Direct identification of non-proportional modal damping matrix for lumped mass system using modal parameters

Journal of Mechanical Science and Technology 26 (4) (2012) 993~1002

www.springerlink.com/content/1738-494x DOI 10.1007/s12206-012-0221-1

Direct identification of non-proportional modal damping matrix for lumped

mass system using modal parameters† Cheonhong Min1, Hanil Park2,* and Sooyong Park3 1Graduate School of Korea Maritime University, Busan, 606-791, Korea

2Department of Ocean Engineering, Korea Maritime University, Busan, 606-791, Korea 3Department of Architecture and Ocean Space, Korea Maritime University, Busan, 606-791, Korea

(Manuscript Received February 23, 2011; Revised September 22, 2011; Accepted October 4, 2011)

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Abstract Several damping materials have been employed to reduce the vibration of marine structures. In this paper, a new method of identifying

system matrices for non-proportional damping structures using modal parameters is proposed. This method has two advantages. First, the mass and stiffness matrices do not need to be calculated using the FEM, so errors due to the inaccuracy of these matrices can be reduced. Second, various indirect methods can be used to identify modal parameters such as natural frequencies, modal damping ratios and mode shapes. Three case studies of lumped mass systems with non-proportional damping are carried out to verify the performance of the pro-posed method in this study.

Keywords: Modal analysis; Non-proportional damping system; Identification of system matrices; Direct curve-fitting method ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- 1. Introduction

Damping materials are increasingly used to reduce the vi-bration of marine structures such as ships, submarines and offshore plants. Therefore, it is important to estimate the damping matrix by using the finite element method (FEM) in the analysis of damped composite structures, but no FEM program yet exists that can correctly estimate the damping matrix. The differences between the simulation results and the experimental results for damped composite structures generate errors, which adversely affect the design, maintenance, and repair of the structures.

The measurement of the vibration parameters of structures in vibration experiments is called experimental modal analysis (EMA). There are two methods of performing EMA: an indi-rect method and a direct method. The indirect method obtains system parameters such as natural frequencies, modal damp-ing ratios and mode shapes as analysis results. Ewins [1] and Maia and Silva [2] introduced various types of early EMAs in the 1980s. Peeters et al. [3] carried out the indirect method in the frequency domain using a transfer function. Richardson and Jose [4] and Shye et al. [5] carried out the global indirect method using multi-references. Allemang and Brown [6]

compared the merits and demerits of various published EMAs. Devriendt and Guillaume [7] applied the indirect method us-ing only output data on a linear system. In addition, Lardies et al. [8] and Erlicher and Argoul [9] carried out modal identifi-cation for a non-proportional damping system. Especially, Zivanovic et al. [10], Arora et al. [11], Jahani and Nobari [12] and Avril et al. [13] studied FEM model updating to increase the accuracy of the FEM model using the system parameters of real structures, which were identified using indirect meth-ods. But these indirect methods generate errors, so mass, stiff-ness and damping matrices should be identified to reduce the errors.

The method of identifying system matrices such as mass, stiffness and damping matrices from experimental data is called a direct method. Woodhouse [14] used linear damping models to simulate structural vibration. Adhikari and Wood-house [15] estimated the damping matrix for a viscous damp-ing system using a transfer function, and they [16] also studied a non-viscous damping system. Lee and Kim [17] studied the identification of the damping matrix in a frequency range us-ing the inverse transfer function. Phani and Woodhouse [18] recently classified the existing direct methods into three groups, and compared their merits and demerits. Phani and Woodhouse [19] identified the damping matrices of a cantile-ver beam and a free-free beam using the direct methods used in their previous study [17]. However, the existing direct me-thods have some problems. First, it has not been verified if

*Corresponding author. Tel.: +82 51 410 4326, Fax.: +82 51 403 4320 E-mail address: [email protected]

† Recommended by Associate Editor Ohseop Song © KSME & Springer 2012

994 C. Min et al. / Journal of Mechanical Science and Technology 26 (4) (2012) 993~1002

they can be applied to a highly damped system. Many highly damped structures have been recently designed to reduce vi-bration. Errors can be generated unless the effects of damping are considered. Second, viscoelastic materials have especially been widely used as damping materials. Their vibration prop-erties change according to temperature and frequency, so it is important to consider the effects of this phenomenon when using them.

