Abstract
Identifying coupling dynamic stiffness of structural connection is often needed in substructural dynamic analysis. To overcome the faultiness of conventional approaches existed, five indirect schemes of inverse substructuring analysis by using tested frequency response functions (FRFs) are provided. And the first indirect scheme is verified by three massrubber models constructed as twolevel substructures with monocoupling, bicoupling and tricoupling connection. Compared to existing direct scheme of inverse substructuring analysis, it shows better performance with acceptable precision of determining the stiffness.
1. Introduction
Dynamic substructuring analysis based on frequency response function (FRF) has been paid attentions widely in mechanical structural dynamics. There were a number of researches reported in parameter identification of coupling dynamic characteristics of connected structures [16]. For instances, to overcome the faultiness of modal analysis that needs to establish massdampingstiffness matrices for modeling, HWANG [3] applied FRFmodel to identify parameters of the connection between structures, and eliminate lowprecision of the identification at highsensitive frequency range by use of statistical averages; Targeting at the difficulty of precisely modeling structural connecting dynamic properties by pure analytical approaches, DAMJAN and MIHA [5] identified the dynamic flexibility of bolted connection of beam structures and proposed modeling method in consideration of both translational degrees of freedom (TDOF) and rotational degrees of freedom (RDOF); MAJID et al. [6] used inverse receptance coupling(IRC) and pointmassconnecting model, which considers the connection as an element of massdampingstiffness, to identify connecting characteristics, and combined FEMsimulation with FRFtests to prove the results. These research works obtained the connecting parameters with errors to different extent for sorts of structural connections. However, they are essentially of “positive” substructuring analysis at all, which needs to predetermine the connecting boundary conditions and or some of structural dynamic metrics. Therefore, the computations and or their algorithms are complicated, testing and computational expenses in application are heavy, resulting in partial identified outcomes affected by modal coupling need to be modified via FEM and or statistical schemes.
Inverse substructuring dynamic analysis based on FRFspectra can be used to determine both substructural and coupling dynamic characteristics of structural system with different sorts of connection. It is simple with less computations, and has no needs of complicated modal tests, synthesis, predetermined constraint boundary conditions, such as structural coupling parameters [7]. LU introduced the direct scheme of inverse substructuring dynamic analysis firstly to fix the dynamic stiffness of couplingunitofpackaging [8]. And he also applied FRFbased inverse substructuring approach to analyze the dynamic characteristics [9] and do eigenvalue analysis on mechanical assembly [10]. Afterwards both direct scheme and indirect schemes of inverse substructuring analysis were introduced to analyze the dynamic quality of a mechanical assembly [11], including identification on its coupling dynamic stiffness [12], but lack of verification experimentally on them.
This study provides five potential indirect schemes of inverse substructuring analysis to determine the coupling dynamic stiffness of structural connection. Compared to existing direct scheme of inverse substructuring analysis, the first indirect scheme is verified experimentally by tested FRFs on three massrubber models of twolevel substructures with monocoupling, bicoupling and tricoupling connection. As result of this study, the first indirect scheme shows better performance with acceptable precision of determining the stiffness of structural connection.
2. Determining coupling dynamic stiffness of structural connection by inverse schemes
Fig. 1 shows a dynamic structural system with twolevel substructures A and B connected via connector. Fig. 1(a) and Fig. 1(b) indicate the dynamic excitationforce ($F$), displacement response ($X$), and representative transferfunction ($H$) before and after connection of A and B, named as “FRF at component level” (without subscript $s$) and “FRF at system level” (with extra subscript $s$), respectively. The connection is characterized by “coupling dynamic stiffness (${K}_{s}$)”. Lowercase $a$, $b$ and $c$ indicate substructures A, B and their coupling connector, and $i$ and $o$ represents excitationforce input and displacementresponse output, respectively. Thus, for examples, FRFs ${H}_{oaca}$ and ${H}_{s,oaib}$_{}represents transferfunctions from excitation force on coupling interface on substructure A ($ca$) to output displacementresponse on A ($oa$) without connecting with B, and from excitation input excitationforce on B to output displacementresponse on A connecting with B, respectively.
Fig. 1FRFs of twolevel substructures at component level a) and system level b)
a) Before connection
b) After connection
The dynamic stiffness matrix, [${K}_{s}$], of structural connection can be inversely determined by [7, 12]:
where, $T$ denotes inverse and transpose operation of matrix. Eq. (1) is often classified as “direct scheme of inverse substructuring dynamic analysis”. Determining [${K}_{s}$] needs the tested FRFs, ${H}_{s,caca}$, ${H}_{s,cbcb}$ and ${H}_{s,cacb}$ on both sides of the coupling interfaces of substructures A and B in connection with each. Sometimes, it is hard to do the tests in practice, even impossible in case of very closeconnection.
