A Model for the Statistical Distribution of Road Vehicle Vibrations

Transcription

1 Proceedngs of the World Congress on Engneerng 7 Vol II WCE 7, July -, 7, London, U.K. A Model for the Statstcal Dstrbuton of Road Vehcle Vbratons Manuel-Alfredo Garca-Romeu-Martnez, Vncent Roullard, Vcente-Agustn Cloquell-Ballester. Abstract Ths paper presents the ntal results of a study amed at mprovng the method by whch the vbratons produced by transport vehcles are charactersed and smulated. More specfcally, ths paper focuses on the rgd body vbratons generated by road transport vehcles n the contet of dstrbuton of packaged goods and produce. The research uses a varety of vbraton data, collected from varous vehcle types and routes n Span and Australa wth hgh-capacty vbraton recorders. Vehcles used range from small transport vehcles to large truck-tralers wth both arbags and steel sprng suspensons whle the routes travelled nclude suburban streets, man roads and motorways. The paper dscusses the sgnfcance and lmtatons of the average power spectral densty (PSD) and eplans why the average PSD s not always adequate as the sole descrptor of road vehcle vbratons as the process generally tends to be non-statonary and non-gaussan. The paper adopts an alternatve analyss method, based on the statstcal dstrbuton of the movng root-mean-square (RMS) vbratons, as a supplementary ndcator of overall rde qualty. The measured data was used to compute the statstcal dstrbuton of each vbraton record, the shape of whch was compared for the entre set of records. The sutablty of varous mathematcal models, based on the Webull and Raylegh dstrbutons were nvestgated for descrbng the probablty dstrbuton functon (PDF) of road vehcle vbraton RMS tme hstory. The paper proposes a sngle mathematcal model that can accurately descrbe the statstcal character of the om vbratons generated by road vehcles n general. It shows that the model can also effectvely descrbe the statstcal parameters of the process namely the mean, medan, standard devaton, skewness and kurtoss. Inde Terms Random vbratons, RMS dstrbuton, Webull dstrbuton. I. INTRODUCTION In order to develop optmum packagng t s mportant that engneers are not only aware, but have a thorough understandng of the epected mechancal hazards to whch packages are subjected durng shppng and handlng. Ths nformaton allows them to engneer the optmum amount of M.A. Garca-Romeu Martnez s wth ITENE, Technologcal Insttute of Packagng, Transportaton and Logstcs. Polígono Industral D Obradors, C/Soguers, Godella Valenca, Span. mgarcaromeu@tene.com V. Roullard s wth Vctora Unversty, Melbourne Australa PO Bo 8 MCMC, Melbourne 8, Australa. Vncent.roullard@vu.edu.au V. Cloquell Ballester s wth Valenca Unversty of Technology, Span. Camno de Vera s/n,, Valenca, Span. cloquell@dp.upv.es protectve packagng needed to sutably protect the consgnment aganst the rsk of damage. To assst desgners n reducng cost, ether by avodng wastng packagng materals due to over-packagng or avodng damage due to under-packagng, dstrbuton vbratons need to be smulated n the laboratory n order to test and valdate protectve package desgns. Because verfcaton of desgn by tral shpments have been shown to be both mpractcal and nadequate [], performance testng of packagng systems n the laboratory has become ncreasngly the more adopted tool n the optmsaton of package desgns. Testng of package desgns under controlled laboratory condtons usually nvolves the smulaton of vbratons epected to be encountered durng transportaton. Assumptons regardng the nature and level of vbratons are sometmes adopted and make t dffcult to optmse protectve packagng wthout epermental verfcaton. These unsophstcated and appromate smulaton methodologes promote the adopton of a conservatve approach to packagng desgn whch, n many cases, lead to over packagng. Vbratons that occur n vehcles durng transportaton are comple and play a sgnfcant role n the level of damage eperenced by products durng shpment. Vehcle vbratons have a om nature and ther character and level s dependent on the type of vehcle, suspenson type, payload, vehcle speed and road condton. Because of these varabltes, t s not always possble to represent transport vbratons wth a smple functon such as the power