqs. �51�–�55� one can obtain the ca value, which wouldmake the slope of the frequency response curve zero at a particu-lar frequency �. The expression for the ca value is as follows:

ca =�− a2 ± �a22 − 4a1a3

2a1�56�

As discussed in Sec. 3, it is necessary to find the values ofabsorber damping that would make the slope of the frequencyresponse curve zero at invariant points. Since there is no absorberdamping value that would make the slope zero at both of theinvariant points, simultaneously, Den Hartog suggests that the av-erage of the two ca values should be evaluated and used as theoptimal damping value. So the optimal damping expression can bewritten as follows:

copt =ca_1 + ca_2

2. �57�

In the above expression ca_1 and ca_2 correspond to two dampingvalues, which make the slope of the frequency response curvezero at invariant points 1 and 2, respectively. The ca_1 and ca_2values are to be calculated using Eqs. �51�–�56�. In those equa-tions there are some terms that require taking the derivative of thefrequency response curve. It is possible to compute the derivativeapproximately by taking the amplitude response at two neighbor-ing frequencies and then employing the finite difference to ap-proximate the derivative. Employing the approach of Den Hartogmay not be straightforward because of the complicated nature ofthe Eqs. �51�–�53�. In Sec. 5.2 a different approach will be dis-cussed to make the optimal damping value calculation less com-plex.

5.2 Employing a New Approach for the Calculation of theOptimal Absorber Damper Value. In this section a less compli-

cated method for the calculation of the absorber damping value

346 / Vol. 127, AUGUST 2005

will be introduced. In this method, the frequency of concern willbe the midfrequency of the two invariant point frequencies. Theobjective is to find the damper value, which would result in thesame amplitude response at the midfrequency point as the re-sponse amplitude at the invariant points.

From Eq. �34� the response of the invariant point xinv is

�xinv� = �xmaink��1��� 1 − �1

2ma�2��1�1 − �1

2ma�1��1�� �58�

The midpoint frequency is defined as

�mid =�1 + �2

2�59�

One can find the amplitude of the response at an invariant point byevaluating Eq. �58�. The response at the midfrequency point �us-ing Eq. �28�� can be set equal to response amplitude of the invari-ant point by choosing the correct damping value. Therefore, usingEqs. �58� and �28�, the expression for the damping value is

ca =� �A − C��Amp_rat��Amp_rat��D2 − B2�

�60�

where

Amp_rat =�xinv�2

�xmaink�2

�61�

Here, the expressions for A, B, C, and D can be extracted fromEqs. �27� and �28�. Note, �xmain k� is the amplitude of the mainsystem response at the midfrequency point ��mid�, and �xinv� is theamplitude at the invariant points. Generally, the damper valuescalculated using this procedure are slightly higher than the opti-mal damper values, but, the procedure for their calculation is sig-nificantly easier.

6 Using Den Hartog’s Vibration-Absorber Method inthe Modal Domain for MDOF Main Systems

The presented new method in this paper is a direct extension ofDen Hartog’s vibration-absorber analysis technique to the case ofan undamped MDOF main system. An indirect way to extend DenHartog’s SDOF absorber design method to MDOF systems is totransform the MDOF system into the modal domain and employthe Den Hartog method for SDOF systems to individual modes.�It will be shown in the following example case study that thisapproach is not as effective as the new approach introduced in thispaper.�

Below is the expression for the equation of motion in the physi-cal coordinates

�M�x + �K�x = F�t� �62�Equation �62� can be carried into the modal domain using itseigenvectors to form the transformation matrix �, and the follow-ing equation can be obtained:

�Md�� + �Kd�� = �TF�t� �63�

In Eq. �63� �Md� and �Kd� are the diagonal modal mass and stiff-ness matrices, respectively. Also, � is the modal coordinates vec-tor. When the equations of motion are decoupled, each modalequation can be considered as a SDOF system that corresponds toa mode of the main system. One can employ Den Hartog’s tech-nique for SDOF systems on this single modal equation. Let usassume that the target is minimizing the kth mode of the system.Using Eq. �47� and the optimal damping expression for Den Har-tog’s technique for SDOF systems �3�, one can write the optimalstiffness and optimal damping expressions as follows:

ka = ma

Kd�k,k�Md�k,k�

1

�1 +ma �2 �64�

Md�k,k�

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tem

ca = 2ma

Md�k,k�� 3maKd�k,k�

8�1 +ma

Md�k,k��3 �65�

The absorber parameters calculated using this method are goodfor minimizing the modal response, but not the actual response ofthe target mass. Also, since the method does not relate to physicalcoordinates, no information is available with regard to which co-ordinate the absorber should be attached. This method will beused as a benchmark for comparison with the new method’s re-sults in Sec. 7.

