2015 SEAOC CONVENTION PROCEEDINGS

1

Fundamentals of Tuned Mass Dampers (TMDs) for Seismic Response Reduction

Julio C. Miranda, P.E., S.E.

Principal Technologist CH2M HILL

San Jose, California

Abstract Modelling the structures incorporating Tuned Mass Dampers

(TMDs) as two-degrees-of-freedom mechanical systems, this

paper studies the salient parameters that induce the efficiency

of these devices for the purpose of seismic response

reduction. Focusing intrinsically on the mechanical systems

themselves, and independently of ground motions, numerical

and analytical expressions are obtained for the modal

damping of the systems. Subsequently, for a highest

efficiency in terms of modal damping allocation, a tuning is

proposed such that modal damping is generated in the same

proportion as the participation factors for the real modes.

Further, some useful properties of the frequencies, tuning,

and participation factors, are analytically demonstrated.

Finally, limited calculations using a spectrum compatible

accelerogram are offered to support the proposed method for

modal damping allocation.

Introduction

A review of the current literature reveals that a consensus is

consolidating, suggesting that using TMDs is a practical and

efficient technique to reduce structural responses due to

strong ground motions. The pioneering work by Villaverde

[3], Feng et al. [1], and Sadek et al. [4], demonstrated the

feasibility of using these devices. Subsequent research has

showed that TMDs can develop substantial damping;

Miranda [5], Moutinho [6], and Miranda [7]. Further, other

studies have identified that using large mass ratio TMDs

results in a more robust definition of the parameters leading

to seismic response reduction; Hoang et al. [2], De Angelis et

al. [14], and Chen et al. [12] [13]. This observation extends

the applicability of TMDs to systems in which portions of the

structure itself may be mobilized to protect the complete

structural assembly. Along this line, Hoang et al [2] discuss

the retrofit of a large bridge in Japan. De Angelis et al. [14],

present information on large mass ratio TMDs used by others,

and report on their own experimental work with large mass

ratios. Finally, other researchers have explored conditions for

which using TMDs may be useful in reducing the response to

impulsive ground motion; Salvi et al. [16], and Domizio et al.

[17].

Two approaches have been followed for studying these

devices: a) Calibration, commonly referred to as

optimization, of the TMD parameters such that given a design

ground motion a certain aspect of the structural response is

minimized, and b) Intrinsic consideration of the TMD

parameters, such that independently of any ground motion,

certain properties of the structural system, such as modal

damping, are calibrated with the expectation of a reduced

structural response. Proponents of the first modality usually

consider ground motions that are assimilated to white noise

random processes, or that can be characterized by Kanai-

Tajima types of power spectra density. These proponents

have presented equations corresponding to the optimum

tuning and damping likely to minimize the response of

structures provided with TMDs. The procedures that use this

modality are versatile, but the calibrated TMD properties,

namely the tuning and damping, become a function of the

ground motion. Thus, the possibility of important variations

between the properties of the design ground motion versus

those of the actual event realization, has to be considered.

Indeed, under such circumstances the expectations

numerically forecasted might be critically affected. Recently,

Salvi et al [11] [15], performed extensive time history

calculations using suites of real earthquakes to analyze

buildings equipped with TMDs, and identified optimum

tuning with a rather wide range of values, which in the view

of the author of the present paper indicates the sensitivity of

such optima to ground motions.

While the behavior of structures provided with TMDs

calibrated per the second modality is clearly a function of the

spectral characteristics of the ground motion in relation to the

dynamic properties of the mechanical systems, the tuning and

damping themselves are preset in a manner that is

independent of the potential earthquake. Along these lines

Villaverde [3] suggested to use small, but highly damped,

resonant masses attached to the top of buildings in order to

induce two complex modes with damping ratios

2

approximately equal to the average of the damping ratios of

the resonant mode of the building and the TMD. Following

on this lead, Sadek et al. [4] suggested to install roof mounted

TMDs proportioned so as to induce two complex modes with

equal frequencies and damping ratios. Accordingly, these

authors made extensive studies of single and multiple-degree-

of-freedom structural systems with a wide range of natural

periods, provided with TMDs calibrated in their prescribed

manner, and subjected to fifty two real earthquakes. Their

results showed substantial reduction of the response, in some

cases of up to 50%. Miranda [5] presented an energy-based

numerical model, and successfully used it to verify the

optimum tuning and damping proposed previously by Sadek

et al. [4]. While this model ignores coupling due to damping,

it provides a very convenient platform to study the systems

under consideration. Moutinho [6] considered systems with

two complex modes having equal modal damping

coefficients, but different frequencies, and with TMDs

parameters in correspondence with the minimum value of

their dynamic amplification factors for harmonic loading.

