PRPgenerator User`s Manual
Contents
1. with c ct This simplification makes the explanation easier 0 Substituting equation D into equation 8 the following expression is obtained R 6 2c nee i n cos 2rn dn 10 And considering expressions 6 and R 8 2 p STN 6 2b Qrna V2 2b ow coser EPan 11 Integrals in expressions and are solved semi analitically as explained in appendix A This solution is computationally faster than the numerical integration Once we have computed the cross correlation function cross PSD is computated by means of Discrete Fourier Transform N 1 Gole 2 X Rela E 0 1 N 1 12 n 0 Since R is unique for a pair of profles Gy is also unique Coherence of a pair of profiles can be expressed in this kind of process as G n 13 given that G n is real 9 Coherence Spatial frequency m Figure 4 Coherence function for different distances between profiles Figure 4 shows coherence functions for different distances between profiles Ac tually coherence function defined in expression is independent of c therefore it will not depend on the road class but only on the distance between profiles Coher ence is higher for small spatial frequencies or high wavelengths and gets significantly lower for high spatial frequencies or reduced wavelengths This is quite feasible since high wavelength imperfections affect to a higher width than small wavelength im perfe2. a function of the road class Table 1 Road class G no m A Very good 16 107 B Good 64 1076 C Medium 256 1076 D Poor 1024 1076 E Very poor 4096 1076 Table 1 Ga no for different road classes according to ISO 8608 This expression is adopted in PRPgenerator Lower and upper spatial frequency limits are set as Minin 0 01 m7 and Nmar 10 0 m t 8 those limits correspond to wavelength limits Amar 100 m and Amin 0 1 m 4 2 The improved way parallel profiles When a four wheeled vehicle runs along a road an imposed history of displacement is applied on all its tyres due to the road imperfections For wheels in the same transverse position for example the two right wheels it can be assumed that both run on the same profile Hence the imposed displacement is the same but there is a time lag because front tyres reache every point a while before rear ones On the other hand tyres that are not transversely aligned that is to say left and right tyres run over different paths and therefore the imperfections and the imposed displacements are different As stated before different longitudinal profiles computed by means of equation are independent profiles which correspond to one line along the road PRPgenerator assumes two main hypotheses in regard to road surface e Homogeneity every straight profile on the road has the same statistical characteristics independently of its position e I
3. frequencies In order to generate the second road section we need to change the values of sample distance 0 02 m and number of frequencies 1000 Then click GENERATE PROFILES The new profiles are depicted and the considered frequencies are shown This is just an example any notation can be adopted 13 Road stratch length m Sample distance m 0 02 Generated profiles Number of frequencies L 4 Eevation fm 50 Distance Im Figure 12 Tutorial when the second pair of profiles is generated it is depicted and also are the values of g n employed in the coherence function Figure 12 notice that the considered frequencies are not the same as in the previous profile generation because we have changed the value of N Values of n G G and g and the new profiles are stored in the corresponding files G_GX_2_ 64 dat prof_2 64 LF dat and prof_2 64 RT dat As we need a third set of parallel profiles we need to protect the just generated information Let s change the file names again e G_GX_2_ 64 dat gt G_GX_2_64 1000 1 dat e prof 2_64_LF dat prof_2_64_LF_1000_ 1 dat e prof 2_64_RT dat gt prof _2_64_ RT 1000 _1 dat where _1000_ 1 indicates that we have used 1000 spatial frequencies and this is the first set of profiles The last task is to obtain a new pair of profiles with the same features as the previous one The o
