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

[119867119883] = [

[119867119883]119900(119909)119894(119909)

[119867119883]119900(119909)119888(119909)

[119867119883]119888(119909)119894(119909)

[119867119883]119888(119909)119888(119909)

]

= minus1205962(minus1205962[119872119883] + 119895120596 [119862

119883] + [119870

119883])minus1

119883 = 119860 119861 119862 119878

(2)

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)

1198671198831198881(119909)1198881(119909)

1205731198671198611198881(119887)119894(119887)]

(3)

where

120572 = 1 119909 = 119886 119883 = 119860

minus1 119909 = 119887 119883 = 119861

120573 = 1 119909 = 119887 119883 = 119861

minus1 119909 = 119886 119883 = 119860

(4)

and119863119860119861

= (1198671198601198881(119886)1198881(119886)+ 1198671198611198881(119887)1198881(119887))minus1

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)

1198671198831198882(119909)1198882(119909)

1205731198671198621198882(119888)119894(119888)]

(5)

where

120572 = 1 119909 = 119889 119883 = 119863

minus1 119909 = 119888 119883 = 119862

120573 = 1 119909 = 119888 119883 = 119862

minus1 119909 = 119889 119883 = 119863

(6)

and119863119863119862

= (1198671198631198882(119887)1198882(119887)+ 1198671198621198882(119888)1198882(119888)minus 1205962119870119861119862

minus1)minus1

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

119904119900(119886)119894(119888)

by the FRFs of the three substructures [18 19]

119867119878119900(119886)119894(119888)

= 119867119860119900(119886)119888

1(119886)1198671198601198881(119886)1198881(119886)+ 1198671198611198881(119887)1198881(119887)minus1

times 1198671198611198881(119887)1198882(119887)

times 1198671198611198882(119887)1198882(119887)minus 1198671198611198882(119887)1198881(119887)

times (1198671198601198881(119886)1198881(119886)+ 1198671198611198881(119887)1198881(119887))minus1

1198671198611198881(119887)1198882(119887)

+1198671198621198882(119888)1198882(119888)minus 1205962119870minus1

119861119862minus1

1198671198621198882(119888)119894(119888)

(7)

Similarly one can obtain the system-level FRF for multi-coordinate coupling case as (see Figure 3)

[119867119878]119900(119886)119894(119888)

= [119867119860]119900(119886)1198881(119886)

4 Shock and Vibration

times [119867119860]1198881(119886)1198881(119886)+ [119867119861]1198881(119887)1198881(119887)minus1

[119867119861]1198881(119887)1198882(119887)

times [119867119861]1198882(119887)1198882(119887)minus [119867119861]1198882(119887)1198881(119887)

times ([119867119860]1198881(119886)1198881(119886)+ [119867119861]1198881(119887)1198881(119887))minus1

[119867119861]1198881(119887)1198882(119887)

+ [119867119862]1198882(119888)1198882(119888)minus 1205962[119870119861119862]minus1minus1

[119867119862]1198882(119888)119894(119888)

(8)

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

1198651198882(119889)

1198651198882(119888)

= [119867minus1

1198631198882(119889)1198882(119889)

minus 120596minus2119870119861119862

120596minus2119870119861119862

120596minus2119870119861119862

119867minus1

1198621198882(119888)1198882(119888)minus 120596minus2119870119861119862

]1198882(119889)

1198882(119888)

(9)

The response vector of the coupling coordinates can then besolved as

1198882(119889)

1198882(119888)

= [Γ]minus1[119867minus1

1198621198882(119888)1198882(119888)minus 120596minus2119870119861119862

minus120596minus2119870119861119862

minus120596minus2119870119861119862

119867minus1

1198631198882(119889)1198882(119889)

minus 120596minus2119870119861119862

]

times 1198651198882(119889)

1198651198882(119888)

(10)

where

[Γ] =

100381610038161003816100381610038161003816100381610038161003816

119867minus1

1198631198882(119889)1198882(119889)

minus 120596minus2119870119861119862

120596minus2119870119861119862

120596minus2119870119861119862

119867minus1

1198621198882(119888)1198882(119888)minus 120596minus2119870119861119862

100381610038161003816100381610038161003816100381610038161003816

(11)

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

119870119861119862

=

minus12059621198671198781198882(119887)1198882(119888)

1198671198781198882(119887)1198882(119887)1198671198781198882(119888)1198882(119888)minus 1198672

1198781198882(119887)1198882(119888)

(13)

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)

make an inverse formulation and then get

119870119861119862

= minus12059621198671198621198882(119888)119894(119888)

119867minus1

119878119900(119886)119894(119888)119867119860119900(119886)119888

1(119886)

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

1198651198882(119889)

1198651198882(119888)

= [[119867119863]minus1

1198882(119889)1198882(119889)

minus 120596minus2[119870119861119862] 120596

minus2[119870119861119862]

120596minus2[119870119861119862] [119867

119862]minus1

1198882(119888)1198882(119888)minus 120596minus2[119870119861119862]]

times 1198882(119889)

1198882(119888)

(15)

which is inverted to express the response vector of thecoupling coordinates as

Shock and Vibration 5

1198882(119889)

1198882(119888) = [

[Γ]minus1([119868] minus 120596

2[119867119862]minus1

1198882(119888)1198882(119888)[119870119861119862]minus1) [Γ]

minus1

[Γ]minus119879

(minus1205962[119870119861119862]minus1[119867119863]minus1

1198882(119889)1198882(119889)

+ [119868]) [Γ]minus1]

times 1198651198882(119889)

1198651198882(119888)

(16)

where [Γ] = minus1205962[119867119862]minus1

1198882(119888)1198882(119888)[119870119861119862]minus1[119867119863]minus1

1198882(119889)1198882(119889)

+

[119867119862]minus1

1198882(119888)1198882(119888)+ [119867119863]minus1

1198882(119889)1198882(119889)

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

1198882(119887)1198882(119888)[119867119878]1198882(119888)1198882(119888)minus [119867119878]1198882(119887)1198882(119888))minus1

(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

2

times [119867119862]1198882(119888)119894(119888)

