Using the Soft-Soil tire model

Transcription

1 Using the Sot-Soil tire model The Adams/Tire Sot Soil tire model oers a basic model to describe the tire-soil interaction orces or any tire on elastic/plastic grounds, such as sand, clay, loam and snow. The model requires a tire property ile with keyword SOFT-SOIL and a road data ile (one o the existing ormats) with additional soil properties. Two tire-soil contact models are oered: Elastic-plastic soil deormation model, USE_MODE = 1 Visco-elastic soil deormation model, USE_MODE = 2 Deinition o Tire Slip Quantities Figure 1 Deinition o the slip velocities in the tire-road contact point The longitudinal slip velocity V sx in the SAE-axis system is deined using the longitudinal speed V x, the wheel rotational velocity and the the eective rolling radius R e : V sx = V z R e (1) The lateral slip velocity is equal to the lateral speed in the contact point with respect to the road plane: V sy = V y (2) The slip quantities (longitudinal slip) and (slip angle) are calculated with these slip velocities in the contact point, or negative V sx they are deined as: V sx V x = and tan = V sy V x (3)

2 2 Adams/Tire Using the Sot-Soil tire model and or positive V sx (driving) as: V sx V r = and tan = V sy V r (4) V r is the rolling speed V r is determined using the eective rolling radius R e : (5) Note that or realistic tire orces the slip angle is limited to degrees and the longitudinal slip 1 to. 90 V r R e Loaded and Eective Tire Rolling Radius The loaded rolling tire radius R l is deined as the unloaded tire radius R 0 minus the tire delection 0 due to the vertical load: R l R 0 0 (6) The eective rolling radius R e (at ree rolling o the tire), which is used to calculate the rotational speed o the tire, is deined by: V Re x (7)

3 Using the Sot-Soil tire model Deinition o Tire Slip Quantities 3 For radial tires, the eective rolling radius is rather independent o load in its load range o operation because o the high stiness o the tire belt circumerence. Only at low loads does the eective tire radius decrease with increasing vertical load due to the tire tread thickness, see the Figure 2. Figure 2 Eective and loaded tire radius as a unction o the vertical load

4 4 Adams/Tire Using the Sot-Soil tire model Eective Rolling Radius and Longitudinal Slip Figure 3 Side view o a rolling tire To represent the eective rolling radius R e, a PAC2002 compatible equation is used: R d d e R 0,F ( De atan( Be ) Fe z ) (8) in which,f is the nominal tire delection at the nominal tire load F z0 : 0 z0 0,F z0 F C z0 z (9) and d 0 is called the dimensionless radial tire delection, deined by:

5 Using the Sot-Soil tire model Elastic-plastic tire-soil contact 5 d 0 0 0,F z 0 (10) Elastic-plastic tire-soil contact The interaction orces or a rigid wheel The static sinkage o a rigid object into a sot soil depends on the load on that object: Bekker [1] ormulated the sinkage h o a lat plate with width b as ollows: p( h ) ( k c / b k ) h n (11) k k in which c and are the cohesive and rictional moduli respectively, n the sinkage exponent. The static stress p is in equilibrium with the vertical orce Fz. Figure 4 Pressure distribution under a lat plate

6 6 Adams/Tire Using the Sot-Soil tire model When applying this approach to a non-rolling wheel the static stress distribution can be estimated as shown in the Figure 5 Figure 5 Static stress distribution under a non-rolling rigid wheel For the dynamic sinkage the wheel rotational speed must be taken into account.

7 Using the Sot-Soil tire model Elastic-plastic tire-soil contact 7 Figure 6 Wheel entry and exit angle when rolling on soil Assume a wheel soil contact with entry angle and exit angle r, see also [2], then these angles can be written as a unction o the total sinkage h and the exit penetration h e as ollows: r acos( 1 h / R ) acos( 1 h e / R ) The exit penetration h e depends on the elastic stiness C s o the soil. (12)

8 8 Adams/Tire Using the Sot-Soil tire model Based on the terramechanical approach as described in [2] the normal and shear stresses can be modeled as shown in the Figure 7. Figure 7 Normal and shear stress modelling o a rotating wheel The wheel normal stress distribution can be deined as unction o the wheel angle [2,3]: or m :