4. it is also a straight profile on the same homogeneous and isotropic surface Therefore Ra R V 26 6 where 0 2b is the distance along AB profile between two points at in x direction b is half the distance between the two profiles PSD function G n that describes the profiles must satisfy some conditions in order to be suitable for describing an isotropic surface it must be monotonically non increasing and have a bounded integral 9 Expression proposed in ISO 8608 equation 3 does not satisfy isotropy conditions and therefore it has to be slightly modified If the function is assumed constant at all spatial frequencies below na we will get a piecewise defined spectral density function that is valid for the description of an isotropic surface G no 22 if NnS Tg G n 7 Gal if n gt na where na is set na 0 01 m t Hence PSD function gets the appearance shown in figure Autocorrelation function can be derived from PSD function as R G n cos 27d6n dn 8 E gt a n Figure 3 ISO 8608 spectral density function modified in order to represent a homogeneous and isotropic process logarithmic scale in both axes And cross PSD G n can be obtained from R by means of Fourier transform Let s see how to get autocorrelation and cross sorrelation functions from G n Given that G no and no are constant expression 7 can be rewritten as n n lt G n 9 cn if n gt 1
5. only difference in the imposed vertical displacement at the tyres arises from the time lag between front and rear tyres This is the common procedure employed in road bridge dynamics so the reader is probably familiar with it It is a commonly assumed hypothesis that the randomness of the road surface roughness can be represented with a normal stationary ergodic random process described by means of its Power Spectral Density PSD also known as G 6 Several PSD functions have been proposed by different authors A survey of different approximations can be found in 7 Once a PSD definition has been chosen road profiles are generated as the sum of a series of harmonics r x x V2G n An cos 2rnix 1 where r is the road elevation G n is the one sided power spectral density for the spatial frequency or wavenumber n and is the random phase angle uniformly distributed from 0 to 27 This expression employs N frequencies between two values Nmin ANd Nmar and the increment value An is set constant Nmar Nmi A Max min 2 n Noa Tmin 2 ISO 8608 specifications 8 propose a distribution for the one sided PSD defined by the following expression Gln G r Z 3 no where G n is the one sided power spectral density for the spatial frequency or wavenumber n and G ng is the one sided power spectral density for the reference spatial frequency no 0 1 m t The value for G ng is prescribed by ISO 8608 as
6. PRPgenerator User s Manual Javier Oliva Quecedo Technical University of Madrid UPM Saint Louis University Madrid Campus Contents 1 Introduction 2 Operating system and hardware requirements 3 Installation of PRPgenerator 4 Theoretical background 4 1 The classic way one only profile for every wheel 2 BI 4 2 The improved way parallel profiles al 5 Overview p 6 Tutorial 12 6 1 Using a new working directory ooo a a 20000 A Semi analytical solution of the correlation functions 1 Introduction PRP stands for Parallel Road Profiles and therefore PRPgenerator generates parallel synthetic road profiles of a given length The main objective of the generation of parallel profiles along a road stretch is to take into consideration that road profile is not constant along road width and therefore imperfections under left and right tyres of a 4 or more wheeled vehicle are different This fact is frequenlty neglected in road vehicle and bridge dynamics This simplification can be found both when road profile is measured in situ and also when synthetic profiles are employed This entails that the road profile is the same under all wheels Nevertheless when a four wheeled vehicle runs over a road left and right wheels do not follow the same path thus the profiles under each side tyres are different but those profiles are not independent Road roughness is a main source of dynamic excitation in road veh
7. al in expression 15 where a 270 is solved as follows f cos 27n0 T cos 27n 0 n Na 276 E Si 2nnad 30 a And integral in expression in which a 27 6 2b T cos 27n 6 20 3 n2 a B cos 2TNa 0 2b _ 27 82 2b 2 E Si 2mna y 26 31 Na 18
8. alue must be large enough if we want to get a good profile description N 2000 is a good choice There are two buttons on the lower left corner of the PRPgenerator window COMPUTE g and GENERATE PROFILES e COMPUTE g calculates G n and the coherence function g n At the end of the computation coherence function is depicted in the correponding region of the Results section see figure g Coherence function is stored in the working directory as an ascii file named g_ Distance between profiles _ Road class label dat The Road class label is the value of G no in table 1 multiplied by 10 that is to say Class A 16 Class B 64 Class C 256 Class D 1024 Class E 4096 For example g 2 2 16 dat is the coherence function for a distance between profiles equal to 2 2 meters and Road class A This file is organized into two columns 1 spatial frequency n m and 2 coherence function g For the computation of g n only Road class and Distance between profiles are needed so the other fields can remain unspecified at this point e GENERATE PROFILES does what it says Road section length sample dis tance and number of frequencies are needed now When this button is clicked profiles are generated and depicted in the lower part of the Results section figure 9 The values of g employed in the generation that will depend on the Number of frequencies considered are marked as black circles over the co herence function obtained before Those
