An efficient finite element method for computing modal damping of laminated composites: Theory and experiment

Accepted Manuscript An efficient finite element method for computing modal damping of laminated composites: theory and experiment Yi He, Yi Xiao, Yanqing Liu, Zhen Zhang PII: DOI: Reference:

S0263-8223(17)31689-6 https://doi.org/10.1016/j.compstruct.2017.10.024 COST 8999

To appear in:

Composite Structures

Received Date: Revised Date: Accepted Date:

29 May 2017 10 September 2017 9 October 2017

Please cite this article as: He, Y., Xiao, Y., Liu, Y., Zhang, Z., An efficient finite element method for computing modal damping of laminated composites: theory and experiment, Composite Structures (2017), doi: https://doi.org/ 10.1016/j.compstruct.2017.10.024

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(Title Page)

An efficient finite element method for computing modal damping of laminated composites: theory and experiment Yi He, Yi Xiao *, Yanqing Liu and Zhen Zhang School of Aerospace Engineering and Applied Mechanics Tongji University, Shanghai 200092, China *Corresponding author: +86 2165983674. Email: [email protected]

Abstract: This paper presents an efficient finite element method (FEM) for computing the modal damping of laminated composites using the general purpose finite element software. The method is based on an extended elastic-viscoelastic correspondence principle, which accounts for the frequency dependence of viscoelastic complex stiffness matrices. The implementation of the proposed model is described as a UMAT subroutine for ABAQUS/Standard. The experimentally determined material complex modulus values for the carbon/epoxy laminated composites combined with the results of cantilever percussion free-decay testing are used as the input parameters for the model. Subsequently, the analyses of modal damping and frequency response for laminated composites are implemented by using the complex eigenvalue method. The computed results from this model are in good agreement with the test data. Thus, the proposed numerical method is quite efficient and accurate, and capable of providing an effective way to determine the modal damping of anisotropic materials by using ABAQUS code. Keywords: laminated composites; damping ratio; FEM analysis; elastic-viscoelastic correspondence principle; frequency dependence

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(Text)

1 Introduction With the recent advances in technology, composite materials are being commonly used in various engineering applications in the aerospace, mechanical, marine, and automotive industries. As a result, there is considerable interest in studying and understanding the dynamic behavior of composite structures. The dynamic analysis of composite structures, i.e., the study of modal parameters based on resonance frequencies, modal loss factors (damping) and modal shapes, has played an important role in structural dynamic characterization, dynamic design, damage detection and condition monitoring [1-[6]. Nowadays, the finite element method provides a very effective way for the engineering community to carry out structural dynamic response analysis. Thus, in order to allow efficient calculation of the damping characteristics of laminated composites, a numerical method with good versatility, perfect function and convenient solution needs to be established based on the general finite element software platform. The complex modulus approach and strain energy method have been extensively used for prediction of damping at micromechanical, macromechanical and structural levels in laminated composites (e.g. [7-8]). The former mainly uses the correspondence principle to describe the damping characteristics of fiber reinforced composite materials [9-10]. The latter combines the total damping of the structure with the energy dissipation and the strain energy fraction of each element to characterize the damping [11-[14]. By comparing the two theories, Salater and his colleagues [15] have theoretically proved that the strain energy method is actually a decoupled form of the complex modulus approach. In order to simplify the solution, the damping matrix is usually expressed in a proportional form to the mass and stiffness matrices, i.e., Rayleigh damping. By using the Rayleigh damping, diagonalization is formed in the damping matrix, which can decouple the primitive equation into several independent principal coordinate differential equations. Therefore, for practical applications, the strain energy method can rapidly provide sufficient accuracy without changing the constitutive equation in the approximate Rayleigh damping system. For complex damping systems, the complex modulus approach can achieve a higher precision solution. A particular area where finite element methods could prove valuable is the prediction of damping properties of vibrating structures. Kyriazoglou and Guild [16] proposed a hybrid method combining the resonance test with equivalent Rayleigh damping to achieve finite element analysis of the vibrational characteristics of the laminated composites. The Rayleigh coefficients were obtained by iterating the first-order damping ratio results of the resonance test. In another study, Berthelot et al. [17] proved that the use of a strain energy approach can be easily integrated into finite element schemes and eventually applied to several types of composite structures. By combining the loss factors deduced from testing beam specimens with the strain energy information gained from finite element analysis, it is possible to obtain 2

good estimations of the energy dissipation taking place in the structure as a whole. Zhang and Chen [18] also applied the strain energy method to ANSYS to analyze the damping characteristics of the two composite sandwich panels [08/d]S and [(45/-45)4/d]S, and quantitatively analyzed the contribution of composite damping in the two kinds of sandwich panels. The calculation results show that for [(45/-45)4/d]S, by considering the composite damping contribution, the result is 58% greater than that by ignoring this contribution, which illustrates the importance of considering the composite damping contribution. However, their studies did not consider the frequency dependence of damping in fiber composites. All of the investigations mentioned above involve the use of strain energy approach to describe composites with a general finite element code, such as ABAQUS or ANSYS. To the best of the authors’ knowledge, none of the previous research has specifically concentrated on the complex modulus approach. As the current finite element software still displays some inadequacies in the damping analysis for laminated composites, this paper established a three-dimensional complex stiffness matrix for anisotropic composites with frequency dependence and compiled it into the UMAT subroutine for ABAQUS software. A detailed discussion on the prediction of damping properties of vibrating laminates is presented, including the basic modal damping and mode shapes, frequency response and frequency-domain analysis or time-domain analysis for the dynamic response of the laminated composites. 2 Theoretical foundation for generalized complex modulus approach 2.1 Undamped vibration system In this paper, the generalized displacement field is derived by the generalized displacement theory [19] and is expressed as:  ∂w0 ( x, y; t ) + q ( z )ϕ ( x, y; t ) u ( x, y, z; t ) = u ( x, y; t ) + p ( z ) ∂x  ∂w0 ( x, y; t )  + q ( z )θ ( x, y; t ) v ( x, y, z; t ) = v ( x, y; t ) + p ( z ) ∂y   w ( x, y, z; t ) = w ( x, y; t )

(1)

If it is assumed that p ( z ) = − z , q ( z ) =0 , the theory will degrade into the Classical Lamination Theory. Also, suppose that p ( z ) =0 , q( z ) = z , the theory will degrade into First-order Shear Deformation Theory [20]. Similarly, if p ( z ) and q ( z ) are given different functions, the theory could be transformed into corresponding High-order Shear Deformation Theory, such as Reddy's Third-order Shear Deformation Theory [21], p ( z ) = − 4z3 / 3h2 , q ( z ) =z − 4 z3 / (3h2 ) ， q ( z ) =z − 4 z 3 / (3h 2 ) .