Vibration properties of viscoelastic materials have been in-vestigated extensively. The vibration properties of viscoelastic damping materials were changed by Nashif et al. [20] and Jones [21]. ASTM [22] proposed an experimental method of identifying the loss factors of these materials. Park et al. [23] investigated the temperature effect on the variation of the vi-bration properties of viscoelastic materials. They [24] also studied the accurate measurement of loss factor and Young’s modulus for a composite structure using a multi degree-of-freedom curve-fitting method.

In this study, a new and simple direct method is proposed. The proposed method estimates mass, stiffness and damping matrices for a non-proportional viscous damping system using modal parameters. This method has several features. First, mass, stiffness and damping matrices are identified using a simple conversion matrix, which is composed of natural fre-quencies, modal damping ratios and mode shapes, which are identified using an indirect method. Second, a wide range of selectable indirect methods is available because there is no limitation in the kinds of methods that can be used. Two val-ues of degrees (three and 30) for a lumped mass system with non-proportional damping were considered to verify the per-formance of the new method. The three-DOF lumped mass system was divided into a high and a low damping system.

2. Theory of system matrix identification

The general equation of motion for a multi-degree-of-freedom (MDOF) system of N degrees of freedom with vis-cous damping is as follows:

[ ]{ } [ ]{ } [ ]{ } { }M x C x K x f+ + = (1)

where [ ]M , [ ]C , and [ ]K are the [ ]N N× mass, damp-ing, and stiffness matrices, and { }x and { }f are the [ ]1N × vectors of the time-varying displacements and forces.

The case without excitation was first considered to deter-mine the natural modes of the system. Next, a new coordinate vector { }y that contained both the displacements { }x and velocities { }x vectors was defined, as follows:

{ }( )2 1

.N

xy

x ×

⎧ ⎫= ⎨ ⎬⎩ ⎭

(2)

Then Eq. (1) can be rewritten for modal analysis in the fol-

lowing form:

[ ] [ ] { } [ ] [ ] { } { }2 1 12: : 0 0 .N NN N

C M y K y× ××

⎡ ⎤ ⎡ ⎤+ =⎣ ⎦ ⎣ ⎦ (3) In this form, however, there are N equations and 2N un-

knowns; thus, it is necessary to add an identification equation of the following type:

[ ] [ ] { } [ ] [ ] { } { }: 0 0 : 0 .M y M y⎡ ⎤ ⎡ ⎤+ =⎣ ⎦ ⎣ ⎦ (4)

Eqs. (3) and (4) were combined to form a set of 2N equa-

tions. [ ] [ ][ ] [ ] { } [ ] [ ]

[ ] [ ] { } { }0

00 0

C M Ky y

M M⎡ ⎤ ⎡ ⎤

+ =⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦ (5)

The above equation leads to a standard eigenvalue problem,

and Eq. (6) can be assumed from it. [ ] [ ] [ ] [ ] [ ]A X B X= Λ (6)

The respective matrices of Eq. (6) are defined as follows:

[ ] [ ] [ ][ ] [ ]

00K

AM

⎡ ⎤= ⎢ ⎥−⎣ ⎦

, [ ] [ ] [ ][ ] [ ]0

C MB

M⎡ ⎤− −

= ⎢ ⎥−⎣ ⎦,

[ ] { }{ } { }1 2 2NX ψ ψ ψ⎡ ⎤= ⎣ ⎦ ,

[ ]1

2

2N

λλ

λ

⎡ ⎤⎢ ⎥⎢ ⎥Λ =⎢ ⎥⎢ ⎥⎣ ⎦

(7)

where { }ψ is the eigenvector that forms self-conjugate sets, and λ is the eigenvalue.

Multiplying each side of Eq. (6) by [ ] 1X − in which super-script -1 denotes inverse matrix.