spectral densty (PSD) functon. Wth the advent of sophstcated vbraton recorders n the past decade, packagng engneers have been able to measure and analyze ncreasng volumes of vbratons that occur n commercal shpments. Recently, numerous studes have been undertaken wth the am of measurng and evaluatng the vbratons n varous dstrbuton envronments around the globe and usng partcular vehcle types to enable packagng engneers to develop packagng solutons to meet world-wde dstrbuton challenges [][3][][][]. The man purpose of these eercsed was to generate effectve laboratory test schedules for evaluatng the performance of package systems when subjected to vehcle vbratons durng dstrbuton. Unfortunately, the prevalng trend s to characterse these comple vbratons wth a sngle functon namely, the average power spectral densty (PSD). ISBN: WCE 7

2 Proceedngs of the World Congress on Engneerng 7 Vol II WCE 7, July -, 7, London, U.K. a(m/s ) crest factor event number RMS Crest Factor Fgure. Typcal road vehcle vbraton record along wth the RMS and crest factor tme hstores - - The now well-establshed and wdely adopted procedure for the laboratory smulaton of vehcle vbratons s to synthesze om vbratons from the average PSD of measured vbraton data (whch s assumed to wholly descrbe the transport envronment) usng om vbraton controllers. These controllers synthesze normally-dstrbuted vbratons by contnually computng the Inverse Fourer Transform of the PSD coupled wth a unformly dstrbuted om phase array. Unfortunately, by vrtue of the fact that these systems use solely the average PSD to synthesze a normally-dstrbuted om sgnal, the smulated vbratons turn out to be statonary hence not capable of emulatng the ecursons n vbraton ampltude whch are found to occur n the feld. One nterestng characterstc of road vehcle vbratons s that the shape of the PSD remans largely unchanged for the duraton of each transport event []. In effect, the non-statonarty of the process s manfested through fluctuatons n RMS level [7]. Most vbraton controllers can be programmed so that the RMS level of the synthessed om sgnal s made to vary as a functon of tme. However, there s no establshed technque to determne how ths modulaton of ampltude should be mplemented. The man objectve of ths paper s to establsh whether a sngle mathematcal model can be used to descrbe the statstcal dstrbuton of the movng RMS of vbratons generated by road vehcles n general, and whether the model can be used to characterse the overall rde qualty as well as be of use n determnng laboratory test schedules that nclude the generaton of om vbratons of varyng RMS levels. II. MODELLING THE RMS DISTRIBUTION Roullard & Sek [8], studed the non-statonary behavour of road vehcle vbratons and proposed a statstcal model for charactersng what they term the vbraton ntensty. Ther model s a modfed verson of the Raylegh dstrbuton that ncludes an eponent parameter and a scale parameter. Ther model appled to the vbraton ntensty whch can only be computed by an elaborated algorthm based on the Hlbert transform. Further work amed at usng the RMS dstrbuton of vehcle vbratons to desgn laboratory tests schedules was undertaken by Roullard & Sek [7]. Ths work shows how more realstc vbratons can be syntheszed by recognzng that road vehcle vbratons are non-statonary and by makng use of the RMS dstrbuton. One of the most elementary approaches to charactersng non-statonartes s to compute the RMS (or mean-square) of the vbraton record over relatvely short segments [9]. The length of the segments and the ncremental step for computng the movng RMS are crtcal to the analyss. The movng RMS of a functon (t) can be wrtten as: + w ˆ () t ( j) for, δ, δ,3 δ...nδ n j Where w s the segment length, δ s the ncremental step and N s the total number of segments n the sample. The effects of the wndow wdth and the ncremental step (overlap) on the movng RMS of non-statonary vbraton sgnals have been llustrated by Roullard []. It shows that care must be taken n selectng the parameters for computng the RMS tme hstory of non-statonary sgnals. In order to valdate the proposed model, a number of sample vbraton records were collected from a wde range of vehcles and routes. The vbraton data were collected usng self-contaned