deterministic and probabilistic serviceability assessment

Research ArticleDeterministic and Probabilistic Serviceability Assessment ofFootbridge Vibrations due to a Single Walker Crossing

Cristoforo Demartino1 Alberto Maria Avossa 2 and Francesco Ricciardelli2

1College of Civil Engineering Nanjing Tech University Nanjing 211816 China2DICDEA Universita della Campania Luigi Vanvitelli Via Roma 9 81031 Aversa Italy

Correspondence should be addressed to Alberto Maria Avossa albertomariaavossaunicampaniait

Received 9 July 2017 Revised 5 September 2017 Accepted 12 November 2017 Published 17 January 2018

Academic Editor Hugo Rodrigues

Copyright copy 2018 Cristoforo Demartino et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

This paper presents a numerical study on the deterministic and probabilistic serviceability assessment of footbridge vibrations dueto a single walker crossing The dynamic response of the footbridge is analyzed by means of modal analysis considering only thefirst lateral and vertical modes Single span footbridges with uniform mass distribution are considered with different values of thespan length natural frequencies mass and structural damping and with different support conditionsThe load induced by a singlewalker crossing the footbridge ismodeled as amoving sinusoidal force either in the lateral or in the vertical directionThe variabilityof the characteristics of the load induced by walkers is modeled using probability distributions taken from the literature defining aStandard Population of walkers Deterministic and probabilistic approaches were adopted to assess the peak response Based on theresults of the simulations deterministic and probabilistic vibration serviceability assessment methods are proposed not requiringnumerical analyses Finally an example of the application of the proposedmethod to a truss steel footbridge is presentedThe resultshighlight the advantages of the probabilistic procedure in terms of reliability quantification

1 Introduction

Vibrations of footbridges due to human loading have recentlyreceived much attention due to the increasing number ofvibrations incidents occurring worldwide The main causeof these large vibrations is the low stiffness and damping ofrecently built footbridges As a matter of fact if only staticdead and live loads are considered in the design processfootbridges can prove unable to meet serviceability require-ments against vibrations

In particular the significant amount of research producedin the last one and a half decades has been triggered by thetwo vibration incidents of the Paris Passerelle Solferino onDecember 15 1999 and of the London Millennium Bridgeon July 10 2000 that played a similar role in the publicopinion and in the scientific community to that played by thecollapse of the TacomaNarrows bridge in November 1940 [1]Nevertheless crowd-related failures of bridges have occurredover the centuries with much more devastating effects thefirst of which being documented is probably that of the bridge

over river Ouse in England in 1154 [2] In spite of the longseries of failures to the authorsrsquo knowledge the first paper tohave appeared dealing with the effects of human movementson structural loading is that of Tilden [3]This is a pioneeringwork where many aspects of the human loading of structuresseem to have already been recognized though not quantified

The vibrations are generated by the ldquoquasirdquo harmonic loadinduced by walkers and joggers If the central frequency ofthe load and a natural vibration frequency of the footbridgeare similar resonant vibrations can occur The latter isa rather common condition [4] However this apparentlysimple mechanism is in fact not easy to quantify First ofall walkers do not induce a perfectly periodic load due tothe intrasubject variability gait and this load differs fromone subject to another (intersubject variability) Finally twoforms of feedback can take place (i) interwalker interactionand (ii) walker-structure interaction [5]Thefirst refers to gaitmodifications due to the presence of neighboring walkerswhereas the second is the adjustment of gait as an effect ofthe floor vibration

HindawiShock and VibrationVolume 2018 Article ID 1917629 26 pageshttpsdoiorg10115520181917629

2 Shock and Vibration

Standards and guidelines have been developed to help thedesigner in the evaluation of the vibration serviceability basedon simplified loading models simulating different possiblescenarios [5 6] In broad terms the design of footbridgesagainst pedestrian-induced vibrations requires the knowl-edge of [7] (i) the characteristics of the pedestrian action (ii) aresponse evaluationmethod and (iii) a comfort criterionTheinterested reader is referred to [5 6 8 9] Design guidelinesoutline procedures for vibration serviceability checks butit is noticeable that most of these assume that the actionis deterministic [10] yet this is stochastic and it wouldbe reasonable to incorporate its variability in the loadingmodels However it should be mentioned that other loads onbridges should be considered in the design procedures suchas seismic wind and impact loads (eg [11ndash16])

Different authors have tried to characterize the random-ness of the pedestrian action in either time or frequencydomain considering both intrasubject and intersubject vari-ability Brownjohn et al [17] for the vertical direction andPizzimenti and Ricciardelli [18] and Ricciardelli and Pizzi-menti [19] for the lateral direction gave Power SpectralDensity Functions (PSDFs) of the load induced by a walkerfor use in the evaluation of the stationary response to a streamof walkers including intrasubject and intersubject variabilityof gait

Subsequently Butz [20] presented a spectral approachfor the evaluation of the peak acceleration induced by unre-stricted pedestrian trafficThe PSDFs of themodal force wereapproximated through a Gaussian function fitting data com-ing fromMonte Carlo simulationsThese were carried out fordifferent bridge geometries and for four different pedestriandensities Step frequency pedestrian mass force amplitudeand pedestrian arrival time were randomly selected fromgiven probability distributions The PSDF of the accelerationwas evaluated from the PSDF of themodal force and the 95thfractile peak modal acceleration was derived as the productof the RMS acceleration and a peak factor The latter wasevaluated to be around 4 All the coefficients required for theapplication of the procedure were expressed in a parametricform This model has then been incorporated into HIVOSSguidelines [21]

Zivanovic et al [22] presented a multiharmonic forcemodel for calculation of the multimode structural responseto a crossing accounting for inter- and intrasubject vari-ability in the walking force The model is again based onMonte Carlo simulations with pedestrian characteristics alsoselected from given distributionsThe intrasubject variabilitywas accounted for describing the force in the frequencydomain and then converting it to the time domain Noparametric form of the peak response as a function of thedifferent parameters and no procedure for the serviceabilityassessment are given

Ingolfsson et al [23] proposed a Response Spectrumapproach inspired by earthquake engineering ThroughMonte Carlo simulations they evaluated a reference verticalacceleration to the action of a flow of 1walkers withprobabilistically modeled characteristics to which empir-ical correction factors are applied to account for returnperiod modal mass mean arrival rate structural damping

footbridge span and mode shape A total of 97 windows300 s long were used to establish the peak acceleration Gen-eralized Extreme Value (GEV) distribution parameters Tworeference populations were considered with step frequenciesof 18 and 20Hz with STD of 01 Hz and Poisson distributedarrivals The input force was modeled as a harmonic loadand the intersubject variability was considered varying thecharacteristics of each pedestrian according to given distri-butions All the parameters contained in the procedure aregiven in parametric form and the GEV distribution of thepeak acceleration is found to tend to a Gumbel distribution(ie the shape parameter tends to 0)

Piccardo and Tubino [24] studied the vertical vibrationserviceability of footbridges based on a probabilistic charac-terization of pedestrian-induced forces taking into accountintersubject variability and considering only one mode ofvibration Only the 95th fractile peak modal acceleration wasderived and expressed in two nondimensional forms (a) theEquivalent Amplification Factor that is the ratio between themaximum dynamic response to a realistic loading scenarioand the maximum dynamic response to a single resonantpedestrian (b) the Equivalent Synchronization Factor that isthe ratio between themaximumdynamic response to a realis-tic loading scenario and the maximum dynamic response touniformly distributed resonant pedestrians Comparison oftheir procedure with similar methods contained in standardsand design guidelines has pointed out that the latter aregenerally conservative (often largely conservative) and canbecome only slightly nonconservative in particular casesThey concluded that further investigations on the evaluationof the PDF of the maximum dynamic response are required

Zivanovic et al [25] reviewed different time-domaindesign procedures for vibration serviceability assessment offootbridges exposed to streams of pedestrians and evalu-ated their performance in predicting the vertical vibrationresponse of two existing footbridges They compared theprocedures contained in Eurocode 5 [26] ISO 10137 [27]Setra [28] BSI [29] Brownjohn et al [30] Butz [20]Ingolfsson et al [23] and Zivanovic et al [22] They foundsome discrepancies between the predicted and measuredvibration levels and discussed their potential causes amongwhich are interwalker and walker-structure interaction

Pedersen and Frier [10] evaluated the effect of the proba-bilistic modeling of the parameters describing walking loadsstep frequency stride length Dynamic Load Factor (DLF)and walker weight A literature review revealed a variety ofprobability distributions through which the walking param-eters can be modeled and these were used for exploringthe sensitivity of the 95th fractile of midspan accelerationThey observed that the step frequency distribution can havea strong influence whereas the DLF the walker weight andstride length have a much lower influence No interactionswere considered in this study

Ingolfsson and Georgakis [31] presented a probabilisticlateral load model in which the forces are given as the sumof an external component and a frequency and amplitude-dependent self-excited component the latter is quantifiedthrough equivalent pedestrian damping andmass coefficientsmeasured from experiments They found that the peak

Shock and Vibration 3

response of a footbridge to a pedestrian flow is very sensitiveto the selection of the pacing rate distribution

Piccardo and Tubino [32] introduced an equivalent spec-tral model for the analysis of the dynamic response of foot-bridges to unrestricted pedestrian traffic (ie no interwalkerinteraction) using a complete probabilistic representation ofpedestrians They provided simple closed-form expressionsfor the evaluation of the maximum dynamic response foruse in vibration serviceability analyses similarly to classicalprocedures adopted in wind engineering These expressionsare based on the definition of the peak factor found byDavenport [33]

Recently Ricciardelli and Demartino [5] comparedbackground hypotheses fields of applicability and resultsobtained through a number of different loading and responseevaluation models In particular they compared singlewalker models multiple walkers models interaction models(interwalker and walker-structure) and instability modelstogether with current design procedures incorporated intostandards and guidelines Avossa et al [34] applied the designprocedures to various steel footbridges highlighting the largedifferences that they bring in the resultsThey concluded thata critical revision of design procedures is needed as theseeven though inspired by the same principles and applyingthe same rules show different results this should also bedone through validation with the available full-scale dataFinally Avossa et al [35] evaluated through Monte Carlosimulations the probability distribution of the footbridgepeak acceleration to single and multiple crossing walkers fortwo specific footbridge configurations

In combination with a probabilistic definition of theload criteria for the probabilistic definition of the structuralcapacity must be set For instance Eurocode 0 [36] requiresthat a structure is designed to have adequate (i) resistance(ii) serviceability and (iii) durability In particular the limitstate of vibrations causing discomfort to people andorlimiting the functional effectiveness of the structure must beconsidered Moreover it establishes that when the structureis prone to significant acceleration dynamic analyses must beperformed Similarly ISO 2394 [37] specifies general princi-ples for the reliability assessment of structures subjected toknown or foreseeable types of actions providing more or lesssimilar requirements for safety serviceability and durabilityIn Annex E of ISO 2394 [37] principles of reliability-baseddesign are given specifying the requirements in terms ofprobability of failure for different limit states

ISO 10137 [27] contains structural acceleration limitsfor different situations ISO 10137 recognizes the vibrationsource path and receiver as three key elements which requirebeing identified when dealing with vibration serviceabilityIn the context of walking-induced vibrations in footbridgesthe walkers are the vibration source the footbridge is thepath and the walkers are again the receivers According toISO 10137 an analysis of the response requires a calculationmodel that incorporates the characteristics of the sourceand of the transmission path which must be solved for thevibration response of the receiver in doing so the dynamicaction of one or more walkers can be described as force timehistories This action varies in time and space as the walkers

move on the footbridge It is recommended that the followingscenarios are considered (i) one person walking across thebridge (ii) an average pedestrian flow (group size of 8 to15 walkers) (iii) streams of walkers (significantly more than15 walkers) and (iv) occasional festive choreographic events(when relevant)

However although many authors have derived proba-bilistic models to describe the vibration response inducedby pedestrian loads a fully probabilistic procedure for theserviceability assessment of footbridge vibrations due to asingle walker crossing and a comparison with deterministicapproaches is not yet available In particular the studiesreviewed above do not allow for variation of the reliabilitylevels as they take as demand parameter the 95th fractile ofthe peak acceleration response It is important to notice thatalthough the research interest is nowadays mainly orientedtowards the multipedestrian case the need for analyzing thesingle pedestrian case stems from at least three differentreasons (i) this case can induce the largest acceleration espe-cially for short low-damped footbridges (ii) many standardsand codes of practices refer to this load scenario and (iii)vibration assessment procedures for multipedestrian loadingare often derived from the single pedestrian case

This study presents criteria for the deterministic andprobabilistic vibration serviceability assessment of foot-bridges to the crossing of one walker In Section 2 the loadinduced by a single walker is modeled as a moving har-monic force having lateral and vertical components whosecharacteristics derive from a Standard Population (SP) ofwalkers The latter is defined based on data available in theliterature concerning the probabilistic distribution of walkercharacteristics and gait parameters In Section 3 the dynamiccharacteristics of a single span footbridge (span length natu-ral frequencies mass structural damping and support con-ditions) are defined and amodal dynamicmodel is presentedIn Section 4 numerical analyses of the transient responseto a moving harmonic load are presented through whichthe peak response is evaluated in both a deterministic andprobabilistic way In Section 5 closed-form deterministic andprobabilistic vibration serviceability methods are proposedwhose applications do not require numerical analyses Theseincorporate the acceleration limits of ISO 10137 [27] and therequired reliability level of ISO 2394 [37] leading to amethodwhich also complies with Eurocode 0 [36] As an examplein Section 6 the deterministic and probabilistic methods areapplied to a prototype truss steel footbridge Finally someconclusions and prospects are drawn (Section 7)

2 Single Walker Behavior

Ground Reaction Forces (GRFs) are defined as the forcesinduced on the ground bywalkersThemeasurement ofGRFshas advanced considerably over the recent decades It firstbecame a useful clinical tool starting from the pioneeringwork of Beely [38] and Elftman [39] Nowadays observa-tional gait analysis is regularly performed by physical ther-apists to determine treatment goals and is used as an evalua-tion tool during rehabilitation [40] In Medical Sciences themain goal is the identification of the kinematic characteristics


of a subject Differently in civil engineering it is of interestto characterize GRFs with the final aim of evaluating thestructural response for comfort assessment [9 41]

GRFs are characterized by different magnitudes andfrequency content in the vertical lateral and longitudinaldirections Based on the existing knowledge of GRFs severalloading models have been developed for footbridges someof which consider the crossing of a single pedestrian [5] Acommon approach is that of periodic loading assuming thata walker generates identical footfalls with constant frequencyneglecting intrasubject variability In this case the dynamicpart of the GRF is expanded in Fourier series

119865119894 (119905) = 119882 sdot infinsum119895=1

DLF119894119895 (sin (119895 sdot 120587 sdot 119891119908119894 sdot 119905) minus 120595119894119895) (1)

where the subscript 119894 = 119881 119871 indicates the vertical or lateraldirection (the longitudinal component is neglected) andwhere119882 is theweight of thewalkerDLF119894119895 is the 119895thDynamicLoad Factor (DLF) that is the 119895th harmonic load amplitudenormalized by the body weight 119891119908119894 = 119891119908 when 119894 = 119881 and119891119908119894 = 05 sdot 119891119908 when 119894 = 119871 119891119908 being the step frequencyand 120595119894119895 is the phase lag of the 119895th harmonic Moreover inthe following only the first harmonic will be retained andaccordingly subscript 119895 will be omitted This representationof the load is consistent with different standards such as UKAnnex to EC1 [29] and ISO 10137 [27]

When the walker crosses a footbridge of span 119871 themodal load associated with the first bending mode is

119891119894 (119905) = int1198710120601119894 (119909) sdot 119865119894 (119905) sdot 120575 (119909 minus V sdot 119905)

sdot [119867 (119905) minus 119867 (119905 minus 119879119901)] d119909(2)

where 120601119894(119909) is first mode shape (Section 3) 120575(∙) is the DiracFunction 119909 defines the position of the walker on the bridge119867(∙) is the Heaviside function and 119879119901 = 119871V is the crossingtime V = 119891119908119897119908 being the walking speed 119871 the span lengthand 119897119908 the step length

21 Standard Population of Walkers The definition of aStandard Population (SP) of walkers is needed to characterizeintersubject variability probabilisticallyThis is not trivial dueto the large scatter of the data available in the literature andone must be aware of the fact that changing the populationwill lead to a different vibration response [10]

The parameters governing the excitation generated by awalker are (i) the walking speed V (ii) the step frequency119891119908 (iii) the Dynamic Load Factors DLF119881 and DLF119871 (iv)the weight of the walker 119882 and (v) the phase angles 120595119881and 120595119871 The data mainly come from the Biomechanics andTransportation fields although recent results have also beenpublished in the area of structural engineering The SPdefined in this section is based on research developed inEuropean countries

Humans can walk up to 4ms [42] but the speed ofroughly 22ms represents a natural transition from walkingto running [43 44] In spite of this the walking speed is

usually considered as normally distributed and a large scatterin the mean value is found in the literature This is due tophysiological and psychological factors such as biometriccharacteristics of the walker (body weight height age andgender) cultural and racial differences travel purpose andtype of walking facility [45] In the following the walkerspeed is assumed as [46]

V = N (141 0224) gt 041 [ms] (3)

Equation (3) is truncated at 041ms as smaller valueslead to negative STDs in (4)

It is agreed that walking occurs at an average step fre-quency of approximately119891119908 = 2Hz (eg [28]) Biomechanicsstudies established that walkers tend to adjust their stepfrequency and therefore step length 119897119908 so to minimizeenergy consumption at a given walking speed [47] Thestep length (and its double the stride length) varies withthe physical characteristics of the subject (height weightetc) and from one country to another due to the differenttraditions and lifestyle Accordingly the correlation betweenthe walking speed and the step frequency has been reportedin literature with a large scatter [45 48ndash50]Many researchershave considered the step frequency as normally distributed[41 48 51] In this study it is assumed that themean and STDof the normal distribution of119891119908 are linearly dependent on thewalking speed [52]

119891119908 = N (07868 sdot V + 07886 00857 sdot V minus 0035)gt 0 [Hz] (4)

Negative values are truncated asmeaninglessThemean valueof 119891119908 associated with the mean walking speed (ie 141ms)is 1898Hz and the standard deviation is 0086Hz

Values of the DLFs have been reported in many publi-cations and have been incorporated into design guidelinesThese are usually derived from force measurements oninstrumented floors or treadmills [53] In this study onlyfixed floor conditions are considered since walker-structureinteraction effects are neglectedThis interaction is significantin the lateral direction [54] and less in the vertical directionDLFs measured on a rigid floor are therefore assumed tobe the same as those that would be measured on a movingfloor if the displacements are small Moreover DLFs can alsobe derived using analytical models that are usually inspiredby biomechanics as the inverted pendulum model [55 56]Zivanovic et al [8] reported a review of the DLFs used insingle walker force models The mean values of DLF119881 areapproximately in the range of 0257 [57] to 05 [58] Generallythe dynamic part of the vertical GRF is found to be dependenton 119891119908 The first study revealing this issue is that of Kajikawa[59] (reported in [60]) It is now widely accepted that DLF119881increases with 119891119908 up to a maximum of approximately 05 [6162] Accordingly in this study the SP is described through aDLF119881 depending on 119891119908 (see (4)) as [62]

0 lt DLF119881 = 037 sdot (119891119908 minus 095) le 05 (5)

According to (5) DLF119881 is described by a normal distri-bution with mean equal to 035 and STD 007


00

1

2PD

F

0

10

20

0

05

1

15

2

25

1 2 3v (ms)

1 2 30fw (Hz)

05 10＄V ()

0

20

40

005 010＄L ()

0

02

04

06

08

1

CDF

0

2

4

500 10000W (N)

times10minus3

Figure 1 Probability Density Function (PDF thin line) and Cumulated Distribution Function (CDF thick line) of the random variablesdefining the SP V 119891119908 DLF119881 DLF119871 and119882

On the other hand the mean values of DLF119871 are approx-imately in the range of 0039 [63] to 01 [58] Accordinglyin this study the SP is assumed to have a DLF119871 describedthrough a normal distribution [52]

DLF119871 = N (003792 001459) gt 0 (6)

In (5) and (6) negative values are truncated as meaning-less Moreover in (5) an upper bound was set at 05

The body weight is very much dependent on heighttherefore in medical applications it is preferred to refer tothe Body Mass Index (BMI) that is the body mass dividedby the square of the height [64] For each country Walpole etal [65] used available data on BMI and height distributionto estimate average adult body mass In particular theyreported the average body mass by world regions as in 2005The average body mass ranges between 577 kg for Asia and807 kg for North America Indeed load models (eg (1))require the definition of the walker weight 119882 that is thebody weight increased by the weight of clothing and otheritems carried by the walker The walker weight is taken equalto 700Nbymany loadingmodels (eg [28 57]) In this studythe weight of the SP is assumed to be normally distributed asin HIVOSS [21]

119882 = N (744 130) ge 0 [N] (7)

The mean value in (7) is 7 larger than that of 695Ncorresponding to the average body mass 708 kg reported byWalpole et al [65] for Europe since the latter lacks clothingand other items carried by the walker Negative values aretruncated as meaningless

Finally the distribution of phase lags120595119894 between walkersis a measure of the correlation of the forces they exert Fora continuous PDF of walking frequencies phase lags arecharacterized by the PDF of the phase spectrum If this isuniformly distributed between 0 and 2120587 then thewalkers andthe walking forces are uncorrelated Correlation increaseswhen the PDF of the phase spectrum is peaky around a given

value as this value approaches 0 the forces tend to be in phaseIn the case of a single walker all this loses itsmeaning andwillbe neglected in the following

In Figure 1 the Probability Density Functions (PDFs)and the Cumulative Distribution Functions (CDFs) of therandom variables defining the SP previously described areshown

3 Footbridge Characteristics andMechanical Model

In this study single span beam footbridges are consideredThe dynamic response of the footbridge is analyzed bymeansof modal analysis considering only the first lateral and verti-calmodesThe latter assumption ismade since (i) in commonfootbridges torsional vibrations are not an issue with theexception of some research aimed at the mitigation oftorsional vibrations on suspended footbridges [66] and (ii)usually only onemode either vertical or lateral is responsiblefor the footbridge liveliness [67]

Referring to Figure 2 the footbridge response is governedby the two uncoupled differential equations (119894 = 119881 119871)

119898119894 (119909 119905) + 119888119894119894 (119909 119905) + 119864119868119894119906119868119881119894 (119909 119905)= 119865119894 (119905) sdot 120575 (119909 minus V sdot 119905) (8)

where119898 is the footbridge uniform mass per unit length 119888119894 isthe viscous damping per unit length and 119864119868119894 is the bendingstiffness and 119906119894(119909 119905) are the displacements in the vertical119906119881(119909 119905) and lateral 119906119871(119909 119905) directions 119865119894(119905) and 120575(119909 minus V sdot119905) have been defined in Section 2 The viscous damping isrelated to the inherent structural damping and to that basedon isolation or supplemental energy dissipation devices (eg[68 69])

Considering only one mode of vibration the vertical andlateral deflection of the footbridge is written as

119906119894 (119909 119905) = 120601119894 (119909) 120578119894 (119905) (9)


Kr1V

Kr1L

m

y

x

z1 2

uL(x t)

uV(x t)

cV cL EIV EIL Kr2V

Kr2L

L

Figure 2 Beam characteristics dynamic model

where 120601119894(119909) is the mode shape and 120578119894(119905) the associated gen-eralized coordinates in the vertical 120578119881(119905) and lateral 120578119871(119905)directions

For a single span beam the mode shape takes the generalform

120601119894 (119909) = 1198881119894 sin(120582119894 119909119871) + 1198882119894 cos(120582119894 119909119871)+ 1198883119894 sinh(120582119894 119909119871) + 1198884119894 cosh (120582119894 119909119871)

(10)

where 120582119894 is the first eigenvalue of the secular equation and1198881119894 1198882119894 1198883119894 and 1198884119894 depend on the supports rotational stiffness1198701199031119894 and1198701199032119894 (Figure 2) For example for a simply supportedbeam (1198701199031119894 = 1198701199032119894 = 0) 1198881119894 = 1 and 1198882119894 = 1198883119894 = 1198884119894 = 0 and120582119894 = 120587 The mode shape is normalized so as to have maxi-mum value equal to one

The modal equations of motion are

120578119894 (119905) + 4120587120585119894119891119894 120578119894 (119905) + 412058721198912119894 120578119894 (119905) = 119898minus1119894 119891119894 (119905) (11)

where 120585119894 is the modal damping ratio 119891119894 is the naturalfrequency and119898119894 is themodalmass in the vertical and lateraldirections

Assuming that the motion starts from rest (ie 120578119894(0) =120578119894(0) = 0) the solution of (11) is

120578119894 (119905) = 1119898119894 int119905

119900119891119894 (120591) sdot ℎ119894 (119905 minus 120591) sdot d120591 (12)

where ℎ119894(119905 minus 120591) is the unit-impulse response functionThe governing parameters of the dynamic response are (i)

the span length 119871 (ii) the natural frequencies 119891119881 and 119891119871 (iii)the structural damping ratios 120585119881 and 120585119871 and (iv) the modalshapes 120601119881(119909) and 120601119871(119909) The footbridge mass per unit length119898 also influences the response However this parameterappears as a linear factor in the equations and accordingly itplays the role of a scale parameter in the results

In the analyses values of the span length 119871 in the widerange of 10m to 200m were considered where the smallestvalue corresponds to a simple road crossing and the largest isassumed as an upper limit for beam footbridges

The vertical and lateral natural frequencies are expressedin terms of frequency ratios 120572119894 = 119891119908119894119891119894 Here and in thefollowing the overbar indicates a mean value of the distribu-tionThe range of interests for vibration serviceability assess-ments is 03 le 120572119894 le 17 which approximately corresponds to

the external boundaries of Range 3 (low risk of resonance forstandard loading situations) as defined in Setra [28]

The damping ratio can vary in the wide range of 01to 20 [70] In particular standards and guidelines suchas Setra [28] Heinemeyer et al [21] ISO 10137 [27] andEurocodes (EC1 [71] EC3 [72] and EC5 [26]) suggestminimum and mean values depending on constructionmaterial (Table 1) The lower values apply to steel bridges(02 divide 05) whereas the largest values are for timberbridges (15 divide 3) In particular the values given by FIB[73] are the result of the combination of material bridge typeand support conditions According to the values given inTable 1 damping ratios in the range of 01 to 15were con-sidered in this study It is worth mentioning that it is dif-ficult to accurately predict (during the design process) andestimate (during the assessment process) this parameter andaccordingly fairly large uncertainties are associated withit

Finally themode shapes (ie the supports rotational stiff-ness values 1198701199031119894 and 1198701199032119894) influence the dynamic responseof the footbridge In particular the mode shapes vary themodal masses 119898119881 or 119898119871 and the modal loads (see (2))Mode shapes (see (10)) in turn depend on the rotationalstiffness of the supports 1198701199031119894 and 1198701199032119894 In this study onlysymmetric support conditions have been considered that is1198701199031119894 = 1198701199032119894 = 119870119903119894

In general the mode shapes are a function of 119870119903119894However when themode shape is evaluated from a FEmodelthe evaluation of the end rotational stiffness can be cumber-some and also a direct comparison to fit (10) is not immediateAccordingly in the following a simplified and approximateprocedure is proposed to define themode shapes of the equiv-alent beammodel (ie (10)) using static analysis from the FEmodel of the footbridge The only reason for this simplifiedprocedure is to provide practitioners with a simple tool toapply the proposedmethodThismethod is accurate if the FEmodel meets the assumptions of (10) that is constant massand stiffness To this aim the support condition is expressedin terms of a nondimensional restrained level (RL) definedas the ratio of the end moment for the particular supportcondition (ie value of 119870119903119894) due to an arbitrary symmetricload 119872119894(119870119903) to that of the clamped beam subjected to thesame load119872119894(119870119903 rarr infin)

RL119894 = 119872119894 (119870119903119894)119872119894 (119870119903119894 rarr infin) (13)


Table 1 Suggested damping ratios 120585 [] for different construction material as given by different standards and guidelines