[119867119878]minus1

119900(119886)119894(119888)[119867119860]119900(119886)1198881(119886)

times ([119867119860]1198881(119886)1198881(119886)+ [119867119861]1198881(119887)1198881(119887))minus1

times [119867119861]1198881(119887)1198882(119887)minus [119867119861]1198882(119887)1198882(119887)+ [119867119861]1198882(119887)1198881(119887)

times ([119867119860]1198881(119886)1198881(119886)+ [119867119861]1198881(119887)1198881(119887))minus1

times [119867119861]1198881(119887)1198882(119887)minus [119867119862]1198882(119888)1198882(119888)minus1

(19)where [Ψ] = [119867

119860]119900(119886)1198881(119886)([119867119860]1198881(119886)1198881(119886)

+

[119867119861]1198881(119887)1198881(119887))minus1[119867119861]1198881(119887)1198882(119887)

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

119860119894119895 119894 119895 = (2) (3) and

119894 = (1) (2) 119895 = (2) (3)119867119861119894119895

119894 119895 = (1) (2) and 119894 119895 = (3) (4)and 119894 = (1) (2) 119895 = (3) (4) and119867

119862119894119895 119894 = (6) (7) 119895 = (7) (8)

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

Mass (kg) Stiffness (Nm) Damping (Nsm)m1 m21 m22 m3 k11 k12 k2 k3 c11 c12 c2 c320 8 8 22 7200 7200 3600 4200 15 15 12 15m4 m5 m6 m7 k4 k5 k6 k7 c4 c5 c6 c718 15 12 16 7900 6800 6000 5800 10 8 22 8m8 m9 k57 k68 c57 c68 16 40 5800 6200 16 12

m1

c12

k12

c11

k11

k2 c2

c4

k4

c3

k3

k5 c5

c68

k68

c57

k57

c7

k7

c6

k6

m8

A B C

Figure 4 A lumped mass-spring-damper model

component 119862

[119872119862] = [

[

1198987

0 0

0 1198988

0

0 0 1198989

]

]

[119870119862] = [

[

1198966

0 minus1198966

0 1198967

minus1198967

minus1198966minus11989671198966+ 1198967

]

]

[119862119862] = [

[

1198886

0 minus1198886

0 1198887

minus1198887

minus1198886minus11988871198886+ 1198887

]

]

(22)

system 119878

[119872119878] =

[[[[[[[[[[

[

1198981

0 0 0 0 0 0 0

0 11989821+ 1198983

0 0 0 0 0 0

0 0 11989822+ 1198984

0 0 0 0 0

0 0 0 1198985

0 0 0 0

0 0 0 0 1198986

0 0 0

0 0 0 0 0 1198987

0 0

0 0 0 0 0 0 1198988

0

0 0 0 0 0 0 0 1198989

]]]]]]]]]]

]

[119870119878] =

[[[[[[[[[[

[

11989611+ 11989612

minus11989611

minus11989612

0 0 0 0 0

minus11989611

11989611+ 1198962+ 1198963

minus1198962

minus1198963

0 0 0 0

minus11989612

minus1198962

11989612+ 1198962+ 1198964

0 minus1198964

0 0 0

0 minus1198963

0 1198963+ 1198965+ 11989657

minus1198965

minus11989657

0 0

0 0 minus1198964

minus1198965

1198964+ 1198965+ 11989668

0 minus11989668

0

0 0 0 minus11989657

0 1198966+ 11989657

0 minus1198966

0 0 0 0 minus11989668

0 1198967+ 11989668

minus1198967

0 0 0 0 0 minus1198966

minus1198967

1198966+ 1198967

]]]]]]]]]]

]

[119862119878] =

[[[[[[[[[[

[

11988811+ 11988812

minus11988811

minus11988812

0 0 0 0 0

minus11988811

11988811+ 1198882+ 1198883

minus1198882

minus1198883

0 0 0 0

minus11988812

minus1198882

11988812+ 1198882+ 1198884

0 minus1198884

0 0 0

0 minus1198883

0 1198883+ 1198885+ 11988857

minus1198885

minus11988857

0 0

0 0 minus1198884

minus1198885

1198884+ 1198885+ 11988868

0 minus11988868

0

0 0 0 minus11988857

0 1198886+ 11988857

0 minus1198886

0 0 0 0 minus11988868

0 1198887+ 11988868

minus1198887

0 0 0 0 0 minus1198886

minus1198887

1198886+ 1198887

]]]]]]]]]]

]

(23)

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

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Shock and Vibration

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DistributedSensor Networks

International Journal of

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

[119867119883] = [

[119867119883]119900(119909)119894(119909)

[119867119883]119900(119909)119888(119909)

[119867119883]119888(119909)119894(119909)

[119867119883]119888(119909)119888(119909)

]

= minus1205962(minus1205962[119872119883] + 119895120596 [119862

119883] + [119870

119883])minus1

119883 = 119860 119861 119862 119878

(2)

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)

1198671198831198881(119909)1198881(119909)

1205731198671198611198881(119887)119894(119887)]

(3)

where

120572 = 1 119909 = 119886 119883 = 119860

minus1 119909 = 119887 119883 = 119861

120573 = 1 119909 = 119887 119883 = 119861

minus1 119909 = 119886 119883 = 119860

(4)

and119863119860119861

= (1198671198601198881(119886)1198881(119886)+ 1198671198611198881(119887)1198881(119887))minus1

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)

1198671198831198882(119909)1198882(119909)

1205731198671198621198882(119888)119894(119888)]

(5)

where

120572 = 1 119909 = 119889 119883 = 119863

minus1 119909 = 119888 119883 = 119862

120573 = 1 119909 = 119888 119883 = 119862

minus1 119909 = 119889 119883 = 119863

(6)

and119863119863119862

= (1198671198631198882(119887)1198882(119887)+ 1198671198621198882(119888)1198882(119888)minus 1205962119870119861119862

minus1)minus1

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

119904119900(119886)119894(119888)

by the FRFs of the three substructures [18 19]

119867119878119900(119886)119894(119888)

