Structural parameters identification using improved normal frequency response function method

ARTICLE IN PRESS Mechanical Systems and Signal Processing Mechanical Systems and Signal Processing 22 (2008) 1858–1868 www.elsevier.com/locate/jnlabr/ymssp

Structural parameters identification using improved normal frequency response function method Kyu-Sik Kima, Yeon June Kanga,, Jeonghoon Yoob a

Advanced Automotive Research Center, School of Mechanical and Aerospace Engineering, Seoul National University, Gwanangno 599, Sillim-9Dong, Gwanak-gu, Seoul 151-744, Republic of Korea b School of Mechanical Engineering, Yonsei University, 134 Sinchon-dong, Seodaemun-gu, Seoul 120-742, Republic of Korea Received 9 November 2007; received in revised form 5 February 2008; accepted 6 February 2008 Available online 10 March 2008

Abstract An improved method that is based on a normal frequency response function (FRF) is proposed in this study in order to identify structural parameters such as mass, stiffness and damping matrices directly from the FRFs of a linear mechanical system. This paper demonstrates that the characteristic matrices may be extracted more accurately by using a weighted equation and by eliminating the matrix inverse operation. The method is verified for a four degrees-of-freedom lumped parameter system and an eight degrees-of-freedom finite element beam. Experimental verification is also performed for a free–free steel beam whose size and physical properties are the same as those of the finite element beam. The results show that the structural parameters, especially the damping matrix, can be estimated more accurately by the proposed method. r 2008 Elsevier Ltd. All rights reserved. Keywords: Structural parameters; Normal FRF; Weighted equation; Matrix inverse elimination; Least squares method

1. Introduction In order to predict accurate dynamic characteristics of a mechanical system, it is necessary to formulate a mathematical model as accurately as possible in terms of structural parameters such as mass, stiffness and damping matrices of the system. However, since most mechanical systems are complicated, it is difficult to accurately estimate their system parameters. For these reasons, many studies have been undertaken using theoretical and experimental methods in order to identify structural parameters. One of the general methods for estimating system parameters is to indirectly use frequency response function (FRF) data. The system parameters are extracted using modal parameters such as natural frequencies and mode shapes derived from the FRF data [1,2]. However, the structural parameters estimated by these methods can be affected by several factors including the inaccuracy of FRFs, the errors in the extracted modal parameters and the incompleteness of the modal information [3].

Corresponding author. Tel.: +82 2 880 1691; fax: +82 2 888 5950.

E-mail address: [email protected] (Y.J. Kang). 0888-3270/$ - see front matter r 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ymssp.2008.02.001

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A more extensively studied method that is theoretically easier and simpler than the indirect method is known as a direct method. Using the direct method, the system matrices can be obtained directly from the measured FRFs of a mechanical system without using modal parameters. Fritzen [4] proposed the Instrument Variable (IV) method that is suitable for the estimation of structural parameters from noisy data. By comparing the estimated results from the IV method with those from the least squares method, Fritzen’s work demonstrates that the IV method is less sensitive to measurement noise than the least squares method. However, the IV method requires numerous data points in order to estimate the structural parameters and it is therefore very time consuming. In order to overcome these problems, Wang [5] developed the weighted FRF that can be combined with the IV method. In Wang’s approach, appropriate configurations of data distribution in FRFs were proposed in order to reduce the number of data points required by the IV method. Chen et al. [6] developed the ‘normal FRF method’, which extracts the normal FRFs from the complex FRFs of a structure. Chen et al. demonstrated that accurate identification results can be obtained when the damping matrix is identified independently from the mass and stiffness matrices. Lee et al. [7] conducted a theoretical validation and related error analysis for the normal FRF method as developed by Chen et al. and developed a method for experimental estimation of damping matrices of a general dynamic system. Lee et al. applied that method to a single-reed beam with a viscous damper attached, and identified two types of damping mechanisms directly from the measured FRFs, i.e. viscous and hysteretic damping matrices. Later, they also developed a much simpler algorithm in numerical operations that identifies the damping matrices directly from the dynamic stiffness matrix (or the inverse of the frequency response matrix) [8]. However, good care must be taken of some technical issues, including a phase-matching technique between force transducer and accelerometers, a sign convention of FRFs, and a conditioning of frequency response matrix that makes the frequency response matrix symmetric. In this paper, an improved method, which itself is based on the normal FRF method developed by Chen et al., is presented so as to identify the structural parameters directly from the FRFs of a mechanical system. Accurate results have been achieved by two modified and additional theoretical procedures, that of the elimination of matrix inverse operation and the imposition of weighting matrix. The advantage of this improved method is validated by two numerical examples, that of a simple four degrees-of-freedom (4 d.o.f.) lumped parameter system and that of an 8 d.o.f. finite element beam structure. Its robustness to noisy data are also illustrated through experimental verification of a steel beam. All the results from the present method have been compared with exact results and those of the normal FRF method. 2. Identification theory The equation of motion for a vibratory system can be written as € þ CxðtÞ _ þ ðK þ jDÞxðtÞ ¼ fðtÞ, MxðtÞ