TypeDamping ratios 120585 []

Setra Hivoss ISO 10137 Eurocodes1 FIBMin Mean Min Mean Mean Mean Min Mean

Reinforced concrete 08 13 08 13 08 15 08 12Prestressed concrete 05 10 05 10 10 07 10Steel 02 04 02 04 05 02042 05 10Composite steel-concrete 03 06 03 06 06 06 06 10Timber 15 30 10 15 - 1153 08 141EC1 [71] EC3 [72] and EC5 [26] 20204 if weldedbolted connection are present 31 if no mechanical joints are present and 15 otherwise

08 09 107

mi (RLi)mi (RLi = 0)

0

02

04

06

08

1

RLi

(a)

10minus1 100 101 10210minus2

Kri middot LEIi

0

02

04

06

08

1

RLi

(b)

1 20 3 4

RDi

0

02

04

06

08

1

RLi

(c)

Figure 3 Restrained level RL119894 as a function of119898119894(RL119894)119898119894(RL119894 = 0) (a) 119870119903119894119871119864119868119894 (b) and RD119894 (c)

Therefore RL119894 is in the range of 0 (supported case 119870119903119894 =0) to 1 (clamped case 119870119903119894 rarr infin) The relationship between119870119903119894 and RL119894 can be simply obtained as

119870119903119894 = 2 sdot RL1198941 minus RL119894sdot 119864119868119894119871 (14)

The relationship between 119870119903119894 and RL119894 is shown in anondimensional form in Figure 3 The derivation of (14) isgiven in Appendix C1

On the other hand the static deformation due to an arbi-trary uniform load of the Finite Element model is expressedas the nondimensional Rotation-to-Deflection ratio (RD119894)

RD119894 = Θ119894 sdot 119871120575119894 (15)

where 120579119894 and 120575119894 are the end rotation and midspan deflectionThe relationship betweenRL119894 andRD119894 can be simply obtainedas

RL119894 = 4 minus 125 sdot RD1198944 minus RD119894(16)

which is also shown in Figure 3 Finally the parameters of(10) are chosen in order to have the same value of RD119894 as thatobtained from the FE model The derivation of (16) is givenin Appendix C2

To conclude the simplified and approximated procedureto evaluate the mode shape using static analysis of the FEmodel of the footbridge requires the following steps (i)evaluation of RD119894 (see (15)) using the FE model of thefootbridge (ie evaluation of the end rotation and midspandeflection 120579119894 and 120575119894 due to a uniform load) (ii) conversion ofRD119894 to an equivalent RL119894 using (16) (Figure 3) (iii) evaluationof 119870119903119894 using (14) (Figure 3) and (iv) evaluation of the modeshapes (see (10)) using as input the obtained 119870119903119894 in theboundary conditions and solving the eigenvalue problemobtaining 120582119894 1198881119894 1198882119894 1198883119894 and 1198884119894 Finally by integrating themass per unit length119898 over themode shape themodalmasscan be obtained The ratio of the modal mass for the genericboundary119898119894(RL119894) to themodalmass of the supported beam119898119894(RL119894 = 0) is shown as a function of RL119894 in Figure 3 Thiswas obtained by numerically solving the eigenvalue problemfor different values of RL119894 As expected the modal massreduces when the rotational stiffness of the supports increases(ie RL119894) The latter ratio will be used in Section 412

4 Response Evaluation

In this section the response evaluation procedure is pre-sentedThis is summarized in Figure 4The aim of the assess-ment is to verify that the acceleration induced by the crossingof a single walker is not causing discomfort to the walker


Modal force

Standard Population (SP) of walkers

fi(t)

Mechanical model

FEM model

Evaluation of the TFRF

Modal characteristicsV

p ()

pW(W)

LL

V Lm

V L

２＄V ２＄L

２V ２L

i(t) + 4ifi i(t) + 42f2i i(t) = mminus1

i fi(t)

Deterministic responsei (２i i L i))

Deterministic checki (２i i L i) le imax

Probabilistic responseP(i (２i i L i))

Probabilistic checkPfaili le PDi

pf(fw)

p＄(＄V)

p＄(＄L)

i (２i L i) = middot1

2i(２i i L i)middot W

mi

( )minus1

＄i i

Figure 4 Deterministic and probabilistic assessment procedures

himselfherself or other people standing on the footbridgeThe engineering demand parameter chosen to assess thisscenario is the peak acceleration as it will be shown inSection 5 A deterministic and a probabilistic approach willbe used

The input parameters are characteristics of the walker(modal force) and the dynamic properties of the footbridge(mechanicalmodel)The SPdescribed in Section 21 is used todescribe the characteristics of the walker and of the inducedmodal force The deterministic response is evaluated refer-ring to the SW (ie the walker having mean characteristicsof the SP)The probabilistic response is evaluated referring tothe SP defined in Section 2

The output is expressed through the Transient FrequencyResponse Function (TFRF) that is the ratio between themodal peak nonstationary response induced by a givenwalker crossing the bridge and the corresponding stationaryresponse induced by the SW In this way the peak modalacceleration 120578119894 can be expressed in terms of a TFRF [5 75]

120578119894 (RL119894 120572119894 119871 120585119894) = DLF119894 sdot 1198822120585119894 sdot 1119898119894 sdot 120593119894 (RL119894 120572119894 119871 120585119894) (17)

where 120593119894 is the TFRF Here and in the following the hat∙ indicates the peak response and the overline ∙ indicatesthe mean value The mechanical model of the footbridge isdeterministic and the response is expressed as a function ofits input parameters

RL119894 997888rarr mode shape120572119894 997888rarr frequency ratio

120585119894 997888rarr damping ratio119871 997888rarr number of loading cycles

(18)

The arrow indicates to which characteristics of the mechan-ical model or modal force the input parameters are referredIn particular the first three are nondimensional parametersdefining the dynamic properties of the structure The last isthe dimensional footbridge span which is a measure of thenumber of loading cycles (see (2)) The number of cycles isthe ratio of 119871 to the wavelength the latter being the ratio ofthe walking speed to the walking frequency Accordingly fora given value for thewavelength increasing of the span lengthwill correspond to larger values of the response tendingto those of a stationary system this also depends on thestructural damping [75ndash77]

The TFRF can be evaluated in both deterministic andprobabilistic forms and the two approaches are discussedin Sections 41 and 42 respectively On the other handthe deterministic and probabilistic check procedures will bepresented in Section 5

The sources of uncertainty and error were discussed inSections 21 and 3 The interested reader is referred to [10]for a discussion on the sensitivity of footbridge vibrations tostochastic walking parameters

41 Deterministic Approach The modal response to a har-monic load crossing a beam at a constant speed can beobtained as the superposition of forced and of free-decayresponses Different authors derived the closed-form solution


005

005

005

005

01

01

0101

02

02 02

0404 04

0707 0909

005005

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005 005

0101

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0404

040707

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0

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1

50 100 150 2000L

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16V

04

06

08

1

12

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16

L

50 100 150 2000L

01

02

V(V L V = 0015) L(L L L = 0015)

Figure 5 120593119881(120572119881 119871 120585119881 = 0015) and 120593119871(120572119871 119871 120585119871 = 0015) for the simply supported beam evaluated using (19)

Table 2 Limits of validity and availability of closed-form peakmodal response from different authors

Reference 120572 RL119894 Peak modal responseAbu-Hilal and Mohsen [74] All AllRicciardelli and Briatico [75] All 0 radicPiccardo and Tubino [76] 1 All radic

to this problem First Abu-Hilal and Mohsen [74] deriveda closed-form solution of the resonant and nonresonantresponse for any support condition Subsequently limited tothe simply supported beam Ricciardelli and Briatico [75]derived an approximated closed-form solution of the reso-nant and nonresonant response together with a closed-formsolution of the peak modal response Then Piccardo andTubino [76] derived an approximated closed-form solution

only of the resonant response but for any support conditionstogether with a closed-form solution of the peak modalresponse A synthesis of the limits of validity and of theavailability of closed-form peak modal response for differentauthors is reported in Table 2

In the following the case of a simply supported beam(RL119894 = 0) will be analyzed in Section 411 while the effectsof the change of the support conditions will be described inSection 412

411 Simply Supported Beam (RL119894 = 0) For the case of asimply supported beam different closed-form solutions forthe TFRF are available in the literature (see Table 2) and acomparison for the resonant case can be found in [5 34]Themost accurate solution proves to be that of Ricciardelli andBriatico [75]

120593119894 (RL119894 = 0 120572119894 119871 120585119894) = min

120585119894 sdot 2120572211989410038161003816100381610038161 minus 1205722119894 1003816100381610038161003816 [1 +1119899119894

210038161003816100381610038161 minus 1205722119894 1003816100381610038161003816 sdot exp [minus120587 sdot 119899119894 (119871) sdot 1205851198942120572119894 ]]

119899119894 (119871) sdot 1205851198941 + 1198992119894 (119871) sdot 1205852119894 [radic1 + 1198992119894 (119871) sdot 1205852119894 + exp [minus119899119894 (119871) sdot 120585119894 (1205872 + arctan 1119899119894 (119871) sdot 120585119894)]]

(19)

where 119899119894(119871) is twice the number of load cycles imposed by thewalker to the footbridge and is defined as

119899119894 (119871) = 2 sdot 119891119908119894V

sdot 119871 = 270mminus1 sdot 119871 for 119894 = 119881135mminus1 sdot 119871 for 119894 = 119871 (20)

Equation (19) for the resonant case is quite consistent withthat of Piccardo and Tubino [76]

An example of the vertical and lateral TFRF for 120585119894 =0015 is shown in Figure 5 where it is visible that themaximum response is obtained for the resonant conditionsAt resonance the TFRF increases increasing the span lengthdue to the larger number of loading cycles In particular theTFRF increases with 119871 only for relatively low values of 119871(around 20m for 119894 = 119881 and 40m for 119894 = 119871) while for largerbeam lengths the TFRF remains constant Out of resonancethe two figures are almost identical


412 Effects of Different Support Conditions (RL119894 gt 0)Change of support conditions influences mode shapes andthus acceleration response In order to account for generalsupport conditions a boundary factor BF119894 is defined as theratio of the modal peak acceleration on beam with arbitraryRL119894 to that of a simply supported beam (ie for RL119894 = 0)

BF119894 (RL119894 120572119894 119871 120585119894) = 120578119894 (RL119894 120572119894 119871 120585119894)120578119894 (RL119894 = 0 120572119894 119871 120585119894) (21)

Substituting (17) in (21) it can be observed that the boundaryfactor can be split into two parts

BF119894 (RL119894 120572119894 119871 120585119894) = BF119898119894 (RL119894) sdot BF120593119894 (RL119894 120572119894 119871 120585119894)= 119898119894 (RL119894 = 0)

119898119894 (RL119894)sdot 120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)

(22)

where 120593119894(RL119894 = 0 120572119894 119871 120585119894) and 119898119894(RL119894 = 0) are the TFRF(see (19)) and the modal mass of a simply supported beamand 120593119894(RL119894 120572119894 119871 120585119894) and119898119894(RL119894) are those for a generic sym-metric support conditions (ie arbitrary value of RL119894) Thederivation of (22) is given inAppendixC3 According to (22)BF119894(RL119894 120572119894 119871 120585119894) accounts for the variation of modal massthrough a mass boundary factor BF119898119894(RL119894) and of modeshape through a TFRF boundary factor BF120593119894(RL119894 120572119894 119871 120585119894)

The variation of modal mass with RL119894 that is BF119898119894(RL119894)was shown in Figure 3 as 119898119894(RL119894)119898119894(RL119894 = 0) that is theinverse of BF119898119894(RL119894) This contribution only depends on themode shape with increasing RL119894 the modal mass reducesThe second term in (22) BF120593119894(RL119894 120572119894 119871 120585119894) depends on allthe variables as in (19)

Using the definition of BF120593119894(RL119894) given in (22) the TFRFfor general support conditions can be expressed as

120593119894 (RL119894 120572119894 119871 120585119894) = BF120593119894 (RL119894 120572119894 119871 120585119894)sdot 120593119894 (RL119894 = 0 120572119894 119871 120585119894)

(23)

At resonance BF119894(RL119894 120572119894 = 1 119871 120585119894) can be evaluatedusing the solution of Piccardo and Tubino [76] while fornonresonant conditions the numerical solution of Abu-Hilaland Mohsen [74] must be used (see Table 2)

In the following the trend of the boundary factor is onlyreported for the vertical direction for the sake of brevityalthough similar conclusions apply to the case of lateralvibrations The variation of BF119881(RL119881 120572119881 = 1 119871 120585119881 = 0015)(solid lines) and BF120593119881(RL119881 120572119881 = 1 119871 120585119881 = 0015) (dashedlines) is shown in Figure 6 for RL119881 equal to 0 (simplysupported beam) 025 05 075 and 1 (clamped beam) Thevalue of BF120593119881 increases with reducing rotational stiffness119870119903119881 (ie reducing RL119881) The increase of the modal peakacceleration with increasing RL119881 (ie increasing119870119903119881) is dueto the choice of considering the mass per unit length 119898and the natural frequency 119891119881 as input parametersThe latterresult derives from having calculated the modal masses (ie

119898119881(RL119881) and 119898119881(RL119881 = 0)) starting from the same valuesof mass per unit length 119898 Accordingly for the same valueof 119898 the increase of RL119881 leads to a decrease of modal mass119898119881 as reported in Figure 3 The reduction of modal massis followed by a reduction of stiffness of the beam 119864119868119881 (see(8)) to keep 119891119881 constant being the frequency proportionalto the ratio of the stiffness and modal mass Accordingly thereduction of stiffness brings an increase of the peak responsethus explaining the increase of BF119881(RL119881 120572119881 = 1 119871 120585119881) withincreasing of RL119881

On the other hand if the modal mass119898119881 (instead of themass per unit length 119898) and the natural frequency 119891119881 areused as input parameters the beam will be characterized bythe same stiffness 119864119868119881 so as to have the same value of 119891119881If the same modal mass is considered (ie 119898119881(RL119881) is thesame for all RL119881) then BF119898119881(RL119881) = 1 Accordingly thepeak response is reduced with increasing RL119881 as intuitivelyit should behave This can be observed looking only at theeffect of RL119881 on the TFRF that is BF120593119881(RL119881 120572119881 = 1 119871 120585119881)(dashed line) in Figure 6 since BF119898119881(RL119881) = 1 In this casethe peak response reduction increases with increasing RL119881

Away from resonance the variation of BF119881(RL119881 =1 120572119881 119871 120585119881 = 0015) is also shown in Figure 6 BF119881 showsa marked dependence on 119871 for 119871 le 50m whereas it is almostconstant for 119871 ge 50m especially away from resonance

Finally BF119894(RL119894 = 1 120572119894 119871 120585119894) for different values of 120585119894 isreported in Appendix A for the vertical and lateral directions

42 Probabilistic Approach ThePDF of the peak accelerationas a function of the footbridge characteristics (ie 120572119894 119871 and120585119894) is obtained through the convolution of the response for thevertical and lateral directions

119901120578119894 (120578119894 (120572119894 119871 120585119894)) = intinfin0

intinfin0

intinfin0

intinfin0

119901120578119894 119882DLF119894 V119891119882119894sdot d119882 sdot 119889DLF119894 sdot 119889V sdot 119889119891119908119894

(24)

where the PDF of the peak acceleration reported in theintegral can be found applying the chain rule

119901120578119894 119882DLF119894 V119891119882119894 = 119901120578119894 (120578119894 (120572119894 119871 120585119894) | 119882DLF119894 V 119891119908119894)sdot 119901119882 (119882) sdot 119901DLF119894 (DLF119894 | 119891119908119894)sdot 119901119891119908119894 (119891119908119894 | V) sdot 119901V (V)

(25)

where 119901119860(119886 | 119887) is the conditional probability of 119886 given119887 All the variables describing the SP (see Section 21) arestatistically independent except for V and 119891119908119881 Accordinglyin the derivation of (25) only the conditional probability of119891119908119894 given V and DLF119881 given 119891119908119881 was considered

The multiple integral in (24) combined with (25) wassolved throughMonte Carlo simulations using the followingprocedure in the vertical and lateral directions

(1) Evaluation of the integral for 03 le 120572119894 le 17 (step 01)10m le 119871 le 200m(step 10m) and 0001 le 120585119894 le 0015(step 0001) and for RL119894 = 0 and 1

(2) Random generation of 10000 walkers whose charac-teristics (119882 DLF119894 V and 119891119908) follow the PDFs of theSP of Section 21


２V = 000２V = 025２V = 050

２V = 075２V = 100

07

075

08

085

09

095

1

105

11

115

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125

13V

50 100 150 2000L

12

12

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125125

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12513

1

105

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135

14

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04

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50 100 150 2000L

V(２V V = 1 L V = 0015) V(２V = 1 V L V = 0015)

Figure 6 Boundary factor BF119881(RL119881 120572119881 = 1 119871 120585119881 = 0015) (solid lines) and TFRF boundary factor BF120593119881(RL119881 120572119881 = 1 119871 120585119881 = 0015)(dashed lines) for different values of RL119881 evaluated using the solution of Piccardo and Tubino [76] BF119881(RL119881 = 1 120572119881 119871 120585119881 = 0015) evaluatednumerically using the solution of Abu-Hilal and Mohsen [74]

(3) Evaluation of the modal vertical and lateral acceler-ation time histories 120578119894(119905) for RL119894 = 0 and RL119894 = 1using the model of Abu-Hilal and Mohsen [74] (see(12)) for the population of walkers generated in (2) (atime step of 001 s was used in the simulations)

(4) For each walker evaluation of the lateral and verticalpeak acceleration and evaluation of 120593119894(RL119894 120572119894 119871 120585119894)(see (17))

(5) Evaluation of the empirical PDF of the TFRF119901120593119894(120593119894(RL119894 120572119894 119871 120585119894))(6) Fit of the empirical PDF to a GEV distribution

The distributions of the TFRF are fitted using the GEVdistribution with the following CDF

119875120593119894 (120593119894 (RL119894 120572119894 119871 120585119894))= expminus[1 + 119896119894 (120593119894 minus 120583119894120590119894 )]minus1119896119894 (26)

where the location parameter 120583119894 the scale parameter 120590119894 andthe shape parameter 119896119894 are all dependent on RL119894120572119894 119871 and 120585119894A preliminary convergence study showed that the number of10000 simulations ensures a good accuracy and repeatabilityof the results

The estimated GEV parameters for lateral and verticalvibrations for supported (RL119894 = 0) and clamped (RL119894 = 1)beams are shown in Appendix B In all the cases investigatedthe GEV parameters were found to depend on 120572119894 and 120585119894

whereas a dependency on 119871 was found only for 119871 le 20mfor lateral vibrations and for 119871 le 40m for vertical vibrationsThe explanation of this difference can be found observing theresults of the deterministic approach reported in Section 41Observing Figure 5 it can be seen that the variation of 120593119894 isnoticeable up to 50m while for larger values it is negligibleAccordingly also the probabilistic model exhibits small vari-ations of the GEV parameters for values of 119871 associated with119871 le 50m In Appendix B the GEV parameters are shown for119871 = 10m 20m 30m and 40m and for 119871 in excess of 50mthe latter are the average value obtained varying 119871 from 50mto 200m In order to verify the accuracy of the mean GEVparameter representation the STD of the GEV parametersin the range of 119871 from 50m to 200m was calculated findingvery low values of the standard deviation (compared withthe values assumed by the variables reported in Appendix B)1 times 10minus2 for 120583119894 2 times 10minus3 for 120590119894 and 2 times 10minus3 for 119896119894

In all the cases investigated 119896119894 broadly ranges from minus04to 12 taking the largest values around resonance Moreoverat resonance 119896119894 slightly increases with decreasing structuraldamping while it is approximately constant away fromresonance The maximum value of 119896119894 is reached for 120572 = 09for the lowest structural damping 119896119894 takes approximately thesame values for RL = 0 and RL = 1 Globally 119896119881 takes largervalues compared with 119896119871 with small exceptions 119896119881 is lessdependent on damping than 119896119871 The variation of 119896119881 and 119896119871with 119871 is in agreement with the previous observations

In all the cases investigated 120590119894 and 120583119894 broadly range from0 to 02 taking larger values at resonance and both increasing


Table 3 Probabilistic design values 120573119863[119875119863] after ISO 2394 [37]

Relative costs of safety measures Consequences of failureSmall Some Moderate Great

High 0 [5 sdot 10minus1] 15 [7 sdot 10minus2] 23 [1 sdot 10minus2] 31 [1 sdot 10minus3]Moderate 13 [1 sdot 10minus1] 23 [1 sdot 10minus2] 31 [1 sdot 10minus3] 38 [7 sdot 10minus5]Low 23 [1 sdot 10minus2] 31 [1 sdot 10minus3] 38 [7 sdot 10minus5] 43 [8 sdot 10minus6]

with damping Globally 120590119871 is larger than 120590119881 for 120572119894 gt 1especially for low damping and the two quantities are quitesimilar for 120572119894 lt 1 For both vertical and lateral directionsboth 120590119894 and 120583119894 are larger for the clamped case than for thesupported case this is in agreement with the variation of BF119894(Figures 7 and 8)The same observations can bemade regard-ing the dependence on 119871 of 120590119894 and 120583119894 as those made for 1198961198945 Reliability Analysis Deterministic andProbabilistic Approaches

In this section first (deterministic) and third (probabilistic)level safety assessments methods will be applied In particu-lar it is assessed that the acceleration induced by the crossingof a walker is not causing loss of comfort for a receiver

In the application of the first level safety assessmentmethod it is assumed that the basic variables are summarizedinto a deterministic value of capacity (themaximum tolerableacceleration) 119862119894 and a deterministic value of demand (thepeak acceleration induced by a single walker) 119863119894 Capacityand demand are expressed in terms of TFRFs

119863119894 = 120593119894 (RL119894 120572119894 119871 120585119894) le 120593119894max = 119862119894 forall119894 (27)

where 119863119894 is expressed as in (23) and the limit acceleration isexpressed in terms of equivalent TFRF

119862119894 = 120593119894max

= (BF119898119894 (RL119894) sdot DLF119894 sdot 1198822120585119894 sdot 1119898119894)minus1 120578119894max

(28)

where119898119894 is the modal masses evaluated considering RL119894 = 0(simply supported beam) BF119898119894(RL119894) is the mass boundaryfactor correcting the modal masses for support conditionand 120578119894max is the maximum tolerable acceleration

Application of the third level method requires eithernumerical integration or approximate analytical methods(such as first- and second-order reliability methods) orsimulation methods [78] the latter was used in this study(Section 42) Given the Joint Probability Density Function(JPDF) 119901119862119863(119888 119889) of capacity and demand then the proba-bility of failure 119875fail119894 is given as

119875fail119894 = int119862minus119863le0

119901119862119894 119863119894 (119888119894 119889119894) sdot d119888119894 sdot d119889119894 (29)

119862119894 has a negligible variability compared to 119863119894 then itis reasonable to consider it as deterministic Under this

assumption 119875fail119894 can be evaluated as the ComplementaryCumulative Distribution Function (CCDF)

119875fail119894 = 1 minus int120593119894max

0119901120593119894 (120593119894 (RL119894 120572119894 119871 120585119894)) sdot d120593119894 (30)

where the integration limit is equal to themaximum tolerableacceleration reported in the capacity value (see (28))

Equivalently the reliability index can be considered

120573119894 = minusΦminus1119880 (119875fail119894) (31)

whereΦminus1119880 denotes the inverse standardized normal distribu-tion function

The acceptance condition is based on the requirementthat the probability of failure does not exceed the designvalue 119875119863119894 or that the reliability index is greater than itscorresponding design value 120573119863119894 = minusΦminus1119880 (119875119863119894) [37]

119875fail119894 le 119875119863119894 lArrrArr120573119894 ge 120573119863119894 (32)

In Eurocode 0 [36] and ISO 2394 [37] reliability require-ments are expressed in terms of the reliability index 120573119863these are related to the expected social and economic conse-quences In particular ISO 2394 [37] gives the probabilisticdesign values as a function of the relative costs of safetymeasures and of the consequence of failure (Table 3)

The consequence classes in Table 3 are quantified throughthe ratio between the failure costs and the costs of construc-tion In the case of footbridge vibrations the consequenceof vibrations is small as no damage to things or peopleoccurs ISO 2394 [37] suggests 120573119863 = 0 [119875119863 = 5 sdot 10minus1] forreversible limit states (Table 3) However based on perfor-mance demand and on the outcome of cost-benefit analysesthe designer may find it appropriate to increase or reducethe probabilistic design values that is to increase or reducereliability As reported by ISO 2394 [37] the probabilisticdesign values are ldquoformal or notional numbers intendedprimarily as a tool for developing consistent design rules ratherthan giving a description of the structural failure frequencyrdquoIn the same manner Eurocode 0 [36] gives a target reliabilityindex 120573119863 of 15 for serviceability irreversible limit state notproviding any indication for the serviceability reversible one

For both levels of assessment (deterministic and prob-abilistic) it is necessary to quantify capacity that is to setthe maximum tolerable acceleration reference values canbe found in the literature [8] The interested reader is alsoreferred to [79 80] for further information ISO 10137 [27]states that ldquothe designer shall decide on the serviceability


11 115115

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V = 0001 V = 0003

V = 0005 V = 0007

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05

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L

05

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V

0 50 200100 150

L

Figure 7 BF119881(RL119881 = 1 120572119881 119871 120585119881) for different values of 120585119881 numerically evaluated using the solution of Abu-Hilal and Mohsen [74]


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L = 0001 L = 0003

L = 0007

L = 0011

L = 0015

L = 0005

L = 0009

L = 0013

50 100 150 2000

L

50 100 150 2000

L

50 100 150 2000

L

50 100 150 2000

L

50 100 150 2000

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criterion and its variabilityrdquo Further ISO 10137 [27] states thatfootbridges ldquoshall be designed so that vibration amplitudesfrom applicable vibration sources do not alarm potential usersrdquoIn particular Annex C to ISO 10137 [27] provides limitingcriteria for human comfort on footbridges in the range offrequency of 1 to 80Hz The capacity is expressed in termsof RMS acceleration evaluated on a 1 s window (runningRMS method ISO 2631 [81]) the limit values are given asa function of the vibration frequency (base curves) for thelateral (side to side and back to chest) and vertical directionsFor footbridges the base curvesmust bemultiplied by a factorof 30 in the vertical directions and by a factor of 60 in thelateral direction The factor of 30 applies only to pedestriansstanding still on the bridge because sensitivity to vibrationdecreases when walking Finally the results of the numericalsimulations show that for the range of the input parametersconsidered the ratio between the peak values and the 1 sRMS peak values of the acceleration (Maximum TransientVibration Value ISO 2631 [81]) is around radic2 (113 to 157 inthe lateral direction and 130 to 151 in the vertical direction)similar to the peak factor of a stationary sinusoidal process

The maximum tolerable acceleration is therefore evalu-ated from the ISO 10137 base curves

120578119881max (120572119881) = radic2 sdot 30 sdot 120578119881rms (120572119881)

= 021 if 120572119881 le 0470140 + 0150 sdot 120572119881 if 120572119881 gt 047


= 1252 minus 2575 sdot 120572119871 if 120572119871 le 047030 if 120572119871 gt 047

(33)

where 120578119881max(120572119881) and 120578119871max(120572119871) are the peak accelerationthresholds and 120578119881rms(120572119881) and 120578119871rms(120572119871) are the correspond-ing 1 s RMS peak acceleration thresholds (base curves) inms2 The acceleration thresholds are given in terms of gen-eralized coordinates (ie 120578119881max(120572119881) and 120578119871max(120572119871)) whichapplies to the case where mode shapes are normalized to one(Section 3)

Equations (33) are shown in Figure 9 limited to thefrequency range of interest for footbridges For120572119871 ge 119891119908119871 (ie119891119871 le 1Hz) the acceleration threshold has been extrapolatedto a constant value For the resonant case where the acceler-ation is larger the vertical and lateral acceleration thresholdsare similar and equal to 120578119881rms(120572119881 = 1) = 029ms2 and120578119871rms(120572119871 = 1) = 030ms2 respectively In the vertical direc-tion 120578119881rms(120572119881) increases in almost the entire range of 120572119881In the lateral direction 120578119871rms(120572119871) is taken as constant for120572119871 ge 047 and strongly increases for 120572119871 lt 047 reducing therisk of loss of comfort

