Transfer function modeling of structural vibration of complex ...

Journal of Mechanical Science and Technology 27 (5) (2013) 1245~1253 www.springerlink.com/content/1738-494x

DOI 10.1007/s12206-013-0308-3

Transfer function modeling of structural vibration of complex aerospace structures based on finite element analysis † Xin Feng Wu, Yong Jun Lei*, Dao Kui Li and Yan Xie College of Aerospace Science and Engineering, National University of Defense Technology, Changsha, 410073, China (Manuscript Received May 2, 2012; Revised October 4, 2012; Accepted January 10, 2013) ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

Abstract An approach to establish the transfer function of complex aerospace structures to express their structural vibration behavior through finite element modeling is proposed. The fundamental idea of the approach is to characterize the vibrations of complicated structures through the transfer function model for an advanced control system design, while the parameters of the model are identified from the response data obtained through finite element analysis. The proposed method comprises four steps, namely, finite element modeling and validation, data preparation based on the finite element analysis, parameter identification of the transfer function, and model validation, and is presented from both frequency and time domains. The developed approach is applied to a cantilever beam, a strap-on launch vehicle, and a local part of the aerospace structure to demonstrate the method’s effectiveness and satisfactory performance. Keywords: Vibration model; Transfer function; Finite element analysis; Parameter identification ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

1. Introduction Structural vibration is a key issue in the control system design of launch vehicle and aerospace systems. Although numerous analysis techniques have been developed to model the structural vibration [1], these methods are probably difficult to apply to large-scale aerospace structures made of new materials and have high flexibility and complex configurations, among others. In practice, approximate mathematical models that contain the main dynamic characteristics of structural vibrations are typically used to reduce the difficulty of modeling and analysis. Two approaches are employed in developing mathematical models of a structural vibration system. The first is to regard the dynamical system as a physical one, expressed in terms of distributed-parameter wave equations. The other approach is to consider the structural vibration from a system identification [2] viewpoint, where state-space system models are employed. In the first modeling method, the vibration model is usually expressed by partial differential equations (PDEs) with infinite-dimension dynamic order. A well-developed example is the continuum modeling method, which has been applied extensively to model vibrations of launch vehicles [3], aerospace systems with lattice structures [4], and flexible structures [5]. However, in practice, a finite dimensional approxi*

Corresponding author. Tel.: +86 731 84572111, Fax.: +86 731 84522027 E-mail address: [email protected] † Recommended by Associate Editor Jun-Sik Kim © KSME & Springer 2013

mate model is relatively easier to use than the continuum model. An example of the former is the widely used experimental modal analysis (EMA) [6] which identifies a modal model through measured dynamic excitations and vibration responses. Modal testing and parameter identification are two main components of the EMA. Many identification methods, such as frequency domain and time domain methods, have been developed and applied to engineering structures [2, 7-11]. Recently, several new identification methods were presented [12, 13] with new applications to nonlinear systems [14, 15]. Though well developed, both classes of vibration modeling approaches have disadvantages in practical applications. PDEs can be difficult to use in solving for large strap-on launch vehicles and complicated aerospace structures, such as satellites and space stations. On the other hand, the EMA method heavily depends on experiments and testing, which require high costs and long preparations for ground tests. In this paper, a new approach to determine the vibration models of complicated aerospace structures is proposed from the viewpoint of the control system design of a large flexible launch vehicle [16, 17], that is, the transfer function (TF) is used to describe lower dynamic characteristics and subsequently appended to the rigid body of the control systems. The data of the vibration response are obtained through finite element analysis (FEA), while the TF model is determined through parameter identification. The proposed method avoids costly testing experiments and has a finite dimension that is easy to solve.

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The paper is organized as follows: the subsequent section contains an overview of the numerical modeling methods to highlight the relationship between the FEA and TF models. In section 3, an FEA-based TF method on modeling structural vibrations of complicated aerospace structures is proposed, and the main steps are presented from the frequency and the time domains; parameter identification used in the underlying TF model is discussed. In section 4, three examples are provided to illustrate the validity and effectiveness of the proposed method. The conclusions are presented in the final section.

