A Generalized Translational-Rotational Tuned Mass Damper ... · a translational mass, translational spring and translational viscous damper attached to a vibrating structure to reduce

A Generalized Translational-Rotational Tuned Mass Damper (T-R TMD) system for passive control of vibrations in structures

E. A. Mashaly1, M. H. El-katt1, A. I. M. AL-Janabi2

& I. M. Abubakar1 1Structural Engineering Department, Alexandria University, Egypt 2Civil Engineering Department, Altahadi University, Libya

Abstract

This paper proposes and studies a generalized tuned mass damper that uses a translational spring damper attached to a rotational spring damper as a means for passive energy dissipation. To obtain the optimum design parameters of T-R TMD attached to an SDOF structure, two methods are used. In the first method, the Den Hartog model is used where the optimum design parameters are defined as those parameters that minimize the maximum displacement of the main structure when subjected to harmonic excitation. In the second method, the Sadek model is used where the optimum design parameters are defined as the parameters that would result in equal and large modal damping in the first two modes of vibration without considering the type of excitation. Numerical computer procedures are developed to obtain the optimum design parameters for the both methods. Comparative studies are made to see the effect of the various T-R TMD properties on the optimum damping ratio and the optimum frequency ratio. Simplified design equations for the optimum parameters of the T-R TMD system are also suggested for each method. Finally, SDOF and MDOF framed structures are selected to investigate the effectiveness of the proposed T-R TMD system in reducing structural response when subjected to different earthquake loadings. Keywords: earthquake vibrations, energy dissipation, passive control, tuned mass dampers, seismic design.

© 2005 WIT Press WIT Transactions on The Built Environment, Vol 81, www.witpress.com, ISSN 1743-3509 (on-line)

Earthquake Resistant Engineering Structures V 661

1 Introduction

Improving of structural resistance in seismically active regions generally involve increasing the capability of the structure to absorb and dissipate energy imparted to it during an earthquake. This can be done by various techniques such as modifying rigidities, masses, damping, or shape, and by providing passive or active counter forces [1]. Several passive and active energy-dissipating mechanisms have been proposed and tested as alternative means for vibration control. Among these vibration control techniques the conventional tuned mass damper (TMD) system which is a passive energy absorbing device consisting of a translational mass, translational spring and translational viscous damper attached to a vibrating structure to reduce undesirable vibrations [2,3]. Tuned mass damper (TMD) system has received significant attention due to its simplicity and its possibility to be installed at existing buildings. There has been a considerable amount of research done on how to design the best TMD system for use in passive control of structures subjected to dynamic excitation forces such as winds and earthquakes. One of the earliest such researches was carried out by Den Hartog [4]. In that study, the optimum design parameters of TMD system are defined as those parameters which minimize the max. displacement of the structure when subjected to harmonic excitation, however it was extensively used for seismic applications. Other studies on the applicability of TMD systems for seismic applications were carried out by Villaverde [5,6] where it was found that a TMD system will perform best when the first two complex modes of vibration of the combined structure and TMD have approximately the same damping ratio as the average of the damping ratios of the structure and the TMD. An improvement to that work was presented by Sadek et al. [7]. In that study, the method used to obtain the optimum parameters of (TMD) is to select, for a given mass ratio, the tuning frequency and damping ratios that would result in equal and large modal damping in the first two modes without considering the type of excitation. In the present work a generalized translational – rotational tuned mass damper (T-R TMD) system is suggested as alternative means for the conventional TMD system. The methods proposed by Den Hartog [4] and Sadek et al. [7] for estimating the properties of conventional TMD is generalized to cover the T-R TMD proposed in this work. The effect of the various properties of the proposed system and its effectiveness in reducing structural response are investigated.

