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Research ArticleIndirect Inverse Substructuring Method for Multibody ProductTransport System with Rigid and Flexible Coupling
Jun Wang123 Li-xin Lu13 Pengjiang Qian4 Li-qiang Huang2
Yan Hua13 and Guang-yi Pu3
1Department of Packaging Engineering Jiangnan University Wuxi 214122 China2Tianjin Key Laboratory of Pulp amp Paper Tianjin University of Science amp Technology Tianjin 300457 China3Jiangsu Key Laboratory of Advanced Food Manufacturing Equipment and Technology Jiangnan University Wuxi 214122 China4School of Digital Media Jiangnan University Wuxi 214122 China
Correspondence should be addressed to Jun Wang wangj 1982jiangnaneducn and Li-xin Lu lulxjiangnaneducn
Received 16 July 2014 Accepted 30 October 2014
Academic Editor De Gao
Copyright copy 2015 Jun Wang et alThis is an open access article distributed under theCreativeCommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
The aim of this paper is to develop a new frequency response function- (FRF-) based indirect inverse substructuring methodwithout measuring system-level FRFs in the coupling DOFs for the analysis of the dynamic characteristics of a three-substructurecoupled product transport systemwith rigid and flexible coupling By enforcing the dynamic equilibrium conditions at the couplingcoordinates and the displacement compatibility conditions a closed-form analytical solution to inverse substructuring analysis ofmultisubstructure coupled product transport system is derived based on the relationship of easy-to-monitor component-level FRFsand the system-level FRFs at the coupling coordinatesThe proposed method is validated by a lumpedmass-spring-damper modeland the predicted coupling dynamic stiffness is compared with the direct computation showing exact agreement The methoddeveloped offers an approach to predict the unknown coupling dynamic stiffness from measured FRFs purely The suggestedmethod may help to obtain the main controlling factors and contributions from the various structure-borne paths for producttransport system
1 Introduction
Packaging dynamics for protection of the packaged producthas attracted increasingly more attention during the lasttwo decades [1ndash9] In distribution process the productpackaging and vehicle constitute a complex built-up producttransport system The packaged product will be damagedwhen the dynamic response due to the environmental vibra-tion and shock exceeds some limit value [10] To protect thepackaged product one needs to isolate the packaged itemwith flexible cushions Hence it is of significant importanceto identify the coupling dynamic stiffness between packagedproduct and vehicle system for cushioning packaging design[11 12] However the interaction between the packaged prod-uct and vehicle is extremely complicated including internaland external cushion container pallet nail and rope It isdifficult to predict the coupling dynamic stiffness accuratelyby reported identification methods [12ndash14] as some of the
frequency response functions (FRFs) required cannot beeasily measured between the complex coupling interfacesfor a product transport system To address this issue Zhenet al [15] proposed an inverse formulation that the FRFsof individual components and dynamic characteristics ofthe coupling elements can be predicted directly from thesystem-level FRFs Although this technique includes severalmathematical operations that might be very sensitive to mea-surement errors andor and inconsistencies it has its uniqueadvantages in various application fields since no component-level spectra response is needed Then it was applied to studythe dynamics of product transport system [16 17] showingits great application prospect in industry In spite of thosepromising successful applications the two-component struc-ture assumption of the inverse substructure theory proposedby Zhen restricts its application for a complex product trans-port systemThe discretely connected system should be takenas a multisubstructure coupled system in order to perform
Hindawi Publishing CorporationShock and VibrationVolume 2015 Article ID 106416 8 pageshttpdxdoiorg1011552015106416
2 Shock and Vibration
i(x)
c(x)
o(x)
[HX]
Figure 1 A general substructure representation
Rigid [KBC]
c1(a) c1(b) c2(b) c2(c)
o(a) i(c)
D
A B C
Figure 2 Single-coordinate coupled three-substructure system
the parametric analysis of each substructure and connectormore efficiently In our recent published work [18 19] theinverse substructuring method of three-component coupledproduct transport system for both single coordinate couplingcase and multicoordinate coupling case was proposed andapplied to perform the parameter analysis of product trans-port system However in the previous work the dynamicstiffness of the coupling interface was predicted purely fromthe system-level FRFs Actually the system-level FRFs fromcoupling degree of freedoms (DOFs) may not be measuredaccurately due to the difficulties of vibration excitation andmeasurement for the coupled interface between packagedproduct and vehicle within the limited space In this paper anindirect inverse substructuring method is proposed to iden-tify the coupling dynamic stiffness Instead of identification ofthe coupling dynamic stiffness only from system-level FRFsthe proposed method aims to obtain the packaging stiffnessaccurately from easy-to-monitor DOFs which will avoid thedifficulties of vibration excitation and measurement due tospace limitation The accuracy of the proposed theory is firstvalidated by a lumped parameter model
2 Indirect Inverse SubstructuringMethod for Multicomponent CoupledProduct Transport System with Rigid andFlexible Coupling
A free substructure 119883 is shown in Figure 1 where 119900(119909)119894(119909) and 119888(119909) are the response excitation and couplingcoordinates respectively At the coupling coordinate the
substructure is connected to the rest of the system Thegoverning equations of motion can be written as [16]
[119872119883] 119883 + [119862
119883] 119883 + [119870
119883] 119884119883 = 119865
119883 (1)
where [119872119883] [119862119883] and [119870
119883] are the mass damping and
stiffness matrices respectively and the subscript 119883 refers tosubstructure 119883(119883 = 119860 119861 119862 119878) Assuming forced periodicexcitation the steady-state vibratory response in frequencydomain can be expressed as
where 120596 = 2120587119891 is the excitation frequency This form isused to generate the necessary frequency response function(FRF) matrix of the substructure 119883 Here 119867
119883119892119896represents
the frequency response function matrices of substructure119883 (response at 119892 coordinate due to an excitation at 119896
coordinate) for example for 119883 = 119860 and 119909 = 119886 119867119860119900(119886)119894(119886)
is the transfer response function between the excitationcoordinate 119894(119886) and response coordinate 119900(119886) of substructure119860
Consider a system with three substructures 119860 119861 and 119862coupled at interface 119888
1by a rigid connector between 119860 and
119861 and at interface 1198882by a flexible connector defined as 119870
119861119862
between 119861 and 119862 (see Figure 2)By applying the substructuring method the system-level
FRFs can be predicted from the component-level FRFs
Shock and Vibration 3
Rigid [KBC]
c1(a) c1(b) c2(b) c2(c)
o(a) i(c)
D
A B C
Figure 3 Multicoordinate coupled three-substructure system
First take the structure 119863 coupled by substructure 119860 andsubstructure 119861 as a system and the FRFs of structure 119863 canbe expressed by the FRFs of substructure 119860 and substructure119861 as [18 19]
[
[
119867119863119900(119886)119894(119886)
119867119863119900(119886)119888
1(119909)
119867119863119900(119886)119894(119887)
1198671198631198881(119909)119894(119886)
1198671198631198881(119909)1198881(119909)
1198671198631198881(119909)119894(119887)
119867119863119900(119887)119894(119886)
119867119863119900(119887)119888
1(119909)
119867119863119900(119887)119894(119887)
]
]
= [
[
119867119860119900(119886)119894(119886)
119867119860119900(119886)119888
1(119909)
0
1198671198831198881(119909)119894(119886)
1198671198831198881(119909)1198881(119909)
1198671198831198881(119909)119894(119887)
0 119867119861119900(119887)119888
1(119909)
119867119861119900(119887)119894(119887)
]
]
minus [
[
120572119867119860119900(119886)119888
1(119886)
1198671198831198881(119909)1198881(119909)
120573119867119861119900(119887)119888
1(119887)
]
]
[119863119860119861]
times [1205721198671198601198881(119886)119894(119886)
Then take the system 119878 as a coupled one by substructure119863 and substructure 119862 (here structure 119863 is treated as acomponent) and the FRFs of the system 119878 can be expressedby the FRFs of substructure119863 and substructure 119862 as
[
[
119867119878119900(119889)119894(119889)
119867119878119900(119889)119888
2(119909)
119867119878119900(119889)119894(119888)
1198671198781198882(119909)119894(119889)
1198671198781198882(119909)1198882(119909)
1198671198781198882(119909)119894(119888)
119867119878119900(119888)119894(119889)
119867119878119900(119888)119888
2(119909)
119867119878119900(119888)119894(119888)
]
]
= [
[
119867119863119900(119889)119894(119889)
119867119863119900(119889)119888
2(119909)
0
1198671198831198882(119909)119894(119889)
1198671198831198882(119909)1198882(119909)
1198671198831198882(119909)119894(119888)
0 119867119862119900(119888)119888
2(119909)
119867119862119900(119888)119894(119888)
]
]
minus [
[
120572119867119863119900(119889)119888
2(119889)
1198671198831198882(119909)1198882(119909)
120573119867119862119900(119888)119888
2(119888)
]
]
[119863119863119862]
times [1205721198671198631198882(119889)119894(119889)
From (5) and (3) it can be seen thatHS is expressed byHDand HC and HD is expressed by HA and HB Substituting (5)into (3) yields the FRFs of the system S expressed by the FRFsof the three substructures A B and C Because the responseat 119900(119886) coordinate of substructure A due to an excitation at119894(119888) coordinate of substructure C is most concerned we givehere the explicit expression of the system-level FRF119867
The above derivation develops a method for a three-substructure coupled system to obtain the system-levelresponses from the prior knowledge of FRFs of the threesubstructures and the dynamic stiffness of the couplinginterfaces However for many complex structure systems thesubstructure-level FRFs may not be easily obtained Besidesthe dynamic stiffness of the coupling interface may beunknown If the physical system exists but is not convenientlyseparable into two or more substructures it is desirable toexpress the problem in terms of measurable system-levelFRFs
21 Closed-Form Solution for Single Coupling The coupledsystem is first decoupled as two substructures D and C asillustrated in Figure 2The force compatibility of the couplingcoordinates between substructureD and substructureC leadsto the following equation
The coefficient matrix in (10) is the transfer matrix HSbetween the coupling coordinates on substructure D andsubstructure C and it can be written in terms of system-levelFRFs as
1198882(119889)
1198882(119888)
= [1198671198781198882(119889)1198882(119889)
1198671198781198882(119889)1198882(119888)
1198671198781198882(119888)1198882(119889)
1198671198781198882(119888)1198882(119888)
]1198651198882(119889)
1198651198882(119888)
(12)
Comparing (10) and (12) and recalling (5) reveal a seriesof equations relating the component-level FRFs to system-level FRFs from which we can obtain the expression of thecoupling dynamic stiffness as depicted in
Equation (13) provides an inverse method to predict thecoupling dynamic stiffness for product transport systemBut the problem remaining is that some system-level FRFscannot be easily measured from the coupling DOFs such as1198671199041198882(119888)1198882(119888) To overcome this shortcoming we recall back (7)
times [1198671198601198881(119886)1198881(119886)+ 1198671198611198881(119887)1198881(119887)]minus1