7 Case StudyA case study is performed to show the applications of the deri-

vations given in Secs. 4 and 5. The system in the case study is a4-DOF linear mass-spring system as shown in Fig. 3. It is differ-ent from the system given in Fig. 1; the purpose of choosing sucha system that has a more complex array of spring connections is toshow that the method is general and does not only apply to thecanonical form depicted in Fig. 1. In the case study the forces areapplied to all of the masses. The target mass �the mass whoseresponse is to be minimized around a particular mode’s resonantfrequency� is mass number 1. In the case study, the effect ofattaching the absorber to different masses will be discussed.

The numerical values of the main system parameters are givenin Table 1. The undamped response of the main system before theabsorber system is attached is given in Fig. 4. Note, the mainsystem is assumed to be undamped; therefore, the peak values ofthe resonant peaks that are depicted in the figure result from thefrequency step sizes used in the simulations.

In the first part of the case study, the absorber, which consists ofa mass, damper, and spring, will be attached to the third mass ofthe main system. The absorber mass that is used will be 0.5 kg inall parts of the case study. The optimal absorber stiffness and theabsorber damping will be calculated using three different meth-ods. The first method is the one derived in this paper, a directextension of Den Hartog’s method to MDOF systems. The formu-lations in Secs. 4.1 and 5.1 are used to calculate the optimal ab-sorber parameters. In the second method the stiffness value will becalculated the same way as it is calculated in the first method andthe damping value will be calculated using the method describedin Sec. 5.2, which is considerably simpler. The third method is themodal domain application of Den Hartog’s method that is de-scribed in Sec. 6.

Figure 5 shows the response of the target mass after eachmethod is applied. It is seen in Fig. 5 that the first method givesthe best results, followed by the second method. The response

Fig. 3 The 4-DOF main sys

Table 1 Numerical values of th

Index # 1 2 3

k �N/m� 30,000 30,000 20,00m �kg� 4 10 4F �N� 1 1 1

Nat. freq. �Hz� 8.196 12.250 22.62

Journal of Vibration and Acoustics

using the alternative damping calculation gives slightly higher re-sponses. The third method, the modal domain application of theDen Hartog method, performs poorly compared to the other twonew methods introduced in this paper. The main reason for thepoorer performance is the application of the method in the modaldomain, not in the physical domain.

Note, for the newly introduced method, the optimal absorberparameters change with the location of the absorber. Therefore,the performance of the absorber changes when it is attached todifferent coordinates. In Fig. 6, it is shown that the effectivenessof the absorber changes as much as tenfold based on the attachedcoordinate. In Table 2, the optimal stiffness and damping valuesare given using two different methods. In the table ka representsthe optimal absorber stiffness, ca is the optimal absorber dampingusing the extended Den Hartog method �Sec. 5.1�, and calt is theoptimal damping using the alternative damping calculationmethod �Sec. 5.2�. From Table 2 it can be seen that the alternativedamping method results in slightly higher damping values. Theexpressions �1 and �2 are the frequency locations of the invariantpoints. From the invariant point data and the maximum amplitudedata in Table 2 one can draw the conclusion that the larger thedifference of the invariant points, the better the attenuation of theresponse. It is seen in Fig. 6 and Table 2 that the absorber givesthe best performance when the absorber is attached to mass 1. Theresponse is slightly higher when it is attached to mass 3, and evenhigher when attached to mass 2. The worst performance is ob-tained when it is attached to mass 4.

One might think that the attenuation is the highest when theabsorber is attached to the first mass because that is the target.However, this is not valid when a similar analysis is done aroundthe first mode. Figure 7 and Table 3 show that the absorber per-forms the best when it is attached not to the first mass, but to thesecond mass. When the mode shape of the different modes arecompared, it is clear why the performance differs when the ab-sorber is attached to different masses. From Fig. 8 and Table 4,one can see that the most active mass around the third mode is thefirst mass and the most active mass around the first mode is thesecond mass. Attachment to mass 1 and 2 gave the best perfor-mance around the first and third modes. So, the more active thebase mass is, the better the absorber will couple to that mode, anda higher reduction of response will be obtained.