2 Theory of Translational-Rotational TMD system

2.1 T-R TMD model attached to a SDOF system

Figure 1 shows a T-R TMD connected to a single degree of freedom structure (with stiffness k1, damping c1 and mass m1). The T-R TMD consists of a rotational spring-damper (with a rotational stiffness kθ and a rotational damping cθ) and a translational spring-damper (with translational stiffness k2 and translational damping c2). The mass of the T-R TMD m2 is attached at the end of


662 Earthquake Resistant Engineering Structures V

the translational spring. As shown in fig. 2, the relationship between the displacement x2 at the top end of the TMD system, and the displacement x1 at the connection point with the structure (using stiffness properties of the T-R TMD system) can be written as

xhxx ∆++= .12 θ (1)

Figure 1: T-R TMD attached to SDOF structure.

Figure 2: Relation between ),,( 21 θxx .

m1



However,θ and x∆ can be eliminated from the above equation using the stiffness properties of the translational spring k2 and rotational spring θk as follows:

hk.kxx.k

hk,

h.k

hMF But,xkF

22

..2

θ∆∆

θθ∆ θθθ =⇒=∴===

Substituting in Equation (1)x∆

( )

2 1 1 22 2

2 12 22 2

. . .. ... ..2 1 2 12 1

. 1 .

Putting and 1 (1 ) ,then (1 ). .

( ) ( ) ( ),(1 ). (1 ). (1 ).

k kx x h x hk h k h

k k x x hk h k h

x x x x x xandh h h

θ θ

θ θ

θθ θ

λ λ λ θ

θ θ θλ λ λ

= + + = + +

= + = + − = +

− − −∴ = = =

+ + +

2.2 Equation of motion for T-R TMD system

The equation of motion for the SDOF system with the T-R TMD under general dynamic excitations F1(t) and F2(t) can be formulated by considering the equilibrium study of the system as follows:

For mass m1: ∑ = 0Fx

( ) ( )

( )

.. .

1 11 2 2 1 2 2 1 1 1 1 1

.. .

1 11 2 1 1 2 2 2 1 2

.

22 12

.

( ) ( ) ( )1 1 1 (1 )

( ) (2)1 (1 )

cm x k x x h c x x h k x c x F th

cm x k k x k x c c xh

cc x F th

θ

θ

θ

θθ θ

λ λ λλ λ λ λ

λλ λ

− − − − − − − + + =

+ + + − + + + + + + +

+ − − = + +

For mass m2: ∑ = 0Fx

( ) ( )

( )

... . . .

2 2 12 2 2 1 2 2

.. .

2 12 1 2 2 2 2 2

.

2 2 22

.

( ) ( ) ( )1 1 1 (1 )

( ) (3)1 (1 )

cm x k x x h c x x h F th

cm x x k x k x ch

cx c F th

θ

θ

θ

θθ θ

λ λ λλ λ λ λ

λλ λ

+ − − + + − − =

− + + − − + + + +

+ + = + +



The above two equations can be written in matrix form as:

.. .1

1 11 2 2

.. .2 2

2 22

1 2 2

2 2

01 1

1 10

1 1

1 1

mx xc c c

c cx xm

k k k

k k

λ ρ λ ρλ λ

λ ρ λ ρλ λ

λ λλ λ

λ λλ λ

+ + + − + + + + + − + +

+ − + + + − + +

1 1

2 2

(4 )

x F

x F

=

where, 22hccθρ = (ratio of rotational damping to translational damping).

On the other hand, the state space matrix [A] for the above two equations can be written as follows using the state space variables concept:

[A]=

1 2 1 22 2

1 1 1 1

2 2 2 2

2 2 2 2

0 0 1 00 0 0 1

1 11 1

1 1 1 1

k k c ck c

m m m m

k k c c

m m m m

λ λ ρλ λ ρλ λλ λ

λ λ λ ρ λ ρλ λ λ λ

+ + − + − + + + + + +

+ + + + + + − −

(5)

2.3 Optimum parameters of the T-R TMD using Den Hartog model

In this model it is assumed that the structure is subjected to harmonic excitation, thus the load vector can be represented as follows:

{ }

=0

tioeFF

ω where ω is the exciting force frequency.