2. Theoretical background This section reviews three kinds of basic modeling methods from which the possibility of establishing the TF model through parameter identification based on FEA can be observed. 2.1 Continuous system model In general, aerospace structures composed of beams, plates, and shells possess continuously distributed mass and elasticity, as well as infinite degrees of freedom. Under the assumptions of homogeneous and isotropic characteristics, along with Hooke’s law, the vibration of aerospace structures can be described by PDEs. For instance, in the Saturn I vehicle, which is a complex ensemble of nine beams [3], each single beam can be equivalent to a Euler beam loaded with a distributed force p ( x, t ) . The general motion equation is given by

ly useful for structural static or dynamic analysis. First, the continuous system is simplified by a discrete model with finite degrees of freedom, which greatly reduces the difficulty of vibration modeling. The resulting equilibrium of such system can be represented by the following differential equation: M {xɺɺ} + C{xɺ} + K{x} = f (t )

(3)

where M , C , and K are the mass, damping, and stiffness matrices of the system, respectively; x , xɺ , and xɺɺ , are the displacement, velocity, and acceleration of each degree of freedom, respectively; and f (t ) is the force acting on the system. Although the FE model has finite degrees of freedom, the process of transforming a continuous system to a discrete model increases the stiffness of the system, thus making it difficult to use the model to simulate structural connections. Therefore, modifications are required for the FE model in practical applications [1, 18]. After necessary modifications, dynamic analysis based on a finite element model can be conducted by using software such as MSC.NASTRAN, ANSYS, and ABAQUS, among others. 2.3 Transfer function model The TF model is commonly used to represent the relationship between the input and output of a linear time-invariant system [19-21]. Through Laplace transformation, the secondorder differential equation given in Eq. (3), can be represented by the following algebraic equation: [ Ms 2 + Cs + K ] X ( s ) = F ( s ) .

4

∂ y ( x, t ) ∂ y ( x, t ) EI + m( x ) = p ( x, t ) ∂x 4 ∂x 2

(1) The transfer function of the system can be denoted by

where E , I , and m( x) are the modulus of elasticity, crosssection moment of inertia, and the mass of the beam, respectively. For the given boundary and initial conditions, the 1-D PDE can be solved by using mode participation methods. Moreover, the mutual coupling among the nine single beams has to be considered when modeling the vibrations [3]. The continuum modeling for flexible structures can be described by a system of PDE [5] m( z )

(4)

2

∂ 2 w(t , z ) ∂w(t , z ) + D0 + A0 w(t , z ) = F (t , z ) . ∂t ∂t 2

H (s) =

X ( s) 1 = F ( s) Ms 2 + Cs + K

where F ( s ) , X ( s ) , and H ( s ) are the Laplace transform of input f (t ) and output { x} , and displacement transfer function of the system, respectively. For a more general case, Eq. (5) can be written as the following zero-pole-gain form: ( N −m ) / 2

m

(2)

Ks H (s) =

The continuous system model elucidates the physical behavior of the system, in which the parameters are strongly correlated to the physical properties of the system [11]. However, in practice, establishing a system of PDEs for large, flexible, and complex aerospace structures is difficult. 2.2 Finite element model The finite element (FE) method has been proven to be vast-

(5)

C (s) = D ( s)

∏

(s + z j )

j =1

∏

( s 2 + f k s + zk2 )

k = m +1

(6)

N

∏( s

2

2

+ 2ξi ωi s + ωi )

i =1

where C ( s) and D( s ) are polynomials, and D( s ) is of higher order than C ( s) ; N is the order of modes; ωi and ξi are the ith natural frequency and damping coefficient, where ωi = ki / mi and ξi = 0.5ci / ki mi , respectively; K s and z j (or zk ) are the gain and the zero point of the TF function, respectively; and f k is a coefficient with respect to

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be directly used in control systems. Conversely, the TF model, the simplest model used in describing the vibration, can be used in control systems, but the coefficients in the transfer function can be acquired only through conducting groundtests. Therefore, by combining TF and FEA, a new approach to establish the vibration model for control system is proposed. Fig. 1 shows the flowchart of the proposed vibration modeling method which consists of two parts: FEA and parameter identification. In the first part, an FE model of the complicated structure is established, and mode analysis and model validation should be performed to guarantee the accuracy of the FE model. Consequently, dynamic analysis, such as transient or frequency analysis, is conducted to obtain the dynamic response data. In the second part, parameter identification methods are applied to establish the TF model based on the excitation and response data obtained through FEA. Correspondingly, the proposed method can be carried out in frequency and time domains. 3.1 Frequency domain modeling The frequency domain modeling method comprises the following four steps:

Fig. 1. Flowchart of the TF modeling approach based on FEA.

zk . In the TF zero-pole-gain form, the key dynamical parameters of the system, such as the gain, natural frequencies, and damping coefficients, can be easily derived. The corresponding frequency response function (FRF) of Eq. (6) can be represented by 2N

C ( jω ) H (ω ) = = D ( jω )

∑ a ( jω )

k

k

k =0

(7)

2 N −1

∑ b ( jω ) k

k

+ b2 N ( jω )

2N

k =0

where ak and bk can be identified from the FRF that is directly obtained through testing or FEA.

