experimental and numerical investigation of the effect of

Research ArticleExperimental and Numerical Investigation of the Effect ofStanding People on Dynamic Properties of a Beam-Like Bridge

Wei He12 Weiping Xie1 and Lisheng Liu2

1School of Civil Engineering and Architecture Wuhan University of Technology Wuhan 430070 China2Department of Mechanics and Engineering Structure School of Science Wuhan University of Technology Wuhan 430070 China

Correspondence should be addressed to Wei He heweismile126com

Received 28 July 2017 Revised 17 October 2017 Accepted 31 October 2017 Published 15 November 2017

Academic Editor Oleg V Gendelman

Copyright copy 2017 Wei He et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper studies the vertical dynamic properties of a beam-like bridge attachedwith standing people A purpose-built lively bridgewas constructed Model properties of the empty structure are obtained using ambient vibration testing method Experimental testsof the bridge attached with standing people were also conducted covering a variety of densities of occupants and different posturesThe considerable number of participants and repetitions makes it possible to take into account inter- and intrasubject variabilityTo illustrate the variations of dynamic properties of the structure a mathematic model of standing people-structure interactionsystem is developed employing the single degree-of-freedom (SDOF) human body model It is shown that the model developedherein can effectively illustrate the experimental observations Based on the model the effect of damping and natural frequencyof the human body on dynamic properties of the occupied structure is scrutinized Results show that the modal properties of thehuman body contribute remarkably to the structural damping but little to the structural natural frequencies

1 Introduction

Human-structure interaction (HSI) first introduced by Grif-fin [1] has become a consensus and frontier research areaboth in biomechanical engineering and in civil engineeringcommunities Extensive research interest hasmounted on themodeling of HSI phenomenon aimed at obtaining a betterillustration of in site or experimental observations in lateral(Fujino et al [2] Dallard et al [3 4] Macdonald [5]) or invertical vibrations (Ellis and Ji [6] Falati [7] Brownjohn [8]Sachse et al [9] Brownjohn et al [10] Zivanovic et al [11]Reynolds and Pavic [12] Silva andThambiratnam [13] Joneset al [14] Salyards et al [15 16] An et al [17] Dey et al [18])of slender flexible structures

Some remarkable experimental work either in laboratoryor in real-life structures has yielded results indicating varia-tions in frequencies (decrease or increase) and an increase instructural damping of the occupied structureThe changes ondynamic properties mainly depend on natural frequency ofthe structure the posture of the occupants and themass ratioof occupants to structure As a matter of fact the interactionbetween human body and structure may effectively change

the characteristics of the system especially for large massratio cases [6] A better understanding of the dynamiccharacteristics of crowd-occupied structures will be helpfulto structural design and vibration control of these structuresespecially if vibration serviceability is controlling the design

Generally there are two representative models of thevertical human bodyOne considers the body as spring-mass-damper (SMD) systemsThismodel is based on experimentalobservations of frequency response functions (FRFs) of ver-tical human body It has been experimentally demonstratedthat an individual or a crowd acts at least as a single degree-of-freedom (SDOF) system when the individual or crowd isstationary such as sitting or standing [19 20] SDOF humanmodel is widely adopted although multi-degree-of-freedommodel may improve the results of simulation [21 22] Theother model is called continuous bar model In 1995 Ji [23]proposed a biomechanical continuous bar model to simulatethe vertical vibration of human body This model gives analternative to modeling HSI and has been adopted by Zhouet al [24 25] in their recent research Compared with theSDOF model the rationality of the bar model has not beenexperimentally verified Moreover the continuous bar model

HindawiMathematical Problems in EngineeringVolume 2017 Article ID 1710820 14 pageshttpsdoiorg10115520171710820

2 Mathematical Problems in Engineering

is more complicated in computation and it is difficult todetermine the distributed stiffness and damping parametersof the human body

Corresponding to the human body model there are tworepresentative methods of modeling the vertical vibration ofHSI One is called separated modeling method [25] whichconsiders the structure and the crowd as independent SDOFsystem and the crowd-occupied structures could be treatedas the damped 2-degree-of-freedom (2DOF) representations[26] This model is efficient in qualitatively illustrating thevariations of dynamic properties of crowd-occupied struc-tures However as the structure is simplified as a SDOFsystem themodel basically ignores the effect of highermodesand cannot consider the effect of distribution of crowdsThe other method of modeling the vertical vibration of HSIis called integrative method [25] that is the body and thestructure are treated as an inseparable whole Zhou et al [25]developed a 2DOF system to describe the coupled vibrationof the body and the structure in which one degree is fromthe human body and the other degree from the structure It isworth noting that this method also ignores the contributionof higher modes of the structure and cannot consider theeffect of distribution of crowds Moreover it is challengingto determine the parameters of the model especially thecoupled mass and stiffness and damping parameters

This paper studies the vertical dynamic characteristicsof a beam-like bridge attached with standing people viaboth experimental and theoretical modeling work A detailedexperimental test was conducted for both the empty bridgeand the bridge occupied with different sizes of participantsTo illustrate the experimental observations a coupled modelof standing people-bridge interaction system is developedin which the bridge is modeled as a simply supported beamand the human body as SDOF spring-mass-damper systemBased on the model the effect of modal properties of thehuman body on dynamic properties of the occupied structureis considered in detail The main objective of this study istwofold (1) part of it is to examine the capability of theproposed model in modeling the HSI (2) the authors hopethe comparatively detailed experimental work can help in abetter understanding of the HSI and enriching the currentexperimental database on this issue especially for lightweightbridges

The paper is structured as follows Section 2 describes thedetails of the experiment The problem concerned herein isformulated in Section 3 The effectiveness of the proposedmodel is examined in Section 4 Based on the model theeffect of the standing people on dynamic properties ofthe structure is investigated through numerical examplescovering a range of parameters

2 Experiment

21 Description of the Test Bridge A flexible lightweightbridge (see Figure 1) was constructed for the experimentalinvestigation The bridge is 105m long in length and 1m inwidth A composite (steel-concrete) cross section is employedin the design The composite cross section consisted of twosteel I-section beams HW100 times 100 times 6 times 8 (depth times flange

widthtimesweb thicknesstimes flange thickness) and a 120mm thickdeck made of class 30 concrete A 6mm thick antiskid steelplate welded on top of the I-profile beam was also utilizedserving as formwork when casting concrete The compositeaction is achieved by means of shear studs (diameter 12mmlength 75mm) welded to the steel plate on top of the twobeams To provide full interaction between the steel andconcrete the studs are spaced at 250mm The span length ofthe bridge is 103m The total mass of the bridge is 3500 kg

211 Modal Properties of the Bridge Modal tests usingambient vibration method [27 28] were performed Previousmodeling of the structure indicates that the first lateralfrequency of the bridge is over 18Hz In the present studyonly the vertical vibration is concerned

Measurement points were chosen to both sides of thebridge and a total of 38 locations sim19 points per side wereselected The accelerometers were installed on the surfaceof the bridge in the vertical directions Figure 2 shows thelocation of the measuring points on the bridge Bruel ampKjaer 4507B piezoelectric accelerometers having a nominalsensitivity of 1mVg and the frequency range from 03Hzto 6000Hz were employed for response measurements Thedata were acquired using PULSE data acquisition softwareThe sampling frequency is 200Hz The duration of therecording period is one hour which ensures efficient lengthof test data

The stochastic subspace identification (SSI) method(Overschee and de Moor [29 30] Peeters and de Roeck[31] Ren et al [32]) was employed for modal parameteridentification The data processing and modal parameteridentification were carried out using Matlab software

Table 1 summarizes the dynamic characteristics of thebridge identified from ambient vibration test data The modeshapes of the bridge are shown in Figure 3 It can be seen fromTable 1 that the bridge has a very low vertical frequency lightdamping and light weight Note that the 2nd and 3rd verticalvibration frequencies are 392 and 837 times respectively ofthat of the 1st mode which indicates that the bridge behavesclosely to an ideal simply supported beam (in that case 4 and9 times respectively)

212 Numerical Modeling of the Bridge Afinite element (FE)model of the bridge (Figure 4) was developed employingbeam and shear elements in ANSYS software for furtherinvestigation of the problem The steel I-profile beams weremodeled using 3D BEAM4 elements The concrete deck andthe supporting steel plate were modeled using orthotropicSHELL63 elements assuming isotropic properties Theseelements are capable of transferring both in-plane and out-of-plane loadsThe steel shear studswere not incorporated in theFE model to reduce the number of elements The beam andthe deck were assumed to be closely bonded during vibrationModeling parameters Youngrsquosmodulus for steel and concreteis 200 and 30GPa respectively material density for steel andconcrete is 7850 and 2600 kgm3 respectively and Poissonrsquosratio for steel and concrete is 03 and 02 respectivelySupports at both ends of the bridge were modeled as pinnedbut with a possibility of sliding free in the longitudinal

Mathematical Problems in Engineering 3

Stiener

Support

HW100 times 100 times 6 times 8

10500

10300

1000 1000 1000 1000 1000 1000 1000 1000

100100

1000 1000250 250

I-prole steel beamAntiskid steel plateminus6 minus6

(a) Plane view

HW100 times 100 times 6 times 8StienerI-prole steel beam

Concrete deck (120 mm)