9. and the generated profiles in the lower one Two buttons COMPUTE g and GENERATE PROFILES are located on the left hand lower corner PRP_main a Help Problem data Results Coherence function d Coherence Road class Very good A gt Distance between profiles m Road stretch length m 0 0 01 Frequency cycles m Sample distance m m Generated profiles Number of frequencies GENERATE PROFILES 01 2 i a Elevation m o 50 Distance m Figure 7 Initial appearance of the main and only window of PRPgenerator The following parameters are set by the user in the Problem data section Road class according to ISO 8608 One option in the drop down list has to be chosen available road classes range from Very good A to Very poor E Distance between profiles in meters In general it will correspond with the ve hicle width or more precisely with the axle track distance between the cen terline of two wheels on the same axle Road stretch length in meters This is the length of the profiles to be generated Sample distance in meters Since 0 1 m is the lower considered wavelength the distance between samples in the generated profiles should not be higher than 0 02 m Number of frequencies or equivalently wavelengths to be used in the profile generation This value corresponds to the N in expression 14 This v
10. ctions that are more localized We can also appreciate that coherence is lower for all wavelengths if the distance between profiles is increased which is also logical Once cross PSD G n has been computed the two parallel profiles are obtained as r x D V2G n An cos 2rn x 14a 2 ra x gt Gni An cos 27n x i r 2 D ni An cos 2rn x di os V2 G ni Gz n Ancos 2an x 6 where and 6 are random phase angles uniformly distributed from 0 to 27 Figure 5 shows parallel Class A profiles at different distances generated with PRPgenerator The PSD of those profiles is computed smoothed according to ISO 8608 and compared with the ISO definition Eq 3p The lower is the distance between profiles the higher is the similarity as it is expected As can be seen PSD agreement is excellent If the road stretch length is higher the computation of PSD will be more accurate as can be seen in figure 6 where the road section considered is 400 meters long Elevation mm Elevation mm Elevation mm Left profile seiseecterien Right profile Power Spectral Density m 100 a Distance 0 1 20 i i 60 80 Distance m 20 Left profile I5 f Right profile Power Spectral Density m 100 b Distance 2 0 0 20 40 60 80 Distance m 15 r a r 7 Left profile PEESI 10 Right profile 4 Power Spectral Den
11. d with PRPgenerator The aim of this work is to get the user acquainted with the software The tasks are the following Generate three pairs of Class B profiles for a 2 meters wide vehicle 1 The first road stretch is 100 meters long sample distance is set as 0 01 m and 2000 spatial frequencies are considered N 2000 2 The second stretch is also 100 meters long but the sample dis tance is 0 02 m and only 1000 frequencies are employed N 1000 3 The third stretch has the same features as the second one In first place coherence function g n for Class B profiles at 2 0 meters has to be computed Select Road class Good B from the drop down list and specify the distance between profiles that is the distance between left and right tyres of the same axle 2 0 meters in this case No other information is needed for the coherence function calculation Now click COMPUTE g and wait this part of the computation is lengthy Once the progress bar reaches the end coherence function is graphed in the upper region of the Results section as shown in Figure 10 The fileg 2 64 dat that contains the computed coherence function is then saved in the working directory 12 PRP main r Results Frequency feyclesin Elevation m d Figure 11 Tutorial after clicking GENERATE PROFILES they are depicted in the bottom right hand window
