Terramechanics. Origin and nature of lunar soil Soil mechanics Wheel-soil interactions Tandem wheel calculations Grousers MARYLAND

Transcription

1 Terramechanics Origin and nature of lunar soil Soil mechanics Wheel-soil interactions Tandem wheel calculations Grousers 1 16 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 1-1 torr (= 1.x1-1 Pa = 1.93x1-14 psi) Meteorites at velocities >1 5 m/sec Galactic cosmic rays, solar particles Temperature range +5 F -5 F

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) 3

4 JSC-1 Simulant Ash vented from Merriam Crater in San Francisco volcano field near Flagstaff, AZ K-Ar dated at 15, years old ± 3, Major constituents SiO, TiO, Al O 3, Fe O 3, FeO, MgO, CaO, Na O, other <1% Represents low-ti regolith from lunar mare MLS-1 simulant (U.Minn.) preferred for simulation of highland material BP-1 (Flagstaff, AZ) is ground basaltic lava - higher fidelity because of angular grain shapes 4

5 Particle Size Distribution in Regolith Rahmatian and Metzger, Soil Test Apparatus for Lunar Surfaces Earth and Space 1, ASCE 5

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 = T A Soil Normal Stress T = W A cross-section A Shear Stress = angle of internal friction 7 c = Cohesion Normal Stress

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 Soil Penetration Depth z Scaled Pressure P k n= n=.5 n=1. n=1.5 9

10 Lunokhod 1 Penetrometer Data Mitchell et.al., Mechanical Properties of Lunar Soil: Density, Porosity, Cohesion, and Angle of Internal Friction Proceedings of the Third Lunar Science Conference, Vol. 3, MIT Press, 197 1

11 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, 4 11

12 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 Pdz = z o kz n dz = k zn+1 o n +1 1

13 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 13

14 Rolling Resistance For n =1:P = kz; E A = k z o ; R = 1 kbz o For n = 1 : P = k z; E A = 3 kz 3 o ; R = 3 kbz 3 o For n =: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 14

15 Soil Displacement Calculations wheel width b z o R W o F z F x df R df = Pbr d W + Z o Z o df sin = df cos = df cos = Pb dx df sin = Pb dz R = Z o Pb dz W = Z o Pb dx In general, P = kx n W = Z zo bkz n dx 15

16 Soil Displacement Calculations wheel width b z o z A B D AB = D (z o z) x = D x AB = D apple D (z o z) = D D + D (z o z) (z o z) x =[D (z o z)](z o z) 16

17 Soil Compression Calculations But D z o z x D(z o z) ) x dx= Ddz so from W = W = Z zo bk bkz n dx we get W = Z zo Z zo z n D p D p dz z o z bkz n D x dz W = bk Z zo z n p! Ddz p dz z o z 17

18 Soil Displacement Calculations Define z o z t ) dz = tdt W = bk p D Z p z o (z o t ) n dt Taylor Series expansion (z o t ) n = z n o nz n 1 o t + W bkp Dz o 3 z n o (3 n) for n =1) W = 3 bkz op Dzo for n = 1 ) W = 5 6 bkz op D for n =) W = bk p Dz o 18

19 Rolling Resistance as f(w) for n =) W = bk p Dz o ) z o = W bk 1 D R = kbz o ) R = kb W (kb) D ) R = W kbd for n = 1 ) W = 5 6 bkz p o D ) zo = 6 5 W bk p D R = 3 kbz 3 o ) R = 3 kb 6 5 W kb p D 3 = W 3 p 3 5 kbd 4 R =.876 W 3 p kbd

20 Rolling Resistance as f(w) for n =1) W = 3 bkz 3 o p D ) z o = R = 1 kbz o ) R = 1 kb 3W kb p D 4 3 = 1 3W kb p D W kbd W 4 R =.859 kbd 1 3

21 Rolling Resistance as f(w) (Generic) W = bkp Dz o 3 z n o (3 z n+ 1 o = n) = bkp D 3 3 (3 n) W bk p D z n+ 1 o (3 n) z n+1 o = 3 3 n W bk p D n+1 n+ 1 = 3 3 n W bk p D (n+1) n+1 R = bk n +1 zn+1 o = bk n +1 R = 1 n n n W bk p D W (n+1) n+1 1 p D bk (n+1) n+1 1 n+1

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

23 Soil Characteristics soil type n k c N m n+1 k N m n+ Dry Sand ,58, Lunar Regolith , Sandy Loam ,515, Sandy Loam (MER-B) Slope Soil (MER-B) 1 8, 7,6, , Clay (Earth).5 13,19 69, 3

24 Terzaghi Analysis of Soil Deformation 4

25 Terzaghi Soil Bearing Capacity Factors N c = cot N q = exp 3 tan cos < exp 3 tan 9 = 1 : cos 4 + ; = cot (N q 1) K p = tan 3 + N = 1 Kp cos 1 tan = Angle of internal resistance of soil 5

26 Parameter Definition K c Modulus of density of soil deformation K c =(N c tan ) cos K Modulus of cohesion of soil deformation apple N K = +1 cos tan Angle of approach of wheel to soil = cos 1 1 N Weight density of soil m 3 6 z D

27 Soil Bearing Limit Safe weight on the soil W s = A cn c + zn q + 1 bn c Soil cohesion (Pa) b Wheel width (m) 7

28 Equations for Compression Resistance z = apple 3W w (3 n)bk p d W w = weight on wheel d = wheel diameter bk R c = n + 1 n+1 z n+1 R c = compression resistance (per wheel) 8