150 700 150

(b) Cross section (c) Real structure

Figure 1 Configuration of the test bridge (all dimensions in mm)

TP1 TP3 TP5 TP7 TP9 TP11 TP13 TP15 TP17 TP19 TP21 TP23 TP25 TP27 TP29 TP31 TP33 TP35 TP37


570 590 570 570 570 570 570 570 590 570 570 570

10500

570570570570

100

570 570

100

1000

Figure 2 Deployment of sensors for modal test (all dimensions in mm)

Table 1 Dynamic characteristics of the bridge

Mode number Test bridge (total weight 3500 kg)Frequency (Hz) Damping ratio () Mode description

1 283 042 1st bending2 1084 043 2nd bending3 1992 053 1st torsion4 2324 060 3rd bending


(a) 1st vertical mode 119891V1 = 283Hz (b) 2nd vertical mode 119891V2 = 1084Hz

(c) 1st torsion mode 1198911199051 = 1992Hz (d) 3rd vertical mode 119891V3 = 2324Hz

Figure 3 Four modal shapes identified from modal test results

(a) (b)

Figure 4 FE model of the bridge (a) 3D model and (b) cross section

direction Specifically all the translational degrees of freedomwere restricted for one support while only the vertical and thelateral translational degrees of freedomwere restricted for theother support The FE model of the bridge has 887 elementsand 780 nodes in totalModal properties of the structure wereobtained by modal analysis using the finite element model(shown in Figure 5) Comparison between Figures 3 and 5indicates a good match between the numerical modeling andthe experimental results

22 Experimental Setup

221 Test Subjects Twenty-three test subjects (TSs) twenty-one males and two females volunteered to participate in theexperiments The general characteristics of the TSs in termsof the average plusmn one standard deviation are age 210 plusmn 12years height 173plusmn 007m andmass 681plusmn 125 kgTheir basicproperties are presented in Table 2

222 Test Cases Two different postures of TSs were consid-ered in the experiment (1) standing with straight knees and(2) standing with bent knees Numbers of TSs including 1

3 5 7 9 13 and 15 persons on the bridge were also takeninto account in the experiments resulting in a range of massratios

The positions of test subjects of various crowd sizes weregiven in Figure 6 TSs were positioned symmetrically withrespect to the mid-span of the bridge at an equal distance of07m so they can behave freely Besides all test subjects wereinvolved in single-person tests to account for intersubjectvariability

To get a stronger excitation of the structure andimprove the quality of test signals the heel-impact methodwas employed Previous studies indicated that heel-impactmethod can successfully reveal the dynamic behavior ofstructures [33 34] For straight-knees posture TSs wereinstructed to raise their heels and stand on tiptoe (Figure 7(a))and then drop their heels suddenly and keep knees straightsimultaneously (Figure 7(b)) while for bent-knees postureTSs were instructed to bend their knees as heels dropping(Figure 7(c)) Ametronomewas used to guide theTSs in samepattern during the tests Before the experiment started priortraining was performed to ensure that TSs were familiar withthe testing process Each test was repeated five times


(a) 1st vertical mode 119891V1 = 282Hz (minus04) (b) 2nd vertical mode 119891V2 = 1099Hz (+15)

(c) 1st torsion mode 1198911199051 = 1917Hz (minus38) (d) 3rd vertical mode 119891V3 = 2404Hz (+34)

Figure 5 Four modal shapes obtained from FE modal analysis Values in parentheses represent errors between analysis and experimentalresults (Table 1)

700 700 700 700 700 700 700 700 700 700 700 700 700 700 350350

16 191 221517 18 2120237 8 5 4

(a)

16 191 221517 18 212023 8 5

(b)

16 191 221517 18 2120

(c)

16 191 221517 18

(d)

16 191 2215

(e)

16 191

(f)

(g)

Figure 6 Positions of test subjects of various crowd sizes (a) 15persons (b) 13 persons (c) 9 persons (d) 7 persons (e) 5 persons(f) 3 persons and (g) single personNote all dimensions are in mm

Table 2 Test subject data

Test subject Gender Age (year) Height (m) Body mass (kg)1 M 21 180 8852 M 20 173 6153 M 20 174 6584 M 20 172 6005 M 20 174 7116 M 22 172 6497 M 21 168 6558 M 21 176 5609 M 21 175 67810 M 20 172 67811 M 22 173 61012 M 21 176 68213 M 22 170 57014 M 22 175 80115 M 21 175 87616 M 20 183 87217 M 20 175 83018 M 21 170 73619 M 22 175 72920 F 20 160 49121 F 21 152 46222 M 21 172 49623 M 25 183 825[Mean STD] [2104 115] [173 007] [6813 1249]Note F = female M = male STD = standard deviation


(a) Heel raised (b) Straight-knees posture (c) Bent-knees posture

Figure 7 Test scenarios for stationary cases using heel-impact method

2575257525752575100

5 3 6

2 41100

Figure 8 Position of themeasuring pointsNote Red square depictslocation of measuring points All dimensions are in mm

Six accelerometers were placed on the lower flange of theI-section beam tomeasure the vertical acceleration responsesof the bridge measuring points 2 and 3 on the mid-spanand measuring points 1 4 5 and 6 on the quarter-spanrespectively that is three points per side as shown inFigure 8

23 Methodology and Results For each test vibration signalswere recorded and only the free decay responseswere selectedto calculate natural frequency and damping As the vibrationfrequencies of the bridge are sparsely spaced (see Table 1)the responses of free vibration signals could be decomposedat each dominating component using a band-pass filteringmethod Each dominating harmonic consisted of a singlefrequency component and can be treated as a generalizedSDOF systemThe damping ratio of the SDOF system can beestimated using the logarithmic decrement method

Figure 9 shows examples of recorded mid-span accel-eration responses of the bridge in the time and frequencydomains from single-person tests for straight-knees andbent-knees postures It is clear that acceleration of the bridgeincreases sharply when the heels of TS suddenly dropped onthe deck then it gradually decreases in a free attenuationmanner when the TS is totally attached on the deck Greaterstructural damping could be observed for the bent-kneesposture compared with the straight-knees posture As theimpact position is in mid-span the responses of the bridgeare dominated by the first vertical frequency component

Figure 10 describes the Fourier spectra of all test plotsobtained on measuring point 1 for case of single TS withstraight-knees posture As shown in Figure 10(b) the funda-mental frequency of the bridge decreases after TSrsquos occupa-tion All plots indicate similar trend even some dispersionoccurs A probable illustration for dispersion in the frequencydomain could be attributed to the intersubject variabilityon biomechanical nature of the human body [11] Besidesthe magnitude of the spectrum varies between individuals

Factors including differences in TSsrsquo body weight and heightof heel raise (and thus leading to different energy input) aswell as human biomechanical properties may all contributeto the discrepancy

As the main harmonics in the frequency domain aresparsely spaced a filtering method is employed in estimationof the damping ratio In this study only the first mode isconsidered All signals are filtered using a band-pass fifth-order Butterworth filter The cut-off frequency of the filter is1 to 4Hz

Table 3 summarizes themean value of 1st vertical bendingfrequency and modal damping ratio of the bridge for all testcasesThe results from the equivalent mass modeling are alsogiven in the table for comparison purpose

It is clear that the experimental results show a decreasein natural frequency and an increase in modal dampingratio of the bridge as the number of TSs increases Asexpected results from the equivalent mass modeling indicatea decrease in natural frequency with increasing numberof occupants The equivalent mass model can capture thevariation in fundamental natural frequency but it cannotillustrate the significant increase in structural damping Theresults from straight-knees and bent-knees postures alsoindicate a decrease in the natural frequency similar to thatof the equivalent mass model For straight-knees posturethe modal damping ratio increases from 094 (for singleTS) to 145 (for 13 TSs) For bent-knees posture the modaldamping ratio of the structure monotonously increases from22 (for single TS) to 957 (for 15 TSs) Moreover onecan observe that the fundamental frequency of the bridge isnot sensitive to the postures of the occupants however it isnoteworthy that the modal damping ratio of the occupiedstructure is very sensitive to the postures of the occupantsMuch bigger values of the structural modal damping ratiocould be observed for bent-knees posture than straight-kneesposture under the same number of occupants This finding isin accordance with the predictions in Figure 14(a) which willbe interpreted later

It should be noted that only the modal damping ratiosof the occupied bridge are presented here The individualcontribution of material damping structural damping andthe influence of reinforcement has not been quantified inthe current study due to the limitation of test equipmentGenerally the damping in a civil engineering structure can becomposed of the following contributions material damping


minus12

minus08

minus04

0

04

08

12Ac

cele

ratio

n (m

Ｍ2)

5 10 15 200Time (s)

(a)

minus12

minus08

minus04

0

04

08

12

Acce

lera

tion

(mＭ

2)

5 10 15 200Time (s)

(b)

0

005

01

015

02

025

Four

ier a

mpl

itude

(mＭ

2)