Miranda [7] proposed an analytical methodology for the

tuning of systems with two equal real mode damping

coefficients, observing that this condition also implies

equality of damping coefficients for the corresponding two

complex modes. Additionally, such proposal was shown to

contain both the methods by Villaverde [3] and by Sadek et al

[4] as particular cases.

The purpose of this paper is to discuss the parameters that

define the structural dynamics of the mechanical systems

composed by the TMDs and the structures to which they are

affixed to. This is done in a manner that examines these

parameters intrinsically, without consideration of ground

motions. To gain insight into such parameters, three methods

are used: a) the modal energy-based method proposed by

Miranda [5], which is approximated since it ignores coupling

due to damping, b) comparison of the system’s characteristic

equations in terms of real modes, and c) comparison of the

system’s characteristic equations in terms of complex modes.

The last two methods stem from the invariant properties of

the system’s frequencies. Emphasis is placed on discussing

the damping properties, as this is the most significant reason

to use TMDs under seismic excitation. The paper will then

discuss a method to calibrate TMDs according to a newly

proposed damping control strategy. Subsequently, the paper

discusses some properties relative to the frequencies, and

participation factors. Finally, limited numerical calculations

using spectrum compatible accelerograms are offered to

support the proposed control strategy.

The theoretical model used in this paper considers a two-

degree-of-freedom system, which although simplistic, has

nevertheless enabled many studies, including the classic

works by Den Hartog [8], and Warburton [9]. The study of

such systems lends itself to complex modal analysis and as

such in this paper, modes, frequencies, and damping, are as

appropriate double sub-indexed to signify derivation through,

or in correspondence with, that methodology. Likewise,

modes, frequencies, and damping, derived utilizing real mode

analyses are affected with single sub-indexes to differentiate

them from the parameters obtained with complex procedures.

As long as it is clear, this paper will refer to “real” or

“complex” parameters, with the understanding that they

pertain to real or complex mode analyses correspondingly.

Theoretical Considerations Consider the two-degree-of-freedom mechanical system

depicted in Figure 1. The upper portion, which constitutes the

TMD, is characterized by its mass MU, its damping constant

CU, and its spring stiffness KU. The lower portion, is

characterized by a mass ML, a damping constant CL, and a

spring with stiffness KL. The latter parameters represent the

effective modal properties for the structure under

consideration, and are presumed to be known.

Figure 1: two-degree-of-freedom mechanical system

The following parameters are defined in order to characterize

the mechanical system:

U

U

U

K

M

2 (1)

LL

L

K

M

2 (2)

3

U

L

(3)

M

M

U

L

(4)

UU

U

UMK

C

2 (5)

LL

LL

MK

C

2 (6)

Equation (1) provides the circular frequency for the upper

portion of the system, ωU, when considered independently.

Equation (2) provides the circular frequency for the lower

portion, ωL, when considered independently. Equation (3)

represents the tuning ratio, Ω, between the circular

frequencies of the upper and lower portions. In equation (4),

µ, is the ratio of the upper portion mass to the lower portion

mass, parameter which is also presumed to be known.

Equation (5) furnishes the coefficient of damping for the

upper part, ξU, whereas equation (6) furnishes the coefficient

of damping for the lower part, ξL.

Since the approximated numerical platform proposed by

Miranda [5] is being used, the reader is referred to that paper.

Per this method, and for given tuning and mass ratios, any

dynamic state of the two-degree-of-freedom system, as

represented by its modal frequencies, modal shapes, modal

participation factors, and modal damping, is defined by

determining just one of the four interrelated modal energy

parameters αj and βj corresponding to the jth mode of

vibration.

Considering the system shown in Figure 1 while undergoing

free vibrations, its characteristic equation, see Miranda [7],

may be written as:

001

2

2

3

3

4 ffff (7)

Where λ represents a complex frequency. The factors fi, are

given in Appendix A equations (A.1.a) to (A.1.d), and are a

function of the mechanical properties as previously defined. Similarly, if the equations of motion under free vibration are

transformed to modal coordinates using the real modes, their

characteristic equation is still written per equation (7).

However, the factors fi are now expressed in terms of real

mode properties. Such factors are also provided in Appendix

A, equations (A.2.a) to (A.2.d). In these equations, Mj, Cj,

and Kj are the generalized mass, damping, and stiffness, for

the jth real mode respectively. Cc is the term responsible for

the coupling of the modal coordinates, while ωj and ξj

represent the circular frequency and coefficient of damping

for the jth mode of vibration.

