Prediction of Radial-Ply Tire Deflection Based on Section Width, Overall Unloaded Diameter, Inflation Pressure and Vertical Load

Transcription

1 World Applied Scieces Joural 1 (1): , 013 ISSN IDOSI Publicatios, 013 DOI: /idosi.wasj Predictio of Radial-Ply Tire Deflectio ased o Sectio Width, Overall Uloaded Diameter, Iflatio Pressure ad Vertical Load Majid Rashidi, Mohammad-Ali Sheikhi, Shahram Razavi, Milad Niyazadeh ad Morteza Arkia Departmet of Agricultural Machiery, Takesta rach, Islamic Azad Uiversity, Takesta, Ira Abstract: Tire deflectio is a key parameter ad may equatios have bee developed based o it to evaluate the tractive performace of bias-ply ad radial-ply tires. As deflectios for a give tire size, iflatio pressure ad vertical load are sigificatly differet betwee bias-ply ad radial-ply tires, this study was coducted to predict deflectio ( ) of radial-ply tire based o sectio width (b), overall uloaded diameter (d), iflatio pressure (P) ad vertical load (W). For this purpose, deflectio of four radial-ply tires with differet sectio width ad/or overall uloaded diameter were measured at five levels of iflatio pressure ad five levels of vertical load. Results of deflectio measuremet for radial-ply tires No. 1, ad 3 were utilized to determie multiple variables regressio model ad results of deflectio measuremet for radial-ply tire No. 4 were used to verify selected model. The paired samples t-test results showed that the differece betwee the deflectio values predicted by model ad measured by test apparatus were ot statistically sigificat ad to predict deflectio of radial-ply tire based o sectio width, overall uloaded diameter, iflatio pressure ad vertical load the multiple variables regressio model = b d P W with R = ca be strogly recommeded. Key words: Radial-ply tire Deflectio Sectio width Overall uloaded diameter Iflatio pressure Vertical load Predictio 0.5 INTRODUCTION L = (d ) () I the case of tracked vehicles, the cotact area betwee machie ad groud surface is relatively d = Overall uloaded diameter (m) costat for varyig sikage i the soil ad is calculated = Deflectio (m) as the legth of track o hard groud times track width. However, a flexible tire has a smaller cotact area o hard Deflectio is a key parameter ad may equatios surface tha it dose o soft groud. A rule of thumb have bee developed based o it to evaluate the tractive which ca be used for estimatio of tire cotact area is performace of bias-ply ad radial-ply tires operatig i show by equatio 1 [1]: cohesive-frictioal soils. Gross tractio, motio resistace, et tractio ad tractive efficiecy are A = bl (1) predicted as a fuctio of soil stregth, tire load, tire slip, tire size ad tire deflectio [4]. The most widely used dimesioal aalysis approach for predictig off-road A = Cotact area (m ) tractio makes use of the followig ratios [4-6]: b = Sectio width (m) L = Cotact legth (m) CI b d C = W (3) Wog [] ad ekker [3] gave a approximate method b WD = (4) for calculatig cotact legth as equatio : d Correspodig Author: Dr. Majid Rashidi, Departmet of Agricultural Machiery, Takesta rach, Islamic Azad Uiversity, Takesta, Ira. 1804

2 World Appl. Sci. J., 1 (1): , 013 DR = (5) h 1+ 5 CI b d h = (6) W b 1+ 3 C d = Wheel umeric (dimesioless) CI - = Coe idex (kpa or knm ) W = Vertical load (kn) = Mobility umber (dimesioless) WD = Sectio width to overall uloaded diameter ratio (dimesioless) The empirical model developed by rixius [4] is DR = Deflectio ratio (dimesioless) widely used for predictio of off-road tire performace. h = Sectio height (m) It has also bee adopted i ASAE stadard D497.4 [8] for predictig tractor performace. I this model, soil Fig. 1 shows the tire dimesios (b, d, ad h) used. coditio is represeted by the coe idex value, which The tire dimesios ca be obtaied from tire data book is the average force per uit area required to force a or by measurig the tire [4]. The sectio width (b) is the coe-shaped probe vertically ito the soil at a steady rate. first umber i a tire size desigatio (i.e., omially 18.4 The average before-traffic coe idex for the top 150 mm iches for a tire). The overall uloaded diameter layer of soil is used i the predictio equatios that follow (d) ca be obtaied from the tire data hadbooks available [5, 7]. ASAE stadards S313.3 [9] ad EP54 [10] describe from off-road tire maufacturers. The tire deflectio ( ) the soil coe peetrometer ad procedures for its use. o a hard surface is equal to d/ mius the measured A average of several coe idex values obtaied at a test static loaded radius. The static loaded radius for the tire s site ofte yields a represetative measure of soil stregth rated load ad iflatio pressure is also stadard tire data [11]. from the tire data hadbooks. It ca also be obtaied by I additio, the coefficiet of gross tractio is measurig the tire. The sectio height (h) is equal to half depedet o the mobility umber ad the tire slip ad is the differece betwee the overall uloaded diameter ad give by equatio 7 [4-7]: the rim diameter. The rim diameter ca i tur be estimated by addig 50 mm to the omial rim diameter, which is the (1 S GT = e )(1 e ) (7) secod umber i a tire size desigatio, i.e. 38 iches for a tire [4, 5]. To further simplify the predictio equatios, rixius µ GT = Coefficiet of gross tractio (dimesioless) [4] combied above three dimesioless ratios ito a S = Tire slip (decimal). sigle product termed the mobility umber, which is give e = Napier s costat (the umerical value of e by equatio 6 [5-7]: trucated to 30 decimal places is ). Additioally, the coefficiet of motio resistace depeds o the mobility umber ad the tire slip ad is give by equatio 8 [4-7]: 1 0.5S = (8) = Coefficiet of motio resistace (dimesioless) y combiig equatios 7 ad 8, we obtai equatio 9 for the coefficiet of et tractio [4-7]: (1 S S NT = e )(1 e ) ( + ) Fig. 1: Tire dimesios, adapted from rixius [4] (9) 1805