According to the geometric relationship, the strain field can be derived from the generalized displacement field as follows:

3

 ∂u ( x, y, z ) = ε x0 + p ε 1x + q ε x2 ε x = ∂ x   ∂v ( x, y, z ) = ε y0 + p ε 1y + q ε y2 ε y = ∂ y   ∂v ( x , y , z ) ∂u ( x , y , z ) + = γ xy0 + p γ 1xy + q γ xy2  γ xy = ∂ x ∂ y   ∂w ( x , y , z ) ∂u ( x , y , z ) ∂p 0 ∂q 1  γ xz = γ xz + = (1 + ) γ xz + ∂x ∂z ∂z ∂z   ∂w ( x , y , z ) ∂ v ( x , y , z ) ∂p 0 ∂q 1  γ yz = γ yz + = (1 + ) γ yz + ∂ y ∂ z ∂ z ∂z 

(2)

where ε x0 =

∂u 1 ∂2 w 2 ∂ϕ 0 ∂v 1 ∂2 w 2 ∂θ 0 ∂v ∂u 1 ∂2w ,ε x = 2 , ε x = , ε y = ,ε y = 2 ,ε y = , γ xy = + , γ xy = 2 , ∂x ∂x ∂x ∂y ∂y ∂y ∂x ∂y ∂x∂y

∂θ ∂ϕ 0 ∂w 1 ∂w + , γ xz = , γ xz = ϕ , γ yz0 = , γ 1yz = θ γ = ∂x ∂y ∂x ∂y

(3)

2 xy

For anisotropic materials, the three-dimensional stress-strain relationship corresponding to the main direction can be derived by Hooke’s law as shown in Eq. (4) below: T {σ1 , σ 2 , σ 3 , τ 23 , τ13 , τ12 } T = [C ]{ε1 , ε 2 , ε 3 , γ 23 , γ 13 , γ 12 }

(4)

The stress-stain relationship of the off-axis is defined as follows:

{σ

T

x

, σ y , σ z , τ yz , τ xz , τ xy }

= C  {ε x , ε y , ε z , γ yz , γ xz , γ xy }

T

(5)

Considering the free vibration of laminated composites, suppose the displacements in x, y and z directions are u, v and w, respectively (see Fig. 1). Based on the Hamiltonian principle, the motion equations of the laminates are written as follows: &&, x + I3ϕ&&  N x , x +N xy , y =I1u&&0 + I 2 w  &&, y + I3θ&&  N y , y +N xy , x =I1v&&0 + I 2 w  1 1 1 1 1 M x , xx +M y , yy +2M xy , xy +S yz, y +S xz , x  &&, xx + w) + I3ϕ&&, x + I4θ&&, y + I 6 w &&, yy v, y + w = I 2 (u&&, x + &&   2 2 2 &&, x + I5ϕ&& M x , x +M xy , y +S xy =I3u&& + I 4 w M 2 +M 2 +S 2 =I v&& + I w && && xy , x yz 3 4 , y + I 5θ  y, y

(6)

The derivation process and the parameters in the equation are described in detail in Appendix I and II. The five displacement variables are approximated as algebraic polynomials. x x y y 1 2 3 4 5 (u , v, w, ϕ ,θ ) = ∑ ∑ (α mn , α mn , α mn , α mn , α mn )( ) m (1 − )c ( ) n (1 − ) d eiωt L L W W m = a n =b

(7)

Substituting equation (7) into equation (6) gives the simplified motion equation as shown below:

4

([ K ] − ω 2 [ M ]) {φ} = {0}

(8)

At present, there are two different methods for solving eigenvalue problems; one is called the transformation method, and the other is called the vector iteration method. The transformation method, also called the direct solution method, directly conducts a series of transformations on the original matrix, which makes it easy to solve the eigenvalue. In the vector iteration method, the eigenvalues and eigenvectors are obtained by a series of matrix vector products. Finally, the natural frequency ωi and the corresponding mode shape

{φ}i

can be obtained. 2.2 Viscoelastic damping vibration system The elastic-viscoelastic correspondence principle is the basis of analyzing the mechanical properties of viscoelastic materials [22]. According to the correspondence principle, if we replace the stress, strain and elasticity modulus under static conditions with the corresponding dynamic stress, dynamic strain and complex modulus, the linear viscoelastic problem can be easily solved by using linear elastic analysis. In the elastic range, the mechanical properties can be simply expressed as:

E = E ( pj )

(9)

Using the correspondence principle, Hashion [9] expressed the property of the viscoelastic composite material as:

E* = E ( p j* )

(10)

p j * =p j ' [1 + iη j ]

(11)

where,

The superscript * indicates that the variables are complex numbers, the same below. In order to consider the frequency dependence of the viscoelastic composite’s dynamic property, the elastic-viscoelastic correspondence principle can be expanded, and the complex modulus can be expressed in the form of frequency function. p j * ( f ) =p j ' ( f )[1 + iη j ( f )]

(12)

Thus, the damping of composite can be derived from the complex stiffness, and the engineering constants can be expressed in the complex form as shown below: EL* = EL' ( f )(1 + iη L ( f ))

* ' ( f ) = GLT ( f )(1 + i η LT ( f )) GLT

ET* = ET' ( f )(1 + iη T ( f ))

* ( f ) = GT' T ( f )(1 + iη TT ( f )) GTT

The elastic constitutive relation of the composite is expressed as follows:

5

(13)

σij = C ε kl

(14)

ijkl

Using the correspondence principle, the viscoelastic constitutive relation of the composite is obtained as shown below: σ ij* = C ijkl *ε kl *

(15)

Cijkl * = C ijkl Real ( f ) + iC ijkl Imag ( f )

(16)

where

Similarly, considering the damping of composite materials, the relevant stiffness *

coefficient can be expressed in the plural form, i.e., [Q* ],[Q ],[ A*],[B1* ],[B2* ],[D1* ],[D1*] and [ E * ] (See Appendix II for details). The complex stiffness matrix [ K * ] of the eigenvalue equation can be written as: [K * ] = [KR ] + i[KI ]

(17)

Thus, the eigenvalue problem is transformed into the complex eigenvalue problem. Equation (8) is transformed into: (  K *  − (ω * ) 2 [ M ]) {φ * } = {0 }

(18)

where

{φ } ={φ } +i {φ } *

R

I

(19)

*

ω = ω 1 + iη The Taylor series of equation (19) is written as follows:

ω* =ω[(1 +

η2 8

−

5η 4 η η 3 7η 5 + L) + i( − + − L)] 128 2 16 256

(20)

