cavacece - identification of modal damping ratios.pdf

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Identification of Modal Damping Ratios of

Four-Flue Chimney of a Thermo-Electrical Plant

using Pseudo-inverse Matrix Method

M. Cavacece1, P.P. Valentini2, L. Vita2

1Department of Structural Engineering - DIMSAT, University of Cassino, 03043

Cassino, Italy

2Department of Mechanical Engineering, University of Rome, Tor Vergata,

00133 Rome, Italy

Abstract

Some computational issues related to the identification of modal pa-

rameters of structures are presented in this paper. Optimal estimation

of modal parameters often requires the solution of overdetermined linear

system of equations. Hence, the computation of a pseudoinverse matrix

is involved. In this paper the numerical performances of different algo-

rithms for MoorePenrose pseudoinverse computation have been tested for

modal analysis of four-flue chimney of a thermo-electrical plant. The com-

putational scheme herein adopted for parameters identification is based on

well known modal properties and has a fast rate of convergence to solution.

The computation of the Rayleigh damping coefficients and is an impor-

tant step in the area of the modal superposition technique. The proposedapproach can predict accurately damping ratios and all the eigenvectors

without evaluating mass and stiffness matrices.

Keywords: Modal analysis; Frequency response; pseudoinverse matrix method;Damped system.

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1 Introduction

Recently analytical and experimental modal analysis, or vibration system iden-tification, has become an increasingly popular technique. Developments in mea-suments have facilitated the acquisition of reliable experimental data. Theseprocedures lead to extraction of the modal properties of a test structure [1, 2, 3,5, 6, 7, 9, 10, 11, 12, 15, 16, 17]. The building response to dynamic excitation,such as wind or earthquake, can be evaluated using techniques, such as the di-

rect integration of equations of motion to modal analysis or the use of responsespectra. In each case the availability of the stiffness, inertia and dissipative pa-rameters is very important.The objective of this paper is to test different methods for the numerical com-putation of pseudoinverse matrix and to discuss their applicability in structuraldynamics applications. The generalized inverse is a well known tool in modernlinear matrix theory. It is usually used to solve a set of overdetermined or unde-termined simultaneous equations [4] through the application of the least squaresoptimization criterion. As numerical test, by means of the proposed method themodal damping ratios and the modal shape of a chimney have been estimated

assuming Rayleigh damping.In the first two sections the basic equations used for the mathematical setting ofthe identification problems are briefly deduced. The third section is dedicated toreview different approaches for computing the MoorePenrose pseudoinverse ma-trix. In the fourth section the numerical results for different cases are discussed.

2 Fundamental equations of structural dynam-

ics

The equations of motion [14] of a linear viscous damped multi degreeoffreedomsystem in matrix form are

[m] {y} + [c] {y} + [k] {y} = {F (t)} . (1)

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By means of Laplace transformation of the response canonical equations of modalmotion [8], assuming zero initial conditions, one obtains the transfer function

H(s) =

Nrr=1

Hr (XiXk)r =

Nrr=1

(XiXk)r[s2 + 2rrs + 2r ]

(2)

between the excitation at the kth location and response at the ith location.Assuming Rayleigh damping, the damping matrix could be expressed as a linear

combination of the mass and stiffness matrices[c] = [m] + [k] (3)

This type of damping is known as proportional damping. This condition is suffi-cient to decouple the equations of motion (1), allowing the use of modal analysis.By means of equation (3) one obtains

+ 2r = 2rr . (4)

From

dr = r

1 2r (5)

equation (4) becomes + 2r = 2

2r

2Er,q

(6)

where is dr = Er,q with Er,q experimental circular frequency of dampedoscillation, obtained from the qth measurement and rth vibration mode.

3 Identification of modal parameters

The equations deduced in the previous sections constitute the main blocks of the

method for the identification of modal parameters (Fig.2). In this section theiruse for such purpose is presented.Since the mathematical approach adopted is essentially a least squares solution ofa redundant system of linear equations, from our treatment nonlinear equationsare excluded.

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3.1 Rayleigh damping

From equation (6), the following system could be obtained

1 211 21

...1 21...1 2r1 2r...1 2r...1 2Nr

1 2Nr...1 2Nr

=

2

21 2E1,1

2

21 2E1,2

...