6 Application to a Steel Truss Footbridge

As an example the proposed procedure is applied to a proto-type steel truss footbridge with an overall length of 10190m

Table 4 Dynamic characteristics of the truss footbridge119898119881 and119898119871are evaluated considering a simply supported beam that is RL119894 = 0Variable Dimension 119894 = 119881 119894 = 119871119871 [m] 90119898 [kgm] 1495119898119894 [kg] 67275 67275120585119894 [] 05 05120572119894 [ndash] 1061 0507RD119894 [ndash] 2694 0357RL119894 [ndash] 048 0975BF119894 [ndash] 1 121BF119898119894 [ndash] 1 125BF120593119894 = (BF119894BF119898119894) [ndash] 1 097

(central span of 9000mwith two lateral cantilevers of 620mand 570m resp) a width of 350m and a height of truss of445m The structural members designed according to theItalian Code provisions [82] are made of grade S355 steel

The footbridge was modeled using the FE software pack-age SAP2000 [83] as a 3D truss beam simply supported atfour nodes In particular fixed bearings were placed on oneside of the beam and longitudinally sliding bearings on theother sideThemeanmass per unit of length of the footbridgeis 1495 kgm corresponding to a modal mass of 67275 kgin both directions considering simply support (RL119894 = 0)conditions Damping ratio was set to 05 in both directionsModal analysis was performed and the first two modes ofvibration were found to be the first vertical bending mode(119891119881 = 1789Hz) and the first lateral bending mode (119891119871 =1873Hz) respectively (Figure 10) The natural frequenciescorrespond to 120572119881 = 106 and 120572119871 = 051 both indicatingpossible walking-induced vibrations

To evaluate the actual support condition in terms of RL119894using (16) evaluation of RD119894 defined through (15) is requiredthis was done by applying a uniform load in lateral andvertical directions to the FE model and evaluating the corre-sponding end rotation 120579119894 andmidspan deflection 120575119894 It shouldbe pointed out that the uniform load coming from the deck isallocated to the nodes of the FE model using a tributary areaload criterion Finally RD119894 was converted into an equivalentRL119894 using (16) A synthesis of the footbridge dynamic charac-teristic is given in Table 4 The RL (Table 4) correspond to aclamped beam in the lateral direction and to an intermediatecondition between the simply supported and clamped beamfor the vertical direction In this application considering thatthe boundary factor corresponding to RL119894 = 05 is quite simi-lar to that corresponding to RL119894 = 0 (Figure 6) the conditionRL119871 = 1 is used for the lateral direction and RL119881 = 0 for thevertical direction The last assumption is also made becausethe procedure proposed in this study is only given for RL119894 = 0and RL119894 = 1 The variation of the support conditions withrespect to the simply supported case in the lateral directionmakes it necessary to evaluate BF119871 BF119898119871 and BF120593119871 In par-ticular from Figure 8 one can find BF119871 = 121 while BF119898119871 =125 was found from Figure 3 Moreover BF120593119871 = 0968 wasestimated dividing BF119871 by BF119898119871 using the definition of (22)


316fL

05606307609512719

0

05

1

15

2

25

3

35

4

45

5

L

max

075 1 12505 15 1703L

633fV

1121271521925338

0

005

01

015

02

025

03

035

04

045

05

V

max

075 1 12505 15 1703V

Figure 9 Threshold vertical and lateral peak acceleration using (33) The red dashed line indicates extrapolated values for 119891119871 le 1Hz Greenareas indicate the region of acceptable comfort

FEM model 1st vertical mode 1st lateral mode

Figure 10 3D SAP2000 Finite Element model of the truss footbridge and first lateral and vertical mode shapes

The limit acceleration was calculated according to ISO10137 [27] using (33) (see Figure 9) and the design valueof 119875119863 and 120573 considering small consequences of failure andmoderate relative costs of safety measures according toISO 2394 [37] (see Table 3) For both vertical and lateraldirections

120578max = 03ms2119875119863 = 01120573119863 = minusΦminus1119880 (119875119863) = 13

(34)

The acceleration capacity is expressed in terms of equiva-lent TFRFs using (28)

119862119881 = (BF119898119881 sdot DLF119881 sdot 11988221205851198811119898119881)minus1

sdot 120578119881max = 0775


sdot 120578119871max = 5723(35)

The acceleration demand TFRFs are evaluated for thesimply supported beam using (19) and these values arecorrected to account for themode shape using BF as reportedin (23)

119863119881 = 120593119881 (RL119881 120572119881 119871 120585119881) = 1 sdot 00904 = 00904119863119871 = 120593119871 (RL119871 120572119871 119871 120585119871) = 097 sdot 00035 = 00034 (36)

Thedeterministic check is performed using the inequalitycondition expressed in (27)

119863119881 = 00904 lt 0775 = 119862119881119863119871 = 00034 lt 5723 = 119862119871 (37)

Using the deterministic approach it is found that max-imum acceleration induced by the SW is lower than theacceptable threshold In the lateral direction the demand ismuch smaller than the capacity indicating large reliabilityConversely in the vertical direction the demand and thecapacity are closer showing lower safety


The probabilistic reliability analysis is based on thedefinition of the GEV parameters (see (26)) these are givenin Appendix B The lateral GEV parameters are evaluatedconsidering RL119871 = 1 and using Figure 11 whereas the verticalGEV parameters are evaluated considering RL119881 = 0 andusing Figure 12

119896119881 (RL119881 = 0 120572119881 119871 120585119881) = 06718120583119881 (RL119881 = 0 120572119881 119871 120585119881) = 005067120590119881 (RL119881 = 0 120572119881 119871 120585119881) = 005588119896119871 (RL119871 = 1 120572119871 119871 120585119871) = minus00129120583119871 (RL119871 = 1 120572119871 119871 120585119871) = 0001828120590119871 (RL119871 = 1 120572119871 119871 120585119871) = 0003309

(38)

The value of the CDF corresponding to TFRF predictedusing the deterministic procedure (ie (36)) is equal to

119875120593119881 (119863119881) = 0565119875120593119871 (119863119871) = 0379 (39)

The PDF and CDF of 120593119894 for the vertical and lateraldirection evaluated using the GEV parameters reported in(38) are shown in Figure 13 together with the deterministicacceleration demand TFRFs (see (36))This result shows thatusing the SW (ie walker having the mean characteristicsof the SP) the cumulative probability is either higher orlower than themedian of the probability distribution In otherwords this means that using mean characteristics of the SPcharacteristics can lead to unconservative evaluations as theobtained probability of exceeding can be different from themedian that is 05119875fail119894 can be evaluated as the Complementary CumulativeDistribution Function (CCDF) associated with a TFRF equalto either 119862119881 or 119862119871

119875fail119881 = 00296 lArrrArr120573119881 = 1887

119875fail119871 997888rarr 00 lArrrArr120573119871 997888rarr infin

(40)

The 0 probability of failure in the lateral direction of(40) is due to the negative values of 119896119871 (see (38)) leading toa reversed Weibull distribution that is characterized by anupper limit Accordingly also the reliability index assumesinfinitive value

The acceptance condition is based on a requirement thatthe probability of failure 119875fail119894 does not exceed the design

value 119875119863119894 or the reliability index 120573119894 is greater than its designvalue 120573119863119894 [37]

119875fail119881 = 00296 lt 01 = 119875119863119881 lArrrArr120573119881 = 1887 gt 13 = 120573119863

119875fail119871 = 00 lt 01 = 119875119863119871 lArrrArr120573119871 = infin gt 13 = 120573119863

(41)

Application of the probabilistic approach shows that theprobability of failure is lower than the design probability Inparticular in the lateral direction the check is satisfied in allcases (it can be considered a deterministic safety condition)while in the vertical direction the ratio between the capacityand the demand is lower Approximately in 296 (see (40))of the cases the vertical acceleration induced by a singlewalker crossing the footbridge is larger than the thresholdvalue

7 Conclusions

Prior work has evaluated the dynamic effects induced by asingle walker crossing a footbridge The research availablefocuses on the definition of force characteristics and on therelated dynamic response No complete procedure is availablefor the probabilistic assessment of footbridges against thecrossing of single walkers

In this paper a procedure for the deterministic andprobabilistic assessment of footbridges against the crossingof single walkers was presented This scenario is consideredby different standards and design guidelines such as ISO10137 The procedure presented complies with Eurocode 0[36] and ISO 2394 [37] and allows controlling the reliabilitylevel The flow diagram of the procedure is given in Figure 4First the definition of the walker and footbridge dynamiccharacteristics is needed a study of the data available in theliterature allowed the definition of a Standard Populationof walkers Only walking conditions were considered Thecommon beam-type footbridge characteristics were evalu-ated and a dynamic modal model according to Abu-Hilaland Mohsen [74] was presented All the variables definingthe footbridge characteristics were chosen as such to obtaina simple procedure suitable for design implementation Thepeak acceleration was chosen as the engineering demandparameter and it was derived with both a deterministic and aprobabilistic approach Using the deterministic approach theserviceability check is carried out comparing the peak accel-eration induced by a walker having the mean characteristicsof the Standard Population with the acceleration thresholdsof ISO 10137 [27] With the probabilistic approach theexceedance probability of the threshold acceleration must belower than the reliability levels defined in ISO 2394 [37]Thisstudy therefore indicates that the use of the deterministicapproach without the knowledge of the real reliability levelscan lead to unconservative evaluations Finally applicationof deterministic and probabilistic approaches to a prototypesteel footbridge showed how it can be easily used in theengineering practice


0

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1505 1 1505 1

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L

kL - L = 50ndash200Ｇ kL - L = 10 Ｇ kL - L = 20 Ｇ

L - L = 10 Ｇ L - L = 20 Ｇ

kL - L = 30 Ｇ kL - L = 40 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ

L - L = 10 Ｇ L - L = 20 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ

L - L = 50ndash200Ｇ


Figure 11 GEV parameters of 119875120593119871(120593119871(RL119871 = 1 120572119871 119871 120585119871)) double clamped case in lateral direction


0

02

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002501 0025

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05 1 15

V ()

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1 1505

V ()1505 1

V ()

1 15051 1505

1 1505 1 1505

1505 115105

1 150505 151

1 1505 105 15

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V

04

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1

12

14

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V

kV - L = 50ndash200Ｇ kV - L = 10 Ｇ kV - L = 20 Ｇ

V - L = 10 Ｇ V - L = 20 Ｇ

kV - L = 30 Ｇ kV - L = 40 Ｇ

V - L = 30 Ｇ V - L = 40 Ｇ

L = 10 Ｇ L = 20 Ｇ

L = 30 Ｇ L = 40 Ｇ

V - L = 50ndash200Ｇ

L = 50ndash200Ｇ V - V -

V -V -

V -

Figure 12 GEV parameters of 119875120593119881(120593119881(RL119881 = 0 120572119881 119871 120585119881)) simply supported case in vertical direction


0

02

04

06

08

1

CDF

0005 001 0015 00200

02

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08

1

0

2

4

6

8

10

PDF

104 06 080200

50

100

150

200

250

V [mdash] L [mdash]

Figure 13 Probability Density Function (PDF thin line) and Cumulated Distribution Function (CDF thick line) of 120593119894 for the vertical andlateral direction (GEV parameters are reported in (38)) The vertical dashed line indicates the deterministic acceleration demand TFRFs119863119881and119863119871 (see (36))

Appendix

A Boundary Factor for the ClampedCase RL = 1

See Figures 7 and 8

B GEV Parameters for RL = 1 and RL = 0See Figures 15 and 11 for RL = 1 See Figures 12 and 14 forRL = 0C Derivation of (14) (16) and (22)

C1 Derivation of (14) Let us consider a beam supportedat both ends by fixed vertical constraints and with variablerotational stiffness (Figure 2) In this study only symmetricsupport conditions will be considered

1198701199031119894 = 1198701199032119894 = 119870119903119894 (C1)

where the subscript 119894 = 119881 119871 indicates the vertical and lateraldirection respectively The following derivation will be donefor a generic direction 119894

If we consider the beam loaded by an arbitrary symmetricload the rotation at the two ends will be the same

Θ1119894 = Θ2119894 = Θ119894 (C2)

Let us define a nondimensional restrained level (RL119894)defined as the ratio of the end moment for the particularsupport condition (ie value of 119870119903119894) due to an arbitrarysymmetric load 119872119894(119870119903119894) to that of the clamped beamsubjected to the same load119872119894(119870119903119894 rarr infin)

RL119894 equiv 119872119894 (119870119903119894)119872119894 (119870119903119894 rarr infin) =119872119894 (RL119894)119872119894 (RL119894 rarr 1) (C3)

Therefore RL119894 is in the range of 0 (supported case 119870119903119894 =0) to 1 (clamped case 119870119903119894 rarr infin) The bending moment atthe two ends can be expressed according to

119872119894 (RL119894) = 119870119903119894 (RL119894) sdot Θ119894 (RL119894)= RL119894 sdot 119872119894 (RL119894 997888rarr 1) (C4)

Using the assumptions of symmetric load and boundaryconditions the rotation at the end for a generic value of RL119894Θ119894(RL119894) can be written as

Θ119894 (RL119894) = Θ119894 (RL119894 = 0) minus 119872119894 (RL119894) 1198713119864119868119894 minus 119872119894 (RL119894) 1198716119864119868119894= Θ119894 (RL119894 = 0) minus 119872119894 (RL119894) 1198712119864119868119894

(C5)

where Θ119894(RL119894 = 0) is the rotation at one end of a simplysupported beam and119872119894(RL119894) is the bending moment at oneend (the two bending moments at the two ends are the samedue to the symmetry in the load and constraints)

Substituting (C4) in (C5) the following can be obtained

Θ119894 (RL119894) = Θ119894 (RL119894 = 0) minus 119872119894 (RL119894) 1198712119864119868119894= Θ119894 (RL119894 = 0) minus 119870119903119894 sdot Θ119894 (RL119894) sdot 1198712119864119868119894

(C6)

Then solving by Θ119894(RL119894) the following relation as afunction of 119870119903119894 can be found

Θ119894 (RL119894) = Θ119894 (RL119894 = 0) sdot (1 + 2119864119868119894119870119903119894 sdot 119871) (C7)



L - L = 10 Ｇ L - L = 20 Ｇ

kL - L = 30 Ｇ kL - L = 40 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ

L - L = 10 Ｇ L - L = 20 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ



0

0

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minus02

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Figure 14 GEV parameters of 119875120593119871(120593119871(RL119871 = 0 120572119871 119871 120585119871)) simply supported case in lateral direction



V - L = 10 Ｇ V - L = 20 Ｇ

kV - L = 30 Ｇ kV - L = 40 Ｇ

V - L = 30 Ｇ V - L = 40 Ｇ

L = 10 Ｇ L = 20 Ｇ

L = 30 Ｇ L = 40 Ｇ


L = 50ndash200Ｇ

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Figure 15 GEV parameters of 119875120593119881(120593119881(RL119881 = 1 120572119881 119871 120585119881)) double clamped case in vertical direction


Substituting (C4) and (C7) in (C3) the restrained levelRL119894 can be expressed as a function of the rotational stiffness119870119903119894

RL119894 = 119870119903119894 sdot Θ119894 (RL119894)119872119894 (RL119894 rarr 1)= 119870119903119894Θ119894 (RL119894) sdot 21198641198681198942119864119868119894 + 119870119903119894 sdot 119871 sdot 1

119872119894 sdot (RL119894 rarr 1) (C8)

Solving (C8) by the rotational stiffness119870119903119894 the followingrelation as a function of the restrained level RL119894 can be found

119870119903119894 = 2 sdot 119864119868119894119871 sdot RL119894sdot 1((Θ119894 (RL119894 = 0) sdot 2119864119868119894) (119872119894 (RL119894 rarr 1) sdot 119871) minus RL119894)

(C9)

For a symmetric load applied on a symmetric constrainedbeam it can be simply demonstrated that

119872119894 (RL119894 997888rarr 1) = Θ119894 (RL119894 = 0) sdot 2119864119868119894119871 (C10)

Using the previous equation in (C9) the rotationalstiffness as a function of RL119894 can be expressed and simplifiedas

119870119903119894 = 2 sdot 119864119868119894119871 sdot RL119894(1 minus RL119894) (C11)

That is the demonstration of (14)

C2 Derivation of (16) Let us define a nondimensionalRotation-to-Deflection ratio (RD119894)

RD119894 = Θ119894 (RL119894) sdot 119871120575119894 (RL119894) (C12)

where Θ119894(RL119894) and 120575119894(RL119894) are the end rotation and themidspan deflection due to a uniform loadThe last deflectionterm can be expressed as sum of simply supported midspanbeam deflection 120575119894(RL119894 = 0) and midspan deflection due tothe end moments119872119894(RL119894 rarr 1) as follows

120575119894 (RL119894) = 120575119894 (RL119894 = 0) minus 119872119894 (RL119894) sdot 11987128119864119868119894 (C13)

Substituting (C7) (C11) and (C13) in (C12) the nondi-mensional Rotation-to-Deflection ratio can be expressed as

RD119894 = Θ119894 (RL119894 = 0) sdot (1 minus RL119894)sdot 119871120575119894 (RL119894 = 0) minus (RL119894 sdot 119872119894 (RL119894 rarr 1) sdot 1198712) 8119864119868119894

(C14)

Solving (C14) by the restrained level RL119894 the followingrelation as a function of the Rotation-to-Deflection ratio RD119894can be derived

RL119894

= (1 minus (RD119894 sdot 120575119894 (RL119894 = 0)) (Θ119894 (RL119894 = 0) sdot 119871))(1 minus (RD119894 sdot 119872119894 (RL119894 rarr 1) sdot 119871) (Θ119894 (RL119894 = 0) sdot 8119864119868119894))

(C15)

Finally substituting the expression of 120575119894(RL119894 = 0)Θ119894(RL119894= 0) and 119872119894(RL119894 rarr 1) due to a uniform load pattern theparameter RD119894 can be expressed more simply as

RL119894 = 4 minus 125 sdot RD1198944 minus RD119894 (C16)


C3 Derivation of (22) The peak modal acceleration 120578119894(RL119894120572119894 119871 120585119894) is expressed in terms of a TFRF (RL119894 120572119894 119871 120585119894)120578119894 (RL119894 120572119894 119871 120585119894) = DLF119894 sdot 1198822120585119894 sdot 1

119898119894sdot 120593119894 (RL119894 120572119894 119871 120585119894)

(C17)

Since change of support conditions influences modeshapes and thus acceleration response a boundary factorBF119894(RL119894 120572119894 119871 120585119894) is defined as

BF119894 (RL119894 120572119894 119871 120585119894) = 120578119894 (RL119894 120572119894 119871 120585119894)120578119894 (RL119894 = 0 120572119894 119871 120585119894) (C18)

The boundary factor is the ratio between the peak modalacceleration evaluated for arbitrary boundary conditions120578119894(RL119894 120572119894 119871 120585119894) and that evaluated for simply supportedconditions 120578119894(RL119894 = 0 120572119894 119871 120585119894) Substituting (C17) in (C18)the following can be obtained

BF119894 (RL119894 120572119894 119871 120585119894)= (DLF119894 sdot 119882) 2120585119894 sdot 1119898119894 (RL119894) sdot 120593119894 (RL119894 120572119894 119871 120585119894)(DLF119894 sdot 119882) 2120585119894 sdot 1119898119894 (RL119894 = 0) sdot 120593119894 (RL119894 = 0 120572119894 119871 120585119894)

(C19)

Rearranging the previous equation

BF119894 (RL119894 120572119894 119871 120585119894)

= (DLF119894 sdot 119882) 2120585119894(DLF119894 sdot 119882) 2120585119894sdot 1119898119894 (RL119894)1119898119894 (RL119894 = 0)

120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)= 119898119894 (RL119894 = 0)

119898119894 (RL119894)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟BF119898119894(RL119894)

sdot 120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟BF120593119894(RL119894 120572119894 119871120585119894)

(C20)

obtaining

BF119894 (RL119894 120572119894 119871 120585119894) = BF119898119894 (RL119894)sdot BF120593119894 (RL119894 120572119894 119871 120585119894)

= 119898119894 (RL119894 = 0)119898119894 (RL119894)

sdot 120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)

(C21)



Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The authors express their personal thanks and appreciationto the Consorzio Interuniversitario di Ricerca MeSE forhaving funded the publication and dissemination of theseresearch results

References

[1] H Petroski To Engineer Is Human The Role of Failure in Suc-cessful Design Vintage books New York New York NY USA1992

[2] B Wolmuth and J Surtees ldquoCrowd-related failure of bridgesrdquoProceedings of the Institution of Civil Engineers Civil Engineer-ing vol 156 no 3 pp 116ndash123 2003

[3] C Tilden ldquoKinetic effects of crowdsrdquoTransactions of the Ameri-can Society of Civil Engineers vol 76 no 1 pp 2107ndash2126 1913

[4] A Cunha and C Moutinho ldquoActive control of vibrations inpedestrian bridgesrdquo in Proceedings of the Conference of theEuropean Association for Structural Dynamics (Eurodyn rsquo99)1999

[5] F Ricciardelli and C Demartino ldquoDesign of footbridges againstpedestrian-induced vibrationsrdquo Journal of Bridge EngineeringArticle ID C4015003 pp 1ndash13 2016

[6] K Van Nimmen G Lombaert G De Roeck and P Van denBroeck ldquoVibration serviceability of footbridges evaluation ofthe current codes of practicerdquo Engineering Structures vol 59pp 448ndash461 2014

[7] J Wheeler ldquoPrediction and control of pedestrian-inducedvibration in footbridgesrdquo Journal of the Structural Division vol108 no 9 pp 2045ndash2065 1982

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

[9] E T Ingolfsson C T Georgakis and J Jonsson ldquoPedestrian-induced lateral vibrations of footbridges a literature reviewrdquoEngineering Structures vol 45 pp 21ndash52 2012

[10] L Pedersen and C Frier ldquoSensitivity of footbridge vibrations tostochastic walking parametersrdquo Journal of Sound and Vibrationvol 329 no 13 pp 2683ndash2701 2010

[11] A Avossa R Di Camillo and P Malangone ldquoPerformance-based retrofit of a prestressed concrete road bridge in seismicarea Life-Cycle and Sustainability of Civil Infrastructure Sys-temsrdquo in Proceedings of the 3rd International Symposium onLife-Cycle Civil Engineering (IALCCE rsquo12) CRC Press ViennaAustria 2012

[12] A M Avossa and P Malangone ldquoSeismic retrofit of a pre-stressed concrete road bridgerdquo Ingegneria Sismica (InternationalJournal of Earthquake Engineering) vol 29 no 4 pp 5ndash25 2012

[13] C Demartino and F Ricciardelli ldquoAerodynamic stability of ice-accreted bridge cablesrdquo Journal of Fluids and Structures vol 52pp 81ndash100 2015

[14] C Demartino H H Koss C T Georgakis and F RicciardellildquoEffects of ice accretion on the aerodynamics of bridge cablesrdquoJournal of Wind Engineering amp Industrial Aerodynamics vol138 pp 98ndash119 2015

[15] C Demartino JWu and Y Xiao ldquoExperimental and numericalstudy on the behavior of circular RC columns under impactloadingrdquo Procedia Engineering vol 199 pp 2457ndash2462 2017

[16] C Demartino J Wu and Y Xiao ldquoResponse of shear-deficientreinforced circular RC columns under lateral impact loadingrdquoInternational Journal of Impact Engineering vol 109 pp 196ndash213 2017

[17] J MW Brownjohn A Pavic and P A Omenzetter ldquoA spectraldensity approach for modelling continuous vertical forces onpedestrian structures due to walkingrdquoCanadian Journal of CivilEngineering vol 31 no 1 pp 65ndash77 2004

[18] A Pizzimenti and F Ricciardelli ldquoExperimental evaluation ofthe dynamic lateral loading of footbridges by walking pedestri-ansrdquo in Proceedings of the EURODYN rsquo05 vol 1 pp 435ndash4402005

[19] F Ricciardelli and A D Pizzimenti ldquoLateral walking-inducedforces on footbridgesrdquo Journal of Bridge Engineering vol 12 no6 pp 677ndash688 2007

[20] C Butz Beitrag zur Berechnung fuszligganger induzierte Bruck-enschwingungen [PhD thesis] Shaker Verlag Aachen AachenGermany 2006

[21] C Heinemeyer C Butz A Keil et al ldquoDesign of lightweightfootbridges for human induced vibrationsrdquo in BackgroundDocument in Support to The Implementation Harmonizationand Further Development of The Eurocodes 2009

[22] S Zivanovic A Pavic and P Reynolds ldquoProbability-basedprediction of multi-mode vibration response to walking exci-tationrdquo Engineering Structures vol 29 no 6 pp 942ndash954 2007

[23] E Ingolfsson C Georgakis and M Svendsen ldquoVertical foot-bridge vibrations details regarding and experimental validationof the response spectrum methodologyrdquo in Proceedings of the3rd international conference on footbridge 2008

[24] G Piccardo and F Tubino ldquoSimplified procedures for vibrationserviceability analysis of footbridges subjected to realistic walk-ing loadsrdquo Computers amp Structures vol 87 no 13-14 pp 890ndash903 2009

[25] S Zivanovic A Pavic and E T Ingolfsson ldquoModeling spatiallyunrestricted pedestrian traffic on footbridgesrdquo Journal of Struc-tural Engineering vol 136 no 10 pp 1296ndash1308 2010

[26] CEN Eurocode 5 Design of timber structures ComiteEuropeen de Normalisation 2005

[27] ISO Bases for design of structures-Serviceability of buildignsand walkways against vibration 2007

[28] Setra Technical guide-Footbridges Assessment of vibrationalbehaviour of footbridges under pedestrian loading 2006

[29] BSI UK national annex to Eurocode 1 2008[30] J Brownjohn P Fok M Roche and P Omenzetter ldquoLong span

steel pedestrian bridge at Singapore Changi Airportmdashpart 2crowd loading tests and vibration mitigation measuresrdquo TheStructural Engineer vol 82 no 16 pp 28ndash34 2004

[31] E T Ingolfsson and C T Georgakis ldquoA stochastic load modelfor pedestrian-induced lateral forces on footbridgesrdquo Engineer-ing Structures vol 33 no 12 pp 3454ndash3470 2011

[32] G Piccardo and F Tubino ldquoEquivalent spectral model andmaximum dynamic response for the serviceability analysis offootbridgesrdquo Engineering Structures vol 40 pp 445ndash456 2012

[33] AGDavenport ldquoNote on the distribution of the largest value ofa random functionwith application to gust loadingrdquo in Proceed-ings of the ICE vol 28 pp 187ndash196 1964

[34] A M Avossa C Demartino and F Ricciardelli ldquoDesign pro-cedures for footbridges subjected to walking loads comparison


and remarksrdquoTheBaltic Journal of Road and Bridge Engineeringvol 12 no 2 pp 94ndash105 2017

[35] A M Avossa C Demartino and F Ricciardelli ldquoProbabilitydistribution of footbridge peak acceleration to single and mult-iple crossing walkersrdquo Procedia Engineering vol 199 pp 2766ndash2771 2017

[36] CEN Eurocode 0 Basis of structural design Comite Europeende Normalisation 2002

[37] ISO 2394 General Principles on Reliability for Structures 1998[38] F Beely ldquoZur Mechanik des Stehensrdquo Longenbeckrsquos Archiv Fur

Klinische Chirurgie vol 27 pp 457ndash471 1882[39] H Elftman ldquoA cinematic study of the distribution of pressure

in the human footrdquo The Anatomical Record vol 59 no 4 pp481ndash491 1934

[40] C J T VanUden andM P Besser ldquoTest-retest reliability of tem-poral and spatial gait characteristics measured with an instru-mented walkway system (GAITRite)rdquo BMC MusculoskeletalDisorders vol 5 no 1 article 13 2004