(1) FE modeling and validation The geometrical and FE models of the complex aerospace structures are established by using MSC.PATRAN, by which several types of element can be selected, including the Bar, Quad, and Hex elements. Errors may result from some necessary assumptions and simplifications, which involve inaccuracy in the FE model discretization, uncertainties in geometry and boundary conditions, and variation in the material properties. Therefore, many updates and validation methods have been developed to increase the accuracy of the FE models, including those used by D. Ribeiro [22], Yuen [23], Mao [24], and Scigliano [25], which are generally based on structural responses and dynamical characteristic parameters (namely, natural frequencies, mode shapes, and frequency response functions) measured in experiments. Essentially, the FE model is updated to determine uncertain parameters in the initial model, minimizing the errors in the experimentations and the FEA results. In practice, several lower-order natural frequencies and mode shapes are generally obtained from ground testing and widely used in FE model updating. Additionally, the classical modal assurance criteria, which measure the correlation between two mode shapes, are defined as follows [26]: MACφA ,φT =

3. TF modeling based on FEA Three kinds of modeling methods of vibration are presented in section 2. Of these methods, the FE model is commonly used in practical applications to analyze the dynamic properties of continuous structures. However, the FEA results cannot

(φTA φT )2 T (φA φA )(φTT φT )

(8)

where φA and φT are the mode shape vector obtained through FEA and experiments, respectively. Experiments and tests are usually costly and require long periods of preparation. Hence, in this paper, the analytical

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results are applied to model validation and modification for economic considerations. In addition, optimization-based updating method is used to denote the objective function based on the discrepancy of the natural frequencies of interest. The uncertain parameters that need to be updated are material properties, such as the modulus of elasticity and density. (2) Data preparation using frequency analysis The data of frequency response function of the complicated aerospace structures can be obtained by using frequency analysis or tests. In this paper, frequency analysis is conducted via MSC.NASTRAN according to the excitation of a flat spectrum. The frequency band of excitation, damping coefficient, and boundary conditions are defined according to practical working environments. In practice, the damping parameter is usually considered suitable for the first two or three global mode damping coefficients identified from the test results or engineering experience. The obtained frequency response data can be used for parameter identification in the next step. (3) TF parameter identification in frequency domain System identification in frequency domain is used in estimating the coefficients of a transfer function of linear systems using experimental frequency response data [2]. Three basic elements are required to implement a system identification problem: models, inputs, and a criterion. The models should be appropriate to the system structures, and the input should have persistent excitation along with the corresponding models. The problem of identification in the frequency domain can be reduced to the fitting of a model, which is an optimization problem based on the given cost function, and can be solved by using either the linear least squares method or maximum likelihood estimation [6, 27-29]. In this paper, the TF parameter identification problem is solved by using the curve fitting command invfreqs and Levy algorithm [30] included in MATLAB. In practice, a large number of identification methods are used in the frequency domain [7, 8, 19, 27], which can be used here. (4) TF model validation Model validation is necessary in TF modeling. As mentioned above, test data are generally difficult to obtain in practice. Thus, the analytical results obtained by the FEA are used as a substitute. To guarantee higher validation accuracy, a comparison between the response results of the TF and FE models under the same excitation conditions was completed. The error ε is defined as ε=

|| xtf − xNas ||2 || xNas − xNas ||2

(9)

where xtf , xNas are the time domain responses of the TF and FE models, respectively, and xNas is the mean value of xNas . The TF model and its parameters should be modified

until the error ε is small enough (e.g., 10% in this paper) to be acceptable. 3.2 Time domain modeling Analogous to the frequency domain modeling method presented above, the TF model can also be established in the time domain through the following steps: (1) FE modeling and validation (2) Data preparation based on transient analysis In the time domain modeling method, transient analysis is performed by using MSC.NASTRAN to prepare the data for parameter identification. Excitation, which includes white noise or pseudo-random binary sequence (PRBS) signal, should cover the whole dynamic response characteristics in the band of interested frequency. The transient analysis methods can be categorized as direct transient analysis and modal transient analysis; the latter is superior in terms of computational efficiency and ability to control the frequency band. (3) TF parameter identification in time domain Many parameter identification methods are used in the time domain, such as auto-regressive (AR), moving average (MA), and auto-regressive and moving average (ARMA) methods, among others. In this paper, the ARMA model [10] is selected to characterize the dynamic response characteristics of the systems. (4) TF model validation To verify the accuracy of the TF model established in the time domain, the frequency response characteristics of the systems are utilized. Based on the FE model, frequency analysis is conducted to obtain the FRF. By comparing the amplitude and phase response curves of the FRF with those given by the TF model, the latter is modified in such a way that the error between the two models (FRF and TF) would be small enough to be acceptable.