(1)

where M, C, K and D represent the mass, viscous damping, stiffness and hysteretic damping matrices of the _ x€ and f are the vectors of displacements, velocities, accelerometers and forces, system, respectively, and x, x, pffiffiffiffiffiffiffi and j ¼ 1. For harmonic excitation, Eq. (1) is expressed as ðo2 M þ KÞXðoÞ þ jðo C þ DÞXðoÞ ¼ FðoÞ. 2

1

Multiplying both sides of Eq. (2) by [o M+K] 2

1

(2) yields

2

ðI þ j½o M þ K ½o C þ DÞXðoÞ ¼ ½o M þ K1 FðoÞ.

(3) N

Here, I denotes the identity matrix. By introducing the normal FRF matrix H (o) and the transformation matrix G(o) as defined in Ref. [6] as HN ðoÞ ¼ ½o2 M þ K1 ,

(4)

GðoÞ ¼ HN ðoÞ½o C þ D,

(5)

and substituting Eqs. (4) and (5) into Eq. (3) yields ½I þ jGðoÞXðoÞ ¼ HN ðoÞFðoÞ.

(6)

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The complex frequency response matrix equation may be expressed as XðoÞ ¼ HC ðoÞFðoÞ, C

(7) C

where H (o) represents the complex FRF matrix. Since H (o) can be decomposed into the real and imaginary parts, Eq. (7) is rewritten as C XðoÞ ¼ ½HC R ðoÞ þ jHI ðoÞFðoÞ,

(8) C

where the subscripts R and I denote the real and imaginary parts of H (o), respectively. Substituting Eq. (8) into Eq. (6) yields C C C HN ðoÞ ¼ ½HC R ðoÞ  GðoÞHI ðoÞ þ j½GðoÞHR ðoÞ þ HI ðoÞ.

(9)

Since HN(o) must be a real matrix, from the imaginary part in the right-hand side of Eq. (9), G(o) is obtained as C 1 GðoÞ ¼ HC I ðoÞ½HR ðoÞ .

(10)

C It is noted here that G(o) would have large errors when HC R ðoÞ is ill-conditioned since small errors in HR ðoÞ 1 C can be amplified by its matrix inverse operation, ½HR ðoÞ . Substituting Eq. (10) into the real part of Eq. (9)

yields 1 C C C HN ðoÞ ¼ HC HI ðoÞ. R ðoÞ þ HI ðoÞ½HR ðoÞ

(11)

N

Once H (o) and G(o) are calculated by Eqs. (10) and (11), Eqs. (4) and (5) can be rewritten as [6]   C N N ðoÞ H ðoÞ oH  ¼ GðoÞ, ½ D   M ¼ I. ½ o2 HN ðoÞ HN ðoÞ  K

(12)

(13)

Generally, the structural parameters can be calculated precisely from Eqs. (12) and (13) for a noise-free case, and many researchers have solved these equations using a least squares method. In a practical case, however, noise is inevitably involved when measuring the FRFs of systems, and thus the damping parameters C and D cannot be identified accurately by Eq. (12), especially when taking into consideration the matrix inverse operation as mentioned previously in Eq. (10). This study proposes an improved formulation for damping parameters identification, which is much less sensitive to noise. The formulation can be obtained from the simple idea of eliminating the matrix inverse operation in Eq. (10). From Eqs. (5) and (10), the equation for estimating the damping matrices is expressed as 1 C HC ¼ HN ðoÞZ1 ðoÞ, I ðoÞ½HR ðoÞ

(14)

where Z1 ðoÞ ¼ o C þ D. The matrix inverse operation can then be removed easily by post-multiplying both sides of Eq. (14) by N C HC I ðoÞ ¼ H ðoÞZ1 ðoÞHR ðoÞ.