Thus, we have, ωd =Re(ω * )=ω (1 +

η2 8

−

5η 4 + L) 128

(21)

The modal damping ratio is given by equation (22):

( ) 2

ηi = Im (ω* ) / Re(ω* )

2

(22)

3 Finite element implementation In this paper, the finite element software-ABAQUS is used to calculate the damping characteristics of laminated composites, which provides theoretical and technical support for rapid and accurate analysis of complex structural problems in engineering. As the dynamic analysis module of ABAQUS software does not support the direct definition of anisotropic viscoelastic materials, it is not convenient for the majority of users. Therefore, with the secondary development interface UMAT provided by ABAQUS, the complex stiffness matrix 6

of viscoelastic composites can be introduced with experimental data. A convenient solution to the dynamics of laminates can be achieved with this approach. 3.1 Material damping definition The complex Jacobian matrix-DDSDDE (NTENS, NTENS, 2) is defined in the UMAT, and the real and imaginary parts of the complex stiffness matrix are defined as follows: DDSDDE ( i , j ,1) = Cij' ( f )

(23)

DDSDDE ( i , j , 2 ) = C ij" ( f )

Thus, we realize the input of anisotropic viscoelastic damping of the composite material. Assuming that the laminates are divided into finite units, then in each cell, five unknown displacement variables (u, v, w,ϕ ,θ ) can be written as follows: n

n

n

n

n

i =1

i =1

i =1

i =1

i =1

u = ∑ uiφi v = ∑ viφi w = ∑ wiφi ϕ = ∑ ϕiφi θ = ∑ θiφi

(24)

Substituting equation (24) into equation (6), we obtain a discrete complex eigenvalue system as follows:

[M ]{u&&} + [K * ]{u} = {F}

(25)

The finite element software ABAQUS is used to solve the complex eigenvalues mainly by vector iteration method. In this paper, an improved form of vector iteration method- the Lanczos method is used to solve the complex eigenvalues of composite laminates. The modal frequencies and modal damping of the structure are calculated from equation (21) and (22). 3.2 ABAQUS implementation Fig. 2 shows the flow chart of the finite element analysis process and the main steps are as follows: （1）Determine the initial fundamental frequency. The former N-order natural frequencies f0r ( r = 1,2L N ) of the undamped laminates are calculated using the ABAQUS frequency extraction analysis step. （2）Iterative solution of the r-order modal damping. i. Solution of modal parameters of damped laminates The basic complex modulus values corresponding to the input frequency f0r and the three-dimensional viscoelastic constitutive equations written by UMAT are called. The former N-order natural frequency f1r and modal damping η1r of the damped laminates are calculated using the ABAQUS complex frequency extraction analysis step. 7

ii. Approximate true damping results Considering the frequency dependence of damping, the natural frequency fir and modal damping ηir of the first N-order are calculated again by the complex frequency extraction analysis step according to the basic complex modulus corresponding to: f1r . ( i = 2,3, 4L) iii. Condition for judging iteration stop When the absolute value of the two natural frequencies is less than 0.1 Hz, the iteration is stopped and the final iteration results fir and ηir are taken as the r-order natural frequency and modal damping of the laminates, respectively. （3）Repeat step (2) until all the Nth-order modal damping ratio are obtained and the analysis is finished. 4 Experiment 4.1 Specimens and experimental methods The raw material used for preparing the composite specimens is a prepreg cloth (Weihai Guang Wei Carbon Fiber Co., Ltd., China) which consists of carbon fiber and epoxy resin (T300/7901). According to certain specifications, 16 prepreg cloths are fabricated into a laminate by a vacuum pressing machine using the molding process. The initial size of the laminate is 360 mm × 360 mm × 2 mm, which is then cut into a 30 mm wide strip sample by a cutting machine. In order to investigate the frequency dependence of the complex stiffness, different lengths of the specimen between the range of 320 mm-150 mm are used in the study. The sample size is shown in Fig. 3 and the test system is shown in Fig. 4. The specimen is placed vertically and the top 50 mm is clamped and tightened by the fixed bracket to provide fixed boundary conditions. A non-contact eddy current displacement sensor (DH910) is placed at the end of the specimen and a data acquisition system (DH5923N) is used to record the time domain signal for the specimen after the end of free-time percussion. Finally, the corresponding kinetic parameters are calculated by a computer. In this paper, the cantilever percussion free-decay method is used to test the damping of the specimen, based on the general method of testing the dynamic characteristics of materials developed by the International Organization for Standardization. Compared with the resonance method, the percussion free-decay method has the advantages of rapidity, accuracy and minimal environmental effect, and is suitable for the small damping test. Fig. 5 shows the free-decay time-domain signal of unidirectional laminate specimen of 0˚ with a length of 250 mm. Based on vibration mechanics, it is known that when the cantilever beam of viscoelastic material vibrates freely, the displacement response of the end can be expressed by Equation (26). 8

A ( t )=e-ηωn t ( c1 cosωd t + c2 sinωd t )

(26)

The method of obtaining the damping is as follows. Firstly, the Fast Fourier Transform is performed on the time-domain signal at the end of free vibration, and the natural frequency of the specimen is determined by the peak point of the transformed curve. Secondly, a suitable MATLAB program is written and the exponential expression of the formula (26) is used to fit the original time-domain signal. Finally, the first-order damping ratio of the specimen is obtained. In order to eliminate the interference of the higher-order signals which decay faster in the free-fall signal, the fitting interval is set to 3.6 s before decay to the environment signal of 0.05 mm. The test procedure is shown in Fig. 6. The initial length of the specimen is 320 mm. The specimen is clamped into the test system. Considering the influence of air damping, the specimen is tested repeatedly for more than 10 times. The intensity of percussion is controlled carefully to ensure that the maximum amplitude of the end is in the range of 0.2-0.8 mm. Then, all the percussive free-decay time-domain signals are processed, and the damping value of the corresponding resonant frequency is obtained. After that, the specimen is removed and its length is adjusted by cutting to obtain the corresponding material damping at other frequencies. The minimum length of the specimen is 150 mm, which is limited by the laboratory equipment. 4.2 Effect of air damping Adams and Bacon [23] studied the effect of air damping during the experiment by comparing the damping test results of the double cantilever beam aluminum alloy specimens in air and vacuum environment, and found that the air damping is proportional to the ratio of amplitude and specimen thickness. Therefore, the damping value of the experimental test includes two parts (not considering the end of the energy-holding side of the dissipation), one is the damping of the specimen itself and the other is air damping. As shown in equation (27), the value ηl is independent of the amplitude and ηa is proportional to the amplitude. In the damping result of the multi-percussion test shown in Fig. 7, the black points denote the damping value of 0˚ unidirectional laminate with the length of 200 mm, which are linearly fitted as shown by the red line. The intersection point of the fitting line and the ordinate gives the value of ηl .