2 4 6 8 100Frequency (Hz)

(c)

0

005

01

015

02

025

Four

ier a

mpl

itude

(mＭ

2)


(d)

Figure 9 Vertical acceleration and its Fourier spectrum at mid-span from straight-knees posture (a c) and bent-knees posture (b d) forsingle-person test

contact-surface damping and structural damping [35ndash39]Material damping is the energy dissipation within a materialdue to deformation andor displacement Its physical causesare heat flows induced by deformation slip effects andmicroplastic deformations [35] Contact-surface damping iscaused by relative motions in the contact surfaces of joinedcomponents such as screwed riveted and clamped joints[35] Structural damping includes the energy release to thesurrounding medium for example bedding damping orslides [35] In the following study the authors will spare moreeffort on the damping issue to improve the results

3 Theoretical Modeling

In this section an analytical model of standing people-structure interaction system is developed

The following assumptions are made before the deriva-tion

(1) The bridge can be treated as a simply supportedBernoulli-Euler beam having a constant cross sec-tion The span length area of the cross sectionflexural rigidity and density are L A EI and 120588respectively


5 10 15 20 250Frequency (Hz)

002

004

006

008

01

012

014

016

018

Four

ier a

mpl

itude

(mＭ

2)

(a)

24 26 28 3 3222Frequency (Hz)

002

004

006

008

01

012

014

016

018

Four

ier a

mpl

itude

(mＭ

2)

(b)

Figure 10 Fourier spectra of all test plots obtained on measuring point 1 for single person with straight-knees posture (a) Frequencycomponents up to 25Hz and (b) enlarged view 22 to 32Hz

Table 3 First vertical bending frequency and modal damping ratio of the bridge

Number ofoccupants

Mass ratio(occupantsbridge)

Equivalent mass Stand with straight knees Stand with bent knees

Frequency (Hz) Frequency(Hz)

Modal damping ratio()

Frequency(Hz)


0 0 282 283 042 283 0421 0025 273 273 094 273 2203 0071 262 260 105 263 5615 0110 254 257 123 259 7557 0155 248 248 143 250 8169 0182 246 242 142 247 83813 0255 243 241 145 244 91915 0291 242 240 144 242 957Note The results of equivalent mass model were obtained using the FE model attached with equivalent mass

(2) The single human body could be modeled as a SDOFspring-mass-damper (SMD) system

(3) The damping of the empty beam is relatively smallcompared with the human body and is ignored in thisstudy

(4) Thebiomechanical properties of human body are verycomplex depending on postures vibration level ofthe surrounding environment and many others [1]Besides it differs among individuals So it is reallychallenging to identify a determined value for a givenposture for each individual A statistical value fora certain posture derived from statistical test datasoundsmore realistic andmeaningful So in this studyan average natural frequency and damping ratio ofeach posture of the standing people are used for allindividuals

The problem considered here can be described by usingFigure 11

For simply supported beam the normalized jth modalshape of the beam is

120593119895 (119909) = 1198621 sin(119895120587119909119871 ) (1)

where 119871 is the span length of the beam and the constant 1198621 isobtained by setting the modal mass to unity

int1198710120588119860120593119895120593119896119889119909 =

1 if 119895 = 1198960 if 119895 = 119896 (2)

The displacement of the beam can be obtained usingmodal superposition method


1 j N

Attachment j

x

m1

k1 c1

mj

xj

kj cj

zjmN

kN cN

Figure 11 The model of human-beam coupled system

119908 (119909 119905) = 119869sum119895=1

120593119895 (119909) 119902119895 (119905) (3)

where J is the number of terms in the seriesThe kinetic and potential energies of the system in

Figure 11 are respectively

119879 = 12 int119871

0120588119860 (119909 119905)2 119889119909 + 12

119873sum119895=1

1198981198952119895 = 12sdot 119869sum119895=1

119869sum119896=1

(int1198710120588119860120593119895120593119896119889119909) 119902119895 119902119896 + 12

119873sum119895=1

1198981198952119895

119881 = 12 int119871

0EI(12059721199081205971199092 )

2 119889119909 + 12119873sum119904=1

119896119904 (119908 (119909119904 119905) minus 119911119904)2

= 12119869sum119895=1

119869sum119896=1

(int1198710EI11988921205931198951198891199092 119889

21205931198961198891199092 119889119909

+ 119873sum119904=1

119896119904120593119895 (119909119904) 120593119896 (119909119904)) 119902119895119902119896 minus 119873sum119904=1

119869sum119896=1

119896119904120593119896 (119909119904) 119902119896119911119904+ 12119873sum119895=1

1198961199041199112119904

(4)

The Rayleigh dissipation function is

119863 = 12119873sum119904=1

119888119904 ( (119909 119905) minus 119904)2

= 12119873sum119904=1

119888119904 [[119869sum119895=1

119869sum119896=1

120593119895 (119909119904) 120593119896 (119909119904) 119902119895 119902119896]]minus 119873sum119904=1

119869sum119896=1

119888119904120593119896 (119909119904) 119902119896119904 + 12119873sum119904=1

1198881199042119904

(5)

Using the representation of T V and D the motionequation of the system in Figure 11 can be obtained byemploying Lagrangersquos equations The equation of motion isgiven as

Mq + Cq + Kq = 0 (6)

whereM C K and q are defined as

M = [[120575119894119895]119869times119869 [0]119869times119873[0]119879119873times119869 [120575119903119904119898119904]119873times119873]

C = [[[[ 119873sum119904=1119888119904120593119894 (119909119904) 120593119895 (119909119904)]

119869times119869

[minus119888119904120593119895 (119909119904)]119869times119873[minus119888119904120593119895 (119909119904)]119879119873times119869 [120575119903119904119888119904]119873times119873

]]]

K

= [[[[1205751198941198951205962119895 + 119873sum

119904=1119896119904120593119894 (119909119904) 120593119895 (119909119904)]

119869times119869


]]]

q = 119902119895119869times1119911119904119873times1

(7)

where 119894 119895 = 1 2 119869 and 119903 119904 = 1 2 119873

4 Model Validation

41 Tracking Dynamic Properties of the System Using State-Space Method For (6) the dynamic characteristics of thesystem could be obtained using the state-space methodEquation (6) can be converted to state-space form as follows[40]

Sx minus Rx = 0 (8)

where S R and x are defined as

S = [minusK 00 M

]

R = [ 0 minusKminusK minusC]

x = qq

(9)

The solution of (8) is obtained by substituting x = 120601119890120582119905into (8) resulting in the symmetric generalized eigenvalueproblem

(R minus 120582119899S) 120601119899 = 0 (10)

in which 120582119899 and 120601119899 are the nth complex eigenvalue and itscorresponding eigenvector of the 2(119873 + 119869) eigensolutions

Eigenvalues 120582119899 of (10) would be obtained using complexmodal analysis method [41] and the natural frequencies 119891119899and damping ratios 120577119899 can be given as [26]

119891119899 = 12120587 10038161003816100381610038161205821198991003816100381610038161003816 120577119899 = minusRe (120582119899)10038161003816100381610038161205821198991003816100381610038161003816

(11)


eoretical-straight kneeseoretical-bent knees

Experimental-straight kneesExperimental-bent knees

3 5 7 9 11 13 151Number of people on the bridge

23

24

25

26

27

28

Nat

ural

freq

uenc

y (H

z)


0

2

4

6

8

10

Dam

ping

ratio

()

Figure 12 Comparison between theoretical and experimental re-sults

42 Comparisons of Theoretical Results with ExperimentalData Parameters used in the numerical study are brieflyillustrated below The natural frequencies of 55Hz in thenormal posture and 275Hz in the legs bent posture forstanding human body suggested by Matsumoto and Griffin[20] are used Due to the complex nature of the human bodyand the availability of experimental data in the literaturean average modal damping ratio 120577119867 = 04 suggestedby Griffin [1] was used for both normal standing postureand knees bent posture The span length of the bridge119871 = 103m and its total mass is 3500 kg Occupationsrsquolocations (119909119895) of each test scenario can be found in Figure 6and the body mass of each individual can be found inTable 2

Figure 12 shows the comparison results of dynamic prop-erties of the bridge between experimental data and simulatedones It is clear that the theoretical results show similar trendto that of the experimental data for both natural frequencyand modal damping ratio A decrease in natural frequencyand an increase in modal damping ratio are observed asthe number of occupants increases The simulated naturalfrequency and modal damping ratio of the occupied bridgeagree well in general with the experimental results even somediscrepancy occurs This discrepancy could be attributed tothe assumption of identical properties of the vertical humanbody of each individual which may differ from the actualcases In general the theoretical model developed in thispaper can give a promising prediction of the variations ofdynamic properties of the structure Hence the model can beused as an alternative to modeling the HSI In the followinganalysis the model is employed to further discuss the effectof some key factors on dynamic properties of the occupiedstructure