Using complex numbers technique and assuming a response in the form of:

{ }

=

= ti

ti

eXeX

txtx

x ω

ω

2

1

2

1)()(

then { } { }

−=

=

=

= ti

ti

ti

ti

eXeX

txtx

xeXeXi

txtx

x ω

ω

ω

ωωω

2

12

2

1

2

1

2

1)()(

,)()(



The equation of motion can be written in matrix form as follows:

++

++

++

−++

+−

++

+++

++

++−

)1

(k )1

(- )1

()1

(-

)1

()1

(- )1

())1

((

2222

22

22212112

λλ

λρλωω

λλ

λρλω

λλ

λρλω

λλ

λρλωω

cimkci

kcikkccim

=

02

1 oF

x

x

(6)

Then, it can be shown that the amplitude of structure displacement (X1) is given by:

22

22

1

01 .

dcba

kF

X+

+= (7)

where:

−

+= 22

1rqa

λλ

++

=λλρξ

12 2rqb

( )

++

+

++

+

+++−=

λλ

λλ

λλρξξ

λλµ

1114

111 2

21224 qqqrrc

+

++

+

+

++

+−

++

++

+=

λλ

λλρξ

λλµ

λλρξξ

λλξ

λλρξ

λλ

11

1122

11.

12

2

2132

12

q

qrrqrqd

1

11 m

k=ω (Natural frequency of the structure)

11

11 2 m

cω

ξ = (Damping ratio of the structure)



22

22 2 m

cω

ξ = (Translational damping ratio of T-R TMD)

1

2

ωω

=q (Ratio of T-R TMD frequency to structure frequency)

1

2

mm

=µ (Ratio of the mass of the T-R TMD to the mass of the structure)

1ωω

=r (Ratio of exciting force frequency to the frequency of structure)

In the Den Hartog model, the optimum design parameters of T-R TMD are defined as those parameters that minimize the maximum displacement of the structure when subjected to harmonic excitation. To obtain the optimum T-R TMD properties qopt and ξ2opt for given values of µ, ξ1, λ and ρ, a special numerical optimization algorithm using Matlab Software is developed. This algorithm consists of nested loops that search for the numerical minimum value of maximum amplitude occurring in the system for the given parameters.

2.4 Optimum parameters of the T-R TMD using Sadek model

The system matrix (Equation 5) can be written in terms of non-dimensional parameters as follows: [ ]

++

−

++

+

−

+

++

+

++

+−

+

+

+

+−

=

λρλξω

λρλξω

λλω

λλω

λρλξµω

λρλξµξω

λλωµ

λλµω

12

12

11

12

12

111

10000100

212122

122

1

2121121

221

qqqq

qqqq

A

(8) The eigenvalue problem [ ] { }[ ]IA γ− results in the following fourth-order equation:

041

3

2

12

3

11

4

1=+

+

+

+

aaaa

ωγ

ωγ

ωγ

ωγ (9)



where 1 2 3 4, ,a a a and a can be obtained by expanding the [ ] [ ]{ }γIA − determinate. The solution of the above equations is in complex conjugate pairs with the following complex eigenvalues:

3,1,12

1, =−±=+ ni nnnnnn ξωζωγ (10)

where nγ is the nth eigenvalue, nω and nξ are the natural frequency and damping ratio of the whole system (structure and T-R TMD) in the nth mode and i is the unit imaginary number ( )1−=i . Sadek et. al. [7] showed that for a TMD to be effective, the damping ratios in the two complex modes of vibration γ1 and γ2 should be approximately equal, i.e 31 ξξ ≅ and the tuning frequencies

in these two modes are also approximately equal, i.e. 31 ωω ≈ . The main advantage in this procedure is that the T-R TMD properties are obtained without considering the type of excitation force and they can be considered as general for any type of loading. To obtain the optimum T-R TMD properties qopt and ξ2opt for given values of µ,ξ 1, λ and ρ, a numerical procedure is developed using MATLAB software. This procedure utilizes the optimization toolbox available in MATLAB.