= 119867119860119900(119886)119888

1(119886)1198671198601198881(119886)1198881(119886)+ 1198671198611198881(119887)1198881(119887)minus1

times 1198671198611198881(119887)1198882(119887)

times 1198671198611198882(119887)1198882(119887)minus 1198671198611198882(119887)1198881(119887)

times (1198671198601198881(119886)1198881(119886)+ 1198671198611198881(119887)1198881(119887))minus1

1198671198611198881(119887)1198882(119887)

+1198671198621198882(119888)1198882(119888)minus 1205962119870minus1

119861119862minus1

1198671198621198882(119888)119894(119888)

(7)

Similarly one can obtain the system-level FRF for multi-coordinate coupling case as (see Figure 3)

[119867119878]119900(119886)119894(119888)

= [119867119860]119900(119886)1198881(119886)

4 Shock and Vibration

times [119867119860]1198881(119886)1198881(119886)+ [119867119861]1198881(119887)1198881(119887)minus1

[119867119861]1198881(119887)1198882(119887)

times [119867119861]1198882(119887)1198882(119887)minus [119867119861]1198882(119887)1198881(119887)

times ([119867119860]1198881(119886)1198881(119886)+ [119867119861]1198881(119887)1198881(119887))minus1

[119867119861]1198881(119887)1198882(119887)

+ [119867119862]1198882(119888)1198882(119888)minus 1205962[119870119861119862]minus1minus1

[119867119862]1198882(119888)119894(119888)

(8)

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

1198651198882(119889)

1198651198882(119888)

= [119867minus1

1198631198882(119889)1198882(119889)

minus 120596minus2119870119861119862

120596minus2119870119861119862

120596minus2119870119861119862

119867minus1

1198621198882(119888)1198882(119888)minus 120596minus2119870119861119862

]1198882(119889)

1198882(119888)

(9)

The response vector of the coupling coordinates can then besolved as

1198882(119889)

1198882(119888)

= [Γ]minus1[119867minus1

1198621198882(119888)1198882(119888)minus 120596minus2119870119861119862

minus120596minus2119870119861119862

minus120596minus2119870119861119862

119867minus1

1198631198882(119889)1198882(119889)

minus 120596minus2119870119861119862

]

times 1198651198882(119889)

1198651198882(119888)

(10)

where

[Γ] =

100381610038161003816100381610038161003816100381610038161003816

119867minus1

1198631198882(119889)1198882(119889)

minus 120596minus2119870119861119862

120596minus2119870119861119862

120596minus2119870119861119862

119867minus1

1198621198882(119888)1198882(119888)minus 120596minus2119870119861119862

100381610038161003816100381610038161003816100381610038161003816

(11)

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

119870119861119862

=

minus12059621198671198781198882(119887)1198882(119888)

1198671198781198882(119887)1198882(119887)1198671198781198882(119888)1198882(119888)minus 1198672

1198781198882(119887)1198882(119888)

(13)

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)

make an inverse formulation and then get

119870119861119862

= minus12059621198671198621198882(119888)119894(119888)

119867minus1

119878119900(119886)119894(119888)119867119860119900(119886)119888

1(119886)

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

1198651198882(119889)

1198651198882(119888)

= [[119867119863]minus1

1198882(119889)1198882(119889)

minus 120596minus2[119870119861119862] 120596

minus2[119870119861119862]

120596minus2[119870119861119862] [119867

119862]minus1

1198882(119888)1198882(119888)minus 120596minus2[119870119861119862]]

times 1198882(119889)

1198882(119888)

(15)

which is inverted to express the response vector of thecoupling coordinates as

Shock and Vibration 5

1198882(119889)

1198882(119888) = [

[Γ]minus1([119868] minus 120596

2[119867119862]minus1

1198882(119888)1198882(119888)[119870119861119862]minus1) [Γ]

minus1

[Γ]minus119879

(minus1205962[119870119861119862]minus1[119867119863]minus1

1198882(119889)1198882(119889)

+ [119868]) [Γ]minus1]

times 1198651198882(119889)

1198651198882(119888)

(16)

where [Γ] = minus1205962[119867119862]minus1

1198882(119888)1198882(119888)[119870119861119862]minus1[119867119863]minus1

1198882(119889)1198882(119889)

+

[119867119862]minus1

1198882(119888)1198882(119888)+ [119867119863]minus1

1198882(119889)1198882(119889)

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

1198882(119887)1198882(119888)[119867119878]1198882(119888)1198882(119888)minus [119867119878]1198882(119887)1198882(119888))minus1

(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

2

times [119867119862]1198882(119888)119894(119888)

[119867119878]minus1

119900(119886)119894(119888)[119867119860]119900(119886)1198881(119886)

times ([119867119860]1198881(119886)1198881(119886)+ [119867119861]1198881(119887)1198881(119887))minus1

times [119867119861]1198881(119887)1198882(119887)minus [119867119861]1198882(119887)1198882(119887)+ [119867119861]1198882(119887)1198881(119887)

times ([119867119860]1198881(119886)1198881(119886)+ [119867119861]1198881(119887)1198881(119887))minus1

times [119867119861]1198881(119887)1198882(119887)minus [119867119862]1198882(119888)1198882(119888)minus1

(19)where [Ψ] = [119867

119860]119900(119886)1198881(119886)([119867119860]1198881(119886)1198881(119886)

+

[119867119861]1198881(119887)1198881(119887))minus1[119867119861]1198881(119887)1198882(119887)

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

119860119894119895 119894 119895 = (2) (3) and

119894 = (1) (2) 119895 = (2) (3)119867119861119894119895

119894 119895 = (1) (2) and 119894 119895 = (3) (4)and 119894 = (1) (2) 119895 = (3) (4) and119867

119862119894119895 119894 = (6) (7) 119895 = (7) (8)

