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Apr 11, 2012 - Abstract. Several damping materials have been employed to reduce the vibration of marine structures. In this paper, a new method of ...
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 1

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

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

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 proposed 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 vibration 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 indirect method and a direct method. The indirect method obtains system parameters such as natural frequencies, modal damping 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] *

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 †

compared the merits and demerits of various published EMAs. Devriendt and Guillaume [7] applied the indirect method using only output data on a linear system. In addition, Lardies et al. [8] and Erlicher and Argoul [9] carried out modal identification 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 methods. But these indirect methods generate errors, so mass, stiffness 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 Woodhouse [15] estimated the damping matrix for a viscous damping 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 using 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 cantilever beam and a free-free beam using the direct methods used in their previous study [17]. However, the existing direct methods have some problems. First, it has not been verified if

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they can be applied to a highly damped system. Many highly damped structures have been recently designed to reduce vibration. Errors can be generated unless the effects of damping are considered. Second, viscoelastic materials have especially been widely used as damping materials. Their vibration properties 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 investigated 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 vibration 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-offreedom 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 frequencies, 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 values of degrees (three and 30) for a lumped mass system with non-proportional damping were considered to verify the performance 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-offreedom (MDOF) system of N degrees of freedom with viscous damping is as follows:

[ M ]{ x} + [C ]{ x} + [ K ]{ x} = { f }

(1)

where [ M ] , [C ] , and [ K ] are the [ N × N ] mass, damping, and stiffness matrices, and { x} and { f } are the [ N × 1] vectors of the time-varying displacements and forces. The case without excitation was first considered to determine 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: ⎧x⎫ . ⎩ x ⎭( 2 N ×1)

{ y} = ⎨  ⎬

(2)

Then Eq. (1) can be rewritten for modal analysis in the following form:

⎣⎡[C ] : [ M ]⎦⎤ N ×2 N { y }2 N ×1 + ⎣⎡[ K ] : [ 0]⎦⎤ { y} = {0} N ×1 .

(3)

In this form, however, there are N equations and 2N unknowns; thus, it is necessary to add an identification equation of the following type: ⎣⎡[ M ] : [ 0]⎦⎤ { y } + ⎣⎡[ 0] : [ M ]⎦⎤ { y} = {0}.

(4)

Eqs. (3) and (4) were combined to form a set of 2N equations. ⎡ [C ] ⎢ ⎣[ M ]

[ M ]⎤ y + ⎡[ K ] [0] ⎤ y = 0 {} ⎢ ⎥{ } { } [0] ⎥⎦ ⎣ [ 0 ] [ − 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: ⎡[ K ] [ 0] ⎤ ⎡ [ −C ] [ − M ]⎤ , [ A] = ⎢ 0 − M ⎥ , [ B ] = ⎢ − M [ ] [ ] ] [0] ⎥⎦ ⎣ ⎦ ⎣[ [ X ] = ⎡⎣{ψ 1}{ψ 2}"{ψ 2 N }⎤⎦ , ⎡ λ1 ⎤ ⎢ ⎥ λ2 ⎥ [ Λ ] = ⎢⎢ ⎥ % ⎢ ⎥ λ 2N ⎦ ⎣

(7)

where {ψ } is the eigenvector that forms self-conjugate sets, and λ is the eigenvalue. −1 Multiplying each side of Eq. (6) by [ X ] in which superscript -1 denotes inverse matrix.

[ A] = [ B ] [ X ] [ Λ ] [ X ]−1

(8)

Also multiplying each side of Eq. (8) by [ B ] , −1

[ B ]−1 [ A] = [ X ] [ Λ ] [ X ]−1 = [ Z ].

(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 −1 matrix in Eq. (9) is only [ B ] [ A] . As a general rule, it is −1 impossible to separate [ A] and [ B ] from [ B ] [ A] . In the case of the linear non-proportional viscous damping system, however, it is possible to use the following two boundary conditions. (1) The mass matrix is diagonal. (2) The damping and stiffness matrices are symmetrical.