3 Results and discussions

3.1 Effect of λ and ρ on T-R TMD behaviour

In this case study the effect of the parameters λ and ρ is investigated. Figures 3,4 show the variation of the optimum damping ratio ξ2opt with T-R TMD mass ratio µ for various values of λ and ρ. The results for both Den Hartog and Sadek models are shown in these figures. The figures show that the Sadek model gives higher value of ξ2opt than the Den Hartag model. The figures also show that for both values of ρ (1 and 100) ξ2opt increases with the increase of the value of λ. However, for very large value of λ (which represents stiff rotational spring), the increase in ξ2opt is negligible and curves will be similar and coincides with the results of conventional TMD. This is true for both Den Hartog and Sadek T-R TMD models. Comparison of figure 3 with figure 4 shows that for the same value of λ, the value of ξ2opt for high value of ρ (high rotational damping) will be less than value of ξ2opt for low value of ρ (low rotational damping) as expected. Figure 5 shows that the parameter ρ has no effect on the optimum frequency ratio qopt. However, the figure shows that qopt decreases with the increase of λ. But for large values of λ the decrease in qopt is negligible and all curves will be similar and coincide with the results of conventional TMD. The figure also shows that for ξ1=0 (no structure damping) both models (Den Hartog and Sadek) yield the same results for qopt.



0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.02 0.04 0.06 0.08 0.1Mass Ratio µ

λ=1E6

ρ=1.0

Dam

ping

ratio

ξ 2

opt

λ=1E6

λ=10

λ=10 λ=1.0

λ=1.0

λ=0.1

λ=0.1

Figure 3: Variation of damping ratio ξ 2opt with mass ratio µ for various values of λ (ρ=1.0).

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.02 0.04 0.06 0.08 0.1Mass Ratio µ

λ=1E6ρ=100

Dam

ping

rat

io ξ

2op

t

λ=1E6

λ=1000

λ=1000λ=100

λ=100

λ=1λ=1

Figure 4: Variation of damping ratio ξ 2opt with mass ratio µ for various

values of λ (ρ=100).



0.90

1.00

1.10

1.20

1.30

1.40

1.50

0 0.02 0.04 0.06 0.08 0.1Mass Ratio µ

Freq

uenc

y ra

tio q

opt

λ=1

λ=1E6 λ=100λ=1

For all values of ρ

Den Hartog and Sadek Models

Figure 5: Variation of frequency ratio qopt with mass ratio µ for various

values of λ (for all values of ρ) and both Den Hartog and Sadek models.

3.2 Suggested design equations

Extensive numerical results for studying the behaviour of T-R TMD model have been carried out by Abubakar [8]. However, for practical design purposes, it may be more convenient to present the optimum T-R TMD parameters by simple equations rather than graphical results. Using trial and error curve fitting procedure, it was found that the following equations give close approximations to the values of qopt and ξ2opt for Den Harog model:

( ) ( )( )( )λ

1λ*µ14

µ9ζ1ξµ

q 11opt+

++−

+

= 11

1 (11)

( ) ( ) ( )( )

( )ρλλλ

µζζ

µµξ

++

+++

+=

11

161518

3

1

12 **

*opt (12)

and for Sadek model:

11 11 .

1 1optq µ λξµ µ λ +

= − + + (13)

( )( )

12opt

1.

1 1λ λµ ξξ

µ µ λ ρ+

= + + + + (14)



Figures 6 and 7 show the close agreement between the result of the above suggested equations with numerical results obtained for ρ= λ = 10 and ξ1 =0.03 as an example.

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0 0.02 0.04 0.06 0.08 0.1

Eq.14 Sadek Model Eq. 12 Denhartog Model Numerical Results

Mass Ratio µ

Dam

ping

ratio

ξ 2o

pt

Figure 6: Variation of damping ratio ξ 2opt with mass ratio µ for ξ1= 0.03 and ρ= λ =10.

0.94

0.96

0.98

1

1.02

1.04

1.06

0 0.02 0.04 0.06 0.08 0.1

Eq. 13 Sadek Model Eq. 11 Den Hartog Model Numerical Results

Mass Ratio µ

Freq

uenc

y ra

tio q

opt

Figure 7: Variation of frequency ratio qopt with mass ratio µ for ξ1=0.03 , and ρ= λ= 10 .