[41] S Zivanovic A Pavic P Reynolds and P Vujovicrsquo ldquoDynamicanalysis of lively footbridge under everyday pedestrian trafficrdquoin Proceedings of the 6th European Conference on StructuralDynamics vol 1 pp 453ndash459 2005

[42] N A Borghese L Bianchi and F Lacquaniti ldquoKinematic defer-minants of human locomotionrdquo The Journal of Physiology vol494 no 3 pp 863ndash879 1996

[43] L Li E C H Van Den Bogert G E Caldwell R E A VanEmmerik and J Hamill ldquoCoordination patterns of walking andrunning at similar speed and stride frequencyrdquo Human Move-ment Science vol 18 no 1 pp 67ndash85 1999

[44] J L Lelas G J Merriman P O Riley and D C Kerrigan ldquoPre-dicting peak kinematic and kinetic parameters from gait speedrdquoGait amp Posture vol 17 no 2 pp 106ndash112 2003

[45] F Venuti and L Bruno ldquoCrowd-structure interaction in livelyfootbridges under synchronous lateral excitation a literaturereviewrdquo Physics of Life Reviews vol 6 no 3 pp 176ndash206 2009

[46] F Ricciardelli C Briatico E Ingolfsson and C GeorgakisldquoExperimental validation and calibration of pedestrian loadingmodels for footbridgesrdquo Experimental Vibration Analysis forCivil Engineering Structures 2006

[47] M Y Zarrugh F N Todd and H J Ralston ldquoOptimization ofenergy expenditure during level walkingrdquo European Journal ofApplied Physiology vol 33 no 4 pp 293ndash306 1974

[48] A Pachi and T Ji ldquoFrequency and velocity of people walkingrdquoThe Structural Engineer vol 83 no 3 pp 36ndash40 2005

[49] S Zivanovic V Racic I El-Bahnasy and A Pavic ldquoStatisticalcharacterisation of parameters defining human walking asobserved on an indoor passerellerdquo in Proceedings of the Experi-mental Vibration Analysis for Civil Engineering StructuresEVACES rsquo07 pp 219ndash225 2007

[50] E T Ingolfsson C T Georgakis F Ricciardelli and J Jons-son ldquoExperimental identification of pedestrian-induced lateralforces on footbridgesrdquo Journal of Sound and Vibration vol 330no 6 pp 1265ndash1284 2011

[51] Y Matsumoto T Nishioka H Shiojiri and K MatsuzakildquoDynamic design of footbridgesrdquo in Proceedings of the IABSEProceeding 1978

[52] C Butz M Feldmann C Heinemeyer et al ldquoAdvanced loadmodels for synchronous pedestrian excitation and optimiseddesign guidelines for steel foot bridges (SYNPEX)rdquo Endberichtund Zwischenberichte zum RFCS Projekt RFS-CR vol 3 article19 2007

[53] V Racic A Pavic and JMW Brownjohn ldquoExperimental iden-tification and analytical modelling of human walking forcesliterature reviewrdquo Journal of Sound and Vibration vol 326 no1-2 pp 1ndash49 2009

[54] F Ricciardelli M Mafrici and E T Ingolfsson ldquoLateral pedes-trian-induced vibrations of footbridges characteristics of walk-ing forcesrdquo Journal of Bridge Engineering vol 19 no 9 ArticleID 04014035 2014

[55] J H Macdonald ldquoLateral excitation of bridges by balancingpedestriansrdquo Proceedings of the Royal Society A MathematicalPhysical and Engineering Sciences 2008

[56] Q S Yang JWQin and S S Law ldquoA three-dimensional humanwalking modelrdquo Journal of Sound and Vibration vol 357 pp437ndash456 2015

[57] J Blanchard B Davies and J Smith ldquoDesign criteria andanalysis for dynamic loading of footbridgesrdquo in Proceedings ofthe Proceeding of a Symposium onDynamic Behaviour of Bridgesat the Transport and Road Research Laboratory 1977

[58] H Bachmann A Pretlove and J Rainer Dynamic forces fromrythmical human body motions

[59] Y Kajikawa ldquoSome considerations on ergonomical serviceabil-ity analysis of pedestrian bridge vibrationsrdquo in Proceedings of theJSCE vol 325 pp 23ndash33 1982 (Japanese)

[60] M Yoneda ldquoA simplified method to evaluate pedestrian-induced maximum response of cable-supported pedestrianbridgesrdquo in Proceedings of the International Conference on theDesign and Dynamic Behaviour of footbridges 2002

[61] S Kerr Human induced loading on staircases [PhD thesis]University of London London UK 1998

[62] P Young ldquoImproved floor vibration predictionmethodologiesrdquoin Proceedings of the ARUP vibration seminar vol 4 2001

[63] H Bachmann and W Ammann ldquoVibrations in structuresinduced byman andmachinesrdquoVibrations in structures inducedby man and machines vol 3 1987

[64] M Fareed andM Afzal ldquoEvidence of inbreeding depression onheight weight and body mass index a population-based childcohort studyrdquo American Journal of Human Biology vol 26 no6 pp 784ndash795 2014

[65] S C Walpole D Prieto-Merino P Edwards J Cleland GStevens and I Roberts ldquoTheweight of nations an estimation ofadult human biomassrdquo BMC Public Health vol 12 no 1 article439 2012

[66] L Bruno F Venuti andVNasce ldquoPedestrian-induced torsionalvibrations of suspended footbridges proposal and evaluation ofvibration countermeasuresrdquo Engineering Structures vol 36 pp228ndash238 2012

[67] H Bachmann Vibration Problems in Structures PracticalGuidelines Springer Berlin Germany 1995

[68] M Ferraioli A M Avossa and F Formato ldquoBase isolation seis-mic retrofit of a hospital building in Italy design and construc-tionrdquo in Proceedings of the Final Conference on COST ActionC26 Urban Habitat Constructions under Catastrophic Eventspp 835ndash840 September 2010

[69] A M Avossa and G Pianese ldquoDamping effects on the seismicresponse of base-isolated structures with LRB devicesrdquo Ingegne-ria Sismica vol 34 no 2 pp 3ndash29 2017

[70] Y Kubota and T Kishimoto ldquoStructural continuity and relativ-ity for footbridge designrdquo in Proceedings of the Footbridges 2008Third International Conference 2008

[71] CEN Eurocode 1 Actions on structures omite Europeen deNormalisation 2003


[72] CEN Eurocode 3 Design of Steel Structures Comite Europeende Normalisation 2005

[73] FIB Guidelines for the design of footbridges Federation Inter-nationale du Beton Lausanne Switzerland 2005

[74] MAbu-Hilal andMMohsen ldquoVibration of beamswith generalboundary conditions due to a moving harmonic loadrdquo Journalof Sound and Vibration vol 232 no 4 pp 703ndash717 2000

[75] F Ricciardelli and C Briatico ldquoTransient response of supportedbeams to moving forces with sinusoidal time variationrdquo Journalof Engineering Mechanics vol 137 no 6 pp 422ndash430 2011

[76] G Piccardo and F Tubino ldquoDynamic response of Euler-Ber-noulli beams to resonant harmonic moving loadsrdquo StructuralEngineering and Mechanics vol 44 no 5 pp 681ndash704 2012

[77] A Palmeri F Ricciardelli GMuscolino and A De Luca ldquoRan-dom vibration of systems with viscoelastic memoryrdquo Journal ofEngineering Mechanics vol 130 no 9 pp 1052ndash1061 2004

[78] H Karadeniz and T Vrouwenvelder Overview reliability meth-ods vol 5 2003

[79] M Griffin Handbook of human vibration 1990[80] N Mansfield Human Response to Vibration CRC Press Boca

Raton Fla USA 2004[81] ISO ISO 2631 Mechanical vibration and shock-Evaluation

of human exposure to whole-body vibration-Part 1 Generalrequirements 1997

[82] MIT Ministerial Decree NTC 2008-Norme tecniche per le cos-truzioni Ministero delle Infrastrutture italy 2008

[83] S CSI Ver 1400 integrated finite element analysis and designof structures basic analysis reference manual 2010

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Standards and guidelines have been developed to help thedesigner in the evaluation of the vibration serviceability basedon simplified loading models simulating different possiblescenarios [5 6] In broad terms the design of footbridgesagainst pedestrian-induced vibrations requires the knowl-edge of [7] (i) the characteristics of the pedestrian action (ii) aresponse evaluationmethod and (iii) a comfort criterionTheinterested reader is referred to [5 6 8 9] Design guidelinesoutline procedures for vibration serviceability checks butit is noticeable that most of these assume that the actionis deterministic [10] yet this is stochastic and it wouldbe reasonable to incorporate its variability in the loadingmodels However it should be mentioned that other loads onbridges should be considered in the design procedures suchas seismic wind and impact loads (eg [11ndash16])

Different authors have tried to characterize the random-ness of the pedestrian action in either time or frequencydomain considering both intrasubject and intersubject vari-ability Brownjohn et al [17] for the vertical direction andPizzimenti and Ricciardelli [18] and Ricciardelli and Pizzi-menti [19] for the lateral direction gave Power SpectralDensity Functions (PSDFs) of the load induced by a walkerfor use in the evaluation of the stationary response to a streamof walkers including intrasubject and intersubject variabilityof gait

Subsequently Butz [20] presented a spectral approachfor the evaluation of the peak acceleration induced by unre-stricted pedestrian trafficThe PSDFs of themodal force wereapproximated through a Gaussian function fitting data com-ing fromMonte Carlo simulationsThese were carried out fordifferent bridge geometries and for four different pedestriandensities Step frequency pedestrian mass force amplitudeand pedestrian arrival time were randomly selected fromgiven probability distributions The PSDF of the accelerationwas evaluated from the PSDF of themodal force and the 95thfractile peak modal acceleration was derived as the productof the RMS acceleration and a peak factor The latter wasevaluated to be around 4 All the coefficients required for theapplication of the procedure were expressed in a parametricform This model has then been incorporated into HIVOSSguidelines [21]

Zivanovic et al [22] presented a multiharmonic forcemodel for calculation of the multimode structural responseto a crossing accounting for inter- and intrasubject vari-ability in the walking force The model is again based onMonte Carlo simulations with pedestrian characteristics alsoselected from given distributionsThe intrasubject variabilitywas accounted for describing the force in the frequencydomain and then converting it to the time domain Noparametric form of the peak response as a function of thedifferent parameters and no procedure for the serviceabilityassessment are given

Ingolfsson et al [23] proposed a Response Spectrumapproach inspired by earthquake engineering ThroughMonte Carlo simulations they evaluated a reference verticalacceleration to the action of a flow of 1walkers withprobabilistically modeled characteristics to which empir-ical correction factors are applied to account for returnperiod modal mass mean arrival rate structural damping

footbridge span and mode shape A total of 97 windows300 s long were used to establish the peak acceleration Gen-eralized Extreme Value (GEV) distribution parameters Tworeference populations were considered with step frequenciesof 18 and 20Hz with STD of 01 Hz and Poisson distributedarrivals The input force was modeled as a harmonic loadand the intersubject variability was considered varying thecharacteristics of each pedestrian according to given distri-butions All the parameters contained in the procedure aregiven in parametric form and the GEV distribution of thepeak acceleration is found to tend to a Gumbel distribution(ie the shape parameter tends to 0)

Piccardo and Tubino [24] studied the vertical vibrationserviceability of footbridges based on a probabilistic charac-terization of pedestrian-induced forces taking into accountintersubject variability and considering only one mode ofvibration Only the 95th fractile peak modal acceleration wasderived and expressed in two nondimensional forms (a) theEquivalent Amplification Factor that is the ratio between themaximum dynamic response to a realistic loading scenarioand the maximum dynamic response to a single resonantpedestrian (b) the Equivalent Synchronization Factor that isthe ratio between themaximumdynamic response to a realis-tic loading scenario and the maximum dynamic response touniformly distributed resonant pedestrians Comparison oftheir procedure with similar methods contained in standardsand design guidelines has pointed out that the latter aregenerally conservative (often largely conservative) and canbecome only slightly nonconservative in particular casesThey concluded that further investigations on the evaluationof the PDF of the maximum dynamic response are required

Zivanovic et al [25] reviewed different time-domaindesign procedures for vibration serviceability assessment offootbridges exposed to streams of pedestrians and evalu-ated their performance in predicting the vertical vibrationresponse of two existing footbridges They compared theprocedures contained in Eurocode 5 [26] ISO 10137 [27]Setra [28] BSI [29] Brownjohn et al [30] Butz [20]Ingolfsson et al [23] and Zivanovic et al [22] They foundsome discrepancies between the predicted and measuredvibration levels and discussed their potential causes amongwhich are interwalker and walker-structure interaction

Pedersen and Frier [10] evaluated the effect of the proba-bilistic modeling of the parameters describing walking loadsstep frequency stride length Dynamic Load Factor (DLF)and walker weight A literature review revealed a variety ofprobability distributions through which the walking param-eters can be modeled and these were used for exploringthe sensitivity of the 95th fractile of midspan accelerationThey observed that the step frequency distribution can havea strong influence whereas the DLF the walker weight andstride length have a much lower influence No interactionswere considered in this study

Ingolfsson and Georgakis [31] presented a probabilisticlateral load model in which the forces are given as the sumof an external component and a frequency and amplitude-dependent self-excited component the latter is quantifiedthrough equivalent pedestrian damping andmass coefficientsmeasured from experiments They found that the peak















119865119894 (119905) = 119882 sdot infinsum119895=1











V = N (141 0224) gt 041 [ms] (3)









00

1

2PD

F

0

10

20

0

05

1

15

2

25

1 2 3v (ms)

1 2 30fw (Hz)

05 10＄V ()

0

20

40

005 010＄L ()

0

02

04

06

08

1

CDF

0

2

4

500 10000W (N)

times10minus3



DLF119871 = N (003792 001459) gt 0 (6)



119882 = N (744 130) ge 0 [N] (7)








119898119894 (119909 119905) + 119888119894119894 (119909 119905) + 119864119868119894119906119868119881119894 (119909 119905)= 119865119894 (119905) sdot 120575 (119909 minus V sdot 119905) (8)



119906119894 (119909 119905) = 120601119894 (119909) 120578119894 (119905) (9)


Kr1V

Kr1L

m

y

x

z1 2

uL(x t)

uV(x t)

cV cL EIV EIL Kr2V

Kr2L

L




120601119894 (119909) = 1198881119894 sin(120582119894 119909119871) + 1198882119894 cos(120582119894 119909119871)+ 1198883119894 sinh(120582119894 119909119871) + 1198884119894 cosh (120582119894 119909119871)

(10)



120578119894 (119905) + 4120587120585119894119891119894 120578119894 (119905) + 412058721198912119894 120578119894 (119905) = 119898minus1119894 119891119894 (119905) (11)



120578119894 (119905) = 1119898119894 int119905










RL119894 = 119872119894 (119870119903119894)119872119894 (119870119903119894 rarr infin) (13)






08 09 107


0

02

04

06

08

1

RLi

(a)

10minus1 100 101 10210minus2

Kri middot LEIi

0

02

04

06

08

1

RLi

(b)

1 20 3 4

RDi

0

02

04

06

08

1

RLi

(c)






RD119894 = Θ119894 sdot 119871120575119894 (15)








Modal force


fi(t)

Mechanical model

FEM model



p ()

pW(W)

LL

V Lm

V L

２＄V ２＄L

２V ２L


i fi(t)





pf(fw)

p＄(＄V)

p＄(＄L)



mi

( )minus1

＄i i









(18)






005

005

005

005

01

01

0101

02

02 02

0404 04

0707 0909

005005

005

005 005

0101

01

02

02

0404

040707

07 0909

0

01

02

03

04

05

06

07

08

09

1

0

01

02

03

04

05

06

07

08

09

1

50 100 150 2000L

04

06

08

1

12

14

16V

04

06

08

1

12

14

16

L

50 100 150 2000L

01

02

V(V L V = 0015) L(L L L = 0015)








120593119894 (RL119894 = 0 120572119894 119871 120585119894) = min

120585119894 sdot 2120572211989410038161003816100381610038161 minus 1205722119894 1003816100381610038161003816 [1 +1119899119894



(19)


119899119894 (119871) = 2 sdot 119891119908119894V






BF119894 (RL119894 120572119894 119871 120585119894) = 120578119894 (RL119894 120572119894 119871 120585119894)120578119894 (RL119894 = 0 120572119894 119871 120585119894) (21)



119898119894 (RL119894)sdot 120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)

(22)




120593119894 (RL119894 120572119894 119871 120585119894) = BF120593119894 (RL119894 120572119894 119871 120585119894)sdot 120593119894 (RL119894 = 0 120572119894 119871 120585119894)

(23)








119901120578119894 (120578119894 (120572119894 119871 120585119894)) = intinfin0

intinfin0

intinfin0

intinfin0


(24)



(25)






２V = 000２V = 025２V = 050

２V = 075２V = 100

07

075

08

085

09

095

1

105

11

115

12

125

13V

50 100 150 2000L

12

12

125

125125

125125

125

12513

1

105

11

115

12

125

13

135

14

145

15

04

06

08

1

12

14

16

V

50 100 150 2000L

V(２V V = 1 L V = 0015) V(２V = 1 V L V = 0015)



















119863119894 = 120593119894 (RL119894 120572119894 119871 120585119894) le 120593119894max = 119862119894 forall119894 (27)


119862119894 = 120593119894max


(28)



119875fail119894 = int119862minus119863le0

119901119862119894 119863119894 (119888119894 119889119894) sdot d119888119894 sdot d119889119894 (29)




0119901120593119894 (120593119894 (RL119894 120572119894 119871 120585119894)) sdot d120593119894 (30)











11 115115

12

12

12

125

11 115115

1212

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125

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12

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125

11 115

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125125

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12

12125

125

12

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125 125

125

V = 0001 V = 0003

V = 0005 V = 0007

V = 0009 V = 0011

V = 0013 V = 0015

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1

0 50 200100 150

L

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V

05

1

15

V

05

1

15

V

05

1

15

V

0 50 200100 150

L

0 50 200100 150

L

05

1

15

V

0 50 200100 150

L

0 50 200100 150

L

05

1

15

V

05

1

15

V

0 50 200100 150

L

05

1

15

V

0 50 200100 150

L



11 115115

12 12

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125

1

11

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13

14

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12

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1

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1212

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1

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12125

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1

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1

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L = 0001 L = 0003

L = 0007

L = 0011

L = 0015

L = 0005

L = 0009

L = 0013

50 100 150 2000

L

50 100 150 2000

L

50 100 150 2000

L

50 100 150 2000

L

50 100 150 2000

L

50 100 150 2000

L

100 200150500

L

100 200150500

L

05

1

15

L

05

1

15

L

05

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15

L

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15

L

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15

L

05

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15

L

05

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15

L

05

1

15

L






= 021 if 120572119881 le 0470140 + 0150 sdot 120572119881 if 120572119881 gt 047



(33)










316fL

05606307609512719

0

05

1

15

2

25

3

35

4

45

5

L

max

075 1 12505 15 1703L

633fV

1121271521925338

0

005

01

015

02

025

03

035

04

045

05

V

max

075 1 12505 15 1703V






(34)



sdot 120578119881max = 0775


sdot 120578119871max = 5723(35)


119863119881 = 120593119881 (RL119881 120572119881 119871 120585119881) = 1 sdot 00904 = 00904119863119871 = 120593119871 (RL119871 120572119871 119871 120585119871) = 097 sdot 00035 = 00034 (36)


119863119881 = 00904 lt 0775 = 119862119881119863119871 = 00034 lt 5723 = 119862119871 (37)





(38)


119875120593119881 (119863119881) = 0565119875120593119871 (119863119871) = 0379 (39)


119875fail119881 = 00296 lArrrArr120573119881 = 1887


(40)






(41)


7 Conclusions




0

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L ()

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L

04

06

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1

12

14

16

L

04

06

08

1

12

14

16

L


L - L = 10 Ｇ L - L = 20 Ｇ

kL - L = 30 Ｇ kL - L = 40 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ

L - L = 10 Ｇ L - L = 20 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ





0

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05 1 15

V ()

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15105

1 1505

V ()1505 1

V ()

1 15051 1505

1 1505 1 1505

1505 115105

1 150505 151

1 1505 105 15

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V

04

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1

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16

V


V - L = 10 Ｇ V - L = 20 Ｇ

kV - L = 30 Ｇ kV - L = 40 Ｇ

V - L = 30 Ｇ V - L = 40 Ｇ

L = 10 Ｇ L = 20 Ｇ

L = 30 Ｇ L = 40 Ｇ


L = 50ndash200Ｇ V - V -

V -V -

V -



0

02

04

06

08

1

CDF

0005 001 0015 00200

02

04

06

08

1

0

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PDF

104 06 080200

50

100

150

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250

V [mdash] L [mdash]


Appendix


See Figures 7 and 8



1198701199031119894 = 1198701199032119894 = 119870119903119894 (C1)



Θ1119894 = Θ2119894 = Θ119894 (C2)







(C5)




(C6)





L - L = 10 Ｇ L - L = 20 Ｇ

kL - L = 30 Ｇ kL - L = 40 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ

L - L = 10 Ｇ L - L = 20 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ



0

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105 15

1 1505

105 15 105 15

105 15105 15

1 1505 1 1505

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105 15 105 15

1 1505

L ()1 1505

L ()1 1505

L ()

05

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1

15




V - L = 10 Ｇ V - L = 20 Ｇ

kV - L = 30 Ｇ kV - L = 40 Ｇ

V - L = 30 Ｇ V - L = 40 Ｇ

L = 10 Ｇ L = 20 Ｇ

L = 30 Ｇ L = 40 Ｇ


L = 50ndash200Ｇ

0

02

02

04

04

06

06

0808

minus04

minus02

0

02

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1 1505

V ()1 1505

V ()

1505 115105

1 1505 1505 1

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V ()

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04

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04

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V

V - V -

V - V -V -





119872119894 sdot (RL119894 rarr 1) (C8)



(C9)


119872119894 (RL119894 997888rarr 1) = Θ119894 (RL119894 = 0) sdot 2119864119868119894119871 (C10)





RD119894 = Θ119894 (RL119894) sdot 119871120575119894 (RL119894) (C12)





(C14)


RL119894


(C15)





119898119894sdot 120593119894 (RL119894 120572119894 119871 120585119894)

(C17)


BF119894 (RL119894 120572119894 119871 120585119894) = 120578119894 (RL119894 120572119894 119871 120585119894)120578119894 (RL119894 = 0 120572119894 119871 120585119894) (C18)



(C19)


BF119894 (RL119894 120572119894 119871 120585119894)


120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)= 119898119894 (RL119894 = 0)

119898119894 (RL119894)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟BF119898119894(RL119894)

sdot 120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟BF120593119894(RL119894 120572119894 119871120585119894)

(C20)

obtaining


= 119898119894 (RL119894 = 0)119898119894 (RL119894)

sdot 120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)

(C21)





Acknowledgments


References

























































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia
















119865119894 (119905) = 119882 sdot infinsum119895=1











V = N (141 0224) gt 041 [ms] (3)









00

1

2PD

F

0

10

20

0

05

1

15

2

25

1 2 3v (ms)

1 2 30fw (Hz)

05 10＄V ()

0

20

40

005 010＄L ()

0

02

04

06

08

1

CDF

0

2

4

500 10000W (N)

times10minus3



DLF119871 = N (003792 001459) gt 0 (6)



119882 = N (744 130) ge 0 [N] (7)








119898119894 (119909 119905) + 119888119894119894 (119909 119905) + 119864119868119894119906119868119881119894 (119909 119905)= 119865119894 (119905) sdot 120575 (119909 minus V sdot 119905) (8)



119906119894 (119909 119905) = 120601119894 (119909) 120578119894 (119905) (9)


Kr1V

Kr1L

m

y

x

z1 2

uL(x t)

uV(x t)

cV cL EIV EIL Kr2V

Kr2L

L




120601119894 (119909) = 1198881119894 sin(120582119894 119909119871) + 1198882119894 cos(120582119894 119909119871)+ 1198883119894 sinh(120582119894 119909119871) + 1198884119894 cosh (120582119894 119909119871)

(10)



120578119894 (119905) + 4120587120585119894119891119894 120578119894 (119905) + 412058721198912119894 120578119894 (119905) = 119898minus1119894 119891119894 (119905) (11)



120578119894 (119905) = 1119898119894 int119905










RL119894 = 119872119894 (119870119903119894)119872119894 (119870119903119894 rarr infin) (13)






08 09 107


0

02

04

06

08

1

RLi

(a)

10minus1 100 101 10210minus2

Kri middot LEIi

0

02

04

06

08

1

RLi

(b)

1 20 3 4

RDi

0

02

04

06

08

1

RLi

(c)






RD119894 = Θ119894 sdot 119871120575119894 (15)








Modal force


fi(t)

Mechanical model

FEM model



p ()

pW(W)

LL

V Lm

V L

２＄V ２＄L

２V ２L


i fi(t)





pf(fw)

p＄(＄V)

p＄(＄L)



mi

( )minus1

＄i i









(18)






005

005

005

005

01

01

0101

02

02 02

0404 04

0707 0909

005005

005

005 005

0101

01

02

02

0404

040707

07 0909

0

01

02

03

04

05

06

07

08

09

1

0

01

02

03

04

05

06

07

08

09

1

50 100 150 2000L

04

06

08

1

12

14

16V

04

06

08

1

12

14

16

L

50 100 150 2000L

01

02

V(V L V = 0015) L(L L L = 0015)








120593119894 (RL119894 = 0 120572119894 119871 120585119894) = min

120585119894 sdot 2120572211989410038161003816100381610038161 minus 1205722119894 1003816100381610038161003816 [1 +1119899119894



(19)


119899119894 (119871) = 2 sdot 119891119908119894V






BF119894 (RL119894 120572119894 119871 120585119894) = 120578119894 (RL119894 120572119894 119871 120585119894)120578119894 (RL119894 = 0 120572119894 119871 120585119894) (21)



119898119894 (RL119894)sdot 120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)

(22)




120593119894 (RL119894 120572119894 119871 120585119894) = BF120593119894 (RL119894 120572119894 119871 120585119894)sdot 120593119894 (RL119894 = 0 120572119894 119871 120585119894)

(23)








119901120578119894 (120578119894 (120572119894 119871 120585119894)) = intinfin0

intinfin0

intinfin0

intinfin0


(24)



(25)






２V = 000２V = 025２V = 050

２V = 075２V = 100

07

075

08

085

09

095

1

105

11

115

12

125

13V

50 100 150 2000L

12

12

125

125125

125125

125

12513

1

105

11

115

12

125

13

135

14

145

15

04

06

08

1

12

14

16

V

50 100 150 2000L

V(２V V = 1 L V = 0015) V(２V = 1 V L V = 0015)



















119863119894 = 120593119894 (RL119894 120572119894 119871 120585119894) le 120593119894max = 119862119894 forall119894 (27)


119862119894 = 120593119894max


(28)



119875fail119894 = int119862minus119863le0

119901119862119894 119863119894 (119888119894 119889119894) sdot d119888119894 sdot d119889119894 (29)




0119901120593119894 (120593119894 (RL119894 120572119894 119871 120585119894)) sdot d120593119894 (30)











11 115115

12

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12

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11 115115

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12125

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125125

125

12

12125

125

12

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125

125

125 125

125

V = 0001 V = 0003

V = 0005 V = 0007

V = 0009 V = 0011

V = 0013 V = 0015

15

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1

0 50 200100 150

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V

05

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V

05

1

15

V

05

1

15

V

0 50 200100 150

L

0 50 200100 150

L

05

1

15

V

0 50 200100 150

L

0 50 200100 150

L

05

1

15

V

05

1

15

V

0 50 200100 150

L

05

1

15

V

0 50 200100 150

L



11 115115

12 12

12

125

1

11

12

13

14

15

115

12

12

12

125

125

1

11

12

13

14

15

115

12

1212

125

125

125

1

11

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14

15

115

12

12

12125

125

1

11

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13

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125

125

1

11

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125

1

11

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125

1

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15

12

12

125

125

1

11

12

13

14

15

L = 0001 L = 0003

L = 0007

L = 0011

L = 0015

L = 0005

L = 0009

L = 0013

50 100 150 2000

L

50 100 150 2000

L

50 100 150 2000

L

50 100 150 2000

L

50 100 150 2000

L

50 100 150 2000

L

100 200150500

L

100 200150500

L

05

1

15

L

05

1

15

L

05

1

15

L

05

1

15

L

05

1

15

L

05

1

15

L

05

1

15

L

05

1

15

L






= 021 if 120572119881 le 0470140 + 0150 sdot 120572119881 if 120572119881 gt 047



(33)