5 Influence of Human ParametersNumerical Results

Previous studies show that the occupants to structure massratios occupants to structure frequency ratios andmany oth-ers will affect the dynamic properties of the occupied struc-tures [11 26] Compared to the above-mentioned factors theinfluence of human body parameters (especially dampingproperties and natural frequencies of standing human body)on dynamic properties of the occupied structure is rarelyconcerned To this end in this section the effect of thehuman parameters on dynamic properties of the structure isdiscussed in detail using the model developed in Section 3

Numerical example related to a real single-span steel-concrete composite bridge [42] is considered The basicproperties of the bridge are as follows span length 162mtotal mass 13432 kg the first two natural frequencies of thebridge 248 and 778Hz

The occupants are assumed to be uniformly distributedon the full length of the bridge The mass stiffness anddamping properties of the human body are assumed to beidentical for each individual The bridge is approximatelytreated as a simply supported beam and the nth mode shapeof the beam is given by 120601119899(119909) = sin(119899120587119909119871) In engineeringpractice we usually focus on the lowermodes of the structureso only the lowest two modes are considered herein

51 Effect of Damping Properties of the Human Body Asaforementioned the damping properties of the human bodyvary depending on postures vibration amplitude of the sur-rounding environment and some other factors An averagedamping ratio ranging from 03 to 05 for standing postureis suggested by Griffin [1] To examine the effect of dampingproperties of human body on dynamic characteristics of thestructure three damping ratios of the standing human body120585119867 = 03 04 and 05 are considered The occupants areassumed to be uniformly distributed on the full length of thebridge with the density of 120588 = 2 pm2 An average body massof 70 kg and natural frequency of 55Hz are used

Figure 13 shows the first two dimensionless natural fre-quencies andmodal damping ratios of the bridgewith respectto different damping properties of the human body It is clearthat the damping properties of the human body contributelittle to the natural frequencies of the occupied bridge Fordifferent body modal damping ratios the frequency of theoccupied bridge remains almost the same for the first mode(09 times of the empty structure ie a minus10 decrease)and the variation for the second mode is also small (097 to101 for the second mode) which indicates that the naturalfrequencies of the occupied bridge are not sensitive to thedamping properties of the human body An increase inmodaldamping ratios could be observed for the first two modes Inthe parameter scope of the present study the variation of thedamping ratio iswithin the scope of 15 to 18 for the firstmodeand 158 to 17 for the second mode

52 Effect of Natural Frequency of the Human Body Peoplewith different postures (normal standing standing with bentknees standing with one leg etc) have different natural


0

05

1r w

1

0

05

1

15

2

r c1

04 0503Damping ratio of the body


(a)

0

5

10

15

20

r c2

0

02

04

06

08

1

12

r w2



(b)

Figure 13 First two natural frequencies and damping ratios of the bridge against different damping properties of the human body first mode(a) and second mode (b)

frequencies [1] To examine the effect of natural frequencyof human body on dynamic characteristics of the occupiedstructure three natural frequencies of the standing humanbody 120596119867 = 275 (bent-knees posture) 35 (one-leg posture)and 55Hz (straight-knees posture) are considered Theoccupants are assumed to be uniformly distributed on the fulllength of the bridge with the density of 20 pm2 An averagebody mass of 70 kg and damping ratio of 04 are used

Figure 14 illustrates the first two dimensionless natu-ral frequencies and modal damping ratios of the bridgewith respect to different natural frequencies of the humanbody

As shown in Figure 14 the lowest two frequencies of theoccupied bridge are not sensitive to the natural frequenciesof the human body However the modal damping ratiosof the occupied bridge are very sensitive to the naturalfrequencies of the human body In the given parameter scopethe frequency of the bridge with occupants is 09 timesthat of the empty structure for the first mode (ie a minus10decrease) and 098 to 1 for the second mode The variationof the modal damping ratio is within the scope of 17 to96 for the first mode and 57 to 17 for the second mode

Comparison of the three body natural frequencies indicatesthat the smaller the body natural frequencies the bigger thefirst modal damping ratios and the smaller the second modaldamping ratiosThis can account for the experimental resultspresented in Table 3 and Figure 12 As shown in Table 3 andFigure 12 the first natural frequency of the occupied bridge isnot sensitive to different postures while much bigger valuesof modal damping ratio were observed for the bent-kneesposture compared with the straight-knees postureMoreovera general decrease in natural frequencies and an increasein modal damping ratios for the first two modes could beobserved from Figure 14 although the variation of the naturalfrequencies of the second mode is very small

6 Conclusions

A purpose-built lively bridge was constructed Model prop-erties of the empty structure are obtained based on ambientvibration testing method Experimental tests of the bridgeattachedwith standing people were also conducted Amathe-matic model of standing people-structure interaction systemis developed and verified


0

05

1r w

1

0

5

10

r c1

35 55275Natural frequency of the body (Hz)


(a)

0

05

1

r w2

0

5

10

15

20

r c2



(b)

Figure 14 First two natural frequencies and damping ratios of the bridge against different natural frequencies of the human body first mode(a) and second mode (b)

It is shown that the model developed in this paper caneffectively illustrate the experimental observations Hencethe model can be used as an alternative to modeling theHSI

Numerical examples show that the modal properties ofthe human body contribute remarkably to the structuraldamping but little to the natural frequencies of the occupiedstructure

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research was sponsored by National Natural ScienceFoundation of China (Grant no 51508431) China Postdoc-toral Science Foundation (Grant no 2015M582288) andscholarship from China Scholarship Council (Grant no201606955008)

References

[1] M J Griffin Handbook of Human Vibration Academic PressLondon 1990

[2] Y Fujino B M Pacheco S-I Nakamura and P WarnitchaildquoSynchronization of human walking observed during lateralvibration of a congested pedestrian bridgerdquo Earthquake Engi-neering amp Structural Dynamics vol 22 no 9 pp 741ndash758 1993

[3] P Dallard A J Fitzpatrick A Flint et al ldquoThe Londonmillennium footbridgerdquo Structural Engineering Internationalvol 79 no 22 pp 17ndash33 2001

[4] P Dallard T Fitzpatrick A Flint et al ldquoLondon millenniumbridge pedestrian-induced lateral vibrationrdquo Journal of BridgeEngineering vol 6 no 6 pp 412ndash417 2001

[5] J H Macdonald ldquoLateral excitation of bridges by balancingpedestriansrdquo Proceedings of the Royal Society A MathematicalPhysical and Engineering Sciences vol 465 no 2104 pp 1055ndash1073 2009

[6] B R Ellis and T Ji ldquoHuman-structure interaction in verticalvibrationsrdquo Proceedings of the Institution of Civil Engineers -Structures and Buildings vol 122 no 1 pp 1ndash9 1997


[7] S Falati The contribution of non-structural components to theoverall dynamic behaviour of concrete floor slabs [PhD thesis]University of Oxford Oxford 1999

[8] J M W Brownjohn ldquoEnergy dissipation from vibrating floorslabs due to human-structure interactionrdquo Shock and Vibrationvol 8 no 6 pp 315ndash323 2001

[9] R Sachse A Pavic and P Reynolds ldquoHuman-structuredynamic interaction in civil engineering dynamics a literaturereviewrdquo Shock and Vibration vol 35 no 1 pp 3ndash18 2003

[10] JMW Brownjohn P FokM Roche and POmenzetter ldquoLongspan steel pedestrian bridge at Singapore Changi Airportmdashpart2 crowd loading tests and vibration mitigation measuresrdquoStructural Engineering International vol 82 no 16 pp 28ndash342004

[11] S Zivanovic A Pavic andP Reynolds ldquoVibration serviceabilityof footbridges under human-induced excitation a literaturereviewrdquo Journal of Sound and Vibration vol 279 no 1-2 pp 1ndash74 2005

[12] P Reynolds and A Pavic ldquoVibration performance of a largecantilever grandstand during an international football matchrdquoJournal of Performance of Constructed Facilities vol 20 no 3pp 202ndash212 2006

[13] S S D Silva and D PThambiratnam ldquoDynamic characteristicsof steel-deck composite floors under human-induced loadsrdquoComputers amp Structures vol 87 no 17-18 pp 1067ndash1076 2009

[14] C A Jones P Reynolds and A Pavic ldquoVibration serviceabilityof stadia structures subjected to dynamic crowd loads aliterature reviewrdquo Journal of Sound and Vibration vol 330 no8 pp 1531ndash1566 2011

[15] K A Salyards and N C Noss ldquoExperimental evaluation ofthe influence of human-structure interaction for vibrationserviceabilityrdquo Journal of Performance of Constructed Facilitiesvol 28 no 3 pp 458ndash465 2014

[16] K A Salyards and Y Hua ldquoAssessment of dynamic propertiesof a crowd model for humanndashstructure interaction modelingrdquoEngineering Structures vol 89 pp 103ndash110 2015

[17] Q An Q Ren H Liu X Yan and Z Chen ldquoDynamicperformance characteristics of an innovative Cable SupportedBeam Structure-Concrete Slab Composite Floor System underhuman-induced loadsrdquo Engineering Structures vol 117 pp 40ndash57 2016

[18] P Dey A Sychterz S Narasimhan and S Walbridge ldquoPer-formance of Pedestrian-Load Models through ExperimentalStudies on Lightweight Aluminum Bridgesrdquo Journal of BridgeEngineering vol 21 no 8 Article ID C4015005 2016