(15) HC R ðoÞ: (16)

Eq. (16) will be transformed into a set of linear equations and then solved by the least squares method. In order to identify mass and stiffness matrices from Eq. (13) by the least squares method, Chen et al. [6] transformed the equation directly into a series of linear equations, in the from of Ax ¼ b, by using an outer product expansion and rearranging matrices. Although this procedure could, to some extent, alleviate numerical inaccuracy in solving the equation by the least squares method, small singular values of A could still cause erroneous results when applying the least squares method. This is most likely because some elements of each row in A become zero owing to the outer product expansion. In order to overcome those problems, a weighting matrix is imposed on Eq. (13) in this paper. Eq. (4) can be rearranged and both of its sides can be

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multiplied by HC R ðoÞ, and a similar form of equation to Eq. (16) can be obtained as N C HC R ðoÞ ¼ H ðoÞZ2 ðoÞHR ðoÞ,

(17)

where Z2 ðoÞ ¼ o2 M þ K.

(18)

Henceforth, Eqs. (16) and (17) will be used as basic equations to estimate the structural parameters. Note that Eqs. (16) and (17) are weighted by the real part of the complex FRF HC R ðoÞ, and both can be written in the same form: (19)

HA ðoÞZS ðoÞHB ðoÞ ¼ HC ðoÞ,

where HA(o), HB(o) and HC(o) are known coefficient matrices from measured or simulated FRFs, and ZS(o) is the structural parameter matrix that is to be estimated. The procedure to calculate ZS(o) in Eq. (19) will be examined in the following section. 3. Calculation procedure When the size of each matrix in Eq. (19) is N  N, Eq. (19) may be expressed as a set of linear equations by using a Kronecker product and a vector-valued function as follows [9]: ½KP N 2 N 2 fzgN 2 1 ¼ frgN 2 1 ,

(20)

in which KP ¼ HA  HTB , and

z ¼ nðZS Þ

(21) r ¼ nðHC Þ.

(22,23)

Here, the notations , n(  ) and superscript T represent the Kronecker product, the vector-valued function and the matrix transpose, respectively. The subscripts N2  N2 and N2  1 denote the sizes of the corresponding matrix and vectors. The detailed explanation of the Kronecker product and vector-valued function is presented in Appendix. Note that KP and r include the information for the FRFs such as HA, HB and HC. Since the vector z is made by rearranging the elements of the matrix ZS, i.e., Z1(o) and Z2(o) as shown in Eqs. (15) and (18), it can be expressed as z ¼ oc þ d

or

z ¼ k  o2 m,

(24,25)

where the structural parameter vectors c, d, m and k consist of the corresponding elements of the viscous damping, hysteretic damping, mass and stiffness matrices, respectively. Again, Eqs. (24) and (25) can be rewritten in the form: z ¼ ½ oI

I f c

d gT

or

z ¼ ½ o2 I

I f m

k gT .

(26,27)

Substituting Eqs. (26) and (27) into Eq. (20) and imposing omax and o2max yields ½KP N 2 N 2 ½EN 2 2N 2 fxg2N 2 1 ¼ frgN 2 1 , where

"

E¼ or

" E¼

 o I omax

o omax

2

(28)

# I

x ¼ f omax c d gT ,

(29,30)

and x ¼ f o2max m k gT .