[ ] [ ] [ ] [ ] [ ] 1A B X X −= Λ (8) Also multiplying each side of Eq. (8) by [ ] 1B − , [ ] [ ] [ ] [ ] [ ] [ ]1 1 .B A X X Z− −= Λ = (9)

Eq. (9) can be estimated using the curve-fitted experimental

data, because the right-hand side of Eq. (9) is defined in terms of eigenvalues and eigenvectors. Therefore, the unknown matrix in Eq. (9) is only [ ] [ ]1B A− . As a general rule, it is impossible to separate [ ]A and [ ]B from [ ] [ ]1B A− . In the case of the linear non-proportional viscous damping system, however, it is possible to use the following two boundary con-ditions.

(1) The mass matrix is diagonal. (2) The damping and stiffness matrices are symmetrical.

C. Min et al. / Journal of Mechanical Science and Technology 26 (4) (2012) 993~1002 995

[ ] 1B − is defined in the following form,

[ ] [ ] [ ][ ] [ ]

1 2

3 4

0 BB

B B− ⎡ ⎤= ⎢ ⎥⎣ ⎦

(10)

where [ ]2B and [ ]3B are the same and diagonal matrices. These matrices are defined by Eq. (11).

11

22

2 3

33

1 0 0 0

10 0 0

[ ] [ ] 10 0 0

10 0 0NN

M

MB B

M

M

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥= = −⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

(11)

[ ]4B is a symmetrical matrix and defined by Eq. (12). In

this equation, ijIC mean ith row and jth column element of 1[ ]C −− .

11 12 13 1

12 22 23 2

4 13 23 33 3

1 2 3

[ ]

N

N

N

N N N NN

IC IC IC ICIC IC IC IC

B IC IC IC IC

IC IC IC IC

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

(12)

Also, [ ]A is defined as in Eq. (13).

[ ] [ ] [ ][ ] [ ]

1

4

00A

AA

⎡ ⎤= ⎢ ⎥⎣ ⎦

(13)

where [ ]1A is symmetrical matrix and defined as Eq. (14).

11 12 13 1

12 22 23 2

1 13 23 33 3

1 2 3

[ ]

N

N

N

N N N NN

K K K KK K K K

A K K K K

K K K K

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

(14)

[ ]4A is diagonal matrix and defined as Eq. (15).

11

22

4 33

0 0 00 0 0

[ ] 0 0 0

0 0 0 NN

MM

A M

M

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥= −⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

(15)

[ ]Z is defined by Eq. (16).

11 12 13 111 11 11 11

12 22 23 222 22 22 22

13 23 33 333 33 33 33

1 2 3

0 0 0 0 1 0 0 00 0 0 0 0 1 0 00 0 0 0 0 0 1 0

0 0 0 0 0 0 0 1

1 1 1 1

[ ]1 1 1 1

1 1 1 1

1 1 1

N

N

N

N N NNN NN NN

K K K KM M M Mz

K K K KM M M M

K K K KM M M M

K K KM M M

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

=

−

11 11 12 22 13 33 1

12 11 22 22 23 33 2

13 11 23 22 33 33 3

1 11 2 22 3 33

1

N NN

N NN

N NN

N N N NN NN

NNNN

IC M IC M IC M IC MIC M IC M IC M IC MIC M IC M IC M IC M

IC M IC M IC M IC M

KM

⎡ ⎤⎢ ⎥⎢⎢⎢⎢⎢⎢⎢ ⎡ ⎤⎢ ⎢ ⎥⎢ ⎢ ⎥⎢ ⎢ ⎥ ⎡ ⎤⎢ ⎢ ⎥ ⎢ ⎥⎢ ⎢ ⎥ ⎢ ⎥⎢ ⎢ ⎥ ⎢ ⎥−⎢ ⎢ ⎥ ⎢ ⎥⎢ ⎢ ⎥ ⎢ ⎥⎢ ⎢ ⎥ ⎢ ⎥⎣ ⎦⎢ ⎢ ⎥⎢ ⎢ ⎥⎢ ⎢ ⎥⎢ ⎣ ⎦⎣ ⎦

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

(16)

Then [ ]A and [ ] 1B − are estimated using the following

process. In this study, the parameters in the formation of ma-trices are defined as follows. A minuscule means an element of a matrix, and a superscript means the name of a matrix. Also, the subscripts i and j are the row and column of the ma-trices, respectively.