data recorders (Saver by Lansmont) confgured to record vbratons for predetermned sub-record lengths of 8 seconds at a samplng rate of Hz. The recorders were confgured to ntate recordng at specfc perods varyng from 9 seconds to one mnute. A total of thrteen measurements were undertaken usng varous vehcles ncludng small utlty, vans, rgd trucks and sem-tralers wth varous suspenson types and payloads. Routes ncluded poorly mantaned local roads, country roads, urban roads, and hghways located n Vctora, Australa and Span as shown n Table. The RMS tme hstory of each vbraton record was computed usng () wth w 8 seconds and no overlap (δ w + /fs, where fs samplng frequency n Hz). A typcal eample of a vbraton record along wth the movng RMS s shown n Fg.. It also () ISBN: WCE 7

3 Proceedngs of the World Congress on Engneerng 7 Vol II WCE 7, July -, 7, London, U.K. ncludes a plot the movng crest factor whch ndcates the non-statonary character of the process; for a Gaussan process of 89 samples, the lkelhood that the crest factor eceeds 3. s.%. Ths s a strong ndcaton that the data recorded are non-gaussan and non-statonary []. Table. Summary of measured vbraton record parameters. Record ID Vehcle type & load Country Route Type DATA A Utlty vehcle ( Tonne cap.). Load: Australa Suburban streets < % cap. DATA B Prme mover + Sem traler (Ar rde Australa Country roads susp.). Load: 9% cap. DATA C Transport van (7 kg cap.). Load: Australa Suburban streets % cap. DATA D Transport van (7 kg cap.). Load: % cap. Australa Man suburban hwy DATA E Transport van (7 kg cap.). Load: Australa Motorway % cap. DATA F Prme mover + Sem traler (Leaf Australa Country roads sprng susp.). Load: < % cap. DATA G Tpper truck ( Tonnes cap., Ar Australa Country roads rde susp.). Load: % capacty. DATA H Small flat bet truck ( Tonne cap., Australa Suburban streets Leaf sprng susp.). Load <% cap. DATA J Flat bed truck ( Tonnes cap., Leaf Australa Country roads sprng susp.). Load >9% cap. DATA K Sedan car. Load: passenger Australa Suburban streets DATA L Prme mover + Sem traler (Ar rde Span Motorway susp.). Load: % cap. DATA M Prme mover + Sem traler (Ar rde Span Motorway susp.). Load: % cap. DATA N Prme mover + Sem traler (Leaf Span Motorway sprng susp.). Load: % cap. DATA O Prme mover + Sem traler (Leaf sprng susp.). Load: < % cap. Span Motorway The Probablty Densty Functon (PDF) of the RMS tme hstory of each of the thrteen vbraton records was computed wth the am of developng a generc mathematcal model that can be used to characterse the statstcal characterstcs of the process regardless of vehcle type, payload or route. A range of statstcal dstrbutons were studed and a model gven n (3) was developed, based on the three-parameter Webull dstrbuton gven n (). γ γ γ P( ) e The proposed modfed Webull dstrbuton model was developed to afford addtonal control over varous aspects of the shape of the dstrbuton functon. It ncludes an eponent parameter,, whch enables the control of the slope of the rght-hand tal of the dstrbuton and ncreases the scope of the model for charactersng a wder range of dstrbuton functons. The model, gven n (3), was found to be generc enough to be able to produce a range of well-known dstrbutons for whch the parameters are gven n Table. () Where s the movng RMS,,, γ and are the modfed Webull parameters and s the left hand doman lmt. P() P() P() P() (Influence of parameter). (Influence 3 of γ parameter) γ γ. 3 7 (Influence of parameter) (Influence of o parameter) X X. γ where ρ Γ, ] ], // P( ) γ ρ e + [ //, [ (3) Fgure. Influence of parameters on the proposed four-parameter modfed Webull dstrbuton. ISBN: WCE 7