[19] J Herterich and J Schnauber ldquoThe effect of vertical mechanicalvibration on standing manrdquo J Low Freq Noise Vib vol 11 pp52ndash60 1992

[20] Y Matsumoto and M J Griffin ldquoDynamic response of thestanding human body exposed to vertical vibration influenceof posture and vibration magnituderdquo Journal of Sound andVibration vol 212 no 1 pp 85ndash107 1998

[21] S Kitazaki and M J Griffin ldquoA modal analysis of whole-bodyvertical vibration using a finite element model of the humanbodyrdquo Journal of Sound and Vibration vol 200 no 1 pp 83ndash102 1997

[22] Y Matsumoto and M J Griffin ldquoMathematical models for theapparent masses of standing subjects exposed to vertical whole-body vibrationrdquo Journal of Sound and Vibration vol 260 no 3pp 431ndash451 2003

[23] T Ji ldquoA continuous model for the vertical vibration of thehuman body in a standing positionrdquo in United Kingdom Infor-mal GroupMeeting onHumanResponse toVibration Silsoe UK1995

[24] D Zhou T Ji and W Liu ldquoDynamic characteristics of astanding human on a SDOF structurerdquo Adv Vib Eng vol 11pp 83ndash96 2012

[25] D Zhou H Han T Ji and X Xu ldquoComparison of twomodels for human-structure interactionrdquoAppliedMathematicalModelling vol 40 no 5-6 pp 3738ndash3748 2016

[26] R Sachse A Pavic and P Reynolds ldquoParametric study ofmodalproperties of damped two-degree-of-freedom crowd-structuredynamic systemsrdquo Journal of Sound and Vibration vol 274 no3-5 pp 461ndash480 2004

[27] S Ivanovic M Trifunac D and M Todorovska I ldquoAmbientvibration test-a reviewrdquo ISET Journal of Eearthquake Technol-ogy vol 37 no 4 pp 165ndash197 2000

[28] B Jaishi andW X Ren ldquoStructural finite element model updat-ing using ambient vibration test resultsrdquo Journal of StructuralEngineering vol 131 no 4 pp 617ndash628 2005

[29] P V Overschee and B de Moor ldquoSubspace algorithms forthe stochastic identification problemrdquo in Proceedings of the30th IEEE Conference on Decision and Control pp 1321ndash1326Brighton England

[30] P van Overschee and B de Moor Subspace Identification forLinear Systems Theory Implementation Applications KluwerAcademic Publishers Dordrecht The Netherlands 1996

[31] B Peeters and G de Roeck ldquoReference-based stochastic sub-space identification for output-onlymodal analysisrdquoMechanicalSystems and Signal Processing vol 13 no 6 pp 855ndash878 1999

[32] W-X Ren X-L Peng and Y-Q Lin ldquoExperimental andanalytical studies on dynamic characteristics of a large spancable-stayed bridgerdquo Engineering Structures vol 27 no 4 pp535ndash548 2005

[33] W D Varela and R C Battista ldquoControl of vibrations inducedby people walking on large span composite floor decksrdquoEngineering Structures vol 33 no 9 pp 2485ndash2494 2011

[34] C M Abeysinghe D P Thambiratnam and N J PereraldquoDynamic performance characteristics of an innovative HybridComposite Floor Plate System under human-induced loadsrdquoComposite Structures vol 96 pp 590ndash600 2013

[35] H Wiechmann ldquoVDI-Richtlinien (VDI 3796 Blatt 1 2 und3) Bestimmung von Thallium in Boden und Pflanzen (VDI3792 Blatt 3) Messen der Immissions-Wirkdosis von Blei inPflanzen mit dem Verfahren der standardisierten GraskulturBeuth-Verlag Postfach 1145 1000 Berlin 30 (1985)rdquo Zeitschriftfur Pflanzenernahrung und Bodenkunde vol 150 no 2 pp 125-125 1987

[36] VDI 3830 Blatt 2 Damping of materials and members-Dampingof solids vol 10 Beuth Verlag Berlin Germany 2004

[37] Beuth Verlag Damping of materials and members-Damping ofassemblies vol 8 Beuth Verlag Berlin Germany 2004

[38] VDI 3830 Blatt 4Damping ofmaterials andmembers-Models fordamped structures Beuth Verlag Berlin Germany 2005

[39] VDI 3830 Blatt 5 Damping of materials and members-Experimental techniques for the determination of damping char-acteristics Beuth Verlag Berlin 2005

[40] M V Drexel and J H Ginsberg ldquoModal overlap and dissipationeffects of a cantilever beam with multiple attached oscillatorsrdquoJournal of Vibration and Acoustics vol 123 no 2 pp 181ndash1872001


[41] K A Foss ldquoCo-ordinates which uncouple the equations ofmotion of damped linear dynamic systemsrdquo vol 25 pp 361ndash3641958

[42] S Zivanovic R P Johnson H V Dang and J Dobric ldquoDesignand construction of a very lively bridgerdquo in Proceedings of the31st IMAC A Conference on Structural Dynamics 2013 pp 371ndash380 USA February 2013

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Journal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


is more complicated in computation and it is difficult todetermine the distributed stiffness and damping parametersof the human body

Corresponding to the human body model there are tworepresentative methods of modeling the vertical vibration ofHSI One is called separated modeling method [25] whichconsiders the structure and the crowd as independent SDOFsystem and the crowd-occupied structures could be treatedas the damped 2-degree-of-freedom (2DOF) representations[26] This model is efficient in qualitatively illustrating thevariations of dynamic properties of crowd-occupied struc-tures However as the structure is simplified as a SDOFsystem themodel basically ignores the effect of highermodesand cannot consider the effect of distribution of crowdsThe other method of modeling the vertical vibration of HSIis called integrative method [25] that is the body and thestructure are treated as an inseparable whole Zhou et al [25]developed a 2DOF system to describe the coupled vibrationof the body and the structure in which one degree is fromthe human body and the other degree from the structure It isworth noting that this method also ignores the contributionof higher modes of the structure and cannot consider theeffect of distribution of crowds Moreover it is challengingto determine the parameters of the model especially thecoupled mass and stiffness and damping parameters

This paper studies the vertical dynamic characteristicsof a beam-like bridge attached with standing people viaboth experimental and theoretical modeling work A detailedexperimental test was conducted for both the empty bridgeand the bridge occupied with different sizes of participantsTo illustrate the experimental observations a coupled modelof standing people-bridge interaction system is developedin which the bridge is modeled as a simply supported beamand the human body as SDOF spring-mass-damper systemBased on the model the effect of modal properties of thehuman body on dynamic properties of the occupied structureis considered in detail The main objective of this study istwofold (1) part of it is to examine the capability of theproposed model in modeling the HSI (2) the authors hopethe comparatively detailed experimental work can help in abetter understanding of the HSI and enriching the currentexperimental database on this issue especially for lightweightbridges

The paper is structured as follows Section 2 describes thedetails of the experiment The problem concerned herein isformulated in Section 3 The effectiveness of the proposedmodel is examined in Section 4 Based on the model theeffect of the standing people on dynamic properties ofthe structure is investigated through numerical examplescovering a range of parameters

2 Experiment

21 Description of the Test Bridge A flexible lightweightbridge (see Figure 1) was constructed for the experimentalinvestigation The bridge is 105m long in length and 1m inwidth A composite (steel-concrete) cross section is employedin the design The composite cross section consisted of twosteel I-section beams HW100 times 100 times 6 times 8 (depth times flange

widthtimesweb thicknesstimes flange thickness) and a 120mm thickdeck made of class 30 concrete A 6mm thick antiskid steelplate welded on top of the I-profile beam was also utilizedserving as formwork when casting concrete The compositeaction is achieved by means of shear studs (diameter 12mmlength 75mm) welded to the steel plate on top of the twobeams To provide full interaction between the steel andconcrete the studs are spaced at 250mm The span length ofthe bridge is 103m The total mass of the bridge is 3500 kg

211 Modal Properties of the Bridge Modal tests usingambient vibration method [27 28] were performed Previousmodeling of the structure indicates that the first lateralfrequency of the bridge is over 18Hz In the present studyonly the vertical vibration is concerned

Measurement points were chosen to both sides of thebridge and a total of 38 locations sim19 points per side wereselected The accelerometers were installed on the surfaceof the bridge in the vertical directions Figure 2 shows thelocation of the measuring points on the bridge Bruel ampKjaer 4507B piezoelectric accelerometers having a nominalsensitivity of 1mVg and the frequency range from 03Hzto 6000Hz were employed for response measurements Thedata were acquired using PULSE data acquisition softwareThe sampling frequency is 200Hz The duration of therecording period is one hour which ensures efficient lengthof test data

The stochastic subspace identification (SSI) method(Overschee and de Moor [29 30] Peeters and de Roeck[31] Ren et al [32]) was employed for modal parameteridentification The data processing and modal parameteridentification were carried out using Matlab software