The first row and first column element of [ ]11B is defined as 1. This make 11M 1, by this, 1~N columns of 1+N row elements of [ ]Z are defined by Eq. (17).

11, 1,j N ja z += ( )1 ~j N= (17)

12 13 1, , , NK K K are identified by Eq. 17.

Diagonal elements of [ ]2B , 22 33

1 1 1, , , ,NNM M M

are calculated by Eq. (17), and defined as Eq. (18).

,12, 1

1,

N ii j

i

zb

a+= (i j= and 2 ~ )i N= (18)

Diagonal elements of [ ]1A , 22 33, , , NNK K K , are defined

as Eq. (19).

,1, 2

,

N i N ji j

i j

za

b+ += (i j= and 2 ~ )i N= (19)

[ ]1A and [ ]2B are defined by Eqs. (17)-(19), and [ ]4A

is [ ] 12B − . Next, [ ]4B is defined as follows:

,4

, 4,

i N j Ni j

j j

zb

a+ += ( )i j≤

4 4, ,i j j ib b= ( )i j> (20)

Eq. (9) can be rewritten in the following form.

[ ] [ ] [ ] [ ][ ] [ ]

[ ] [ ][ ] [ ]

11 21

2 4 2

00 10

ABB A

B B Bα

α−

−

⎡ ⎤⎡ ⎤= ⎢ ⎥⎢ ⎥

⎢ ⎥⎣ ⎦ ⎣ ⎦ (21)

In this study, Eq. (21) is defined as a conversion matrix, and

α is defined as an amplitude ratio, which can be calculated

996 C. Min et al. / Journal of Mechanical Science and Technology 26 (4) (2012) 993~1002

by comparing the amplitude of the original transfer function with the amplitude of the estimated transfer function. Finally, through Eq. (21), we can identify [ ]M , [ ]C , and [ ]K .

3. Indirect curve-fitting method

In this study, a non-linear least squares (NLLS) method of Min et al. [25] and a multi-degree-of-freedom (MDOF) curve-fitting method, were used to calculate Eq. (9). This method can be applied to a non-proportional highly viscous damping system.

The accelerance transfer function of the non-proportional damping system is defined by Eq. (22) below:

( ) ( ) ( )2 2

1

( ) ( )Nr r r r

dr r dr rr

U jV U jVLj jω ωωω ω σ ω ω σ

=

⎧ ⎫− + − −⎪ ⎪= +⎨ ⎬− + + +⎪ ⎪⎩ ⎭

∑ (22)

in which ω is the frequency, r nr rσ ω ς= , nrω is the natu-ral frequency, rς is the modal damping ratio, j is the imaginary unit and r r ri rlU jV ψ ψ+ = , ψ is the eigenvector. Usually, the frequency range of measurement is limited. Therefore, it is necessary to consider residual terms as given below:

( ) ( ) ( )2 2

12

( ) ( )Nr r r r

dr r dr rr

U jV U jVLj j

C jD E j F

ω ωωω ω σ ω ω σ

ω ω=

⎧ ⎫− + − −⎪ ⎪= +⎨ ⎬− + + +⎪ ⎪⎩ ⎭

+ + − −

∑ (23)

in which parameter C is the real part of the residual mass, D is the imaginary part of the residual mass, E is the real part of the residual stiffness and F is the imaginary part of the residual stiffness.

The coefficients , , , , , , ,dr r r rU V C D E Fω σ are unknown factors and are denoted by parameter ( )1 ~ 4 4h h Nγ = + . Eq. (23) is not linear and thus the parameters cannot be directly obtained from it. If drω and rσ are known, however, Eq. (23) can be solved directly. In this study, the approximate values of drω and rσ were obtained using a half-power bandwidth method. Then the NLLS method was used to de-fine the other parameters such as , , , , ,r rU V C D E F . In addi-tion, the initial values of hγ were denoted by parameter hsγ .

h hs hγ γ γ= + Δ (24)

Eq. (25) below is the Tayler series of Eq. (21) expanded by

hγΔ .