316fL

05606307609512719

0

05

1

15

2

25

3

35

4

45

5

L

max

075 1 12505 15 1703L

633fV

1121271521925338

0

005

01

015

02

025

03

035

04

045

05

V

max

075 1 12505 15 1703V






(34)



sdot 120578119881max = 0775


sdot 120578119871max = 5723(35)


119863119881 = 120593119881 (RL119881 120572119881 119871 120585119881) = 1 sdot 00904 = 00904119863119871 = 120593119871 (RL119871 120572119871 119871 120585119871) = 097 sdot 00035 = 00034 (36)


119863119881 = 00904 lt 0775 = 119862119881119863119871 = 00034 lt 5723 = 119862119871 (37)





(38)


119875120593119881 (119863119881) = 0565119875120593119871 (119863119871) = 0379 (39)


119875fail119881 = 00296 lArrrArr120573119881 = 1887


(40)






(41)


7 Conclusions




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L ()

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L ()

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L

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12

14

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L

04

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08

1

12

14

16

L


L - L = 10 Ｇ L - L = 20 Ｇ

kL - L = 30 Ｇ kL - L = 40 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ

L - L = 10 Ｇ L - L = 20 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ





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V ()

105 15

15105

1 1505

V ()1505 1

V ()

1 15051 1505

1 1505 1 1505

1505 115105

1 150505 151

1 1505 105 15

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V

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1

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14

16

V

04

06

08

1

12

14

16

V


V - L = 10 Ｇ V - L = 20 Ｇ

kV - L = 30 Ｇ kV - L = 40 Ｇ

V - L = 30 Ｇ V - L = 40 Ｇ

L = 10 Ｇ L = 20 Ｇ

L = 30 Ｇ L = 40 Ｇ


L = 50ndash200Ｇ V - V -

V -V -

V -



0

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1

CDF

0005 001 0015 00200

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PDF

104 06 080200

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100

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V [mdash] L [mdash]


Appendix


See Figures 7 and 8



1198701199031119894 = 1198701199032119894 = 119870119903119894 (C1)



Θ1119894 = Θ2119894 = Θ119894 (C2)







(C5)




(C6)





L - L = 10 Ｇ L - L = 20 Ｇ

kL - L = 30 Ｇ kL - L = 40 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ

L - L = 10 Ｇ L - L = 20 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ



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L

105 15

1 1505

105 15 105 15

105 15105 15

1 1505 1 1505

1 15051 1505

105 15 105 15

1 1505

L ()1 1505

L ()1 1505

L ()

05

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1

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05

1

15




V - L = 10 Ｇ V - L = 20 Ｇ

kV - L = 30 Ｇ kV - L = 40 Ｇ

V - L = 30 Ｇ V - L = 40 Ｇ

L = 10 Ｇ L = 20 Ｇ

L = 30 Ｇ L = 40 Ｇ


L = 50ndash200Ｇ

0

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minus04

minus02

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0001

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002501

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002501

1 1505

V ()1 1505

V ()

1505 115105

1 1505 1505 1

151051 1505

151051505 1

1 1505 1 1505

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1

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V ()

1 1505

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V

04

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1

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V

04

06

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1

12

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16

V

V - V -

V - V -V -





119872119894 sdot (RL119894 rarr 1) (C8)



(C9)


119872119894 (RL119894 997888rarr 1) = Θ119894 (RL119894 = 0) sdot 2119864119868119894119871 (C10)





RD119894 = Θ119894 (RL119894) sdot 119871120575119894 (RL119894) (C12)





(C14)


RL119894


(C15)





119898119894sdot 120593119894 (RL119894 120572119894 119871 120585119894)

(C17)


BF119894 (RL119894 120572119894 119871 120585119894) = 120578119894 (RL119894 120572119894 119871 120585119894)120578119894 (RL119894 = 0 120572119894 119871 120585119894) (C18)



(C19)


BF119894 (RL119894 120572119894 119871 120585119894)


120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)= 119898119894 (RL119894 = 0)

119898119894 (RL119894)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟BF119898119894(RL119894)

sdot 120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟BF120593119894(RL119894 120572119894 119871120585119894)

(C20)

obtaining


= 119898119894 (RL119894 = 0)119898119894 (RL119894)

sdot 120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)

(C21)





Acknowledgments


References

























































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



00

1

2PD

F

0

10

20

0

05

1

15

2

25

1 2 3v (ms)

1 2 30fw (Hz)

05 10＄V ()

0

20

40

005 010＄L ()

0

02

04

06

08

1

CDF

0

2

4

500 10000W (N)

times10minus3



DLF119871 = N (003792 001459) gt 0 (6)



119882 = N (744 130) ge 0 [N] (7)








119898119894 (119909 119905) + 119888119894119894 (119909 119905) + 119864119868119894119906119868119881119894 (119909 119905)= 119865119894 (119905) sdot 120575 (119909 minus V sdot 119905) (8)



119906119894 (119909 119905) = 120601119894 (119909) 120578119894 (119905) (9)


Kr1V

Kr1L

m

y

x

z1 2

uL(x t)

uV(x t)

cV cL EIV EIL Kr2V

Kr2L

L




120601119894 (119909) = 1198881119894 sin(120582119894 119909119871) + 1198882119894 cos(120582119894 119909119871)+ 1198883119894 sinh(120582119894 119909119871) + 1198884119894 cosh (120582119894 119909119871)

(10)



120578119894 (119905) + 4120587120585119894119891119894 120578119894 (119905) + 412058721198912119894 120578119894 (119905) = 119898minus1119894 119891119894 (119905) (11)



120578119894 (119905) = 1119898119894 int119905










RL119894 = 119872119894 (119870119903119894)119872119894 (119870119903119894 rarr infin) (13)






08 09 107


0

02

04

06

08

1

RLi

(a)

10minus1 100 101 10210minus2

Kri middot LEIi

0

02

04

06

08

1

RLi

(b)

1 20 3 4

RDi

0

02

04

06

08

1

RLi

(c)






RD119894 = Θ119894 sdot 119871120575119894 (15)








Modal force


fi(t)

Mechanical model

FEM model



p ()

pW(W)

LL

V Lm

V L

２＄V ２＄L

２V ２L


i fi(t)





pf(fw)

p＄(＄V)

p＄(＄L)



mi

( )minus1

＄i i









(18)






005

005

005

005

01

01

0101

02

02 02

0404 04

0707 0909

005005

005

005 005

0101

01

02

02

0404

040707

07 0909

0

01

02

03

04

05

06

07

08

09

1

0

01

02

03

04

05

06

07

08

09

1

50 100 150 2000L

04

06

08

1

12

14

16V

04

06

08

1

12

14

16

L

50 100 150 2000L

01

02

V(V L V = 0015) L(L L L = 0015)








120593119894 (RL119894 = 0 120572119894 119871 120585119894) = min

120585119894 sdot 2120572211989410038161003816100381610038161 minus 1205722119894 1003816100381610038161003816 [1 +1119899119894



(19)


119899119894 (119871) = 2 sdot 119891119908119894V






BF119894 (RL119894 120572119894 119871 120585119894) = 120578119894 (RL119894 120572119894 119871 120585119894)120578119894 (RL119894 = 0 120572119894 119871 120585119894) (21)



119898119894 (RL119894)sdot 120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)

(22)




120593119894 (RL119894 120572119894 119871 120585119894) = BF120593119894 (RL119894 120572119894 119871 120585119894)sdot 120593119894 (RL119894 = 0 120572119894 119871 120585119894)

(23)








119901120578119894 (120578119894 (120572119894 119871 120585119894)) = intinfin0

intinfin0

intinfin0

intinfin0


(24)



(25)






２V = 000２V = 025２V = 050

２V = 075２V = 100

07

075

08

085

09

095

1

105

11

115

12

125

13V

50 100 150 2000L

12

12

125

125125

125125

125

12513

1

105

11

115

12

125

13

135

14

145

15

04

06

08

1

12

14

16

V

50 100 150 2000L

V(２V V = 1 L V = 0015) V(２V = 1 V L V = 0015)



















119863119894 = 120593119894 (RL119894 120572119894 119871 120585119894) le 120593119894max = 119862119894 forall119894 (27)


119862119894 = 120593119894max


(28)



119875fail119894 = int119862minus119863le0

119901119862119894 119863119894 (119888119894 119889119894) sdot d119888119894 sdot d119889119894 (29)




0119901120593119894 (120593119894 (RL119894 120572119894 119871 120585119894)) sdot d120593119894 (30)











11 115115

12

12

12

125

11 115115

1212

125

125

115

12

12

12

125

125

11 115

12

1212

125

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115

12

1212

125

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12

12125

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115

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1212

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12125

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12125

125125

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12125

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12125

125

125

12

12125

125

12

12

125

125125

125

12

12125

125

12

12

125

125

125 125

125

V = 0001 V = 0003

V = 0005 V = 0007

V = 0009 V = 0011

V = 0013 V = 0015

15

14

13

12

11

1

15

14

13

12

11

1

15

14

13

12

11

1

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11

1

0 50 200100 150

L

0 50 200100 150

L

05

1

15

V

05

1

15

V

05

1

15

V

05

1

15

V

0 50 200100 150

L

0 50 200100 150

L

05

1

15

V

0 50 200100 150

L

0 50 200100 150

L

05

1

15

V

05

1

15

V

0 50 200100 150

L

05

1

15

V

0 50 200100 150

L



11 115115

12 12

12

125

1

11

12

13

14

15

115

12

12

12

125

125

1

11

12

13

14

15

115

12

1212

125

125

125

1

11

12

13

14

15

115

12

12

12125

125

1

11

12

13

14

15

12

12

125

125

1

11

12

13

14

15

12

12

125

125

1

11

12

13

14

15

12

12

125

125

1

11

12

13

14

15

12

12

125

125

1

11

12

13

14

15

L = 0001 L = 0003

L = 0007

L = 0011

L = 0015

L = 0005

L = 0009

L = 0013

50 100 150 2000

L

50 100 150 2000

L

50 100 150 2000

L

50 100 150 2000

L

50 100 150 2000

L

50 100 150 2000

L

100 200150500

L

100 200150500

L

05

1

15

L

05

1

15

L

05

1

15

L

05

1

15

L

05

1

15

L

05

1

15

L

05

1

15

L

05

1

15

L






= 021 if 120572119881 le 0470140 + 0150 sdot 120572119881 if 120572119881 gt 047



(33)










316fL

05606307609512719

0

05

1

15

2

25

3

35

4

45

5

L

max

075 1 12505 15 1703L

633fV

1121271521925338

0

005

01

015

02

025

03

035

04

045

05

V

max

075 1 12505 15 1703V






(34)



sdot 120578119881max = 0775


sdot 120578119871max = 5723(35)


119863119881 = 120593119881 (RL119881 120572119881 119871 120585119881) = 1 sdot 00904 = 00904119863119871 = 120593119871 (RL119871 120572119871 119871 120585119871) = 097 sdot 00035 = 00034 (36)


119863119881 = 00904 lt 0775 = 119862119881119863119871 = 00034 lt 5723 = 119862119871 (37)





(38)


119875120593119881 (119863119881) = 0565119875120593119871 (119863119871) = 0379 (39)


119875fail119881 = 00296 lArrrArr120573119881 = 1887


(40)






(41)


7 Conclusions




0

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002501

1 1505

L ()1 1505

L ()

1 1505105 15

15105 15105

105 15 105 15

1505 1 1505 1

1 15051 1505

05

1

15

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1

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1

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105 15

L ()

15105

1 1505

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1

12

14

16

L

04

06

08

1

12

14

16

L

04

06

08

1

12

14

16

L


L - L = 10 Ｇ L - L = 20 Ｇ

kL - L = 30 Ｇ kL - L = 40 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ

L - L = 10 Ｇ L - L = 20 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ





0

02

02

04

04

06

06

0808

minus04

minus02

0

02

04

06

08

1

12

0 0

0

0404

0

04

04

0808

0

04

04

0808

0

04

04

0808

0001

0025

0025

005

005

01

0

002

004

006

008

01

012

014

016

018

02

0001

002501

0001

002501

0001

002501

0001

002501

0001

0025

0025

005

00501

0

002

004

006

008

01

012

014

016

018

02

002501 0025

01

002501 0025

01

05 1 15

V ()

105 15

15105

1 1505

V ()1505 1

V ()

1 15051 1505

1 1505 1 1505

1505 115105

1 150505 151

1 1505 105 15

05

1

15

05

1

15

05

1

15

05

1

15

05

1

15

05

1

15

05

1

15

05

1

15

05

1

15

05

1

15

05

1

15

05

1

15

04

06

08

1

12

14

16

V

04

06

08

1

12

14

16

V

04

06

08

1

12

14

16

V


V - L = 10 Ｇ V - L = 20 Ｇ

kV - L = 30 Ｇ kV - L = 40 Ｇ

V - L = 30 Ｇ V - L = 40 Ｇ

L = 10 Ｇ L = 20 Ｇ

L = 30 Ｇ L = 40 Ｇ


L = 50ndash200Ｇ V - V -

V -V -

V -



0

02

04

06

08

1

CDF

0005 001 0015 00200

02

04

06

08

1

0

2

4

6

8

10

PDF

104 06 080200

50

100

150

200

250

V [mdash] L [mdash]


Appendix


See Figures 7 and 8



1198701199031119894 = 1198701199032119894 = 119870119903119894 (C1)



Θ1119894 = Θ2119894 = Θ119894 (C2)







(C5)




(C6)





L - L = 10 Ｇ L - L = 20 Ｇ

kL - L = 30 Ｇ kL - L = 40 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ

L - L = 10 Ｇ L - L = 20 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ



0

0

02

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008

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002501

002501

002501

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1

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L

04

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08

1

12

14

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L

105 15

1 1505

105 15 105 15

105 15105 15

1 1505 1 1505

1 15051 1505

105 15 105 15

1 1505

L ()1 1505

L ()1 1505

L ()

05

1

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1

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1

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1

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05

1

15




V - L = 10 Ｇ V - L = 20 Ｇ

kV - L = 30 Ｇ kV - L = 40 Ｇ

V - L = 30 Ｇ V - L = 40 Ｇ

L = 10 Ｇ L = 20 Ｇ

L = 30 Ｇ L = 40 Ｇ


L = 50ndash200Ｇ

0

02

02

04

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06

0808

minus04

minus02

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002501 02

002501

002501

002501

1 1505

V ()1 1505

V ()

1505 115105

1 1505 1505 1

151051 1505

151051505 1

1 1505 1 1505

05

1

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1

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1

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1 1505

V ()

1 1505

1 1505

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1

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V

04

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08

1

12

14

16

V

04

06

08

1

12

14

16

V

V - V -

V - V -V -





119872119894 sdot (RL119894 rarr 1) (C8)



(C9)


119872119894 (RL119894 997888rarr 1) = Θ119894 (RL119894 = 0) sdot 2119864119868119894119871 (C10)





RD119894 = Θ119894 (RL119894) sdot 119871120575119894 (RL119894) (C12)





(C14)


RL119894


(C15)





119898119894sdot 120593119894 (RL119894 120572119894 119871 120585119894)

(C17)


BF119894 (RL119894 120572119894 119871 120585119894) = 120578119894 (RL119894 120572119894 119871 120585119894)120578119894 (RL119894 = 0 120572119894 119871 120585119894) (C18)



(C19)


BF119894 (RL119894 120572119894 119871 120585119894)


120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)= 119898119894 (RL119894 = 0)

119898119894 (RL119894)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟BF119898119894(RL119894)

sdot 120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟BF120593119894(RL119894 120572119894 119871120585119894)

(C20)

obtaining


= 119898119894 (RL119894 = 0)119898119894 (RL119894)

sdot 120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)

(C21)





Acknowledgments


References

























































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



Kr1V

Kr1L

m

y

x

z1 2

uL(x t)

uV(x t)

cV cL EIV EIL Kr2V

Kr2L

L




120601119894 (119909) = 1198881119894 sin(120582119894 119909119871) + 1198882119894 cos(120582119894 119909119871)+ 1198883119894 sinh(120582119894 119909119871) + 1198884119894 cosh (120582119894 119909119871)

(10)



120578119894 (119905) + 4120587120585119894119891119894 120578119894 (119905) + 412058721198912119894 120578119894 (119905) = 119898minus1119894 119891119894 (119905) (11)



120578119894 (119905) = 1119898119894 int119905










RL119894 = 119872119894 (119870119903119894)119872119894 (119870119903119894 rarr infin) (13)






08 09 107


0

02

04

06

08

1

RLi

(a)

10minus1 100 101 10210minus2

Kri middot LEIi

0

02

04

06

08

1

RLi

(b)

1 20 3 4

RDi

0

02

04

06

08

1

RLi

(c)






RD119894 = Θ119894 sdot 119871120575119894 (15)








Modal force


fi(t)

Mechanical model

FEM model



p ()

pW(W)

LL

V Lm

V L

２＄V ２＄L

２V ２L


i fi(t)





pf(fw)

p＄(＄V)

p＄(＄L)



mi

( )minus1

＄i i









(18)






005

005

005

005

01

01

0101

02

02 02

0404 04

0707 0909

005005

005

005 005

0101

01

02

02

0404

040707

07 0909

0

01

02

03

04

05

06

07

08

09

1

0

01

02

03

04

05

06

07

08

09

1

50 100 150 2000L

04

06

08

1

12

14

16V

04

06

08

1

12

14

16

L

50 100 150 2000L

01

02

V(V L V = 0015) L(L L L = 0015)








120593119894 (RL119894 = 0 120572119894 119871 120585119894) = min

120585119894 sdot 2120572211989410038161003816100381610038161 minus 1205722119894 1003816100381610038161003816 [1 +1119899119894



(19)


119899119894 (119871) = 2 sdot 119891119908119894V






BF119894 (RL119894 120572119894 119871 120585119894) = 120578119894 (RL119894 120572119894 119871 120585119894)120578119894 (RL119894 = 0 120572119894 119871 120585119894) (21)



119898119894 (RL119894)sdot 120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)

(22)




120593119894 (RL119894 120572119894 119871 120585119894) = BF120593119894 (RL119894 120572119894 119871 120585119894)sdot 120593119894 (RL119894 = 0 120572119894 119871 120585119894)

(23)








119901120578119894 (120578119894 (120572119894 119871 120585119894)) = intinfin0

intinfin0

intinfin0

intinfin0


(24)



(25)






２V = 000２V = 025２V = 050

２V = 075２V = 100

07

075

08

085

09

095

1

105

11

115

12

125

13V

50 100 150 2000L

12

12

125

125125

125125

125

12513

1

105

11

115

12

125

13

135

14

145

15

04

06

08

1

12

14

16

V

50 100 150 2000L

V(２V V = 1 L V = 0015) V(２V = 1 V L V = 0015)



















119863119894 = 120593119894 (RL119894 120572119894 119871 120585119894) le 120593119894max = 119862119894 forall119894 (27)


119862119894 = 120593119894max


(28)



119875fail119894 = int119862minus119863le0

119901119862119894 119863119894 (119888119894 119889119894) sdot d119888119894 sdot d119889119894 (29)




0119901120593119894 (120593119894 (RL119894 120572119894 119871 120585119894)) sdot d120593119894 (30)











11 115115

12

12

12

125

11 115115

1212

125

125

115

12

12

12

125

125

11 115

12

1212

125

125

115

12

1212

125

125

125

115

12

12125

125

115

12

1212

125

125

12

12125

125

125

1212

125

125

12

12125

125125

12

12125

125

12

12125

125

125

12

12125

125

12

12

125

125125

125

12

12125

125

12

12

125

125

125 125

125

V = 0001 V = 0003

V = 0005 V = 0007

V = 0009 V = 0011

V = 0013 V = 0015

15

14

13

12

11

1

15

14

13

12

11

1

15

14

13

12

11

1

15

14

13

12

11

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14

13

12

11

1

15

14

13

12

11

1

15

14

13

12

11

1

15

14

13

12

11

1

0 50 200100 150

L

0 50 200100 150

L

05

1

15

V

05

1

15

V

05

1

15

V

05

1

15

V

0 50 200100 150

L

0 50 200100 150

L

05

1

15

V

0 50 200100 150

L

0 50 200100 150

L

05

1

15

V

05

1

15

V

0 50 200100 150

L

05

1

15

V

0 50 200100 150

L



11 115115

12 12

12

125

1

11

12

13

14

15

115

12

12

12

125

125

1

11

12

13

14

15

115

12

1212

125

125

125

1

11

12

13

14

15

115

12

12

12125

125

1

11

12

13

14

15

12

12

125

125

1

11

12

13

14

15

12

12

125

125

1

11

12

13

14

15

12

12

125

125

1

11

12

13

14

15

12

12

125

125

1

11

12

13

14

15

L = 0001 L = 0003

L = 0007

L = 0011

L = 0015

L = 0005

L = 0009

L = 0013

50 100 150 2000

L

50 100 150 2000

L

50 100 150 2000

L

50 100 150 2000

L

50 100 150 2000

L

50 100 150 2000

L

100 200150500

L

100 200150500

L

05

1

15

L

05

1

15

L

05

1

15

L

05

1

15

L

05

1

15

L

05

1

15

L

05

1

15

L

05

1

15

L






= 021 if 120572119881 le 0470140 + 0150 sdot 120572119881 if 120572119881 gt 047



(33)










316fL

05606307609512719

0

05

1

15

2

25

3

35

4

45

5

L

max

075 1 12505 15 1703L

633fV

1121271521925338

0

005

01

015

02

025

03

035

04

045

05

V

max

075 1 12505 15 1703V






(34)



sdot 120578119881max = 0775


sdot 120578119871max = 5723(35)


119863119881 = 120593119881 (RL119881 120572119881 119871 120585119881) = 1 sdot 00904 = 00904119863119871 = 120593119871 (RL119871 120572119871 119871 120585119871) = 097 sdot 00035 = 00034 (36)


119863119881 = 00904 lt 0775 = 119862119881119863119871 = 00034 lt 5723 = 119862119871 (37)





(38)


119875120593119881 (119863119881) = 0565119875120593119871 (119863119871) = 0379 (39)


119875fail119881 = 00296 lArrrArr120573119881 = 1887


(40)






(41)


7 Conclusions




0

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1 1505

L ()1 1505

L ()

1 1505105 15

15105 15105

105 15 105 15

1505 1 1505 1

1 15051 1505

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15

105 15

L ()

15105

1 1505

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1

12

14

16

L

04

06

08

1

12

14

16

L

04

06

08

1

12

14

16

L


L - L = 10 Ｇ L - L = 20 Ｇ

kL - L = 30 Ｇ kL - L = 40 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ

L - L = 10 Ｇ L - L = 20 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ





0

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01

002501 0025

01

05 1 15

V ()

105 15

15105

1 1505

V ()1505 1

V ()

1 15051 1505

1 1505 1 1505

1505 115105

1 150505 151

1 1505 105 15

05

1

15

05

1

15

05

1

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05

1

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1

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1

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12

14

16

V

04

06

08

1

12

14

16

V

04

06

08

1

12

14

16

V


V - L = 10 Ｇ V - L = 20 Ｇ

kV - L = 30 Ｇ kV - L = 40 Ｇ

V - L = 30 Ｇ V - L = 40 Ｇ

L = 10 Ｇ L = 20 Ｇ

L = 30 Ｇ L = 40 Ｇ


L = 50ndash200Ｇ V - V -

V -V -

V -



0

02

04

06

08

1

CDF

0005 001 0015 00200

02

04

06

08

1

0

2

4

6

8

10

PDF

104 06 080200

50

100

150

200

250

V [mdash] L [mdash]


Appendix


See Figures 7 and 8



1198701199031119894 = 1198701199032119894 = 119870119903119894 (C1)



Θ1119894 = Θ2119894 = Θ119894 (C2)







(C5)




(C6)





L - L = 10 Ｇ L - L = 20 Ｇ

kL - L = 30 Ｇ kL - L = 40 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ

L - L = 10 Ｇ L - L = 20 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ



0

0

02

02

04

04

06

06

minus04

minus02

0

02

04

06

08

1

12

0

00

0

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0404

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0

0

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0001

0025

0025

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002

004

006

008

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012

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016

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02

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0025

0001

002501

0001

002501

0001

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0001

0025

0025

005

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002

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006

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01

012

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002501

04

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1

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L

04

06

08

1

12

14

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L

04

06

08

1

12

14

16

L

105 15

1 1505

105 15 105 15

105 15105 15

1 1505 1 1505

1 15051 1505

105 15 105 15

1 1505

L ()1 1505

L ()1 1505

L ()

05

1

15

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1

15

05

1

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1

15

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1

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1

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05

1

15

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1

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05

1

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05

1

15

05

1

15

05

1

15




V - L = 10 Ｇ V - L = 20 Ｇ

kV - L = 30 Ｇ kV - L = 40 Ｇ

V - L = 30 Ｇ V - L = 40 Ｇ

L = 10 Ｇ L = 20 Ｇ

L = 30 Ｇ L = 40 Ｇ


L = 50ndash200Ｇ

0

02

02

04

04

06

06

0808

minus04

minus02

0

02

04

06

08

1

12

00

0

0404

0

04

04

0808

0

04

04

0808

0

04

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0808

0001

0025

0025

005

00501

0

002

004

006

008

01

012

014

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02

002501 0025

01

002501 0025

01

0025

0025

005

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015

0

002

004

006

008

01

012

014

016

018

02

002501 02

002501

002501

002501

1 1505

V ()1 1505

V ()

1505 115105

1 1505 1505 1

151051 1505

151051505 1

1 1505 1 1505

05

1

15

05

1

15

05

1

15

05

1

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05

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1 1505

V ()

1 1505

1 1505

04

06

08

1

12

14

16

V

04

06

08

1

12

14

16

V

04

06

08

1

12

14

16

V

V - V -

V - V -V -





119872119894 sdot (RL119894 rarr 1) (C8)



(C9)


119872119894 (RL119894 997888rarr 1) = Θ119894 (RL119894 = 0) sdot 2119864119868119894119871 (C10)





RD119894 = Θ119894 (RL119894) sdot 119871120575119894 (RL119894) (C12)





(C14)


RL119894


(C15)





119898119894sdot 120593119894 (RL119894 120572119894 119871 120585119894)

(C17)


BF119894 (RL119894 120572119894 119871 120585119894) = 120578119894 (RL119894 120572119894 119871 120585119894)120578119894 (RL119894 = 0 120572119894 119871 120585119894) (C18)



(C19)


BF119894 (RL119894 120572119894 119871 120585119894)


120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)= 119898119894 (RL119894 = 0)

119898119894 (RL119894)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟BF119898119894(RL119894)

sdot 120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟BF120593119894(RL119894 120572119894 119871120585119894)

(C20)

obtaining


= 119898119894 (RL119894 = 0)119898119894 (RL119894)

sdot 120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)

(C21)





Acknowledgments


References

























































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia







08 09 107


0

02

04

06

08

1

RLi

(a)

10minus1 100 101 10210minus2

Kri middot LEIi

0

02

04

06

08

1

RLi

(b)

1 20 3 4

RDi

0

02

04

06

08

1

RLi

(c)






RD119894 = Θ119894 sdot 119871120575119894 (15)








Modal force


fi(t)

Mechanical model

FEM model



p ()

pW(W)

LL

V Lm

V L

２＄V ２＄L

２V ２L


i fi(t)





pf(fw)

p＄(＄V)

p＄(＄L)



mi

( )minus1

＄i i









(18)






