Appendix B Tractive Limitations of Rigid Wheels on Soil
B.3 Parametric Analysis based on Terramechanics Model
B.3.2 Results and Discussions
Effects of Slip and Sinkage Ratio on Stress Distributions
Simulations are first conducted to analyze dependence properties of slip swand sinkage ratio hw/rwto the stress distributions. As an example of these simulations, the numerical results under sw=0.2,0.8 and hw/rw=0.2,0.8 are depicted in Figure B.5. From these, the tendency thatθwm
is proportional to swis confirmed as expressed in Eq. (B.9). The maximum values of the stresses do not change with sw in these results. In the meantime, hw/rw affects notably the maximum values, and then they increase more than quadrupled with hw/rw=0.2→0.8. Therefore, it is
B.3 Parametric Analysis based on Terramechanics Model
thus it is concluded that the wheel’s geometric radius is a key factor for the stress distributions.
Here the other stress distributions under sw=0.2∼0.8 and hw/w=0.2∼0.8 can be summarized in the intermediaries between the states shown in Figure B.5.
Effects of Slip and Wheel Geometry on Shear Function
Simulation analyses with respect to the shear function expressed in Eq. (B.10) are demon-strated. The shear function τ/τmax, which is defined as the ratio of the shear stress over the shear strength, is an important index for evaluating the drawbar pull. Figure B.6 plots the effects of swand rw/K on the shear function with constant hw/rw (constantθw f). Seen from this graph, rw/K is more effective in the change of the shear function than sw. Given rw/K affects expo-nentially the shear function, the increase of the shear function becomes lower with the increase of rw/K. Thus, since K is an uncontrollable soil parameter, larger rw should be designed to obtain the enough shear stress on the soil. In contrast with this, it is confirmed thatτ does not essentially depend much on switself.
0 20 40 60 80
0 30 60 90 120
Wheel Angle, θw [deg]
Stress [kPa]
sw = 0.2, hw/rw = 0.2
0 20 40 60 80
0 30 60 90 120
Stress [kPa]
Wheel Angle, θw [deg]
sw = 0.2, hw/rw = 0.4
0 20 40 60 80
0 30 60 90 120
Wheel Angle, θw [deg]
Stress [kPa]
sw = 0.2, hw/rw = 0.6
0 20 40 60 80
0 30 60 90 120
Wheel Angle, θw [deg]
Stress [kPa]
sw = 0.2, hw/rw = 0.8
0 20 40 60 80
0 30 60 90 120
Wheel Angle, θw [deg]
Stress [kPa]
sw = 0.4, h w/r
w = 0.2
0 20 40 60 80
0 30 60 90 120
Wheel Angle, θw [deg]
Stress [kPa]
sw = 0.4, h w/r
w = 0.4
0 20 40 60 80
0 30 60 90 120
Wheel Angle, θw [deg]
Stress [kPa]
sw = 0.4, h w/r
w = 0.6
0 20 40 60 80
0 30 60 90 120
Wheel Angle, θw [deg]
Stress [kPa]
sw = 0.4, h w/r
w = 0.8
0 20 40 60 80
0 30 60 90 120
Wheel Angle, θw [deg]
Stress [kPa]
sw = 0.6, h w/r
w = 0.2
0 20 40 60 80
0 30 60 90 120
Wheel Angle, θw [deg]
Stress [kPa]
sw = 0.6, h w/r
w = 0.4
0 20 40 60 80
0 30 60 90 120
Wheel Angle, θw [deg]
Stress [kPa]
sw = 0.6, h w/r
w = 0.6
0 20 40 60 80
0 30 60 90 120
Wheel Angle, θw [deg]
Stress [kPa]
sw = 0.6, h w/r
w = 0.8
0 20 40 60 80
0 30 60 90 120
Wheel Angle, θw [deg]
Stress [kPa]
sw = 0.8, h w/r
w = 0.2
0 20 40 60 80
0 30 60 90 120
Wheel Angle, θw [deg]
Stress [kPa]
sw = 0.8, h w/r
w = 0.4
0 20 40 60 80
0 30 60 90 120
Wheel Angle, θw [deg]
Stress [kPa]
sw = 0.8, h w/r
w = 0.6
0 20 40 60 80
0 30 60 90 120
Wheel Angle, θw [deg]
Stress [kPa]
sw = 0.8, h w/r
w = 0.8
Figure B.5 : Stress distributions alongθw: —σ, —τ.