2

21 2E1,Nq...

2

22 2Er,1

2

22 2Er,2

...

2

22 2Er,Nq

...

2

23 2ENr,1

2

23 2ENr,2

...

2

23 2ENr,Nq

(7)

for q = 1, 2, . . . , N q and mode r = 1, 2, . . . , N r. By applying least-squares op-timality criterion, the redundant Nq Nr equations system (7) is solved withrespect to constants and . In this phase computation of the Moore-Penrosepseudoinverse matrix has its main role.

3.2 Transmissibility function

From equation (2), being 1 = (2f1)2, 2 = (2f2)

2, . . . , r = (2fr)

2, . . . , Nr =(2fNr)

2 the square of the eigenvalues, the real component [Hik (f)] of the trans-

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missibility function at response location and excitation location k of chimneyis

[Hik (f)] =

(XiXk)11

1

ffd1

22

1

ffd1

22+

21f

fd1

2 +(XiXk)2

2

1

ffd2

22

1

ffd2

22+

22f

fd2

2 +

. . . +

(XiXk)rr

1

ffdr

22

1

ffdr

22+

2rf

fdr

2 + . . . +(

XiX

k)NrNr

1

ffdNr

22

1

f

fdNr

22+

2Nr

ffdNr

2 (8)

the imaginary component [Hik (f)] of the transmissibility function at responselocation i and excitation location k of chimney is

[Hik (f)] =

(XiXk)11

21

ffd1

1 f

fd122

+

21f

fd12 +

(XiXk)22

22

ffd2

1 f

fd222

+

22f

fd22 + . . . +

+

(XiXk)rNr

2Nr

ffdr

1

ffdr

22+

2rf

fdr

2 + . . . +(XiXk)Nr

Nr

2Nr

ffdNr

1

f

fdNr

22+

2Nr

ffdNr

2(9)

Then, the modulus of the transmissibility function is

Hik (f) =

[Hik (f)]

2

+ [Hik (f)]

2

(10)

From equations (8) and (9) with f = f1, f2, ..., fNp, the following system could beobtained

[A] {X} = {b} (11)

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with

[A] =

11

1

f1

fd1

22

1

f1

fd1

22

+

21 f1

fd1

2. . .

1Nr

1

f1

fdNr

22

1

f1

fdNr

22

+

2Nr ffdNr

2

11

21

f2fd1

1

f2

fd1

22

+

21 f2

fd1

2. . .

1Nr

2Nr

f2fdNr

1

f2

fdNr

22

+

2Nr f2fdNr

2

......

...

11

21 fp

fd1

1 fp

fd1

22

+21

fp

fd1

2. . .

1Nr

2Nr fpfdNr

1 fNp

fdNr

22

+2Nr

fp

fdNr

2

......

...

11

21 fNp

fd1

1

fNpfd1

22

+

21 fNp

fd1

2. . .

1Nr

2Nr fNpfdNr

1

fNpfdNr

22

+

2Nr fNpfdNr

2

(12)

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{b} =

Hik

f1

Hik

f2

...

Hik

fp

...

Hik

fNp

, {X} =

(XiXk)1(XiXk)2

...(XiXk)r

...(XiXk)Nr

(13)

By applying least-squares optimality criterion, the redundant Np Nr equationssystem (11) is solved with respect to constants (XiXk)1 , (XiXk)2, . . ., (XiXk)r,. . ., (XiXk)Nr . In this phase computation of the Moore-Penrose pseudoinversematrix has its main role.

4 The Moore-Penrose pseudoinverse matrix

This section, for completeness, summarizes the main properties of the Moore-Penrose pseudoinverse matrix and the steps of the different algorithms testedduring the dynamic simulations [18].

