Terramechanics V Note - I haven t posted the slides from Tuesday because there were a number of typos (and outright mistakes) that were (almost) all corrected on Thursday. This set of slides are the corrected slides from both Tuesday and Thursday in the appropriate order to make a full set of slides on terramechanics. I know this is late (and I apologize for that), so if you would like extra time on the problem set, take it with no penalty. I d rather you understand and do it right than worry about grades at this point. U N I V E R S I T Y O F MARYLAND 1 Terramechanics V ENAE 788X - Planetary Surface Robotics
LRV Three-View and Dimensions U N I V E R S I T Y O F MARYLAND 2 Terramechanics V ENAE 788X - Planetary Surface Robotics
Location of LRV Center of Gravity U N I V E R S I T Y O F MARYLAND 3 Terramechanics V ENAE 788X - Planetary Surface Robotics
Rover Climbing/Descending Slope ` h T 2 N 2 mg a T 1 r N 1 h a ` r T 1 N 1 mg T 2 N 2 U N I V E R S I T Y O F MARYLAND 4 Terramechanics V ENAE 788X - Planetary Surface Robotics
Static Equilibrium Conditions X Forces? to surface X Forces k to surface N 1 + N 2 = mg cos T 1 + T 2 = mg sin X Torques about rear axle T 1 r + T 2 r + N 1` = mg [(` a) cos h sin ] Friction forces proportional to force into surface U N I V E R S I T Y O F MARYLAND T 1 = T 2 N 1 N 2 Four equations, four unknowns 5 Terramechanics V ENAE 788X - Planetary Surface Robotics
Static Equilibrium Solutions apple a h N 1 = mg 1 cos ` ` + r` sin apple a h N 2 = mg ` cos + ` + r` sin T 2 = T 1 = N 2 N 1 + N 2 mg sin N 1 N 1 + N 2 mg sin U N I V E R S I T Y O F MARYLAND 6 Terramechanics V ENAE 788X - Planetary Surface Robotics
Loaded LRV Weight Distribution r N 1 a h ` N T 2 1 mg T 2 m = 690 kg mg = 1120 N ` = 228.6 cm r = 40.6 cm a = 114.3 cm U N I V E R S I T Y O F MARYLAND 7 h = 50.8 cm N 1 = N 2 = 279.8 N b = 22.9 cm Terramechanics V ENAE 788X - Planetary Surface Robotics
Normal and Shear Wheel Force w/slope 0.9" 0.8" 0.7" 0.6" 0.5" 0.4" 0.3" 0.2" 0.1" 0" N 1 mg N 2 mg 0" 10" 20" 30" 40" 50" 60" N1" N2" T1" T2" T 2 mg N 2 mg U N I V E R S I T Y O F MARYLAND 8 Terramechanics V ENAE 788X - Planetary Surface Robotics
Longitudinal Dynamic Conditions X Forces? to surface X Forces k to surface N 1 + N 2 = mg cos T 1 + T 2 = mg sin + ma x X Torques about rear axle T 1 r + T 2 r + N 1` = mg [(` a) cos h sin ] Friction forces proportional to force into surface U N I V E R S I T Y O F MARYLAND T 1 = T 2 N 1 N 2 Four equations, four unknowns 9 Terramechanics V ENAE 788X - Planetary Surface Robotics
Longitudinal Dynamic Solutions N 1 = mg apple 1 N 2 = mg a ` cos apple a ` cos + h ` + r` sin h ` + r` sin + a x g a x g T 2 = T 1 = N 2 N 1 + N 2 (mg sin + ma x ) N 1 N 1 + N 2 (mg sin + ma x ) U N I V E R S I T Y O F MARYLAND 10 Terramechanics V ENAE 788X - Planetary Surface Robotics
Overturn Limits 1 a ` cos limit h` + r` sin limit = a x g 1 For the static case (a x = 0) a h cos limit = ` ` + r` sin limit U N I V E R S I T Y O F MARYLAND tan limit = 1 a ` h ` + r` 11 = ` a h + r Limiting accel on flat ground a x,limit = g 1 a ` Terramechanics V ENAE 788X - Planetary Surface Robotics