1198671198611198881(119887)1198882(119887)
minus 1198671198611198882(119887)1198882(119887)+ 1198671198611198882(119887)1198881(119887)
times [1198671198601198881(119886)1198881(119886)+ 1198671198611198881(119887)1198881(119887)]minus1
times 1198671198611198881(119887)1198882(119887)minus 1198671198621198882(119888)1198882(119888)minus1
(14)
From (14) the coupling dynamic stiffness of the interfacebetween packaged product and vehicle can be predicted fromboth system-level FRFs and component-level FRFs all of theFRFs from DOFs of coupling c
2are measured in component
level which avoids the difficulties of vibration excitationand pickup As the coupling dynamic stiffness is predictedfrom both system-level FRFs and component-level FRFs wecall the method proposed indirect inverse substructuringmethod
22 Closed-Form Solution for Multicoordinate Coupling Asdepicted in Figure 3 the components are connected by a setof [KC] of dimension 119901 gt 1 In many cases [KC] is nearlydiagonal especially in product transport system Similarto the single coupling problem the dynamic equilibriumconditions are applied to the coupling coordinates on bothsides of the connecting springs to obtain
The coefficient matrix in (16) is the transfer matrix [HS]between the coupling coordinates on substructure D andsubstructure C and it can be written in terms of system-levelFRFs as
1198882(119889)
1198882(119888) = [
[119867119878]1198882(119889)1198882(119889)
[119867119878]1198882(119889)1198882(119888)
[119867119878]1198882(119888)1198882(119889)
[119867119878]1198882(119888)1198882(119888)
]1198651198882(119889)
1198651198882(119888) (17)
Comparing (16) and (17) and recalling (8) reveal a seriesof equations relating the component-level FRFs to system-level FRFs from which we can obtain the expression of thecoupling dynamic stiffness between B and C as in (18) after alengthy derivation
[119870119861119862]
= minus1205962
times ([119867119878]1198882(119887)1198882(119887)[119867119878]minus119879
(18)Equation (18) provides an inverse method to predict the
coupling dynamic stiffness for product transport systemBut the remaining problem is that some system-level FRFscannot be easily measured from the coupling DOFs such as[119867119904]1198882(119888)1198882(119888) To overcome this shortcoming we recall back (8)
and make an inverse formulation[119870119861119862] = minus120596
Equation (19) can be used to predict the coupling dynamicstiffness from both system-level FRFs and component-levelFRFs avoiding the difficulties of vibration excitation andpickup
3 Numerical Validation Using a LumpedParameter Model
To verify the indirect inverse substructure proposed abovea lumped mass-spring-damper model with three substruc-tures shown in Figure 4 is taken as an example The specificparameters of the system are listed in Table 1 The necessaryFRFs of the free substructures and the coupled system aregenerated from (2) and the computed system response func-tions as well as the necessary component response functionsare used to predict the coupling dynamic stiffness applying(19) for packaging interface Then the results are comparedto direct calculations as shown in Figure 5 Specifically for119867119878119894119895 119894 = (1) (2) 119895 = (7) (8) and 119867
and 119894 119895 = (6) (7) All necessary FRFs are calculated from (2)The matrices for components 119860 119861 119862 and 119878 are expressedexplicitly as follows
component 119860
[119872119860] = [
[
1198981
0 0
0 11989821
0
0 0 11989822
]
]
[119870119860] = [
[
11989611+ 11989612
minus11989611
minus11989612
minus11989611
11989611
0
minus11989612
0 11989612
]
]
[119862119860] = [
[
11988811+ 11988812
minus11988811
minus11988812
minus11988811
11988811
0
minus11988812
0 11988812
]
]
(20)
component 119861
[119872119861] =
[[[
[
1198983
0 0 0
0 1198984
0 0
0 0 1198985
0
0 0 0 1198986
]]]
]
[119870119861] =
[[[
[
1198962+ 1198963
minus1198962
minus1198963
0
minus1198962
1198962+ 1198964
0 minus1198964
minus1198963
0 1198963+ 1198965
minus1198965
0 minus1198964
minus1198965
1198964+ 1198965
]]]
]
[119862119861] =
[[[
[
1198882+ 1198883
minus1198882
minus1198883
0
minus1198882
1198882+ 1198884
0 minus1198884
minus1198883
0 1198883+ 1198885
minus1198885
0 minus1198884
minus1198885
1198884+ 1198885
]]]
]
(21)
6 Shock and Vibration
Table 1 Model parameters for lumped parameter model shown in Figure 4
From Figure 5 we can see that the predicted dynamicstiffness is in exact agreement with the direct computationHence the proposed indirect inverse substructuring method
demonstrates its validity for identifying the dynamic stiffnessat coupling interfaces It should be noted that the formulationmay be sensitive to input random errors As illustrated
Shock and Vibration 7
0 10 20 30 40 505800
6000
6200
6400
6600
6800
7000
7200
7400
7600
7800
Mag
nitu
de (N
m)
Given |K57|
Predicted |K57|
Given |K68|
Predicted |K68|
Figure 5 Comparison of predicted and exact coupling dynamicstiffness here 119870
57= 11989657+ 119895120596119888
57 11987068= 11989668+ 119895120596119888
68
0 5 10 15 20Frequency (Hz)
Mag
nitu
de (m
N)
No noise1 noise
2 noise5 noise
100
10minus5
10minus10
10minus15
Figure 6 Effect of random noise of measured system-level FRFs onthe predicted frequency response function from119898
9to1198981
in Figure 6 the predicted system-level frequency responsefunction is affected remarkably by the input random errors ofmeasured system-level frequency response functions imply-ing that the formulation for prediction of coupling stiffness isextremely sensitive to the experiment errors A further studyon the error propagation as well as the elimination approachwill be conducted in the future
4 Conclusions
Three sets of equations for multicomponent coupled prod-uct transport system are obtained including the dynamicequilibrium conditions the displacement compatibility con-ditions in addition to the relationship of the system-level
FRFs and the selected easy-to-monitor component-levelFRFs Then the closed-form analytical solution to inversesubstructuring analysis ofmultisubstructure coupled producttransport system with rigid and flexible coupling is derivedThe method developed offers an approach to predict theunknown coupling dynamic stiffness from system-level FRFsand easy-to-monitor component-level FRFs avoiding thedifficulties of vibration excitation andor measurement in thecoupling interface Then the proposed method is validatedby a lumped mass-spring-damper model and the couplingdynamic stiffness is compared with the direct computationshowing exact agreement The method developed offers anapproach to predict the unknown coupling dynamic stiffnessfrom system-level FRFs and component-level FRFs whichcan be easily measured The suggested method may help toobtain themain controlling factors and contribution from thevarious structure-borne paths for product transport systemwhich may certainly facilitate the cushioning packagingdesign
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to appreciate the financial supportof National Natural Science Foundation of China (Grantno 51205167) Research Fund for the Doctoral Program ofHigher Education of China (Grant no 20120093120014) andFundamental Research Funds for the Central Universities(Grant no JUSRP51403A) the project was supported by theFoundation (Grant no 2013010) of Tianjin Key Laboratoryof Pulp amp Paper (Tianjin University of Science amp Technol-ogy) and by the Foundation of Jiangsu Key Laboratory ofAdvanced Food Manufacturing Equipment and Technology(Grant no FM-201403) and the research was also sponsoredby Qing Lan Project of Jiangsu Province (2014)
References
[1] R E Newton Fragility Assessment Theory and Practice Mon-terey Research Laboratory Monterey Calif USA 1968
[2] G J Burgess ldquoProduct fragility and damage boundary theoryrdquoPackaging Technology and Science vol 15 no 10 pp 5ndash10 1988
[3] Z Wang C Wu and D Xi ldquoDamage boundary of a packagingsystem under rectangular pulse excitationrdquo Packaging Technol-ogy and Science vol 11 no 4 pp 189ndash202 1998
[4] ZWWang and C Y Hu ldquoShock spectra and damage boundarycurves for nonlinear package cushioning systemrdquo PackagingTechnology and Science vol 12 no 5 pp 207ndash217 1999
[5] C Ge ldquoModel of accelerated vibration testrdquo Packaging Technol-ogy and Science vol 13 no 1 pp 7ndash11 2000
[6] J Wang Z-W Wang L-X Lu Y Zhu and Y-G WangldquoThree-dimensional shock spectrum of critical component fornonlinear packaging systemrdquo Shock and Vibration vol 18 no 3pp 437ndash445 2011
8 Shock and Vibration
[7] J Wang F Duan J-H Jiang L-X Lu and Z-WWang ldquoDrop-ping damage evaluation for a hyperbolic tangent cushioningsystem with a critical componentrdquo Journal of Vibration andControl vol 18 no 10 pp 1417ndash1421 2012
[8] F-D Lu W-M Tao and D Gao ldquoVirtual mass method forsolution of dynamic response of composite cushion packagingsystemrdquo Packaging Technology and Science vol 26 no S1 pp32ndash42 2013
[9] H Kitazawa K Saito and Y Ishikawa ldquoEffect of difference inacceleration and velocity change on product damage due torepetitive shockrdquo Packaging Technology and Science vol 27 no3 pp 221ndash230 2014
[10] D Shires ldquoOn the time compression (test acceleration) ofbroadband random vibration testsrdquo Packaging Technology andScience vol 24 no 2 pp 75ndash87 2011
[11] M A Sek V Rouillard and A J Parker ldquoA study of nonlineareffects in a cushion-product system on its vibration transmis-sibility estimates with the reverse multiple input-single outputtechniquerdquo Packaging Technology and Science vol 26 no 3 pp125ndash135 2013
[12] Y Ren and C F Beards ldquoIdentification of joint properties of astructure using frf datardquo Journal of Sound and Vibration vol186 no 4 pp 567ndash587 1995
[13] T Yang S-H Fan and C-S Lin ldquoJoint stiffness identificationusing FRF measurementsrdquo Computers and Structures vol 81no 28-29 pp 2549ndash2556 2003
[14] D Celic and M Boltezar ldquoIdentification of the dynamic prop-erties of joints using frequency-response functionsrdquo Journal ofSound and Vibration vol 317 no 1-2 pp 158ndash174 2008
[15] J Zhen T C Lim and G Lu ldquoDetermination of systemvibratory response characteristics applying a spectral-basedinverse sub-structuring approach Part I analytical formula-tionrdquo International Journal of Vehicle Noise and Vibration vol1 no 1-2 pp 1ndash30 2004
[16] Z-WWang JWang Y-B Zhang C-Y Hu and Y Zhu ldquoAppli-cation of the inverse substructuremethod in the investigation ofdynamic characteristics of product transport systemrdquoPackagingTechnology and Science vol 25 no 6 pp 351ndash362 2012
[17] JWang L Lu and ZWang ldquoModeling the complex interactionbetween packaged product and vehiclerdquo Advanced ScienceLetters vol 4 no 6-7 pp 2207ndash2212 2011
[18] Z-W Wang and J Wang ldquoInverse substructure method ofthree-substructures coupled system and its application inproduct-transport-systemrdquo Journal of Vibration and Controlvol 17 no 6 pp 943ndash952 2011
[19] J Wang X Hong Y Qian Z-W Wang and L-X Lu ldquoInversesub-structuring method for multi-coordinate coupled producttransport systemrdquo Packaging Technology and Science vol 27 no5 pp 385ndash408 2014
Figure 2 Single-coordinate coupled three-substructure system
the parametric analysis of each substructure and connectormore efficiently In our recent published work [18 19] theinverse substructuring method of three-component coupledproduct transport system for both single coordinate couplingcase and multicoordinate coupling case was proposed andapplied to perform the parameter analysis of product trans-port system However in the previous work the dynamicstiffness of the coupling interface was predicted purely fromthe system-level FRFs Actually the system-level FRFs fromcoupling degree of freedoms (DOFs) may not be measuredaccurately due to the difficulties of vibration excitation andmeasurement for the coupled interface between packagedproduct and vehicle within the limited space In this paper anindirect inverse substructuring method is proposed to iden-tify the coupling dynamic stiffness Instead of identification ofthe coupling dynamic stiffness only from system-level FRFsthe proposed method aims to obtain the packaging stiffnessaccurately from easy-to-monitor DOFs which will avoid thedifficulties of vibration excitation and measurement due tospace limitation The accuracy of the proposed theory is firstvalidated by a lumped parameter model