9 Using the Sot-Soil tire model Elastic-plastic tire-soil contact 9 n kc ( ) R0 k cos( ) cos( b or r m : ) n (13) ( ) R n 0 kc b k cos r m r ( m ) cos( ) n with b the wheel width and R 0 the wheel radius. The angle m is the angle at which the maximum normal stress occurs [4]: m a0 a1 ) ( (14) The shear stress [5,6] in longitudinal direction is: ( ) ( c ( )tan( ))( 1 e x x x and in lateral direction yields: j ( ) / k ) (15) ( ) ( c ( )tan( ))( 1 e y jy( ) / k y ) (16) In equations 15 and 16 c represents the cohesion stress o the soil, the riction angle o the soil and k x and k y the shear deormation moduli. Assuming that the wheel has a longitudinal slip, the longitudinal shear displacement along the contact area j x in equation 16 can be estimated [5,6] by using the longitudinal slip and wheel radius R 0 : j ( ) R [ ( 1 )(sin( x 0 ) sin( ))] (17) Similar the lateral shear displacement j y will depend on the slip angle and the wheel radius R 0 : j ( ) R ( 1 )( y 0 )tan( ) (18)

10 10 Adams/Tire Using the Sot-Soil tire model Figure 8 illustrates the shear stress as a unction o soil deormation. Figure 8 Measured shear stress compared to itted stress (equation 15) The longitudinal shear deormation modulus k x is deined as: k x k k 1 x0 x (19) and the lateral shear deormation modulus k y : k y k k 1 y0 y Having the normal and shear stress or the rotating wheel, the tire-soil interaction contact orces and moments can be calculated: Longitudinal orce:

11 Using the Sot-Soil tire model Elastic-plastic tire-soil contact 11 F x br r { ( )cos( ) ( )sin( )}d x (20) Lateral orce: F y br ( )d r y (21) Vertical load: F z br r { ( )sin( ) ( )sin( )}d x (22) Overturning moment: M x 0 (23) Rolling resistance moment: M y 2 br r 0 ( )sin( )d c rol F z (24) with c rol the tire (internal) rolling resistance coeicient. Aligning moment: M z 2 br ( )sin( )d r y (25) Tire deormation In order to take the tire delection into account the substitution circle approach is taken as was suggested by Bekker [7]:

12 12 Adams/Tire Using the Sot-Soil tire model Figure 9 Substitution circle to account or tire delection At a certain penetration o the tire into the soil the tire delection and sinkage can be determined by an iteration process based on the act that the vertical tire orce and the orce due to the sinkage must be equal. The tire orce can be calculated with the tire stiness C z and tire delection 0 by: F z,tire C z 0 (26) while the tire sinkage orce is deined by equation (22), however, replacing the unloaded tire radius R 0 by the radius o the substitution circle R*. Bekker [7] derived ollowing relation in between the tire delection 0 and tire sinkage h:

13 Using the Sot-Soil tire model Elastic-plastic tire-soil contact 13 * R 0 1 h h R 0 0 (27) Elastic and Plastic deormation Depending on the soil properties one part o the deormation is elastic and the remaining part is nonirreversible (plastic deormation). The elastic deormation is calculated with by the soil stiness C s at the maximum normal stress max : ( he C m s ) (28) Multi-pass eect When a tire has passed a certain spot o soil, a second tire will experience dierent soil properties when rolling over that spot due to the plastic deormation o the soil by the irst tire. Thereore this Sot Soil tire model stores the elastic and plastic deormation o each tire as a unction o the contact point x,y coordinates. When a tire passes a point with plastic deormation caused by a previous tire, the normal pressure calculation will account or the plastic deormation history. Figure 10 explains the mechanism applied in this tire model [8]: Assume two tires rolling ater each other over the same spot o soil. The irst tire will have a total deormation h 1 existing o a plastic part h p1 and an elastic part h e1. When a second tire passes the same spot, the soil will irst have an elastic deormation rom A to B (= h e1 ) and then continue to ollow the normal pressure characteristic to point C. The plastic deormation o the second tire h p2 will be equal to the total deormation h 2 subtracted with the elastic deormation h e2.

14 14 Adams/Tire Using the Sot-Soil tire model Figure 10 Normal pressure characteristic or multi-pass approach Note: The tire model stores the x, y coordinates, the elastic and plastic deormation and tire width o each tire. Because o the one-point o contact approach used in this Sot-Soil tire model, the total stored plastic deormation will be applied or a next tire when its contact point comes into the rut o a previous tire. Visco-elastic tire-soil contact Next to elastic-plastic deormation models or sot soil, also visco-elastic modeling approaches exist. Wanjii e.o. [9] derived a visco-elastic model or the normal stress along the contact line in between the tire and the soil. A three element Maxwell approach is used or a rigid wheel, see Figure 11.