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[ B ]−1

is defined in the following form,

⎡ [ 0 ] [ B ]⎤ [ B ]−1 = ⎢ B B2 ⎥ ⎣[ 3 ] [ 4 ]⎦

(10)

where [ B2 ] and [ B3 ] are the same and diagonal matrices. These matrices are defined by Eq. (11). ⎡ 1 ⎢M ⎢ 11 ⎢ ⎢ 0 ⎢ [ B2 ] = [ B3 ] = − ⎢ ⎢ 0 ⎢ ⎢ # ⎢ ⎢ ⎢ 0 ⎣

0

0

"

1 M 22

0

"

0

1 M 33

"

#

#

%

0

0

"

⎤ 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ # ⎥ ⎥ 1 ⎥ M NN ⎥⎦

Also,

[ A]

IC12 IC22 IC23 # IC2 N

" " " % "

IC13 IC23 IC33 # IC3 N

IC1N ⎤ IC2 N ⎥⎥ IC3 N ⎥ ⎥ # ⎥ IC NN ⎥⎦

(11)

(12)

[ A4 ]

K12 K 22 K 23 # K2N

(13)

" " " % "

K1N ⎤ K 2 N ⎥⎥ K3 N ⎥ ⎥ # ⎥ K NN ⎥⎦

(14)

0 0 M 33 # 0

is defined by Eq. (16).

0 ⎤ " 0 ⎥⎥ " 0 ⎥ " ⎥ % # ⎥ " M NN ⎥⎦

"

1 K23 " M22 1 K33 " M33 # % 1 K3N " MNN

1 ⎤ K1N ⎥ M11 ⎥ ⎥ ⎡ IC M 1 K2N ⎥ ⎢ 11 11 M22 ⎥ ⎢ IC12M11 ⎥ −⎢ IC M 1 K3N ⎥ ⎢ 13 11 M33 ⎥ ⎢ # ⎥ ⎢IC M # ⎥ ⎣ 1N 11 1 ⎥ KNN ⎥ MNN ⎦

0 1 0 # 0

IC12M22 IC22M22 IC23M22 # IC2N M22

0 0 1 # 0

" " " % "

0⎤ 0⎥⎥ 0⎥ ⎥ #⎥ 1⎥⎦

IC13M33 IC23M33 IC33M33 # IC3N M33

" " " % "

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ IC1N MNN ⎤⎥ ⎥ ⎥ IC2NMNN ⎥⎥ IC3NMNN ⎥⎥ ⎥⎥ # ⎥⎥ ⎥ ICNN MNN ⎥⎦⎥ ⎥ ⎥ ⎥⎦

( j =1~ N)

(17)

K12 , K13 ,", K1N are identified by Eq. 17. Diagonal elements of [ B2 ] , 1 M , 1 M ,", 1 M , 22 33 NN are calculated by Eq. (17), and defined as Eq. (18).

z N + i ,1 a1,1 i

(i = j and i = 2 ~ N )

Diagonal elements of as Eq. (19). ai1, j =

zN +i, N + j bi2, j

[ A1 ] ,

zi + N , j + N a 4j , j

(18)

K 22 , K 33 ,", K NN , are defined

(i = j and i = 2 ~ N )

[ A1 ] −1and [ B2 ] are defined by Eqs. (17)-(19), and is [ B2 ] . Next, [ B4 ] is defined as follows: bi4, j =

(19)

[ A4 ]

(i ≤ j )

bi4, j = b 4j ,i ( i > j )

is diagonal matrix and defined as Eq. (15).