2 Indirect Inverse SubstructuringMethod for Multicomponent CoupledProduct Transport System with Rigid andFlexible Coupling
A free substructure 119883 is shown in Figure 1 where 119900(119909)119894(119909) and 119888(119909) are the response excitation and couplingcoordinates respectively At the coupling coordinate the
substructure is connected to the rest of the system Thegoverning equations of motion can be written as [16]
[119872119883] 119883 + [119862
119883] 119883 + [119870
119883] 119884119883 = 119865
119883 (1)
where [119872119883] [119862119883] and [119870
119883] are the mass damping and
stiffness matrices respectively and the subscript 119883 refers tosubstructure 119883(119883 = 119860 119861 119862 119878) Assuming forced periodicexcitation the steady-state vibratory response in frequencydomain can be expressed as
where 120596 = 2120587119891 is the excitation frequency This form isused to generate the necessary frequency response function(FRF) matrix of the substructure 119883 Here 119867
119883119892119896represents
the frequency response function matrices of substructure119883 (response at 119892 coordinate due to an excitation at 119896
coordinate) for example for 119883 = 119860 and 119909 = 119886 119867119860119900(119886)119894(119886)
is the transfer response function between the excitationcoordinate 119894(119886) and response coordinate 119900(119886) of substructure119860
Consider a system with three substructures 119860 119861 and 119862coupled at interface 119888
1by a rigid connector between 119860 and
119861 and at interface 1198882by a flexible connector defined as 119870
119861119862
between 119861 and 119862 (see Figure 2)By applying the substructuring method the system-level
FRFs can be predicted from the component-level FRFs
Shock and Vibration 3
Rigid [KBC]
c1(a) c1(b) c2(b) c2(c)
o(a) i(c)
D
A B C
Figure 3 Multicoordinate coupled three-substructure system
First take the structure 119863 coupled by substructure 119860 andsubstructure 119861 as a system and the FRFs of structure 119863 canbe expressed by the FRFs of substructure 119860 and substructure119861 as [18 19]
[
[
119867119863119900(119886)119894(119886)
119867119863119900(119886)119888
1(119909)
119867119863119900(119886)119894(119887)
1198671198631198881(119909)119894(119886)
1198671198631198881(119909)1198881(119909)
1198671198631198881(119909)119894(119887)
119867119863119900(119887)119894(119886)
119867119863119900(119887)119888
1(119909)
119867119863119900(119887)119894(119887)
]
]
= [
[
119867119860119900(119886)119894(119886)
119867119860119900(119886)119888
1(119909)
0
1198671198831198881(119909)119894(119886)
1198671198831198881(119909)1198881(119909)
1198671198831198881(119909)119894(119887)
0 119867119861119900(119887)119888
1(119909)
119867119861119900(119887)119894(119887)
]
]
minus [
[
120572119867119860119900(119886)119888
1(119886)
1198671198831198881(119909)1198881(119909)
120573119867119861119900(119887)119888
1(119887)
]
]
[119863119860119861]
times [1205721198671198601198881(119886)119894(119886)
Then take the system 119878 as a coupled one by substructure119863 and substructure 119862 (here structure 119863 is treated as acomponent) and the FRFs of the system 119878 can be expressedby the FRFs of substructure119863 and substructure 119862 as
[
[
119867119878119900(119889)119894(119889)
119867119878119900(119889)119888
2(119909)
119867119878119900(119889)119894(119888)
1198671198781198882(119909)119894(119889)
1198671198781198882(119909)1198882(119909)
1198671198781198882(119909)119894(119888)
119867119878119900(119888)119894(119889)
119867119878119900(119888)119888
2(119909)
119867119878119900(119888)119894(119888)
]
]
= [
[
119867119863119900(119889)119894(119889)
119867119863119900(119889)119888
2(119909)
0
1198671198831198882(119909)119894(119889)
1198671198831198882(119909)1198882(119909)
1198671198831198882(119909)119894(119888)
0 119867119862119900(119888)119888
2(119909)
119867119862119900(119888)119894(119888)
]
]
minus [
[
120572119867119863119900(119889)119888
2(119889)
1198671198831198882(119909)1198882(119909)
120573119867119862119900(119888)119888
2(119888)
]
]
[119863119863119862]
times [1205721198671198631198882(119889)119894(119889)
From (5) and (3) it can be seen thatHS is expressed byHDand HC and HD is expressed by HA and HB Substituting (5)into (3) yields the FRFs of the system S expressed by the FRFsof the three substructures A B and C Because the responseat 119900(119886) coordinate of substructure A due to an excitation at119894(119888) coordinate of substructure C is most concerned we givehere the explicit expression of the system-level FRF119867
The above derivation develops a method for a three-substructure coupled system to obtain the system-levelresponses from the prior knowledge of FRFs of the threesubstructures and the dynamic stiffness of the couplinginterfaces However for many complex structure systems thesubstructure-level FRFs may not be easily obtained Besidesthe dynamic stiffness of the coupling interface may beunknown If the physical system exists but is not convenientlyseparable into two or more substructures it is desirable toexpress the problem in terms of measurable system-levelFRFs
21 Closed-Form Solution for Single Coupling The coupledsystem is first decoupled as two substructures D and C asillustrated in Figure 2The force compatibility of the couplingcoordinates between substructureD and substructureC leadsto the following equation
The coefficient matrix in (10) is the transfer matrix HSbetween the coupling coordinates on substructure D andsubstructure C and it can be written in terms of system-levelFRFs as
1198882(119889)
1198882(119888)
= [1198671198781198882(119889)1198882(119889)
1198671198781198882(119889)1198882(119888)
1198671198781198882(119888)1198882(119889)
1198671198781198882(119888)1198882(119888)
]1198651198882(119889)
1198651198882(119888)
(12)
Comparing (10) and (12) and recalling (5) reveal a seriesof equations relating the component-level FRFs to system-level FRFs from which we can obtain the expression of thecoupling dynamic stiffness as depicted in
Equation (13) provides an inverse method to predict thecoupling dynamic stiffness for product transport systemBut the problem remaining is that some system-level FRFscannot be easily measured from the coupling DOFs such as1198671199041198882(119888)1198882(119888) To overcome this shortcoming we recall back (7)
times [1198671198601198881(119886)1198881(119886)+ 1198671198611198881(119887)1198881(119887)]minus1
1198671198611198881(119887)1198882(119887)
minus 1198671198611198882(119887)1198882(119887)+ 1198671198611198882(119887)1198881(119887)
times [1198671198601198881(119886)1198881(119886)+ 1198671198611198881(119887)1198881(119887)]minus1
times 1198671198611198881(119887)1198882(119887)minus 1198671198621198882(119888)1198882(119888)minus1
(14)
From (14) the coupling dynamic stiffness of the interfacebetween packaged product and vehicle can be predicted fromboth system-level FRFs and component-level FRFs all of theFRFs from DOFs of coupling c
2are measured in component
level which avoids the difficulties of vibration excitationand pickup As the coupling dynamic stiffness is predictedfrom both system-level FRFs and component-level FRFs wecall the method proposed indirect inverse substructuringmethod
22 Closed-Form Solution for Multicoordinate Coupling Asdepicted in Figure 3 the components are connected by a setof [KC] of dimension 119901 gt 1 In many cases [KC] is nearlydiagonal especially in product transport system Similarto the single coupling problem the dynamic equilibriumconditions are applied to the coupling coordinates on bothsides of the connecting springs to obtain
The coefficient matrix in (16) is the transfer matrix [HS]between the coupling coordinates on substructure D andsubstructure C and it can be written in terms of system-levelFRFs as
1198882(119889)
1198882(119888) = [
[119867119878]1198882(119889)1198882(119889)
[119867119878]1198882(119889)1198882(119888)
[119867119878]1198882(119888)1198882(119889)
[119867119878]1198882(119888)1198882(119888)
]1198651198882(119889)
1198651198882(119888) (17)
Comparing (16) and (17) and recalling (8) reveal a seriesof equations relating the component-level FRFs to system-level FRFs from which we can obtain the expression of thecoupling dynamic stiffness between B and C as in (18) after alengthy derivation
[119870119861119862]
= minus1205962
times ([119867119878]1198882(119887)1198882(119887)[119867119878]minus119879
(18)Equation (18) provides an inverse method to predict the
coupling dynamic stiffness for product transport systemBut the remaining problem is that some system-level FRFscannot be easily measured from the coupling DOFs such as[119867119904]1198882(119888)1198882(119888) To overcome this shortcoming we recall back (8)
and make an inverse formulation[119870119861119862] = minus120596
Equation (19) can be used to predict the coupling dynamicstiffness from both system-level FRFs and component-levelFRFs avoiding the difficulties of vibration excitation andpickup
3 Numerical Validation Using a LumpedParameter Model
To verify the indirect inverse substructure proposed abovea lumped mass-spring-damper model with three substruc-tures shown in Figure 4 is taken as an example The specificparameters of the system are listed in Table 1 The necessaryFRFs of the free substructures and the coupled system aregenerated from (2) and the computed system response func-tions as well as the necessary component response functionsare used to predict the coupling dynamic stiffness applying(19) for packaging interface Then the results are comparedto direct calculations as shown in Figure 5 Specifically for119867119878119894119895 119894 = (1) (2) 119895 = (7) (8) and 119867
and 119894 119895 = (6) (7) All necessary FRFs are calculated from (2)The matrices for components 119860 119861 119862 and 119878 are expressedexplicitly as follows
component 119860
[119872119860] = [
[
1198981
0 0
0 11989821
0
0 0 11989822
]
]
[119870119860] = [
[
11989611+ 11989612
minus11989611
minus11989612
minus11989611
11989611
0
minus11989612
0 11989612
]
]
[119862119860] = [
[
11988811+ 11988812
minus11988811
minus11988812
minus11988811
11988811
0
minus11988812
0 11988812
]
]
(20)
component 119861
[119872119861] =
[[[
[
1198983
0 0 0
0 1198984
0 0
0 0 1198985
0
0 0 0 1198986
]]]
]
[119870119861] =
[[[
[
1198962+ 1198963
minus1198962
minus1198963
0
minus1198962
1198962+ 1198964
0 minus1198964
minus1198963
0 1198963+ 1198965
minus1198965
0 minus1198964
minus1198965
1198964+ 1198965
]]]
]
[119862119861] =
[[[
[
1198882+ 1198883
minus1198882
minus1198883
0
minus1198882
1198882+ 1198884
0 minus1198884
minus1198883
0 1198883+ 1198885
minus1198885
0 minus1198884
minus1198885
1198884+ 1198885
]]]
]
(21)
6 Shock and Vibration
Table 1 Model parameters for lumped parameter model shown in Figure 4
From Figure 5 we can see that the predicted dynamicstiffness is in exact agreement with the direct computationHence the proposed indirect inverse substructuring method
demonstrates its validity for identifying the dynamic stiffnessat coupling interfaces It should be noted that the formulationmay be sensitive to input random errors As illustrated
Shock and Vibration 7
0 10 20 30 40 505800
6000
6200
6400
6600
6800
7000
7200
7400
7600
7800
Mag
nitu
de (N
m)
Given |K57|
Predicted |K57|
Given |K68|
Predicted |K68|
Figure 5 Comparison of predicted and exact coupling dynamicstiffness here 119870
57= 11989657+ 119895120596119888
57 11987068= 11989668+ 119895120596119888
68
0 5 10 15 20Frequency (Hz)
Mag
nitu
de (m
N)
No noise1 noise
2 noise5 noise
100
10minus5
10minus10
10minus15
Figure 6 Effect of random noise of measured system-level FRFs onthe predicted frequency response function from119898
9to1198981