4.1 Definitions

The main properties of the MoorePenrose pseudoinverse matrix [A]+ of a matrix[A] are:

[A] [A]+

T

= [A] [A]+

[A]+ [A]T

= [A]+ [A]

[A] [A]+ [A] = [A]

[A]+ [A] [A]+ = [A]+

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When [A] is a square matrix with full rank, then its pseudoinverse coincide withthe inverse. The MoorePenrose pseudoinverse matrix is associated with the leastsquares solution of the linear system of equations

[A] {x} = {b} (14)

where the number m of equations is not equal to the number n of unknownsand [A] does not necessarily has full rank. In particular, the following cases are

distinguished:Overdetermined system of equations (m > n)By requiring that [B] is

h [A] {x} = {b}22 (15)

is a minimum, one obtains

[A]T [A] {x} = [A]T {b} (16)

Therefore, the solution of (1) can be stated as

{x} = [A]T {b} (17)

is the right pseudoinverse matrix.Undetermined system of equations (m < n) The solution is obtained impos-ing the minimum of the Euclidean norm

g x22 (18)

with {x} subjected to (14). Thus, introduced the new objective function,

g g + {x}T ([A] {x} {b}) (19)

the solution is achieved solving the systemI AT

AT 0

x

(20)

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or

{x} = [A]+ {b} (21)

where

[A]+ = [A]T

[A] [A]T1

(22)

is the left pseudoinverse matrix.

4.2 The least squares method

Since there is abundance of software procedures for computing the least squaressolution of a system of algebraic equations

[A] {x} = {b} (23)

the computation of the pseudoinverse can be reduced to such solution. These pro-cedures are often based on QR decomposition by means of Householder reflections

or GS orthogonalization. Let

{b1} =

1 0 0 . . . 0T

,

{b2} =

0 1 0 . . . 0T

,

{b3} =

0 0 1 . . . 0T

,

... (24)

{bm} =

0 0 0 . . . 1T

, (25)

The procedure differs according to the dimensions of [A].Case m > n

1. Solve m times the system

[A]T [A] {x} = [A]T {b} (26)

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2. From the m solutions {x1}, {x2} , . . . , {xm} one can form the pseudoinversematrix

[A]+ =

{x1} {x2} {x2} . . . {xm}

(27)

Case n < m

1. Solve m times the system I AT

AT 0

x

(28)

Also in this case the pseudoinverse is given by equation (27). The squarematrices in (26) and (28) are singular or ill conditioned. Thus their solutionrequires special care. As mentioned, an appropriate way to solve leastsquares problems is by means of GS orthogonalization or Householder QRfactorization.

4.3 Grevilles method1. Decompose the matrix [A]

mninto row vectors {ai}, (i = 1, 2, . . . , m)

[A]+ =

aT1 aT2 a

T3 . . . a

Tm

(29)

2. Let matrix

[Ai]in =

Ai1

ai

(30)

with [A1]1m = {a1}1m.

3. For i = 2, . . . , m compute the matrices [A]+ as

[Ai]+ni

= [Ai1]+ {bi}

T {di} | {bi}T (31)

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where

{di}+1(i1) = {ai} [Ai1]

+

{ci}1n = {ai} {di} [Ai1]

{bi}1n ={ci}

{ci} {ci}T

(ci = 0)

{bi}1

n

={di} A

+i1

T

1 + {di} {di}T(ci = 0)

[A1]+ =

{a1}

{a1} {a1}T

(a1 = 0)

[A1]+ = ai (a1 = 0)

(32)

4. After m repetitions [Am]+ gives the pseudoinverse [Am]

+mn of matrix [A]

5 Modal testing of chimneyThe height of the studied chimney, w.r.t. ground placed at 4 m above sea level,is 200 m (Fig.1). The building is composed of the following parts:

An external shell, made of reinforced concrete, with an height of 195 .8 mand with diameter of section variable from 23.0 m (at the bottom) to 19.1m. The foundation is a cylindrical structure (with a diameter of 55 .0 m and3.0 m thickness) and a series of poles (with a diameter of 1 m each) drilledin the ground to a depth of 56.0 m above sea level.

Four internal flues, with a diameter of 5.0 m each, made of acid-repellent

and insulated bricks. Each flue is divided into 10 portions of 17.0 m, exceptthe highest one, which is substantially shorter, and each one is placed on ahorizontal platform made of reinforced concrete that distributes the weightto external shell. At the top of chimney there is a cover platform at a heightof 197.0 m.