4. Examples In this section, three examples are provided to demonstrate the validity and effectiveness of the proposed TF modeling method, which include a cantilever beam, a strap-on launch vehicle, and a local part of an aerospace structure. 4.1 Example 1: cantilever beam A cantilever beam made of 1Cr18Ni9Ti is considered, which has a length of 1 m and a cross-section diameter of 0.005 m. As shown in Fig. 2, the cantilever beam consists of 20 beam elements. The excitation is applied to node 2, and the response is obtained from node 21.

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Table 1. Natural frequencies and damping coefficients of the TF models.

NASTRAN Transient Response Time-domain Identification Frequency-domain Identification

60

40

Natural frequency (Hz) Damping coefficients

TF model (10) Value

Error (%)

TF model (11) Value

Error (%)

3.82

3.819

-0.033

3.819

-0.033

23.89

23.889

-0.004

23.889

-0.004

66.75

66.267

-0.724

66.270

-0.718

0.01

0.00938

-6.226

0.01000

0.000

0.01

0.01073

7.262

0.00999

-0.067

0.01

0.01082

8.198

0.01010

0.987

Node 2

Node 21

Input

Acceleration (m/s2)

FE model

20

0

-20

-40

-60 0.05

0.1

0.15

0.2

0.25 Time (s)

0.3

0.35

0.4

0.45

0.5

Fig. 3. Acceleration response curves of a cantilever beam obtained through different models.

Output NASTRAN Frequency Response Time-domain Identification Frequency-domain Identification

70

Fig. 2. FE model of a cantilever beam.

HTime _ D ( s ) = -0.91× 2

(s-4618)(s-355.7)(s+67.2)(s-72.2)(s +421.4s+84420) (s 2 +0.45s+575.7)(s 2 +3.22s+22530)(s 2 +9.01s+173360)

50

40

30

20

10

0

10

1

10 Frequency （ Hz ）

2

10

Fig. 4. Magnitude of the cantilever beam’s frequency response curves obtained by using different models. 200 150

NASTRAN Frequency Response Time-domain Identification Frequency-domain Identification

100 Phase angle（ deg）

First, the mode analysis with fixed node 2 is conducted to obtain the dynamic characteristics of the cantilever beam. The first three bending frequencies in the Y direction are 3.82, 23.89, and 66.75 Hz. Subsequently, the transient analysis with the PRBS excitation and frequency analysis with the flat spectrum excitation (frequency band 0 Hz to 80 Hz, mode damping coefficient 1%) are implemented by using MSC.NASTRAN. After the above analysis, the data to be used in identification and validation are obtained. With the time domain identification modeling method, the ARMA model is applied, while the results of the transient and frequency analyses are used for identification and validation, respectively. In the frequency domain identification modeling method, the Levy method is employed, while the results of frequency and transient analyses are used for identification and validation, respectively. The TF models are given by

Magnitude（ DB）

60

50 0 -50 -100 -150

(10) H Fre _ D ( s ) = -0.87 × (s-73.7)(s+68.3)(s 2 -592.7s+122200） (s 2 +593.3s+124000) . 2 2 (s +0.48s+575.8)(s +3.01s+22530)(s 2 + 8.41s+173380)

-200

0

10

1

10 Frequency（ Hz ）

2

10

Fig. 5. Phase of frequency response curves of a cantilever beam obtained by using different models.

(11) Using the TF models expressed in Eqs. (10) and (11), the first three bending frequencies and mode damping coefficient can be obtained by using Eq. (6). As shown in Table 1, the errors of natural frequencies are less than 1%, whereas the errors of mode damping coefficients identified from Eq. (10) are larger than that of Eq. (11). These errors are due to the objective function used in the identification process and can be reduced by using advanced identification methods in future work. Fig. 3 shows the acceleration response of the cantilever

beam obtained by using NASTRAN, where the TF models expressed in Eqs. (10) and (11) have the same excitation levels. The result indicates that the responses of the TF models are as accurate as the result derived by using the FE method. The magnitude and phase frequency response curves are shown in Figs. 4 and 5, where similar conclusions can be drawn. Nonetheless, the accuracy of the TF model depends on the applied identification methods. The time domain modeling method has been proven to build a TF model with higher accuracy in the time domain (96.2%), but with lower accuracy in the frequency domain (90.1%). Similar results can be found in

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Table 2. Geometric parameter values of the strap-on launch vehicle.