in Figure 6 the predicted system-level frequency responsefunction is affected remarkably by the input random errors ofmeasured system-level frequency response functions imply-ing that the formulation for prediction of coupling stiffness isextremely sensitive to the experiment errors A further studyon the error propagation as well as the elimination approachwill be conducted in the future
4 Conclusions
Three sets of equations for multicomponent coupled prod-uct transport system are obtained including the dynamicequilibrium conditions the displacement compatibility con-ditions in addition to the relationship of the system-level
FRFs and the selected easy-to-monitor component-levelFRFs Then the closed-form analytical solution to inversesubstructuring analysis ofmultisubstructure coupled producttransport system with rigid and flexible coupling is derivedThe method developed offers an approach to predict theunknown coupling dynamic stiffness from system-level FRFsand easy-to-monitor component-level FRFs avoiding thedifficulties of vibration excitation andor measurement in thecoupling interface Then the proposed method is validatedby a lumped mass-spring-damper model and the couplingdynamic stiffness is compared with the direct computationshowing exact agreement The method developed offers anapproach to predict the unknown coupling dynamic stiffnessfrom system-level FRFs and component-level FRFs whichcan be easily measured The suggested method may help toobtain themain controlling factors and contribution from thevarious structure-borne paths for product transport systemwhich may certainly facilitate the cushioning packagingdesign
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to appreciate the financial supportof National Natural Science Foundation of China (Grantno 51205167) Research Fund for the Doctoral Program ofHigher Education of China (Grant no 20120093120014) andFundamental Research Funds for the Central Universities(Grant no JUSRP51403A) the project was supported by theFoundation (Grant no 2013010) of Tianjin Key Laboratoryof Pulp amp Paper (Tianjin University of Science amp Technol-ogy) and by the Foundation of Jiangsu Key Laboratory ofAdvanced Food Manufacturing Equipment and Technology(Grant no FM-201403) and the research was also sponsoredby Qing Lan Project of Jiangsu Province (2014)
References
[1] R E Newton Fragility Assessment Theory and Practice Mon-terey Research Laboratory Monterey Calif USA 1968
[2] G J Burgess ldquoProduct fragility and damage boundary theoryrdquoPackaging Technology and Science vol 15 no 10 pp 5ndash10 1988
[3] Z Wang C Wu and D Xi ldquoDamage boundary of a packagingsystem under rectangular pulse excitationrdquo Packaging Technol-ogy and Science vol 11 no 4 pp 189ndash202 1998
[4] ZWWang and C Y Hu ldquoShock spectra and damage boundarycurves for nonlinear package cushioning systemrdquo PackagingTechnology and Science vol 12 no 5 pp 207ndash217 1999
[5] C Ge ldquoModel of accelerated vibration testrdquo Packaging Technol-ogy and Science vol 13 no 1 pp 7ndash11 2000
[6] J Wang Z-W Wang L-X Lu Y Zhu and Y-G WangldquoThree-dimensional shock spectrum of critical component fornonlinear packaging systemrdquo Shock and Vibration vol 18 no 3pp 437ndash445 2011
8 Shock and Vibration
[7] J Wang F Duan J-H Jiang L-X Lu and Z-WWang ldquoDrop-ping damage evaluation for a hyperbolic tangent cushioningsystem with a critical componentrdquo Journal of Vibration andControl vol 18 no 10 pp 1417ndash1421 2012
[8] F-D Lu W-M Tao and D Gao ldquoVirtual mass method forsolution of dynamic response of composite cushion packagingsystemrdquo Packaging Technology and Science vol 26 no S1 pp32ndash42 2013
[9] H Kitazawa K Saito and Y Ishikawa ldquoEffect of difference inacceleration and velocity change on product damage due torepetitive shockrdquo Packaging Technology and Science vol 27 no3 pp 221ndash230 2014
[10] D Shires ldquoOn the time compression (test acceleration) ofbroadband random vibration testsrdquo Packaging Technology andScience vol 24 no 2 pp 75ndash87 2011
[11] M A Sek V Rouillard and A J Parker ldquoA study of nonlineareffects in a cushion-product system on its vibration transmis-sibility estimates with the reverse multiple input-single outputtechniquerdquo Packaging Technology and Science vol 26 no 3 pp125ndash135 2013
[12] Y Ren and C F Beards ldquoIdentification of joint properties of astructure using frf datardquo Journal of Sound and Vibration vol186 no 4 pp 567ndash587 1995
[13] T Yang S-H Fan and C-S Lin ldquoJoint stiffness identificationusing FRF measurementsrdquo Computers and Structures vol 81no 28-29 pp 2549ndash2556 2003
[14] D Celic and M Boltezar ldquoIdentification of the dynamic prop-erties of joints using frequency-response functionsrdquo Journal ofSound and Vibration vol 317 no 1-2 pp 158ndash174 2008
[15] J Zhen T C Lim and G Lu ldquoDetermination of systemvibratory response characteristics applying a spectral-basedinverse sub-structuring approach Part I analytical formula-tionrdquo International Journal of Vehicle Noise and Vibration vol1 no 1-2 pp 1ndash30 2004
[16] Z-WWang JWang Y-B Zhang C-Y Hu and Y Zhu ldquoAppli-cation of the inverse substructuremethod in the investigation ofdynamic characteristics of product transport systemrdquoPackagingTechnology and Science vol 25 no 6 pp 351ndash362 2012
[17] JWang L Lu and ZWang ldquoModeling the complex interactionbetween packaged product and vehiclerdquo Advanced ScienceLetters vol 4 no 6-7 pp 2207ndash2212 2011
[18] Z-W Wang and J Wang ldquoInverse substructure method ofthree-substructures coupled system and its application inproduct-transport-systemrdquo Journal of Vibration and Controlvol 17 no 6 pp 943ndash952 2011
[19] J Wang X Hong Y Qian Z-W Wang and L-X Lu ldquoInversesub-structuring method for multi-coordinate coupled producttransport systemrdquo Packaging Technology and Science vol 27 no5 pp 385ndash408 2014
Figure 3 Multicoordinate coupled three-substructure system
First take the structure 119863 coupled by substructure 119860 andsubstructure 119861 as a system and the FRFs of structure 119863 canbe expressed by the FRFs of substructure 119860 and substructure119861 as [18 19]
[
[
119867119863119900(119886)119894(119886)
119867119863119900(119886)119888
1(119909)
119867119863119900(119886)119894(119887)
1198671198631198881(119909)119894(119886)
1198671198631198881(119909)1198881(119909)
1198671198631198881(119909)119894(119887)
119867119863119900(119887)119894(119886)
119867119863119900(119887)119888
1(119909)
119867119863119900(119887)119894(119887)
]
]
= [
[
119867119860119900(119886)119894(119886)
119867119860119900(119886)119888
1(119909)
0
1198671198831198881(119909)119894(119886)
1198671198831198881(119909)1198881(119909)
1198671198831198881(119909)119894(119887)
0 119867119861119900(119887)119888
1(119909)
119867119861119900(119887)119894(119887)
]
]
minus [
[
120572119867119860119900(119886)119888
1(119886)
1198671198831198881(119909)1198881(119909)
120573119867119861119900(119887)119888
1(119887)
]
]
[119863119860119861]
times [1205721198671198601198881(119886)119894(119886)
Then take the system 119878 as a coupled one by substructure119863 and substructure 119862 (here structure 119863 is treated as acomponent) and the FRFs of the system 119878 can be expressedby the FRFs of substructure119863 and substructure 119862 as
[
[
119867119878119900(119889)119894(119889)
119867119878119900(119889)119888
2(119909)
119867119878119900(119889)119894(119888)
1198671198781198882(119909)119894(119889)
1198671198781198882(119909)1198882(119909)
1198671198781198882(119909)119894(119888)
119867119878119900(119888)119894(119889)
119867119878119900(119888)119888
2(119909)
119867119878119900(119888)119894(119888)
]
]
= [
[
119867119863119900(119889)119894(119889)
119867119863119900(119889)119888
2(119909)
0
1198671198831198882(119909)119894(119889)
1198671198831198882(119909)1198882(119909)
1198671198831198882(119909)119894(119888)
0 119867119862119900(119888)119888
2(119909)
119867119862119900(119888)119894(119888)
]
]
minus [
[
120572119867119863119900(119889)119888
2(119889)
1198671198831198882(119909)1198882(119909)
120573119867119862119900(119888)119888
2(119888)
]
]
[119863119863119862]
times [1205721198671198631198882(119889)119894(119889)
From (5) and (3) it can be seen thatHS is expressed byHDand HC and HD is expressed by HA and HB Substituting (5)into (3) yields the FRFs of the system S expressed by the FRFsof the three substructures A B and C Because the responseat 119900(119886) coordinate of substructure A due to an excitation at119894(119888) coordinate of substructure C is most concerned we givehere the explicit expression of the system-level FRF119867
The above derivation develops a method for a three-substructure coupled system to obtain the system-levelresponses from the prior knowledge of FRFs of the threesubstructures and the dynamic stiffness of the couplinginterfaces However for many complex structure systems thesubstructure-level FRFs may not be easily obtained Besidesthe dynamic stiffness of the coupling interface may beunknown If the physical system exists but is not convenientlyseparable into two or more substructures it is desirable toexpress the problem in terms of measurable system-levelFRFs
21 Closed-Form Solution for Single Coupling The coupledsystem is first decoupled as two substructures D and C asillustrated in Figure 2The force compatibility of the couplingcoordinates between substructureD and substructureC leadsto the following equation
The coefficient matrix in (10) is the transfer matrix HSbetween the coupling coordinates on substructure D andsubstructure C and it can be written in terms of system-levelFRFs as
1198882(119889)
1198882(119888)
= [1198671198781198882(119889)1198882(119889)
1198671198781198882(119889)1198882(119888)
1198671198781198882(119888)1198882(119889)
1198671198781198882(119888)1198882(119888)
]1198651198882(119889)
1198651198882(119888)
(12)
Comparing (10) and (12) and recalling (5) reveal a seriesof equations relating the component-level FRFs to system-level FRFs from which we can obtain the expression of thecoupling dynamic stiffness as depicted in
Equation (13) provides an inverse method to predict thecoupling dynamic stiffness for product transport systemBut the problem remaining is that some system-level FRFscannot be easily measured from the coupling DOFs such as1198671199041198882(119888)1198882(119888) To overcome this shortcoming we recall back (7)
times [1198671198601198881(119886)1198881(119886)+ 1198671198611198881(119887)1198881(119887)]minus1
1198671198611198881(119887)1198882(119887)
minus 1198671198611198882(119887)1198882(119887)+ 1198671198611198882(119887)1198881(119887)
times [1198671198601198881(119886)1198881(119886)+ 1198671198611198881(119887)1198881(119887)]minus1
times 1198671198611198881(119887)1198882(119887)minus 1198671198621198882(119888)1198882(119888)minus1
(14)
From (14) the coupling dynamic stiffness of the interfacebetween packaged product and vehicle can be predicted fromboth system-level FRFs and component-level FRFs all of theFRFs from DOFs of coupling c
2are measured in component
level which avoids the difficulties of vibration excitationand pickup As the coupling dynamic stiffness is predictedfrom both system-level FRFs and component-level FRFs wecall the method proposed indirect inverse substructuringmethod
22 Closed-Form Solution for Multicoordinate Coupling Asdepicted in Figure 3 the components are connected by a setof [KC] of dimension 119901 gt 1 In many cases [KC] is nearlydiagonal especially in product transport system Similarto the single coupling problem the dynamic equilibriumconditions are applied to the coupling coordinates on bothsides of the connecting springs to obtain
The coefficient matrix in (16) is the transfer matrix [HS]between the coupling coordinates on substructure D andsubstructure C and it can be written in terms of system-levelFRFs as
1198882(119889)
1198882(119888) = [
[119867119878]1198882(119889)1198882(119889)
[119867119878]1198882(119889)1198882(119888)
[119867119878]1198882(119888)1198882(119889)