( ) ( ) ( )4 4

1

Re Im

, , ,N

h hs hs hhh

LL L

A jA

ω γ ω γ ω γ γγ

+

=

∂≅ + • Δ

∂

= +

∑ (25)

in which h

Lγ∂∂

is denoted by:

( )( ){ }

( )( ){ }

( )

( )( ){ }

( )( ){ }

( )

( ) ( ) ( )

( ) ( ) ( )

2 2

2 2

2 2

2 2

2 2

2 2

2

1 ~

1 ~

1 ~

1 ~

1, , ,

r r r r

dr dr r dr r

r r r r

r dr r dr r

r dr r dr r

r dr r dr r

j U V j U VL r Nj j

U jV U jVL r Nj j

L r NU j j

L j j r NV j jL L L LjC D e F

ω ωω ω ω σ ω ω σ

ω ωσ ω ω σ ω ω σ

ω ωω ω σ ω ω σ

ω ωω ω σ ω ω σ

ω

− + −∂= + =

∂ − + + +

+ −∂= + =

∂ − + + +

∂ − −= + =

∂ − + + +

∂ −= + =

∂ − + + +

∂ ∂ ∂ ∂= = = − = −

∂ ∂ ∂ ∂2 .jω

(26) The measurement data are denoted by ReL and ImL . On

the other hand, the theoretical data are denoted by ReA and ImA . The error function is defined as τ as follows:

( ) ( ){ }2 2Re Re Im Im

1

.m

i i i ii

L A L Aτ=

= − + −∑ (27)

The error function needs to be minimized. This can be

achieved by differentiating Eq. (27) by hγΔ .

( ) ( )Re ImRe Re Im Im

1

2

0

mi i

i i i ih h hi

A AA L A Lτγ γ γ

=

⎧ ⎫∂ ∂∂ ⎪ ⎪= − + −⎨ ⎬∂Δ ∂Δ ∂Δ⎪ ⎪⎩ ⎭

=

∑ (28)

Eq. (28) is transposed simultaneously with

[ ](4 4) (4 4)N N+ × + . Finally, the values of hγ are obtained by simultaneously solving the equations.

4. Numerical examples

4.1 3-DOF non-proportional low-viscosity damping system

Fig. 1 shows a 3DOF non-proportional viscous damping system, which is defined by the lumped masses 1m , 2m , and

3m of 10, 14 and 12 kg ; the spring constants 1k , 2k , and 3k of 2000, 3000, and 2500 /N m ; and the damping coeffi-

cients 1c , 2c , and 3c of 2.1, 3.2, and 2.5 /Ns m . This sys-tem has low damping coefficients, and its system matrices (mass, stiffness, and damping matrices) are shown in Table 1.

The NLLS method was utilized as follows. First, the num-ber of peak points was selected. Since the computation time and the accuracy of the results are greatly affected by how many peak points are selected, the number of peak points selected is important. The second step in the NLLS method is to obtain unknown parameters such as ,dr rω σ . The reso-nance frequencies were assumed to be the approximated val-ues of drω , after which the approximate values of rσ were obtained using SDOF-curve-fitting methods, such as the cir-cle-fitting method and the half-power bandwidth method. In the third step, other parameters such as , , , , ,r rU V C D E F are obtained using the linear least squares (LLS) method. The

C. Min et al. / Journal of Mechanical Science and Technology 26 (4) (2012) 993~1002 997

results of the LLS method are used as the initial values in the NLLS method. Finally, all the parameters are precisely ob-tained using the NLLS method. The solid lines in Fig. 2 show the results of the accelerance transfer functions of the 3-DOF system calculated by Eq. (22). The calculated accelerance transfer function data are used as the simulation data. The circles in Fig. 2 show the curve-fitted results of the virtual simulation data using the NLLS method. The eigenvalues and eigenvectors are calculated using the natural frequencies, the modal damping ratios, and the mode shapes that are identified by the NLLS method. Table 2 shows the simulation and curve-fitted complex eigenvalues of 3-DOF non-proportional low-viscosity damping system.

The solid lines in Fig. 3 show the results of the simulation accelerance transfer functions of the 3-DOF system calculated by NLLS method, and the dotted lines show the results of the accelerance transfer function of the estimated matrices using the proposed method. Table 3 shows the estimated matrices. The amplitude ratio is α calculated to compare the original data and the estimated data, and is 0.1. This result is correct because the magnitude of the element that was placed in the first row and first column of the original mass matrix was 10. As a result, the circles in Fig. 3 show the compensation results, and the solid lines and circles are in good agreement.