Proceeding with the third method used to obtain the desired

parameters, it is recalled that for non-proportional damping,

the equations of motion can be exactly resolved using

complex modal analysis, through a state-space decomposition

as described, for example, by Hurty and Rubinstein [10]. It

may be shown then, that it is possible to transform to

complex modal coordinates the equations of motion of the

system shown in Figure 1 while undergoing free vibrations,

and write the following two equations:

0)()()( 111111111111 tqKtqCtqM (8.a)

0)()()( 222222222222 tqKtqCtqM (8.b)

wherein the qjj are complex modal coordinates, and Mjj, Cjj,

and Kjj, are the generalized mass, damping, and stiffness,

corresponding to the jth complex mode, respectively. The

characteristic equation of the last two expressions can again

be written per equation (7), but with the fi factors written as a

function of complex modes, as presented in Appendix A,

equations (A.4.a) to (A.4.d). The parameters ωjj and ξjj

represent the circular frequency and coefficient of damping

for the jth complex mode of vibration.

It is noted that since the eigenvalues of the system are

invariants, the three set of factors fi , either derived from the

mechanical properties, the real modal, or the complex modal

equations, are by definition equal to each other respectively.

This fundamental property provides the base for the

discussion to be established below.

Discussion of Parameters

Numerical examples are used to discuss the salient

parameters of the systems under study. To highlight TMDs

with large mass ratios, values of µ from 0.25 to 1.0, with

increments of 0.25 are considered. Since TMDs are usually

applied to weakly damped structures, a structural coefficient

of damping ξL equal to 0.03 is considered, while the TMD

coefficient of damping ξU is set at 0.3. The latter number

corresponds to an upper bound than can currently be readily

achieved. Values of Ω ranging from 0.1 up to higher than 1.0

are chosen. For the purpose of illustration, some calculation

tables are presented in Appendix B, which will be referred to

as required. For example, Table I shows the modal energy

4

coefficients αj and βj, calculated per the procedures shown in

Miranda [5], for the dynamic states corresponding to each

chosen tuning and for a mass ratio equal to 0.5. It is noted

that this table also provides coefficients corresponding to

equal modal damping, real or complex, see Miranda [7]. In

addition, the table provides coefficients corresponding to the

condition of perfect balance of real mode strain energy, of

resonance between the TMD and structure, and perfect

balance of real mode kinetic energy, respectively, see

Miranda [5]. Table II uses these energy coefficients to

calculate the real circular frequencies normalized with respect

to the structural frequency, the modal participation factors,

and modal damping. In this table the participation factors, γj,

are defined, as usual, equal to j

T

j

T

j MMr / where M is the

system’s mass matrix, φj is the vector corresponding to the jth

real mode, and r is a vector of ones. It is noted that the mode

components corresponding to the lower mass have been

conveniently normalized to one.

Table III uses a complex modal procedure to calculate the

exact frequencies, once again normalized with respect to the

structural frequency, and the exact coefficients of modal

damping corresponding to the chosen tuning and mass ratio,

and for the same damping of the lower and upper portions of

the mechanical system as considered in the preceding tables.

(a)

(b)

(c)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 0.5 1 1.5

NO

RM

ALI

ZED

FR

EQ

TUNING

µ=0.25, ξU=0.3, ξL=0.03

ω1/ωL ω2/ωL

ω11/ωL ω22/ωL

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 0.5 1 1.5

NO

RM

ALI

ZED

FR

EQU

ENC

IES

TUNING

µ=0.5, ξU=0.3, ξL=0.03

ω1/ωL ω2/ωLω11/ωL ω22/ωL

0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2 2.5

NO

RM

ALI

ZED

FR

EQ

TUNING

µ=0.75, ξU=0.3, ξL=0.03

ω1/ωL ω2/ωLω11/ωL ω22/ωL

5

(d)

Figure 2: Approximated versus exact normalized frequencies

Figure 2, above, depicts the approximated normalized

frequencies, and the exact normalized frequencies, as a

function of the tuning ratios, and for various mass ratios. It

may be appreciated that the exact and approximated

frequencies, for all mass ratios, and for both modes, agree

very well. The largest difference is about 4.7% for the second

mode, and for a mass ratio of 0.25. Analytical expression for

the approximated normalized frequencies are given in

Miranda [18].

Figure 3 depicts the approximated modal damping, and the

exact modal damping, as a function of the tuning ratios, and

for various mass ratios. It may be appreciated that the

approximated and exact damping, for all mass ratios, and for

both modes, agree well. The largest difference is about 5.0%

for the second mode, and for a mass ratio of 0.25. Later,

analytical expressions for the approximate and exact modal

damping will be obtained, as well as an expression furnishing

the product of the real mode damping coefficients.