[119867119878]1198882(119888)1198882(119888)
]1198651198882(119889)
1198651198882(119888) (17)
Comparing (16) and (17) and recalling (8) reveal a seriesof equations relating the component-level FRFs to system-level FRFs from which we can obtain the expression of thecoupling dynamic stiffness between B and C as in (18) after alengthy derivation
[119870119861119862]
= minus1205962
times ([119867119878]1198882(119887)1198882(119887)[119867119878]minus119879
(18)Equation (18) provides an inverse method to predict the
coupling dynamic stiffness for product transport systemBut the remaining problem is that some system-level FRFscannot be easily measured from the coupling DOFs such as[119867119904]1198882(119888)1198882(119888) To overcome this shortcoming we recall back (8)
and make an inverse formulation[119870119861119862] = minus120596
Equation (19) can be used to predict the coupling dynamicstiffness from both system-level FRFs and component-levelFRFs avoiding the difficulties of vibration excitation andpickup
3 Numerical Validation Using a LumpedParameter Model
To verify the indirect inverse substructure proposed abovea lumped mass-spring-damper model with three substruc-tures shown in Figure 4 is taken as an example The specificparameters of the system are listed in Table 1 The necessaryFRFs of the free substructures and the coupled system aregenerated from (2) and the computed system response func-tions as well as the necessary component response functionsare used to predict the coupling dynamic stiffness applying(19) for packaging interface Then the results are comparedto direct calculations as shown in Figure 5 Specifically for119867119878119894119895 119894 = (1) (2) 119895 = (7) (8) and 119867
and 119894 119895 = (6) (7) All necessary FRFs are calculated from (2)The matrices for components 119860 119861 119862 and 119878 are expressedexplicitly as follows
component 119860
[119872119860] = [
[
1198981
0 0
0 11989821
0
0 0 11989822
]
]
[119870119860] = [
[
11989611+ 11989612
minus11989611
minus11989612
minus11989611
11989611
0
minus11989612
0 11989612
]
]
[119862119860] = [
[
11988811+ 11988812
minus11988811
minus11988812
minus11988811
11988811
0
minus11988812
0 11988812
]
]
(20)
component 119861
[119872119861] =
[[[
[
1198983
0 0 0
0 1198984
0 0
0 0 1198985
0
0 0 0 1198986
]]]
]
[119870119861] =
[[[
[
1198962+ 1198963
minus1198962
minus1198963
0
minus1198962
1198962+ 1198964
0 minus1198964
minus1198963
0 1198963+ 1198965
minus1198965
0 minus1198964
minus1198965
1198964+ 1198965
]]]
]
[119862119861] =
[[[
[
1198882+ 1198883
minus1198882
minus1198883
0
minus1198882
1198882+ 1198884
0 minus1198884
minus1198883
0 1198883+ 1198885
minus1198885
0 minus1198884
minus1198885
1198884+ 1198885
]]]
]
(21)
6 Shock and Vibration
Table 1 Model parameters for lumped parameter model shown in Figure 4
From Figure 5 we can see that the predicted dynamicstiffness is in exact agreement with the direct computationHence the proposed indirect inverse substructuring method
demonstrates its validity for identifying the dynamic stiffnessat coupling interfaces It should be noted that the formulationmay be sensitive to input random errors As illustrated
Shock and Vibration 7
0 10 20 30 40 505800
6000
6200
6400
6600
6800
7000
7200
7400
7600
7800
Mag
nitu
de (N
m)
Given |K57|
Predicted |K57|
Given |K68|
Predicted |K68|
Figure 5 Comparison of predicted and exact coupling dynamicstiffness here 119870
57= 11989657+ 119895120596119888
57 11987068= 11989668+ 119895120596119888
68
0 5 10 15 20Frequency (Hz)
Mag
nitu
de (m
N)
No noise1 noise
2 noise5 noise
100
10minus5
10minus10
10minus15
Figure 6 Effect of random noise of measured system-level FRFs onthe predicted frequency response function from119898
9to1198981
in Figure 6 the predicted system-level frequency responsefunction is affected remarkably by the input random errors ofmeasured system-level frequency response functions imply-ing that the formulation for prediction of coupling stiffness isextremely sensitive to the experiment errors A further studyon the error propagation as well as the elimination approachwill be conducted in the future
4 Conclusions
Three sets of equations for multicomponent coupled prod-uct transport system are obtained including the dynamicequilibrium conditions the displacement compatibility con-ditions in addition to the relationship of the system-level
FRFs and the selected easy-to-monitor component-levelFRFs Then the closed-form analytical solution to inversesubstructuring analysis ofmultisubstructure coupled producttransport system with rigid and flexible coupling is derivedThe method developed offers an approach to predict theunknown coupling dynamic stiffness from system-level FRFsand easy-to-monitor component-level FRFs avoiding thedifficulties of vibration excitation andor measurement in thecoupling interface Then the proposed method is validatedby a lumped mass-spring-damper model and the couplingdynamic stiffness is compared with the direct computationshowing exact agreement The method developed offers anapproach to predict the unknown coupling dynamic stiffnessfrom system-level FRFs and component-level FRFs whichcan be easily measured The suggested method may help toobtain themain controlling factors and contribution from thevarious structure-borne paths for product transport systemwhich may certainly facilitate the cushioning packagingdesign
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to appreciate the financial supportof National Natural Science Foundation of China (Grantno 51205167) Research Fund for the Doctoral Program ofHigher Education of China (Grant no 20120093120014) andFundamental Research Funds for the Central Universities(Grant no JUSRP51403A) the project was supported by theFoundation (Grant no 2013010) of Tianjin Key Laboratoryof Pulp amp Paper (Tianjin University of Science amp Technol-ogy) and by the Foundation of Jiangsu Key Laboratory ofAdvanced Food Manufacturing Equipment and Technology(Grant no FM-201403) and the research was also sponsoredby Qing Lan Project of Jiangsu Province (2014)
References
[1] R E Newton Fragility Assessment Theory and Practice Mon-terey Research Laboratory Monterey Calif USA 1968
[2] G J Burgess ldquoProduct fragility and damage boundary theoryrdquoPackaging Technology and Science vol 15 no 10 pp 5ndash10 1988
[3] Z Wang C Wu and D Xi ldquoDamage boundary of a packagingsystem under rectangular pulse excitationrdquo Packaging Technol-ogy and Science vol 11 no 4 pp 189ndash202 1998
[4] ZWWang and C Y Hu ldquoShock spectra and damage boundarycurves for nonlinear package cushioning systemrdquo PackagingTechnology and Science vol 12 no 5 pp 207ndash217 1999
[5] C Ge ldquoModel of accelerated vibration testrdquo Packaging Technol-ogy and Science vol 13 no 1 pp 7ndash11 2000
[6] J Wang Z-W Wang L-X Lu Y Zhu and Y-G WangldquoThree-dimensional shock spectrum of critical component fornonlinear packaging systemrdquo Shock and Vibration vol 18 no 3pp 437ndash445 2011
8 Shock and Vibration
[7] J Wang F Duan J-H Jiang L-X Lu and Z-WWang ldquoDrop-ping damage evaluation for a hyperbolic tangent cushioningsystem with a critical componentrdquo Journal of Vibration andControl vol 18 no 10 pp 1417ndash1421 2012
[8] F-D Lu W-M Tao and D Gao ldquoVirtual mass method forsolution of dynamic response of composite cushion packagingsystemrdquo Packaging Technology and Science vol 26 no S1 pp32ndash42 2013
[9] H Kitazawa K Saito and Y Ishikawa ldquoEffect of difference inacceleration and velocity change on product damage due torepetitive shockrdquo Packaging Technology and Science vol 27 no3 pp 221ndash230 2014
[10] D Shires ldquoOn the time compression (test acceleration) ofbroadband random vibration testsrdquo Packaging Technology andScience vol 24 no 2 pp 75ndash87 2011
[11] M A Sek V Rouillard and A J Parker ldquoA study of nonlineareffects in a cushion-product system on its vibration transmis-sibility estimates with the reverse multiple input-single outputtechniquerdquo Packaging Technology and Science vol 26 no 3 pp125ndash135 2013
[12] Y Ren and C F Beards ldquoIdentification of joint properties of astructure using frf datardquo Journal of Sound and Vibration vol186 no 4 pp 567ndash587 1995
[13] T Yang S-H Fan and C-S Lin ldquoJoint stiffness identificationusing FRF measurementsrdquo Computers and Structures vol 81no 28-29 pp 2549ndash2556 2003
[14] D Celic and M Boltezar ldquoIdentification of the dynamic prop-erties of joints using frequency-response functionsrdquo Journal ofSound and Vibration vol 317 no 1-2 pp 158ndash174 2008
[15] J Zhen T C Lim and G Lu ldquoDetermination of systemvibratory response characteristics applying a spectral-basedinverse sub-structuring approach Part I analytical formula-tionrdquo International Journal of Vehicle Noise and Vibration vol1 no 1-2 pp 1ndash30 2004
[16] Z-WWang JWang Y-B Zhang C-Y Hu and Y Zhu ldquoAppli-cation of the inverse substructuremethod in the investigation ofdynamic characteristics of product transport systemrdquoPackagingTechnology and Science vol 25 no 6 pp 351ndash362 2012
[17] JWang L Lu and ZWang ldquoModeling the complex interactionbetween packaged product and vehiclerdquo Advanced ScienceLetters vol 4 no 6-7 pp 2207ndash2212 2011
[18] Z-W Wang and J Wang ldquoInverse substructure method ofthree-substructures coupled system and its application inproduct-transport-systemrdquo Journal of Vibration and Controlvol 17 no 6 pp 943ndash952 2011
[19] J Wang X Hong Y Qian Z-W Wang and L-X Lu ldquoInversesub-structuring method for multi-coordinate coupled producttransport systemrdquo Packaging Technology and Science vol 27 no5 pp 385ndash408 2014
The above derivation develops a method for a three-substructure coupled system to obtain the system-levelresponses from the prior knowledge of FRFs of the threesubstructures and the dynamic stiffness of the couplinginterfaces However for many complex structure systems thesubstructure-level FRFs may not be easily obtained Besidesthe dynamic stiffness of the coupling interface may beunknown If the physical system exists but is not convenientlyseparable into two or more substructures it is desirable toexpress the problem in terms of measurable system-levelFRFs
21 Closed-Form Solution for Single Coupling The coupledsystem is first decoupled as two substructures D and C asillustrated in Figure 2The force compatibility of the couplingcoordinates between substructureD and substructureC leadsto the following equation
The coefficient matrix in (10) is the transfer matrix HSbetween the coupling coordinates on substructure D andsubstructure C and it can be written in terms of system-levelFRFs as
1198882(119889)
1198882(119888)
= [1198671198781198882(119889)1198882(119889)
1198671198781198882(119889)1198882(119888)
1198671198781198882(119888)1198882(119889)
1198671198781198882(119888)1198882(119888)
]1198651198882(119889)
1198651198882(119888)
(12)
Comparing (10) and (12) and recalling (5) reveal a seriesof equations relating the component-level FRFs to system-level FRFs from which we can obtain the expression of thecoupling dynamic stiffness as depicted in
Equation (13) provides an inverse method to predict thecoupling dynamic stiffness for product transport systemBut the problem remaining is that some system-level FRFscannot be easily measured from the coupling DOFs such as1198671199041198882(119888)1198882(119888) To overcome this shortcoming we recall back (7)
times [1198671198601198881(119886)1198881(119886)+ 1198671198611198881(119887)1198881(119887)]minus1
1198671198611198881(119887)1198882(119887)
minus 1198671198611198882(119887)1198882(119887)+ 1198671198611198882(119887)1198881(119887)
times [1198671198601198881(119886)1198881(119886)+ 1198671198611198881(119887)1198881(119887)]minus1
times 1198671198611198881(119887)1198882(119887)minus 1198671198621198882(119888)1198882(119888)minus1
(14)
From (14) the coupling dynamic stiffness of the interfacebetween packaged product and vehicle can be predicted fromboth system-level FRFs and component-level FRFs all of theFRFs from DOFs of coupling c
2are measured in component