0 ⎡ M 11 ⎢ 0 M 22 ⎢ [ A4 ] = − ⎢ 0 0 ⎢ # # ⎢ ⎢ 0 0 ⎣

[Z ]

K13 K 23 K 33 # K3 N

1 K13 M11

⎡1 ⎢0 ⎢ ⎢0 ⎢ ⎢# ⎢⎣0

−1

bi2, j =

where [ A1 ] is symmetrical matrix and defined as Eq. (14). ⎡ K11 ⎢K ⎢ 12 [ A1 ] = ⎢ K13 ⎢ ⎢ # ⎢K ⎣ 1N

0⎤ 0⎥⎥ 0⎥ ⎥ #⎥ 0⎥⎦

Then [ A] and [ B ] are estimated using the following process. In this study, the parameters in the formation of matrices 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 matrices, respectively. The first row and first column element of [ B11 ] is defined as 1. This make M 11 1, by this, 1~N columns of 1+N row elements of [ Z ] are defined by Eq. (17). a1,1 j = z N +1, j

is defined as in Eq. (13).

⎡[ A ] [ 0] ⎤ [ A] = ⎢ 01 A ⎥ ⎣ [ ] [ 4 ]⎦

" " " % "

0 0 0 # 0

(16)

[ B4 ] is a symmetrical matrix and defined by Eq. (12). In this equation, ICij mean ith row and jth column element of −[C ]−1 . ⎡ IC11 ⎢ IC ⎢ 12 [ B4 ] = ⎢ IC13 ⎢ ⎢ # ⎢ IC ⎣ 1N

⎡ ⎡0 0 ⎢ ⎢0 0 ⎢ ⎢ ⎢ ⎢0 0 ⎢ ⎢ ⎢ ⎢# # ⎢ ⎢⎣0 0 ⎢ ⎢ ⎡ 1 1 K11 K12 ⎢ ⎢ M M11 [z] = ⎢ ⎢ 11 ⎢ ⎢ 1 1 K12 K22 ⎢ ⎢ M22 ⎢ ⎢ M22 ⎢−⎢ 1 1 ⎢ ⎢ K13 K23 M33 ⎢ ⎢ M33 ⎢ ⎢ # # ⎢ ⎢ ⎢ ⎢ 1 1 K1N K2N ⎢ ⎢ MNN ⎣⎢ ⎣ MNN

(20)

Eq. (9) can be rewritten in the following form. (15)

[ 0] ⎤ ⎡ [ 0] [ B ]⎤ 1 ⎡[ A1 ] [ B ]−1 [ A] = α ⎢ B B2 ⎥ ⎢ −1 ⎥ ⎣[ 2 ] [ 4 ]⎦ α ⎢⎣ [ 0] [ B2 ] ⎥⎦

(21)

In this study, Eq. (21) is defined as a conversion matrix, and α is defined as an amplitude ratio, which can be calculated

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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) curvefitting 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: L (ω ) =

N

⎧ −ω 2 (U r + jVr ) −ω 2 (U r − jVr ) ⎫⎪ + ⎬ j (ω + ωdr ) + σ r ⎪⎭ dr ) + σ r r =1

∑ ⎪⎨⎪⎩ j (ω − ω

(22)

in which ω is the frequency, σ r = ωnrς r , ωnr is the natural frequency, ς r is the modal damping ratio, j is the imaginary unit and U r + jVr = ψ riψ rl , ψ is the eigenvector. Usually, the frequency range of measurement is limited. Therefore, it is necessary to consider residual terms as given below: L (ω ) =

N

⎧ −ω 2 (U r + jVr ) −ω 2 (U r − jVr ) ⎫⎪ + ⎬ j (ω + ωdr ) + σ r ⎪⎭ dr ) + σ r r =1

∑ ⎪⎨⎪⎩ j (ω − ω

− jω 2 (U r + Vr ) jω 2 (U r − Vr ) ∂L = + 2 ∂ωdr { j (ω − ω ) + σ } { j (ω + ω ) + σ }2 dr r dr r

ω 2 (U r + jVr ) ω 2 (U r − jVr ) ∂L = + 2 ∂σ r { j (ω − ω ) + σ } { j (ω + ω ) + σ }2 dr r dr r

(r = 1 ~ N )

∂L − jω 2 jω 2 = + ∂Vr j (ω − ωdr ) + σ r j (ω + ωdr ) + σ r

(r = 1 ~ N )