12. ed in the upper region of the Results section using a logarithmic scale for the x axis and a linear scale for the y axis PRP main JE Help m Results Coherence function 1 Computed g os Employed values i Problem data 3 Road class Veryaoed a iz aa Distance between o2 profiles m Road stretch _ length m Frequeney feyclesitn Sample distance m 0 02 Number of frequencies GENERATE PROFILES L O a Left profile Fight profile Figure 9 At the end of the profile generation the two parallel profiles are graphed in the lower part of the Results section coherence function values for the spatial frequencies considered are marked in the coherence function graph 11 PRP main PRP sera Problem data 2 7 Road class Good Distance between profiles m Road stretch Jength m Frequency leyclesin Sample distance m Generated profiles Number of frequencies oos E o 005 GENERATE PROFILES oo s Distance fn L a Figure 10 Tutorial coherence function for Road class B and 2 0 meters between profiles is computed and depicted in PRPgenerator 6 Tutorial In this tutorial some parallel road profiles are generate
13. erator are saved to that folder References 1 J Oliva J M Goicolea M A Astiz and P Antolin Fully three dimensional vehicle dynamics over rough pavement Proceedings of the ICE Transport 166 3 144 157 2013 2 J Oliva J M Goicolea P Antolin and M A Astiz Relevance of a complete road surface description in vehicle bridge interaction dynamics Engineering Structutres 56 466 476 2013 3 J Oliva J M Goicolea P Antolin and M A Astiz Dynamic behaviour of underspanned suspension road bridges under traffic loads Journal of the South African Institution of Civil Engineering 56 3 77 87 2014 4 L Deng and C S Cai Development of dynamic impact factor for performance evaluation of existing multi girder concrete bridges Engineering Structures 32 21 31 2010 L Ding H Hao and X Zhu Evaluation of dynamic vehicle axle loads on bridges with different surface conditions Journal of Sound and Vibration 323 826 848 2009 ao 15 6 A Belay E O Brien and D Kroese Truck fleet model for design and assessment of flexible pavements Journal of Sound and Vibration 311 1161 1174 2008 7 P Andr n Power spectral density approximations of longitudinal road profiles Int J Vehicle Design 40 2 14 2006 8 ISO 8608 Mechanical vibration Road surface profiles Reporting of measured data 1995 9 K M A Kamash and J D Robson Implications of isotropy in random surfaces J
14. icle bridge dynamics and has an influence on both systems the vehicle and the bridge A correct and complete definition of the road surface is then a key point in vehicle bridge dynamic analysis Neglecting L R difference has an effeect on both the vehicle and the bridge With respect to Vehicle dynamics vertical acceleration is overestimated body roll is not gathered forces under wheels are not accurately determined In regard to Bridge dynamic effects are overestimated accelerations deflection Dynamic Amplification Factors Influence in vehicle dynamic behaviour is addressed in I effects on bridges in 2 PRPgenerator is implemented in MATLAB and it is a stand alone application so no MATLAB license or installation is required The software is introduced in 3 even though its theroretical background is explained in I B this background is contained in this manual as well 2 Operating system and hardware requirements Current version of PRPgenerator works in Windows 7 XP Vista 8 Minimum re quirements are expected to be e 4GB RAM e 64 bit processor 3 Installation of PRPgenerator 1 Download PRPgenerator rar to your computer 2 Extract its content at any folder A new folder called PRPgenerator should appear It contains one folder called License and two files PRPgenera tor Manual pdf and PRPgenerator_ pkg exe 3 Double click PRPgenerator_pkg exe and follow the instructions on the screen 4 Once the in
15. nly thing we have to do is to click GENERATE PROFILES again A new pair of profiles is depicted and stored Figure 13 If we want to keep a consistent file notation we shall rename the new files e G_GX_2_ 64 dat gt G_GX_2_64 1000 2 dat e prof_2_ 64 _LF dat prof_2_64_LF 1000 _ 2 dat e prof_2_64_RT dat gt prof_2_64_RT 1000 _2 dat where 1000 _ 2 indicates that we have used 1000 spatial frequencies and this is the second set of profiles It is important to notice that G_GX_2_64 1000 2 dat and G_GX_2_ 64 1000 2 dat will be exactly the same file since the Spatial fre quencies considered have been the same in both calculations 14 PRP main r Results Coherence function PRP sansa Problem data Coherence Road class Good Distance between profiles m Road strech length m se Sample distance m 0 02 Generated profiles Number of frequencies e PROFILES Elevation Im Figure 13 Tutorial a ner pair of profiles with the same properties as the previous one is generated and graphed if GENERATE PROFILES is clicked again 6 1 Using a new working directory If you want to work in another directory just copy PRPgenerator exe not PRPgenerator_pkg exe and the License folder into that directory and double click PRPgenerator exe on that new working directory Thus the software is run there and files created by PRPgen