3.3 Effectiveness of the T-R TMD

In this case study two framed structures are examined to investigate the effectiveness of the proposed T-R TMD system in reducing structural response due to seismic excitations. The dimensions and properties of both frames are shown in Figure 8. The first frame represents a SDOF system subjected to El Centro 1940 earthquake. The second frame investigated is a 5-storey, two bay frame (i.e. MDOF system) subjected to El Centro and Parkfield California earthquakes, with the peak ground acceleration scaled to 0.30g. The analysis results with and without T-R TMDs for max. relative displacement, and max. base column moments are shown in Tables 1 and 2. The results for both frames show that the conventional TMD and T-R TMD give similar magnitude of reduction in displacements and moments. For example, this reduction is in the range of (35% - 45 %) for El Centro earthquake and (12% - 24%) for Parkfield California earthquake. It is observed that with the use of T-R TMD, similar efficiency in reduction of structural response can be obtained using different values of λ and ρ. This may have great practical importance in designing TMD systems by providing various alternative values for the properties of T-R TMD system.

Five storey R.C. frame. Single storey R.C. frame.

Figure 8: Dimensions and properties of investigated frames.

Mass per floor = 50 ton T(period) without T-R TMD= 0.925s Height per floor= 3.5 m Span = 5 m

Mass per floor = 15 ton T(period) without T-R TMD= 0.3099s Beam very stiff. Height of floor = 4 m Span = 6 m



Tabl

e 1:

Res

pons

e of

the

sing

le st

orey

fram

e ( µ

=0.0

5, h

=1.5

m, ω

1=20

.274

rad/

sec,

ξ1=

0.00

).

El C

entro

194

0 D

escr

iptio

n of

TM

D

mod

el

λ ρ

q opt

ξ 2

opt

k 2

kN/m

k θ

kN

.m

c2

kN.se

c/m

c θ

kN.m

/sec

x max

m

M

c max

kN

.m

No.

TM

D

- -

- -

- -

- -

0.01

167

72.0

7 C

onv.

Den

Har

tog

- -

0.95

24

0.13

3627

9.64

-

3.87

0 -

0.00

670

41.4

0 T-

R D

en H

arto

g 1.

E+10

0 0.

9524

0.

1336

279.

64

6.29

E+12

3.

870

0.00

0 0.

0067

041

.37

T-R

Den

Har

tog

100

100

0.95

71

0.06

7128

2.43

63

547.

68

1.95

5 43

9.78

0.

0063

739

.33

T-R

Den

Har

tog

1 1

1.34

69

0.09

4555

9.28

12

58.3

7 3.

870

8.70

9 0.

0063

739

.33

T-R

Den

Har

tog

1 0

1.34

69

0.18

9055

9.28

12

58.3

7 7.

741

0.00

0 0.

0063

739

.33

Con

v. S

adek

-

- 0.

9524

0.

2182

279.

64

- 6.

320

- 0.

0071

43

.97

T-R

Sad

ek

1.E+

100

0.95

24

0.21

8227

9.64

6.

29E+

12

6.32

0 0.

000

0.00

71

44.0

2 T-

R S

adek

10

0 10

00.

9571

0.

1097

282.

43

6354

7.68

3.

192

718.

163

0.00

66

40.6

9 T-

R S

adek

1

1 1.

3469

0.

1543

559.

28

1258

.37

6.32

0 14

.221

0.

0066

40

.71

T-R

Sad

ek

1 0

1.34

69

0.30

8655

9.28

12

58.3

7 12

.641

0.

000

0.00

66

40.7

3

Tabl

e 2:

Res

pons

e of

the

5 st

orey

fram

e ( µ

=0.0

5, h

=1.5

m, ω

1= 6

.789

rad/

sec,

ξ1=

0.05

).

El C

entro

194

0 Pa

rkfie

ld

Des

crip

tion

of T

MD

m

odel

λ

ρ q o

pt

ξ 2op

t k 2

kN/m

k θ

kN.m

c2

kN

.sec/

m

c θ

kN.m

/sec

x m

ax

m

Mc m

ax

kN.m

x m

ax

m

Mc m

ax

kN.m

N

o. T

MD

-

- -

- -

- -

- 0.

0427

434

5.00

0.

0152

17

6.96

C

onv.