(31,32)

and

# I I ;

Here, omax is a maximum measured (or simulated) angular frequency that will be used as a reference frequency. If omax and o2max are not imposed on Eqs. (29)–(32), the magnitudes of elements of E vary

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significantly by a factor of o for the viscous and hysteretic damping vectors and by o2 for the mass and stiffness vectors. This may cause numerical problems in solving Eq. (28). In order to avoid that problem, omax and o2max are imposed on Eqs. (29) and (31) [10]. Note here that omax and o2max play a role of scaling factors, which make the maximum value in E become unity. The least squares solution of Eq. (28) can be affected significantly by differences in magnitudes of the coefficients of KP, since the magnitudes of the FRFs vary considerably as the frequency changes. Therefore, the solution will be overwhelmed by the information at the resonance frequencies of a system. Moreover, the coefficients of each individual equation are also different in magnitudes, which may cause a faulty solution. In order to obtain accurate results, the available information needs to be used effectively. This can be achieved by imposing appropriate weighting factors on KP. In this paper, these factors are selected so that the maximum element of each equation in Eq. (28) is simply scaled to unity [10]. After applying the weighting technique, Eq. (28) takes the form ½SðoÞN 2 2N 2 fxg2N 2 1 ¼ fr0 ðoÞgN 2 1 ,

(33)

where ½SðoÞN 2 2N 2 ¼ ½WðoÞN 2 N 2 ½KP ðoÞN 2 N 2 ½EðoÞN 2 2N 2 ,

(34)

fr0 ðoÞgN 2 1 ¼ ½WðoÞN 2 N 2 frðoÞgN 2 1 .

(35)

Here, W is a weighting diagonal matrix whose diagonal elements are made up of the weighting factors. If the measured data have n spectral lines, Eq. (33) becomes 9 8 0 3 2 Sðo1 Þ r ðo1 Þ > > > > > > > > 7 6 > = < r0 ðo2 Þ > 6 Sðo2 Þ 7 7 6 fxg2N 2 1 ¼ . (36) 6 .. 7 .. > > 6 . 7 > > > . > 5 4 > > > ; : r0 ðon Þ > Sðon Þ 2 2 2 nN 2N

nN 1

Therefore, the solution of Eq. (36) can be obtained using the least squares method. 4. Numerical simulations and experimental verification In this section, two numerical simulation examples will be described, that of a 4 d.o.f. lumped parameter system and that of an 8 d.o.f. finite element beam. In addition, an experimental verification will be shown for a free–free steel beam. 4.1. Numerical simulation 1 A four d.o.f. system as shown in Fig. 1 is used, and the system matrices of the equations of motion are listed in Table 1. In this case, the structural damping loss factors for the first to fourth modes are 0.002, 0.0017, 0.0021 and 0.0005, respectively. A total number of 16 receptance FRFs were generated from the matrices in Table 1, and the frequency range of interest and its frequency resolution for them was set to be 1–1000 and 1 Hz, respectively. In order to simulate practical situations, all FRFs were artificially contaminated by random noises. In this section, 20% random noise is imposed on the FRFs. Here, the FRF with 20% random noise

Fig. 1. Four d.o.f. lumped parameter system. mi values (kg): m1, 150; m2, 160; m3, 130; m4, 100. ki values (N/m): k1 ¼ k2 ¼ k4 ¼ k5 ¼ 5  108; k3 ¼ 1.5  109. di values (N/m): d1 ¼ d3 ¼ d4 ¼ 5  105; d2, 1.0  106; d5, 2.0  106.

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Table 1 System matrices of the 4 d.o.f. lumped parameter system Stiffness matrix K (  107 N/m)

Mass matrix M (kg) 150 0 0 0

0 160 0 0

0 0 130 0

0 0 0 100

100 50 0 0

0 150 200 50

50 200 150 0

Hysteretic damping matrix D (  104 N/m) 0 0 50 100

150 100 0 0

100 150 50 0

0 50 100 50

0 0 50 250

Table 2 Comparison of identified structural parameters from FRFs of the 4 d.o.f. lumped parameter system Proposed method

Normal FRF method

Estimated matrix M (kg) 149.2 0.8 0.3 0.0

Absolute error (%)

Estimated matrix

Absolute error (%)