It may also be possible to attenuate the neighboring mode whenthe absorber is attached to a mass that is also active around theneighboring mode. Figure 9 shows the effect of attaching the ab-sorber to the third and fourth coordinates. Looking at the modeshapes, one can see that mass 3 is almost 15 times more activethan mass 4 around the fourth mode and more than five times

used in the case studies

arameters of the main system

4 5 6 7

50,000 20,000 30,000 45,00081

33.279

e p

0

1

AUGUST 2005, Vol. 127 / 347

active around the third mode, Therefore, it reduces the third�tuned� and the fourth �untuned� modes better than the configura-tion where the absorber is attached to the fourth mass.

8 ConclusionsIn this study, derivations that directly extend Den Hartog’s

tuned mass absorber method to multi-degree-of-freedom mainsystems are given. In order to find the expressions that wouldyield the invariant points of an undamped multi-degree-of-freedom system, the Sherman-Morrison matrix inversion methodhas been used. Using this method the response of a particularmass after the introduction of the absorber system is expressed interms of two components, the response of the original system�without the absorber system� and the effect of the absorber sys-tem. Using the obtained expression the optimal absorber stiffness

Fig. 4 Undamped response of mass 1

Fig. 5 Response of the first mass when the absorber is at-tached to the third mass. The absorber parameter calculationmethod is _ extended Den Hartog method for MDOF systems,ka=10,319 N/m, ca=19.2 Ns/m; _ _ _ _ extended Den Hartogmethod for MDOF systems with alternative damping calcula-tion method, ka=10,319 N/m, ca=22.0 Ns/m; — — —Den Har-tog method applied via modal domain, ka=8129 N/m, ca

=25.0 Ns/m.

348 / Vol. 127, AUGUST 2005

value can be calculated by numerically solving the given equa-tions. The optimal damping required for the absorber can be cal-culated through derived analytical equations. An alternative ex-

Fig. 6 Response of the target mass around the third modewhen it is attached to different coordinates. The absorber isattached to _ mass 1, ---- mass 2, _ _ _ mass 3, and __ _ __mass # 4.

Table 2 Optimal absorber parameters, the invariant frequency,and maximum amplitude data for attenuation around the thirdmode

Absorberattached to

mass 1

Absorberattached to

mass 2

Absorberattached to

mass 3

Absorberattached to

mass 4

ka �N/m� 9401.6 9391.5 10320.0 9237.4ca �Ns/m� 21.4 6.2 19.2 2.7calt �Ns/m� 25.0 7.2 22.0 3.2�1 �Hz� 20.08 21.97 20.78 22.34�2 �Hz� 24.06 23.16 23.70 22.88Max. amplitude �m� 4.8�10−5 1.7�10−4 6.4�10−5 3.9�10−4

Fig. 7 Response of the target mass around the first modewhen it is attached to different coordinates. The absorber isattached to _ _ _ mass 1,___ mass 2, ---- mass # 3, — – — –

mass # 4.

Transactions of the ASME

pression for the optimal damping is also derived. This expressiondetermines the damping value that creates the same response am-plitude as the response amplitude of the invariant points at a fre-quency that is the arithmetic mean of the invariant points.

The effect of mounting the absorber to different masses and theeffect of the absorber location on the invariant points, and optimaldamping and stiffness values have been discussed. It has beenshown that having the invariant points more separated increasesthe performance of the absorber. Also, it has been shown that theselection of the absorber location can affect neighboring modes.Even if the absorber is not tuned to attenuate a particular mode, ifthe mass to which the absorber is attached is active for that par-ticular mode, the absorber will attenuate the system response atthat mode by dissipating energy through its damper �similar to aLanchester damper�.

AcknowledgmentThe financial support of the National Science Foundation

�Grant No. 9733565 is acknowledged.

Table 3 Optimal absorber parameters, the invariant frequency,and maximum amplitude data for attenuation around the firstmode

Absorberattached to

mass 1

Absorberattached to

mass 2

Absorberattached to

mass 3

Absorberattached to

mass 4

ka �N/m� 1301.1 1225.3 1315.2 1375.9ca �Ns/m� 3.2 6.1 2.6 2.0calt �Ns/m� 3.7 7.0 3.0 2.3�1 �Hz� 7.85 7.43 7.92 8.01�2 �Hz� 8.44 8.55 8.40 8.34Max. amplitude �m� 4.9�10−4 3.4�10−4 6.0�10−4 8.7�10−4

Fig. 8 Mode-shape drawings of the main system modes

Table 4 Mode-shape values of each mode

MODE 1 MODE 2 MODE 3 MODE 4

Mass 1 0.51 −0.1 1 −0.77Mass 2 1 −0.25 −0.30 0.008Mass 3 0.41 0.13 0.77 1Mass 4 0.29 1 −0.14 −0.067

Journal of Vibration and Acoustics

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