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

Mass (kg) Stiffness (Nm) Damping (Nsm)m1 m21 m22 m3 k11 k12 k2 k3 c11 c12 c2 c320 8 8 22 7200 7200 3600 4200 15 15 12 15m4 m5 m6 m7 k4 k5 k6 k7 c4 c5 c6 c718 15 12 16 7900 6800 6000 5800 10 8 22 8m8 m9 k57 k68 c57 c68 16 40 5800 6200 16 12

m1

c12

k12

c11

k11

k2 c2

c4

k4

c3

k3

k5 c5

c68

k68

c57

k57

c7

k7

c6

k6

m8

A B C

Figure 4 A lumped mass-spring-damper model

component 119862

[119872119862] = [

[

1198987

0 0

0 1198988

0

0 0 1198989

]

]

[119870119862] = [

[

1198966

0 minus1198966

0 1198967

minus1198967

minus1198966minus11989671198966+ 1198967

]

]

[119862119862] = [

[

1198886

0 minus1198886

0 1198887

minus1198887

minus1198886minus11988871198886+ 1198887

]

]

(22)

system 119878

[119872119878] =

[[[[[[[[[[

[

1198981

0 0 0 0 0 0 0

0 11989821+ 1198983

0 0 0 0 0 0

0 0 11989822+ 1198984

0 0 0 0 0

0 0 0 1198985

0 0 0 0

0 0 0 0 1198986

0 0 0

0 0 0 0 0 1198987

0 0

0 0 0 0 0 0 1198988

0

0 0 0 0 0 0 0 1198989

]]]]]]]]]]

]

[119870119878] =

[[[[[[[[[[

[

11989611+ 11989612

minus11989611

minus11989612

0 0 0 0 0

minus11989611

11989611+ 1198962+ 1198963

minus1198962

minus1198963

0 0 0 0

minus11989612

minus1198962

11989612+ 1198962+ 1198964

0 minus1198964

0 0 0

0 minus1198963

0 1198963+ 1198965+ 11989657

minus1198965

minus11989657

0 0

0 0 minus1198964

minus1198965

1198964+ 1198965+ 11989668

0 minus11989668

0

0 0 0 minus11989657

0 1198966+ 11989657

0 minus1198966

0 0 0 0 minus11989668

0 1198967+ 11989668

minus1198967

0 0 0 0 0 minus1198966

minus1198967

1198966+ 1198967

]]]]]]]]]]

]

[119862119878] =

[[[[[[[[[[

[

11988811+ 11988812

minus11988811

minus11988812

0 0 0 0 0

minus11988811

11988811+ 1198882+ 1198883

minus1198882

minus1198883

0 0 0 0

minus11988812

minus1198882

11988812+ 1198882+ 1198884

0 minus1198884

0 0 0

0 minus1198883

0 1198883+ 1198885+ 11988857

minus1198885

minus11988857

0 0

0 0 minus1198884

minus1198885

1198884+ 1198885+ 11988868

0 minus11988868

0

0 0 0 minus11988857

0 1198886+ 11988857

0 minus1198886

0 0 0 0 minus11988868

0 1198887+ 11988868

minus1198887

0 0 0 0 0 minus1198886

minus1198887

1198886+ 1198887

]]]]]]]]]]

]

(23)

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

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DistributedSensor Networks

International Journal of

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)

1198671198831198881(119909)1198881(119909)

1205731198671198611198881(119887)119894(119887)]

(3)

where

120572 = 1 119909 = 119886 119883 = 119860

minus1 119909 = 119887 119883 = 119861

120573 = 1 119909 = 119887 119883 = 119861

minus1 119909 = 119886 119883 = 119860

(4)

and119863119860119861

= (1198671198601198881(119886)1198881(119886)+ 1198671198611198881(119887)1198881(119887))minus1

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)

1198671198831198882(119909)1198882(119909)

1205731198671198621198882(119888)119894(119888)]

(5)

where

120572 = 1 119909 = 119889 119883 = 119863

minus1 119909 = 119888 119883 = 119862

120573 = 1 119909 = 119888 119883 = 119862

minus1 119909 = 119889 119883 = 119863

(6)

and119863119863119862

= (1198671198631198882(119887)1198882(119887)+ 1198671198621198882(119888)1198882(119888)minus 1205962119870119861119862

minus1)minus1

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

119904119900(119886)119894(119888)

by the FRFs of the three substructures [18 19]

119867119878119900(119886)119894(119888)

= 119867119860119900(119886)119888

1(119886)1198671198601198881(119886)1198881(119886)+ 1198671198611198881(119887)1198881(119887)minus1

times 1198671198611198881(119887)1198882(119887)

times 1198671198611198882(119887)1198882(119887)minus 1198671198611198882(119887)1198881(119887)

times (1198671198601198881(119886)1198881(119886)+ 1198671198611198881(119887)1198881(119887))minus1

1198671198611198881(119887)1198882(119887)

+1198671198621198882(119888)1198882(119888)minus 1205962119870minus1

119861119862minus1

1198671198621198882(119888)119894(119888)

(7)

Similarly one can obtain the system-level FRF for multi-coordinate coupling case as (see Figure 3)

[119867119878]119900(119886)119894(119888)

= [119867119860]119900(119886)1198881(119886)

4 Shock and Vibration

times [119867119860]1198881(119886)1198881(119886)+ [119867119861]1198881(119887)1198881(119887)minus1

[119867119861]1198881(119887)1198882(119887)

times [119867119861]1198882(119887)1198882(119887)minus [119867119861]1198882(119887)1198881(119887)

times ([119867119860]1198881(119886)1198881(119886)+ [119867119861]1198881(119887)1198881(119887))minus1

[119867119861]1198881(119887)1198882(119887)

+ [119867119862]1198882(119888)1198882(119888)minus 1205962[119870119861119862]minus1minus1

[119867119862]1198882(119888)119894(119888)

(8)

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

1198651198882(119889)

1198651198882(119888)

= [119867minus1

1198631198882(119889)1198882(119889)

minus 120596minus2119870119861119862

120596minus2119870119861119862

120596minus2119870119861119862

119867minus1

1198621198882(119888)1198882(119888)minus 120596minus2119870119861119862

]1198882(119889)

1198882(119888)

(9)

The response vector of the coupling coordinates can then besolved as

1198882(119889)

1198882(119888)