Den

Har

tog

- -

0.93

60

0.14

1050

5.29

8 -

22.4

0 -

0.02

594

227.

37

0.01

26

137.

91

T-R

Den

Har

tog

1.E+

100

0.93

60

0.14

1050

5.29

8 1.

14E+

1322

.40

0.00

0 0.

0256

722

4.24

0.

0124

13

5.12

T-

R D

en H

arto

g 10

0 10

00.

9407

0.

0708

510.

351

1148

28.9

11.3

1 25

45.7

63

0.02

468

215.

14

0.01

29

133.

02

T-R

Den

Har

tog

1 1

1.32

37

0.09

9710

10.5

9522

73.8

4 22

.40

50.4

11

0.02

468

215.

12

0.01

29

133.

00

T-R

Den

Har

tog

1 0

1.32

37

0.19

9310

10.5

9522

73.8

4 44

.81

0.00

0 0.

0246

821

5.10

0.

0129

13

3.02

C

onv.

Sad

ek

- -

0.94

20

0.26

5851

1.76

94-

42.5

2 -

0.02

74

249.

9 0.

0134

14

8.71

T-

R S

adek

1.

E+10

0 0.

9420

0.

2658

511.

7694

1.15

E+13

42.5

2 0

0.02

713

243.

95

0.01

33

146.

07

T-R

Sad

ek

100

100

0.94

67

0.13

3651

6.88

7111

6299

.621

.47

4831

.82

0.02

553

223.

21

0.01

24

135.

22

T-R

Sad

ek

1 1

1.33

22

0.18

8010

23.5

3823

02.9

6 42

.52

95.6

7 0.

0255

22

3.15

0.

0124

13

5.44

T-

R S

adek

1

0 1.

3322

0.

3760

1023

.5

2302

.96

85.0

5 0

0.02

549

223.

08

0.01

24

135.

1



4 Conclusions

A generalized translational-rotational mass damper system has been proposed and investigated. In deriving the optimum parameters of the T-R TMD system, Den Hartog and Sadek conventional models are used. Numerical optimization algorithms for computing the optimum parameters of T-R TMD attached to SDOF structure has been developed. Numerical case studies have been carried out to investigate the effect of the properties of the proposed system and its efficiency in reducing structural response. The results have shown that the T-R TMD system may be an efficient alternative to the conventional TMD model. As expected, it was found that the present study model reduces to conventional model in case of very high value of stiffness parameter λ and for any value of damping parameter ρ. Finally, the optimum design parameters of T-R TMD for SDOF structure are presented in equation forms suitable for practical design purposes.

References

[1] Housner, G.W., Bergman, L.A., Caughey, T.K., Chassiakos, A.G., Claus, R.O., Masri, S.F., Skelton, R.E., Soong, T.T., Spencer, B.F., Yao, J.T.P. , Structural control: past, present and future. Journal of Engineering Mechanics, ASCE, 123 (9), pp. 897 – 924, 1997.

[2] McNamara, R. J., Tuned mass dampers for building. Journal of Structural Division, ASCE, 103(9), pp.1785 – 1798, 1977.

[3] Luft, R. W., Optimal tuned mass dampers for building, Journal of Structural Division, ASCE, 105(12), pp. 2766 – 2772, 1979.

[4] Den Hartog, J. P., Mechanical vibrations, McGraw – Hill, New York,1956. [5] Villaverde, R., Reduction in seismic response with heavily – damped

vibration absorbers. Earthquake Engineering and Structural Dynamics, 13, pp. 33-42,1985.

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[7] Sadek, F., Mohraz, B., Taylor, A. W., Chung, R. M., A method of estimating the parameters of tuned mass dampers for seismic applications. Earthquake Engineering and Structural Dynamics, 26, pp. 617 – 635, 1997.

[8] Abubakar, I. M., Control of R.C. buildings vibrations using tuned mass dampers. A thesis to be submitted in partial fulfilment of the requirements for the degree of doctor of philosophy in structural engineering. Alexandria University, Egypt.



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A Generalized Translational-Rotational Tuned Mass Damper ... · a translational mass, translational spring and translational viscous damper attached to a vibrating structure to reduce