To overcome the difficulty of the direct scheme for FRFtests in engineering applications, Five indirect schemes of inverse substructuring analysis are formulated here. The FRFs at system level can be related to the FRFs at component level by [7]:
$\left[\begin{array}{c}\alpha \left[{H}_{oaca}\right]\\ \left[{H}_{cxcx}\right]\\ \beta \left[{H}_{obcb}\right]\end{array}\right]\left[C\right]\left[\alpha \left[{H}_{caia}\right]\left[{H}_{cxcx}\right]\beta \left[{H}_{cbib}\right]\right],$
where, $\left[D\right]=\left[{H}_{caca}\right]+\left[{H}_{cbcb}\right]$, $\alpha =$ +1, −1 for $x=a$ and $b$, and $\beta =$ +1, −1 for $x=b$ and $a$, respectively.
Considering all independent equalities in Eq. (2), and substructure A can be mutually exchanged in practice, if choose the number of input excitationforce ($ia$, $ib$) to be the same as those of both displacementresponse ouputs ($oa$, $ob$) and the connection points at coupling interfaces ($ca$, $cb$), then, five indirect schemes of inverse substructuring analysis to determine coupling dynamic stiffness matrix [${K}_{s}$] of a structural connection can be formulated as follow:
where:
$\left[{I}_{Ac}\right]=\left[I\right]{H}_{s,caca}\left]\right[{H}_{caca}{]}^{1},\left[{I}_{Aci}\right]=\left[I\right]\left[{H}_{s,caia}\right]{\left[{H}_{caia}\right]}^{1},$
and [$I$] is unity matrix with same size of the FRFmatrices. For monocoupling connection, there is only one entry in the matrices of Eqs. (4)(8), i.e., they can be expressed in algebraic form as follow:
where:
$\u2206{H}_{caca}={H}_{caca}{H}_{s,caca},\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\u2206{H}_{caia}={H}_{caia}{H}_{s,caia}.$
Eqs. (4)(8) and or Eqs. (9)(13) are named here as the first to fifth indirect schemes of inverse substructuring analysis for determining coupling dynamic stiffness of structural connection, due to use of FRFs at both system level and component level in computation. FRFs involved in Eqs. (4)(13) are needed to be tested before computing the stiffness.
3. Experimentally verification on the first indirect scheme
To verify the effectiveness of the indirect schemes of inverse substructuring analysis in determining coupling dynamic stiffness of a structural connection, three experimental models of twolevel substructures are established and constructed as shown in Fig. 2. Where, single, two and three point(s) of structural connection are named as “mono, bi and tricoupling”, respectively, in this study. Their lumpedparameters are measured and listed in Table 1.
Using analytical lumpedparameter study, all of the five indirect schemes are analytically validated at first by applying them to the tricoupling connection, as shown in Fig. 2(c). The moduli of coupling dynamic stiffness $\left[{K}_{s}\right]$ are computed by Eqs. (4)(8) using the lumpedparameters in Table 1. The results are shown in Fig. 3, compared to that by direct scheme Eq. (1) and the measured ones (calculated by ${k}_{s}=k+j\omega c$, $j=$ pure imaginary unit, and $\omega =2\pi f$, $f=$ analytical frequency in Hertz). From Fig. 3, it can be seen that, the dynamic stiffness computed by both direct and indirect schemes are exactly identical, and they are also right the same as the measured ones. Therefore, the five indirect schemes are analytically valid in determining coupling dynamic stiffness of structural connection, as it does by direct scheme.
Fig. 2Experimental models of twolevel substructures
a) Monocoupled connection
b) Bicoupled connection
c) Tricoupled connection
To experimentally verify the indirect schemes, here target only the first one as an example. Using the tested FRFs involved in the direct scheme Eq. (1) and the 1st indirect scheme Eqs. (4) and (9) on the three models in Fig. 2, the coupling dynamic stiffness are computed by the equations respectively, and shown in Fig. 4, compared with the measured ones. The computed stiffness are also fitted by LS curve fittings. Table 2 lists the fitted stiffness, compared with the measured ones, including their relative errors.
It can be seen that, (i) both direct and indirect schemes can be applied to determine the coupling dynamic stiffness of structural connection with acceptable precision for the three models, especially the mean relative error for static stiffness can be lower than 10 %; (ii) The 1st indirect scheme shows better performance with mean relative errors of 1.9 % and 46 % in determining static stiffness $k$ and damping factor $c$, compared to that of direct scheme with the errors of 8.1 % and 68.1 % respectively; (iii) The relative errors of determining damping factors are much greater than that of determining static stiffness. They are probably resulted from LS curve fitting, damping ratio tests and or nonlinear effects of both static stiffness and damping factor.