Table 1 summarizes the dynamic characteristics of thebridge identified from ambient vibration test data The modeshapes of the bridge are shown in Figure 3 It can be seen fromTable 1 that the bridge has a very low vertical frequency lightdamping and light weight Note that the 2nd and 3rd verticalvibration frequencies are 392 and 837 times respectively ofthat of the 1st mode which indicates that the bridge behavesclosely to an ideal simply supported beam (in that case 4 and9 times respectively)

212 Numerical Modeling of the Bridge Afinite element (FE)model of the bridge (Figure 4) was developed employingbeam and shear elements in ANSYS software for furtherinvestigation of the problem The steel I-profile beams weremodeled using 3D BEAM4 elements The concrete deck andthe supporting steel plate were modeled using orthotropicSHELL63 elements assuming isotropic properties Theseelements are capable of transferring both in-plane and out-of-plane loadsThe steel shear studswere not incorporated in theFE model to reduce the number of elements The beam andthe deck were assumed to be closely bonded during vibrationModeling parameters Youngrsquosmodulus for steel and concreteis 200 and 30GPa respectively material density for steel andconcrete is 7850 and 2600 kgm3 respectively and Poissonrsquosratio for steel and concrete is 03 and 02 respectivelySupports at both ends of the bridge were modeled as pinnedbut with a possibility of sliding free in the longitudinal


Stiener

Support


10500

10300

1000 1000 1000 1000 1000 1000 1000 1000

100100

1000 1000250 250


(a) Plane view



150 700 150





570 590 570 570 570 570 570 570 590 570 570 570

10500

570570570570

100

570 570

100

1000









(a) (b)













700 700 700 700 700 700 700 700 700 700 700 700 700 700 350350

16 191 221517 18 2120237 8 5 4

(a)

16 191 221517 18 212023 8 5

(b)

16 191 221517 18 2120

(c)

16 191 221517 18

(d)

16 191 2215

(e)

16 191

(f)

(g)







2575257525752575100

5 3 6

2 41100












minus12

minus08

minus04

0

04

08

12Ac

cele

ratio

n (m

Ｍ2)

5 10 15 200Time (s)

(a)

minus12

minus08

minus04

0

04

08

12

Acce

lera

tion

(mＭ

2)

5 10 15 200Time (s)

(b)

0

005

01

015

02

025

Four

ier a

mpl

itude

(mＭ

2)


(c)

0

005

01

015

02

025

Four

ier a

mpl

itude

(mＭ

2)


(d)








5 10 15 20 250Frequency (Hz)

002

004

006

008

01

012

014

016

018

Four

ier a

mpl

itude

(mＭ

2)

(a)

24 26 28 3 3222Frequency (Hz)

002

004

006

008

01

012

014

016

018

Four

ier a

mpl

itude

(mＭ

2)

(b)



Number ofoccupants





Frequency(Hz)








120593119895 (119909) = 1198621 sin(119895120587119909119871 ) (1)


int1198710120588119860120593119895120593119896119889119909 =

1 if 119895 = 1198960 if 119895 = 119896 (2)



1 j N

Attachment j

x

m1

k1 c1

mj

xj

kj cj

zjmN

kN cN


119908 (119909 119905) = 119869sum119895=1

120593119895 (119909) 119902119895 (119905) (3)



119879 = 12 int119871

0120588119860 (119909 119905)2 119889119909 + 12

119873sum119895=1

1198981198952119895 = 12sdot 119869sum119895=1

119869sum119896=1

(int1198710120588119860120593119895120593119896119889119909) 119902119895 119902119896 + 12

119873sum119895=1

1198981198952119895

119881 = 12 int119871

0EI(12059721199081205971199092 )

2 119889119909 + 12119873sum119904=1

119896119904 (119908 (119909119904 119905) minus 119911119904)2

= 12119869sum119895=1

119869sum119896=1

(int1198710EI11988921205931198951198891199092 119889

21205931198961198891199092 119889119909

+ 119873sum119904=1

119896119904120593119895 (119909119904) 120593119896 (119909119904)) 119902119895119902119896 minus 119873sum119904=1

119869sum119896=1

119896119904120593119896 (119909119904) 119902119896119911119904+ 12119873sum119895=1

1198961199041199112119904

(4)


119863 = 12119873sum119904=1

119888119904 ( (119909 119905) minus 119904)2

= 12119873sum119904=1

119888119904 [[119869sum119895=1

119869sum119896=1

120593119895 (119909119904) 120593119896 (119909119904) 119902119895 119902119896]]minus 119873sum119904=1

119869sum119896=1

119888119904120593119896 (119909119904) 119902119896119904 + 12119873sum119904=1

1198881199042119904

(5)


Mq + Cq + Kq = 0 (6)



C = [[[[ 119873sum119904=1119888119904120593119894 (119909119904) 120593119895 (119909119904)]

119869times119869


]]]

K

= [[[[1205751198941198951205962119895 + 119873sum

119904=1119896119904120593119894 (119909119904) 120593119895 (119909119904)]

119869times119869


]]]

q = 119902119895119869times1119911119904119873times1

(7)

where 119894 119895 = 1 2 119869 and 119903 119904 = 1 2 119873

4 Model Validation


Sx minus Rx = 0 (8)


S = [minusK 00 M

]


x = qq

(9)


(R minus 120582119899S) 120601119899 = 0 (10)



119891119899 = 12120587 10038161003816100381610038161205821198991003816100381610038161003816 120577119899 = minusRe (120582119899)10038161003816100381610038161205821198991003816100381610038161003816

(11)





23

24

25

26

27

28

Nat

ural

freq

uenc

y (H

z)


0

2

4

6

8

10

Dam

ping

ratio

()












0

05

1r w

1

0

05

1

15

2

r c1



(a)

0

5

10

15

20

r c2

0

02

04

06

08

1

12

r w2



(b)






6 Conclusions



0

05

1r w

1

0

5

10

r c1



(a)

0

05

1

r w2

0

5

10

15

20

r c2



(b)






Acknowledgments


References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





Stiener

Support


10500

10300

1000 1000 1000 1000 1000 1000 1000 1000

100100

1000 1000250 250


(a) Plane view



150 700 150





570 590 570 570 570 570 570 570 590 570 570 570

10500

570570570570

100

570 570

100

1000









(a) (b)













700 700 700 700 700 700 700 700 700 700 700 700 700 700 350350

16 191 221517 18 2120237 8 5 4

(a)

16 191 221517 18 212023 8 5

(b)

16 191 221517 18 2120

(c)

16 191 221517 18

(d)

16 191 2215

(e)

16 191

(f)

(g)







2575257525752575100

5 3 6

2 41100












minus12

minus08

minus04

0

04

08

12Ac

cele

ratio

n (m

Ｍ2)

5 10 15 200Time (s)

(a)

minus12

minus08

minus04

0

04

08

12

Acce

lera

tion

(mＭ

2)

5 10 15 200Time (s)

(b)

0

005

01

015

02

025

Four

ier a

mpl

itude

(mＭ

2)


(c)

0

005

01

015

02

025

Four

ier a

mpl

itude

(mＭ

2)


(d)








5 10 15 20 250Frequency (Hz)

002

004

006

008

01

012

014

016

018

Four

ier a

mpl

itude

(mＭ

2)

(a)

24 26 28 3 3222Frequency (Hz)

002

004

006

008

01

012

014

016

018

Four

ier a

mpl

itude

(mＭ

2)

(b)



Number ofoccupants





Frequency(Hz)








120593119895 (119909) = 1198621 sin(119895120587119909119871 ) (1)


int1198710120588119860120593119895120593119896119889119909 =

1 if 119895 = 1198960 if 119895 = 119896 (2)



1 j N

Attachment j

x

m1

k1 c1

mj

xj

kj cj

zjmN

kN cN


119908 (119909 119905) = 119869sum119895=1

120593119895 (119909) 119902119895 (119905) (3)



119879 = 12 int119871

0120588119860 (119909 119905)2 119889119909 + 12

119873sum119895=1

1198981198952119895 = 12sdot 119869sum119895=1

119869sum119896=1

(int1198710120588119860120593119895120593119896119889119909) 119902119895 119902119896 + 12

119873sum119895=1

1198981198952119895

119881 = 12 int119871

0EI(12059721199081205971199092 )

2 119889119909 + 12119873sum119904=1

119896119904 (119908 (119909119904 119905) minus 119911119904)2

= 12119869sum119895=1

119869sum119896=1

(int1198710EI11988921205931198951198891199092 119889

21205931198961198891199092 119889119909

+ 119873sum119904=1

119896119904120593119895 (119909119904) 120593119896 (119909119904)) 119902119895119902119896 minus 119873sum119904=1

119869sum119896=1

119896119904120593119896 (119909119904) 119902119896119911119904+ 12119873sum119895=1

1198961199041199112119904

(4)


119863 = 12119873sum119904=1

119888119904 ( (119909 119905) minus 119904)2

= 12119873sum119904=1

119888119904 [[119869sum119895=1

119869sum119896=1

120593119895 (119909119904) 120593119896 (119909119904) 119902119895 119902119896]]minus 119873sum119904=1