The experimental data frequently include noise data be-cause the experimental conditions in the field have various error factors, such as the characteristics of the experiment equipment, climate and temperature. In this case, the accuracy of the analysis results is rapidly reduced if the method used is sensitive to noise. Thus, it is necessary to measure the method’s sensitivity to noise. The solid lines in Fig. 4 show the results that were obtained when 1% noise, which was gen-

Table 1. System matrices of 3-DOF non-proportional low-viscosity damping system.

Mass matrix (kg)

10 0 0

0 14 0

0 0 12

Viscous damping matrix ( /Ns m )

5.3 -3.2 0

-3.2 5.7 -2.5

0 -2.5 2.5

Stiffness matrix ( /N m )

5000 -3000 0

-3000 5500 -2500

0 -2500 2500

Fig. 1. 3-DOF non-proportional viscous damping system.

Table 2. Complex eigenvalues of 3-DOF non-proportional low-viscosity damping.

Mode Complex eigenvalues of simulation data

Complex eigenvalues of curve-fitted data

1th mode -0.0194 + 6.0788i -0.0194 + 6.0788i

2nd mode -0.1668 + 18.1195i -0.1668 + 18.1195i

3rd mode -0.3866 + 27.1246i -0.3866 + 27.1246i Table 3. Estimated system matrices of 3-DOF non-proportional low-viscosity damping system.

Mass matrix (kg)

1 0 0

0 1.4 0

0 0 1.2

Viscous damping matrix ( /Ns m )

0.5 -0.32 0

-0.32 0.57 -0.25

0 -0.25 0.25

Stiffness matrix ( /N m )

500 -300 0

-300 550 -250

0 -250 250

Fig. 2. Accelerance transfer functions plot for the 3-DOF non-proportional low-viscosity damping system at the first point.

Fig. 3. Overlay plot of the simulation, estimated and corrected accelerance transfer functions for the 3-DOF non-proportional low-viscosity damping system at the first point.

998 C. Min et al. / Journal of Mechanical Science and Technology 26 (4) (2012) 993~1002

erated by MATLAB, was added to the simulation accelerance transfer functions. Despite the added 1% noise, the circles in Fig. 4 show correct curve-fitted results by the NLLS method. Also, the eigenvalues and eigenvectors were calculated using the natural frequencies, the modal damping ratios and the mode shapes that were identified using the NLLS method. Table 4 shows the simulation and curve-fitted complex eigen-values of a 3-DOF non-proportional low-viscosity damping system added with 1% noise.

The solid and dotted lines in Fig. 5 show the simulation ac-celerance transfer functions resulting from the additional 1% noise and the results of the accelerance transfer function of the

estimated matrices in the proposed method, respectively. In addition, the circles in Fig. 5 show the compensation results. Table 5 shows the estimated matrices. The comparison of Tables 3 and 5 shows that an over 2% error was generated for several elements, but an under 1% error was generated for most elements. Considering the added 1% noise, these errors are negligible.

4.2 3-DOF non-proportional highly viscous damping system

The 3-DOF non-proportional highly viscous damping sys-tem was defined by the mass and stiffness matrices of the previous low damping system and the damping coefficients

1c , 2c , and 3c of 21, 32, and 25 /Ns m , respectively. The mass, stiffness, and damping matrices of the high damping system are shown in Table 6.

Fig. 6 shows the overlay plot of the accelerance transfer functions calculated in Eq. (22) and the curve-fitted acceler-ance transfer functions using the NLLS method. In particular, the resonance points could not be distinguished due to the effect of high damping. In this case, it is very hard to correctly

Table 4. Complex eigenvalues of 3-DOF non-proportional low-viscosity damping system added with 1% noise.

Mode Complex eigenvalues of simulation data

Complex eigenvalues of curve-fitted data

1th mode -0.0194 + 6.0788i -0.0196 + 6.0790i

2nd mode -0.1668 + 18.1195i -0.1667 + 18.1188i

3rd mode -0.3866 + 27.1246i -0.3874 + 27.1272i

Fig. 4. Accelerance transfer functions plot for the 3-DOF non-proportional low-viscosity damping system added with 1% noise at the first point.