∂L = 1, ∂C

γ h = γ hs + Δγ h

(24)

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

h =1

= ARe + jAIm

in which

∂L

∑ ∂γ

∂L is denoted by: ∂γ h

∂L = −ω 2 , ∂e

∂L = − jω 2 . ∂F

The measurement data are denoted by LRe and LIm . On the other hand, the theoretical data are denoted by ARe and AIm . The error function is defined as τ as follows: m

τ=

∑{( L

}

− ARe i ) + ( LIm i − AIm i ) . 2

Re i

i =1

2

(27)

The error function needs to be minimized. This can be achieved by differentiating Eq. (27) by Δγ h . ∂τ =2 ∂Δγ h i =1

⎧ ∂ARe i

∑ ⎪⎨⎪⎩ ∂Δγ

( ARe i − LRe i ) +

h

⎫ ∂AIm i ( AIm i − LIm i )⎪⎬ (28) ∂Δγ h ⎪⎭

=0

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 ωdr ,σ r ,U r ,Vr , C , D, E , F are unknown factors and are denoted by parameter γ h ( h = 1 ~ 4 N + 4 ) . Eq. (23) is not linear and thus the parameters cannot be directly obtained from it. If ωdr and σ r are known, however, Eq. (23) can be solved directly. In this study, the approximate values of ωdr and σ r were obtained using a half-power bandwidth method. Then the NLLS method was used to define the other parameters such as U r ,Vr , C , D, E , F . In addition, the initial values of γ h were denoted by parameter γ hs .

4N +4

∂L = j, ∂D

(26)

+C + jD − ω E − jω 2 F

L (ω , γ h ) ≅ L (ω , γ hs ) +

(r = 1 ~ N )

∂L −ω 2 −ω 2 = + ∂U r j (ω − ωdr ) + σ r j (ω + ωdr ) + σ r

m

(23)

(r = 1 ~ N )

h

(ω , γ hs ) • Δγ h

(25)

Eq.

(28)

is

transposed

simultaneously with γ h are obtained by simultaneously solving the equations.

[(4 N + 4) × (4 N + 4)] . Finally, the values of

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 m1 , m2 , and m3 of 10, 14 and 12 kg ; the spring constants k1 , k2 , and k3 of 2000, 3000, and 2500 N / m ; and the damping coefficients c1 , c2 , and c3 of 2.1, 3.2, and 2.5 Ns / m . This system 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 number 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 ωdr ,σ r . The resonance frequencies were assumed to be the approximated values of ωdr , after which the approximate values of σ r were obtained using SDOF-curve-fitting methods, such as the circle-fitting method and the half-power bandwidth method. In the third step, other parameters such as U r ,Vr , C , D, E , F are obtained using the linear least squares (LLS) method. The

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Table 1. System matrices of 3-DOF non-proportional low-viscosity damping system. Mass matrix (kg)

1th mode

Complex eigenvalues of simulation data -0.0194 + 6.0788i

Complex eigenvalues of curve-fitted data -0.0194 + 6.0788i

2nd mode

-0.1668 + 18.1195i

-0.1668 + 18.1195i

-0.3866 + 27.1246i

-0.3866 + 27.1246i

Mode

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

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

-3000

0

-3000

5500

-2500

0

-2500

2500

rd

3 mode

Table 3. Estimated system matrices of 3-DOF non-proportional lowviscosity 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

-0.25

0.25

0

Stiffness matrix ( N / m ) Fig. 1. 3-DOF non-proportional viscous damping system.

results of the LLS method are used as the initial values in the NLLS method. Finally, all the parameters are precisely obtained 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 because 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-

500

-300

0

-300

550

-250

0

-250

250

Fig. 2. Accelerance transfer functions plot for the 3-DOF nonproportional 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 lowviscosity damping system at the first point.

