Quasi-static data recovery for dynamic analyses of structural systems

Finite Elements in Analysis and Design 37 (2001) 825–841 www.elsevier.com/locate/"nel

Quasi-static data recovery for dynamic analyses of structural systems Jianmin Gua , Zheng-Dong Mab , Gregory M. Hulbertb; ∗ a

Research and Vehicle Technology, Ford Motor Company, 2400 Village Road, Dearborn, MI 48124, USA b Department of Mechanical Engineering, 2250 G.G. Brown Laboratory, University of Michigan, 2350 Hayward Street, Ann Arbor, MI 48109-2125, USA

Abstract Data recovery is a post-processing procedure employed in dynamic analyses to recover the physical response from the modal responses obtained using the mode-displacement method. An enhanced data recovery process can improve accuracy by compensating for the modal truncation errors. The mode-acceleration method is a well-known data recovery technique. However, its domain of application is restricted to problems having a low-frequency band of interest. In this paper, new data recovery methods, the quasi-static data recovery methods, are presented. These new methods are based upon a quasi-static compensation technique that is applicable for any banded frequency domain of interest, with both high- and low-frequency modes truncated. Numerical examples are given to demonstrate the improvements on accuracy and applicability of the proposed method compared with traditional data recovery. ? 2001 Elsevier Science B.V. All rights reserved. Keywords: Structural dynamics; Modal analysis; Data recovery; Mode-displacement method; Mode-acceleration method; Quasi-static compensation

1. Introduction Mode-superposition techniques often are used in structural dynamic analyses for reducing the problem size, decoupling the equations, and providing physical intuition for interpreting results. The use of the classical mode-superposition method, known as the mode-displacement (M-D) method, however, may result in a loss in accuracy due to the truncation of a large number of modes. The resultant errors in the computed stresses, accelerations, and acoustic pressures may become quite unacceptable [1]. Data recovery is a procedure that can be applied at the ∗

Corresponding author. Tel.: +1-313-763-4456; fax: +1-313-747-3170. E-mail address: [email protected] (G.M. Hulbert). 0168-874X/01/$ - see front matter ? 2001 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 8 7 4 X ( 0 1 ) 0 0 0 7 0 - 1

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post-processing stage to recover dynamic responses and improve accuracy of the responses obtained by the M-D [2–5]. A well-known approach for data recovery is the mode-acceleration (M-A) method, which was "rst proposed by Williams in 1945 [2]. The M-A method has been implemented into the general "nite element analysis code, MSC=NASTRAN, as a standard option for the transient and frequency response analyses of structural and structural–acoustic systems [3]. This method has also been extended to deal with the sensitivity analysis problem [4] and dynamic simulation of Hexible multibody systems [5]. The M-A method can be used to compensate for the truncation errors that arise from ignoring the high-frequency modes in the modal response calculation process. The compensation is accomplished by re-solving the static equations of the structural system while considering the inertial loads calculated from the mode-displacement results. This compensation procedure assumes that the eIects of the truncated high-frequency 1 modes’ responses can be approximated by a static response. Therefore, the truncation errors can be compensated by solving an appropriate static problem. The major drawback of the M-A method is that it cannot be used when the low-frequency modes of the structure are also truncated. In such cases, the M-A method turns out to provide worse results than the no compensation case. Assuming eMD to be the i truncation error in the M-D method due to ignoring the ith mode, the truncation error in the M-A method due to ignoring the same mode is [1]
2  MA ei = eMD (1) i ; !i where  is the excitation frequency, !i is the natural frequency of the ith mode, and for the sake of simplicity, we consider here only an undamped frequency response problem. As shown in Eq. (1), if the frequency of the truncated mode !i is higher than the excitation frequency , then the M-A method is more accurate than the M-D method, otherwise the M-A method’s accuracy is worse. To insure the accuracy of the M-A method, it is necessary to retain all of the modes below the highest excitation frequency of interest; this is ineMcient for cases where only the midor high-frequency responses are important. In this paper, a new data recovery method, the quasi-static data recovery method (QSDR), is proposed to improve the M-A method. The QSDR method is based on a quasi-static compensation technique [1], and can be used when both lowand high-frequency modes are truncated; thus it extends the M-A method for more general applications. It will be shown that even for low-frequency problems, accuracy can be improved by using the QSDR method compared with the use of the M-A method. Quasi-static compensation techniques have been applied for dynamic analysis problems [1], component mode synthesis [6,7], and load-dependent Ritz vectors [8], by including the compensation terms (e.g., quasi-static modes) in the dynamic equations of the reduced system. In this paper, we investigate an alternate way, namely, how to apply the quasi-static compensation technique at the post-processing stage. The bene"ts from applying the quasi-static compensation technique at the post-processing stage is that one can have the full advantages of the standard mode-displacement method, i.e., minimal size of the reduced system, decoupled equations, and easy physical interpretation of results, as well as have the option to apply or not apply 1