Limiting Slope Under Acceleration Maximum'Slope'(deg)' 60" 50" 40" 30" 20" 10" 0" limit 0" 0.1" 0.2" 0.3" 0.4" 0.5" 0.6" Accelera4on' a x g U N I V E R S I T Y O F MARYLAND 12 Terramechanics V ENAE 788X - Planetary Surface Robotics
Compression Resistance (Lunar Soil) R c = 1 n + 1 (k c + bk ) 1 2n+1 n = 1 3W w (3 n) d 2(n+1) 2n+1 k c =0.14 N/cm 2 k =0.827 N/cm 3 R c = 1 2 (k c + bk ) 1 3 3W w 2 d 4 3 R c = 20.89 N/wheel = 83.57 N (total) U N I V E R S I T Y O F MARYLAND 13 Terramechanics V ENAE 788X - Planetary Surface Robotics
Lunar Regolith Soil Properties Symbol Description Value Used in n Exponent of sinkage 1.0 EVERYTHING k Cohesive modulus of soil deformation 1400 N/m Rc, Rb, H k Frictional modulus of soil deformation 820,000 N/m Rc, Rb, H θ Slope angle Rg Cf Coefficient of friction 0.05 Rr γ Soil density 1.6 g/cm Rb Co Cohesive strength of soil 0.017 N/cm H ф Angle of internal friction 30-40 degrees Rb K Coefficient of soil Slip 1.8 cm H s slip 0.02 0.05 H U N I V E R S I T Y O F MARYLAND 14 Terramechanics V ENAE 788X - Planetary Surface Robotics
Equations for Compression Resistance z = U N I V E R S I T Y O F MARYLAND apple 3W w (3 n)bk p d W w = weight on wheel d = wheel diameter bk R c = n + 1 z n+1 R c = compression resistance (per wheel) 15 2 2n+1 Correction - Page 25 of L03 (Corrected 141117) Terramechanics V ENAE 788X - Planetary Surface Robotics
LRV Sinkage in Lunar Regolith z = " 3W w 2(k c + bk ) p d z =1.812 cm # 2 3 Back wheel is a tandem wheel W w = 2 3 (k c + bk ) p dz 2 3 U N I V E R S I T Y O F MARYLAND 16 Terramechanics V ENAE 788X - Planetary Surface Robotics
Tandem Wheels, n=1 N i = 2 3 (k c + bk ) p d p z i z i 1 (2z i z i 1 ) We already calculated z 1 =1.812 cm N 2 = 2 3 (k c + bk ) p d p z 2 z 1 (2z 2 z 1 Solve for z 2 (numerically would be easiest) z 2 =2.443 cm This is the total depth for wheel 2 U N I V E R S I T Y O F MARYLAND 17 Terramechanics V ENAE 788X - Planetary Surface Robotics
Compaction Resistance, n=1 R c = 1 2 (k c + bk )z 2 R c = 56.92 N (per side) =) R c = 113.8 N U N I V E R S I T Y O F MARYLAND 18 Terramechanics V ENAE 788X - Planetary Surface Robotics
Or, You Could Cheat Approximation formulas for n=1 Two wheels in tandem R c = 1.7 2 (k c + bk )z 2 Three wheels in tandem R c = 2.3 2 (k c + bk )z 2 z is the sinkage depth of the front wheel in both cases R g, R f are straightforward U N I V E R S I T Y O F MARYLAND 19 Terramechanics V ENAE 788X - Planetary Surface Robotics
Bulldozing Resistance General case: R b = b sin ( + ) 2sin cos 2zcK c + z 2 K +`3o 3 2 All angles in radians! + c`2o apple 1 + tan 4 + 2 For tracked vehicles, only the first term applies: R b = U N I V E R S I T Y O F MARYLAND `o = z tan 2 4 2 `o soil disruption depth, and is not the same as contact length ` b sin ( + ) 2sin cos 20 2zcK c + z 2 K Terramechanics V ENAE 788X - Planetary Surface Robotics