η

test

A( L) =η + η = η + k l a l h

(27)

4.3 Basic parameter determination When calculating the modal parameters of laminates using the finite element method 9

based on the extended correspondence principle, it is necessary to input the complex engineering modulus of the composite material, which is obtained by performing a percussion experiment on the 0˚, 90˚ and 45˚ unidirectional laminates. The experimental results are fitted with the logarithmic expression, and the relationship between the complex modulus and the modal parameters is derived. Fig. 8 shows the variation of the 0˚, 90˚ and 45˚ unidirectional plate damping with the frequency; the black points are the test results of different lengths, and the red curve is the logarithmic fitting curve. The damping increases with the increase in frequency, and the damping increases rapidly at frequencies around 250 Hz. After 250 Hz, the damping increases slowly and gradually becomes stable. The experimental results of the 0˚ unidirectional plate show the frequency dependence of the longitudinal complex modulus, and the corresponding damping varies from 0.03% to 0.11% in the range of 30-500 Hz. The results of the 90˚ unidirectional plate show the frequency dependence of the transverse complex modulus, and the corresponding damping varies from 0.27% to 0.43% in the range of 30-500 Hz. The results of the 45˚ unidirectional plate show the frequency dependence of the shear modulus, and the corresponding damping varies from 0.26% to 0.42% in the range of 30-500 Hz. The results show that the frequency dependence of energy dissipation in the carbon fiber composites is mainly caused by viscoelastic matrix. Table 1 shows the input material parameters for the finite element calculation. The expression of complex engineering modulus is shown in formula (28). E1* =16 × e10 × (1 + i (2.29 × e −4 ln( f ) − 3.25 × e −4 ) × 2π ) Pa E2* =8.7 × e9 × (1 + i (3.965 × e −4 ln( f )+1.63 × e −3 ) × 2π ) Pa * 12

9

−4

(28)

−4

G =6.5 × e × (1 + i (3.99 × e ln( f )+17.4 × e ) × 2π ) Pa

5

Verification and discussion

5.1 Frequency dependence of damping of laminated composites In order to verify the effectiveness of the finite element method, frequency dependence of the damping of three kinds of laminated plates-cross-ply, quasi-isotropic and 30˚ symmetry angle-ply, is calculated in the range of 30-250 Hz (For plates-cross-ply, quasi-isotropic and 30˚ symmetry angle-ply, the length ranges from 250mm to 121mm, 235mm to 92mm and 240mm to 94mm, respectively.), and the results are compared with the corresponding experimental results. Fig. 9 (a) presents the experimental and finite element calculation results of the cantilever cross-ply laminates, in which the black points represent the experimental values, and the red points represent the finite element calculation results. It can be seen that, compared to the experimental damping results, the average error is 2.2% for the calculated values. The damping ratio varies by 0.025%, from 0.065% to 0.09%. Fig. 9 (b) shows the experimental and finite element results of the cantilever beam with quasi-isotropic laminates. The average error is 3.3% compared with the experimental data. The damping ratio varies by 10

0.043%, from 0.072% to 0.115%. Fig. 9 (c) shows the experimental and finite element results of the cantilever beam with 30˚ angle-ply laminates. The average error is 3.0% compared with the experimental data. The damping ratio varies by 0.02%, from 0.17% to 0.19%. The damping of the 30˚ angle-ply laminates is the largest, nearly twice that of the orthotropic and quasi-directional laminates. Also, the damping of the quasi-isotropic laminates is slightly larger than that of the orthotropic laminates. This is mainly due to the fact that the 0˚ ply contained in the laminates significantly limits the energy dissipation while increasing the rigidity. The damping range of the quasi-isotropic laminates is the largest, which is mainly affected by the 45˚ layer. 5.2 Time-domain free-decay curves and frequency response curves Fig. 10 shows the time-decay curves for the three kinds of laminates with the length of 193 mm as example. Similar to the experiment, the test piece is fully fixed at one end and the other end is free. A unit pulse is applied to the base of the specimen and the displacement response at the end of the specimen is calculated. Finally, the initial maximum displacement is normalized. Then, the results of the finite element simulation of the time domain are compared with the results of the percussion experiment, as shown in Fig. 10. The solid line in the figure represents the contour line of the attenuation curve calculated by the finite element method, and the dotted line represents the result of the percussion experiment. The red, blue and green curves correspond to the orthotropic, quasi-isotropic and 30˚ symmetry angle laminations, respectively. It can be seen that the time-domain attenuation curve calculated by this finite element method is in good agreement with the experimental results for these three commonly used laminates. In general, the degree of decay is reflected by the logarithmic decay rate δ. The 30˚ angle-ply laminate shows the fastest decay at this length, and the logarithmic decay rate is

11.62 ×10−3 . The next fastest decay is shown by the quasi-isotropic laminates, with the −3

logarithmic decay rate of 5.655×10 , while the cross-ply laminates show the slowest −3

attenuation, with the logarithmic decay rate of 4.398×10 .Fig. 11 shows the corresponding frequency response curves obtained by processing the time-domain signal by Fourier transform. The finite element results are in good agreement with the experimental data, and the peak point of the frequency response curve corresponds well to the resonance frequency of the specimen. At this length, the resonant frequencies of the 30˚ symmetry angle-ply, quasi-isotropic and cross-ply laminates are 58.54 Hz, 71.83 Hz and 75.49 Hz, respectively. Then, using the finite element steady-state analysis, the frequency response curve of the laminated plate is calculated. As shown in Fig. 12, the third-order result is larger than the experimental one, but the first two results are in good agreement with the experimental values. What's more, the experimental frequency response of [0/90]4s beam shows a split mode and an 11

anti-resonance in the third-order mode about 1000 Hz, which seems to exhibit a high damping trend. Thus, it can be noticed that the finite element steady-state analysis can't effectively simulate this situation, and a method based on unsteady-state needs to be proposed. 5.3 Modal damping and mode shapes In order to discuss the effect of different ply angles on modal damping, the symmetry angle-ply specimens with length of 193mm and interval angle of 15˚ are calculated. Fig. 13 shows the variation of the modal properties with the ply angle θ for symmetry angle-ply laminates: (a) the modal damping, (b) the bending/torsional loss factor, and (c) the first four modes , which correspond to I-bending mode, II-bending mode, I-in-plane bending mode and I-torsional mode denoted by black, red, green and blue points, respectively. The experimental data for I-bending mode damping given in the figure is derived from the literature [25]. It can be seen that the modal damping corresponding to the first three bending modes (including in-plane bending) exhibits a law that increases significantly as the angle θ increases, with a more rapid increase rate between 15˚ and 60˚ and a slow one between 60˚ and 90˚. The first-order results of this finite element analysis are in good agreement with the experimental values in the literature [25]. The maximum deviation (12%) occurs when the ply angle is 60˚. The modal damping corresponding to the torsional mode (fourth order modal) shows a law that decreases first and then increases with the angle, reaching the minimum near 45 ˚. According to elastic-viscoelastic correspondence principle (see section 2.2), the *