(a)

(b)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 0.2 0.4 0.6 0.8 1 1.2

NO

RM

ALI

ZED

FR

EQ

TUNING

µ=1.0, ξU=0.3, ξL=0.03

ω1/ωL ω2/ωLω11/ωL ω22/ωL

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 0.5 1 1.5

MO

DA

L D

AM

PIN

GTUNING

µ=0.25, ξU=0.3, ξL=0.03

ξ1 ξ2 ξ11 ξ22

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 0.5 1 1.5

MO

DA

L D

AM

PIN

G

TUNING

µ=0.5, ξU=0.3, ξL=0.03

ξ1 ξ2 ξ11 ξ22

6

(c)

(d)

Figure 3: Approximated versus exact modal damping

Figure 4 depicts the mode participation factors as a function

of tuning, and for the selected values of the mass ratio. These

expressions were derived using the expressions provided by

Miranda [5] based on modal energy parameters. Later,

analytical expressions for the product of the participation

factors will be obtained.

(a)

(b)

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 0.5 1 1.5 2 2.5

MO

DA

L D

AM

PIN

G

TUNING

µ=0.75, ξU=0.3, ξL=0.03

ξ1 ξ2 ξ11 ξ22

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 0.2 0.4 0.6 0.8 1 1.2

MO

DA

L D

AM

PIN

G

TUNING

µ=1.0, ξU=0.3, ξL=0.03

ξ1 ξ2 ξ11 ξ22

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5P

AR

TIC

IPA

TIO

N F

AC

TOR

STUNING

µ=0.25, ξU=0.3, ξL=0.03

FIRST MODE SECOND MODE

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5

PA

RTI

CIP

ATI

ON

FA

CTO

RS

TUNING

µ=0.5, ξU=0.3, ξL=0.03

FIRST MODE SECOND MODE

7

(c)

(d)

Figure 4: Participation Factors

Discussion of modal damping

The dynamic behavior of the two-degree-of-freedom systems

comprised by a TMD affixed to a structure is obviously

fundamentally different from that of the original single-

degree-of-freedom structure. Modal damping aside, and as a

function of the spectral distribution of seismic energy, the

response of the mechanical system may be, but not

necessarily so, reduced by the addition of an un-damped

upper mass. Thus, it will be assumed in this paper that the

effect of enhanced damping due to the use of TMDs will, at a

minimum, offset any potentially negative effects that could

arise from the modified dynamics of the mechanical systems.

Other than this consideration, this paper will focus on the

damping generating properties of TMDs.

It may be shown, Miranda [5], that the coefficients of

damping for the real modes can be written as a linear

combination of the damping furnished separately by the TMD

and the structure, per the following equation:

LjUjj BA (9)

The factors Aj and Bj , corresponding to the jth mode, may be

written as functions of the modal energy coefficients, as

follows:

11

11

11

A (10)

11

111

1

B (11)

22

22

11

A (12)

22

211

1

B (13)

with:

21

212

2

2

(14)

Equation (9) indicates the fundamental reason for using

TMDs: for structures with low damping ξL, significant modal

damping can be induced by implementing TMDs delivering

high values of the coefficient ξU. Alternatively, an analytical

form of equation (9), using real mode properties, can be

obtained through the equality of the f0, f1, and f3 factors, per

equations (A.1.a), (A.1.c), (A.1.d) with (A.2.a), (A.2.c),

(A2.d), as follows:

2122

12

1

)()(

)1(

LL

LLA

(15)

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2 2.5

PA

RTI

CIP

ATI

ON

FA

CTO

RS

TUNING

µ=0.75, ξU=0.3, ξL=0.03

FIRST MODE SECOND MODE

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1 1.2

PA

RTI

CIP

ATI

ON

FA

CTO

RS

TUNING

µ=1.0, ξU=0.3, ξL=0.03

FIRST MODE SECOND MODE

8

2122

12

1

)()(LL

LLB

(16)

2122

12

2

)()(

)1(

LL

LLA

(17)

2122

12

2

)()(LL

LLB

(18)

It is seen that the following expression applies:

21

212

AA

BB (19)

Further, for the complex modes a corresponding analytical

expression can be obtained for the exact modal coefficients of

damping through the equality of the f0 , f1 , and f3 factors, per

equations (A.1.a), (A.1.c), (A.1.d), with (A.4.a), (A.4.c) and

(A.4.d), as follows:

LjjUjjjj BA (20)

It can be therefore written that:

211222

1122

11

)()(

)1(

LL

LLA

(21)

211222

1122

11

)()(LL

LLB

(22)

211222

1122

22

)()(

)1(

LL

LLA

(23)

211222

1122

22

)()(LL

LLB

(24)

The correctness of the three formats for modal damping

equations furnished above, may be verified using the data

provided in the tables in appendix, for both the real, and

complex modes.