= [Γ]minus1[119867minus1

1198621198882(119888)1198882(119888)minus 120596minus2119870119861119862

minus120596minus2119870119861119862

minus120596minus2119870119861119862

119867minus1

1198631198882(119889)1198882(119889)

minus 120596minus2119870119861119862

]

times 1198651198882(119889)

1198651198882(119888)

(10)

where

[Γ] =

100381610038161003816100381610038161003816100381610038161003816

119867minus1

1198631198882(119889)1198882(119889)

minus 120596minus2119870119861119862

120596minus2119870119861119862

120596minus2119870119861119862

119867minus1

1198621198882(119888)1198882(119888)minus 120596minus2119870119861119862

100381610038161003816100381610038161003816100381610038161003816

(11)

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

119870119861119862

=

minus12059621198671198781198882(119887)1198882(119888)

1198671198781198882(119887)1198882(119887)1198671198781198882(119888)1198882(119888)minus 1198672

1198781198882(119887)1198882(119888)

(13)

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)

make an inverse formulation and then get

119870119861119862

= minus12059621198671198621198882(119888)119894(119888)

119867minus1

119878119900(119886)119894(119888)119867119860119900(119886)119888

1(119886)

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

1198651198882(119889)

1198651198882(119888)

= [[119867119863]minus1

1198882(119889)1198882(119889)

minus 120596minus2[119870119861119862] 120596

minus2[119870119861119862]

120596minus2[119870119861119862] [119867

119862]minus1

1198882(119888)1198882(119888)minus 120596minus2[119870119861119862]]

times 1198882(119889)

1198882(119888)

(15)

which is inverted to express the response vector of thecoupling coordinates as

Shock and Vibration 5

1198882(119889)

1198882(119888) = [

[Γ]minus1([119868] minus 120596

2[119867119862]minus1

1198882(119888)1198882(119888)[119870119861119862]minus1) [Γ]

minus1

[Γ]minus119879

(minus1205962[119870119861119862]minus1[119867119863]minus1

1198882(119889)1198882(119889)

+ [119868]) [Γ]minus1]

times 1198651198882(119889)

1198651198882(119888)

(16)

where [Γ] = minus1205962[119867119862]minus1

1198882(119888)1198882(119888)[119870119861119862]minus1[119867119863]minus1

1198882(119889)1198882(119889)

+

[119867119862]minus1

1198882(119888)1198882(119888)+ [119867119863]minus1

1198882(119889)1198882(119889)

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

1198882(119887)1198882(119888)[119867119878]1198882(119888)1198882(119888)minus [119867119878]1198882(119887)1198882(119888))minus1

(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

2

times [119867119862]1198882(119888)119894(119888)

[119867119878]minus1

119900(119886)119894(119888)[119867119860]119900(119886)1198881(119886)

times ([119867119860]1198881(119886)1198881(119886)+ [119867119861]1198881(119887)1198881(119887))minus1

times [119867119861]1198881(119887)1198882(119887)minus [119867119861]1198882(119887)1198882(119887)+ [119867119861]1198882(119887)1198881(119887)

times ([119867119860]1198881(119886)1198881(119886)+ [119867119861]1198881(119887)1198881(119887))minus1

times [119867119861]1198881(119887)1198882(119887)minus [119867119862]1198882(119888)1198882(119888)minus1

(19)where [Ψ] = [119867

119860]119900(119886)1198881(119886)([119867119860]1198881(119886)1198881(119886)

+

[119867119861]1198881(119887)1198881(119887))minus1[119867119861]1198881(119887)1198882(119887)

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

119860119894119895 119894 119895 = (2) (3) and

119894 = (1) (2) 119895 = (2) (3)119867119861119894119895

119894 119895 = (1) (2) and 119894 119895 = (3) (4)and 119894 = (1) (2) 119895 = (3) (4) and119867

119862119894119895 119894 = (6) (7) 119895 = (7) (8)

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

Mass (kg) Stiffness (Nm) Damping (Nsm)m1 m21 m22 m3 k11 k12 k2 k3 c11 c12 c2 c320 8 8 22 7200 7200 3600 4200 15 15 12 15m4 m5 m6 m7 k4 k5 k6 k7 c4 c5 c6 c718 15 12 16 7900 6800 6000 5800 10 8 22 8m8 m9 k57 k68 c57 c68 16 40 5800 6200 16 12

m1

c12

k12

c11

k11

k2 c2

c4

k4

c3

k3

k5 c5

c68

k68

c57

k57

c7

k7

c6

k6

m8

A B C

Figure 4 A lumped mass-spring-damper model

component 119862

[119872119862] = [

[

1198987

0 0

0 1198988

0

0 0 1198989

]

]

[119870119862] = [

[

1198966

0 minus1198966

0 1198967

minus1198967

minus1198966minus11989671198966+ 1198967

]

]

[119862119862] = [

[

1198886

0 minus1198886

0 1198887

minus1198887

minus1198886minus11988871198886+ 1198887

]

]

(22)

system 119878

[119872119878] =

[[[[[[[[[[

[

1198981

0 0 0 0 0 0 0

0 11989821+ 1198983

0 0 0 0 0 0

0 0 11989822+ 1198984

0 0 0 0 0

0 0 0 1198985

0 0 0 0

0 0 0 0 1198986

0 0 0

0 0 0 0 0 1198987

0 0

0 0 0 0 0 0 1198988

0

0 0 0 0 0 0 0 1198989

]]]]]]]]]]

]

[119870119878] =

[[[[[[[[[[

[

11989611+ 11989612

minus11989611

minus11989612

0 0 0 0 0

minus11989611

11989611+ 1198962+ 1198963

minus1198962

minus1198963

0 0 0 0

minus11989612

minus1198962

11989612+ 1198962+ 1198964

0 minus1198964

0 0 0

0 minus1198963

0 1198963+ 1198965+ 11989657

minus1198965

minus11989657

0 0

0 0 minus1198964

minus1198965

1198964+ 1198965+ 11989668

0 minus11989668

0

0 0 0 minus11989657

0 1198966+ 11989657

0 minus1198966

0 0 0 0 minus11989668

0 1198967+ 11989668

minus1198967

0 0 0 0 0 minus1198966

minus1198967

1198966+ 1198967

]]]]]]]]]]