0.9 159.1 1.1 0.1

0.3 1.4 128.1 0.5

0.3 0.5 0.7 100.0

0.4 0 0 0

0 0.5 0 0

0 0 1.4 0

0 0 0 0.0

143.3 0.4 0.1 2.1

3.6 145.2 15.8 13.7

5.6 11.5 131.8 9.9

0.9 1.2 1.4 98.7

4.4 0 0 0

0 9.2 0 0

0 0 1.4 0

0 0 0 4.2

K (  107 N/m) 99.7 50.3 50.1 200.4 0.7 151.3 0.1 1.1

0.4 150.5 199.0 50.2

0.2 1.7 51.6 100.6

0.1 0.3 0 0

0.6 0.2 0.8 0

0 0.3 0.4 0.4

0 0 3.2 0.6

94.3 40.7 5.2 3.6

45.3 167.6 134.2 27.2

11.5 114.5 185.3 27.4

2.1 8.8 46.3 89.9

7.1 8.7 0 0

14.8 8.6 5.8 0.0

0 16.6 4.1 31.9

0 0 0.2 5.1

D (  104 N/m) 154.7 101.4 101.4 154.1 0.5 50.6 0.0 0.8

1.3 52.3 100.8 50.6

0.9 0.7 50.6 253.6

3.1 1.4 0 0

1.4 2.7 1.4 0

0 4.7 0.8 1.2

0 0 1.2 1.4

156.6 120.5 9.0 5.4

22.2 105.5 14.3 11.9

55.5 103.0 139.3 70.4

28.7 91.7 21.4 234.1

4.4 20.5 0 0

77.8 29.6 71.3 0

0 106.0 39.3 40.9

0 0 57.1 6.3

implies that the real and imaginary parts of the noise-free FRF at each frequency are separately multiplied by uniformly distributed random numbers varying from 0.8 to 1.2 (SNR ratio is 13.9 dB). The structural parameters of the 4 d.o.f. system were identified using the proposed method and the normal FRF method [6]. The identification results from FRFs with 20% random noise are shown in Table 2 where the errors are taken in absolute value. From the results of each method in this table, while the maximum errors in the mass, stiffness and damping parameters identified from the normal FRF method are approximately 9.2%, 31.9% and 106.0%, respectively, the maximum errors in the mass, stiffness and damping parameters from the present method are about 1.4%, 3.2% and 4.7%, respectively. As can be seen, the proposed method gives more accurate results than the normal FRF method. Although the present method gives the results that are in good agreement with the exact values, the normal FRF method is affected significantly by noises, especially in the identification of damping parameters. The damping parameters are more difficult to identify than the mass and stiffness matrices. Nevertheless, it may be concluded that the proposed method can identify all structural parameters with reasonably good accuracy by alleviating errors caused by noises. In Fig. 2, the FRFs H12 reconstructed from the identified structural parameters for 20% random noises are compared with the corresponding true FRF. Natural frequencies from the proposed method in the range of interest are found to be 186, 418, 525 and 802 Hz, which are same as the exact natural frequencies of this system. On the contrary, those frequencies from the normal FRF method are obtained to be 174, 420, 520 and 793 Hz. 4.2. Numerical simulation 2 As shown in Fig. 3, a free–free beam having the elastic modulus of E ¼ 210  109 N/m2 and mass density of r ¼ 7800 kg/m3 was modeled as an 8 d.o.f. finite element beam. In this case, 8  8 mass and stiffness matrices

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Receptance (dB ref 1m/N)

-120 -140 -160 -180 -200 -220 -240 -260

0

200

400 600 Frequency (Hz)

800

1000

Fig. 2. Comparison of reconstructed FRFs H12 of the 4 d.o.f. lumped parameter system. —, exact FRF; - - -, FRF by the proposed method; y, FRF by the normal FRF method.

1

133.33 mm

3

4 m

m

2

133.33 mm

35

14.7 mm

133.33 mm

400 mm Fig. 3. Configuration for 8 d.o.f. finite element beam model.

were condensed to 4  4 matrices by the condensation method [11] in order to later compare them with the experimental results. The condensed mass and stiffness matrices are: 2 3 0:159 0:101 0:028 0:014 6 0:101 0:523 0:021 0:028 7 6 7 M¼6 (37a) 7ðkgÞ, 4 0:028 0:021 0:523 0:101 5 2

0:014

0:028

0:101

0:159

1:217

3:022

2:080

0:275

6 3:022 6 K¼6 4 2:080 0:275

9:193 8:251 2:080

3

8:251 2:080 7 7 7  106 ðN=mÞ. 9:193 3:022 5 3:022

(37b)

1:217

It was assumed that the hysteretic damping matrix is linearly proportional to the stiffness and mass matrices. Thus the proportional damping matrix is D ¼ a  K þ b  M ðN=mÞ.