From Figure 3, it may be seen that for the first mode,

damping decreases with increasing tuning. Conversely, it

may be seen that the damping for the second mode increases

with increased tuning. Further, from Figure 4, it may be seen

that for the first mode the participation factor increases with

increased tuning, whereas for the second mode the

participation factor decreases with increased tuning. It is seen

thus, that as represented by the listed values of damping,

tuning, and mass ratios, the allocation of damping to the

modes, per equations (9) or (20), is discordant with respect to

the importance of the participation factors. In other words,

there is an inefficient modal damping allocation with respect

to the importance of the modes providing the seismic

response. Tuning a TMD that excessively over or under

damps the modes of a mechanical system in an opposite sense

to the importance of the participation factors, might result in

systems that do not experience seismic response reduction, or

that require inordinate amounts of damping in order to

experience a desired level of response reduction.

Economically and technically, this is not a rational situation.

An improvement to this condition occurs when tuning is

achieved at a value that results in equal damping for both

modes, as under such circumstance both participation factors

will be equally damped, and this provides some measure of

control to the over or under damping of the modes. This

equality of energy dissipation potential explains the success

that tuning for equal mode damping has in decreasing the

seismic response for single or multi-degree-of-freedom

structures as demonstrated by Villaverde [3] and by Sadek et

al [4]. Further in-depth discussion of mechanical systems

tuned in such manner can be found in Miranda [7].

There appears to exist a lingering perception, that TMDs

require tuning at, or near, resonance in order to be effective.

Such idea is flawed, as can be inferred from the discussion

above. However, as shown by Miranda [5], for systems with

significant exchange of energy between the upper and lower

parts, as the mass ratio decreases, say to a range of 0.01, then

tuning in general approaches values closer to one, resulting in

quasi-resonant conditions. Likewise, for such small mass

ratios, both modal damping coefficients are close to each

9

other, and with a magnitude roughly equal to the average of

the damping furnished independently by the TMD and the

structure, as demonstrated by Villaverde [3].

Using the method proposed by Miranda [5], the following

expression yielding the product of the real mode coefficients

of damping can be obtained:

)2(

)(

21

22

21

2

21

LLUU

(25)

Given defined mechanical properties, the right term of

equation (25) is a constant. This mean, for instance, that

small fundamental mode damping requires correspondingly

high second mode damping in order to maintain the equality,

and vice versa.

The discussion regarding discrepancy between modal

damping allocation and modal participation factors is retaken

next.

Discussion on modal damping allocation

It was observed above, that while high modal damping can be

obtained per equations (9) or (20), the manner in which such

damping is effectively conveyed to the modes requires proper

calibration of the system parameters for due efficiency. It can

be argued that for a rational assignment, and given the

normalization of the modes, the ratio of the modal damping

should be equal to the ratio of the participation factors, that

is, the modes are to be damped in proportion to their

corresponding importance for the response, as reflected by

the participation factors. Using the approximated method

proposed by Miranda [5], it can be readily demonstrated that

this approach leads to:

1

1

2

1

2

(26)

It is understood that while this method of tuning involves

parameter that are exact within the consideration of real

modes, it becomes approximated when dealing with complex

modes. Additional calculations are prepared to illustrate the

results deriving from equation (26). This time, it will be

considered that the mass ratio µ varies from 0.1 to 1.0, with

increments of 0.1, with a TMD damping ξU equal to 0.3, and

with a structural damping ξL equal to 0.03. Table IV presents

the corresponding modal energy parameters for the chosen

mass ratios, and damping, whereas Tables V contain the real

normalized frequencies and damping coefficients, all

calculated with the constraint per equation (26). Table VI

uses a complex modal procedure to calculate the exact

normalized frequencies and modal damping corresponding to

the chosen mass ratios and constrained tuning.

Figure 5: Approximated versus exact constrained normalized

frequencies

Using the data from these tables, Figure 5 indicates that

except for mass ratios smaller than approximately 0.15, the

agreement between the constrained approximated normalized

frequencies and the exact ones is very good for the first

mode, and slightly less good for the second mode having a

maximum difference of 9%.