* equivalent bending complex stiffness Ebending and equivalent torsional stiffness Etorsion of

the laminate can be expressed as follows [26]: * ' " Ebending = Ebending + iEbending =

12 12 * ' " ，Etorsion = Etorsion + iEtorsion = 3 * −1 h 3 D11* −1 h D66

(29)

Where

D11* =

1 N * (k ) 3 3 1 N * (k) 3 3 * C z z D − , = ( C66 ) ( zk − zk −1 ) ∑( 11 ) ( k k−1 ) 66 3 ∑ 3 k =1 k =1

(30)

Then, we have

ηbending =

′′ Ebending ′ Ebending

, ηtorsion =

′′ Etorsion ′ Etorsion

(31)

Therefore, comparing Fig.13 (a) and Fig.13 (b), it is easy to see that the law showing the change of the modal damping with angle precisely reflects the corresponding the law showing the change of bending and torsional loss factor with the angle. Fig.14 shows the variation of the modal frequency (a) and the normalized equivalent bending/equivalent torsional stiffness (b) with the ply angle θ for symmetry angle-ply laminates. In contrast to the modal damping variation shown in Fig.13 (a), the modal frequency corresponding to the first three bending modes (including in-plane bending) 12

decreases with increasing angle θ , and the modal frequency corresponding to the torsional mode increases first and then decreases with the angle, reaching the maximum near 45 ˚. Similarly, comparing Fig.14 (a) and Fig.14 (b), it shows that the variation of the equivalent bending / equivalent torsional stiffness with the ply angle is consistent with the change of the modal frequency. The above shows that the modal damping and the modal frequency are clearly toward the opposite trend and satisfied with the change of the actual stiffness and energy dissipation, respectively. 6 Conclusions This paper provides a finite element method for predicting modal response of composite structures. This method is based on the extended elastic-viscoelastic correspondence principle. A complex stiffness matrix of viscoelastic composites with frequency dependence is defined based on experimental results and it is integrated into the solver of ABAQUS by the secondary development interface UMAT. The prediction and analysis of the modal parameters of any laminate can thus be realized by this approach. The main conclusions of this study are as follows: (1) Based on the development of ABAQUS user subroutine UMAT, the constitutive equation for anisotropic viscoelastic materials considering frequency dependence is successfully introduced, which improves the function of general finite element analysis software. (2) An effective material damping test method considering the effect of air damping is established, in which a cantilever beam is subjected to multiple striking tests and the measured damping varies with amplitude in a linear manner. The efficiency of the experiment is improved after the related operation is suitably programmed. (3) The frequency dependence of the damping of orthotropic, quasi-isotropic and 30˚ symmetric laminates is analyzed by using the finite element method. The average error is less than 3.3%. Moreover, the simulation results of time-domain attenuation curve and amplitude-frequency response curve are in good agreement with the experimental results. (4) The relationship between the modal damping and the angle of the symmetrical angle ply is explored by the finite element method. The experimental results of the first-order bending mode damping show good agreement with the calculated results. The modal damping of the bending mode is found to increase with the angle. The modal damping corresponding to the torsional vibration mode decreases initially up to the minimum value of around 45˚ and then increases with the angle, which is consistent with the theoretical deduction. (5) The finite element method is very effective for the analysis of the dynamic characteristics of various laminates, and it is a practical numerical tool to analyze and predict the vibrational response of composite structures. The method is expected 13

to find a wide range of applications in engineering. The novelty of this study is that it can effectively introduce classical theory into ABAQUS software using a user-defined material model, which will be specifically targeting integration of composites designing, dynamic mechanics analysis, and damage progression modeling codes to address composite material development and application issues. The benchmarks evaluation for the comparison of this method and existing methods has been demonstrated in early study [27], where showed the validity and the correctness for this model. This together with the application of this method in sandwich composite structures [28] and bolted composite joints [29] for the vibration damping analysis are the outlook and the major directions of our ongoing investigations.

Acknowledgements This research was performed as a part of project HSSF- High strength steel fatigue characteristics (1330-239-0009). The authors gratefully acknowledge the support provided by Hitachi Construction Machinery Co. Ltd. (Japan).

AppendixⅠ Ⅰ: Motion equations for laminates Using the Hamilton variational principle, the motion equations of the laminated composites are derived. The total energy in the laminate is expressed as follows: (I-1) Π =U − V − T Variation of the total energy is given by the below equation;

∫

t2

t1

(δ U − δ V − δ T )dt = 0

(I-2)

There is no external force in the free vibration of the laminate. Equation (I-2) is developed and simplified as follows: t2

∫ [∫ ((σ δε x

x

&&δ w))dV ]dt = 0 + σ yδε y + σ zδε z +τ xyδγ xy +τ yzδγ yz +τ xzδγ xz ) − ρ(u&&δ u + v&&δ v + w

(I-3)

t1 V

Substituting equation (2) and equation (5) into equation (I-3) gives the below equation: t2

∫[∫(δuN

x,x

+δwM1x,xx +δϕMx2,x +δvNy, y +δ wM1y, yy +δθMy2, y +δvNxy,x +δuNxy, y + 2δ wM1xy,xy +δθMxy2 , x

t1 A

&&, x + I3ϕ&&) +δv(I1&& &&, y + I3θ&&) + +δϕMxy2 , y +δ wS1yz, y +δθSyz2 +δ wSxz1 ,x +δϕSxy2 )dA− ∫{δu(I1u&& + I2w v + I2w A

&&,xx + w) + I3ϕ&&,x + I4θ&&, y + I6w &&, yy ) +δϕ(I3u&&+ I4w &&, x + I5ϕ&&) +δθ(I3&& &&, y + I5θ&&)}dA]dt = 0 v + I4w δ w(I2 (u&&,x + &&v, y + w

(I-4) N, M, and S are defined as follows:

14

( N x , N y , N xy ) = ∫

h /2

− h/ 2

( M 1x , M 1y , M 1xy ) = ∫

h /2

− h/ 2

( M x2 , M y2 , M xy2 ) = ∫

N

(σ x ,σ y ,τ xy )dz = ∑ ∫ k =1

zk +1

zk

(σ x ,σ y ,τ xy )dz N

(σ x ,σ y ,τ xy ) p( z )dz = ∑ ∫

h/ 2

− h /2

k =1 N

(σ x ,σ y ,τ xy )q( z )dz = ∑ ∫ k =1

zk +1

zk zk +1

zk

(σ x ,σ y ,τ xy ) p( z )dz (σ x ,σ y ,τ xy ) q( z )dz

(I-5)

N zk +1 ∂p( z ) ∂p( z ) ) dz = ∑ ∫ (τ xz ,τ yz )(1 + ) dz − h/ 2 z ∂z ∂z k =1 k N h /2 zk +1 ∂q( z ) ∂q( z ) ( S xz2 , S yz2 ) = ∫ (τ xz ,τ yz ) dz = ∑ ∫ (τ xz ,τ yz ) dz − h/ 2 z ∂z ∂z k =1 k

( S1xz , S 1yz ) = ∫

h /2

(τ xz ,τ yz )(1 +

( I1, I 2 , I 3 , I 4 , I 5 , I 6 ) =∫

h /2

− h/ 2

(I-6)

ρ (1, p ( z ), q ( z ), p 2 ( z ), p ( z )q ( z ), q 2 ( z ))dz

AppendixⅡ Ⅱ: Parameter Expression Substitute equation (4) into equation (I-5), and integrate along the thickness. Then, the internal force and the internal torque in the unit area can be expressed as follows: N  A  1  1 M  = B 2  2  M   B

B 2  ε0    D 2   ε1  E  ε 2 

B1 D1 D

 S1   F  2 =   S  G

2

(II-1)

G  γ 0    H   γ 1 

(II-2)

where T

N = {N x

Ny

N xy } ，M 1 = {M 1x

S 1 = {S 1yz

S xz1 } , S 2 = {S yz2

T

T

M xy2 } ,

T

T

T

T

T

ε xy1 } , ε 2 = {ε x2 ε y2 ε xy2 } ,

T

γ 0 = {γ yz0

γ xz0 } , γ1 = {γ 1yz γ 1xz } ;

 A11 A =  A12  A16

A12 A22

 F44 F =  F45

M y2

S xz2 } ,

ε 0 = {ε x0 ε y0 ε xy0 } , ε1 = {ε x1 ε 1y

 D111  D1 =  D121  D161 

T

M 1xy } , M 2 = {M x2

M 1y

 B111 A16   A26  , B1 =  B121  B161 A66  

 B112 B122 B162  B161    2 1 2 2 2  B26  , B =  B12 B22 B26 ， 1 1   B162 B262 B662  A26 B26 B66    1 1 1 1 1  D11 D12 D16  D12 D16   E11 E12 E16   1 1 1  2 1 1  D22 D26  , D =  D12 D22 D26  , E =  E12 E22 E26 ， 1 1  1 1   D161 D26  E16 E26 E66  D26 D66 D66    F45  G44 G45   H 44 H 45  ， G= ，H =    ; F55  G45 G55   H 45 H 55  B121 1 B22

where, the stiffness coefficient is defined as follows: 15

(II-3)

h/ 2

( Aij , Bij1 , Bij2 , Dij1 , Dij2 , Eij ) =

∫

C ij (1, p ( z ), q ( z ), p 2 ( z ), p ( z ) q ( z ), q 2 ( z ))dz ,(i, j ) = (1, 2, 6)

− h/ 2 h/ 2

∂p ( z ) 2 ∂q ( z ) ∂p ( z ) ∂q ( z ) 2 ( Fij , Gij , H ij ) = ∫ C ij ((1 + ) , (1 + ), ( ) ) dz， (i , j ) = (4,5) ∂z ∂z ∂z ∂z − h/ 2

(II-4)

Combine the same stiffness coefficient in equation (I-4) to obtain the motion equation (5) of the laminate. Appendix III: Effect of Boundary Conditions In reality, there are many factors that affect modal damping of laminated composites, where boundary conditions are one of the major factors. Therefore, 5 comparisons are made between the results calculated by proposed method and the original ones in reference [30] for the cross-ply plate with 5 boundary conditions (i.e., FFFF, CCCC, CCFC, CFCF and CFFF, C represents the Cantilever boundary condition and F represents the Free boundary condition), which are shown in Table A1 to Table A5. From these comparisons, it can be noticed that the results calculated by proposed method are in good agreement with the original ones in FFFF and CCCC, however there are obvious differences under CCFC, CFCF and CFFF. In this paper, as the focus is not on this issue, the reasonable explanation is not able to be given here, but the future research can concentrate on that. Reference [1] Zhang, Z., Xiao, Y., Liu, Y. Q., & Su, Z. Q. A quantitative investigation on vibration durability of viscoelastic relaxation in bolted composite joints. Journal of Composite Materials, 2016, 50(29): 4041-4056. [2] Doebling, Scott W., Charles R. Farrar, and Michael B. Prime. "A summary review of vibration-based damage identification methods." Shock and vibration digest 30.2 (1998): 91-105. [3] Fan W, Qiao P. Vibration-based damage identification methods: a review and comparative study. Structural Health Monitoring, 2011, 10(1): 83-111. [4] Flor F R, de Medeiros R, Tita V. Numerical and Experimental Damage Identification in Metal-Composite Bonded Joint. The Journal of Adhesion, 2015, 91(10-11): 863-882. [5] Yang C, Oyadiji S O. Detection of delamination in composite beams using frequency deviations due to concentrated mass loading. Composite Structures, 2016, 146: 1-13. [6] Zhang, Z., Xu, H., Liao, Y., Su, Z., Xiao, Y., Vibro-Acoustic Modulation (VAM)-inspired Structural Integrity Monitoring and Its Applications to Bolted Composite Joints, Composite Structures(2017), doi: http://dx.doi.org/10.1016/j.compstruct.2017.05.043 [7] Chandra, R., S. P. Singh, and K. Gupta. "Damping studies in fiber-reinforced composites–a review." Composite structures 46.1 (1999): 41-51. [8] Treviso A, Van Genechten B, Mundo D, et al. Damping in composite materials: properties and models. Composites Part B: Engineering, 2015, 78: 144-152. 16