“High” is compared with the highest excitation frequency of interest.

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compensation after "rst examining the mode-displacement results. Furthermore, for certain problems (e.g., dynamic simulation of a Hexible multibody system), including the compensation terms (e.g., static modes) in the dynamics equations may result in a stiI set of diIerentialalgebraic equations, which can cause numerical integration problems (e.g., too small integration time step, convergence problems, etc.). Applying the compensation technique after the dynamic response calculations are completed can avoid these kinds of problems. Multi-step QSDR methods are also investigated in this paper for further improving the accuracy and applicability of the method. The eIectiveness of the QSDR methods is demonstrated using a 2-D truss structure and an automotive lower control arm. 2. Theoretical background The dynamics equations emanating from a "nite element discretization of a structure can be written as M u P + C u ˙ + Ku = f;

(2)

where M , C and K are the mass, damping and stiIness matrices of size N × N ; u and f are the nodal displacement and loading vectors of size N × 1; N is the number of degrees of freedom used in the "nite element model. The corresponding undamped eigenvalue problem of Eq. (2) is (K − !i2 M )i = 0

(i = 1; 2; : : : ; N );

(3)

where !i and i are the eigenfrequency and eigenvector for the ith mode, with the following orthonormalization conditions: Ti Kj = !i2 ij

and

Ti M j = ij

(i; j = 1; 2; : : : ; N );

(4)

where ij is the Kronecker delta function. Using the classical mode-superposition technique, i.e., the M-D method, an approximated solution of Eq. (2) can be written as n  u T= i qi ; (5) i=1

where u T is the response of the system approximated by a linear combination of the "rst n normal modes of the structure; n is usually a much smaller number than N , i.e., nN ; qi denotes the ith modal coordinate. Substituting Eq. (5) into Eq. (2) and employing Eqs. (3) and (4) results in qPi + 2i !i q˙i + !i2 qi = Ti f

(i = 1; 2; : : : ; n);

(6)

where i is the ith modal damping coeMcient, and it is assumed here that the damping matrix C can also be decoupled by the normal √ modes. For frequency response analysis, let  be the it excitation frequency f = Fe (i = −1), then the following solutions can be obtained: n  T = i Qi ; (7) U i=1

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T eit and where u T = U Qi =

Ti F : !i2 − 2 + 2ii !i 

(8)

Since, if n = N , Eq. (7) gives the exact solution 2 of Eq. (2) for the frequency response, the truncation error due to ignoring the ith mode (i ¿ n) in the M-D method is eMD = i Qi : i

(9)

As mentioned in Section 1, the total truncation error (the summation of Eq. (9) from i = n+1 to N ) may become excessive due to the large number (N −n) of modes truncated. To compensate for the truncation errors in the M-D method, it is seen that when !i  , Eq. (8) can be approximated as Qi ≈

Ti F !i2

(for !i  ):

(10)

Thus the total residual displacement (the total truncation error) due to ignoring the (N − n) high-frequency modes can be approximated as U residual ≈

N  i Ti F

!i2

i=n+1

= G MA F;