Bulldozing Resistance (Rb) Symbol Description Value Angle of internal friction γ Soil density 1.6 gm 0.002595 N/cm Co Cohesive strength of soil 0.017 Lo (degrees) Kc (degrees) K (degrees) Distance of rupture Modulus of cohesion of soil deformation Modulus of density of soil deformation 30-40 degrees for Lunar Regolith Nc (radians) Coefficient of passive earth pressure α (degrees) U N I V E R S I T Y O F MARYLAND Angle of approach of the wheel 21 Terramechanics V ENAE 788X - Planetary Surface Robotics
Bulldozing Example (1) = cos 1 1 2z d = 33 o =0.576 rad = cos 1 1 `o = z 1 tan 2 2 4 2(1.812) 81.2 = 17.18 o =0.2999 rad =0.5341 N q = U N I V E R S I T Y O F MARYLAND e(1.5 )tan 2 cos 2 4 + 2 N c = N q 1 tan 22 = 32.23 = 48.09 Corrected 141123 Terramechanics V ENAE 788X - Planetary Surface Robotics
Bulldozing Example (2) N = 2(N q + 1) tan 1+0.4sin4 = 33.27 K c =(N c tan ) cos 2 = 33.37 2N K = tan +1 cos 2 = 72.77 ` = d 1 2 cos 1 2z = 12.18 cm d =0.002595 N cm 3 ; c o =0.017 N cm 2 U N I V E R S I T Y O F MARYLAND 23 Terramechanics V ENAE 788X - Planetary Surface Robotics
Terzhagi Parameters Red lines represent values calculated for this example U N I V E R S I T Y O F MARYLAND 24 Terramechanics V ENAE 788X - Planetary Surface Robotics
Bulldozing Example (3) R b = b sin ( + ) 2sin cos R b i = cm cm 2zcK c + +`3o N cm 2 + 3 z 2 K 2 N cm 3 cm2 + c`2o apple 1 + tan 4 + 2 + cm 3 N cm 3 + N cm 2 cm2 R b = 94.98 + 0.000131 + 0.014 = 95.00 N per leading wheel R b,total = 190.0 N U N I V E R S I T Y O F MARYLAND 25 Terramechanics V ENAE 788X - Planetary Surface Robotics
Tractive Force per Wheel (No Grousers) H =[AC b + W w tan b ] A = area of contact = b` C b = coe cient of soil/wheel cohesion K = coe U N I V E R S I T Y O F MARYLAND b = wheel/soil friction angle s = wheel slip ratio cient of soil slip = length of contact patch 26 1 K H (s=0) = 251.6 N 1 e s K Terramechanics V ENAE 788X - Planetary Surface Robotics
Tractive Force per Wheel (With Grousers) H = b C b 1 + 2h b N g + W tan b 1 + 0.64 h b arctan b 1 h K s 1 e K A = area of contact = b C b = soil/wheel cohesion = 0.017 N/cm 2 b = wheel/soil friction angle = 35 s = wheel slip ratio (typ. 0.02-0.05) K = coe cient of soil slip = 1.8 cm = length of contact patch = D 2 cos 1 1 h = height of grouser U N I V E R S I T Y O F MARYLAND All values typical for lunar soil 27 2z D Terramechanics V ENAE 788X - Planetary Surface Robotics
Effect of Soil Thrust Fraction 1.2 Soil Thrust Fraction 1 K 1 e s K Soil Thrust Fraction 1 0.8 0.6 0.4 0.2 K/l=0.1 0.25 0.5 1 2 4 10 0 0 0.2 0.4 0.6 0.8 1 Slip Ratio U N I V E R S I T Y O F MARYLAND 28 Terramechanics V ENAE 788X - Planetary Surface Robotics
Basic Equation of Vehicle Propulsion DP = H (R c + R b + R g + R r ) DP: Drawbar pull (residual drive force) H: Maximum tractive force of wheels R c : Compaction resistance R b : Bulldozing resistance R g : Gravitational resistance R r : Rolling resistance (internal) U N I V E R S I T Y O F MARYLAND 29 Terramechanics V ENAE 788X - Planetary Surface Robotics
Terramechanics References Bekker, M.G, Introduction to Terrain-Vehicle Systems, Ann Arbor, University of Michigan, 1969 Terzaghi, Theoretical Soil Mechanics, 1943 Wong, J.Y. Theory of Ground Vehicles 2 nd Edition, Canada U N I V E R S I T Y O F MARYLAND 79 Terramechanics V ENAE 788X - Planetary Surface Robotics