]

[119862119878] =

[[[[[[[[[[

[

11988811+ 11988812

minus11988811

minus11988812

0 0 0 0 0

minus11988811

11988811+ 1198882+ 1198883

minus1198882

minus1198883

0 0 0 0

minus11988812

minus1198882

11988812+ 1198882+ 1198884

0 minus1198884

0 0 0

0 minus1198883

0 1198883+ 1198885+ 11988857

minus1198885

minus11988857

0 0

0 0 minus1198884

minus1198885

1198884+ 1198885+ 11988868

0 minus11988868

0

0 0 0 minus11988857

0 1198886+ 11988857

0 minus1198886

0 0 0 0 minus11988868

0 1198887+ 11988868

minus1198887

0 0 0 0 0 minus1198886

minus1198887

1198886+ 1198887

]]]]]]]]]]

]

(23)

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

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Shock and Vibration

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DistributedSensor Networks

International Journal of

4 Shock and Vibration

times [119867119860]1198881(119886)1198881(119886)+ [119867119861]1198881(119887)1198881(119887)minus1

[119867119861]1198881(119887)1198882(119887)

times [119867119861]1198882(119887)1198882(119887)minus [119867119861]1198882(119887)1198881(119887)

times ([119867119860]1198881(119886)1198881(119886)+ [119867119861]1198881(119887)1198881(119887))minus1

[119867119861]1198881(119887)1198882(119887)

+ [119867119862]1198882(119888)1198882(119888)minus 1205962[119870119861119862]minus1minus1

[119867119862]1198882(119888)119894(119888)

(8)

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

1198651198882(119889)

1198651198882(119888)

= [119867minus1

1198631198882(119889)1198882(119889)

minus 120596minus2119870119861119862

120596minus2119870119861119862

120596minus2119870119861119862

119867minus1

1198621198882(119888)1198882(119888)minus 120596minus2119870119861119862

]1198882(119889)

1198882(119888)

(9)

The response vector of the coupling coordinates can then besolved as

1198882(119889)

1198882(119888)

= [Γ]minus1[119867minus1

1198621198882(119888)1198882(119888)minus 120596minus2119870119861119862

minus120596minus2119870119861119862

minus120596minus2119870119861119862

119867minus1

1198631198882(119889)1198882(119889)

minus 120596minus2119870119861119862

]

times 1198651198882(119889)

1198651198882(119888)

(10)

where

[Γ] =

100381610038161003816100381610038161003816100381610038161003816

119867minus1

1198631198882(119889)1198882(119889)

minus 120596minus2119870119861119862

120596minus2119870119861119862

120596minus2119870119861119862

119867minus1

1198621198882(119888)1198882(119888)minus 120596minus2119870119861119862

100381610038161003816100381610038161003816100381610038161003816

(11)

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

119870119861119862

=

minus12059621198671198781198882(119887)1198882(119888)

1198671198781198882(119887)1198882(119887)1198671198781198882(119888)1198882(119888)minus 1198672

1198781198882(119887)1198882(119888)

(13)

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)

make an inverse formulation and then get

119870119861119862

= minus12059621198671198621198882(119888)119894(119888)

119867minus1

119878119900(119886)119894(119888)119867119860119900(119886)119888

1(119886)

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

1198651198882(119889)

1198651198882(119888)

= [[119867119863]minus1

1198882(119889)1198882(119889)

minus 120596minus2[119870119861119862] 120596

minus2[119870119861119862]

120596minus2[119870119861119862] [119867

119862]minus1

1198882(119888)1198882(119888)minus 120596minus2[119870119861119862]]

times 1198882(119889)

1198882(119888)

(15)

which is inverted to express the response vector of thecoupling coordinates as

Shock and Vibration 5

1198882(119889)

1198882(119888) = [

[Γ]minus1([119868] minus 120596

2[119867119862]minus1

1198882(119888)1198882(119888)[119870119861119862]minus1) [Γ]

minus1

[Γ]minus119879

(minus1205962[119870119861119862]minus1[119867119863]minus1

1198882(119889)1198882(119889)

+ [119868]) [Γ]minus1]

times 1198651198882(119889)

1198651198882(119888)

(16)

where [Γ] = minus1205962[119867119862]minus1

1198882(119888)1198882(119888)[119870119861119862]minus1[119867119863]minus1

1198882(119889)1198882(119889)

+

[119867119862]minus1

1198882(119888)1198882(119888)+ [119867119863]minus1

1198882(119889)1198882(119889)

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

1198882(119887)1198882(119888)[119867119878]1198882(119888)1198882(119888)minus [119867119878]1198882(119887)1198882(119888))minus1

(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

2

times [119867119862]1198882(119888)119894(119888)

[119867119878]minus1

119900(119886)119894(119888)[119867119860]119900(119886)1198881(119886)

times ([119867119860]1198881(119886)1198881(119886)+ [119867119861]1198881(119887)1198881(119887))minus1

times [119867119861]1198881(119887)1198882(119887)minus [119867119861]1198882(119887)1198882(119887)+ [119867119861]1198882(119887)1198881(119887)

times ([119867119860]1198881(119886)1198881(119886)+ [119867119861]1198881(119887)1198881(119887))minus1

times [119867119861]1198881(119887)1198882(119887)minus [119867119862]1198882(119888)1198882(119888)minus1

(19)where [Ψ] = [119867

119860]119900(119886)1198881(119886)([119867119860]1198881(119886)1198881(119886)

+

[119867119861]1198881(119887)1198881(119887))minus1[119867119861]1198881(119887)1198882(119887)

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

119860119894119895 119894 119895 = (2) (3) and

119894 = (1) (2) 119895 = (2) (3)119867119861119894119895

119894 119895 = (1) (2) and 119894 119895 = (3) (4)and 119894 = (1) (2) 119895 = (3) (4) and119867

119862119894119895 119894 = (6) (7) 119895 = (7) (8)