3 µ = Coefficiet of et tractio (dimesioless) NT World Appl. Sci. J., 1 (1): , 013 The empirical models of rixius [4] were origially developed for peumatic bias-ply tires with deflectio ratio ( /h) ragig from 0.1 to 0.3. Although extrapolatio of empirical equatios may be ot valid, the use of rixius models with a rigid wheel is a extrapolatio of these equatios, where /h = 0 [7]. Also, as radial-ply tire usage has expaded, the eed for radial-ply tire predictio equatios has icreased. Radial-ply tire equatios should be similar to the equatios for bias-ply tires, with appropriate adjustmet i several of the equatio costats. The costat 7.5 i equatios 7 ad 9 should be icreased to This term accouts for the sigificat improvemet i grippig the soil. The costat 0.04 should be reduced to , which accouts for the reduced motio resistace o a hard surface. The 0.1 term, combied with 0.88 cotrols the maximum torque that ca be applied at high tire slip. This value seems to be the same for bias-ply ad radial-ply tires. Further aalysis may show a slightly higher value is eeded, but o lower. The 1/ term i equatios 8 ad 9 should be chaged to 0.9/. This reflects lower motio resistace due to less soil compactio ad tire sikage for radial-ply tires. Further ivestigatio is eeded regardig the effect of tire deflectio. The costat cotrollig deflectio i equatio 6 possibly eeds revisio [4, 5]. Therefore, the followig modified forms of the above three equatios are suggested for radial-ply tires [5, 6]: Fig. : Tire deflectio test apparatus Fig. 3: Measurig static loaded radius S 0.88(1 GT = e )(1 e ) S = (10) levels of iflatio pressure ad vertical load. As deflectio o a hard surface is equal to d/ mius the measured static loaded radius [4, 5], the static loaded radius was (11) obtaied by measurig as show i Fig. 3. Experimetal Procedure: For this purpose, deflectio of (1 S S NT = e )(1 e ) ( + ) (1) four radial-ply tires with differet dimesios were measured at five levels of iflatio pressure ad five As deflectios for a give tire size, iflatio pressure levels of vertical load. The dimesios of four radial-ply ad vertical load are sigificatly differet betwee tires are give i Table 1. Results of deflectio bias-ply ad radial-ply tires [4], this study was coducted measuremet for radial-ply tires No. 1, ad 3 (Tables, to predict deflectio ( ) of radial-ply tire based o sectio 3 ad 4) were utilized to determie multiple variables width (b), overall uloaded diameter (d), iflatio pressure regressio models ad results of deflectio measuremet (P) ad vertical load (W). for radial-ply tire No. 4 (Table 5) were used to verify selected model. MATERIALS AND METHODS Regressio Model: A typical multiple variables regressio Tire Deflectio Test Apparatus: A tire deflectio test model is show i equatio 13: apparatus (Fig. ) was desiged ad costructed to measure deflectio of tires with differet sizes at diverse Y = C + C X + C X + + C X (13)

4 World Appl. Sci. J., 1 (1): , 013 Table 1: Dimesios of the four radial-ply tires used i this study Tire No. Tire size desigatio Sectio width b (mm) Overall uloaded diameter d (mm) 1 R13-165/ R14-185/ R15-185/ R16-16/ Table : Sectio width, overall uloaded diameter, iflatio pressure, vertical load ad deflectio for radial-ply tire No. 1 Tire No. Sectio width b (mm) Overall uloaded diameter d (mm) Iflatio pressure P (kpa) Vertical load W (kn) Deflectio (mm) Table 3: Sectio width, overall uloaded diameter, iflatio pressure, vertical load ad deflectio for radial-ply tire No. Tire No. Sectio width b (mm) Overall uloaded diameter d (mm) Iflatio pressure P (kpa) Vertical load W (kn) Deflectio (mm)