level which avoids the difficulties of vibration excitationand pickup As the coupling dynamic stiffness is predictedfrom both system-level FRFs and component-level FRFs wecall the method proposed indirect inverse substructuringmethod
22 Closed-Form Solution for Multicoordinate Coupling Asdepicted in Figure 3 the components are connected by a setof [KC] of dimension 119901 gt 1 In many cases [KC] is nearlydiagonal especially in product transport system Similarto the single coupling problem the dynamic equilibriumconditions are applied to the coupling coordinates on bothsides of the connecting springs to obtain
The coefficient matrix in (16) is the transfer matrix [HS]between the coupling coordinates on substructure D andsubstructure C and it can be written in terms of system-levelFRFs as
1198882(119889)
1198882(119888) = [
[119867119878]1198882(119889)1198882(119889)
[119867119878]1198882(119889)1198882(119888)
[119867119878]1198882(119888)1198882(119889)
[119867119878]1198882(119888)1198882(119888)
]1198651198882(119889)
1198651198882(119888) (17)
Comparing (16) and (17) and recalling (8) reveal a seriesof equations relating the component-level FRFs to system-level FRFs from which we can obtain the expression of thecoupling dynamic stiffness between B and C as in (18) after alengthy derivation
[119870119861119862]
= minus1205962
times ([119867119878]1198882(119887)1198882(119887)[119867119878]minus119879
(18)Equation (18) provides an inverse method to predict the
coupling dynamic stiffness for product transport systemBut the remaining problem is that some system-level FRFscannot be easily measured from the coupling DOFs such as[119867119904]1198882(119888)1198882(119888) To overcome this shortcoming we recall back (8)
and make an inverse formulation[119870119861119862] = minus120596
Equation (19) can be used to predict the coupling dynamicstiffness from both system-level FRFs and component-levelFRFs avoiding the difficulties of vibration excitation andpickup
3 Numerical Validation Using a LumpedParameter Model
To verify the indirect inverse substructure proposed abovea lumped mass-spring-damper model with three substruc-tures shown in Figure 4 is taken as an example The specificparameters of the system are listed in Table 1 The necessaryFRFs of the free substructures and the coupled system aregenerated from (2) and the computed system response func-tions as well as the necessary component response functionsare used to predict the coupling dynamic stiffness applying(19) for packaging interface Then the results are comparedto direct calculations as shown in Figure 5 Specifically for119867119878119894119895 119894 = (1) (2) 119895 = (7) (8) and 119867
and 119894 119895 = (6) (7) All necessary FRFs are calculated from (2)The matrices for components 119860 119861 119862 and 119878 are expressedexplicitly as follows
component 119860
[119872119860] = [
[
1198981
0 0
0 11989821
0
0 0 11989822
]
]
[119870119860] = [
[
11989611+ 11989612
minus11989611
minus11989612
minus11989611
11989611
0
minus11989612
0 11989612
]
]
[119862119860] = [
[
11988811+ 11988812
minus11988811
minus11988812
minus11988811
11988811
0
minus11988812
0 11988812
]
]
(20)
component 119861
[119872119861] =
[[[
[
1198983
0 0 0
0 1198984
0 0
0 0 1198985
0
0 0 0 1198986
]]]
]
[119870119861] =
[[[
[
1198962+ 1198963
minus1198962
minus1198963
0
minus1198962
1198962+ 1198964
0 minus1198964
minus1198963
0 1198963+ 1198965
minus1198965
0 minus1198964
minus1198965
1198964+ 1198965
]]]
]
[119862119861] =
[[[
[
1198882+ 1198883
minus1198882
minus1198883
0
minus1198882
1198882+ 1198884
0 minus1198884
minus1198883
0 1198883+ 1198885
minus1198885
0 minus1198884
minus1198885
1198884+ 1198885
]]]
]
(21)
6 Shock and Vibration
Table 1 Model parameters for lumped parameter model shown in Figure 4
From Figure 5 we can see that the predicted dynamicstiffness is in exact agreement with the direct computationHence the proposed indirect inverse substructuring method
demonstrates its validity for identifying the dynamic stiffnessat coupling interfaces It should be noted that the formulationmay be sensitive to input random errors As illustrated
Shock and Vibration 7
0 10 20 30 40 505800
6000
6200
6400
6600
6800
7000
7200
7400
7600
7800
Mag
nitu
de (N
m)
Given |K57|
Predicted |K57|
Given |K68|
Predicted |K68|
Figure 5 Comparison of predicted and exact coupling dynamicstiffness here 119870
57= 11989657+ 119895120596119888
57 11987068= 11989668+ 119895120596119888
68
0 5 10 15 20Frequency (Hz)
Mag
nitu
de (m
N)
No noise1 noise
2 noise5 noise
100
10minus5
10minus10
10minus15
Figure 6 Effect of random noise of measured system-level FRFs onthe predicted frequency response function from119898
9to1198981
in Figure 6 the predicted system-level frequency responsefunction is affected remarkably by the input random errors ofmeasured system-level frequency response functions imply-ing that the formulation for prediction of coupling stiffness isextremely sensitive to the experiment errors A further studyon the error propagation as well as the elimination approachwill be conducted in the future
4 Conclusions
Three sets of equations for multicomponent coupled prod-uct transport system are obtained including the dynamicequilibrium conditions the displacement compatibility con-ditions in addition to the relationship of the system-level
FRFs and the selected easy-to-monitor component-levelFRFs Then the closed-form analytical solution to inversesubstructuring analysis ofmultisubstructure coupled producttransport system with rigid and flexible coupling is derivedThe method developed offers an approach to predict theunknown coupling dynamic stiffness from system-level FRFsand easy-to-monitor component-level FRFs avoiding thedifficulties of vibration excitation andor measurement in thecoupling interface Then the proposed method is validatedby a lumped mass-spring-damper model and the couplingdynamic stiffness is compared with the direct computationshowing exact agreement The method developed offers anapproach to predict the unknown coupling dynamic stiffnessfrom system-level FRFs and component-level FRFs whichcan be easily measured The suggested method may help toobtain themain controlling factors and contribution from thevarious structure-borne paths for product transport systemwhich may certainly facilitate the cushioning packagingdesign
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to appreciate the financial supportof National Natural Science Foundation of China (Grantno 51205167) Research Fund for the Doctoral Program ofHigher Education of China (Grant no 20120093120014) andFundamental Research Funds for the Central Universities(Grant no JUSRP51403A) the project was supported by theFoundation (Grant no 2013010) of Tianjin Key Laboratoryof Pulp amp Paper (Tianjin University of Science amp Technol-ogy) and by the Foundation of Jiangsu Key Laboratory ofAdvanced Food Manufacturing Equipment and Technology(Grant no FM-201403) and the research was also sponsoredby Qing Lan Project of Jiangsu Province (2014)
References
[1] R E Newton Fragility Assessment Theory and Practice Mon-terey Research Laboratory Monterey Calif USA 1968
[2] G J Burgess ldquoProduct fragility and damage boundary theoryrdquoPackaging Technology and Science vol 15 no 10 pp 5ndash10 1988
[3] Z Wang C Wu and D Xi ldquoDamage boundary of a packagingsystem under rectangular pulse excitationrdquo Packaging Technol-ogy and Science vol 11 no 4 pp 189ndash202 1998
[4] ZWWang and C Y Hu ldquoShock spectra and damage boundarycurves for nonlinear package cushioning systemrdquo PackagingTechnology and Science vol 12 no 5 pp 207ndash217 1999
[5] C Ge ldquoModel of accelerated vibration testrdquo Packaging Technol-ogy and Science vol 13 no 1 pp 7ndash11 2000
[6] J Wang Z-W Wang L-X Lu Y Zhu and Y-G WangldquoThree-dimensional shock spectrum of critical component fornonlinear packaging systemrdquo Shock and Vibration vol 18 no 3pp 437ndash445 2011
8 Shock and Vibration
[7] J Wang F Duan J-H Jiang L-X Lu and Z-WWang ldquoDrop-ping damage evaluation for a hyperbolic tangent cushioningsystem with a critical componentrdquo Journal of Vibration andControl vol 18 no 10 pp 1417ndash1421 2012
[8] F-D Lu W-M Tao and D Gao ldquoVirtual mass method forsolution of dynamic response of composite cushion packagingsystemrdquo Packaging Technology and Science vol 26 no S1 pp32ndash42 2013
[9] H Kitazawa K Saito and Y Ishikawa ldquoEffect of difference inacceleration and velocity change on product damage due torepetitive shockrdquo Packaging Technology and Science vol 27 no3 pp 221ndash230 2014
[10] D Shires ldquoOn the time compression (test acceleration) ofbroadband random vibration testsrdquo Packaging Technology andScience vol 24 no 2 pp 75ndash87 2011
[11] M A Sek V Rouillard and A J Parker ldquoA study of nonlineareffects in a cushion-product system on its vibration transmis-sibility estimates with the reverse multiple input-single outputtechniquerdquo Packaging Technology and Science vol 26 no 3 pp125ndash135 2013
[12] Y Ren and C F Beards ldquoIdentification of joint properties of astructure using frf datardquo Journal of Sound and Vibration vol186 no 4 pp 567ndash587 1995
[13] T Yang S-H Fan and C-S Lin ldquoJoint stiffness identificationusing FRF measurementsrdquo Computers and Structures vol 81no 28-29 pp 2549ndash2556 2003
[14] D Celic and M Boltezar ldquoIdentification of the dynamic prop-erties of joints using frequency-response functionsrdquo Journal ofSound and Vibration vol 317 no 1-2 pp 158ndash174 2008
[15] J Zhen T C Lim and G Lu ldquoDetermination of systemvibratory response characteristics applying a spectral-basedinverse sub-structuring approach Part I analytical formula-tionrdquo International Journal of Vehicle Noise and Vibration vol1 no 1-2 pp 1ndash30 2004
[16] Z-WWang JWang Y-B Zhang C-Y Hu and Y Zhu ldquoAppli-cation of the inverse substructuremethod in the investigation ofdynamic characteristics of product transport systemrdquoPackagingTechnology and Science vol 25 no 6 pp 351ndash362 2012
[17] JWang L Lu and ZWang ldquoModeling the complex interactionbetween packaged product and vehiclerdquo Advanced ScienceLetters vol 4 no 6-7 pp 2207ndash2212 2011
[18] Z-W Wang and J Wang ldquoInverse substructure method ofthree-substructures coupled system and its application inproduct-transport-systemrdquo Journal of Vibration and Controlvol 17 no 6 pp 943ndash952 2011
[19] J Wang X Hong Y Qian Z-W Wang and L-X Lu ldquoInversesub-structuring method for multi-coordinate coupled producttransport systemrdquo Packaging Technology and Science vol 27 no5 pp 385ndash408 2014
The coefficient matrix in (16) is the transfer matrix [HS]between the coupling coordinates on substructure D andsubstructure C and it can be written in terms of system-levelFRFs as
1198882(119889)
1198882(119888) = [
[119867119878]1198882(119889)1198882(119889)
[119867119878]1198882(119889)1198882(119888)
[119867119878]1198882(119888)1198882(119889)
[119867119878]1198882(119888)1198882(119888)
]1198651198882(119889)
1198651198882(119888) (17)
Comparing (16) and (17) and recalling (8) reveal a seriesof equations relating the component-level FRFs to system-level FRFs from which we can obtain the expression of thecoupling dynamic stiffness between B and C as in (18) after alengthy derivation
[119870119861119862]
= minus1205962
times ([119867119878]1198882(119887)1198882(119887)[119867119878]minus119879
(18)Equation (18) provides an inverse method to predict the
coupling dynamic stiffness for product transport systemBut the remaining problem is that some system-level FRFscannot be easily measured from the coupling DOFs such as[119867119904]1198882(119888)1198882(119888) To overcome this shortcoming we recall back (8)
and make an inverse formulation[119870119861119862] = minus120596
Equation (19) can be used to predict the coupling dynamicstiffness from both system-level FRFs and component-levelFRFs avoiding the difficulties of vibration excitation andpickup
3 Numerical Validation Using a LumpedParameter Model