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Figure 1: Chimney

The material of the chimney is isotropic with Youngs modulus E = 36 GPa,density = 1500 kg/m3 and Poisson coefficient = 0.2.Experimental tests for measurement of the dynamic response of the chimneyhave been performed using a shacker. The shaker used is a lowfrequencyrangemachine (ISMES BF50/5), with the following features:

maximum amplitude of generated force: 50 KN;

maximum frequency of the generated force: 5 Hz;

structural mass (without inertial masses): 1840 kg.

The shaker has been placed at the top of chimney. At a position of 197.0 ma.s.l. the shaker excited the chimney by means of a sinusoidal load with variable

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Experimental response in time domain

Discrete Fourier Transform

Circular frequencies

of damped oscillation

Rayleigh coefficients damping

Modal shapes

Experimental transmissibility function

Natural circular frequencies

Finite Element Method

Moore - Penrose

pseudoinverse matrix

Transmissibility function by proposed method

Moore - Penrose pseudoinverse matrix

Figure 2: Flow chart of the method for the identification of modal parameters

frequency, along two orthogonal directions [13]. The response of the structurehas been monitored at 30 different locations along the chimney. In particular,the 22 piezoelectric accelerometers and 8 eightdynamical seismometers have thefollowing features:

Accelerometer type: Endeveco mod. 2262C-25, frequency response: linearfrom 0 up to 500 Hz, sensitivity: 9.81 cm/s/V;

Seismometer type: Teledyne mod.Geotech S-13, frequency response: linearfrom 1 up to 50 Hz, sensitivity: 0.00170 cm/s/V.

Experimental setup allows us to measure circular frequencies of damped oscilla-tion and transmissibility functions. The measured circular frequencies of dampedoscillation are summarized in table 1. Natural circular frequencies r are obtained

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by finite element method [13] (Table 2). The model of chimney has been meshedusing four-node standard shell elements with 5 d.o.f. per node. Referring to table3, we can calculate

2r Er,q , right side of system (7) with Nr = Nq = 3.

The transmissibility functions have been computed for the nodal displacement atmeasurement locations. The experimental analysis leads to the values of the realpart [H(f)] and of the imaginary part [H(f)] of the frequency response atthe top of chimney. Experimental results of the frequency response at the top ofchimney have been evaluated referring to the values of frequency of Table 4.

Mode rth Er,1 [rad/s] Er,2 [rad/s] Er,3 [rad/s]1 1.2 1.213 1.2382 10.04 10.034 10.0223 21.960 21.953 21.972

Table 1: Experimental circular frequencies of damped oscillation

Mode rrth [rad/s]1 1.2572 10.0533 22.054

Table 2: Natural circular frequencies

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Mode 2

2r 2Er,q

2

2r 2Er,q

2

2r 2Er,q

rth [rad/s] [rad/s] [rad/s]1 0.748 0.659 0.4352 1.022 1.235 1.5773 4.068 4.216 3.800

Table 3: Right side of system

Measure Frequency Frequency responseqth [Hz] [ms2N1]1 0.01 [H] = 0.012 0.1597 [H] = 0.253 0.194 [H] = 0.724 3.0 [H] = 0.00015 3.495 [H] = 0.1

Table 4: Experimental values of frequency response

6 Complete modal data

Lets suppose that the modal data are complete in terms of the number of mea-sured modes. For this application we have an overdetermined system of equations,where the number of equations m = 9 is greater than the number of unknownsn = 2. Referring to Table 3, system (7), with Nr = Nq = 3, provides the followingsolutions

= 0.583 , = 7.065 103 (33)

Damped oscillation frequencies and modal damping ratios are summarized intable 5.

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Mode rth dr [rad/s] r1 1.219 2.366 101

2 10.032 6.453 102

3 21.962 9.113 102

Table 5: Circular frequencies of damped oscillation and modal damping ratio

6.1 Modal shapes

Substituting modal damping ratio r (Table 5) and the ones of frequency response(Table 4) into system (11), we obtain the following solutions:

X2T1

= 0.432 ,

X2T2

= 2.746 ,

X2T3

= 3.744 (34)

at the top of chimney (i = k = T) by means of the pseudoinverse matrix (Fig.1).