(11)

where G MA is called the residual Hexibility matrix, which can be written as G

MA

N  i Ti

=

i=n+1

!i2

= K

−1

−

n  i T i=1

!i2

i

:

(12)

From Eqs. (11) and (12), we see that the residual displacement can be approximated by a static response (when the lowest truncated frequency !n+1 is much higher than the highest excitation frequency, , of interest). Thus, an improved solution can be obtained using T + U residual : U = U

(13)

Substituting Eqs. (7), (11), and (12) into Eq. (13) U = K

−1

F +

n  2 − 2ij !j  i=1

!i2

i Qi :

(14)

Eq. (14) can also be converted back to the time domain, i.e., u = K

−1

f−

n  1 i=1

2

!i2

i qPi −

n  2i i=1

!i

i q˙i :

For simplicity here we neglect degenerated systems.

(15)

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Eq. (15) is the original form of the M-A method. Equivalent forms may be written as T˙ Ku = f − M uTP − C u

(16)

Kuresidual = fresidual ;

(17)

or T is the residual displacement in the time domain, fresidual = f − M u TP − where uresidual = u − u C u T˙ − K u T is the residual force, and uT is calculated by Eq. (5). For all three forms, all terms on the right hand side are known; thus, solving a linear algebraic problem produces uresidual and u. Using Eq. (14), the truncation error due to ignoring the ith mode in the M-A method is  4 + (2i !i )2 MD MA |ei | = |ei |: (18) !i2 If



22i

!i ¿

+



1 + 44i 

(19)

then the M-A method decreases the truncation error obtained by the M-D method. Otherwise, it ampli"es the truncation error. Note that, for an undamped system, Eq. (18) is reduced to Eq. (1), and condition Eq. (19) becomes !i ¿ :

(20)

3. Quasi-static data recovery method The results of the previous section show that the M-A method cannot be used to truncate modes whose frequencies are lower than the (highest) excitation frequency. This drawback has limited the applicability of the M-A method to only low-frequency analysis problems. Quasi-static compensation [1] is based on representing the modal response (Eq. (8)) as a Taylor series: Qi =

Ti F (1 + zi + zi2 + · · ·); !i2 − !c2 + 2ii !i !c

(21)

where zi =

2 − !c2 − 2ii !i ( − !c ) !i2 − !c2 + 2ii !i !c

(22)

and !c is a user-speci"ed frequency, denoted as the centering frequency. To "rst order, the modal response can be approximated as Qi ≈

Ti F : !i2 − !c2 + 2ii !i !c

(23)

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Convergence of the Taylor series in Eq. (21) requires |zi | ¡ 1;

where

 |zi | =

(24) (2 − !c2 )2 + [2i !i ( − !c )]2 : (!i2 − !c2 )2 + (2i !i !c )2

(25)

Assuming an undamped system, from Eqs. (24) and (25), convergence requires |!i2 − !c2 | ¿ |2 − !c2 |

(26)

which implies that if the distance between the frequency of the truncated mode and the centering frequency is larger than the distance between the excitation frequency and the centering frequency, the Taylor series in Eq. (21) will converge. Eq. (26) should be compared with its M-A counterpart, Eq. (20). It is seen that the centering frequency, !c , makes it possible to truncate modes that are both lower and higher in frequency than the excitation frequency, provided that !c is chosen properly. Let us assume that the mode-displacement solution is computed by omitting both high- and low-frequency modes: n  T = U i Qi : (27) i=m

From Eq. (23), the residual displacement can be approximated as U residual = G QSDR F; where



G or

QSDR

=

m  i=1

+

N 

(28)

i=n+1

!i2

i Ti F − !c2 + 2ii !i !c

G QSDR = (K + i!c C − !c2 M )−1 −

n 

i Ti : 2 − !2 + 2i ! ! ! i i c c i i=m

Substituting Eqs. (27), (28), and (30) into Eq. (13), we obtain n  zi i Qi : U = (K + i!c C − !c2 M )−1 F +

(29)

(30)

(31)

i=m

Note that the eIects of the truncated modes are approximated by a quasi-static response that enhances the approximation used in the M-A method. In the time domain we have n  u = (K + i!c C − !c2 M )−1 f + i (ai qi + bi q˙i + ci qPi ); (32) i=m

where ai = (!c2 − 2ii !i !c )ci ; bi = 2i !i ci , and ci = (!i2 − !c2 + 2ii !i !c )−1 .