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

1th mode

Complex eigenvalues of simulation data -0.0194 + 6.0788i

Complex eigenvalues of curve-fitted data -0.0196 + 6.0790i

2nd mode

-0.1668 + 18.1195i

-0.1667 + 18.1188i

3rd mode

-0.3866 + 27.1246i

-0.3874 + 27.1272i

Mode

Table 5. Estimated system matrixes of 3-DOF non-proportional lowviscosity 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)

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

10

0

0

14

0 0

0

0

12

Viscous damping matrix ( Ns / m ) 53

-32

0

-32

57

-25

-25

25

0

Stiffness matrix ( N / m ) 5000

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

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 eigenvalues 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 accelerance transfer functions resulting from the additional 1% noise and the results of the accelerance transfer function of the

-3000

0

-3000

5500

-2500

0

-2500

2500

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 system was defined by the mass and stiffness matrices of the previous low damping system and the damping coefficients c1 , c2 , and c3 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 accelerance 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

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

1th mode

Complex eigenvalues of simulation data -0.1941 + 6.0757i

Complex eigenvalues of curve-fitted data -0.1941 + 6.0757i

2nd mode

-1.6676 + 18.0436i

-1.6676 + 18.0436i

-3.8658 + 26.8501i

-3.8658 + 26.8501i

Mode

rd

3 mode

Table 8. Complex eigenvalues of 3-DOF non-proportional highlyviscosity damping. Mass matrix (kg) 1

0

0

0

1.4

0

0

1.2

0

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 highlyviscosity damping system added with 1% noise.

1th mode

Complex eigenvalues of simulation data -0.1941 + 6.0757i

Complex eigenvalues of curve-fitted data -0.1920 + 6.0773i

2nd mode

-1.6676 + 18.0436i

-1.7147 + 18.0258i

3rd mode

-3.8658 + 26.8501i

-3.6813 + 26.7802i

Mode Fig. 6. Accelerance transfer functions plot for the 3-DOF nonproportional 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 highlyviscosity damping system at the first point.

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 accelerance 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 compensation 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 added to the simulation accelerance transfer functions. Also, the

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

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 3DOF non-proportional highly-viscosity damping system added with 1% noise. The dotted lines in Fig. 9 show the eigenvalues and eigenvectors that were calculated using the estimated mass, stiffness, 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.

1000

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

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

Table 11. Estimated system matrixes of 3-DOF non-proportional highly-viscosity damping system added 1% noise. Mass matrix (kg)

Mass matrix (kg) 0

1

0

0

1.48

0

0

1

0

0

1.17

0

0

1

#

#

#

0

0

1

0

0 0

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 )

0

" " " % "

0 0 0

# 1

Viscous damping matrix ( Ns / m ) 21

-11

0

-11

23

-12

% # "

500.95

-296.75

0.64

0

-12

-296.75

550.76

-247.07

#

#

0

-247.07

236.68

0

0

10

-5

0

-5

10

-5

0

-5

10

#

#

0

0

# "

" " "

0

77

-39

-39

39

" " " %

0

-5

-5

10

0

#

Stiffness matrix ( kN / m ) 0

#

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

Fig. 11. Accelerance transfer functions plot for the 30-DOF nonproportional viscous damping system at the sixth point. Fig. 10. 30-DOF non-proportional viscous damping system.

4.3 30-DOF non-proportional viscous damping system Fig. 10 shows the 30-DOF non-proportional viscous damping system that is defined by the lumped masses m1 , m2 , …, and m30 of 1 kg ; the spring constants k1 , k2 , …, and k30 of 5000 N / m ; and the damping coefficients c1 , c2 , …, and c30 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-

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.

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

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 estimated 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 correct 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 addition, the mass and stiffness matrices do not need to be calculated using FEM in this method. Cases studies confirmed that the proposed method can correctly identify the system matrices 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 lumpedmass system.

Acknowledgment This work is financially supported by the fund from Underwater Vehicle Research Center, Agency for Defense Development, 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 dynamics. He is now the president of Korean Society of Ocean Engineering.