005

005

005

005

01

01

0101

02

02 02

0404 04

0707 0909

005005

005

005 005

0101

01

02

02

0404

040707

07 0909

0

01

02

03

04

05

06

07

08

09

1

0

01

02

03

04

05

06

07

08

09

1

50 100 150 2000L

04

06

08

1

12

14

16V

04

06

08

1

12

14

16

L

50 100 150 2000L

01

02

V(V L V = 0015) L(L L L = 0015)








120593119894 (RL119894 = 0 120572119894 119871 120585119894) = min

120585119894 sdot 2120572211989410038161003816100381610038161 minus 1205722119894 1003816100381610038161003816 [1 +1119899119894



(19)


119899119894 (119871) = 2 sdot 119891119908119894V






BF119894 (RL119894 120572119894 119871 120585119894) = 120578119894 (RL119894 120572119894 119871 120585119894)120578119894 (RL119894 = 0 120572119894 119871 120585119894) (21)



119898119894 (RL119894)sdot 120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)

(22)




120593119894 (RL119894 120572119894 119871 120585119894) = BF120593119894 (RL119894 120572119894 119871 120585119894)sdot 120593119894 (RL119894 = 0 120572119894 119871 120585119894)

(23)








119901120578119894 (120578119894 (120572119894 119871 120585119894)) = intinfin0

intinfin0

intinfin0

intinfin0


(24)



(25)






２V = 000２V = 025２V = 050

２V = 075２V = 100

07

075

08

085

09

095

1

105

11

115

12

125

13V

50 100 150 2000L

12

12

125

125125

125125

125

12513

1

105

11

115

12

125

13

135

14

145

15

04

06

08

1

12

14

16

V

50 100 150 2000L

V(２V V = 1 L V = 0015) V(２V = 1 V L V = 0015)



















119863119894 = 120593119894 (RL119894 120572119894 119871 120585119894) le 120593119894max = 119862119894 forall119894 (27)


119862119894 = 120593119894max


(28)



119875fail119894 = int119862minus119863le0

119901119862119894 119863119894 (119888119894 119889119894) sdot d119888119894 sdot d119889119894 (29)




0119901120593119894 (120593119894 (RL119894 120572119894 119871 120585119894)) sdot d120593119894 (30)











11 115115

12

12

12

125

11 115115

1212

125

125

115

12

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12

125

125

11 115

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1212

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125125

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12125

125

12

12

125

125125

125

12

12125

125

12

12

125

125

125 125

125

V = 0001 V = 0003

V = 0005 V = 0007

V = 0009 V = 0011

V = 0013 V = 0015

15

14

13

12

11

1

15

14

13

12

11

1

15

14

13

12

11

1

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11

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15

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11

1

15

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13

12

11

1

15

14

13

12

11

1

0 50 200100 150

L

0 50 200100 150

L

05

1

15

V

05

1

15

V

05

1

15

V

05

1

15

V

0 50 200100 150

L

0 50 200100 150

L

05

1

15

V

0 50 200100 150

L

0 50 200100 150

L

05

1

15

V

05

1

15

V

0 50 200100 150

L

05

1

15

V

0 50 200100 150

L



11 115115

12 12

12

125

1

11

12

13

14

15

115

12

12

12

125

125

1

11

12

13

14

15

115

12

1212

125

125

125

1

11

12

13

14

15

115

12

12

12125

125

1

11

12

13

14

15

12

12

125

125

1

11

12

13

14

15

12

12

125

125

1

11

12

13

14

15

12

12

125

125

1

11

12

13

14

15

12

12

125

125

1

11

12

13

14

15

L = 0001 L = 0003

L = 0007

L = 0011

L = 0015

L = 0005

L = 0009

L = 0013

50 100 150 2000

L

50 100 150 2000

L

50 100 150 2000

L

50 100 150 2000

L

50 100 150 2000

L

50 100 150 2000

L

100 200150500

L

100 200150500

L

05

1

15

L

05

1

15

L

05

1

15

L

05

1

15

L

05

1

15

L

05

1

15

L

05

1

15

L

05

1

15

L






= 021 if 120572119881 le 0470140 + 0150 sdot 120572119881 if 120572119881 gt 047



(33)










316fL

05606307609512719

0

05

1

15

2

25

3

35

4

45

5

L

max

075 1 12505 15 1703L

633fV

1121271521925338

0

005

01

015

02

025

03

035

04

045

05

V

max

075 1 12505 15 1703V






(34)



sdot 120578119881max = 0775


sdot 120578119871max = 5723(35)


119863119881 = 120593119881 (RL119881 120572119881 119871 120585119881) = 1 sdot 00904 = 00904119863119871 = 120593119871 (RL119871 120572119871 119871 120585119871) = 097 sdot 00035 = 00034 (36)


119863119881 = 00904 lt 0775 = 119862119881119863119871 = 00034 lt 5723 = 119862119871 (37)





(38)


119875120593119881 (119863119881) = 0565119875120593119871 (119863119871) = 0379 (39)


119875fail119881 = 00296 lArrrArr120573119881 = 1887


(40)






(41)


7 Conclusions




0

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002501

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014

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002501

002501

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002501

1 1505

L ()1 1505

L ()

1 1505105 15

15105 15105

105 15 105 15

1505 1 1505 1

1 15051 1505

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105 15

L ()

15105

1 1505

04

06

08

1

12

14

16

L

04

06

08

1

12

14

16

L

04

06

08

1

12

14

16

L


L - L = 10 Ｇ L - L = 20 Ｇ

kL - L = 30 Ｇ kL - L = 40 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ

L - L = 10 Ｇ L - L = 20 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ





0

02

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04

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06

0808

minus04

minus02

0

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0 0

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0808

0001

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002501

0001

002501

0001

002501

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0001

0025

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005

00501

0

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006

008

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002501 0025

01

002501 0025

01

05 1 15

V ()

105 15

15105

1 1505

V ()1505 1

V ()

1 15051 1505

1 1505 1 1505

1505 115105

1 150505 151

1 1505 105 15

05

1

15

05

1

15

05

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1

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1

12

14

16

V

04

06

08

1

12

14

16

V

04

06

08

1

12

14

16

V


V - L = 10 Ｇ V - L = 20 Ｇ

kV - L = 30 Ｇ kV - L = 40 Ｇ

V - L = 30 Ｇ V - L = 40 Ｇ

L = 10 Ｇ L = 20 Ｇ

L = 30 Ｇ L = 40 Ｇ


L = 50ndash200Ｇ V - V -

V -V -

V -



0

02

04

06

08

1

CDF

0005 001 0015 00200

02

04

06

08

1

0

2

4

6

8

10

PDF

104 06 080200

50

100

150

200

250

V [mdash] L [mdash]


Appendix


See Figures 7 and 8



1198701199031119894 = 1198701199032119894 = 119870119903119894 (C1)



Θ1119894 = Θ2119894 = Θ119894 (C2)







(C5)




(C6)





L - L = 10 Ｇ L - L = 20 Ｇ

kL - L = 30 Ｇ kL - L = 40 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ

L - L = 10 Ｇ L - L = 20 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ



0

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L

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L

04

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L

105 15

1 1505

105 15 105 15

105 15105 15

1 1505 1 1505

1 15051 1505

105 15 105 15

1 1505

L ()1 1505

L ()1 1505

L ()

05

1

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1

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05

1

15




V - L = 10 Ｇ V - L = 20 Ｇ

kV - L = 30 Ｇ kV - L = 40 Ｇ

V - L = 30 Ｇ V - L = 40 Ｇ

L = 10 Ｇ L = 20 Ｇ

L = 30 Ｇ L = 40 Ｇ


L = 50ndash200Ｇ

0

02

02

04

04

06

06

0808

minus04

minus02

0

02

04

06

08

1

12

00

0

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0

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0808

0

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0808

0001

0025

0025

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002501 0025

01

002501 0025

01

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002501 02

002501

002501

002501

1 1505

V ()1 1505

V ()

1505 115105

1 1505 1505 1

151051 1505

151051505 1

1 1505 1 1505

05

1

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05

1

15

05

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V ()

1 1505

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V

04

06

08

1

12

14

16

V

04

06

08

1

12

14

16

V

V - V -

V - V -V -





119872119894 sdot (RL119894 rarr 1) (C8)



(C9)


119872119894 (RL119894 997888rarr 1) = Θ119894 (RL119894 = 0) sdot 2119864119868119894119871 (C10)





RD119894 = Θ119894 (RL119894) sdot 119871120575119894 (RL119894) (C12)





(C14)


RL119894


(C15)





119898119894sdot 120593119894 (RL119894 120572119894 119871 120585119894)

(C17)


BF119894 (RL119894 120572119894 119871 120585119894) = 120578119894 (RL119894 120572119894 119871 120585119894)120578119894 (RL119894 = 0 120572119894 119871 120585119894) (C18)



(C19)


BF119894 (RL119894 120572119894 119871 120585119894)


120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)= 119898119894 (RL119894 = 0)

119898119894 (RL119894)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟BF119898119894(RL119894)

sdot 120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟BF120593119894(RL119894 120572119894 119871120585119894)

(C20)

obtaining


= 119898119894 (RL119894 = 0)119898119894 (RL119894)

sdot 120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)

(C21)





Acknowledgments


References

























































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



Modal force


fi(t)

Mechanical model

FEM model



p ()

pW(W)

LL

V Lm

V L

２＄V ２＄L

２V ２L


i fi(t)





pf(fw)

p＄(＄V)

p＄(＄L)



mi

( )minus1

＄i i









(18)






005

005

005

005

01

01

0101

02

02 02

0404 04

0707 0909

005005

005

005 005

0101

01

02

02

0404

040707

07 0909

0

01

02

03

04

05

06

07

08

09

1

0

01

02

03

04

05

06

07

08

09

1

50 100 150 2000L

04

06

08

1

12

14

16V

04

06

08

1

12

14

16

L

50 100 150 2000L

01

02

V(V L V = 0015) L(L L L = 0015)








120593119894 (RL119894 = 0 120572119894 119871 120585119894) = min

120585119894 sdot 2120572211989410038161003816100381610038161 minus 1205722119894 1003816100381610038161003816 [1 +1119899119894



(19)


119899119894 (119871) = 2 sdot 119891119908119894V






BF119894 (RL119894 120572119894 119871 120585119894) = 120578119894 (RL119894 120572119894 119871 120585119894)120578119894 (RL119894 = 0 120572119894 119871 120585119894) (21)



119898119894 (RL119894)sdot 120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)

(22)




120593119894 (RL119894 120572119894 119871 120585119894) = BF120593119894 (RL119894 120572119894 119871 120585119894)sdot 120593119894 (RL119894 = 0 120572119894 119871 120585119894)

(23)








119901120578119894 (120578119894 (120572119894 119871 120585119894)) = intinfin0

intinfin0

intinfin0

intinfin0


(24)



(25)






２V = 000２V = 025２V = 050

２V = 075２V = 100

07

075

08

085

09

095

1

105

11

115

12

125

13V

50 100 150 2000L

12

12

125

125125

125125

125

12513

1

105

11

115

12

125

13

135

14

145

15

04

06

08

1

12

14

16

V

50 100 150 2000L

V(２V V = 1 L V = 0015) V(２V = 1 V L V = 0015)



















119863119894 = 120593119894 (RL119894 120572119894 119871 120585119894) le 120593119894max = 119862119894 forall119894 (27)


119862119894 = 120593119894max


(28)



119875fail119894 = int119862minus119863le0

119901119862119894 119863119894 (119888119894 119889119894) sdot d119888119894 sdot d119889119894 (29)




0119901120593119894 (120593119894 (RL119894 120572119894 119871 120585119894)) sdot d120593119894 (30)











11 115115

12

12

12

125

11 115115

1212

125

125

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12

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V = 0001 V = 0003

V = 0005 V = 0007

V = 0009 V = 0011

V = 0013 V = 0015

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L

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0 50 200100 150

L

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0 50 200100 150

L



11 115115

12 12

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1

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125

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L = 0001 L = 0003

L = 0007

L = 0011

L = 0015

L = 0005

L = 0009

L = 0013

50 100 150 2000

L

50 100 150 2000

L

50 100 150 2000

L

50 100 150 2000

L

50 100 150 2000

L

50 100 150 2000

L

100 200150500

L

100 200150500

L

05

1

15

L

05

1

15

L

05

1

15

L

05

1

15

L

05

1

15

L

05

1

15

L

05

1

15

L

05

1

15

L






= 021 if 120572119881 le 0470140 + 0150 sdot 120572119881 if 120572119881 gt 047



(33)










316fL

05606307609512719

0

05

1

15

2

25

3

35

4

45

5

L

max

075 1 12505 15 1703L

633fV

1121271521925338

0

005

01

015

02

025

03

035

04

045

05

V

max

075 1 12505 15 1703V






(34)



sdot 120578119881max = 0775


sdot 120578119871max = 5723(35)


119863119881 = 120593119881 (RL119881 120572119881 119871 120585119881) = 1 sdot 00904 = 00904119863119871 = 120593119871 (RL119871 120572119871 119871 120585119871) = 097 sdot 00035 = 00034 (36)


119863119881 = 00904 lt 0775 = 119862119881119863119871 = 00034 lt 5723 = 119862119871 (37)





(38)


119875120593119881 (119863119881) = 0565119875120593119871 (119863119871) = 0379 (39)


119875fail119881 = 00296 lArrrArr120573119881 = 1887


(40)






(41)


7 Conclusions




0

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002501

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1 1505

L ()1 1505

L ()

1 1505105 15

15105 15105

105 15 105 15

1505 1 1505 1

1 15051 1505

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1

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L ()

15105

1 1505

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L

04

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1

12

14

16

L

04

06

08

1

12

14

16

L


L - L = 10 Ｇ L - L = 20 Ｇ

kL - L = 30 Ｇ kL - L = 40 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ

L - L = 10 Ｇ L - L = 20 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ





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012

014

016

018

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002501 0025

01

002501 0025

01

05 1 15

V ()

105 15

15105

1 1505

V ()1505 1

V ()

1 15051 1505

1 1505 1 1505

1505 115105

1 150505 151

1 1505 105 15

05

1

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1

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05

1

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1

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V

04

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1

12

14

16

V

04

06

08

1

12

14

16

V


V - L = 10 Ｇ V - L = 20 Ｇ

kV - L = 30 Ｇ kV - L = 40 Ｇ

V - L = 30 Ｇ V - L = 40 Ｇ

L = 10 Ｇ L = 20 Ｇ

L = 30 Ｇ L = 40 Ｇ


L = 50ndash200Ｇ V - V -

V -V -

V -



0

02

04

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08

1

CDF

0005 001 0015 00200

02

04

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1

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PDF

104 06 080200

50

100

150

200

250

V [mdash] L [mdash]


Appendix


See Figures 7 and 8



1198701199031119894 = 1198701199032119894 = 119870119903119894 (C1)



Θ1119894 = Θ2119894 = Θ119894 (C2)







(C5)




(C6)





L - L = 10 Ｇ L - L = 20 Ｇ

kL - L = 30 Ｇ kL - L = 40 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ

L - L = 10 Ｇ L - L = 20 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ



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L

105 15

1 1505

105 15 105 15

105 15105 15

1 1505 1 1505

1 15051 1505

105 15 105 15

1 1505

L ()1 1505

L ()1 1505

L ()

05

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1

15




V - L = 10 Ｇ V - L = 20 Ｇ

kV - L = 30 Ｇ kV - L = 40 Ｇ

V - L = 30 Ｇ V - L = 40 Ｇ

L = 10 Ｇ L = 20 Ｇ

L = 30 Ｇ L = 40 Ｇ


L = 50ndash200Ｇ

0

02

02

04

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06

0808

minus04

minus02

0

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1

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00

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0808

0

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0

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0808

0001

0025

0025

005

00501

0

002

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006

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01

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002501 0025

01

002501 0025

01

0025

0025

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00501

015

0

002

004

006

008

01

012

014

016

018

02

002501 02

002501

002501

002501

1 1505

V ()1 1505

V ()

1505 115105

1 1505 1505 1

151051 1505

151051505 1

1 1505 1 1505

05

1

15

05

1

15

05

1

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1 1505

V ()

1 1505

1 1505

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V

04

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08

1

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16

V

04

06

08

1

12

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V

V - V -

V - V -V -





119872119894 sdot (RL119894 rarr 1) (C8)



(C9)


119872119894 (RL119894 997888rarr 1) = Θ119894 (RL119894 = 0) sdot 2119864119868119894119871 (C10)





RD119894 = Θ119894 (RL119894) sdot 119871120575119894 (RL119894) (C12)





(C14)


RL119894


(C15)





119898119894sdot 120593119894 (RL119894 120572119894 119871 120585119894)

(C17)


BF119894 (RL119894 120572119894 119871 120585119894) = 120578119894 (RL119894 120572119894 119871 120585119894)120578119894 (RL119894 = 0 120572119894 119871 120585119894) (C18)



(C19)


BF119894 (RL119894 120572119894 119871 120585119894)


120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)= 119898119894 (RL119894 = 0)

119898119894 (RL119894)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟BF119898119894(RL119894)

sdot 120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟BF120593119894(RL119894 120572119894 119871120585119894)

(C20)

obtaining


= 119898119894 (RL119894 = 0)119898119894 (RL119894)

sdot 120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)

(C21)





Acknowledgments


References

























































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



005

005

005

005

01

01

0101

02

02 02

0404 04

0707 0909

005005

005

005 005

0101

01

02

02

0404

040707

07 0909

0

01

02

03

04

05

06

07

08

09

1

0

01

02

03

04

05

06

07

08

09

1

50 100 150 2000L

04

06

08

1

12

14

16V

04

06

08

1

12

14

16

L

50 100 150 2000L

01

02

V(V L V = 0015) L(L L L = 0015)








120593119894 (RL119894 = 0 120572119894 119871 120585119894) = min

120585119894 sdot 2120572211989410038161003816100381610038161 minus 1205722119894 1003816100381610038161003816 [1 +1119899119894



(19)


119899119894 (119871) = 2 sdot 119891119908119894V






BF119894 (RL119894 120572119894 119871 120585119894) = 120578119894 (RL119894 120572119894 119871 120585119894)120578119894 (RL119894 = 0 120572119894 119871 120585119894) (21)



119898119894 (RL119894)sdot 120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)

(22)




120593119894 (RL119894 120572119894 119871 120585119894) = BF120593119894 (RL119894 120572119894 119871 120585119894)sdot 120593119894 (RL119894 = 0 120572119894 119871 120585119894)

(23)








119901120578119894 (120578119894 (120572119894 119871 120585119894)) = intinfin0

intinfin0

intinfin0

intinfin0


(24)



(25)






２V = 000２V = 025２V = 050

２V = 075２V = 100

07

075

08

085

09

095

1

105

11

115

12

125

13V

50 100 150 2000L

12

12

125

125125

125125

125

12513

1

105

11

115

12

125

13

135

14

145

15

04

06

08

1

12

14

16

V

50 100 150 2000L

V(２V V = 1 L V = 0015) V(２V = 1 V L V = 0015)



















119863119894 = 120593119894 (RL119894 120572119894 119871 120585119894) le 120593119894max = 119862119894 forall119894 (27)


119862119894 = 120593119894max


(28)



119875fail119894 = int119862minus119863le0

119901119862119894 119863119894 (119888119894 119889119894) sdot d119888119894 sdot d119889119894 (29)




0119901120593119894 (120593119894 (RL119894 120572119894 119871 120585119894)) sdot d120593119894 (30)











11 115115

12

12

12

125

11 115115

1212

125

125

115

12

12

12

125

125

11 115

12

1212

125

125

115

12

1212

125

125

125

115

12

12125

125

115

12

1212

125

125

12

12125

125

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1212

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125

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12125

125125

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12125

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12125

125

125

12

12125

125

12

12

125

125125

125

12

12125

125

12

12

125

125

125 125

125

V = 0001 V = 0003

V = 0005 V = 0007

V = 0009 V = 0011

V = 0013 V = 0015

15

14

13

12

11

1

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13

12

11

1

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0 50 200100 150

L

0 50 200100 150

L

05

1

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V

05

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05

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05

1

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V

0 50 200100 150

L

0 50 200100 150

L

05

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0 50 200100 150

L

0 50 200100 150

L

05

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V

05

1

15

V

0 50 200100 150

L

05

1

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V

0 50 200100 150

L



11 115115

12 12

12

125

1

11

12

13

14

15

115

12

12

12

125

125

1

11

12

13

14

15

115

12

1212

125

125

125

1

11

12

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115

12

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12125

125

1

11

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125

1

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125

1

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125

1

11

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13

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125

125

1

11

12

13

14

15

L = 0001 L = 0003

L = 0007

L = 0011

L = 0015

L = 0005

L = 0009

L = 0013

50 100 150 2000

L

50 100 150 2000

L

50 100 150 2000

L

50 100 150 2000

L

50 100 150 2000

L

50 100 150 2000

L

100 200150500

L

100 200150500

L

05

1

15

L

05

1

15

L

05

1

15

L

05

1

15

L

05

1

15

L

05

1

15

L

05

1

15

L

05

1

15

L






= 021 if 120572119881 le 0470140 + 0150 sdot 120572119881 if 120572119881 gt 047



(33)










316fL

05606307609512719

0

05

1

15

2

25

3

35

4

45

5

L

max

075 1 12505 15 1703L

633fV

1121271521925338

0

005

01

015

02

025

03

035

04

045

05

V

max

075 1 12505 15 1703V






(34)



sdot 120578119881max = 0775


sdot 120578119871max = 5723(35)


119863119881 = 120593119881 (RL119881 120572119881 119871 120585119881) = 1 sdot 00904 = 00904119863119871 = 120593119871 (RL119871 120572119871 119871 120585119871) = 097 sdot 00035 = 00034 (36)


119863119881 = 00904 lt 0775 = 119862119881119863119871 = 00034 lt 5723 = 119862119871 (37)





(38)


119875120593119881 (119863119881) = 0565119875120593119871 (119863119871) = 0379 (39)


119875fail119881 = 00296 lArrrArr120573119881 = 1887


(40)






(41)


7 Conclusions




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1 1505

L ()1 1505

L ()

1 1505105 15

15105 15105

105 15 105 15

1505 1 1505 1

1 15051 1505

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L

04

06

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1

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14

16

L


L - L = 10 Ｇ L - L = 20 Ｇ

kL - L = 30 Ｇ kL - L = 40 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ

L - L = 10 Ｇ L - L = 20 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ





0

02

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minus04

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0808

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0808

0001

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002501

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01

002501 0025

01

05 1 15

V ()

105 15

15105

1 1505

V ()1505 1

V ()

1 15051 1505

1 1505 1 1505

1505 115105

1 150505 151

1 1505 105 15

05

1

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05

1

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05

1

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1

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16

V

04

06

08

1

12

14

16

V

04

06

08

1

12

14

16

V


V - L = 10 Ｇ V - L = 20 Ｇ

kV - L = 30 Ｇ kV - L = 40 Ｇ

V - L = 30 Ｇ V - L = 40 Ｇ

L = 10 Ｇ L = 20 Ｇ

L = 30 Ｇ L = 40 Ｇ


L = 50ndash200Ｇ V - V -

V -V -

V -



0

02

04

06

08

1

CDF

0005 001 0015 00200

02

04

06

08

1

0

2

4

6

8

10

PDF

104 06 080200

50

100

150

200

250

V [mdash] L [mdash]


Appendix


See Figures 7 and 8



1198701199031119894 = 1198701199032119894 = 119870119903119894 (C1)



Θ1119894 = Θ2119894 = Θ119894 (C2)







(C5)




(C6)





L - L = 10 Ｇ L - L = 20 Ｇ

kL - L = 30 Ｇ kL - L = 40 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ

L - L = 10 Ｇ L - L = 20 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ



0

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minus02

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0404

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0025

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008

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016

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002501

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002501

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0001

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006

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002501

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1

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L

04

06

08

1

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14

16

L

04

06

08

1

12

14

16

L

105 15

1 1505

105 15 105 15

105 15105 15

1 1505 1 1505

1 15051 1505

105 15 105 15

1 1505

L ()1 1505

L ()1 1505

L ()

05

1

15

05

1

15

05

1

15

05

1

15

05

1

15

05

1

15

05

1

15

05

1

15

05

1

15

05

1

15

05

1

15

05

1

15




V - L = 10 Ｇ V - L = 20 Ｇ

kV - L = 30 Ｇ kV - L = 40 Ｇ

V - L = 30 Ｇ V - L = 40 Ｇ

L = 10 Ｇ L = 20 Ｇ

L = 30 Ｇ L = 40 Ｇ


L = 50ndash200Ｇ

0

02

02

04

04

06

06

0808

minus04

minus02

0

02

04

06

08

1

12

00

0

0404

0

04

04

0808

0

04

04

0808

0

04

04

0808

0001

0025

0025

005

00501

0

002

004

006

008

01

012

014

016

018

02

002501 0025

01

002501 0025

01

0025

0025

005

00501

015

0

002

004

006

008

01

012

014

016

018

02

002501 02

002501

002501

002501

1 1505

V ()1 1505

V ()

1505 115105

1 1505 1505 1

151051 1505

151051505 1

1 1505 1 1505

05

1

15

05

1

15

05

1

15

05

1

15

05

1

15

05

1

15

05

1

15

05

1

15

05

1

15

05

1

15

05

1

15

05

1

15

1 1505

V ()

1 1505

1 1505

04

06

08

1

12

14

16

V

04

06

08

1

12

14

16

V

04

06

08

1

12

14

16

V

V - V -

V - V -V -





119872119894 sdot (RL119894 rarr 1) (C8)



(C9)


119872119894 (RL119894 997888rarr 1) = Θ119894 (RL119894 = 0) sdot 2119864119868119894119871 (C10)





RD119894 = Θ119894 (RL119894) sdot 119871120575119894 (RL119894) (C12)





(C14)


RL119894


(C15)





119898119894sdot 120593119894 (RL119894 120572119894 119871 120585119894)

(C17)


BF119894 (RL119894 120572119894 119871 120585119894) = 120578119894 (RL119894 120572119894 119871 120585119894)120578119894 (RL119894 = 0 120572119894 119871 120585119894) (C18)



(C19)


BF119894 (RL119894 120572119894 119871 120585119894)


120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)= 119898119894 (RL119894 = 0)

119898119894 (RL119894)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟BF119898119894(RL119894)

sdot 120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟BF120593119894(RL119894 120572119894 119871120585119894)

(C20)

obtaining


= 119898119894 (RL119894 = 0)119898119894 (RL119894)

sdot 120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)

(C21)





Acknowledgments


References

























































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




BF119894 (RL119894 120572119894 119871 120585119894) = 120578119894 (RL119894 120572119894 119871 120585119894)120578119894 (RL119894 = 0 120572119894 119871 120585119894) (21)



119898119894 (RL119894)sdot 120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)

(22)




120593119894 (RL119894 120572119894 119871 120585119894) = BF120593119894 (RL119894 120572119894 119871 120585119894)sdot 120593119894 (RL119894 = 0 120572119894 119871 120585119894)

(23)








119901120578119894 (120578119894 (120572119894 119871 120585119894)) = intinfin0

intinfin0

intinfin0

intinfin0


(24)



(25)






２V = 000２V = 025２V = 050

２V = 075２V = 100

07

075

08

085

09

095

1

105

11

115

12

125

13V

50 100 150 2000L

12

12

125

125125

125125

125

12513

1

105

11

115

12

125

13

135

14

145

15

04

06

08

1

12

14

16

V

50 100 150 2000L

V(２V V = 1 L V = 0015) V(２V = 1 V L V = 0015)



















119863119894 = 120593119894 (RL119894 120572119894 119871 120585119894) le 120593119894max = 119862119894 forall119894 (27)


119862119894 = 120593119894max


(28)



119875fail119894 = int119862minus119863le0

119901119862119894 119863119894 (119888119894 119889119894) sdot d119888119894 sdot d119889119894 (29)




0119901120593119894 (120593119894 (RL119894 120572119894 119871 120585119894)) sdot d120593119894 (30)