Figure 6: Approximated versus exact constrained modal

damping

Likewise, for the same damping coefficient, and tuning,

Figure 6 indicates that except for mass ratios smaller than

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

NO

RM

ALI

ZED

FR

EQU

ENC

IES

MASS RATIO

ξU =0.3, ξL=0.03

ω1/ωL ω2/ωL ω11/ωL ω22/ωL

0

0.05

0.1

0.15

0.2

0.25

0.3

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

MO

DA

L D

AM

PIN

G

MASS RATIO

ξU=0.3, ξL=0.03

ξ1 ξ2 ξ11 ξ22

10

approximately 0.18, the agreement between the constrained

real modal damping and the exact one, is very good for the

first mode, and good for the second mode, with a maximum

difference of 9%. It is also observed that while the damping

for the first mode remains fairly constant, the damping for the

second mode is higher, and increases with the mass ratio.

Similar calculations with higher TMD damping, say equal to

0.6, appear to indicate that the exact frequencies and damping

cannot reliably be predicted using the approximate ones.

Discussion of system frequencies and participation factors

Per equality of the three f0 factors in Appendix A, it may be

written that:

))(())(( 221121

LLLL

(27)

Given a defined set of mechanical properties, the right term

of equation (27) is fixed. Hence, low fundamental frequencies

will require correspondingly high second mode frequencies in

order to maintain the equality, and vice versa. The

frequencies of the system will bracket the original frequency

from above and below, conforming to the following

expression:

)()(1)()( 222111

LLLL

(28)

Using the method proposed by Miranda [5], it may be shown

that the product of the participation factors yields:

2

2

2

2

2

1

2

121

)1()1())((

(29.a)

or alternatively:

2

2

2

2

1

121

)1()1())((

(29.b)

Given sets of defined structural properties, the second terms

of equations (29) will be fixed. Then, a low fundamental

mode participation factor will necessitate a high second mode

participation factor in order to maintain the equalities in the

last two equations, and vice versa.

Some numerical results for seismic excitation

Assume a 300 tons mass having a circular frequency of

5.1302 radians per second, and a damping coefficient of

0.0245, is divided into a lower mass of 200 tons having a

circular frequency of 2π radians per second and a damping

coefficient ξL of 0.03, plus an upper mass of 100 tons with a

damping coefficient of ξU of 0.3, and a circular frequency to

be tuned so as to reduce the displacement of the lower mass

during an earthquake represented by the response spectrum

depicted in Figure 9. After generating a spectrum compatible

time history, the response of the two-degree-of-freedom

systems shown in Table VII hase been obtained. In this table,

xU and xL represent the displacements of the upper and lower

mass respectively. The values for Ω that are considered

correspond in increasing order to; tuning for equal modal

damping; tuning per equation (26); tuning for perfect balance

of modal strain energy; tuning at resonance; and tuning for

perfect balance of modal kinetic energy.

Figure 9: Response Spectrum

Figure 10: Maximum Response Upper and Lower masses

In Figure 10, the lower curve represents the displacements of

the lower mass when excited by the ground motion implied

by Figure 9. It may be seen that tuning using parameters per

02468

1012141618

0 0.5 1 1.5

Mas

s D

isp

lacm

en

ts (

cm)

TUNING

UPPER MASS LOWER MASS

11

equation (26) correctly leads to a minimum of the response of

about 6.11cm. It is noted that for this particular example,

tuning for exact modal balance of strain energy results in a

1% lower displacement of about 6.03 cm, which for practical

purposes would be the same response. As seen, for the range

under consideration the upper mass displacement

monotonically increases as tuning increases. As tuning is

increased, the lower mass displacement increases as shown

both in Table VII and Figure 10. For the case of the large

single mass, the maximum displacement is calculated at

17.06 cm, and therefore the use of the TMD with the

properties above, may reduce that displacement by about

65%.

CONCLUSIONS

This paper discusses the properties that render TMDs

efficient for seismic response reduction. Focusing of

damping, it is demonstrated that modal damping can be

expressed as a linear combination of the damping furnished

separately by the TMD itself and by the structure. One

numerical expression of such modal damping is provided,

along with two analytical forms based on real mode, and

complex mode, methodologies.

It is shown via examples, that the exact frequencies and

damping, for sufficiently low TMD damping, may be

forecasted using real mode methodologies. It is shown also

via examples, that the modal damping thus induced is

discordant in that it is furnished in the wrong proportions to

the participation factors. Under such circumstances the modes

are over or under damped, a condition which reduces the

efficiency of the TMDs, and that could lead to no response

reduction, or to require inordinate amounts of damping in

order to achieve some response benefit. This observation lead

to the proposal of using modal damping in the same

proportion as the participation factors. Providing damping

along this modality is believed to be the most rational use of

the damping provided by the TMD to the mechanical system.