To verify the indirect inverse substructure proposed abovea lumped mass-spring-damper model with three substruc-tures shown in Figure 4 is taken as an example The specificparameters of the system are listed in Table 1 The necessaryFRFs of the free substructures and the coupled system aregenerated from (2) and the computed system response func-tions as well as the necessary component response functionsare used to predict the coupling dynamic stiffness applying(19) for packaging interface Then the results are comparedto direct calculations as shown in Figure 5 Specifically for119867119878119894119895 119894 = (1) (2) 119895 = (7) (8) and 119867
and 119894 119895 = (6) (7) All necessary FRFs are calculated from (2)The matrices for components 119860 119861 119862 and 119878 are expressedexplicitly as follows
component 119860
[119872119860] = [
[
1198981
0 0
0 11989821
0
0 0 11989822
]
]
[119870119860] = [
[
11989611+ 11989612
minus11989611
minus11989612
minus11989611
11989611
0
minus11989612
0 11989612
]
]
[119862119860] = [
[
11988811+ 11988812
minus11988811
minus11988812
minus11988811
11988811
0
minus11988812
0 11988812
]
]
(20)
component 119861
[119872119861] =
[[[
[
1198983
0 0 0
0 1198984
0 0
0 0 1198985
0
0 0 0 1198986
]]]
]
[119870119861] =
[[[
[
1198962+ 1198963
minus1198962
minus1198963
0
minus1198962
1198962+ 1198964
0 minus1198964
minus1198963
0 1198963+ 1198965
minus1198965
0 minus1198964
minus1198965
1198964+ 1198965
]]]
]
[119862119861] =
[[[
[
1198882+ 1198883
minus1198882
minus1198883
0
minus1198882
1198882+ 1198884
0 minus1198884
minus1198883
0 1198883+ 1198885
minus1198885
0 minus1198884
minus1198885
1198884+ 1198885
]]]
]
(21)
6 Shock and Vibration
Table 1 Model parameters for lumped parameter model shown in Figure 4
From Figure 5 we can see that the predicted dynamicstiffness is in exact agreement with the direct computationHence the proposed indirect inverse substructuring method
demonstrates its validity for identifying the dynamic stiffnessat coupling interfaces It should be noted that the formulationmay be sensitive to input random errors As illustrated
Shock and Vibration 7
0 10 20 30 40 505800
6000
6200
6400
6600
6800
7000
7200
7400
7600
7800
Mag
nitu
de (N
m)
Given |K57|
Predicted |K57|
Given |K68|
Predicted |K68|
Figure 5 Comparison of predicted and exact coupling dynamicstiffness here 119870
57= 11989657+ 119895120596119888
57 11987068= 11989668+ 119895120596119888
68
0 5 10 15 20Frequency (Hz)
Mag
nitu
de (m
N)
No noise1 noise
2 noise5 noise
100
10minus5
10minus10
10minus15
Figure 6 Effect of random noise of measured system-level FRFs onthe predicted frequency response function from119898
9to1198981
in Figure 6 the predicted system-level frequency responsefunction is affected remarkably by the input random errors ofmeasured system-level frequency response functions imply-ing that the formulation for prediction of coupling stiffness isextremely sensitive to the experiment errors A further studyon the error propagation as well as the elimination approachwill be conducted in the future
4 Conclusions
Three sets of equations for multicomponent coupled prod-uct transport system are obtained including the dynamicequilibrium conditions the displacement compatibility con-ditions in addition to the relationship of the system-level
FRFs and the selected easy-to-monitor component-levelFRFs Then the closed-form analytical solution to inversesubstructuring analysis ofmultisubstructure coupled producttransport system with rigid and flexible coupling is derivedThe method developed offers an approach to predict theunknown coupling dynamic stiffness from system-level FRFsand easy-to-monitor component-level FRFs avoiding thedifficulties of vibration excitation andor measurement in thecoupling interface Then the proposed method is validatedby a lumped mass-spring-damper model and the couplingdynamic stiffness is compared with the direct computationshowing exact agreement The method developed offers anapproach to predict the unknown coupling dynamic stiffnessfrom system-level FRFs and component-level FRFs whichcan be easily measured The suggested method may help toobtain themain controlling factors and contribution from thevarious structure-borne paths for product transport systemwhich may certainly facilitate the cushioning packagingdesign
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to appreciate the financial supportof National Natural Science Foundation of China (Grantno 51205167) Research Fund for the Doctoral Program ofHigher Education of China (Grant no 20120093120014) andFundamental Research Funds for the Central Universities(Grant no JUSRP51403A) the project was supported by theFoundation (Grant no 2013010) of Tianjin Key Laboratoryof Pulp amp Paper (Tianjin University of Science amp Technol-ogy) and by the Foundation of Jiangsu Key Laboratory ofAdvanced Food Manufacturing Equipment and Technology(Grant no FM-201403) and the research was also sponsoredby Qing Lan Project of Jiangsu Province (2014)
References
[1] R E Newton Fragility Assessment Theory and Practice Mon-terey Research Laboratory Monterey Calif USA 1968
[2] G J Burgess ldquoProduct fragility and damage boundary theoryrdquoPackaging Technology and Science vol 15 no 10 pp 5ndash10 1988
[3] Z Wang C Wu and D Xi ldquoDamage boundary of a packagingsystem under rectangular pulse excitationrdquo Packaging Technol-ogy and Science vol 11 no 4 pp 189ndash202 1998
[4] ZWWang and C Y Hu ldquoShock spectra and damage boundarycurves for nonlinear package cushioning systemrdquo PackagingTechnology and Science vol 12 no 5 pp 207ndash217 1999
[5] C Ge ldquoModel of accelerated vibration testrdquo Packaging Technol-ogy and Science vol 13 no 1 pp 7ndash11 2000
[6] J Wang Z-W Wang L-X Lu Y Zhu and Y-G WangldquoThree-dimensional shock spectrum of critical component fornonlinear packaging systemrdquo Shock and Vibration vol 18 no 3pp 437ndash445 2011
8 Shock and Vibration
[7] J Wang F Duan J-H Jiang L-X Lu and Z-WWang ldquoDrop-ping damage evaluation for a hyperbolic tangent cushioningsystem with a critical componentrdquo Journal of Vibration andControl vol 18 no 10 pp 1417ndash1421 2012
[8] F-D Lu W-M Tao and D Gao ldquoVirtual mass method forsolution of dynamic response of composite cushion packagingsystemrdquo Packaging Technology and Science vol 26 no S1 pp32ndash42 2013
[9] H Kitazawa K Saito and Y Ishikawa ldquoEffect of difference inacceleration and velocity change on product damage due torepetitive shockrdquo Packaging Technology and Science vol 27 no3 pp 221ndash230 2014
[10] D Shires ldquoOn the time compression (test acceleration) ofbroadband random vibration testsrdquo Packaging Technology andScience vol 24 no 2 pp 75ndash87 2011
[11] M A Sek V Rouillard and A J Parker ldquoA study of nonlineareffects in a cushion-product system on its vibration transmis-sibility estimates with the reverse multiple input-single outputtechniquerdquo Packaging Technology and Science vol 26 no 3 pp125ndash135 2013
[12] Y Ren and C F Beards ldquoIdentification of joint properties of astructure using frf datardquo Journal of Sound and Vibration vol186 no 4 pp 567ndash587 1995
[13] T Yang S-H Fan and C-S Lin ldquoJoint stiffness identificationusing FRF measurementsrdquo Computers and Structures vol 81no 28-29 pp 2549ndash2556 2003
[14] D Celic and M Boltezar ldquoIdentification of the dynamic prop-erties of joints using frequency-response functionsrdquo Journal ofSound and Vibration vol 317 no 1-2 pp 158ndash174 2008
[15] J Zhen T C Lim and G Lu ldquoDetermination of systemvibratory response characteristics applying a spectral-basedinverse sub-structuring approach Part I analytical formula-tionrdquo International Journal of Vehicle Noise and Vibration vol1 no 1-2 pp 1ndash30 2004
[16] Z-WWang JWang Y-B Zhang C-Y Hu and Y Zhu ldquoAppli-cation of the inverse substructuremethod in the investigation ofdynamic characteristics of product transport systemrdquoPackagingTechnology and Science vol 25 no 6 pp 351ndash362 2012
[17] JWang L Lu and ZWang ldquoModeling the complex interactionbetween packaged product and vehiclerdquo Advanced ScienceLetters vol 4 no 6-7 pp 2207ndash2212 2011
[18] Z-W Wang and J Wang ldquoInverse substructure method ofthree-substructures coupled system and its application inproduct-transport-systemrdquo Journal of Vibration and Controlvol 17 no 6 pp 943ndash952 2011
[19] J Wang X Hong Y Qian Z-W Wang and L-X Lu ldquoInversesub-structuring method for multi-coordinate coupled producttransport systemrdquo Packaging Technology and Science vol 27 no5 pp 385ndash408 2014
From Figure 5 we can see that the predicted dynamicstiffness is in exact agreement with the direct computationHence the proposed indirect inverse substructuring method
demonstrates its validity for identifying the dynamic stiffnessat coupling interfaces It should be noted that the formulationmay be sensitive to input random errors As illustrated
Shock and Vibration 7
0 10 20 30 40 505800
6000
6200
6400
6600
6800
7000
7200
7400
7600
7800
Mag
nitu
de (N
m)
Given |K57|
Predicted |K57|
Given |K68|
Predicted |K68|
Figure 5 Comparison of predicted and exact coupling dynamicstiffness here 119870
57= 11989657+ 119895120596119888
57 11987068= 11989668+ 119895120596119888
68
0 5 10 15 20Frequency (Hz)
Mag
nitu
de (m
N)
No noise1 noise
2 noise5 noise
100
10minus5
10minus10
10minus15
Figure 6 Effect of random noise of measured system-level FRFs onthe predicted frequency response function from119898
9to1198981
in Figure 6 the predicted system-level frequency responsefunction is affected remarkably by the input random errors ofmeasured system-level frequency response functions imply-ing that the formulation for prediction of coupling stiffness isextremely sensitive to the experiment errors A further studyon the error propagation as well as the elimination approachwill be conducted in the future
4 Conclusions
Three sets of equations for multicomponent coupled prod-uct transport system are obtained including the dynamicequilibrium conditions the displacement compatibility con-ditions in addition to the relationship of the system-level
FRFs and the selected easy-to-monitor component-levelFRFs Then the closed-form analytical solution to inversesubstructuring analysis ofmultisubstructure coupled producttransport system with rigid and flexible coupling is derivedThe method developed offers an approach to predict theunknown coupling dynamic stiffness from system-level FRFsand easy-to-monitor component-level FRFs avoiding thedifficulties of vibration excitation andor measurement in thecoupling interface Then the proposed method is validatedby a lumped mass-spring-damper model and the couplingdynamic stiffness is compared with the direct computationshowing exact agreement The method developed offers anapproach to predict the unknown coupling dynamic stiffnessfrom system-level FRFs and component-level FRFs whichcan be easily measured The suggested method may help toobtain themain controlling factors and contribution from thevarious structure-borne paths for product transport systemwhich may certainly facilitate the cushioning packagingdesign
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to appreciate the financial supportof National Natural Science Foundation of China (Grantno 51205167) Research Fund for the Doctoral Program ofHigher Education of China (Grant no 20120093120014) andFundamental Research Funds for the Central Universities(Grant no JUSRP51403A) the project was supported by theFoundation (Grant no 2013010) of Tianjin Key Laboratoryof Pulp amp Paper (Tianjin University of Science amp Technol-ogy) and by the Foundation of Jiangsu Key Laboratory ofAdvanced Food Manufacturing Equipment and Technology(Grant no FM-201403) and the research was also sponsoredby Qing Lan Project of Jiangsu Province (2014)
References
[1] R E Newton Fragility Assessment Theory and Practice Mon-terey Research Laboratory Monterey Calif USA 1968
[2] G J Burgess ldquoProduct fragility and damage boundary theoryrdquoPackaging Technology and Science vol 15 no 10 pp 5ndash10 1988
[3] Z Wang C Wu and D Xi ldquoDamage boundary of a packagingsystem under rectangular pulse excitationrdquo Packaging Technol-ogy and Science vol 11 no 4 pp 189ndash202 1998