11 115115

12

12

12

125

11 115115

1212

125

125

115

12

12

12

125

125

11 115

12

1212

125

125

115

12

1212

125

125

125

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12125

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1212

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12

12125

125

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1212

125

125

12

12125

125125

12

12125

125

12

12125

125

125

12

12125

125

12

12

125

125125

125

12

12125

125

12

12

125

125

125 125

125

V = 0001 V = 0003

V = 0005 V = 0007

V = 0009 V = 0011

V = 0013 V = 0015

15

14

13

12

11

1

15

14

13

12

11

1

15

14

13

12

11

1

15

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13

12

11

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11

1

15

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12

11

1

15

14

13

12

11

1

15

14

13

12

11

1

0 50 200100 150

L

0 50 200100 150

L

05

1

15

V

05

1

15

V

05

1

15

V

05

1

15

V

0 50 200100 150

L

0 50 200100 150

L

05

1

15

V

0 50 200100 150

L

0 50 200100 150

L

05

1

15

V

05

1

15

V

0 50 200100 150

L

05

1

15

V

0 50 200100 150

L



11 115115

12 12

12

125

1

11

12

13

14

15

115

12

12

12

125

125

1

11

12

13

14

15

115

12

1212

125

125

125

1

11

12

13

14

15

115

12

12

12125

125

1

11

12

13

14

15

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12

125

125

1

11

12

13

14

15

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125

1

11

12

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14

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12

125

125

1

11

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13

14

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12

125

125

1

11

12

13

14

15

L = 0001 L = 0003

L = 0007

L = 0011

L = 0015

L = 0005

L = 0009

L = 0013

50 100 150 2000

L

50 100 150 2000

L

50 100 150 2000

L

50 100 150 2000

L

50 100 150 2000

L

50 100 150 2000

L

100 200150500

L

100 200150500

L

05

1

15

L

05

1

15

L

05

1

15

L

05

1

15

L

05

1

15

L

05

1

15

L

05

1

15

L

05

1

15

L






= 021 if 120572119881 le 0470140 + 0150 sdot 120572119881 if 120572119881 gt 047



(33)










316fL

05606307609512719

0

05

1

15

2

25

3

35

4

45

5

L

max

075 1 12505 15 1703L

633fV

1121271521925338

0

005

01

015

02

025

03

035

04

045

05

V

max

075 1 12505 15 1703V






(34)



sdot 120578119881max = 0775


sdot 120578119871max = 5723(35)


119863119881 = 120593119881 (RL119881 120572119881 119871 120585119881) = 1 sdot 00904 = 00904119863119871 = 120593119871 (RL119871 120572119871 119871 120585119871) = 097 sdot 00035 = 00034 (36)


119863119881 = 00904 lt 0775 = 119862119881119863119871 = 00034 lt 5723 = 119862119871 (37)





(38)


119875120593119881 (119863119881) = 0565119875120593119871 (119863119871) = 0379 (39)


119875fail119881 = 00296 lArrrArr120573119881 = 1887


(40)






(41)


7 Conclusions




0

0

02

02

04

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06

minus04

minus02

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0025 002501

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006

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002501

002501

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002501

1 1505

L ()1 1505

L ()

1 1505105 15

15105 15105

105 15 105 15

1505 1 1505 1

1 15051 1505

05

1

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1

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1

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105 15

L ()

15105

1 1505

04

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08

1

12

14

16

L

04

06

08

1

12

14

16

L

04

06

08

1

12

14

16

L


L - L = 10 Ｇ L - L = 20 Ｇ

kL - L = 30 Ｇ kL - L = 40 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ

L - L = 10 Ｇ L - L = 20 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ





0

02

02

04

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06

06

0808

minus04

minus02

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0 0

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0808

0001

0025

0025

005

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006

008

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012

014

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0001

002501

0001

002501

0001

002501

0001

0025

0025

005

00501

0

002

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006

008

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012

014

016

018

02

002501 0025

01

002501 0025

01

05 1 15

V ()

105 15

15105

1 1505

V ()1505 1

V ()

1 15051 1505

1 1505 1 1505

1505 115105

1 150505 151

1 1505 105 15

05

1

15

05

1

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05

1

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05

1

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1

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V

04

06

08

1

12

14

16

V

04

06

08

1

12

14

16

V


V - L = 10 Ｇ V - L = 20 Ｇ

kV - L = 30 Ｇ kV - L = 40 Ｇ

V - L = 30 Ｇ V - L = 40 Ｇ

L = 10 Ｇ L = 20 Ｇ

L = 30 Ｇ L = 40 Ｇ


L = 50ndash200Ｇ V - V -

V -V -

V -



0

02

04

06

08

1

CDF

0005 001 0015 00200

02

04

06

08

1

0

2

4

6

8

10

PDF

104 06 080200

50

100

150

200

250

V [mdash] L [mdash]


Appendix


See Figures 7 and 8



1198701199031119894 = 1198701199032119894 = 119870119903119894 (C1)



Θ1119894 = Θ2119894 = Θ119894 (C2)







(C5)




(C6)





L - L = 10 Ｇ L - L = 20 Ｇ

kL - L = 30 Ｇ kL - L = 40 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ

L - L = 10 Ｇ L - L = 20 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ



0

0

02

02

04

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06

06

minus04

minus02

0

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1

12

0

00

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0001

0025

0025

005

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01

0

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004

006

008

01

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0001

0025

0001

002501

0001

002501

0001

002501

0001

0025

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005

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006

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014

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002501

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1

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L

04

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08

1

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14

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L

04

06

08

1

12

14

16

L

105 15

1 1505

105 15 105 15

105 15105 15

1 1505 1 1505

1 15051 1505

105 15 105 15

1 1505

L ()1 1505

L ()1 1505

L ()

05

1

15

05

1

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05

1

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05

1

15

05

1

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05

1

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1

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05

1

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05

1

15

05

1

15

05

1

15

05

1

15




V - L = 10 Ｇ V - L = 20 Ｇ

kV - L = 30 Ｇ kV - L = 40 Ｇ

V - L = 30 Ｇ V - L = 40 Ｇ

L = 10 Ｇ L = 20 Ｇ

L = 30 Ｇ L = 40 Ｇ


L = 50ndash200Ｇ

0

02

02

04

04

06

06

0808

minus04

minus02

0

02

04

06

08

1

12

00

0

0404

0

04

04

0808

0

04

04

0808

0

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0808

0001

0025

0025

005

00501

0

002

004

006

008

01

012

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002501 0025

01

002501 0025

01

0025

0025

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015

0

002

004

006

008

01

012

014

016

018

02

002501 02

002501

002501

002501

1 1505

V ()1 1505

V ()

1505 115105

1 1505 1505 1

151051 1505

151051505 1

1 1505 1 1505

05

1

15

05

1

15

05

1

15

05

1

15

05

1

15

05

1

15

05

1

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05

1

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1

15

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1

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1

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1 1505

V ()

1 1505

1 1505

04

06

08

1

12

14

16

V

04

06

08

1

12

14

16

V

04

06

08

1

12

14

16

V

V - V -

V - V -V -





119872119894 sdot (RL119894 rarr 1) (C8)



(C9)


119872119894 (RL119894 997888rarr 1) = Θ119894 (RL119894 = 0) sdot 2119864119868119894119871 (C10)





RD119894 = Θ119894 (RL119894) sdot 119871120575119894 (RL119894) (C12)





(C14)


RL119894


(C15)





119898119894sdot 120593119894 (RL119894 120572119894 119871 120585119894)

(C17)


BF119894 (RL119894 120572119894 119871 120585119894) = 120578119894 (RL119894 120572119894 119871 120585119894)120578119894 (RL119894 = 0 120572119894 119871 120585119894) (C18)



(C19)


BF119894 (RL119894 120572119894 119871 120585119894)


120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)= 119898119894 (RL119894 = 0)

119898119894 (RL119894)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟BF119898119894(RL119894)

sdot 120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟BF120593119894(RL119894 120572119894 119871120585119894)

(C20)

obtaining


= 119898119894 (RL119894 = 0)119898119894 (RL119894)

sdot 120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)

(C21)





Acknowledgments


References

























































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



２V = 000２V = 025２V = 050

２V = 075２V = 100

07

075

08

085

09

095

1

105

11

115

12

125

13V

50 100 150 2000L

12

12

125

125125

125125

125

12513

1

105

11

115

12

125

13

135

14

145

15

04

06

08

1

12

14

16

V

50 100 150 2000L

V(２V V = 1 L V = 0015) V(２V = 1 V L V = 0015)



















119863119894 = 120593119894 (RL119894 120572119894 119871 120585119894) le 120593119894max = 119862119894 forall119894 (27)


119862119894 = 120593119894max


(28)



119875fail119894 = int119862minus119863le0

119901119862119894 119863119894 (119888119894 119889119894) sdot d119888119894 sdot d119889119894 (29)




0119901120593119894 (120593119894 (RL119894 120572119894 119871 120585119894)) sdot d120593119894 (30)











11 115115

12

12

12

125

11 115115

1212

125

125

115

12

12

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125

11 115

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12125

125

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12

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125125

125

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12125

125

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125

125 125

125

V = 0001 V = 0003

V = 0005 V = 0007

V = 0009 V = 0011

V = 0013 V = 0015

15

14

13

12

11

1

15

14

13

12

11

1

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11

1

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1

0 50 200100 150

L

0 50 200100 150

L

05

1

15

V

05

1

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V

05

1

15

V

05

1

15

V

0 50 200100 150

L

0 50 200100 150

L

05

1

15

V

0 50 200100 150

L

0 50 200100 150

L

05

1

15

V

05

1

15

V

0 50 200100 150

L

05

1

15

V

0 50 200100 150

L



11 115115

12 12

12

125

1

11

12

13

14

15

115

12

12

12

125

125

1

11

12

13

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1212

125

125

125

1

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125

1

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125

1

11

12

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15

L = 0001 L = 0003

L = 0007

L = 0011

L = 0015

L = 0005

L = 0009

L = 0013

50 100 150 2000

L

50 100 150 2000

L

50 100 150 2000

L

50 100 150 2000

L

50 100 150 2000

L

50 100 150 2000

L

100 200150500

L

100 200150500

L

05

1

15

L

05

1

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L

05

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L

05

1

15

L

05

1

15

L

05

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L

05

1

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L

05

1

15

L






= 021 if 120572119881 le 0470140 + 0150 sdot 120572119881 if 120572119881 gt 047



(33)










316fL

05606307609512719

0

05

1

15

2

25

3

35

4

45

5

L

max

075 1 12505 15 1703L

633fV

1121271521925338

0

005

01

015

02

025

03

035

04

045

05

V

max

075 1 12505 15 1703V






(34)



sdot 120578119881max = 0775


sdot 120578119871max = 5723(35)


119863119881 = 120593119881 (RL119881 120572119881 119871 120585119881) = 1 sdot 00904 = 00904119863119871 = 120593119871 (RL119871 120572119871 119871 120585119871) = 097 sdot 00035 = 00034 (36)


119863119881 = 00904 lt 0775 = 119862119881119863119871 = 00034 lt 5723 = 119862119871 (37)





(38)


119875120593119881 (119863119881) = 0565119875120593119871 (119863119871) = 0379 (39)


119875fail119881 = 00296 lArrrArr120573119881 = 1887


(40)






(41)


7 Conclusions




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1 1505

L ()1 1505

L ()

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1 15051 1505

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L ()

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1

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04

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L


L - L = 10 Ｇ L - L = 20 Ｇ

kL - L = 30 Ｇ kL - L = 40 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ

L - L = 10 Ｇ L - L = 20 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ





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V ()

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15105

1 1505

V ()1505 1

V ()

1 15051 1505

1 1505 1 1505

1505 115105

1 150505 151

1 1505 105 15

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04

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1

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14

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V


V - L = 10 Ｇ V - L = 20 Ｇ

kV - L = 30 Ｇ kV - L = 40 Ｇ

V - L = 30 Ｇ V - L = 40 Ｇ

L = 10 Ｇ L = 20 Ｇ

L = 30 Ｇ L = 40 Ｇ


L = 50ndash200Ｇ V - V -

V -V -

V -



0

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1

CDF

0005 001 0015 00200

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PDF

104 06 080200

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V [mdash] L [mdash]


Appendix


See Figures 7 and 8



1198701199031119894 = 1198701199032119894 = 119870119903119894 (C1)



Θ1119894 = Θ2119894 = Θ119894 (C2)







(C5)




(C6)





L - L = 10 Ｇ L - L = 20 Ｇ

kL - L = 30 Ｇ kL - L = 40 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ

L - L = 10 Ｇ L - L = 20 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ



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105 15

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1 1505 1 1505

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105 15 105 15

1 1505

L ()1 1505

L ()1 1505

L ()

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15




V - L = 10 Ｇ V - L = 20 Ｇ

kV - L = 30 Ｇ kV - L = 40 Ｇ

V - L = 30 Ｇ V - L = 40 Ｇ

L = 10 Ｇ L = 20 Ｇ

L = 30 Ｇ L = 40 Ｇ


L = 50ndash200Ｇ

0

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002501

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002501

1 1505

V ()1 1505

V ()

1505 115105

1 1505 1505 1

151051 1505

151051505 1

1 1505 1 1505

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1

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V ()

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04

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V - V -

V - V -V -





119872119894 sdot (RL119894 rarr 1) (C8)



(C9)


119872119894 (RL119894 997888rarr 1) = Θ119894 (RL119894 = 0) sdot 2119864119868119894119871 (C10)





RD119894 = Θ119894 (RL119894) sdot 119871120575119894 (RL119894) (C12)





(C14)


RL119894


(C15)





119898119894sdot 120593119894 (RL119894 120572119894 119871 120585119894)

(C17)


BF119894 (RL119894 120572119894 119871 120585119894) = 120578119894 (RL119894 120572119894 119871 120585119894)120578119894 (RL119894 = 0 120572119894 119871 120585119894) (C18)



(C19)


BF119894 (RL119894 120572119894 119871 120585119894)


120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)= 119898119894 (RL119894 = 0)

119898119894 (RL119894)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟BF119898119894(RL119894)

sdot 120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟BF120593119894(RL119894 120572119894 119871120585119894)

(C20)

obtaining


= 119898119894 (RL119894 = 0)119898119894 (RL119894)

sdot 120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)

(C21)





Acknowledgments


References

























































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia









119863119894 = 120593119894 (RL119894 120572119894 119871 120585119894) le 120593119894max = 119862119894 forall119894 (27)


119862119894 = 120593119894max


(28)



119875fail119894 = int119862minus119863le0

119901119862119894 119863119894 (119888119894 119889119894) sdot d119888119894 sdot d119889119894 (29)




0119901120593119894 (120593119894 (RL119894 120572119894 119871 120585119894)) sdot d120593119894 (30)











11 115115

12

12

12

125

11 115115

1212

125

125

115

12

12

12

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125

11 115

12

1212

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1212

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12125

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125125

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125125

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12125

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125 125

125

V = 0001 V = 0003

V = 0005 V = 0007

V = 0009 V = 0011

V = 0013 V = 0015

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1

0 50 200100 150

L

0 50 200100 150

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0 50 200100 150

L

0 50 200100 150

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0 50 200100 150

L

0 50 200100 150

L

05

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V

05

1

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V

0 50 200100 150

L

05

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V

0 50 200100 150

L



11 115115

12 12

12

125

1

11

12

13

14

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12

12

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125

125

1

11

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1212

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1

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12125

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1

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1

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1

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125

1

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125

1

11

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15

L = 0001 L = 0003

L = 0007

L = 0011

L = 0015

L = 0005

L = 0009

L = 0013

50 100 150 2000

L

50 100 150 2000

L

50 100 150 2000

L

50 100 150 2000

L

50 100 150 2000

L

50 100 150 2000

L

100 200150500

L

100 200150500

L

05

1

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L

05

1

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L






= 021 if 120572119881 le 0470140 + 0150 sdot 120572119881 if 120572119881 gt 047



(33)










316fL

05606307609512719

0

05

1

15

2

25

3

35

4

45

5

L

max

075 1 12505 15 1703L

633fV

1121271521925338

0

005

01

015

02

025

03

035

04

045

05

V

max

075 1 12505 15 1703V






(34)



sdot 120578119881max = 0775


sdot 120578119871max = 5723(35)


119863119881 = 120593119881 (RL119881 120572119881 119871 120585119881) = 1 sdot 00904 = 00904119863119871 = 120593119871 (RL119871 120572119871 119871 120585119871) = 097 sdot 00035 = 00034 (36)


119863119881 = 00904 lt 0775 = 119862119881119863119871 = 00034 lt 5723 = 119862119871 (37)





(38)


119875120593119881 (119863119881) = 0565119875120593119871 (119863119871) = 0379 (39)


119875fail119881 = 00296 lArrrArr120573119881 = 1887


(40)






(41)


7 Conclusions




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L - L = 10 Ｇ L - L = 20 Ｇ

kL - L = 30 Ｇ kL - L = 40 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ

L - L = 10 Ｇ L - L = 20 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ





0

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05 1 15

V ()

105 15

15105

1 1505

V ()1505 1

V ()

1 15051 1505

1 1505 1 1505

1505 115105

1 150505 151

1 1505 105 15

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04

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V


V - L = 10 Ｇ V - L = 20 Ｇ

kV - L = 30 Ｇ kV - L = 40 Ｇ

V - L = 30 Ｇ V - L = 40 Ｇ

L = 10 Ｇ L = 20 Ｇ

L = 30 Ｇ L = 40 Ｇ


L = 50ndash200Ｇ V - V -

V -V -

V -



0

02

04

06

08

1

CDF

0005 001 0015 00200

02

04

06

08

1

0

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4

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8

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PDF

104 06 080200

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100

150

200

250

V [mdash] L [mdash]


Appendix


See Figures 7 and 8



1198701199031119894 = 1198701199032119894 = 119870119903119894 (C1)



Θ1119894 = Θ2119894 = Θ119894 (C2)







(C5)




(C6)





L - L = 10 Ｇ L - L = 20 Ｇ

kL - L = 30 Ｇ kL - L = 40 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ

L - L = 10 Ｇ L - L = 20 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ



0

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105 15

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105 15 105 15

105 15105 15

1 1505 1 1505

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105 15 105 15

1 1505

L ()1 1505

L ()1 1505

L ()

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1

15




V - L = 10 Ｇ V - L = 20 Ｇ

kV - L = 30 Ｇ kV - L = 40 Ｇ

V - L = 30 Ｇ V - L = 40 Ｇ

L = 10 Ｇ L = 20 Ｇ

L = 30 Ｇ L = 40 Ｇ


L = 50ndash200Ｇ

0

02

02

04

04

06

06

0808

minus04

minus02

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1

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002501

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1 1505

V ()1 1505

V ()

1505 115105

1 1505 1505 1

151051 1505

151051505 1

1 1505 1 1505

05

1

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1 1505

V ()

1 1505

1 1505

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V

04

06

08

1

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14

16

V

04

06

08

1

12

14

16

V

V - V -

V - V -V -





119872119894 sdot (RL119894 rarr 1) (C8)



(C9)


119872119894 (RL119894 997888rarr 1) = Θ119894 (RL119894 = 0) sdot 2119864119868119894119871 (C10)





RD119894 = Θ119894 (RL119894) sdot 119871120575119894 (RL119894) (C12)





(C14)


RL119894


(C15)





119898119894sdot 120593119894 (RL119894 120572119894 119871 120585119894)

(C17)


BF119894 (RL119894 120572119894 119871 120585119894) = 120578119894 (RL119894 120572119894 119871 120585119894)120578119894 (RL119894 = 0 120572119894 119871 120585119894) (C18)



(C19)


BF119894 (RL119894 120572119894 119871 120585119894)


120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)= 119898119894 (RL119894 = 0)

119898119894 (RL119894)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟BF119898119894(RL119894)

sdot 120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟BF120593119894(RL119894 120572119894 119871120585119894)

(C20)

obtaining


= 119898119894 (RL119894 = 0)119898119894 (RL119894)

sdot 120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)

(C21)





Acknowledgments


References

























































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



11 115115

12

12

12

125

11 115115

1212

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11 115

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125125

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12

12125

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125 125

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V = 0001 V = 0003

V = 0005 V = 0007

V = 0009 V = 0011

V = 0013 V = 0015

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0 50 200100 150

L

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11 115115

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1

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1

11

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125

1

11

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14

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L = 0001 L = 0003

L = 0007

L = 0011

L = 0015

L = 0005

L = 0009

L = 0013

50 100 150 2000

L

50 100 150 2000

L

50 100 150 2000

L

50 100 150 2000

L

50 100 150 2000

L

50 100 150 2000

L

100 200150500

L

100 200150500

L

05

1

15

L

05

1

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L

05

1

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L

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1

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L

05

1

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L

05

1

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L

05

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L

05

1

15

L






= 021 if 120572119881 le 0470140 + 0150 sdot 120572119881 if 120572119881 gt 047



(33)










316fL

05606307609512719

0

05

1

15

2

25

3

35

4

45

5

L

max

075 1 12505 15 1703L

633fV

1121271521925338

0

005

01

015

02

025

03

035

04

045

05

V

max

075 1 12505 15 1703V






(34)



sdot 120578119881max = 0775


sdot 120578119871max = 5723(35)


119863119881 = 120593119881 (RL119881 120572119881 119871 120585119881) = 1 sdot 00904 = 00904119863119871 = 120593119871 (RL119871 120572119871 119871 120585119871) = 097 sdot 00035 = 00034 (36)


119863119881 = 00904 lt 0775 = 119862119881119863119871 = 00034 lt 5723 = 119862119871 (37)





(38)


119875120593119881 (119863119881) = 0565119875120593119871 (119863119871) = 0379 (39)


119875fail119881 = 00296 lArrrArr120573119881 = 1887


(40)






(41)


7 Conclusions




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L ()1 1505

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1 1505105 15

15105 15105

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1 15051 1505

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L ()

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L

04

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1

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L


L - L = 10 Ｇ L - L = 20 Ｇ

kL - L = 30 Ｇ kL - L = 40 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ

L - L = 10 Ｇ L - L = 20 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ





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V ()

105 15

15105

1 1505

V ()1505 1

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1 15051 1505

1 1505 1 1505

1505 115105

1 150505 151

1 1505 105 15

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04

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1

12

14

16

V


V - L = 10 Ｇ V - L = 20 Ｇ

kV - L = 30 Ｇ kV - L = 40 Ｇ

V - L = 30 Ｇ V - L = 40 Ｇ

L = 10 Ｇ L = 20 Ｇ

L = 30 Ｇ L = 40 Ｇ


L = 50ndash200Ｇ V - V -

V -V -

V -



0

02

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08

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CDF

0005 001 0015 00200

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PDF

104 06 080200

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100

150

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V [mdash] L [mdash]


Appendix


See Figures 7 and 8



1198701199031119894 = 1198701199032119894 = 119870119903119894 (C1)



Θ1119894 = Θ2119894 = Θ119894 (C2)







(C5)




(C6)





L - L = 10 Ｇ L - L = 20 Ｇ

kL - L = 30 Ｇ kL - L = 40 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ

L - L = 10 Ｇ L - L = 20 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ



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105 15

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L ()1 1505

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L ()

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1

15




V - L = 10 Ｇ V - L = 20 Ｇ

kV - L = 30 Ｇ kV - L = 40 Ｇ

V - L = 30 Ｇ V - L = 40 Ｇ

L = 10 Ｇ L = 20 Ｇ

L = 30 Ｇ L = 40 Ｇ


L = 50ndash200Ｇ

0

02

02

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minus04

minus02

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0

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0001

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00501

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002501 0025

01

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002501

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1 1505

V ()1 1505

V ()

1505 115105

1 1505 1505 1

151051 1505

151051505 1

1 1505 1 1505

05

1

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05

1

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V ()

1 1505

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04

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V - V -

V - V -V -





119872119894 sdot (RL119894 rarr 1) (C8)



(C9)


119872119894 (RL119894 997888rarr 1) = Θ119894 (RL119894 = 0) sdot 2119864119868119894119871 (C10)





RD119894 = Θ119894 (RL119894) sdot 119871120575119894 (RL119894) (C12)





(C14)


RL119894


(C15)





119898119894sdot 120593119894 (RL119894 120572119894 119871 120585119894)

(C17)


BF119894 (RL119894 120572119894 119871 120585119894) = 120578119894 (RL119894 120572119894 119871 120585119894)120578119894 (RL119894 = 0 120572119894 119871 120585119894) (C18)



(C19)


BF119894 (RL119894 120572119894 119871 120585119894)


120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)= 119898119894 (RL119894 = 0)

119898119894 (RL119894)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟BF119898119894(RL119894)

sdot 120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟BF120593119894(RL119894 120572119894 119871120585119894)

(C20)

obtaining


= 119898119894 (RL119894 = 0)119898119894 (RL119894)

sdot 120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)

(C21)





Acknowledgments


References

























































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



11 115115

12 12

12

125

1

11

12

13

14

15

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1

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15

L = 0001 L = 0003

L = 0007

L = 0011

L = 0015

L = 0005

L = 0009

L = 0013

50 100 150 2000

L

50 100 150 2000

L

50 100 150 2000

L

50 100 150 2000

L

50 100 150 2000

L

50 100 150 2000

L

100 200150500

L

100 200150500

L

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1

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L






= 021 if 120572119881 le 0470140 + 0150 sdot 120572119881 if 120572119881 gt 047



(33)










316fL

05606307609512719

0

05

1

15

2

25

3

35

4

45

5

L

max

075 1 12505 15 1703L

633fV

1121271521925338

0

005

01

015

02

025

03

035

04

045

05

V

max

075 1 12505 15 1703V






(34)



sdot 120578119881max = 0775


sdot 120578119871max = 5723(35)


119863119881 = 120593119881 (RL119881 120572119881 119871 120585119881) = 1 sdot 00904 = 00904119863119871 = 120593119871 (RL119871 120572119871 119871 120585119871) = 097 sdot 00035 = 00034 (36)


119863119881 = 00904 lt 0775 = 119862119881119863119871 = 00034 lt 5723 = 119862119871 (37)





(38)


119875120593119881 (119863119881) = 0565119875120593119871 (119863119871) = 0379 (39)


119875fail119881 = 00296 lArrrArr120573119881 = 1887


(40)






(41)


7 Conclusions




0

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1 1505

L ()1 1505

L ()

1 1505105 15

15105 15105

105 15 105 15

1505 1 1505 1

1 15051 1505

05

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105 15

L ()

15105

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1

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L

04

06

08

1

12

14

16

L

04

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1

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16

L


L - L = 10 Ｇ L - L = 20 Ｇ

kL - L = 30 Ｇ kL - L = 40 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ

L - L = 10 Ｇ L - L = 20 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ





0

02

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002501

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002501

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002501

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05 1 15

V ()

105 15

15105

1 1505

V ()1505 1

V ()

1 15051 1505

1 1505 1 1505

1505 115105

1 150505 151

1 1505 105 15

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04

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04

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V


V - L = 10 Ｇ V - L = 20 Ｇ

kV - L = 30 Ｇ kV - L = 40 Ｇ

V - L = 30 Ｇ V - L = 40 Ｇ

L = 10 Ｇ L = 20 Ｇ

L = 30 Ｇ L = 40 Ｇ


L = 50ndash200Ｇ V - V -

V -V -

V -



0

02

04

06

08

1

CDF

0005 001 0015 00200

02

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4

6

8

10

PDF

104 06 080200

50

100

150

200

250

V [mdash] L [mdash]


Appendix


See Figures 7 and 8



1198701199031119894 = 1198701199032119894 = 119870119903119894 (C1)



Θ1119894 = Θ2119894 = Θ119894 (C2)







(C5)




(C6)





L - L = 10 Ｇ L - L = 20 Ｇ

kL - L = 30 Ｇ kL - L = 40 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ

L - L = 10 Ｇ L - L = 20 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ



0

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105 15

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L ()

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1

15

05

1

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05

1

15

05

1

15

05

1

15

05

1

15




V - L = 10 Ｇ V - L = 20 Ｇ

kV - L = 30 Ｇ kV - L = 40 Ｇ

V - L = 30 Ｇ V - L = 40 Ｇ

L = 10 Ｇ L = 20 Ｇ

L = 30 Ｇ L = 40 Ｇ


L = 50ndash200Ｇ

0

02

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minus02

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0808

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002501 02

002501

002501

002501

1 1505

V ()1 1505

V ()