16. ot this approximation figure we can see that the sine integral tends to 5 when x tends to oo Hence 25 n 2 25 This expression is known as Dirichlet integral after the German mathematician Johann Peter Gustav Lejeune Dirichle lim Si x f any dn w Oo 0 If we substitute sin n with sin an where a is constant it is easy to demonstrate the following 3 if a gt 0 f em y sae 0 ifan 0 n 26 m a if a lt 0 Actually there are several integrals with this name this is one of them 17 It is trivial to prove that a will always be positive or null Let s go back to the expression f sin an in a sin an mn f sin an PA Na n 0 n 0 n The first term in the right hand is solved by means of equation The second term can be approximated by using McLaurin series expansion e sin an ana ang _a ig ee ae oe pe a al Substituting equations 26 and 27 in expression ii sin an Pa Z Sifana si a gt 0 m a s 0 Siana si a 0 where Si ana is computed by means of McLaurin series expansion With expression we can solve the initial integral equation 18 T n dn Sna a 5 Si an 29 n Na a It is important to notice that the piecewise expression in equation vanished in equation 29 the reason is that the second term in the right hand side will be always zero if a 0 so it is not necessary to make a distinction and the expression can be simplified Therefore integr
17. ournal of Sound and Vibration 54 131 145 1977 A Semi analytical solution of the correlation func tions Second term integral in expression 10 5 n cos 2rn dn 15 and second term integral in expression 11 ia n cos 2mnv 62 2b 2 dn 16 are same kind integrals for each value of i j costar dn 17 Na where a is a 270 in the first case and a 27 6 2b in the second one Therefore we need to integrate expression 17 and the solution will be applied to expressions and 16 Firstly we integrate by parts eu din e a f sin an p A n no fa n n _ cos ana af sin an 18 na n Na So now the integral to be solved is o as n a that can be divided into two parts as follows a anu e a U ig ay 20 n n n a 16 Si x Figure 14 Dirichlet Integral Let s stop here to study the integral of function sin n Indefinite integral of this function does not exist in closed form Nevertheless the definite integral with upper and lower limits equal to 0 and x respectively defines a new function of x known as Sine Integral Sine f PUN i 21 21 We can approximate this function by using McLaurin series expansion of sin n n n sin n n zy gr To 22 The integrand turns into sin n n n 1 4 2 n 3 5 3 And the integral becomes f sin n 7 n n E x x a series that converges for all x If we pl
18. sity m Figure 80 100 Distance m le 02 le 03 le 04 le O5 F le 06 F le O7 F le 08 F le 09 Class A ISO 8608 Left profile Right profile 0 01 m le 02 0 1 Spatial frequency m 1 le 03 le 04 le 05 p le 06 p le O7 F le 08 F le 09 Class A ISO 8608 Left profile Right profile 0 01 le 02 0 1 Spatial frequency mt le 03 le 04 le O5 p le 06 F le O7 p le 08 F le 09 Class A ISO 8608 Left profile Right profile 0 01 c Distance 4 0 m le 02 r r on Class A ISO 8608 amp le 03 Left PSD J gt Right PSD 3 le 04 J 5 Q le 05 3 le 06 F a le 07 F d Z le 08 a le 09 0 01 0 1 1 Spatial frequency mt 0 1 Spatial frequency mt 5 Parallel Class A Road Profiles and their Power Spectral Densities Figure 6 Power Spectral Density of two 400 meters long parallel road profiles class A 5 Overview PRPgenerator is quite simple and intuitive Its main and only window in shown in figure We can distinguish the following parts PRPgenerator logo on the left hand upper corner Problem data section on the left side Here is where the user defines the properties of the profiles that are to be generated Results section on the right hand side In this section some results are depicted coherence function in the upper window
19. sotropy every straight profile on the road has the same statistical charac teristics independently of its direction Therefore in a homogeneous and isotropic road surface every straight profile has the same statistical characteristics independently of its direction or position This entails that the surface could be completely described with the PSD of any straight profile Consider any surface r x y where r is the distance to the Oxy plane and two parallel profiles rz x y rr x Left and Right that are the intersections of the surface with the vertical planes y b and y b respectively figure 2 This two profiles are defined as functions of x which is the road longitudinal coordinate The autocorrelation functions of these profiles Rz and Re d in this kind of process fulfill the following condition Rr Rp d R 4 Cross correlation functions for these profiles will be also equal Rrr 6 Rrx 6 Rz 6 5 4 Derecha Figure 2 Parallel profiles in a homogeneous and isotropic surface The cross correlation of the two profiles R d is obtained with the elevation of two points one in each profile at a distance from each other equal to 6 in direction x that is to say rz p and rgR xa Those points are also related in the autocorrelation function of the straight profile taken along line AB the autocorrelation of AB profile is the same as that of the Left and Rigth profiles rz and rp since