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

Mass (kg) Stiffness (Nm) Damping (Nsm)m1 m21 m22 m3 k11 k12 k2 k3 c11 c12 c2 c320 8 8 22 7200 7200 3600 4200 15 15 12 15m4 m5 m6 m7 k4 k5 k6 k7 c4 c5 c6 c718 15 12 16 7900 6800 6000 5800 10 8 22 8m8 m9 k57 k68 c57 c68 16 40 5800 6200 16 12

m1

c12

k12

c11

k11

k2 c2

c4

k4

c3

k3

k5 c5

c68

k68

c57

k57

c7

k7

c6

k6

m8

A B C

Figure 4 A lumped mass-spring-damper model

component 119862

[119872119862] = [

[

1198987

0 0

0 1198988

0

0 0 1198989

]

]

[119870119862] = [

[

1198966

0 minus1198966

0 1198967

minus1198967

minus1198966minus11989671198966+ 1198967

]

]

[119862119862] = [

[

1198886

0 minus1198886

0 1198887

minus1198887

minus1198886minus11988871198886+ 1198887

]

]

(22)

system 119878

[119872119878] =

[[[[[[[[[[

[

1198981

0 0 0 0 0 0 0

0 11989821+ 1198983

0 0 0 0 0 0

0 0 11989822+ 1198984

0 0 0 0 0

0 0 0 1198985

0 0 0 0

0 0 0 0 1198986

0 0 0

0 0 0 0 0 1198987

0 0

0 0 0 0 0 0 1198988

0

0 0 0 0 0 0 0 1198989

]]]]]]]]]]

]

[119870119878] =

[[[[[[[[[[

[

11989611+ 11989612

minus11989611

minus11989612

0 0 0 0 0

minus11989611

11989611+ 1198962+ 1198963

minus1198962

minus1198963

0 0 0 0

minus11989612

minus1198962

11989612+ 1198962+ 1198964

0 minus1198964

0 0 0

0 minus1198963

0 1198963+ 1198965+ 11989657

minus1198965

minus11989657

0 0

0 0 minus1198964

minus1198965

1198964+ 1198965+ 11989668

0 minus11989668

0

0 0 0 minus11989657

0 1198966+ 11989657

0 minus1198966

0 0 0 0 minus11989668

0 1198967+ 11989668

minus1198967

0 0 0 0 0 minus1198966

minus1198967

1198966+ 1198967

]]]]]]]]]]

]

[119862119878] =

[[[[[[[[[[

[

11988811+ 11988812

minus11988811

minus11988812

0 0 0 0 0

minus11988811

11988811+ 1198882+ 1198883

minus1198882

minus1198883

0 0 0 0

minus11988812

minus1198882

11988812+ 1198882+ 1198884

0 minus1198884

0 0 0

0 minus1198883

0 1198883+ 1198885+ 11988857

minus1198885

minus11988857

0 0

0 0 minus1198884

minus1198885

1198884+ 1198885+ 11988868

0 minus11988868

0

0 0 0 minus11988857

0 1198886+ 11988857

0 minus1198886

0 0 0 0 minus11988868

0 1198887+ 11988868

minus1198887

0 0 0 0 0 minus1198886

minus1198887

1198886+ 1198887

]]]]]]]]]]

]

(23)

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

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Advances inOptoElectronics

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Volume 2014

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

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DistributedSensor Networks

International Journal of

Shock and Vibration 5

1198882(119889)

1198882(119888) = [

[Γ]minus1([119868] minus 120596

2[119867119862]minus1

1198882(119888)1198882(119888)[119870119861119862]minus1) [Γ]

minus1

[Γ]minus119879

(minus1205962[119870119861119862]minus1[119867119863]minus1

1198882(119889)1198882(119889)

+ [119868]) [Γ]minus1]

times 1198651198882(119889)

1198651198882(119888)

(16)

where [Γ] = minus1205962[119867119862]minus1

1198882(119888)1198882(119888)[119870119861119862]minus1[119867119863]minus1

1198882(119889)1198882(119889)

+

[119867119862]minus1

1198882(119888)1198882(119888)+ [119867119863]minus1

1198882(119889)1198882(119889)

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

1198882(119887)1198882(119888)[119867119878]1198882(119888)1198882(119888)minus [119867119878]1198882(119887)1198882(119888))minus1

(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

2

times [119867119862]1198882(119888)119894(119888)

[119867119878]minus1

119900(119886)119894(119888)[119867119860]119900(119886)1198881(119886)

times ([119867119860]1198881(119886)1198881(119886)+ [119867119861]1198881(119887)1198881(119887))minus1

times [119867119861]1198881(119887)1198882(119887)minus [119867119861]1198882(119887)1198882(119887)+ [119867119861]1198882(119887)1198881(119887)

times ([119867119860]1198881(119886)1198881(119886)+ [119867119861]1198881(119887)1198881(119887))minus1

times [119867119861]1198881(119887)1198882(119887)minus [119867119862]1198882(119888)1198882(119888)minus1

(19)where [Ψ] = [119867

119860]119900(119886)1198881(119886)([119867119860]1198881(119886)1198881(119886)

+

[119867119861]1198881(119887)1198881(119887))minus1[119867119861]1198881(119887)1198882(119887)

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

119860119894119895 119894 119895 = (2) (3) and

119894 = (1) (2) 119895 = (2) (3)119867119861119894119895

119894 119895 = (1) (2) and 119894 119895 = (3) (4)and 119894 = (1) (2) 119895 = (3) (4) and119867

119862119894119895 119894 = (6) (7) 119895 = (7) (8)

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

Mass (kg) Stiffness (Nm) Damping (Nsm)m1 m21 m22 m3 k11 k12 k2 k3 c11 c12 c2 c320 8 8 22 7200 7200 3600 4200 15 15 12 15m4 m5 m6 m7 k4 k5 k6 k7 c4 c5 c6 c718 15 12 16 7900 6800 6000 5800 10 8 22 8m8 m9 k57 k68 c57 c68 16 40 5800 6200 16 12

m1

c12

k12

c11

k11

k2 c2

c4

k4

c3

k3

k5 c5

c68

k68

c57

k57

c7

k7

c6

k6

m8

A B C

Figure 4 A lumped mass-spring-damper model

component 119862

[119872119862] = [

[

1198987

0 0

0 1198988

0

0 0 1198989

]