(37c)

Here, coefficients a and b are given by 0.0001 and 100, respectively, and the structural damping loss factor for the first mode is then 0.0001. As mentioned before, system matrices of the FE beam structure are condensed to the 4 by 4 matrices. Thus, 16 FRFs were used to simulate the system. Moreover, it is assumed that FRFs were contaminated by the 10% random noise in the frequency range of 1–1000 Hz with 1 Hz increment, the structural parameters were identified using the proposed method. As can be seen in Table 3 where the errors are taken in absolute value,

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Table 3 Comparison of identified structural parameters from FRFs of the FE beam model Proposed method

Normal FRF method

Estimated matrix

Absolute error (%)

Estimated matrix

Absolute error (%)

M (kg) 0.157 0.099 0.028 0.015 0.100 0.513 0.017 0.029 0.031 0.017 0.511 0.100 0.014 0.029 0.098 0.158

0.8 1.0 9.0 0.9

1.6 1.9 16.5 2.4

1.1 16.2 2.3 2.3

5.7 0.6 0.9 0.5

0.139 0.183 0.142 0.033 0.094 0.546 0.022 0.037 0.024 0.059 0.539 0.102 0.032 0.123 0.229 0.137

12.2 6.3 15.7 121.7

81.4 4.4 177.2 327.7

393.6 204.2 3.0 127.1

123.2 29.7 0.5 13.8

K(  106 N/m) 1.207 2.998 2.058 0.267 2.971 9.023 8.098 2.054 2.014 8.051 8.968 2.983 0.267 2.062 2.998 1.218

0.8 1.6 3.1 2.9

0.7 1.8 2.4 0.8

1.0 1.8 2.4 0.7

3.0 1.2 1.2 0.1

1.121 2.786 1.343 0.150 2.978 8.986 6.166 1.711 1.958 7.671 6.945 2.583 0.198 1.781 2.350 1.089

7.9 1.4 5.8 27.4

7.8 2.2 7.0 14.3

35.4 25.2 24.4 22.2

45.2 17.7 14.5 10.5

D (N/m) 137.1 292.6 204.5 26.5 290.6 969.3 820.4 206.0 205.3 831.2 971.8 295.3 24.7 202.4 287.9 136.9

0.3 0.5 0.1 4.9

0.1 0.2 0.4 1.3

0.2 0.8 0.0 1.4

2.0 125.8 281.8 195.5 35.7 0.4 292.5 988.6 803.5 212.6 1.1 208.2 818.6 977.7 286.4 0.4 16.4 179.3 288.5 137.8