119869sum119896=1

119888119904120593119896 (119909119904) 119902119896119904 + 12119873sum119904=1

1198881199042119904

(5)


Mq + Cq + Kq = 0 (6)



C = [[[[ 119873sum119904=1119888119904120593119894 (119909119904) 120593119895 (119909119904)]

119869times119869


]]]

K

= [[[[1205751198941198951205962119895 + 119873sum

119904=1119896119904120593119894 (119909119904) 120593119895 (119909119904)]

119869times119869


]]]

q = 119902119895119869times1119911119904119873times1

(7)

where 119894 119895 = 1 2 119869 and 119903 119904 = 1 2 119873

4 Model Validation


Sx minus Rx = 0 (8)


S = [minusK 00 M

]


x = qq

(9)


(R minus 120582119899S) 120601119899 = 0 (10)



119891119899 = 12120587 10038161003816100381610038161205821198991003816100381610038161003816 120577119899 = minusRe (120582119899)10038161003816100381610038161205821198991003816100381610038161003816

(11)





23

24

25

26

27

28

Nat

ural

freq

uenc

y (H

z)


0

2

4

6

8

10

Dam

ping

ratio

()












0

05

1r w

1

0

05

1

15

2

r c1



(a)

0

5

10

15

20

r c2

0

02

04

06

08

1

12

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6 Conclusions



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Acknowledgments


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Volume 2014




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Journal of


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Algebra





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(a) (b)













700 700 700 700 700 700 700 700 700 700 700 700 700 700 350350

16 191 221517 18 2120237 8 5 4

(a)

16 191 221517 18 212023 8 5

(b)

16 191 221517 18 2120

(c)

16 191 221517 18

(d)

16 191 2215

(e)

16 191

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120593119895 (119909) = 1198621 sin(119895120587119909119871 ) (1)


int1198710120588119860120593119895120593119896119889119909 =

1 if 119895 = 1198960 if 119895 = 119896 (2)



1 j N

Attachment j

x

m1

k1 c1

mj

xj

kj cj

zjmN

kN cN


119908 (119909 119905) = 119869sum119895=1

120593119895 (119909) 119902119895 (119905) (3)



119879 = 12 int119871

0120588119860 (119909 119905)2 119889119909 + 12

119873sum119895=1

1198981198952119895 = 12sdot 119869sum119895=1

119869sum119896=1

(int1198710120588119860120593119895120593119896119889119909) 119902119895 119902119896 + 12

119873sum119895=1

1198981198952119895

119881 = 12 int119871

0EI(12059721199081205971199092 )

2 119889119909 + 12119873sum119904=1

119896119904 (119908 (119909119904 119905) minus 119911119904)2

= 12119869sum119895=1

119869sum119896=1

(int1198710EI11988921205931198951198891199092 119889

21205931198961198891199092 119889119909

+ 119873sum119904=1

119896119904120593119895 (119909119904) 120593119896 (119909119904)) 119902119895119902119896 minus 119873sum119904=1

119869sum119896=1

119896119904120593119896 (119909119904) 119902119896119911119904+ 12119873sum119895=1

1198961199041199112119904

(4)


119863 = 12119873sum119904=1

119888119904 ( (119909 119905) minus 119904)2

= 12119873sum119904=1

119888119904 [[119869sum119895=1

119869sum119896=1

120593119895 (119909119904) 120593119896 (119909119904) 119902119895 119902119896]]minus 119873sum119904=1

119869sum119896=1

119888119904120593119896 (119909119904) 119902119896119904 + 12119873sum119904=1

1198881199042119904

(5)


Mq + Cq + Kq = 0 (6)



C = [[[[ 119873sum119904=1119888119904120593119894 (119909119904) 120593119895 (119909119904)]

119869times119869


]]]

K

= [[[[1205751198941198951205962119895 + 119873sum

119904=1119896119904120593119894 (119909119904) 120593119895 (119909119904)]

119869times119869


]]]

q = 119902119895119869times1119911119904119873times1

(7)

where 119894 119895 = 1 2 119869 and 119903 119904 = 1 2 119873

4 Model Validation


Sx minus Rx = 0 (8)


S = [minusK 00 M

]


x = qq

(9)


(R minus 120582119899S) 120601119899 = 0 (10)



119891119899 = 12120587 10038161003816100381610038161205821198991003816100381610038161003816 120577119899 = minusRe (120582119899)10038161003816100381610038161205821198991003816100381610038161003816

(11)





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6 Conclusions



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Acknowledgments


References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of








700 700 700 700 700 700 700 700 700 700 700 700 700 700 350350

16 191 221517 18 2120237 8 5 4

(a)

16 191 221517 18 212023 8 5

(b)

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(c)

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(e)

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120593119895 (119909) = 1198621 sin(119895120587119909119871 ) (1)


int1198710120588119860120593119895120593119896119889119909 =

1 if 119895 = 1198960 if 119895 = 119896 (2)



1 j N

Attachment j

x

m1

k1 c1

mj

xj

kj cj

zjmN

kN cN


119908 (119909 119905) = 119869sum119895=1

120593119895 (119909) 119902119895 (119905) (3)



119879 = 12 int119871

0120588119860 (119909 119905)2 119889119909 + 12

119873sum119895=1

1198981198952119895 = 12sdot 119869sum119895=1

119869sum119896=1

(int1198710120588119860120593119895120593119896119889119909) 119902119895 119902119896 + 12

119873sum119895=1

1198981198952119895

119881 = 12 int119871

0EI(12059721199081205971199092 )

2 119889119909 + 12119873sum119904=1

119896119904 (119908 (119909119904 119905) minus 119911119904)2

= 12119869sum119895=1

119869sum119896=1

(int1198710EI11988921205931198951198891199092 119889

21205931198961198891199092 119889119909

+ 119873sum119904=1

119896119904120593119895 (119909119904) 120593119896 (119909119904)) 119902119895119902119896 minus 119873sum119904=1

119869sum119896=1

119896119904120593119896 (119909119904) 119902119896119911119904+ 12119873sum119895=1

1198961199041199112119904

(4)


119863 = 12119873sum119904=1

119888119904 ( (119909 119905) minus 119904)2

= 12119873sum119904=1

119888119904 [[119869sum119895=1

119869sum119896=1

120593119895 (119909119904) 120593119896 (119909119904) 119902119895 119902119896]]minus 119873sum119904=1

119869sum119896=1

119888119904120593119896 (119909119904) 119902119896119904 + 12119873sum119904=1

1198881199042119904

(5)


Mq + Cq + Kq = 0 (6)



C = [[[[ 119873sum119904=1119888119904120593119894 (119909119904) 120593119895 (119909119904)]

119869times119869


]]]

K

= [[[[1205751198941198951205962119895 + 119873sum

119904=1119896119904120593119894 (119909119904) 120593119895 (119909119904)]

119869times119869


]]]

q = 119902119895119869times1119911119904119873times1

(7)

where 119894 119895 = 1 2 119869 and 119903 119904 = 1 2 119873

4 Model Validation


Sx minus Rx = 0 (8)


S = [minusK 00 M

]


x = qq

(9)


(R minus 120582119899S) 120601119899 = 0 (10)



119891119899 = 12120587 10038161003816100381610038161205821198991003816100381610038161003816 120577119899 = minusRe (120582119899)10038161003816100381610038161205821198991003816100381610038161003816

(11)





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6 Conclusions



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Acknowledgments


References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of







2575257525752575100

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120593119895 (119909) = 1198621 sin(119895120587119909119871 ) (1)


int1198710120588119860120593119895120593119896119889119909 =

1 if 119895 = 1198960 if 119895 = 119896 (2)



1 j N

Attachment j

x

m1

k1 c1

mj

xj

kj cj

zjmN

kN cN


119908 (119909 119905) = 119869sum119895=1

120593119895 (119909) 119902119895 (119905) (3)



119879 = 12 int119871

0120588119860 (119909 119905)2 119889119909 + 12

119873sum119895=1

1198981198952119895 = 12sdot 119869sum119895=1

119869sum119896=1

(int1198710120588119860120593119895120593119896119889119909) 119902119895 119902119896 + 12

119873sum119895=1

1198981198952119895

119881 = 12 int119871

0EI(12059721199081205971199092 )

2 119889119909 + 12119873sum119904=1

119896119904 (119908 (119909119904 119905) minus 119911119904)2

= 12119869sum119895=1

119869sum119896=1

(int1198710EI11988921205931198951198891199092 119889

21205931198961198891199092 119889119909

+ 119873sum119904=1

119896119904120593119895 (119909119904) 120593119896 (119909119904)) 119902119895119902119896 minus 119873sum119904=1

119869sum119896=1

119896119904120593119896 (119909119904) 119902119896119911119904+ 12119873sum119895=1

1198961199041199112119904

(4)


119863 = 12119873sum119904=1

119888119904 ( (119909 119905) minus 119904)2

= 12119873sum119904=1

119888119904 [[119869sum119895=1

119869sum119896=1

120593119895 (119909119904) 120593119896 (119909119904) 119902119895 119902119896]]minus 119873sum119904=1

119869sum119896=1

119888119904120593119896 (119909119904) 119902119896119904 + 12119873sum119904=1

1198881199042119904

(5)