Method CPU (time ratios)

Least Squares Householder 1.0Least Squares Gram-Schmidt 2.0

Grevilles Method 3.0

Table 6: CPU time

In Fig.3 there is the comparison among the transmissibility functions obtainedby proposed methodology, experimental results and FEM ones. The Fig.4 showsthe real and the imaginary part of the transmissibility fucntion deduced by thepseudoinverse matrix. Referring to Table 6, there is the comparison among theelapsed CPU time obtained by proposed methods.

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0 , 2 5 0 , 5 0 0 , 7 5 1 , 0 0 1 , 2 5 1 , 5 0 1 , 7 5 2 , 0 0 2 , 2 5 2 , 5 0 2 , 7 5 3 , 0 0

T

r

a

n

s

m

i

s

s

i

b

i

l

i

t

y

(

m

/

s

2

/

N

)

1

0

-

5

F r e q u e n c y [ H z ]

E x p e r i m e n t a l

F E M

P r o p o s e d M e t h o d

Figure 3: Transmissibility function

0 , 0 0 , 5 1 , 0 1 , 5 2 , 0 2 , 5 3 , 0

- 0 , 6

- 0 , 5

- 0 , 4

- 0 , 3

- 0 , 2

- 0 , 1

T

r

a

n

s

m

i

s

s

i

b

i

l

i

t

y

(

m

/

s

2

/

N

)

1

0

-

5

F r e q u e n c y [ H z ]

R e a l P a r t

I m a g i n a r y P a r t

Figure 4: Real and imaginary part of the transmissibility function

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7 Incomplete modal data

Lets suppose that the modal data are incomplete only in terms of the number ofmeasured modes. If there is noise, structural identification will be ill-posed. Inthis context, the word noise has wide implications, as for example: any causal orrandom factors which should not or cannot be modelled; further information isnot available; some information of the system has been lost; some vibration datacannot be analyzed; the modal data are incomplete in terms of the number of

measured modes.When the second mode of the eigensolution is omitted, we have an overdeterminedsystem of equations, where the number of equations m = 6 is greater than thenumber of unknowns n = 2. Referring to table 3, we obtain the following solutions

= 0.602 , = 7.037 103 (35)

Damped oscillation frequencies and modal damping ratios are summarized inTable 7.

Mode rth dr [rad/s] r1 1.219 2.438 101

2 10.032 6.529 102

3 21.962 9.124 102

Table 7: Circular frequencies of damped oscillation and modal damping ratio forincomplete modal data

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Mode Complete modal data Incomplete modal data errorrth r r %1 2.366 101 2.438 101 3.02 6.453 102 6.529 102 1.13 9.113 102 9.124 102 0.1

Table 8: Modal damping ratio

8 Conclusions

The generalized inverse, an elegant mathematical technique, provides only one so-lution to solve an overdetermined problem with or without noise. In this paper,the application of the generalized inverse allows us to determine modal dampingratios and modal shapes of chimney.The investigation is divided in two parts. In the first part, we determine themodal damping ratios assuming Rayleigh damping. In the second part, themodal shapes and the frequency response of displacement are computed. It is

shown that Rayleigh damping can predict accurately the response of chimney ina large frequency range, because the chimney has a symmetrical structure andthe material of chimney has linear elastic properties.The generalized inverse, very useful tool in linear matrix theory, has the followingfeatures:

1. By using an overdetermined system of equations with complete modal data,we obtain modal damping ratios. In addition, the acquired signals andthe computed results of frequency response of displacement show a goodagreement.

2. When the second mode of the eigensolution is omitted, the physical param-eters, obtained by using the generalized inverse method, are quite similar tophysical ones, obtained by using overdetermined system of equations withcomplete modal data (Table 8). Therefore, generalized inverse minimizesthe effect of noise in this mechanical system.

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3. Application of the generalized inverse in structural identification is not com-pletely successful by using a set of simultaneous equations which is unde-termined.

Finally, the generalized inverse is a means for determining the least squares solu-tion of a set of simultaneous equations which is overdetermined. We demonstratethat this method leads to reliable results for modal damping ratios of a chim-ney with negligible damping. In fact, the response of chimney obtained by thismethod show good agreement with experimental response of the chimney sub-jected to a generic wind periodic excitation and the response obtained by finiteelements method.