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Eqs. (31) and (32) are the basic equations for the QSDR method, which can be used to improve the accuracy for frequency response and transient response analyses. An equivalent matrix form of Eq. (32) is (K − !c2 M + i!c C)u = f − M (u TP + !c2 u) T − C(uT˙ − i!c u): T

(33)

If !c = 0, then Eq. (33) reduces to Eq. (16). Therefore, the M-A method can be considered as a special case of the QSDR method. In the more general case where i = 0, the convergence condition (Eq. (24)) becomes !i4 − 2(!c2 − 42i !c  + 22i 2 )!i2 + 2!c2 2 − 4 ¿ 0: Solving Eq. (34) for !i2 , !i2 ¿ !c2 + 22i ( − 2!c ) +



(2 − !c2 )2 + 42i ( − 2!c )(i  − !c )2

(for !i ¿ !c ); !i2 ¡ !c2 + 22i ( − 2!c ) −

(34)

(35) 

(2 − !c2 )2 + 42i ( − 2!c )(i  − !c )2

(for !i ¡ !c ):

(36)

Thus, these equations provide the necessary criteria for truncating both low-frequency (Eq. (36)) and high-frequency (Eq. (35)) modes. Since one is usually interested in the excitation within a frequency range,  in Eq. (35) should be the upper bound of the frequency range, and  in Eq. (36) should be the lower bound of the frequency range. From Eq. (31), the truncation error in the QSDR method due to ignoring the ith mode is |eQSDR | = |zi ||eMD i |: i

(37)

If the convergence conditions, Eq. (24) (Eqs. (35) and (36)), are satis"ed, the QSDR method will be more accurate than the M-D method. For the undamped system

2

 − !c2 MD QSDR

|e |: |ei | =

2 (38) !i − !c2 i 4. Multi-step QSDR methods The QSDR method can be extended to a multi-step procedure with multiple centering frequencies. For example, Eq. (33) can be modi"ed as 2 (K − !ck M + i!ck C)u(k) 2 = f − M (u P (k−1) + !ck u(k−1) ) − C(u ˙ (k−1) − i!ck u(k−1) );

(39)

where !ck is the centering frequency used in the kth step of the process, u(k) is the improved displacement obtained at the kth step, u(0) = u, T and k = 1; 2; : : : . For example, for an undamped

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system and with a two-step QSDR, we have 2 2 u(1) = (K − !c1 M )−1 [f − M (u TP + !c1 u)] T

(40)

2 2 u(2) = (K − !c2 M )−1 [f − M (u P (1) + !c2 u(1) )]:

(41)

and

Substituting Eq. (40) into Eq. (41) results in 2 2 2 2 P + !c2 u(2) = (K − !c2 M )−1 f − (K − !c2 M )−1 M (K − !c1 M )−1 {f f 2 2 2 2 −M [u T [4] + (!c1 + !c2 )u TP + !c1 !c2 u] T };

(42)

T For frequency response analysis, Eq. (42) becomes where u T [4] = (@4 =@t 4 )u. 2 2 2 2 U (2) = (K − !c2 M )−1 F + (2 − !c2 )(K − !c2 M )−1 M (K − !c1 M )−1 F n  2 )(2 − !2 )  (2 − !c2 c1 + i Qi : 2 − !2 )(!2 − !2 ) (! i i c2 c1 i=m

The truncation error in the two-step QSDR method due to ignoring the ith mode is

2

2 )(2 − !2 )

( − !c2 QSDR c1

|eMD |= 2 |ei i |: 2 )(!2 − !2 )

(!i − !c2 i c1

(43)

(44)

Comparing Eq. (38) and Eq. (44), we observe that the second QSDR step further reduces the 2 )=(!2 − !2 )| (assuming z (2) ¡ 1). Clearly, these truncation error by a factor of zi(2) = |(2 − !c2 i i c2 results can be further extended for more general cases of the multi-step QSDR methods.

