Terramechanics. Origin and nature of lunar soil Soil mechanics Rigid wheel mechanics MARYLAND U N I V E R S I T Y O F

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1 Terramechanics Origin and nature of lunar soil Soil mechanics Rigid wheel mechanics David L. Akin - All rights reserved

2 Lunar Regolith Broken down from larger pieces over time Major constituents Rock fragments Mineral fragments Glassy particles Local environment torr Meteorites at >10 5 m/sec Galactic cosmic rays, solar particles Temperature range +250 F -250 F 2

3 Regolith Creation Process Only weathering phenomenon on the moon is meteoritic impact! Weathering processes Comminution: breaking rocks and minerals into smaller particles Agglutination: welding fragments together with molten glass formed by impact energy Solar wind spallation and implantation (miniscule) Fire fountaining (dormant)

4 JSC-1 Simulant Ash vented from Merriam Crater in San Francisco volcano field near Flagstaff, AZ K-Ar dated at 150,000 years old ± 0,000 Major constituents SiO 2, TiO 2, Al 2 O, Fe 2 O, FeO, MgO, CaO, Na 2 O, other <1% Represents low-ti regolith from lunar mare MLS-1 simulant (U.Minn.) preferred for simulation of highland material 4

5 Wheel-Soil Interaction Wheel rolling over soil does work Compression Bulldozing from Gibbesch and Schafer, Advanced and Simulation Methods of Planetary Rover Mobility on Soft Terrain 8th ESA Workshop on Advanced Space Technologies for Robotics and Automation, Noordwijk, The Netherlands, November,

6 Soil Testing Apparatus Internal friction angle ϕ Bevameter (force vs. displacement) Shear deformation modulus K 6

7 Soil Characterization Direct Shear W Shear Stress τ = T A Soil Normal Stress σ = W T A cross-section A Shear Stress τ φ = angle of internal friction 7 c = Cohesion Normal Stress σ

8 Modeling Soil Reaction to a Wheel Assume soil reaction is like a (nonlinear) spring P = kz n P = applied pressure z = compression depth k, n = heuristic parameters P z 8

9 Effects of Soil Mechanics 0 Soil Penetration Depth z Scaled Pressure P k n=0 n=0.5 n=1.0 n=

10 Wheel-Soil Interactions wheel width b distance rolled d R z o W area of compressed soil A = bd Displacement Energy E A = F A dz = Pdz E A = zo 0 Pdz = zo 0 kz n dz = k zn+1 o n +1 10

11 Rolling Resistance Total Energy E A A = E A Given a force resisting rolling R, the energy required to roll a distance d is E roll = Rd zn+1 o bd = k n +1 bd E roll = E displacement Rd = E A bd 11

12 Rolling Resistance For n =1:P = kz; E A = k z2 o 2 ; R = 1 2 kbz2 o For n = 1 2 : P 2 = k 2 z; E A = 2 kz 2 o ; R = 2 kbz 2 o For n =0:P = k; E A = kz o; R = kbz o Generic case: P = kz n ; E A = k zn+1 o n +1 ; R = kb zn+1 o n +1 12

13 Soil Displacement Calculations wheel width b z o R W θ θ o F z F x df R W + θo 0 θo 0 df sin θ =0 df cos θ =0 df = Pb df cos θ = Pb dx df sin θ = Pb dz R = θo 0 Pb dz W = θo 0 Pb dx In general, P = kx n W = zo 0 bkz n dx 1

14 Soil Displacement Calculations wheel width b AB = D 2 (z o z) z o z x 2 = A B θ x 2 D 2 D 2 AB 2 = D D 2 (z o z) = D 2 2 D D 2 (z o z) (z o z) 2 x 2 =[D (z o z)](z o z) 14

15 Soil Compression Calculations But D z o z so from W = x 2 D(z o z) 2x dx= D dz zo 0 W = bk bkz n dx we get W = zo 0 zo z n D 2 D dz z o z 0 bkz n D 2x dz W = bk zo 0 Ddz z n 2 z o z 15

16 Soil Displacement Calculations Define z o z t 2 dz = 2tdt W = bk D z o 0 (z o t 2 ) n dt Taylor Series expansion (z o t 2 ) n = z n o nz n 1 o t 2 + W bk Dz o z n o ( n) for n =1 W = 2 bkz o Dzo for n = 1 2 W = 5 6 bkz o D for n =0 W = bk Dz o 16

17 Rolling Resistance as f(w) for n =0 W = bk Dz o z o = W bk 2 1 D R = kbz o R = kb W 2 (kb) 2 D R = W 2 kbd for n = 1 2 W = 5 6 bkz o D zo = 6 5 W bk D R = 2 kbz 2 o R = 2 kb 6 5 W kb D 2 = W 2 5 kbd 4 R =0.876 W 2 kbd 4 17

18 Rolling Resistance as f(w) for n =1 W = 2 bkz 2 o D z 2 o = R = 1 2 kbz2 o R = 1 2 kb W 2kb D 4 = 1 2 W 2kb D 4 4 W kbd 2 W 4 R =0.859 kbd