5 World Appl. Sci. J., 1 (1): , 013 Table 4: Sectio width, overall uloaded diameter, iflatio pressure, vertical load ad deflectio for radial-ply tire No. 3 Tire No. Sectio width b (mm) Overall uloaded diameter d (mm) Iflatio pressure P (kpa) Vertical load W (kn) Deflectio (mm) Table 5: Sectio width, overall uloaded diameter, iflatio pressure, vertical load ad deflectio for radial-ply tire No. 4 Tire No. Sectio width b (mm) Overall uloaded diameter d (mm) Iflatio pressure P (kpa) Vertical load W (kn) Deflectio (mm)

6 World Appl. Sci. J., 1 (1): , 013 Table 6: Seve multiple variables regressio models ad their relatios Model No. Model Relatio 1 = C 0 + C 1 b + C d + C 3 P + C 4 W = b d P W = C 0 + C 1 b + C P + C 3 W = b P W 3 = C 0 + C 1 d + C P + C 3 W = d P W 4 = C 0 + C 1 (bd) + C P + C 4 W = (bd) P W 5 = C 0 + C 1 (b/d) + C P + C 3 W = (b/d) P W 6 = C 0 + C 1 (d/b) + C P + C 3 W = (d/b) P W = C 0 + C 1 (bd) + C P + C 3 W 0.5 = (bd) P W Y = Depedet variable, for example deflectio of radial-ply tire X 1, X,, X = Idepedet variables, for example sectio width, overall uloaded diameter, iflatio pressure ad vertical load C 0, C 1, C,, C = Regressio coefficiets I order to predict deflectio of radial-ply tire from sectio width, overall uloaded diameter, iflatio pressure ad vertical load, seve multiple variables regressio models were suggested ad all the data were subjected to regressio aalysis usig the Microsoft Excel 007. All the multiple variables regressio models ad their relatios are show i Table 6. Statistical Aalysis: A paired samples t-test ad the mea differece cofidece iterval approach were used to compare the deflectio values predicted by selected model with the deflectio values measured by test apparatus. The lad-altma approach [1] was also used to plot the agreemet betwee the deflectio values measured by test apparatus with the deflectio values predicted by selected model. The statistical aalyses were also performed usig Microsoft Excel 007. RESULTS AND DISCUSSION Predicted deflectio (mm) Differece of measured ad predicted deflectio (mm) Measured deflectio (mm) 1.0: 1.0 Fig. 4: Measured deflectio usig test apparatus ad predicted deflectio usig model No. 1 for radial-ply tire No. 4 with the lie of equality (1.0: 1.0) Average of measured ad predicted deflectio (mm) The p-value of idepedet variables ad coefficiet of determiatio (R ) for the seve multiple variables Fig. 5: lad-altma plot for the compariso of regressio models are show i Table 7. Amog the measured deflectio usig test apparatus ad seve models, model No. 1 had the highest R value predicted deflectio usig model No. 1 for radial- (0.986). Moreover, this model totally had the lowest ply tire No. 4; the outer lies idicate the 95% p-value of idepedet variables amog the seve limits of agreemet (-.,.68) ad the ceter lie models. ased o the statistical results model No. 1 shows the average differece (0.3) was selected as the best model, which is give by equatio 14: Deflectio of radial-ply tire No. 4 was the predicted at five levels of iflatio pressure ad five levels of = b d P W (14) vertical load usig the multiple variables regressio model 1809

7 World Appl. Sci. J., 1 (1): , 013 Table 7: The p-value of idepedet variables ad coefficiet of determiatio (R ) for the seve multiple variable regressio models p-value Model No. b d bd b/d d/b 0.5 (bd) P W R E E E E E-15.05E E E- 1.64E E E E E E E E E E E Table 8: Sectio width, overall uloaded diameter, iflatio pressure, vertical load ad deflectio for radial-ply tire No. 4 used i evaluatig model No. 1 Deflectio (mm) Sectio width Overall uloaded Iflatio pressure Vertical load Measured by Predicted by Average of measured ad Differece of measured ad b (mm) diameter d (mm) P (kpa) W (kn) test apparatus model No. 1 predicted deflectio (mm) predicted deflectio (mm) Table 9: Paired samples t-test aalyses o comparig deflectio determiatio methods Determiatio methods Average differece (mm) Stadard deviatio of differece (mm) p-value 95% cofidece itervals for the differece i meas (mm) Test apparatus vs. model No , 0.74 No. 1. The deflectio values predicted by model No. 1 were compared with the deflectio values measured by test apparatus ad are show i Table 8. A plot of the deflectio values predicted by model No. 1 ad the deflectio values measured by test apparatus with the lie of equality (1.0: 1.0) is show i Fig. 4. Also, a paired samples t-test ad the mea differece iterval approach were used to compare the deflectio values predicted by model No. 1 with the deflectio values measured by test apparatus. The lad-altma approach [1] was also used to plot the agreemet betwee the deflectio values measured by test apparatus with deflectio values 1810