[9] Hashin Z V I. Complex moduli of viscoelastic composites—I. General theory and application to particulate composites. International Journal of Solids and Structures, 1970, 6(5): 539-552. [10] Gibson R F, Plunkett R. Dynamic mechanical behavior of fiber-reinforced composites: Measurement and analysis. Journal of Composite Materials, 1976, 10(4): 325-341. [11] Adams, R. D., and D. G. C. Bacon. "Effect of fibre orientation and laminate geometry on the dynamic properties of CFRP." Journal of Composite Materials 7.4 (1973): 402-428. [12] Hwang, S. Jimmy, and Ronald F. Gibson. "The use of strain energy-based finite element techniques in the analysis of various aspects of damping of composite materials and structures." Journal of Composite Materials 26.17 (1992): 2585-2605. [13] Ohta Y, Narita Y, Nagasaki K. On the damping analysis of FRP laminated composite plates. Composite Structures, 2002, 57(1–4):169-175. [14] Tsai J L, Chang N R. 2-D analytical model for characterizing flexural damping responses of composite laminates. Composite Structures, 2009, 89(3):443-447. [15] Slater, Joseph C., W. Keith Belvin and Daniel J. Inman. "A comparison of viscoelastic damping models." 5th NASA/DOD Controls-Structures Interaction Technology Conference. Vol. 1. 1993. [16] Kyriazoglou C, Guild F J. Finite element prediction of damping of composite GFRP and CFRP laminates–a hybrid formulation–vibration damping experiments and Rayleigh damping. Composites Science and Technology, 2007, 67(11): 2643-2654. [17] Berthelot J-M, Assarar M, Sefrani Y, El-Mahi A. Damping analysis of composite materials and structures. Compos Struct 2008; 85(3):189-204. [18] Zhang S H, Chen H L. A study on the damping characteristics of laminated composites with integral viscoelastic layers. Composite Structures, 2006, 74(1): 63-69. [19] Qu Y, Long X, Li H, et al. A variational formulation for dynamic analysis of composite laminated beams based on a general higher-order shear deformation theory. Composite Structures, 2013, 102: 175-192. [20] Altenbach H, Eremeyev V A, Naumenko K. On the use of the first order shear deformation plate theory for the analysis of three‐layer plates with thin soft core layer. ZAMM‐Journal of Applied Mathematics and Mechanics/Zeitschriftfür Angewandte Mathematik und Mechanik, 2015, 95(10): 1004-1011. [21] Jin G, Yang C, Liu Z. Vibration and damping analysis of sandwich viscoelastic-core beam using Reddy’s higher-order theory. Composite Structures, 2016, 140: 390-409. [22] Lakes R S. Viscoelastic materials. Cambridge University Press, 2009. [23] Adams R D, Bacon D G C. Measurement of the flexural damping capacity and dynamic Young's modulus of metals and reinforced plastics. Journal of Physics D: Applied Physics, 1973, 6(1): 27. [24] Hu B G, Dokainish M A. Damped vibrations of laminated composite plates—modeling and finite element analysis. Finite elements in analysis and design, 1993, 15(2): 103-124. 17

[25] Mahi, A. E., Assarar, M., Sefrani, Y., & Berthelot, J. M. Damping analysis of orthotropic composite materials and laminates. Composites Part B Engineering. 2008, 39(7–8), 1069-1076. [26] Crane R M, Gillespie J W. Analytical model for prediction of the damping loss factor of composite materials. Polymer Composites, 1992, 13(3): 179-190. [27]

LIU YQ, XIAO Y, ZHANG Z and HE Y. Modal damping prediction of laminated

composites using elastic-viscoelastic correspondence principle: Theory and finite element implementation. Acta Materiae Compositae Sinica, 2017, 34(7): 1478-1488 (in Chinese). [28]

Yanqing Liu. Finite Element and Experimental Study on Damping Properties of

Composite Materials and Structures Using a Complex Modulus-Based Approach. Diss. Tongji University, 2017. [29]

Z. Zhang, Y. Xiao, Y. Z. Shen and Z.Q. Su: A MULTISCALE MODEL FOR MODAL

ANALYSIS OF COMPOSITE STRUCTURES WITH BOLTED JOINTS. 21st International Conference on Composite Materials. Xi’an, 20-25th August 2017. [30]

Maheri M R. The effect of layup and boundary conditions on the modal damping of

FRP composite panels. Journal of Composite Materials, 2010, 45(13):1411-1422.

18

Nomenclature u ,v ,w ϕ ,θ

p(z) , q (z)

εx , ε y γ xy , γ xz , γ yz

{σ} , {ε }

Shear strain

[C]

Stress and strain vector of principal direction Stiffness matrix in local coordinate system

C   

Stiffness matrix in global coordinate system

U V

T N , Μ1 , Μ2 , S 1 , S2 A, B ,C , D, E ,F ,G , H ρ

a ,b , c ,d L W i (i = 1L5) αmn

[ K] ω

[M ] {φ } ωi {φ }i E pj p j* p 'j ( f )

ηj ( f ) E L* E T*

Displacement in x , y , z directions Displacement parameters influenced by transverse shear effect Shape functions of transverse shear strain and stress distribution along z direction Normal strain

(f) (f)

* (f) GLT * GTT ( f ) *

[K ]

[KR ] [ KI ]

Total strain energy External work Total kinetic energy Internal force and internal moment per unit area Stiffness coefficient Density Coefficients associated with boundary condition Plate length Plate width Unknown coefficients of displacement Total stiffness matrix Eigenvalue Total mass matrix Displacement vector First i -order natural frequency First i -order modal shape Elastic modulus Elastic modulus of composite j th sub-phase Complex modulus of composite j th sub-phase Real part of p j * Loss factor of complex modulus Longitudinal complex modulus Horizontal complex modulus In-plane shear complex modulus Out-plane shear complex modulus Complex stiffness matrix Real part of [ K * ] Imaginary part of [ K * ]

ω*

complex eigenvalue

{φ }

complex eigenvector

*

19

η

ωd ωn ηi n φi ui , vi , wi , ϕi , θi {F }

c1 , c2

η test ηl ηa k A( L )

h

Loss factor Damped natural frequency Undamped natural frequency Modal damping radio Node number in element Interpolation function Node displacement value External force Unknown coefficients associated with initial incentive condition Experimental damping value Damping of laminated plates Air damping Scale factor Amplitude peak of cantilever beam Specimen thickness

20

Table and Figure Captions:

Figure 1 Schematic of the laminated plates and loading coordinate systems. Figure 2

Figure 3

Flowchart of the numerical analysis process.

Dimensions of the composite specimen.

Figure 4 Experimental system set up. Figure 5 Data processing of the free-decay time-domain signal. Figure 6

Flow chart of the experimental procedure.

Figure 7 Plot of air damping vs amplitude. Figure 8 Experimentally determined 1st frequency-dependent damping for unidirectional laminates (a) 0 ˚; (b) 90 ˚; (c) 45 ˚. Figure 9 Calculated vs. experimentally determined frequency-dependent damping for different laminated plates (a) cross-ply; (b) quasi-isotropic; (c) 30˚ angle-ply. Figure 10 Time-decay curves for the three laminates (a) cross-ply; (b) quasi-isotropic; (c) 30˚ angle-ply. Figure 11

Amplitude-frequency curves of three laminates of 1st order.