Table 1Measured lumpedparameters of the three experimental models
Mass (kg)  Static stiffness (kN/m)  Samping factor (kNs/m)  Mass (kg)  Static stiffness (kN/m)  Samping factor (kNs/m)  
${m}_{1}$  1.86  ${k}_{1}$  799  ${c}_{1}$  0.365  ${m}_{9}$  1.17  ${k}_{9}$  680  ${c}_{9}$  0.286 
${m}_{2}$  1.64  ${k}_{2}$  646  ${c}_{2}$  0.346  ${m}_{10}$  1.62  ${k}_{10}$  663  ${c}_{10}$  0.356 
${m}_{3}$  1.75  ${k}_{3}$  655  ${c}_{3}$  0.242  ${k}_{11}$  189  ${c}_{11}$  0.163  
${m}_{4}$  1.76  ${k}_{4}$  198  ${c}_{4}$  0.135  ${k}_{12}$  188  ${c}_{12}$  0.146  
${m}_{5}$  1.41  ${k}_{5}$  682  ${c}_{5}$  0.495  ${k}_{13}$  206  ${c}_{13}$  0.201  
${m}_{6}$  1.28  ${k}_{6}$  698  ${c}_{6}$  0.465  ${k}_{14}$  202  ${c}_{14}$  0.188  
${m}_{7}$  1.18  ${k}_{7}$  691  ${c}_{7}$  0.437  ${k}_{15}$  698  ${c}_{15}$  0.302  
${m}_{8}$  1.25  ${k}_{8}$  671  ${c}_{8}$  0.369 
Fig. 3Measured and computed [Ks] using analytical FRFs of tricoupled model Fig. 2(c) by direct scheme, Eq. (1), and five indirect schemes, Eqs. (4)(8)
Table 2Measured and fitted Ks=ki kN/m+jωci(kNs/m) and their relative errors (%)
Connection  Monocoupled  Bicoupled  Tricoupled  
${K}_{s}$  ${k}_{4}$  ${c}_{4}$  ${k}_{4}$  ${c}_{4}$  ${k}_{11}$  ${c}_{11}$  ${k}_{12}$  ${c}_{12}$  ${k}_{13}$  ${c}_{13}$  ${k}_{14}$  ${c}_{14}$ 
Measured  198  0.135  198  0.135  189  0.163  188  0.146  206  0.201  202  0.188 
Fitted by Direct scheme  188.4  0.246  197.2  0.300  140.7  0.249  177.5  0.239  187.1  0.264  196.5  0.294 
Relative error  4.8  82.2  0.4  122.2  25.6  52.8  5.6  63.7  9.2  31.3  2.7  56.4 
Fitted by 2nd Indirect scheme  197.5  0.229  192.5  0.201  179.4  0.191  185.8  0.233  201.8  0.247  196.5  0.296 
Relative error  0.3  69.6  0.25  48.9  5.1  17.2  1.2  59.6  2.0  22.9  2.7  57.5 
Mean relative error  Scheme  Direct  1st Indirect  
${k}_{i}$  8.1  1.9  
${c}_{i}$  68.1  46.0 
Fig. 4Measured and computed [Ks] using tested FRFs on mono, bi and tricoupling models in Fig. 2 by direct and the 1st indirect scheme
a) Mono coupling
b) Bi coupling
c) Tricoupling
4. Conclusions
To determine coupling dynamic stiffness of substructural connection, this study provides five indirect schemes of inverse substructuring analysis. Compared to existing direct scheme, they are well validated analytically, and as example, the first indirect scheme is further verified experimentally. Results from the study are concluded as follow:
1) The five indirect schemes are analytically valid in determining coupling dynamic stiffness, for which they are right identical to the direct scheme.
2) The first indirect scheme has better precision in determining coupling dynamic stiffness than the direct scheme. The relative errors for static stiffness and damping factor of the models are 1.9 % and 46 % respectively, much less than that of direct scheme. The verifications on the rest four indirect schemes are to be completed by next work.
3) The relative error for determining damping factor is much greater than that for static stiffness. This may be resulted from LS curve fitting, damping test, and or nonlinear effect of the stiffness, and is to be investigated and improved in further study.
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About this article
The authors acknowledge gratefully National Natural Science Foundation of China for supporting research Project (#51475211), and Graduates Innovation Training Projects of Guangdong (#1055915007) and Jinan University (#CX15040), including the study of this paper.