Acknowledgements

The authors wish to acknowledge Prof. Ettore Pennestr for his helpful adviceduring the development of this work.

References

[1] Ewins, D. J. and Gleeson, P. T. 1982. A Method for Modal Identification ofLightly Damped Structures. Journal of Sound and Vibration. 84(1):57-79.

[2] Luk, W.L. 1987. Identification of physical mass, stiffness and damping ma-trices using pseudo-inverse. 5th IMAC. 679-685.

[3] Lagomarsino, S. 1993. Forecast models for damping and vibration periodsfor duildings. Journal of Wind Engineering and Industrial Aerodynamics. 48:221 - 239.

[4] To, W. M. and Ewins, D.J. 1995. The Role of the Generalized Inverse inStructural Dynamics. Journal of Sound and Vibration. 186(2): 185-195.

[5] Watanabe, Y., Isyumov, N., Davemport, A.G. 1997. Empirical aerodynamicdamping function for tall building. Journal of Wind Engineering and Indus-trial Aerodynamics. 72: 313-321.

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[6] Mohammad, D. R., Khan, N.U. and Ramamurti, V. 1995. On the Role ofRayleigh Damping. Journal of Sound and Vibration. 185(2): 207-218.

[7] Dovstam, K. 1997. Recpetance Model based on Isotropic Damping Functionand Elastic Displacement Model. Int. J. Solids Structures. 34 (21):2733-2754.

[8] Clarence W. de Silva, 2000. Vibration Fundamentals and Practice. CRCPress: New York.

[9] Prells, U. and Friswell, M.I. 2000. A Measure of non - proportional Damping.Mechanical Systems and Signal Processing. 14(2): 125-137.

[10] Adhikari, S. and Woodhouse, J. 2001. Identification of Damping: Part 1,Viscous Damping. Journal of Sound and Vibration. 243(1): 43-61.

[11] Gaylard, M. E. 2001. Identification of Proportional and other sorts of Damp-ing Matrices using a Weighted Response - Integral Method. Mechanical Sys-tems and Signal Processing. 15(2): 245-256.

[12] Liu Kefu, Kujath Marek R., Zheng Wanping 2001. Evaluation of Damping

Non-Proportional Using Identified Modal Information. Mechanical Systemsand Signal Processing. 15(1): 227-242.

[13] Cavacece M., Valentini P.P. 2003. An Experimental And Numerical Ap-proach to Investigate the Dynamic Response Of A Four-Flue Chimney Of AThermoelectrical-Electrical Plant. The Structural Design of Tall and SpecialBuildings. 12: 283 - 291.

[14] Singiresu S. Rao 2004. Mechanical Vibrations. Pearson Education: Singa-pore.

[15] Sang-Hyun Lee, Kyung-Won Min, Jae-Seung Hwang, Jinkoo Kim 2004. Eval-uation of equivalent damping ratio of a structure with added dampers. En-gineering Structures. 26: 335-346.

[16] He, J. and Fu, Z.-F. 2004. Modal Analysis. Butterworth Heinemann: Oxford.

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[17] Trombetti T., Silvestri, S. 2006. On the modal damping ratios of shear-typestructures equipped with Rayleigh damping systems. Journal of Sound andVibration. 292: 21-58.

[18] Pennestri, E. and Cheli F. 2006. Cinematica e Dinamica dei Sistemi Multi-body. 1. Casa Editrice Ambrosiana: Milano.

Symbol Description[m] mass (inertia) matrix[c] damping matrix[k] stiffness matrix

, Rayleigh damping coefficients{y} displacement response vector

{F (t)} forcing excitation vectorq = 1, 2,...,Nq measurer = 1, 2,...,Nr modal shape

(XiXk)r rth modal shape at response location i and excitation location k

f frequencyr r

th natural circular frequencyfr r

th natural frequencydr r

th circular frequency of damped oscillationfdr r

th frequency of damped oscillationEr,q experimental circular frequency, acquired for measure q and mode r

i modal damping ratio for the ith normal mode[H(s)] transfer function matrix

s Laplace variable: = 2f in frequency domain

Table 9: Symbols

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