8.5 0.1 1.5 36.8

3.5 1.7 1.0 12.5

4.6 2.8 0.6 1.2

37.1 3.6 1.9 0.2

the estimated results from both methods have higher errors in mass parameters than in other parameters, since the mass parameters are much smaller in magnitude than other parameters. Note that, from the results of the present method, the maximum error of estimated mass is only approximately 16.5%, and that the estimated stiffness and damping parameters are very close to the real system matrices of the FE beam structure. On the contrary, the results from the normal FRF method are erroneous, since the maximum errors in the mass, stiffness and damping parameters are about 393%, 45% and 37%, respectively. In Fig. 4, the FRFs that were reconstructed from the identified structural parameters are compared with the exact FRF H34. It is shown that the estimated natural frequencies from the proposed and Normal FRF methods are 488 and 490 Hz, respectively, which are almost same as the exact frequency, 489 Hz. However, major discrepancies between the exact and estimated FRFs appeared near the anti-resonance frequencies, when the identified matrices from the normal FRF method were used. 4.3. Experimental verification A free–free steel beam (which has the same dimensional size and physical properties as the finite element beam model of the numerical simulation 2) was used to validate the proposed method. As shown in Fig. 5, the beam was hung by two elastic cords during measurement so as to provide free boundary conditions at both of its ends. In order to approximate the beam as a 4 d.o.f. structure, 16 FRFs of the beam structure were obtained by using a PCB 352C22 uni-axial accelerometer, an ENDEVCO 2302-10 impact hammer and an LMS analyzer system (Test.Lab and SCADAS-III). The frequency range and resolution of the measurements were set to 0–1000 and 1 Hz, respectively. In Table 4, the structural identification results from the measurements are compared with those from the condensed matrices of the FE beam model in numerical simulation 2. Note that absolute errors in damping matrices are not shown here, since the exact damping matrix is not known in this case. As can be seen in Table 4, maximum absolute errors in the mass and stiffness identified from the normal FRF method are up to 208.3% and 42.9%, respectively, when comparing them with the condensed matrices of the FE beam mode. The identified mass and stiffness matrices from the present method are more than qualitative, although the identified mass from the present method also has relatively higher percentage errors compared with numerical simulation 2. It can be assumed that poor percentage errors result from the correlation errors between measured and simulated FRFs, and the numerical errors of matrix condensation for the FE beam model.

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Receptance (dB ref 1m/N)

-60 -80 -100 -120 -140 -160 -180 -200 -220

0

200

400 600 Frequency (Hz)

800

1000

Fig. 4. Comparison of reconstructed FRFs H34 of the finite element beam model. —, exact FRF; - - -, FRF by the proposed method; y, FRF by the normal FRF method.

Computer

Data-acquisition front-end (SCADAS III)

Impact hammer

Beam structure

Accelerometer

Fig. 5. Schematic of the experimental set-up (a) and photograph (b) of the steel beam.

Nevertheless, as can be seen in Fig. 6, the regenerated FRF from the proposed method is in very good agreement with the measured FRF. The reconstructed FRF from the normal FRF method has erroneous values near anti-resonance frequencies as already illustrated in simulation 2. 5. Conclusions An improved method is presented in order to identify the structural parameters of the mechanical system; for the examples herein shown, it has been demonstrated that this method is much less sensitive to measurement noise in the frequency response functions when compared with other methods. Two simulation

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Table 4 Comparison of identified structural parameters from measured FRFs Proposed method

Normal FRF method

Estimated matrix

Absolute error (%)

M (kg) 0.189 0.109 0.035 0.014

0.115 0.039 0.015 0.548 0.033 0.035 0.027 0.523 0.104 0.038 0.105 0.177

19.3 7.8 23.4 3.8

13.9 4.8 27.7 31.8

38.3 54.6 0.0 4.5

3.7 21.5 2.9 11.6

0.182 0.095 0.020 0.022

0.122 0.040 0.515 0.023 0.026 0.503 0.045 0.114

0.008 0.068 0.119 0.177

14.8 5.9 27.8 51.1

20.5 1.6 22.5 56.0

39.5 208.3 3.9 12.9

39.9 137.7 18.1 11.8

K(  106 N/m) 1.358 2.993 2.014 0.208 3.197 9.244 8.404 2.173 2.158 8.217 9.274 3.176 0.242 2.077 3.122 1.377

11.6 5.2 3.7 11.9

0.9 0.5 0.4 0.1

3.1 1.8 0.8 3.3

24.3 4.4 5.0 13.1

1.301 3.511 2.501 0.159

2.705 1.822 8.559 7.015 8.086 8.439 1.807 2.769

0.157 1.372 2.666 1.268

6.9 16.1 20.2 42.2

10.5 6.8 2.0 13.1

12.5 14.9 8.2 8.3

42.9 34.0 11.7 4.2

D (  103 N/m) 24.6 16.2 25.8 20.9 288.8 273.9 22.6 250.6 124.8 4.5 53.2 9.2

Estimated matrix

13.8 39.0 55.7 6.5

7.9 61.9 11.2 10.4

41.6 205.5 208.3 73.1

Absolute error (%)

31.5 136.3 88.3 38.8

7.0 18.7 9.6 34.2

800

1000

-60

Receptance (dB ref 1m/N)

-80 -100 -120 -140 -160 -180 -200 -220 0

200

400 600 Frequency (Hz)