Figure 12 Amplitude-frequency curves of three laminates of first three orders (a) cross-ply; (b) quasi-isotropic; (c) 30˚ angle-ply. Figure 13

Modal damping vs. angle of symmetric angle-ply laminates (top) and the corresponding modal shapes (bottom).

Figure 14

Modal frequency vs. angle of symmetric angle-ply laminates.

Table 1 Material properties of T300/7901 Carbon/epoxy composites. Table A1 The results from FFFF boundary conditions Table A2 The results from CCCC boundary conditions

21

Table A3

The results from CCFC boundary conditions

Table A4

The results from CFCF boundary conditions

Table A5

The results from CFFF boundary conditions

Figure 1

Schematic of the laminated plates and loading coordinate systems.

Figure 2

Flowchart of the numerical analysis process. 22

23

Figure 3

Dimensions of the composite specimen.

Figure 4

Figure 5

Experimental system set up.

Data processing of the free-decay time-domain signal.

24

Figure 6

Flow chart of the experimental procedure.

Figure 7

Plot of air damping vs amplitude.

25

Figure 8 Experimentally determined 1st frequency-dependent damping for unidirectional laminates (a) 0 ˚; (b) 90 ˚; (c) 45 ˚.

26

Figure 9 Calculated vs. experimentally determined frequency-dependent damping for different laminated plates (a) cross-ply; (b) quasi-isotropic; (c) 30˚ angle-ply. 27

Figure 10 Time-decay curves for the three laminates. (a) cross-ply; (b) quasi-isotropic; (c) 30˚ angle-ply.

28

Figure 11

Amplitude-frequency curves of three laminates of 1st order.

29

(a)

(b)

(c) Figure 12 Amplitude-frequency curves of three laminates of first three orders. (a) cross-ply; (b) quasi-isotropic; (c) 30˚ angle-ply.

30

(c)

(d)

Mode 1

Mode 2

(f )

(e) Mode 3

Figure 13

Mode 4

Modal damping vs. angle of symmetric angle-ply laminates (top) and the corresponding modal shapes (bottom).

31

Figure 14 Modal frequency vs. angle of symmetric angle-ply laminates.

Table 1 Material properties of T300/7901 Carbon/epoxy composites. E11

E22

G12

G23

(GPa)

(GPa)

(GPa)

(GPa)

160.0

8.7

6.5

3.25

v12

v23

ρ 3

(kg/m ) 0.3 0.45

1690

η11

η22

η12

(%)

(%)

(%)

2.29 × 10−2 ln ( f − 3.25 × 10−2

)

3.965 ×10−2 ln ( f ) 3.99 × 10−2 ln ( f ) +0.163

+17.4 × 10 −2

Notes: In order to simplify the problem, it is assumed that order G23 = G12 / 2 based on previous work [24].

32

Table A1 The results from FFFF boundary conditions Mode

1

st

2

nd

3

rd

4

th

Ref. [30] Theo. f=27.6Hz

Theo.f=71.9Hz

Theo.f=91.1Hz

Theo.f=98.9Hz

Theo.SDC=5.74%

Theo.SDC=1.21%

Theo.SDC=2.90%

Theo.SDC=0.71%

Expr. f=28.6Hz

Expr.f=69.6Hz

Expr.f=88.3Hz

Expr.f=97.4Hz

Expr. SDC=5.73%

Expr.SDC=1.47%

Expr.SDC=3.69%

Expr.SDC=0.64%

f=27.6Hz; SDC=5.75%

f=74.6Hz; SDC=1.52%

f=93.3Hz; SDC=3.03%

f=101.1Hz; SDC=0.95%

Proposed method

*: Modal shape：Good match Frequency：Min. error=0.14%; Max. error=3.7%; Ave. error=2.10% SDC: Min. error=0.17%; Max. error=33.8%; Ave. error=10.46%

Table A2 The results from CCCC boundary conditions Mode

1

st

2

nd

3

rd

4

th

Ref. [30]

f=127Hz; SDC=1.17%

f=231Hz; SDC=1.43%

f=289Hz; SDC=0.95%

f=358Hz; SDC=1.38%

f=130.4Hz; SDC=1.43%

f=239.4Hz; SDC=1.72%

f=295.6Hz; SDC=1.19%

f=369.8Hz; SDC=1.63%

Proposed method

*: Modal shape：Good match Frequency：Min. error=2.29%; Max. error=3.63%; Ave. error=2.98% SDC: Min. error=18.12%; Max. error=25.26%; Ave. error=21.47%

33

Table A3 The results from CCFC boundary conditions Mode

1

st

2

nd

3

rd

4

th

Ref. [30]

f=101Hz; SDC=0.85%

f=131Hz; SDC=1.53%

f=235Hz; SDC=1.59%

f=274Hz; SDC=0.77%

f=89.4Hz; SDC=1.72%

f=156.6Hz; SDC=1.7%

f=238.3Hz; SDC=1.6%

f=285.8Hz; SDC=2.0%

Proposed method

*:Modal shape：Bad match Frequency：Min. error=1.40%; Max. error=19.51%; Ave. error=9.18% SDC: Min. error=3.14%; Max. error=163.64%; Ave. error=70.55%

Table A4 The results from CFCF boundary conditions Mode

1

st

2

nd

3

rd

4

th

Ref. [30]

f=136Hz; SDC=1.79%

f=250Hz; SDC=2.07%

f=296Hz; SDC=1.18%

f=380Hz; SDC=1.86%

f=101Hz; SDC=0.94%

f=105Hz; SDC=1.30%

f=139Hz; SDC=1.99%

f=248Hz; SDC=1.98%

Proposed method

*:Modal shape：Bad match Frequency：Min. error=25.83%; Max. error=58.06%; Ave. error=42.88% SDC: Min. error=34.78%; Max. error=68.64%; Ave. error=47.03%

34

Table A5 The results from CFFF boundary conditions Mode

1

st

2

nd

3

rd

4

th

Ref. [30]

f=104Hz; SDC=1.12%

f=152Hz; SDC=2.54%

f=265Hz; SDC=2.46%

f=276Hz; SDC=0.83%

f=15.9Hz; SDC=0.93%

f=23.1Hz; SDC=3.37%

f=82.8Hz; SDC=2.07%

f=99.5Hz; SDC=0.96%

Proposed method

*:Modal shape：Bad match Frequency：Min. error=63.95%; Max. error=84.83%; Ave. error=75.57% SDC: Min. error=15.66%; Max. error=32.68%; Ave. error=20.29%

35