8 World Appl. Sci. J., 1 (1): , 013 predicted by model No. 1. The average deflectio 4. rixius, W.W., Tractio predictio equatios differece betwee two methods was 0.3 mm (95% for bias ply tires. ASAE Paper No St. Joseph, cofidece iterval: -0.8 mm ad 0.74 mm; P = ). Mich.: ASAE. The stadard deviatio of the deflectio differece was 5. Goerig, C.E., M.L. Stoe, D.W. Smith ad 1.5 mm (Table 9). The paired samples t-test results P.K. Turquist, 006. Off-Road Vehicle Egieerig showed that the deflectio values predicted by model Priciples. St. Joseph, Mich.: ASAE. No. 1 were ot sigificatly differet tha the deflectio 6. Srivastava, A.K., C.E. Goerig, R.P. Rohrbach ad values measured by test apparatus. The deflectio D.R. uckmaster, 006. Egieerig Priciples of differece values betwee two methods were ormally Agricultural Machies. St. Joseph, Mich.: ASAE. distributed ad 95% of these differeces were expected to 7. Asaf, Z., I. Shmulevich ad D. Rubistei, 006. lie betwee µ-1.96 ad µ+1.96, kow as 95% limits of Predictig soil-rigid wheel performace usig distict agreemet [13-17]. The 95% limits of agreemet for elemet methods. Trasactios of the ASAE, compariso of the deflectio values determied by test 49(3): apparatus ad model No. 1 was calculated at -. mm ad 8. ASAE, 003. Agricultural machiery maagemet.68 mm (Fig. 5). Thus, the deflectio values predicted by data. ASAE Stadard D ASAE Stadards, St. model No. 1 for radial-ply tire No. 4 may be. mm lower Joseph, Mich.: ASAE. or.68 mm higher tha the deflectio values measured by 9. ASAE, Soil coe peetrometer. ASAE Stadard test apparatus for this radial-ply tire. The average S ASAE Stadards, St. Joseph, Mich.: ASAE. percetage differece for the deflectio values predicted 10. ASAE, Procedures for usig ad reportig by model No. 1 ad measured by test apparatus was data obtaied with the soil coe peetrometer..9%. Egieerig Practice EP54. ASAE Stadards, St. Joseph, Mich.: ASAE. CONCLUSIONS 11. Schmid, I.C., Iteractio of vehicle ad terrai results from 10 years research at IKK. J. It ca be cocluded that the multiple variables Terramechaics, 3(1): 3-6. regressio model = b d lad, J.M. ad D.G. Altma, Measurig P W with R = ca be strogly suggested to agreemet i method compariso studies. Statistical predict deflectio of radial-ply tire based o sectio width, Method i Medical Research, 8: overall uloaded diameter, iflatio pressure ad vertical 13. Seilsepour, M. ad M. Rashidi, 008. Modelig of load. soil catio exchage capacity based o soil colloidal matrix. Am-Euras. J. Agric. ad Eviro. Sci., ACKNOWLEDGMENTS 3(3): Seilsepour, M. ad M. Rashidi, 008. Predictio of The authors ask God s favor for their late fried ad soil catio exchage capacity based o some soil studet, Egieer Hadi Khalkhali, who desiged ad physical ad chemical properties. World Appl. Sci. J., costructed the tire deflectio test apparatus. 3(): Seilsepour, M., M. Rashidi ad.g. Khabbaz, 009. REFERENCES Predictio of soil exchageable sodium percetage based o soil sodium adsorptio ratio. Am-Euras. 1. McKyes, E., Soil Cuttig ad Tillage. Elsevier J. Agric. ad Eviro. Sci., 5(1): Sciece Publishig Compay Ic., New York, USA. 16. Rashidi, M., I. Rajbar, M. Gholami ad S. Abbassi,. Wog, J.Y., Theory of Groud Vehicles. Joh 010. Predictio of carrot firmess based o carrot Wiley ad Sos, New York, USA. water cotet. Am-Euras. J. Agric. ad Eviro. Sci., 3. ekker, M.G., The effect of tire tread i 7(4): parametric aalyses of tire-soil systems. NRCC 17. Rashidi, M. ad M. Seilsepour, 011. Predictio of Report No. 4146, Natioal Research Coucil of soil sodium adsorptio ratio based o soil electrical Caada. coductivity. Middle-East J. Sci. Res., 8():