Fig. 5. Overlay plot of the simulation, estimated and corrected acceler-ance transfer functions for the 3-DOF non-proportional low-viscosity damping system added with 1% noise at the first point.

Table 5. Estimated system matrixes of 3-DOF non-proportional low-viscosity damping system added with 1% noise.

Mass matrix (kg)

1 0 0

0 1.39 0

0 0 1.2

Viscous damping matrix ( /Ns m )

0.44 -0.47 -0.11

-0.47 0.70 -0.22

-0.11 -0.22 0.24

Stiffness matrix ( /N m )

502.63 -298.04 2.73

-298.04 536.95 -249.63

2.73 -249.63 252.37 Table 6. System matrixes of 3-DOF non-proportional highly-viscosity damping.

Mass matrix (kg)

10 0 0

0 14 0

0 0 12

Viscous damping matrix ( /Ns m )

53 -32 0

-32 57 -25

0 -25 25

Stiffness matrix ( /N m )

5000 -3000 0

-3000 5500 -2500

0 -2500 2500

C. Min et al. / Journal of Mechanical Science and Technology 26 (4) (2012) 993~1002 999

identify system parameters. Nevertheless, Fig. 6 shows the correct curve-fitted results by the NLLS method. Table 7 shows the simulation and curve-fitted complex eigenvalues of the 3-DOF non-proportional highly-viscosity damping system.

The solid and dotted lines in Fig. 7 show the simulation ac-celerance transfer functions and the results of the accelerance transfer functions of the estimated matrices from the proposed method, respectively. The circles in Fig. 7 show the compen-sation results, and the solid lines and circles show good agreement. Table 8 shows the estimated matrices.

The solid lines in Fig. 8 show the results that were obtained when 1% noise, which was generated by MATLAB, was add-ed to the simulation accelerance transfer functions. Also, the

eigenvalues and eigenvectors were calculated using the natural frequencies, the modal damping ratios, and the mode shapes that were identified using the NLLS method. Table 9 shows the simulation and curve-fitted complex eigenvalues of the 3-DOF non-proportional highly-viscosity damping system added with 1% noise.

The dotted lines in Fig. 9 show the eigenvalues and eigen-vectors that were calculated using the estimated mass, stiff-ness, and damping matrices from the proposed method. The circles in Fig. 9 show the compensation results. Table 10 shows the estimated matrices. The comparison of Tables 8 and 10 showed that errors were generated by the added noise.

Table 8. Complex eigenvalues of 3-DOF non-proportional highly-viscosity damping.

Mass matrix (kg)

1 0 0

0 1.4 0

0 0 1.2

Viscous damping matrix ( /Ns m )

5.3 -3.2 0

-3.2 5.7 -2.5

0 -2.5 2.5

Stiffness matrix ( /N m )

500 -300 0

-300 550 -250

0 -250 250 Table 9. Complex eigenvalues of 3-DOF non-proportional highly-viscosity damping system added with 1% noise.

Mode Complex eigenvalues of simulation data

Complex eigenvalues of curve-fitted data

1th mode -0.1941 + 6.0757i -0.1920 + 6.0773i

2nd mode -1.6676 + 18.0436i -1.7147 + 18.0258i

3rd mode -3.8658 + 26.8501i -3.6813 + 26.7802i

Fig. 8. Accelerance transfer functions plot for the 3-DOF non-proportional highly -viscosity damping system added with 1% noise at the first point.

Table 7. Complex eigenvalues of 3-DOF non-proportional highly-viscosity damping.

Mode Complex eigenvalues of simulation data

Complex eigenvalues of curve-fitted data

1th mode -0.1941 + 6.0757i -0.1941 + 6.0757i

2nd mode -1.6676 + 18.0436i -1.6676 + 18.0436i

3rd mode -3.8658 + 26.8501i -3.8658 + 26.8501i

Fig. 6. Accelerance transfer functions plot for the 3-DOF non-proportional highly -viscosity damping system at the first point.

Fig. 7. Overlay plot of the simulation, estimated and corrected accelerance transfer functions for the 3-DOF non-proportional highly-viscosity damping system at the first point.