Fig. 6. Comparison of reconstructed FRFs H34 of the free–free beam. —, measured FRF; - - -, FRF by the proposed method; y, FRF by the normal FRF method.

examples and an experimental example were used for analyzing the proposed method. These examples have demonstrated that the structural parameters estimated by the proposed method were more accurate than the normal FRF when the FRFs were polluted by random noise. It was also illustrated that the damping matrix can be estimated accurately using the method even in high noise level situations. Moreover, it was also verified that system parameters of a heavily damped system (e.g., up to 101 order of magnitude in the simulation 1) could be identified accurately by using the proposed method. However, those results are not shown here due to page limit. If frequencies at which FRFs are less contaminated by the random noises in FRF are known and the associated data with those frequencies are used to identify the structural parameters, then more accurate results can be obtained. Therefore, the present results recommend that further studies are needed on the selection of good frequencies for estimation, together with the analysis of more complicated test articles.

ARTICLE IN PRESS K.-S. Kim et al. / Mechanical Systems and Signal Processing 22 (2008) 1858–1868

1868

Acknowledgments This work was supported by the SNU-IAMD and Brain Korea 21 Projects of the Ministry of Education & Human Resources Development, Korea. Appendix. Kronecker product and vector-valued function To explain the Kronecker product and vector-valued function, four matrices are used here as an example. If each matrix in Eq. (19) has elements as follows " # " # " # " # a11 a12 b11 b12 c11 c12 z11 z12 HA ¼ ; HB ¼ ; HC ¼ ; ZS ¼ , (A.1) a21 a22 b21 b22 c21 c22 z21 z22 the Kronecker product and vector-valued function can be expressed as 2 3 a11 b11 a11 b21 a12 b11 a12 b21 6 a11 b12 a11 b22 a12 b12 a12 b22 7 6 7 HA  HTB ¼ 6 7, 4 a21 b11 a21 b21 a22 b11 a22 b21 5 a21 b12 a21 b22 a22 b12 a22 b22 vðHC Þ ¼ ½ c11

c12

c21

c22 T ;

vðZ C Þ ¼ ½ z11

z12

z21

z22 T ,

(A.2)

(A.3, A.4)

where the symbols  and v(  ) represent the Kronecker product and vector-valued function, respectively. References [1] J. Beliveau, Identification of viscous damping in structures from modal information, American Society of Mechanical Engineers Applied Mechanics 43 (1976) 335–338. [2] S. Burak, Y.M. Ram, The construction of physical parameters from modal data, Mechanical System and Signal Processing 15 (1) (2001) 3–10. [3] C. Minas, D.J. Inman, Identification of non-proportional damping matrix from incomplete modal information, American Society of Mechanical Engineers Journal of Vibration and Acoustics 113 (1991) 219–224. [4] C.P. Fritzen, Identification of mass, damping, and stiffness matrices of mechanical system, American Society of Mechanical Engineers Journal of Vibration, Acoustics, Stress, and Reliability in Design 108 (1986) 9–16. [5] J.H. Wang, Mechanical parameters identification with special consideration of noise effects, Journal of Sound and Vibration 125 (1) (1988) 151–167. [6] S.Y. Chen, M.S. Ju, Y.G. Tsuei, Estimation of mass, stiffness and damping matrices from frequency response functions, American Society of Mechanical Engineers Journal of Vibration and Acoustics 118 (1995) 78–82. [7] J.H. Lee, J. Kim, Identification of damping matrices from measured frequency response functions, Journal of Sound and Vibration 240 (3) (2001) 545–565. [8] J.H. Lee, J. Kim, Development and validation of a new experimental method to identify damping matrices of a dynamic system, Journal of Sound and Vibration 246 (3) (2001) 505–524. [9] P. Lancaster, M. Tismenetsky, The Theory of Matrices Second Edition with Applications, Academic Press, New York, 1985. [10] Y. Ren, C.F. Beards, Identification of joint properties of a structure using FRF data, Journal of Sound and Vibration 186 (4) (1995) 567–587. [11] R.J. Guyan, Reduction of stiffness and mass matrix, AIAA Journal 3 (1965) 380.