Mq + Cq + Kq = 0 (6)



C = [[[[ 119873sum119904=1119888119904120593119894 (119909119904) 120593119895 (119909119904)]

119869times119869


]]]

K

= [[[[1205751198941198951205962119895 + 119873sum

119904=1119896119904120593119894 (119909119904) 120593119895 (119909119904)]

119869times119869


]]]

q = 119902119895119869times1119911119904119873times1

(7)

where 119894 119895 = 1 2 119869 and 119903 119904 = 1 2 119873

4 Model Validation


Sx minus Rx = 0 (8)


S = [minusK 00 M

]


x = qq

(9)


(R minus 120582119899S) 120601119899 = 0 (10)



119891119899 = 12120587 10038161003816100381610038161205821198991003816100381610038161003816 120577119899 = minusRe (120582119899)10038161003816100381610038161205821198991003816100381610038161003816

(11)





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6 Conclusions



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Acknowledgments


References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





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120593119895 (119909) = 1198621 sin(119895120587119909119871 ) (1)


int1198710120588119860120593119895120593119896119889119909 =

1 if 119895 = 1198960 if 119895 = 119896 (2)



1 j N

Attachment j

x

m1

k1 c1

mj

xj

kj cj

zjmN

kN cN


119908 (119909 119905) = 119869sum119895=1

120593119895 (119909) 119902119895 (119905) (3)



119879 = 12 int119871

0120588119860 (119909 119905)2 119889119909 + 12

119873sum119895=1

1198981198952119895 = 12sdot 119869sum119895=1

119869sum119896=1

(int1198710120588119860120593119895120593119896119889119909) 119902119895 119902119896 + 12

119873sum119895=1

1198981198952119895

119881 = 12 int119871

0EI(12059721199081205971199092 )

2 119889119909 + 12119873sum119904=1

119896119904 (119908 (119909119904 119905) minus 119911119904)2

= 12119869sum119895=1

119869sum119896=1

(int1198710EI11988921205931198951198891199092 119889

21205931198961198891199092 119889119909

+ 119873sum119904=1

119896119904120593119895 (119909119904) 120593119896 (119909119904)) 119902119895119902119896 minus 119873sum119904=1

119869sum119896=1

119896119904120593119896 (119909119904) 119902119896119911119904+ 12119873sum119895=1

1198961199041199112119904

(4)


119863 = 12119873sum119904=1

119888119904 ( (119909 119905) minus 119904)2

= 12119873sum119904=1

119888119904 [[119869sum119895=1

119869sum119896=1

120593119895 (119909119904) 120593119896 (119909119904) 119902119895 119902119896]]minus 119873sum119904=1

119869sum119896=1

119888119904120593119896 (119909119904) 119902119896119904 + 12119873sum119904=1

1198881199042119904

(5)


Mq + Cq + Kq = 0 (6)



C = [[[[ 119873sum119904=1119888119904120593119894 (119909119904) 120593119895 (119909119904)]

119869times119869


]]]

K

= [[[[1205751198941198951205962119895 + 119873sum

119904=1119896119904120593119894 (119909119904) 120593119895 (119909119904)]

119869times119869


]]]

q = 119902119895119869times1119911119904119873times1

(7)

where 119894 119895 = 1 2 119869 and 119903 119904 = 1 2 119873

4 Model Validation


Sx minus Rx = 0 (8)


S = [minusK 00 M

]


x = qq

(9)


(R minus 120582119899S) 120601119899 = 0 (10)



119891119899 = 12120587 10038161003816100381610038161205821198991003816100381610038161003816 120577119899 = minusRe (120582119899)10038161003816100381610038161205821198991003816100381610038161003816

(11)





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(b)






6 Conclusions



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Acknowledgments


References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





5 10 15 20 250Frequency (Hz)

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(b)



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120593119895 (119909) = 1198621 sin(119895120587119909119871 ) (1)


int1198710120588119860120593119895120593119896119889119909 =

1 if 119895 = 1198960 if 119895 = 119896 (2)



1 j N

Attachment j

x

m1

k1 c1

mj

xj

kj cj

zjmN

kN cN


119908 (119909 119905) = 119869sum119895=1

120593119895 (119909) 119902119895 (119905) (3)



119879 = 12 int119871

0120588119860 (119909 119905)2 119889119909 + 12

119873sum119895=1

1198981198952119895 = 12sdot 119869sum119895=1

119869sum119896=1

(int1198710120588119860120593119895120593119896119889119909) 119902119895 119902119896 + 12

119873sum119895=1

1198981198952119895

119881 = 12 int119871

0EI(12059721199081205971199092 )

2 119889119909 + 12119873sum119904=1

119896119904 (119908 (119909119904 119905) minus 119911119904)2

= 12119869sum119895=1

119869sum119896=1

(int1198710EI11988921205931198951198891199092 119889

21205931198961198891199092 119889119909

+ 119873sum119904=1

119896119904120593119895 (119909119904) 120593119896 (119909119904)) 119902119895119902119896 minus 119873sum119904=1

119869sum119896=1

119896119904120593119896 (119909119904) 119902119896119911119904+ 12119873sum119895=1

1198961199041199112119904

(4)


119863 = 12119873sum119904=1

119888119904 ( (119909 119905) minus 119904)2

= 12119873sum119904=1

119888119904 [[119869sum119895=1

119869sum119896=1

120593119895 (119909119904) 120593119896 (119909119904) 119902119895 119902119896]]minus 119873sum119904=1

119869sum119896=1

119888119904120593119896 (119909119904) 119902119896119904 + 12119873sum119904=1

1198881199042119904

(5)


Mq + Cq + Kq = 0 (6)



C = [[[[ 119873sum119904=1119888119904120593119894 (119909119904) 120593119895 (119909119904)]

119869times119869


]]]

K

= [[[[1205751198941198951205962119895 + 119873sum

119904=1119896119904120593119894 (119909119904) 120593119895 (119909119904)]

119869times119869


]]]

q = 119902119895119869times1119911119904119873times1

(7)

where 119894 119895 = 1 2 119869 and 119903 119904 = 1 2 119873

4 Model Validation


Sx minus Rx = 0 (8)


S = [minusK 00 M

]


x = qq

(9)


(R minus 120582119899S) 120601119899 = 0 (10)



119891119899 = 12120587 10038161003816100381610038161205821198991003816100381610038161003816 120577119899 = minusRe (120582119899)10038161003816100381610038161205821198991003816100381610038161003816

(11)





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1 j N

Attachment j

x

m1

k1 c1

mj

xj

kj cj

zjmN

kN cN


119908 (119909 119905) = 119869sum119895=1

120593119895 (119909) 119902119895 (119905) (3)



119879 = 12 int119871

0120588119860 (119909 119905)2 119889119909 + 12

119873sum119895=1

1198981198952119895 = 12sdot 119869sum119895=1

119869sum119896=1

(int1198710120588119860120593119895120593119896119889119909) 119902119895 119902119896 + 12

119873sum119895=1

1198981198952119895

119881 = 12 int119871

0EI(12059721199081205971199092 )

2 119889119909 + 12119873sum119904=1

119896119904 (119908 (119909119904 119905) minus 119911119904)2

= 12119869sum119895=1

119869sum119896=1

(int1198710EI11988921205931198951198891199092 119889

21205931198961198891199092 119889119909

+ 119873sum119904=1

119896119904120593119895 (119909119904) 120593119896 (119909119904)) 119902119895119902119896 minus 119873sum119904=1

119869sum119896=1

119896119904120593119896 (119909119904) 119902119896119911119904+ 12119873sum119895=1

1198961199041199112119904

(4)


119863 = 12119873sum119904=1

119888119904 ( (119909 119905) minus 119904)2

= 12119873sum119904=1

119888119904 [[119869sum119895=1

119869sum119896=1

120593119895 (119909119904) 120593119896 (119909119904) 119902119895 119902119896]]minus 119873sum119904=1

119869sum119896=1

119888119904120593119896 (119909119904) 119902119896119904 + 12119873sum119904=1

1198881199042119904

(5)


Mq + Cq + Kq = 0 (6)



C = [[[[ 119873sum119904=1119888119904120593119894 (119909119904) 120593119895 (119909119904)]

119869times119869


]]]

K

= [[[[1205751198941198951205962119895 + 119873sum

119904=1119896119904120593119894 (119909119904) 120593119895 (119909119904)]

119869times119869


]]]

q = 119902119895119869times1119911119904119873times1

(7)

where 119894 119895 = 1 2 119869 and 119903 119904 = 1 2 119873

4 Model Validation


Sx minus Rx = 0 (8)


S = [minusK 00 M

]


x = qq

(9)


(R minus 120582119899S) 120601119899 = 0 (10)



119891119899 = 12120587 10038161003816100381610038161205821198991003816100381610038161003816 120577119899 = minusRe (120582119899)10038161003816100381610038161205821198991003816100381610038161003816

(11)





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Algebra





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Algebra





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Volume 2014




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Journal of


Function Spaces






Algebra





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experimental and numerical investigation of the effect of

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