1000 C. Min et al. / Journal of Mechanical Science and Technology 26 (4) (2012) 993~1002

4.3 30-DOF non-proportional viscous damping system

Fig. 10 shows the 30-DOF non-proportional viscous damp-ing system that is defined by the lumped masses 1m , 2m , …, and 30m of 1 kg ; the spring constants 1k , 2k , …, and 30k of 5000 /N m ; and the damping coefficients 1c , 2c , …, and 30c of 10, 11, …, 39 /Ns m , respectively. The mass, stiffness, and damping matrices are shown in Table 11.

The solid line in Fig. 11 shows the results of the accelerance transfer functions of the 30-DOF system calculated in Eq. (22). The calculated accelerance transfer function data were used as the virtual experimental data. The circles in Fig. 11 show the curve-fitted results of the virtual experimental data using the NLLS method. The eigenvalues and eigenvectors were calcu-

Table 11. Estimated system matrixes of 3-DOF non-proportional highly-viscosity damping system added 1% noise.

Mass matrix (kg)

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

Viscous damping matrix ( /Ns m )

21 -11 0 0

-11 23 -12 0

0 -12

77 -39

0 0 -39 39

Stiffness matrix ( /kN m )

10 -5 0 0

-5 10 -5 0

0 -5 10

-5

0 0 -5 10

Fig. 11. Accelerance transfer functions plot for the 30-DOF non-proportional viscous damping system at the sixth point.

Fig. 12. Overlay plot of the original and estimated accelerance transfer functions for the 30-DOF non-proportional viscous damping system at the sixth point.

Table 10. Estimated system matrixes of 3-DOF non-proportional highly-viscosity damping system added 1% noise.

Mass matrix (kg)

1 0 0

0 1.48 0

0 0 1.17

Viscous damping matrix ( /Ns m )

6.06 -2.78 0

-2.48 4.02 -3.77

0 -3.77 3.02

Stiffness matrix ( /N m )

500.95 -296.75 0.64

-296.75 550.76 -247.07

0 -247.07 236.68

Fig. 9. Overlay plot of the simulation, estimated and corrected acceler-ance transfer functions for the 3-DOF non-proportional highly-viscosity damping system added with 1% noise at the first point.

Fig. 10. 30-DOF non-proportional viscous damping system.

C. Min et al. / Journal of Mechanical Science and Technology 26 (4) (2012) 993~1002 1001

lated using the natural frequencies, the modal damping ratios, and the mode shapes that were identified using the NLLS method.

The solid and dotted lines in Fig. 12 show the results of the original accelerance transfer functions of the 30-DOF system and the results of the accelerance transfer function of the esti-mated matrices using the proposed method, respectively. The amplitude ratio, which α was calculated to compare the original data with the estimated data, was 1. This result is cor-rect because the magnitude of the element placed in the first row and first column of the original mass matrix was 1. As a result, the solid line and the dotted lines in Fig. 12 fall in line.

5. Conclusions

In this study, a new direct method of identifying the mass, stiffness, and damping matrices of damped structures using experimental vibration data was proposed. Various indirect methods can be used to identify system parameters. In addi-tion, the mass and stiffness matrices do not need to be calcu-lated using FEM in this method. Cases studies confirmed that the proposed method can correctly identify the system matri-ces of 3-DOF low and high damping systems as well as those of 30-DOF damping systems. Moreover, the proposed method showed low sensitivity to noise.

Several conclusions are made based on the results of the numerical examples. First, noise did not have an effect on the proposed method. Second, various indirect methods can be used to identify system parameters. Finally, the new method can correctly identify the system matrices of a highly damped system, and is a perfect mathematical model for a lumped-mass system.

Acknowledgment

This work is financially supported by the fund from Un-derwater Vehicle Research Center, Agency for Defense De-velopment, Korea.

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Cheonhong Min is a graduate student in Department of Ocean Engineering at Korea Maritime University. He has studied experimental vibration analysis.

Hanil Park is a professor in Department of Ocean Engineering, Korea Maritime University. He had studied at University College London for his Ph.D. He has researched on offshore structural dy-namics. He is now the president of Ko-rean Society of Ocean Engineering.