Frequency-domain Identification NASTRAN Frequency Response

-150

Parameter

Value

Parameter

Value Parameter Value

Rh1 (m)

2.371

D_A (m)

1.724

T_A (m)

0.008

Rh2 (m)

7.302

D_B (m)

4.200

T_B (m)

0.002

Rh3 (m)

0.897

D_C (m)

3.350

T_C (m)

0.003

Rh4 (m)

10.563

D_D (m)

1.109

T_D (m)

0.004

Rh5 (m)

2.203

D_E (m)

2.250

T_E (m)

0.003

Rh6 (m)

16.141

D_F (m)

0.150

Rh7 (m)

2.121

Total height (m)

41.6

Lift-off mass (t)

50.3

Magnitude（ DB）

-160

-170

-180

-190

-200 0

1

10

10 Frequency（ Hz ）

Fig. 7. Magnitude response curves of Z-direction angular velocity. Input

Output Z

200

●

X

Frequency-domain Identification NASTRAN Frequency Response

150

A

B

B

100

DA

Phase angle（ deg）

A

Rh1 A-A DB

Rh2

50 0 -50

DB -100

Rh3 C

-150

B-B

C

DC

D D

-200

Rh5

0

1

10

Rh4

10 Frequency（ Hz ）

Fig. 8. Phase response curves of Z-direction angular velocity.

C-C DD

E

E

D-D

DE

Rh6 E-E

F F

DF

Rh7 F-F

Fig. 6. Strap-on launch vehicle model and the details of the FE model.

the frequency domain modeling method (frequency domain 99.8%; time domain 91.5%). Therefore, a proper modeling method should be chosen, with consideration of the accuracy in the underlying domain. In practice, the dynamic characteristics of the frequency domain are easily obtained and validated, and thus, the frequency modeling method is chosen in the next two examples.

the position of inertial guidance equipment is assumed to be the response node. The angular velocity of the response node is obtained by the FEA, while the TF model is established in the frequency domain. Generally, the frequency band of interest in the launch vehicles’ control system is 10 Hz, which involves 4-6 orders of global bending modes. In addition, modal damping is used as the mode damping coefficients of the first two bending modes of the strap-on launch vehicle. In this case, the mode damping coefficients are experimentally defined as 4% (0.01 Hz to 4.0 Hz) and 2% (4.01 Hz to 15.0 Hz). The Z-direction angular velocity TF model is given as follows: H rocket _ Z ( s ) = -5.37 ×10-4 ×

4.2 Example 2: strap-on launch vehicle In this example, a more complex structure—a strap-on launch vehicle—is taken into consideration. In the control system design of a rocket, the primary factor to be considered is the frequency of vibration. Hence, the frequency domain modeling method is the appropriate approach to build the TF model. Fig. 6 shows the structures of a strap-on launch vehicle and its corresponding FE model, which consists of 3489 elements (including mass points, Beams, and MPC elements), while Table 2 gives the geometric parameter values. The FEA is applied to simulate the vehicle’s vibration in flight. The thrust data of the engine is regarded as the excitation, while

(s 2 +0.64s+24.12)(s 2 -9.33s+159)(s 2 +15.3s+823.8) (s 2 +0.63s+78.1)(s 2 +1.04s+150.1)(s 2 +1.66s+405.6) ×

(12)

(s 2 -24.29s+2766)(s 2 +5.16s+5247)(s 2 -104.1s+9068) . (s 2 + 2.37s+4000)(s 2 +2.99s+5618)(s 2 + 3.40s+8232)

Through the TF model expressed in Eq. (12), the natural frequencies and damping coefficients can be obtained. The natural frequencies of the launch vehicle are 1.35, 1.95, 3.21, 10.07, 11.93, and 14.44 Hz, while the corresponding mode damping coefficients are 3.72%, 4.24%, 4.12%, 1.87%, 1.99% and 1.87%, respectively. The maximum error of natural frequencies compared with FEA is 2.3% and that of the damping

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Table 3. Geometric parameter values of the local part of an aerospace structure.