Additionally, several useful relationships between

frequencies, tuning, and participation factors are

demonstrated. Finally, the efficiency of the new tuning

proposed in this paper is shown via limited calculations for

systems under ground motion excitation.

APPENDIX A: CHARACTERISTIC EQUATION IN

TERMS OF STRUCTURAL AND MODAL

PROPERTIES

The factor fi for characteristic equation (7) written in terms of

structural properties are:

LLUUf 2)1(23 (A.1.a)

LULULUf 4)1( 22

2 (A.1.b)

22

1 22 ULLLUUf (A.1.c)

22

0 LUf (A.1.d)

Alternatively, the factors fi may be written in terms of modal

properties as:

22113 22 f (A.2.a)

21

2

2121

2

2

2

12 4MM

Cf C (A.2.b)

2

122

2

2111 22 f (A.2.c)

2

2

2

10 f (A.2.d)

where:

1

12

1M

K (A.3.a)

2

22

2M

K (A.3.b)

11

11

2 MK

C (A.3.c)

22

22

2 MK

C (A.3.d)

Additionally, the factors fi may be written in terms of

complex modal properties as:

222211113 22 f (A.4.a)

22112211

2

22

2

112 4 f (A.4.b)

12

2

112222

2

2211111 22 f (A.4.c)

2

22

2

110 f (A.4.d)

where ξjj and ωjj are the exact damping and circular

frequency corresponding to the jth complex mode of the

mechanical system, which are written as:

11

112

11M

K (A.5.a)

22

222

22M

K (A.5.b)

1111

1111

2 MK

C (A.5.c)

2222

2222

2 MK

C (A.5.d)

13

APPENDIX B: TABLES

Table I. Modal energy parameters for equal to 0.5, and Ω variable

as shown, with U =0.3 and L=0.03.

Table II. Normalized frequencies, participation factors, and real

mode damping coefficients, for equal to 0.5, and Ω variable as

shown on Table I, with U =0.3 and L=0.03.

(1) (2) (3) (4) (5) (6)

(1) µ Ω α 1 α 2 β 1 β 2

(2) 0.5 0.1000 19800.49994 0.051E-03 196.029850 5.101E-03

(3) 0.5 0.2000 1202.499168 0.000832 46.158335 0.021665

(4) 0.5 0.3000 227.186956 0.004402 18.573382 0.053841

(5) 0.5 0.4000 68.110318 0.014682 9.110233 0.109767

(6) 0.5 0.5000 26.462210 0.037790 4.921823 0.203177

(7) 0.5 0.6429 9.260441 0.107986 2.255428 0.443375

(8) 0.5 0.8165 3.732051 0.267949 1.000000 1.000000

(9) 0.5 1.0000 2.000000 0.500000 0.500000 2.000000

(10) 0.5 1.4142 1.000000 1.000000 0.171573 5.828427

(1) (2) (3) (4) (5) (6) (7) (8)

(1) µ Ω ω1/ωL ω2/ωL γ1 γ2 ξ1 ξ2

(2) 0.5 0.1000 0.0998 1.0025 0.0051 0.9949 0.2977 0.0452

(3) 0.5 0.2000 0.1980 1.0104 0.0212 0.9788 0.2908 0.0618

(4) 0.5 0.3000 0.2929 1.0243 0.0511 0.9489 0.2784 0.0815

(5) 0.5 0.4000 0.3825 1.0458 0.0989 0.9011 0.2596 0.1059

(6) 0.5 0.5000 0.4644 1.0767 0.1689 0.8311 0.2339 0.1359

(7) 0.5 0.6429 0.5633 1.1414 0.3072 0.6928 0.1873 0.1873

(8) 0.5 0.8165 0.6501 1.2559 0.5000 0.5000 0.1292 0.2496

(9) 0.5 1.0000 0.7071 1.4142 0.6667 0.3333 0.0849 0.2970

(10) 0.5 1.4142 0.7654 1.8478 0.8536 0.1464 0.0434 0.3427

14

Table III. Complex mode normalized frequencies, and complex

modal damping coefficients for equal to 0.5, and Ω variable as

shown on Table I, with U =0.3 and L=0.03.

Table IV. Energy parameters for varying as shown, with U =0.3

and L=0.03.