1505 115105

1 1505 1505 1

151051 1505

151051505 1

1 1505 1 1505

05

1

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15

05

1

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1 1505

V ()

1 1505

1 1505

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V

04

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1

12

14

16

V

04

06

08

1

12

14

16

V

V - V -

V - V -V -





119872119894 sdot (RL119894 rarr 1) (C8)



(C9)


119872119894 (RL119894 997888rarr 1) = Θ119894 (RL119894 = 0) sdot 2119864119868119894119871 (C10)





RD119894 = Θ119894 (RL119894) sdot 119871120575119894 (RL119894) (C12)





(C14)


RL119894


(C15)





119898119894sdot 120593119894 (RL119894 120572119894 119871 120585119894)

(C17)


BF119894 (RL119894 120572119894 119871 120585119894) = 120578119894 (RL119894 120572119894 119871 120585119894)120578119894 (RL119894 = 0 120572119894 119871 120585119894) (C18)



(C19)


BF119894 (RL119894 120572119894 119871 120585119894)


120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)= 119898119894 (RL119894 = 0)

119898119894 (RL119894)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟BF119898119894(RL119894)

sdot 120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟BF120593119894(RL119894 120572119894 119871120585119894)

(C20)

obtaining


= 119898119894 (RL119894 = 0)119898119894 (RL119894)

sdot 120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)

(C21)





Acknowledgments


References

























































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia






= 021 if 120572119881 le 0470140 + 0150 sdot 120572119881 if 120572119881 gt 047



(33)










316fL

05606307609512719

0

05

1

15

2

25

3

35

4

45

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L

max

075 1 12505 15 1703L

633fV

1121271521925338

0

005

01

015

02

025

03

035

04

045

05

V

max

075 1 12505 15 1703V






(34)



sdot 120578119881max = 0775


sdot 120578119871max = 5723(35)


119863119881 = 120593119881 (RL119881 120572119881 119871 120585119881) = 1 sdot 00904 = 00904119863119871 = 120593119871 (RL119871 120572119871 119871 120585119871) = 097 sdot 00035 = 00034 (36)


119863119881 = 00904 lt 0775 = 119862119881119863119871 = 00034 lt 5723 = 119862119871 (37)





(38)


119875120593119881 (119863119881) = 0565119875120593119871 (119863119871) = 0379 (39)


119875fail119881 = 00296 lArrrArr120573119881 = 1887


(40)






(41)


7 Conclusions




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1 1505105 15

15105 15105

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1 15051 1505

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L ()

15105

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1

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L

04

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1

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L


L - L = 10 Ｇ L - L = 20 Ｇ

kL - L = 30 Ｇ kL - L = 40 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ

L - L = 10 Ｇ L - L = 20 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ





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05 1 15

V ()

105 15

15105

1 1505

V ()1505 1

V ()

1 15051 1505

1 1505 1 1505

1505 115105

1 150505 151

1 1505 105 15

05

1

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1

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05

1

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04

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04

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1

12

14

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V


V - L = 10 Ｇ V - L = 20 Ｇ

kV - L = 30 Ｇ kV - L = 40 Ｇ

V - L = 30 Ｇ V - L = 40 Ｇ

L = 10 Ｇ L = 20 Ｇ

L = 30 Ｇ L = 40 Ｇ


L = 50ndash200Ｇ V - V -

V -V -

V -



0

02

04

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08

1

CDF

0005 001 0015 00200

02

04

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08

1

0

2

4

6

8

10

PDF

104 06 080200

50

100

150

200

250

V [mdash] L [mdash]


Appendix


See Figures 7 and 8



1198701199031119894 = 1198701199032119894 = 119870119903119894 (C1)



Θ1119894 = Θ2119894 = Θ119894 (C2)







(C5)




(C6)





L - L = 10 Ｇ L - L = 20 Ｇ

kL - L = 30 Ｇ kL - L = 40 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ

L - L = 10 Ｇ L - L = 20 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ



0

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105 15

1 1505

105 15 105 15

105 15105 15

1 1505 1 1505

1 15051 1505

105 15 105 15

1 1505

L ()1 1505

L ()1 1505

L ()

05

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1

15




V - L = 10 Ｇ V - L = 20 Ｇ

kV - L = 30 Ｇ kV - L = 40 Ｇ

V - L = 30 Ｇ V - L = 40 Ｇ

L = 10 Ｇ L = 20 Ｇ

L = 30 Ｇ L = 40 Ｇ


L = 50ndash200Ｇ

0

02

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minus04

minus02

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008

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012

014

016

018

02

002501 02

002501

002501

002501

1 1505

V ()1 1505

V ()

1505 115105

1 1505 1505 1

151051 1505

151051505 1

1 1505 1 1505

05

1

15

05

1

15

05

1

15

05

1

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1

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1 1505

V ()

1 1505

1 1505

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1

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V

04

06

08

1

12

14

16

V

04

06

08

1

12

14

16

V

V - V -

V - V -V -





119872119894 sdot (RL119894 rarr 1) (C8)



(C9)


119872119894 (RL119894 997888rarr 1) = Θ119894 (RL119894 = 0) sdot 2119864119868119894119871 (C10)





RD119894 = Θ119894 (RL119894) sdot 119871120575119894 (RL119894) (C12)





(C14)


RL119894


(C15)





119898119894sdot 120593119894 (RL119894 120572119894 119871 120585119894)

(C17)


BF119894 (RL119894 120572119894 119871 120585119894) = 120578119894 (RL119894 120572119894 119871 120585119894)120578119894 (RL119894 = 0 120572119894 119871 120585119894) (C18)



(C19)


BF119894 (RL119894 120572119894 119871 120585119894)


120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)= 119898119894 (RL119894 = 0)

119898119894 (RL119894)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟BF119898119894(RL119894)

sdot 120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟BF120593119894(RL119894 120572119894 119871120585119894)

(C20)

obtaining


= 119898119894 (RL119894 = 0)119898119894 (RL119894)

sdot 120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)

(C21)





Acknowledgments


References

























































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



316fL

05606307609512719

0

05

1

15

2

25

3

35

4

45

5

L

max

075 1 12505 15 1703L

633fV

1121271521925338

0

005

01

015

02

025

03

035

04

045

05

V

max

075 1 12505 15 1703V






(34)



sdot 120578119881max = 0775


sdot 120578119871max = 5723(35)


119863119881 = 120593119881 (RL119881 120572119881 119871 120585119881) = 1 sdot 00904 = 00904119863119871 = 120593119871 (RL119871 120572119871 119871 120585119871) = 097 sdot 00035 = 00034 (36)


119863119881 = 00904 lt 0775 = 119862119881119863119871 = 00034 lt 5723 = 119862119871 (37)





(38)


119875120593119881 (119863119881) = 0565119875120593119871 (119863119871) = 0379 (39)


119875fail119881 = 00296 lArrrArr120573119881 = 1887


(40)






(41)


7 Conclusions




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L ()1 1505

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1

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L

04

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1

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L


L - L = 10 Ｇ L - L = 20 Ｇ

kL - L = 30 Ｇ kL - L = 40 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ

L - L = 10 Ｇ L - L = 20 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ





0

02

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minus04

minus02

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0808

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0808

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0025

0025

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006

008

01

012

014

016

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0001

002501

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002501

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002501

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002501

0001

0025

0025

005

00501

0

002

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006

008

01

012

014

016

018

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002501 0025

01

002501 0025

01

05 1 15

V ()

105 15

15105

1 1505

V ()1505 1

V ()

1 15051 1505

1 1505 1 1505

1505 115105

1 150505 151

1 1505 105 15

05

1

15

05

1

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05

1

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1

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1

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04

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14

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04

06

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1

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14

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V


V - L = 10 Ｇ V - L = 20 Ｇ

kV - L = 30 Ｇ kV - L = 40 Ｇ

V - L = 30 Ｇ V - L = 40 Ｇ

L = 10 Ｇ L = 20 Ｇ

L = 30 Ｇ L = 40 Ｇ


L = 50ndash200Ｇ V - V -

V -V -

V -



0

02

04

06

08

1

CDF

0005 001 0015 00200

02

04

06

08

1

0

2

4

6

8

10

PDF

104 06 080200

50

100

150

200

250

V [mdash] L [mdash]


Appendix


See Figures 7 and 8



1198701199031119894 = 1198701199032119894 = 119870119903119894 (C1)



Θ1119894 = Θ2119894 = Θ119894 (C2)







(C5)




(C6)





L - L = 10 Ｇ L - L = 20 Ｇ

kL - L = 30 Ｇ kL - L = 40 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ

L - L = 10 Ｇ L - L = 20 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ



0

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105 15

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105 15 105 15

105 15105 15

1 1505 1 1505

1 15051 1505

105 15 105 15

1 1505

L ()1 1505

L ()1 1505

L ()

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V - L = 10 Ｇ V - L = 20 Ｇ

kV - L = 30 Ｇ kV - L = 40 Ｇ

V - L = 30 Ｇ V - L = 40 Ｇ

L = 10 Ｇ L = 20 Ｇ

L = 30 Ｇ L = 40 Ｇ


L = 50ndash200Ｇ

0

02

02

04

04

06

06

0808

minus04

minus02

0

02

04

06

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1

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0808

0

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0808

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0808

0001

0025

0025

005

00501

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002

004

006

008

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002501 0025

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002501 0025

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00501

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0

002

004

006

008

01

012

014

016

018

02

002501 02

002501

002501

002501

1 1505

V ()1 1505

V ()

1505 115105

1 1505 1505 1

151051 1505

151051505 1

1 1505 1 1505

05

1

15

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1

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1 1505

V ()

1 1505

1 1505

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1

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V

04

06

08

1

12

14

16

V

04

06

08

1

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14

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V

V - V -

V - V -V -





119872119894 sdot (RL119894 rarr 1) (C8)



(C9)


119872119894 (RL119894 997888rarr 1) = Θ119894 (RL119894 = 0) sdot 2119864119868119894119871 (C10)





RD119894 = Θ119894 (RL119894) sdot 119871120575119894 (RL119894) (C12)





(C14)


RL119894


(C15)





119898119894sdot 120593119894 (RL119894 120572119894 119871 120585119894)

(C17)


BF119894 (RL119894 120572119894 119871 120585119894) = 120578119894 (RL119894 120572119894 119871 120585119894)120578119894 (RL119894 = 0 120572119894 119871 120585119894) (C18)



(C19)


BF119894 (RL119894 120572119894 119871 120585119894)


120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)= 119898119894 (RL119894 = 0)

119898119894 (RL119894)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟BF119898119894(RL119894)

sdot 120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟BF120593119894(RL119894 120572119894 119871120585119894)

(C20)

obtaining


= 119898119894 (RL119894 = 0)119898119894 (RL119894)

sdot 120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)

(C21)





Acknowledgments


References

























































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





(38)


119875120593119881 (119863119881) = 0565119875120593119871 (119863119871) = 0379 (39)


119875fail119881 = 00296 lArrrArr120573119881 = 1887


(40)






(41)


7 Conclusions




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1 1505105 15

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L

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L


L - L = 10 Ｇ L - L = 20 Ｇ

kL - L = 30 Ｇ kL - L = 40 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ

L - L = 10 Ｇ L - L = 20 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ





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15105

1 1505

V ()1505 1

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1 15051 1505

1 1505 1 1505

1505 115105

1 150505 151

1 1505 105 15

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V


V - L = 10 Ｇ V - L = 20 Ｇ

kV - L = 30 Ｇ kV - L = 40 Ｇ

V - L = 30 Ｇ V - L = 40 Ｇ

L = 10 Ｇ L = 20 Ｇ

L = 30 Ｇ L = 40 Ｇ


L = 50ndash200Ｇ V - V -

V -V -

V -



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104 06 080200

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V [mdash] L [mdash]


Appendix


See Figures 7 and 8



1198701199031119894 = 1198701199032119894 = 119870119903119894 (C1)



Θ1119894 = Θ2119894 = Θ119894 (C2)







(C5)




(C6)





L - L = 10 Ｇ L - L = 20 Ｇ

kL - L = 30 Ｇ kL - L = 40 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ

L - L = 10 Ｇ L - L = 20 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ



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105 15

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105 15105 15

1 1505 1 1505

1 15051 1505

105 15 105 15

1 1505

L ()1 1505

L ()1 1505

L ()

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V - L = 10 Ｇ V - L = 20 Ｇ

kV - L = 30 Ｇ kV - L = 40 Ｇ

V - L = 30 Ｇ V - L = 40 Ｇ

L = 10 Ｇ L = 20 Ｇ

L = 30 Ｇ L = 40 Ｇ


L = 50ndash200Ｇ

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012

014

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018

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002501 02

002501

002501

002501

1 1505

V ()1 1505

V ()

1505 115105

1 1505 1505 1

151051 1505

151051505 1

1 1505 1 1505

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1

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V ()

1 1505

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V - V -

V - V -V -





119872119894 sdot (RL119894 rarr 1) (C8)



(C9)


119872119894 (RL119894 997888rarr 1) = Θ119894 (RL119894 = 0) sdot 2119864119868119894119871 (C10)





RD119894 = Θ119894 (RL119894) sdot 119871120575119894 (RL119894) (C12)





(C14)


RL119894


(C15)





119898119894sdot 120593119894 (RL119894 120572119894 119871 120585119894)

(C17)


BF119894 (RL119894 120572119894 119871 120585119894) = 120578119894 (RL119894 120572119894 119871 120585119894)120578119894 (RL119894 = 0 120572119894 119871 120585119894) (C18)



(C19)


BF119894 (RL119894 120572119894 119871 120585119894)


120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)= 119898119894 (RL119894 = 0)

119898119894 (RL119894)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟BF119898119894(RL119894)

sdot 120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟BF120593119894(RL119894 120572119894 119871120585119894)

(C20)

obtaining


= 119898119894 (RL119894 = 0)119898119894 (RL119894)

sdot 120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)

(C21)





Acknowledgments


References

























































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



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L - L = 10 Ｇ L - L = 20 Ｇ

kL - L = 30 Ｇ kL - L = 40 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ

L - L = 10 Ｇ L - L = 20 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ





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15105

1 1505

V ()1505 1

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1 15051 1505

1 1505 1 1505

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1 150505 151

1 1505 105 15

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14

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V


V - L = 10 Ｇ V - L = 20 Ｇ

kV - L = 30 Ｇ kV - L = 40 Ｇ

V - L = 30 Ｇ V - L = 40 Ｇ

L = 10 Ｇ L = 20 Ｇ

L = 30 Ｇ L = 40 Ｇ


L = 50ndash200Ｇ V - V -

V -V -

V -



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104 06 080200

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V [mdash] L [mdash]


Appendix


See Figures 7 and 8



1198701199031119894 = 1198701199032119894 = 119870119903119894 (C1)



Θ1119894 = Θ2119894 = Θ119894 (C2)







(C5)




(C6)





L - L = 10 Ｇ L - L = 20 Ｇ

kL - L = 30 Ｇ kL - L = 40 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ

L - L = 10 Ｇ L - L = 20 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ



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105 15

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105 15 105 15

105 15105 15

1 1505 1 1505

1 15051 1505

105 15 105 15

1 1505

L ()1 1505

L ()1 1505

L ()

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V - L = 10 Ｇ V - L = 20 Ｇ

kV - L = 30 Ｇ kV - L = 40 Ｇ

V - L = 30 Ｇ V - L = 40 Ｇ

L = 10 Ｇ L = 20 Ｇ

L = 30 Ｇ L = 40 Ｇ


L = 50ndash200Ｇ

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014

016

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002501 02

002501

002501

002501

1 1505

V ()1 1505

V ()

1505 115105

1 1505 1505 1

151051 1505

151051505 1

1 1505 1 1505

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1

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V ()

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V - V -

V - V -V -





119872119894 sdot (RL119894 rarr 1) (C8)



(C9)


119872119894 (RL119894 997888rarr 1) = Θ119894 (RL119894 = 0) sdot 2119864119868119894119871 (C10)





RD119894 = Θ119894 (RL119894) sdot 119871120575119894 (RL119894) (C12)





(C14)


RL119894


(C15)





119898119894sdot 120593119894 (RL119894 120572119894 119871 120585119894)

(C17)


BF119894 (RL119894 120572119894 119871 120585119894) = 120578119894 (RL119894 120572119894 119871 120585119894)120578119894 (RL119894 = 0 120572119894 119871 120585119894) (C18)



(C19)


BF119894 (RL119894 120572119894 119871 120585119894)


120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)= 119898119894 (RL119894 = 0)

119898119894 (RL119894)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟BF119898119894(RL119894)

sdot 120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟BF120593119894(RL119894 120572119894 119871120585119894)

(C20)

obtaining


= 119898119894 (RL119894 = 0)119898119894 (RL119894)

sdot 120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)

(C21)





Acknowledgments


References

























































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



0

02

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04

06

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minus02

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0

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0808

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05 1 15

V ()

105 15

15105

1 1505

V ()1505 1

V ()

1 15051 1505

1 1505 1 1505

1505 115105

1 150505 151

1 1505 105 15

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V


V - L = 10 Ｇ V - L = 20 Ｇ

kV - L = 30 Ｇ kV - L = 40 Ｇ

V - L = 30 Ｇ V - L = 40 Ｇ

L = 10 Ｇ L = 20 Ｇ

L = 30 Ｇ L = 40 Ｇ


L = 50ndash200Ｇ V - V -

V -V -

V -



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0005 001 0015 00200

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PDF

104 06 080200

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250

V [mdash] L [mdash]


Appendix


See Figures 7 and 8



1198701199031119894 = 1198701199032119894 = 119870119903119894 (C1)



Θ1119894 = Θ2119894 = Θ119894 (C2)







(C5)




(C6)





L - L = 10 Ｇ L - L = 20 Ｇ

kL - L = 30 Ｇ kL - L = 40 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ

L - L = 10 Ｇ L - L = 20 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ



0

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minus02

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L

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1 1505

105 15 105 15

105 15105 15

1 1505 1 1505

1 15051 1505

105 15 105 15

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L ()1 1505

L ()1 1505

L ()

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V - L = 10 Ｇ V - L = 20 Ｇ

kV - L = 30 Ｇ kV - L = 40 Ｇ

V - L = 30 Ｇ V - L = 40 Ｇ

L = 10 Ｇ L = 20 Ｇ

L = 30 Ｇ L = 40 Ｇ


L = 50ndash200Ｇ

0

02

02

04

04

06

06

0808

minus04

minus02

0

02

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1

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006

008

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016

018

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002501 02

002501

002501

002501

1 1505

V ()1 1505

V ()

1505 115105

1 1505 1505 1

151051 1505

151051505 1

1 1505 1 1505

05

1

15

05

1

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1 1505

V ()

1 1505

1 1505

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14

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V

04

06

08

1

12

14

16

V

04

06

08

1

12

14

16

V

V - V -

V - V -V -





119872119894 sdot (RL119894 rarr 1) (C8)



(C9)


119872119894 (RL119894 997888rarr 1) = Θ119894 (RL119894 = 0) sdot 2119864119868119894119871 (C10)





RD119894 = Θ119894 (RL119894) sdot 119871120575119894 (RL119894) (C12)





(C14)


RL119894


(C15)





119898119894sdot 120593119894 (RL119894 120572119894 119871 120585119894)

(C17)


BF119894 (RL119894 120572119894 119871 120585119894) = 120578119894 (RL119894 120572119894 119871 120585119894)120578119894 (RL119894 = 0 120572119894 119871 120585119894) (C18)



(C19)


BF119894 (RL119894 120572119894 119871 120585119894)


120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)= 119898119894 (RL119894 = 0)

119898119894 (RL119894)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟BF119898119894(RL119894)

sdot 120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟BF120593119894(RL119894 120572119894 119871120585119894)

(C20)

obtaining


= 119898119894 (RL119894 = 0)119898119894 (RL119894)

sdot 120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)

(C21)





Acknowledgments


References

























































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



0

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CDF

0005 001 0015 00200

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104 06 080200

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250

V [mdash] L [mdash]


Appendix


See Figures 7 and 8



1198701199031119894 = 1198701199032119894 = 119870119903119894 (C1)



Θ1119894 = Θ2119894 = Θ119894 (C2)







(C5)




(C6)





L - L = 10 Ｇ L - L = 20 Ｇ

kL - L = 30 Ｇ kL - L = 40 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ

L - L = 10 Ｇ L - L = 20 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ



0

0

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06

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minus02

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L

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1

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L

105 15

1 1505

105 15 105 15

105 15105 15

1 1505 1 1505

1 15051 1505

105 15 105 15

1 1505

L ()1 1505

L ()1 1505

L ()

05

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15




V - L = 10 Ｇ V - L = 20 Ｇ

kV - L = 30 Ｇ kV - L = 40 Ｇ

V - L = 30 Ｇ V - L = 40 Ｇ

L = 10 Ｇ L = 20 Ｇ

L = 30 Ｇ L = 40 Ｇ


L = 50ndash200Ｇ

0

02

02

04

04

06

06

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minus04

minus02

0

02

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1

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006

008

01

012

014

016

018

02

002501 02

002501

002501

002501

1 1505

V ()1 1505

V ()

1505 115105

1 1505 1505 1

151051 1505

151051505 1

1 1505 1 1505

05

1

15

05

1

15

05

1

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V ()

1 1505

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V

04

06

08

1

12

14

16

V

04

06

08

1

12

14

16

V

V - V -

V - V -V -





119872119894 sdot (RL119894 rarr 1) (C8)



(C9)


119872119894 (RL119894 997888rarr 1) = Θ119894 (RL119894 = 0) sdot 2119864119868119894119871 (C10)





RD119894 = Θ119894 (RL119894) sdot 119871120575119894 (RL119894) (C12)





(C14)


RL119894


(C15)





119898119894sdot 120593119894 (RL119894 120572119894 119871 120585119894)

(C17)


BF119894 (RL119894 120572119894 119871 120585119894) = 120578119894 (RL119894 120572119894 119871 120585119894)120578119894 (RL119894 = 0 120572119894 119871 120585119894) (C18)



(C19)


BF119894 (RL119894 120572119894 119871 120585119894)


120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)= 119898119894 (RL119894 = 0)

119898119894 (RL119894)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟BF119898119894(RL119894)

sdot 120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟BF120593119894(RL119894 120572119894 119871120585119894)

(C20)

obtaining


= 119898119894 (RL119894 = 0)119898119894 (RL119894)

sdot 120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)

(C21)





Acknowledgments


References

























































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




L - L = 10 Ｇ L - L = 20 Ｇ

kL - L = 30 Ｇ kL - L = 40 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ

L - L = 10 Ｇ L - L = 20 Ｇ

L - L = 30 Ｇ L - L = 40 Ｇ



0

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L

105 15

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105 15 105 15

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105 15 105 15

1 1505

L ()1 1505

L ()1 1505

L ()

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V - L = 10 Ｇ V - L = 20 Ｇ

kV - L = 30 Ｇ kV - L = 40 Ｇ

V - L = 30 Ｇ V - L = 40 Ｇ

L = 10 Ｇ L = 20 Ｇ

L = 30 Ｇ L = 40 Ｇ


L = 50ndash200Ｇ

0

02

02

04

04

06

06

0808

minus04

minus02

0

02

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1

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00

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0

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0001

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006

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016

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02

002501 02

002501

002501

002501

1 1505

V ()1 1505

V ()

1505 115105

1 1505 1505 1

151051 1505

151051505 1

1 1505 1 1505

05

1

15

05

1

15

05

1

15

05

1

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1

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1

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1

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1

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1

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1

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1

15

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1

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1 1505

V ()

1 1505

1 1505

04

06

08

1

12

14

16

V

04

06

08

1

12

14

16

V

04

06

08

1

12

14

16

V

V - V -

V - V -V -





119872119894 sdot (RL119894 rarr 1) (C8)



(C9)


119872119894 (RL119894 997888rarr 1) = Θ119894 (RL119894 = 0) sdot 2119864119868119894119871 (C10)





RD119894 = Θ119894 (RL119894) sdot 119871120575119894 (RL119894) (C12)





(C14)


RL119894


(C15)





119898119894sdot 120593119894 (RL119894 120572119894 119871 120585119894)

(C17)


BF119894 (RL119894 120572119894 119871 120585119894) = 120578119894 (RL119894 120572119894 119871 120585119894)120578119894 (RL119894 = 0 120572119894 119871 120585119894) (C18)



(C19)


BF119894 (RL119894 120572119894 119871 120585119894)


120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)= 119898119894 (RL119894 = 0)

119898119894 (RL119894)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟BF119898119894(RL119894)

sdot 120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟BF120593119894(RL119894 120572119894 119871120585119894)

(C20)

obtaining


= 119898119894 (RL119894 = 0)119898119894 (RL119894)

sdot 120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)

(C21)





Acknowledgments


References

























































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




V - L = 10 Ｇ V - L = 20 Ｇ

kV - L = 30 Ｇ kV - L = 40 Ｇ

V - L = 30 Ｇ V - L = 40 Ｇ

L = 10 Ｇ L = 20 Ｇ

L = 30 Ｇ L = 40 Ｇ


L = 50ndash200Ｇ

0

02

02

04

04

06

06

0808

minus04

minus02

0

02

04

06

08

1

12

00

0

0404

0

04

04

0808

0

04

04

0808

0

04

04

0808

0001

0025

0025

005

00501

0

002

004

006

008

01

012

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016

018

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002501 0025

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002501 0025

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0025

0025

005

00501

015

0

002

004

006

008

01

012

014

016

018

02

002501 02

002501

002501

002501

1 1505

V ()1 1505

V ()

1505 115105

1 1505 1505 1

151051 1505

151051505 1

1 1505 1 1505

05

1

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05

1

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05

1

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05

1

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1

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1

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1

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1

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1

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1

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1 1505

V ()

1 1505

1 1505

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06

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1

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14

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V

04

06

08

1

12

14

16

V

04

06

08

1

12

14

16

V

V - V -

V - V -V -





119872119894 sdot (RL119894 rarr 1) (C8)



(C9)


119872119894 (RL119894 997888rarr 1) = Θ119894 (RL119894 = 0) sdot 2119864119868119894119871 (C10)





RD119894 = Θ119894 (RL119894) sdot 119871120575119894 (RL119894) (C12)





(C14)


RL119894


(C15)





119898119894sdot 120593119894 (RL119894 120572119894 119871 120585119894)

(C17)


BF119894 (RL119894 120572119894 119871 120585119894) = 120578119894 (RL119894 120572119894 119871 120585119894)120578119894 (RL119894 = 0 120572119894 119871 120585119894) (C18)



(C19)


BF119894 (RL119894 120572119894 119871 120585119894)


120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)= 119898119894 (RL119894 = 0)

119898119894 (RL119894)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟BF119898119894(RL119894)

sdot 120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟BF120593119894(RL119894 120572119894 119871120585119894)

(C20)

obtaining


= 119898119894 (RL119894 = 0)119898119894 (RL119894)

sdot 120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)

(C21)





Acknowledgments


References

























































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





119872119894 sdot (RL119894 rarr 1) (C8)



(C9)


119872119894 (RL119894 997888rarr 1) = Θ119894 (RL119894 = 0) sdot 2119864119868119894119871 (C10)





RD119894 = Θ119894 (RL119894) sdot 119871120575119894 (RL119894) (C12)





(C14)


RL119894


(C15)





119898119894sdot 120593119894 (RL119894 120572119894 119871 120585119894)

(C17)


BF119894 (RL119894 120572119894 119871 120585119894) = 120578119894 (RL119894 120572119894 119871 120585119894)120578119894 (RL119894 = 0 120572119894 119871 120585119894) (C18)



(C19)


BF119894 (RL119894 120572119894 119871 120585119894)


120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)= 119898119894 (RL119894 = 0)

119898119894 (RL119894)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟BF119898119894(RL119894)

sdot 120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟BF120593119894(RL119894 120572119894 119871120585119894)

(C20)

obtaining


= 119898119894 (RL119894 = 0)119898119894 (RL119894)

sdot 120593119894 (RL119894 120572119894 119871 120585119894)120593119894 (RL119894 = 0 120572119894 119871 120585119894)

(C21)





Acknowledgments


References

























































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





Acknowledgments


References

























































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia
























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


deterministic and probabilistic serviceability assessment

Documents