5. Numerical examples In this section, two numerical examples are given to demonstrate the improved performance of the proposed QSDR methods. In both examples, the reference solution was computed by using the direct method. Comparison is made between the use of M-D, M-A, and QSDR methods. Matlab was employed for the "rst example, and MSC=NASTRAN was employed for the second example. In the second example, a DMAP alter was written to implement the QSRD methods in MSC=NASTRAN. 5.1. Cantilever truss Fig. 1 depicts a cantilever truss structure in a 2-D plane with 25 repeated structures excited by two separate external loadings. The response was measured at the driving point, f. Table 1

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Fig. 1. Model of 2D cantilever truss structure. Table 1 Eigenfrequencies of truss structure for the frequency bands of interest Mode no. 1 2 3 4 5 .. . 32 33 34 35 36 37 38 39 40 .. .

Frequency (Hz) 2.04 12.5 34.0 50.5 64.1 847.5 860.6 868.5 875.0 875.9 889.4 910.3 937.2 941.1

lists the relevant structural eigenfrequencies. Frequency responses were calculated for both lowand mid-frequency range to show the eIectiveness of the new data recovery methods. For all cases, the reference solution was obtained by direct solution of the underlying problem. The relative response error is de"ned using |Rdr − Rref | Erel = × 100%; (45) max(|Rref |; |Rconst |) where Rdr is the response predicted using one of the data recovery methods, and Rref is the reference response. To prevent biasing of the relative error when the reference response is close to zero, Rconst is set to be equal to 0.01 times the static response for frequency response analyses or the time-averaged response for transient response analyses. The "rst frequency range considered is from 0 to 50 Hz. All "ve system normal modes under 75 Hz were included in the mode-displacement calculations. Then, both the M-A and

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Fig. 2. Frequency response errors of truss structure for diIerent data recovery methods in 0 –50 Hz band.

QSDR methods were employed in the data recovery process to improve the accuracy of the response obtained by the M-D method. For the one-step QSDR method, the centering frequency (CF, to be short) was chosen as CF = 25 Hz; for the two-step QSDR method, CF1 = 10 Hz and CF2 = 40 Hz; for the three-step QSDR method, CF1 = 0 Hz; CF2 = 25 Hz, and CF3 = 50 Hz. Fig. 2 shows the response errors at the driving point of the diIerent methods. Both the M-A and QSDR methods improve the accuracy of the response obtained by the M-D method for the case in which only low-frequency responses are of interest. Note that the M-A method gives more accurate results near 0 Hz, while the one-step QSDR method gives better results in other frequency ranges, typically, around 18 Hz. Both two- and three-step QSDR methods engender more accurate results over the frequency domain than the one-step method. An “exact” response is obtained when the response frequency is close to the centering frequency. Therefore, if the responses around a certain frequency are of principal concern, the classical M-A method can be improved by choosing a centering frequency close to the frequency of interest. The second frequency range considered was from 850 to 900 Hz. All 40 normal modes of the structure under 950 Hz were retained in the M-D computation (which is labeled as M-D standard). The M-A method was employed using all 40 normal modes to recover the frequency responses. On the other hand, only 12 normal modes within the frequency range of 800 –950 Hz were employed in the QSDR methods. For the one-step QSDR, CF = 875 Hz; for the two-step QSDR, CF1 = 860 Hz; CF2 = 890 Hz; for the three-step QSDR, CF1 = 850 Hz; CF2 = 875 Hz; CF3 = 900 Hz. Fig. 3 shows the errors at the driving point for the frequency range considered. The improvement in the responses around the centering frequencies is clear when using the QSDR

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Fig. 3. Frequency response errors of truss structure for diIerent data recovery methods in 850 –900 Hz band.