(1) (2) (3) (4) (5) (6)

(1) µ Ω α 1 α 2 β 1 β 2

(2) 0.1 0.929214 2.214999 0.451467 1.186126 0.843081

(3) 0.2 0.869627 3.034512 0.329542 1.267804 0.788766

(4) 0.3 0.818667 3.830100 0.261090 1.331210 0.751196

(5) 0.4 0.774525 4.630111 0.215978 1.384732 0.722161

(6) 0.5 0.735860 5.443809 0.183695 1.431792 0.698426

(7) 0.6 0.701670 6.274864 0.159366 1.474158 0.678354

(8) 0.7 0.671184 7.124913 0.140353 1.512919 0.660974

(9) 0.8 0.643803 7.994700 0.125083 1.548807 0.645658

(10) 0.9 0.619054 8.884297 0.112558 1.582310 0.631988

(11) 1.0 0.596556 9.793650 0.102107 1.613803 0.619654

(1) (2) (3) (4) (5) (6)

(1) µ Ω ω11/ωL ω22/ωL ξ11 ξ22

(2) 0.5 0.1 0.0998 1.0016 0.2980 0.0452

(3) 0.5 0.2 0.1987 1.0066 0.2918 0.0616

(4) 0.5 0.3 0.2954 1.0155 0.2808 0.0808

(5) 0.5 0.4 0.3886 1.0292 0.2637 0.1045

(6) 0.5 0.5000 0.4763 1.0497 0.2391 0.1344

(7) 0.5 0.6429 0.5848 1.0994 0.1896 0.1896`

(8) 0.5 0.8165 0.6731 1.2131 0.1239 0.2589

(9) 0.5 1.0000 0.7221 1.3849 0.0791 0.3054

(10) 0.5 1.4142 0.7696 1.8377 0.0418 0.3451

15

Table V. Normalized frequencies, participation factors, and real

modal damping coefficients with µ, and Ω per Table IV, with U

=0.3 and L=0.03.

Table VI. Complex mode normalized frequencies, and complex

modal damping coefficients, with µ, and Ω per Table IV, with U

=0.3 and L=0.03.

(1) (2) (3) (4) (5) (6) (7) (8)

(1) µ Ω ω1/ωL ω2/ωL γ1 γ2 ξ1 ξ2

(2) 0.1 0.929214 0.8246 1.1269 0.4574 0.5426 0.1558 0.1848

(3) 0.2 0.869627 0.7497 1.1599 0.4410 0.5590 0.1545 0.1959

(4) 0.3 0.818667 0.6947 1.1784 0.4290 0.5710 0.1543 0.2054

(5) 0.4 0.774525 0.6508 1.1901 0.4193 0.5807 0.1546 0.2140

(6) 0.5 0.735860 0.6143 1.1979 0.4112 0.5888 0.1550 0.2220

(7) 0.6 0.701670 0.5832 1.2032 0.4042 0.5958 0.1556 0.2294

(8) 0.7 0.671184 0.5561 1.2069 0.3979 0.6021 0.1563 0.2364

(9) 0.8 0.643803 0.5323 1.2094 0.3923 0.6077 0.1570 0.2432

(10) 0.9 0.619054 0.5111 1.2111 0.3873 0.6127 0.1577 0.2496

(11) 1.0 0.596556 0.4921 1.2123 0.3826 1.6174 0.1584 0.2557

(1) (2) (3) (4) (5) (6)

(1) µ Ω ω11/ωL ω22/ωL ξ11 ξ22

(2) 0.1 0.929214 0.8979 1.0349 0.1399 0.2039

(3) 0.2 0.869627 0.7923 1.0976 0.1492 0.2048

(4) 0.3 0.818667 0.7275 1.1253 0.1508 0.2129

(5) 0.4 0.774525 0.6784 1.1418 0.1518 0.2210

(6) 0.5 0.735860 0.6385 1.1525 0.1527 0.2287

(7) 0.6 0.701670 0.6050 1.1598 0.1537 0.2361

(8) 0.7 0.671184 0.5761 1.1650 0.1546 0.2431

(9) 0.8 0.643803 0.5509 1.1686 0.1554 0.2499

(10) 0.9 0.619054 0.5285 1.1713 0.1563 0.2563

(11) 1.0 0.596556 0.5085 1.1731 0.1572 0.2626

16

Table VII. Response of two-degree-of-freedom system for

tuning and mass ratio as noted, with U =0.3 and L=0.03.

ACKNOWLEDGEMENTS

The findings, procedures, and opinions expressed in this

paper are the sole responsibility of the author, and do not

necessarily represent the practice of his employer, CH2M

HILL.

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(1) (2) (3) (4)

(1) µ Ω xU (cm) xL (cm)

(2) 0.5 0.6429 12.32 6.24

(3) 0.5 0.7359 12.80 6.11

(4)

(4)(4)

0.5 0.8165 13.46 6.03

(5) 0.5 1.0000 15.31 8.77

(6) 0.5 1.4142 16.75 12.39