15 Using the Sot-Soil tire model Visco-elastic tire-soil contact 15 Figure 11 Three element Maxwell model or a rigid wheel on visco-elastic soil For this model the normal stress in the contact in between tire and ground is: x A x G1 2 2 G2VxTr V T ( x ) ( xa x ) ( xa VxTr ) 1 e x r xa 2R0 R0 x (29) With x a R 0 sin( ) x R 0 sin( ) T r / G 2 In which T r is the relaxation time is the viscosity o the soil

16 16 Adams/Tire Using the Sot-Soil tire model V x is the orward velocity o the tire G 1 is the irst elastic modulus G 2 is the second elastic modulus The longitudinal and lateral shear stresses are calculated using the equations 15 until and including 19 as used or the elastic-plastic tire-soil model. Similar or the tire-ground interaction orces equation (20-24) are used. For the multi-pass eect, the road deormation at the exit o the tire-soil contact (point B) and the time o deormation occurrence is stored. When a second tire passes the same spot, the road deormation corrected with the relaxation eect is taken to correct the road height input. Reerences: 1. Bekker, M.G., O-the-road-locomotion, Ann Arbor, The University o Michigan Press, G. Ishigami, A. Miwa, K. Nagatani, K. Yoshida, Terramechanics - Based Model or Steering Maneuver o Planetary Exploration Rovers on Loose Soil, Journal o Field robotics 24(3), (2007), Wiley Periodicals, Inc. 3. Yoshida, K., Watanabe, T., Mizuna, N., Ishigami, G., Terramechanics - based analysis and traction control o a lunar/planetary rover. In Proceedings o the Int. Con. O Field and Service Robotics (FSR '03), Yamanashi, Japan. 4. Wong, J.Y., Reece, A., Prediction o rigid wheel perormance based on the analysis o soil-wheel stresses part I, perormance driving rigid wheels, Journal o Terramechanics, 4, Janosi, Z. Hanamoto, B., The analytical determination o drawbar pull as a unction o slip or tracked vehicle in deormable soils, In proceedings o the 1 st Int. con. on Terrain-Vehicle systems, Torino, Italy. 6. Wong, J.Y., Theory o Ground Vehicles, John Wiley & Sons, Inc., second edition, Bekker, M.G., Introduction to terrain-vehicle systems, Ann Arbor, The University o Michigan Press, AS 2 TM User's Guide, version 1.12, AESCO GbR, Hamburg. 9. S. Wanjii, T. Hiroma, Y. Ota, T. Kataoka, Predicition o Wheel Perormance by Analysis o Normal and Tangential Stress Distributions under the Wheel-Soil Interace, Journal o Terramechanics, Vol. 34, No. 3, pp , Schmid, I.C., Interaction o Vehicle and Terrain Results rom 10 Years Research at IKK, Journal o Terramechanics, Vol. 32, No. 1, pp. 3-26, Schmid, I.C., Aubel, Th., Der elastische Reien au nachgiebiger Fahrbahn - Rechenmodell im Hinblick au Reiendruckregelung, VDI Berichte nr. 916, Faßbender, F., Simulation der Vertikaldynamik von Fahrzeugen au Geländeböden mit STINA - SOIL TIRE INTERFACE TO ADAMS einem Zusatzmodul ür das Mehrkörperprogramm ADAMS. Number 521 in Fortschritt-Berichte VDI Reihe 12. VDI Verlag, Düsseldor, Dissertation Universität der Bundeswehr Hamburg.

17 Using the Sot-Soil tire model Feature and property overview o the Adams/Tire Sot Soil Tire model 17 Feature and property overview o the Adams/Tire Sot Soil Tire model Two tire-road contact models: Elastic-plastic contact model elastic tire: tire delection is taken into account multi-pass eect: road plastic deormation history is stored and taken into account when another tire passes the same spot Visco-elastic contact model rigid tire: no tire delection multi-pass eect: road viscous deormation is stored. The stored deormation reduced by the relaxation eect is taken into account when another tire passes the same spot Tire eective rolling radius is deined similar to pac2002 tire model Tire properties are very basic (tire vertical stiness and damping, unloaded radius, width and eective rolling radius parameters) The existing Adams/Tire roads can be used, just an additional section with the soil properties is required. These soil properties are valid or the whole road. Linearization o - F x characteristic during q-statics to ensure robust q-statics Linear vertical tire stiness can be replaced by a (non-linear) delection-load curve Scaling actors o road riction, tire cornering and longitudinal stiness' are supported SMP (multi-thread, C++ solver) is supported Tire-road contact is a one-point contact Camber eects are not taken into account Overturning moment is not calculated No bulldozing eects Example o the tire property ile or the Sot-Soil Tire model: $ MDI_HEADER [MDI_HEADER] FILE_TYPE = 'tir' FILE_VERSION = 2.0 FILE_FORMAT = 'ASCII' (COMMENTS) {comment_string} 'Tire - XXXXXX' 'Pressure - XXXXXX' 'Test Date - XXXXXX' 'Test tire' 'New File Format v2.1' $ units