methods. The QSDR methods with multiple steps provide better results than the others. For both M-D standard and M-A methods, accuracy is quite poor although many more normal modes were employed in these methods. A signi"cant advantage of the QSDR methods is that both low- and high-frequency modes can be omitted in the modal response calculations since only the mid-frequency response is of interest. While the accuracy improvement obtained by the one-step QSDR method requires no additional costs (or less costs) compared to the M-A method, when using the multi-step QSDR methods, each additional step of the multi-step methods requires more computation; therefore, a balance needs to be reached between the required accuracy and computational eMciency. 5.2. Automotive lower control arm Fig. 4 shows the top view of an FE model of an automotive lower control arm (LCA). The model consists of 1441 elements (shells, beams, and rigid elements) and 1344 nodes. The excitation was given on the ball joint in both X and Y directions, indicated by fx and fy . Constraints were added at the two bushing centers and the Z direction of the ball joint to remove the 6 rigid body motions. Mid-frequency excitation on the ball joint was considered for the transient analysis of the LCA. Fig. 5 shows the power spectral density of the excitation, having scattered energy input in the frequency range 1000 –3500 Hz, with peak energy density around 1750 Hz. For this example, there are 6 normal modes of the LCA between 1200 and 2300 Hz (see Table 2), which are

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Fig. 4. Finite element model of automotive lower control arm.

Fig. 5. FFT of LCA ball joint loads in X and Y directions.

included in the mode-displacement calculations, followed by the recovery methods. For the one-step QSDR method, CF = 1750 Hz.

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Table 2 Eigenfrequencies of LCA for the frequency band of interest Mode no.

Frequency (Hz)

.. . 5 6 7 8 9 10 .. .

1264.2 1398.2 1651.7 1957.2 2160.3 2279.4

Fig. 6. FFT of LCA displacement at ball joint, in X and Y directions, for diIerent data recovery methods.

Figs. 6 and 7 show the power spectral density of the ball joint displacement and front bushing reaction force, in X and Y directions, respectively. The improved accuracy of the power spectral density outputs using the QSDR method is clear, compared with the classical M-A and M-D methods, particularly in the frequency range of 1200 –2300 Hz. Figs. 8–11 show the time history plots of the LCA response using the diIerent recovery methods. Excellent agreement is observed in Fig. 8 for the X direction displacement recovered by the QSDR method. Due to the choice of single centering frequency and the broader frequency content, both the Y direction displacement (Fig. 9) and the X direction bushing reaction force (Fig. 10), recovered by the QSDR method, are not as accurate. As anticipated from Fig. 7, both

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Fig. 7. FFT of reaction force on LCA front bushing, in X and Y directions, for diIerent data recovery methods.

Fig. 8. LCA displacement at ball joint, in the X direction, for diIerent data recovery methods between 0.01 and 0:013 s.

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Fig. 9. LCA displacement at ball joint, in the Y direction, for diIerent data recovery methods between 0.01 and 0:013 s.

Fig. 10. Reaction force on LCA front bushing, in the X direction, for diIerent data recovery methods between 0.01 and 0:013 s.

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Fig. 11. Reaction force on LCA front bushing, in the Y direction, for diIerent data recovery methods between 0.01 and 0:013 s.

the QSDR and M-D banded methods show reasonable agreement with the reference solution for the Y direction bushing reaction force in Fig. 11. 6. Conclusions An eMcient dynamic data recovery procedure for improving the accuracy of dynamic solutions was presented. The new quasi-static data recovery methods combine the computational eMciency of the M-D method with the high accuracy of the quasi-static compensation technique. The QSDR methods proposed can be used to truncate both high- and low-frequency modes in the mode-displacement calculation process. Compared with the classical M-A method, there is no cost penalty in using the (one-step) QSDR method at the post-processing stage. Numerical examples demonstrated the performance improvements of the new methods over the classical M-A and M-D methods, in terms of computing frequency responses and transient responses. Acknowledgements The authors gratefully acknowledge support of this research by the U.S. Army TankAutomotive Research, Development and Engineering Center, through the Automotive Research Center, a U.S. Army Center of Excellence, under Department of Defense contract number DAAE07-98-3-0022.

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