4 Proceedngs of the World Congress on Engneerng 7 Vol II WCE 7, July -, 7, London, U.K. Table. Parameters values for typcal dstrbutons. The sngle-parameter statstcs for the model, namely, the mean, µ, the medan, mdn, the standard devaton, σ, the skewness, ν (Sk) and the kurtoss, Kt, were derved and are gven as: [] µ + Ψ () γ mdn γ Γ, Γ, [ ] σ E µ Ψ + µ where E 3 σ [] 3 [] 3 3 ν Sk E 3µ E µ Ψ + 3 Ψ + 3 µ where E () () (7) 3 [ ] [ 3] [ ] 3 Kt E µ E + µ E µ σ 3 3 Ψ + Ψ + Ψ + µ where E 3, and γ + j o Γ, [ j] Ψ γ o Γ, In the case of the RMS dstrbuton, the left hand doman lmt,, was chosen as greater than zero snce the RMS tme hstory s, by defnton, always postve. For the purpose of ths study, n whch only rgd body vbratons are of nterest, o. Ths has the effect of dscountng the sustaned, resdual low level vbratons that are not caused by road pavement nteractons []. Therefore (3) can be wrtten as follows, to characterse the movng RMS PDF of road vehcle vbratons: ] [, P γ e γ + Γ [, [ The nfluence of each of the four parameters on the shape of the dstrbuton functon are llustrated n Fg. whch shows that each parameter alters dfferent aspects of the dstrbuton shape. Further analyses, undertaken to nvestgate the cross-correlaton between the parameters, showed that there s no sgnfcant nter-parameter dependence. (8) (9) S n ma_ter k, ε, _mn nn+ Random ntal condtons ε + () ε + () γ ε + () + () o Matlab functon to ft an equaton to a data by least squares optmsaton ( ' _ ',' [,, γ, o ] ', '[ ]' ) Inlne Eq Model Datato ft [ Datato ft ] Least squares ft γ,,, o [ ] False γ>,, and? True E Best ft γ,,, o [ ] True < _ mn? Calculate the of the statstc parameters mean _ model_sp data_sp data_sp ( model_sp mean _ ) std _ mean _ + std _ Calculate statstc parameters for the model model_sp mean model_sp medan model_sp std 3 model_sp Sk model_sp Kt True n < ma_ter? False _ mn k _ mn n Fgure 3. Algorthm for optmsaton of the ft based on the s of the fve statstcs parameters. ISBN: WCE 7

5 Proceedngs of the World Congress on Engneerng 7 Vol II WCE 7, July -, 7, London, U.K. III. RESULTS A computer program (coded n Matlab ) was developed to determne the optmum parameter values that yeld the best ft for the proposed model wth respect to the PDF of measured vbraton data. Results usng the sum-of-squared (least squares) optmsaton were found to produce unstable results. Ths was attrbuted to the relatvely large number (four) of ndependent parameters whch was found to acheve least square s for several combnatons of parameter values. In order to address ths dffculty, code was modfed (Fg. 3) to nclude optmsaton based on the mean and standard devaton of the s between the ftted and measured data for fve statstcal parameters namely the mean, medan, standard devaton, skewness, and kurtoss. Ths curve fttng algorthm was used to subject the proposed four-parameter modfed Webull model to valdaton tests usng all thrteen vbraton records (Table ) and was found to offer good agreement as shown n Fg. whch shows four typcal eamples. The goodness of ft between the dstrbuton of the measured data and the model are best revealed graphcally as shown n Fg. whch shows plots of the man statstcal parameters for all thrteen cases. It can be seen that very good agreement s acheved (R.99) for the frst and second order statstcs (mean, medan and standard devaton) whle reasonably good agreement s acheved (R.9) for the thrd and fourth order statstcs, represented here by the skewness and Kurtoss. The analyss of all thrteen vbraton records show that the model s capable of representng RMS dstrbutons consstng of varous values of kurtoss, skewness and standard devatons as shown n Fg.. IV. CONCLUSIONS Ths paper has presented the ntal results of a study amed at mprovng the method by whch the rgd body vbratons produced by road transport vehcles are charactersed. Vbraton data, collected from varous vehcle types and routes n Span and Australa, was used to develop and valdate a mathematcal model, based on the Webull dstrbuton, to descrbe the probablty densty functon of the movng RMS tme hstores of the process. The paper has addressed the lmtatons of the average power spectral densty (PSD) and eplans why the average PSD s not always adequate as the sole descrptor of road vehcle vbratons as the process generally tends to be non-statonary and non-gaussan Data L (Ar Rde - Load - Hghway) µ Data / Ft:.38 /.3 Mdn Data / Ft:.33 /.3 σ Data / Ft:.38 /.37 Sk Data / Ft:.3 /. Kt Data / Ft:.3 /.8 µ/σ Data / Ft:.7 / γ.87 o Data RMS F(m/s ) (Leaf Sprng Truck - No Load - Local Roads). Data L Ft Mean Data Mean Ft Medan Data µ Medan Data / Ft Ft: 7.7 / Mdn Data / Ft: 7.7 / 7.97 σ Data / Ft:.8 /.7. Sk Data / Ft:.8 /. Kt Data / Ft: 3. / 3.. µ/σ Data / Ft:.37 / γ.89 o Data RMS N(m/s ) (Leaf Sprng - Load - Hghway).8 Data F Ft Mean Data Mean Ft Medan Data µ Medan Data / Ft:.9 /.7 Mdn Data / Ft:. /.9.7 σ Data / Ft:.7 /.77 Sk Data / Ft:.3 /.. Kt Data / Ft:. /.9 µ/σ Data / Ft:.7 /..... γ..3 o Data RMS O(m/s ) Data (Leaf N Sprng - No Load Ft - Hghway) Mean Data. Mean Ft Medan Data µ Medan Data / Ft: 3.73 /.3.3 Mdn Data / Ft: 3.7 / 3. σ Data / Ft:.99 /. Sk Data / Ft:.7 /..3 Kt Data / Ft:. /.7 µ/σ Data / Ft:. / γ.. o RMS (m/s ) Data O Ft Mean Data Mean Ft Medan Data Medan Ft Fgure. Valdaton of four-parameter modfed Webull model for four typcal cases ISBN: WCE 7