18 18 Adams/Tire Using the Sot-Soil tire model [UNITS] LENGTH = 'mm' FORCE = 'newton' ANGLE = 'degree' MASS = 'kg' TIME = 'sec' $ model! use mode 1 2! ! lexible wheel/tire with elastic-plastic road X! rigid wheel/tire with visco-elastic road X! PROPERTY_FILE_FORMAT = 'SOFT-SOIL' USE_MODE = 1.0 $ dimension [DIMENSION] UNLOADED_RADIUS = WIDTH = ASPECT_RATIO = 0.45 $ parameter [PARAMETER] NOMINAL_TIRE_LOAD = 4000 VERTICAL_STIFFNESS = VERTICAL_DAMPING = 0.5 ROLLING_RESISTANCE = 0.01 BREFF = 8.4 DREFF = 0.27 FREFF = 0.07 $ shape [SHAPE] {radial width} $ load_curve $ For a non-linear tire vertical stiness $ Maximum o 100 points [DEFLECTION_LOAD_CURVE] {pen z}

19 Using the Sot-Soil tire model Example o the required Soil properties in the Road Data File: 19 Example o the required Soil properties in the Road Data File: Existing road data iles can be used, but a 'SOIL_PROPERTIES' section has to be added: $ SOIL_PROPERTIES [SOIL_PROPERTIES] FRICTION_ANGLE = 37.2 $units: degree COHESION_STRESS = 8.0E-4 $units: N/mm**2 SOIL_DEFORM_MOD_KX0 = 43.0 $units: mm SOIL_DEFORM_MOD_KX1 = $units: mm/deg SOIL_DEFORM_MOD_KY0 = 20.0 $units: mm SOIL_DEFORM_MOD_KY1 = $units: mm/deg!visco-elastic tire: ELASTIC_MODULUS_G1 = 0.071E-3 $units: N/mm**3 ELASTIC_MODULUS_G2 = 1.072E-3 $units: N/mm**3 SOIL_VISCOSITY = 7.14E-3 $units: Ns/mm**3!plastic-elastic tire: PRESSURE_SINKAGE_KC = 1.37E-3 $units: N/mm**(n+1) PRESSURE_SINKAGE_KFI = 8.14E-4 $units: N/mm**(n+2) SINKAGE_EXPONENT = 1 $units: = n SOIL_INTERACTION_A0 = 0.4 $units: - SOIL_INTERACTION_A1 = 0.15 $units: - SOIL_STIFFNESS = 8.14E-3 $units: N/mm**3 Symbols B e b c c rol D e C z C s 0 d 0 0,F z0 G 1 G 2 F e F x eective rolling radius actor tire/wheel width cohesion tire rolling resistance coeicient eective rolling radius actor tire vertical stiness soil stiness tire delection dimensionless tire delection nominal tire delection elastic modulus elastic modulus eective rolling radius actor longitudinal orce

20 20 Adams/Tire Using the Sot-Soil tire model F y F z F z0 h h e h p k c k lateral orce vertical load nominal tire load sinkage elastic deormation plastic deormation cohesive modulus rictional modulus k x soil deormation modulus k y soil deormation modulus M x overturning moment M y rolling resistance moment M z aligning moment n sinkage component p static stress R e eective rolling radius R 0 unloaded (ree) tire/wheel radius R l tire loaded radius R* radius o substitution circle T r relaxation time V total tire/wheel speed V r tire rolling velocity V x tire/wheel orward speed (parallel to wheel plane) V sx longitudinal slip speed V sy lateral slip speed slip angle longitudinal slip riction angle normal stress tire/wheel rotational speed

21 Using the Sot-Soil tire model Symbols 21 x y r longitudinal shear stress lateral shear stress wheel angle wheel soil entry angle wheel soil entry angle viscosity o the soil

22 22 Adams/Tire Using the Sot-Soil tire model