-50

Frequency-domain Identification NASTRAN Frequency Response

-60

Value

Parameter

Value

H1 (m)

1.000

D1 (m)

0.100

H2 (m)

1.400

D2 (m)

0.090

H3 (m)

1.500

W (m)

0.105

H4 (m)

0.450

H (m)

0.105

D (m)

3.350

t (m)

0.004

θ (degree)

15

t_s (m)

0.0065

Total height (m)

4.83

Total mass (t)

0.58

-70 Magnitude（ DB）

Parameter

-80 -90 -100 -110 -120 -130 -140 0 10

1

10 Frequency （ Hz）

2

10

Fig. 10. Magnitude response curves in Y-direction acceleration. Mass point RBE3 40

Frequency-domain Identification NASTRAN Frequency Response

20 0 RBE2

Phase angle (deg)

A-A A-A

Y B-B X

-20

C-C D2 D1

-40 -60 -80 -100 -120 -140

B-B

-160 0 10

t

H

W C-C

Fig. 9. FE model of a local part of an aerospace structure.

coefficients is 7.06%, thus indicating that the natural frequencies identified by the Levy method have a higher accuracy than the damping coefficients. Therefore, future work must focus on the identification of damping coefficients. Figs. 7 and 8 demonstrate the magnitude and phase response curves obtained by the FEA and the TF, respectively, which indicate that the proposed method has successfully modeled the frequency response. In practice, the acceleration (velocity or displacement) response of the node at any position can be obtained by using this approach, and consequently, the TF models can be established in the control system. 4.3 Example 3: local part of an aerospace structure In practice, numerous complicated structures, such as satellites and aerospace stations, cannot be simplified into the beam model; thus, the existing modeling methods cannot be easily employed. In such cases, a 3D FE model can be estab-

1

10 Frequency （ Hz ）

2

10

Fig. 11. Phase response curves in Y-direction acceleration.

lished to reduce the difficulties encountered in modeling. In this example, a local part of an aerospace structure which cannot be simplified to the beam model is considered. In the FE model illustrated in Fig. 9, the elements Beam2, Quad4, and RBE2 are used to simulate the physical properties and connections between the components. The geometric parameter values of a local part of the aerospace structure are given in Table 3, and the material uses 1Cr18Ni9Ti. The frequency band of interest measures up to 100 Hz for the local vibration here, while the damping coefficient is 4%. The response of the acceleration in the y-direction is established by using the frequency domain modeling method in which the TF model is given by H aY ( s ) = -0.00044 × (s 2 +5.2s+7029)(s 2 +56.1s+127200)(s 2 42.3s+346800) . (s +20.8s+70440)(s 2 +43.4s+288300)(s 2 +48.2s+375500) 2

(13) The natural frequencies of the structure in Eq. (13) are 42.24, 85.46, and 97.53 Hz, while the corresponding damping coefficients are 3.92%, 4.04%, and 3.93%. The maximum errors of the natural frequencies and damping coefficients are both less than 3%, which proves that the TF model follows the FE model well. The magnitude and phase response curves are shown in Figs. 10 and 11, respectively. The order of the TF model is

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expected to be as low as possible in practice. With such considerations, several order reduction methods can be applied to the established TF model.

5. Conclusion A transfer function modeling method for complicated aerospace structures based on finite element analysis is proposed. The paper’s main focus is to describe the vibrations of the aerospace structures by using the TF model, whose parameters can be identified through the response data obtained by using FEA. The modeling methods in the time domain and in the frequency domain are presented, while three examples are provided to illustrate the satisfactory accuracy of the proposed modeling method.

Acknowledgment The author would like to thank the anonymous reviewers who provided helpful suggestions for the improvement of this paper, which was supported by the National Natural Science Foundation of China (Grant Nos. 11272348 and 11132012).

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Xinfeng Wu obtained his M.S. degree from the National University of Defense Technology (NUDT), China in 2009. Currently, Mr. Wu is a Ph.D. candidate of the College of Aerospace Science and Engineering at the NUDT. His research interests include structural dynamics and the control of aerospace structures. Yongjun Lei received his Ph.D. in solid mechanics at the National University of Defense Technology (NUDT), China in 1998. From 2004 to 2005, he worked as a visiting scholar in the Department of Mechanical Engineering at the University of Bristol in the U.K. Dr. Lei is currently a professor in the College of Aerospace Science and Engineering at the NUDT. His research interests include theoretical and applied computational solid mechanics.