6 Proceedngs of the World Congress on Engneerng 7 Vol II WCE 7, July -, 7, London, U.K. The paper adopts an alternatve analyss method, based on the statstcal dstrbuton of the movng root-mean-square (RMS) vbratons, as a supplementary ndcator of overall rde qualty. The paper proposes a sngle mathematcal model that can accurately descrbe the statstcal character of the om vbratons generated by road vehcles n general. Ft Data, RMS (m/s ) Ft Data, Kurtoss 8 8 Epermental Data, RMS (m/s ) 8 : correlaton : correlaton : correlaton Mean Medan Std. Devaton : correlaton Kurtoss Skewness. 8 Epermental Data Fgure. Goodness of ft plots for the man statstcal parameters Ft Data, σ (m/s ) Ft Data, Skewness Ths result s relevant not only for the charactersaton of rde qualty but also for the accurate synthess of road vehcle vbratons n the laboratory. The results can be used to assst n developng a novel method for smulatng non-statonary (modulated) vbraton n the laboratory. The RMS dstrbuton functon can be used to create an RMS level schedule that wll enable the synthess of om vbratons wth varyng RMS level to better represent the road transport vbraton process. REFERENCES [] M. A. Sek, Optmsaton of Packagng Desgn Through an Integrated Approach to the Measurement and Laboratory Smulaton of Transportaton Hazards, Proceedngs of the th Internatonal Conference on Packagng, Internatonal Assocaton of Packagng Research Insttutes, Warsaw, Poland, [] S. P. Sngh, E. Joneson and J. Sngh, Measurement and analyss of US truck vbraton for leaf sprng and ar rde suspensons, and development of tests to smulate these condtons, Journal of Packagng Technology and Scence,, 9: [3] S. P. Sngh and J. Marcondes, Vbraton levels n comercal truck shpments as a functon of suspenson and payload, Journal of Testng and Evaluaton, 99, Vol, No., -9 [] C. Perce, S. P. Sngh and G. A. Burgess, Comparson of leaf sprng to ar cushon traler suspensons n the transportaton envronment, Journal of Packagng Technology and Scence, 99, Vol, - [] S. P. Sngh, J. Antle and G. A. Burgess, Comparson between lateral, longtudnal and vertcal vbraton levels n commercal truck shpments, Journal of Packagng Technology and Scence, 99, Vol., 7-7 [] M. A. Garca-Romeu-Martnez and S.P. Sngh, Developng Vbraton Smulaton Methods for Truck Transport n Span as a Functon of Payload, Suspenson and Truck Speed, Proceedngs of the th Internatonal IAPRI World Conference on Packagng, Tokyo, Japan,, 9- [7] V. Roullard. and M. A. Sek, Generatng road vbraton test schedules from pavement profles for packagng optmzaton, Proceedngs of the st IAPRI Symposum on Packagng, Valenca, Span, 3 [8] V. Roullard and M. A. Sek, Statstcal modellng of predcted nonstatonary vehcle vbratons, Internatonal Journal of Packagng Technology and Scence,, Vol (), pp 93- [9] J. S. Bendat and A. G. Persol, Random data analyss and measurement procedures, John Wley and Sons, New York, 98 [] V. Roullard, On the Laboratory Synthess of Non-statonary Road Vehcle Vbratons, Ph.D Thess, Monash Unversty, Melbourne, Australa (n press), The proposed modfed Webull dstrbuton model was developed to afford addtonal control over varous aspects of the shape of the dstrbuton functon. The model was found to be generc enough to be able to produce a range of well-known dstrbutons. Curve fttng results usng the sum-of-squared (least squares) optmsaton were found to produce unstable results whch requred ncluson of the mean, medan, standard devaton, skewness, and kurtoss n the optmsaton algorthm. Valdaton tests usng all thrteen sample vbraton records and was found to offer good agreement n general. The paper also shows how the model s capable of accurately descrbng the statstcal parameters of the process namely the mean, medan, standard devaton, skewness and kurtoss. ISBN: WCE 7