29 Soil Compression Reece Formulation Shear Stress c = Cohesion = angle of internal friction Normal Stress P = kc b + k z n Problem is that k c and k have variable dimensions, based on n k c units < N/m (n+1) > k units < N/m (n+) > 9

30 Compression Resistance (Lunar Soil) R c = 1 n + 1 (k c + bk ) 1 n+1 n = 1 3W w (3 n) d (n+1) n+1 k c =.14 N/cm k =.87 N/cm 3 R c = 1 (k c + bk ) 1 3 3W w d 4 3 3

31 Compression Resistance (Lunar Soil) R c = 1 n + 1 (k c + bk ) 1 n+1 n = 1 3W w (3 n) d (n+1) n+1 k c =.14 N/cm k =.87 N/cm 3 R c = 1 (k c + bk ) W w d 4 3 R c =.89 N/wheel = N (total)

32 Lunar Regolith Soil Properties Symbol Description Value Used in n Exponent of sinkage 1. EVERYTHING k c Cohesive modulus of soil deformation 14 N/m Rc, Rb, H k ф Frictional modulus of soil deformation 8, N/m 3 Rc, Rb, H θ Slope angle Rg Cf Coefficient of friction.5 Rr γ Soil density 1.6 g/cm 3 Rb Co Cohesive strength of soil.17 N/cm H ф Angle of internal friction 3-4 degrees Rb K Coefficient of soil Slip 1.8 cm H s slip..5 H 3

33 Apollo Lunar Roving Vehicle Example z = 3 53 ( ) 8 3 =.3 cm R c = 1 check units ( ) N 1/3 cm /3 N 4/3 cm / = N 4 3 = 9.8 N 33

34 Rolling and Gravitation Resistance Rolling resistance (tires, bearings, etc.) R r = W v c f W v = weight of vehicle c f = coe cient of friction (typ..5) Gravitational resistance R g = W v sin slope LRV examples (15 slope) R r = 51 N R g = 6 N 34

35 Bulldozing Resistance General case: R b = b sin ( + ) sin cos zck c + 35 z K For tracked vehicles, only the first term applies: R b = +`3o 3 b sin ( + ) sin cos + c`o = angle of attack of wheel in soil cos 1 1 = density of soil kg m 3 All angles in radians! `o = z tan 4 zck c + apple 1 + tan 4 + z K z D

36 Tandem Wheels W W 1 D D 1 1 z 1 z z z = z 1 + z Assume n = 1 =) P = kp z P 1 = k p z 1 P = k p z 1 + z 36

37 Soil Weight Bearing Analysis In general, W = Z df cos = Z Pb dscos W = Z bk p z cos ds ds = rd z = r(cos cos ) W = Z bkr p r(cos cos ) cos d 37

38 Generic Wheel Soil Suspension Assuming small sinkage, z! small,! small, cos 1 cos d d cos 1 W = bkr3/ p 1 W = bkr3/ p + (higher order terms) Z q apple sin 1 d + q 38

39 Weight on the Front Wheel W = bkr3/ 4 p Front wheel: z 1 = r 1 r 1 cos z 1 = r 1 r W 1 = bkz 1 p r1 p =) = z 1 r 1 39

40 Weight on Back Wheel Change to limits of integration:!, r! r W = bkr3/ p Z q = 1 q 1 d + Z q d = = = 1 4

41 Weight on Back Wheel W = bkr3/ p 1 z = r = r z r 1 = z = 1 r r z r W = bk p r p z z 41

42 Track Depth of Tandem Wheels Much algebra then ensues... Front: z 1 = p W 1 bk p r 1 apple W Back: z = bk p 1 p r z z = z 1 + z = p W 1 bk p r 1 + W (bk) r 1 z z p W 1 bk p r 1 z + W (bk) r = 4

43 Rolling Resistance of Tandem Wheels Solve the quadratic equation to get z = W1 bk p + r 1 s W 1 r 1 + W r 1 A This was all done for n = 1 =) R = 3 bkz3/ s R = 3 R = 3 1 W1 bk p + r 1 1 W 1 bk p + D 1 s W 1 r 1 + W r 4W 1 D 1 + W D 1 A 1 A 3/ 3/ 43

44 Nondimensional Forms Total wheel load W = W 1 + W Wheel weight ratio a W 1 W For W 1 = W = W =) a =1 R = 3 W 1 = a 1+a W W = 1 1+a W 1 W 3/ a (a + 1) 3/ bk p + D 1 s 4a D D 1 A 3/ Define wheel diameter ratio D 1 D 44

45 Nondimensional Forms R = 3 1 W 3/ (a + 1) 3/ D 3/4 p bk Let 3 a p + 1 a (a + 1) 3/ p + s s 1+ 4a 1+ 4a!3/!3/ R = p bk W 3/ D 3/4 45

46 Simple Example Case Consider =1 (D 1 = D = D) a =1 W 1 = W = W For tandem wheels, R =.58 p bk W 3/ D 3/4 For single wheel (n=1/), R =.876 p bk W 3/ D 3/4 Tandem wheels reduce rolling resistance by 34% 46

47 Dual Wheels Equivalent to single wheel case twice as wide b =) b R =.876 p 1 p bk W 3/ D 3/4 R dual =.619 p bk W 3/ D 3/4 Dual wheel rolling resistance 9% less than single, 7% higher than tandem 47