[4] ZWWang and C Y Hu ldquoShock spectra and damage boundarycurves for nonlinear package cushioning systemrdquo PackagingTechnology and Science vol 12 no 5 pp 207ndash217 1999
[5] C Ge ldquoModel of accelerated vibration testrdquo Packaging Technol-ogy and Science vol 13 no 1 pp 7ndash11 2000
[6] J Wang Z-W Wang L-X Lu Y Zhu and Y-G WangldquoThree-dimensional shock spectrum of critical component fornonlinear packaging systemrdquo Shock and Vibration vol 18 no 3pp 437ndash445 2011
8 Shock and Vibration
[7] J Wang F Duan J-H Jiang L-X Lu and Z-WWang ldquoDrop-ping damage evaluation for a hyperbolic tangent cushioningsystem with a critical componentrdquo Journal of Vibration andControl vol 18 no 10 pp 1417ndash1421 2012
[8] F-D Lu W-M Tao and D Gao ldquoVirtual mass method forsolution of dynamic response of composite cushion packagingsystemrdquo Packaging Technology and Science vol 26 no S1 pp32ndash42 2013
[9] H Kitazawa K Saito and Y Ishikawa ldquoEffect of difference inacceleration and velocity change on product damage due torepetitive shockrdquo Packaging Technology and Science vol 27 no3 pp 221ndash230 2014
[10] D Shires ldquoOn the time compression (test acceleration) ofbroadband random vibration testsrdquo Packaging Technology andScience vol 24 no 2 pp 75ndash87 2011
[11] M A Sek V Rouillard and A J Parker ldquoA study of nonlineareffects in a cushion-product system on its vibration transmis-sibility estimates with the reverse multiple input-single outputtechniquerdquo Packaging Technology and Science vol 26 no 3 pp125ndash135 2013
[12] Y Ren and C F Beards ldquoIdentification of joint properties of astructure using frf datardquo Journal of Sound and Vibration vol186 no 4 pp 567ndash587 1995
[13] T Yang S-H Fan and C-S Lin ldquoJoint stiffness identificationusing FRF measurementsrdquo Computers and Structures vol 81no 28-29 pp 2549ndash2556 2003
[14] D Celic and M Boltezar ldquoIdentification of the dynamic prop-erties of joints using frequency-response functionsrdquo Journal ofSound and Vibration vol 317 no 1-2 pp 158ndash174 2008
[15] J Zhen T C Lim and G Lu ldquoDetermination of systemvibratory response characteristics applying a spectral-basedinverse sub-structuring approach Part I analytical formula-tionrdquo International Journal of Vehicle Noise and Vibration vol1 no 1-2 pp 1ndash30 2004
[16] Z-WWang JWang Y-B Zhang C-Y Hu and Y Zhu ldquoAppli-cation of the inverse substructuremethod in the investigation ofdynamic characteristics of product transport systemrdquoPackagingTechnology and Science vol 25 no 6 pp 351ndash362 2012
[17] JWang L Lu and ZWang ldquoModeling the complex interactionbetween packaged product and vehiclerdquo Advanced ScienceLetters vol 4 no 6-7 pp 2207ndash2212 2011
[18] Z-W Wang and J Wang ldquoInverse substructure method ofthree-substructures coupled system and its application inproduct-transport-systemrdquo Journal of Vibration and Controlvol 17 no 6 pp 943ndash952 2011
[19] J Wang X Hong Y Qian Z-W Wang and L-X Lu ldquoInversesub-structuring method for multi-coordinate coupled producttransport systemrdquo Packaging Technology and Science vol 27 no5 pp 385ndash408 2014
Figure 5 Comparison of predicted and exact coupling dynamicstiffness here 119870
57= 11989657+ 119895120596119888
57 11987068= 11989668+ 119895120596119888
68
0 5 10 15 20Frequency (Hz)
Mag
nitu
de (m
N)
No noise1 noise
2 noise5 noise
100
10minus5
10minus10
10minus15
Figure 6 Effect of random noise of measured system-level FRFs onthe predicted frequency response function from119898
9to1198981
in Figure 6 the predicted system-level frequency responsefunction is affected remarkably by the input random errors ofmeasured system-level frequency response functions imply-ing that the formulation for prediction of coupling stiffness isextremely sensitive to the experiment errors A further studyon the error propagation as well as the elimination approachwill be conducted in the future
4 Conclusions
Three sets of equations for multicomponent coupled prod-uct transport system are obtained including the dynamicequilibrium conditions the displacement compatibility con-ditions in addition to the relationship of the system-level
FRFs and the selected easy-to-monitor component-levelFRFs Then the closed-form analytical solution to inversesubstructuring analysis ofmultisubstructure coupled producttransport system with rigid and flexible coupling is derivedThe method developed offers an approach to predict theunknown coupling dynamic stiffness from system-level FRFsand easy-to-monitor component-level FRFs avoiding thedifficulties of vibration excitation andor measurement in thecoupling interface Then the proposed method is validatedby a lumped mass-spring-damper model and the couplingdynamic stiffness is compared with the direct computationshowing exact agreement The method developed offers anapproach to predict the unknown coupling dynamic stiffnessfrom system-level FRFs and component-level FRFs whichcan be easily measured The suggested method may help toobtain themain controlling factors and contribution from thevarious structure-borne paths for product transport systemwhich may certainly facilitate the cushioning packagingdesign
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to appreciate the financial supportof National Natural Science Foundation of China (Grantno 51205167) Research Fund for the Doctoral Program ofHigher Education of China (Grant no 20120093120014) andFundamental Research Funds for the Central Universities(Grant no JUSRP51403A) the project was supported by theFoundation (Grant no 2013010) of Tianjin Key Laboratoryof Pulp amp Paper (Tianjin University of Science amp Technol-ogy) and by the Foundation of Jiangsu Key Laboratory ofAdvanced Food Manufacturing Equipment and Technology(Grant no FM-201403) and the research was also sponsoredby Qing Lan Project of Jiangsu Province (2014)
References
[1] R E Newton Fragility Assessment Theory and Practice Mon-terey Research Laboratory Monterey Calif USA 1968
[2] G J Burgess ldquoProduct fragility and damage boundary theoryrdquoPackaging Technology and Science vol 15 no 10 pp 5ndash10 1988
[3] Z Wang C Wu and D Xi ldquoDamage boundary of a packagingsystem under rectangular pulse excitationrdquo Packaging Technol-ogy and Science vol 11 no 4 pp 189ndash202 1998
[4] ZWWang and C Y Hu ldquoShock spectra and damage boundarycurves for nonlinear package cushioning systemrdquo PackagingTechnology and Science vol 12 no 5 pp 207ndash217 1999
[5] C Ge ldquoModel of accelerated vibration testrdquo Packaging Technol-ogy and Science vol 13 no 1 pp 7ndash11 2000
[6] J Wang Z-W Wang L-X Lu Y Zhu and Y-G WangldquoThree-dimensional shock spectrum of critical component fornonlinear packaging systemrdquo Shock and Vibration vol 18 no 3pp 437ndash445 2011
8 Shock and Vibration
[7] J Wang F Duan J-H Jiang L-X Lu and Z-WWang ldquoDrop-ping damage evaluation for a hyperbolic tangent cushioningsystem with a critical componentrdquo Journal of Vibration andControl vol 18 no 10 pp 1417ndash1421 2012
[8] F-D Lu W-M Tao and D Gao ldquoVirtual mass method forsolution of dynamic response of composite cushion packagingsystemrdquo Packaging Technology and Science vol 26 no S1 pp32ndash42 2013
[9] H Kitazawa K Saito and Y Ishikawa ldquoEffect of difference inacceleration and velocity change on product damage due torepetitive shockrdquo Packaging Technology and Science vol 27 no3 pp 221ndash230 2014
[10] D Shires ldquoOn the time compression (test acceleration) ofbroadband random vibration testsrdquo Packaging Technology andScience vol 24 no 2 pp 75ndash87 2011
[11] M A Sek V Rouillard and A J Parker ldquoA study of nonlineareffects in a cushion-product system on its vibration transmis-sibility estimates with the reverse multiple input-single outputtechniquerdquo Packaging Technology and Science vol 26 no 3 pp125ndash135 2013
[12] Y Ren and C F Beards ldquoIdentification of joint properties of astructure using frf datardquo Journal of Sound and Vibration vol186 no 4 pp 567ndash587 1995
[13] T Yang S-H Fan and C-S Lin ldquoJoint stiffness identificationusing FRF measurementsrdquo Computers and Structures vol 81no 28-29 pp 2549ndash2556 2003
[14] D Celic and M Boltezar ldquoIdentification of the dynamic prop-erties of joints using frequency-response functionsrdquo Journal ofSound and Vibration vol 317 no 1-2 pp 158ndash174 2008
[15] J Zhen T C Lim and G Lu ldquoDetermination of systemvibratory response characteristics applying a spectral-basedinverse sub-structuring approach Part I analytical formula-tionrdquo International Journal of Vehicle Noise and Vibration vol1 no 1-2 pp 1ndash30 2004
[16] Z-WWang JWang Y-B Zhang C-Y Hu and Y Zhu ldquoAppli-cation of the inverse substructuremethod in the investigation ofdynamic characteristics of product transport systemrdquoPackagingTechnology and Science vol 25 no 6 pp 351ndash362 2012
[17] JWang L Lu and ZWang ldquoModeling the complex interactionbetween packaged product and vehiclerdquo Advanced ScienceLetters vol 4 no 6-7 pp 2207ndash2212 2011
[18] Z-W Wang and J Wang ldquoInverse substructure method ofthree-substructures coupled system and its application inproduct-transport-systemrdquo Journal of Vibration and Controlvol 17 no 6 pp 943ndash952 2011
[19] J Wang X Hong Y Qian Z-W Wang and L-X Lu ldquoInversesub-structuring method for multi-coordinate coupled producttransport systemrdquo Packaging Technology and Science vol 27 no5 pp 385ndash408 2014
[7] J Wang F Duan J-H Jiang L-X Lu and Z-WWang ldquoDrop-ping damage evaluation for a hyperbolic tangent cushioningsystem with a critical componentrdquo Journal of Vibration andControl vol 18 no 10 pp 1417ndash1421 2012
[8] F-D Lu W-M Tao and D Gao ldquoVirtual mass method forsolution of dynamic response of composite cushion packagingsystemrdquo Packaging Technology and Science vol 26 no S1 pp32ndash42 2013
[9] H Kitazawa K Saito and Y Ishikawa ldquoEffect of difference inacceleration and velocity change on product damage due torepetitive shockrdquo Packaging Technology and Science vol 27 no3 pp 221ndash230 2014
[10] D Shires ldquoOn the time compression (test acceleration) ofbroadband random vibration testsrdquo Packaging Technology andScience vol 24 no 2 pp 75ndash87 2011
[11] M A Sek V Rouillard and A J Parker ldquoA study of nonlineareffects in a cushion-product system on its vibration transmis-sibility estimates with the reverse multiple input-single outputtechniquerdquo Packaging Technology and Science vol 26 no 3 pp125ndash135 2013
[12] Y Ren and C F Beards ldquoIdentification of joint properties of astructure using frf datardquo Journal of Sound and Vibration vol186 no 4 pp 567ndash587 1995
[13] T Yang S-H Fan and C-S Lin ldquoJoint stiffness identificationusing FRF measurementsrdquo Computers and Structures vol 81no 28-29 pp 2549ndash2556 2003
[14] D Celic and M Boltezar ldquoIdentification of the dynamic prop-erties of joints using frequency-response functionsrdquo Journal ofSound and Vibration vol 317 no 1-2 pp 158ndash174 2008
[15] J Zhen T C Lim and G Lu ldquoDetermination of systemvibratory response characteristics applying a spectral-basedinverse sub-structuring approach Part I analytical formula-tionrdquo International Journal of Vehicle Noise and Vibration vol1 no 1-2 pp 1ndash30 2004
[16] Z-WWang JWang Y-B Zhang C-Y Hu and Y Zhu ldquoAppli-cation of the inverse substructuremethod in the investigation ofdynamic characteristics of product transport systemrdquoPackagingTechnology and Science vol 25 no 6 pp 351ndash362 2012
[17] JWang L Lu and ZWang ldquoModeling the complex interactionbetween packaged product and vehiclerdquo Advanced ScienceLetters vol 4 no 6-7 pp 2207ndash2212 2011
[18] Z-W Wang and J Wang ldquoInverse substructure method ofthree-substructures coupled system and its application inproduct-transport-systemrdquo Journal of Vibration and Controlvol 17 no 6 pp 943ndash952 2011
[19] J Wang X Hong Y Qian Z-W Wang and L-X Lu ldquoInversesub-structuring method for multi-coordinate coupled producttransport systemrdquo Packaging Technology and Science vol 27 no5 pp 385ndash408 2014