19 Rolling Resistance as f(w) (Generic) W = bk Dz o z n o ( n) = bk D z n+ 1 2 o = ( n) W bk D z n+ 1 2 o ( n) z n+1 o = n W bk D n+1 n+ 1 2 = n W bk D 2(n+1) 2n+1 R = bk n +1 zn+1 o = bk n +1 R = 1 n +1 n 19 n W bk D W 2(n+1) 2n+1 1 D bk 2(n+1) 2n+1 1 2n+1

20 More Detailed Soil Compression Equation k = k c b + k φ k c = modulus of cohesion of soil deformation k c units < N/m (n+1) > k φ = modulus of friction of soil deformation k φ units < N/m (n+2) > b = wheel width kc P = b + k φ z n 20

21 Soil Characteristics soil type n k c N m n+1 k φ N m n+2 Dry Sand ,528,000 Lunar Regolith ,000 Sandy Loam ,515,000 Sandy Loam (MER-B) Slope Soil (MER-B) ,000 7,600, ,000 Clay (Earth) 0.5 1, ,200 21

22 Compression Resistance (Lunar Soil) R c = 1 n + 1 (k c + bk φ ) 1 2n+1 Ww ( n) d n = 1 k c = 0.14 N/cm 2 2(n+1) 2n+1 k φ = N/cm R c = 1 2 (k c + bk φ ) 1 Ww 2 d 4 22

23 Apollo Lunar Roving Vehicle Example z = 25 2( ) 82 2 = 2.0 cm R c = ( ) check units - N 1/ cm 2/ N 4/ cm 2/ = N 4 = 29.8 N 2

24 Rolling and Gravitation Resistance Rolling resistance (tires, bearings, etc.) R r = W v c f W v = weight of vehicle c f = coefficient of friction (typ. 0.05) Gravitational resistance R g = W v sin θ slope LRV examples (15 slope) R r = 51 N R g = 262 N 24

25 Bulldozing Resistance Bulldozing is the process of pushing soil up ahead of the wheel Ranges from a small factor to a huge one, depending on soil and wheel factors Will be covered in detail in a later lecture 25

26 Tractive Force per Wheel (No Grousers) H =[AC b + W w tan φ b ] A = area of contact C b = coefficient of soil/wheel cohesion s = wheel slip ratio 1 K φ b = wheel/soil friction angle K = coefficient of soil slip = length of contact patch 1 e s K 26

27 Tractive Force per Wheel (With Grousers) H = bc b 1 + 2h b N g + W tan φ b h b arctan b 1 K h 1 e s K A = area of contact = b C b = soil/wheel cohesion = N/cm 2 φ b = wheel/soil friction angle = 5 s = wheel slip ratio (typ ) K = coefficient of soil slip = 1.8 cm = length of contact patch = D 2 cos 1 1 2z h = height of grouser All values typical for lunar soil 27 D

28 Effect of Soil Thrust Fraction Soil Thrust Fraction 1 K 1 e s K 1.2 Soil Thrust Fraction K/l= Slip Ratio 28

29 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) 29

30 Example: Wheelbarrow (Single) Wheel R = (k c + k φ b) 1 2(n+1) 2n+1 W 2n+1 1 n + 1 n 2(n+1) 2n+1 D (n+1) 2n+1 Compaction Resistance (N) Drawbar Resistance (N) b=0.1 m D=0. m Dry Sand Sandy Loam Clay Lunar MER-B Sandy Loam MER-B Slope Soil Wheel Weight (N) 0

31 Effects of Wheel Parameters 200 Compaction Drawbar Resistance (N) W=500 N b=0.1 b=0.25 b= Wheel Diameter (m) 1

32 Effect of Soil Spring Constant on R/W Resistance/Weight n=0 n=0.5 n= Wheel 1 m diameter x 0.2 m width Soil "k" value (N/m) 2

33 Soil Type and Wheel Load Drawbar Compaction Resistance (N) Wheel 1 m diameter x 0.2 m width Weight on Wheel (N) Clay Dry Sand Snow

34 Soil Type and Specific Resistance 10 Resistance/Weight Clay Dry Sand Snow 0.1 Wheel 1 m diameter x 0.2 m width Weight on Wheel (N) 4

35 Effect of Wheel Diameter and Width Wheel Load = 1000 N Compaction Resistance (N) Resistance (N) Resistance (N) 1000 D=1 m D=1 D=2 mm D=2 D=4 mm D=4 Dual m Quad Wheel Width (m) Wheel Width (m) 5

36 Effect of Slope 600 Gravitational Drawbar Resistance (N) Slope (deg) 6

37 Wheel Test Apparatus Wheel testing done at MIT Field and Space Robotics Laboratory Independent control of motion and wheel velocity provides controllable slip s = 1 V ωr 7

38 Wheel Torque vs. Time 9 grousers 18 grousers ϕ=0.24 8

39 Sinkage vs. Slip Ratio 9

40 Drawbar Pull vs. Slip Ratio 40

41 Motor Torque vs. Slip Ratio 41