]

[119870119862] = [

[

1198966

0 minus1198966

0 1198967

minus1198967

minus1198966minus11989671198966+ 1198967

]

]

[119862119862] = [

[

1198886

0 minus1198886

0 1198887

minus1198887

minus1198886minus11988871198886+ 1198887

]

]

(22)

system 119878

[119872119878] =

[[[[[[[[[[

[

1198981

0 0 0 0 0 0 0

0 11989821+ 1198983

0 0 0 0 0 0

0 0 11989822+ 1198984

0 0 0 0 0

0 0 0 1198985

0 0 0 0

0 0 0 0 1198986

0 0 0

0 0 0 0 0 1198987

0 0

0 0 0 0 0 0 1198988

0

0 0 0 0 0 0 0 1198989

]]]]]]]]]]

]

[119870119878] =

[[[[[[[[[[

[

11989611+ 11989612

minus11989611

minus11989612

0 0 0 0 0

minus11989611

11989611+ 1198962+ 1198963

minus1198962

minus1198963

0 0 0 0

minus11989612

minus1198962

11989612+ 1198962+ 1198964

0 minus1198964

0 0 0

0 minus1198963

0 1198963+ 1198965+ 11989657

minus1198965

minus11989657

0 0

0 0 minus1198964

minus1198965

1198964+ 1198965+ 11989668

0 minus11989668

0

0 0 0 minus11989657

0 1198966+ 11989657

0 minus1198966

0 0 0 0 minus11989668

0 1198967+ 11989668

minus1198967

0 0 0 0 0 minus1198966

minus1198967

1198966+ 1198967

]]]]]]]]]]

]

[119862119878] =

[[[[[[[[[[

[

11988811+ 11988812

minus11988811

minus11988812

0 0 0 0 0

minus11988811

11988811+ 1198882+ 1198883

minus1198882

minus1198883

0 0 0 0

minus11988812

minus1198882

11988812+ 1198882+ 1198884

0 minus1198884

0 0 0

0 minus1198883

0 1198883+ 1198885+ 11988857

minus1198885

minus11988857

0 0

0 0 minus1198884

minus1198885

1198884+ 1198885+ 11988868

0 minus11988868

0

0 0 0 minus11988857

0 1198886+ 11988857

0 minus1198886

0 0 0 0 minus11988868

0 1198887+ 11988868

minus1198887

0 0 0 0 0 minus1198886

minus1198887

1198886+ 1198887

]]]]]]]]]]

]

(23)

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

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

6 Shock and Vibration

Table 1 Model parameters for lumped parameter model shown in Figure 4

Mass (kg) Stiffness (Nm) Damping (Nsm)m1 m21 m22 m3 k11 k12 k2 k3 c11 c12 c2 c320 8 8 22 7200 7200 3600 4200 15 15 12 15m4 m5 m6 m7 k4 k5 k6 k7 c4 c5 c6 c718 15 12 16 7900 6800 6000 5800 10 8 22 8m8 m9 k57 k68 c57 c68 16 40 5800 6200 16 12

m1

c12

k12

c11

k11

k2 c2

c4

k4

c3

k3

k5 c5

c68

k68

c57

k57

c7

k7

c6

k6

m8

A B C

Figure 4 A lumped mass-spring-damper model

component 119862

[119872119862] = [

[

1198987

0 0

0 1198988

0

0 0 1198989

]

]

[119870119862] = [

[

1198966

0 minus1198966

0 1198967

minus1198967

minus1198966minus11989671198966+ 1198967

]

]

[119862119862] = [

[

1198886

0 minus1198886

0 1198887

minus1198887

minus1198886minus11988871198886+ 1198887

]

]

(22)

system 119878

[119872119878] =

[[[[[[[[[[

[

1198981

0 0 0 0 0 0 0

0 11989821+ 1198983

0 0 0 0 0 0

0 0 11989822+ 1198984

0 0 0 0 0

0 0 0 1198985

0 0 0 0

0 0 0 0 1198986

0 0 0

0 0 0 0 0 1198987

0 0

0 0 0 0 0 0 1198988

0

0 0 0 0 0 0 0 1198989

]]]]]]]]]]

]

[119870119878] =

[[[[[[[[[[

[

11989611+ 11989612

minus11989611

minus11989612

0 0 0 0 0

minus11989611

11989611+ 1198962+ 1198963

minus1198962

minus1198963

0 0 0 0

minus11989612

minus1198962

11989612+ 1198962+ 1198964

0 minus1198964

0 0 0

0 minus1198963

0 1198963+ 1198965+ 11989657

minus1198965

minus11989657

0 0

0 0 minus1198964

minus1198965

1198964+ 1198965+ 11989668

0 minus11989668

0

0 0 0 minus11989657

0 1198966+ 11989657

0 minus1198966

0 0 0 0 minus11989668

0 1198967+ 11989668

minus1198967

0 0 0 0 0 minus1198966

minus1198967

1198966+ 1198967

]]]]]]]]]]

]

[119862119878] =

[[[[[[[[[[

[

11988811+ 11988812

minus11988811

minus11988812

0 0 0 0 0

minus11988811

11988811+ 1198882+ 1198883

minus1198882

minus1198883

0 0 0 0

minus11988812

minus1198882

11988812+ 1198882+ 1198884

0 minus1198884

0 0 0

0 minus1198883

0 1198883+ 1198885+ 11988857

minus1198885

minus11988857

0 0

0 0 minus1198884

minus1198885

1198884+ 1198885+ 11988868

0 minus11988868

0

0 0 0 minus11988857

0 1198886+ 11988857

0 minus1198886

0 0 0 0 minus11988868

0 1198887+ 11988868

minus1198887

0 0 0 0 0 minus1198886

minus1198887

1198886+ 1198887

]]]]]]]]]]

]

(23)

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

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

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

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

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

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of