20. stallation has finished new files appear in the folder PRPgenerator see figure 1 5 Double click PRPgenerator exe not PRPgenerator_ pkg exe to run the soft ware Nombre Fecha de modifica Tipo Tamano d License a MCRinstaller exe t 423 277 KB PRPgenerator exe 01 02 2013 14 16 Aplicaci n 2 054 KB m PRPgenerator_pkg exe 01 02 2013 14 18 Aplicacion 424 957 KB T readme tet 01 02 2013 14 16 Archivo TXT 1 KB Carpeta de archivos Apli Figure 1 PRPgenerator folder with installation completed 4 Theoretical background In site specific problems the actual road profile can be measured and employed in the calculations When the actual profile of a particular road stretch is not appropriate but a set of profiles that are representative of a certain sort of roads stochastic definitions for the generation of synthetic profiles are used This section in divided in two parts Firstly a brief explanation of the generation of simple profiles is presented this is important because it helps to introduce some concepts and assumptions that will be used in the generation of parallel profiles In second place the generation of parallel road profiles is explained 4 1 The classic way one only profile for every wheel In this subsection we are interested in how to create rough profiles along a road these profiles will be independent of each other Profiles generated in this way are prescribed at every vehicle wheel so the
21. values are saved in G_GX__ Distance between profiles _ Road class label dat along with the values of G and Gx This file is organized into 4 columns 1 Frequency n mt 2 G m 3 Gx m and 4 g Profiles are saved in two files named prof_ Distance be tween profiles _ Road class label Side label dat where Side label is LF for the left profile and RT for the right one Profile files are organized in two columns 1 Distance to the origin m and 2 Profile elevation m If GENERATE PROFILES is clicked again a new pair of profiles is generated depicted and saved They will be stored in files with the same names as the previous profiles so these will be overwritten and therefore lost If we want to create several profiles as is usual the procedure is 1 Generate 2 Protect the files by changing their names or moving them to another folder 3 Generate another pair of profiles 4 Repeat this actions as many times as necessary 10 PRP main eae Help Results Coherence function Problem data Road class Verygood 4 ir Distance between profiles m at Road strech m length m o 1 oo on 1 10 Frequency yelesi Sample distance m p Generated profiles Number of frequencies GENERATE PROFILES Elevation mn d Figure 8 Once the coherence function is calculated by clicking COMPUTE g it is depict
22. values of the coherence function employed in the generation are marked in the coherence function window Notice that your upper graph must be the same as the one in this figure but not the lower one due to the randomness of the synthetic profiles Now you can generate your pairs of profiles Let s start with the first profile Set Road stretch length equal to 100 meters Sample distance equal to 0 01 meters and a Number of frequencies of 2000 Click GENERATE PROFILES Profile generation is much faster than coherence function computation so you do not have to be as patient as in the previous step When the calculation is finished profiles are depicted in the lower part of the Results section and the employed values of g n are marked over the coherence function graph Figure 1ip The software creates G_GX_2_ 64 dat in which the 2000 considered frequencies and the corresponding values of G Gy and g are stored Left profile is saved in prof 2 64 LF dat and right profile in prof 2 64 RT dat Since we have been told to generate another pair of profiles we need to protect the just created files otherwise they would be overwritten and the generated profiles lost We may change their names or move them to another directory Let s change their names as followd e G_GX_2_ 64 dat gt G_GX_2_64 2000 dat e prof_2 64 _LF dat gt prof_2_64_LF 2000 dat e prof_2_64_RT dat prof _2_64_RT 2000 dat where _ 2000 indicates that we have used 2000 spatial
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