B.3 Parametric Analysis based on Terramechanics Model
0 15 30 45 60 75
0 0.2 0.4 0.6 0.8 1
Wheel Angle, θw [deg]
Shear Function
sw = 0.2 sw = 0.4 sw = 0.6 sw = 0.8
(a) Effect of sw with rw/K=1.
0 15 30 45 60 75
0 0.2 0.4 0.6 0.8 1
Wheel Angle, θw [deg]
Shear Function
rw / K = 1 rw / K = 2 rw / K = 3 rw / K = 4
(b) Effect of rw/K with sw=0.5.
Figure B.6 : Shear function alongθwwith constant hw/rw.
Effects of Sinkage Ratio on Drawbar Pull and Vertical Force
The drawbar pull DPw and the vertical force Fz defined by Eqs. (B.1) and (B.2) are funda-mental integrated indexes for discussing tractive performance. So this appendix analyzes the dependence of the sinkage ratio h/r to these forces. Figures B.7 and B.8 show the simulation results. While a certain amount of change in DPw and Fz by sw can be confirmed, hw/rw is obviously a dominant parameter for the forces. The main reason for these results would be that magnitudes of sine and cosine components are reversed once θw turns 45deg. These compo-nents affect angle compocompo-nents ofσ andτ for DPw, and therefore, these depend much on hw/rw
because of the relationship in Figure B.3. Likewise, negative DPw is just the serious state be-ing stuck, and some sort of external forces are needed to move in this situation. Assumbe-ing the conventional relationship that Fz is equal to W at any slips in Eq. (B.2), hw/rw must slightly decrease with the increase of sw. However, in the light of the slip sinkage effect in Eq. (B.12), the increase of hw/rw pertaining to sw should become larger. Therefore, the conventional force equilibrium in the vertical direction would contradict this result. At the same time, these simula-tion analyses reveal the fact that the vertical force balance exerting the sinkage hwis inconsistent with the empirical outcomes. Further, it is indicated that hw/rw depends much on the load W . Then, these results eventually provide an emphasis of wheel’s mechanical design, especially the wheel radius rw and the weight W .
Effects of Slip Sinkage on Drawbar Pull
As the next step, simulations of drawbar pull DP with practical sinkage behavior are
ana-B.3 Parametric Analysis based on Terramechanics Model
0 0.2 0.4 0.6 0.8 1
−5 0 5 10 15 20 25
Slip
Forces [N]
Drawbar Pull
Vertical Forces hw / r
w = 0.1
0 0.2 0.4 0.6 0.8 1
−20 0 20 40 60 80
Slip
Forces [N] Drawbar Pull
Vertical Forces hw / r
w = 0.2
0 0.2 0.4 0.6 0.8 1
−20 0 40 80 120 160
Slip
Forces [N] Drawbar Pull
Vertical Forces hw / r
w = 0.3
0 0.2 0.4 0.6 0.8 1
−40 0 80 160 240
Slip
Forces [N]
Vertical Forces Drawbar Pull
hw / rw = 0.4
Figure B.7 : Integrated forces vs. slip with constant hw/rw.
0.1 0.2
0.3 0.4
0.2 0 0.60.4 10.8
−40
−20 0 20
Sinkage Ratio, Slip, h/r
s
Drawbar Pull, DP [N]
−30
−20
−10 0
(a) Simulated drawbar pull.
0.1 0.2
0.3 0.4
0.2 0 0.60.4 10.8 0 50 100 150 200
Sinkage Ratio, Slip, h/r
s Vertical Force, F z [N]
50 100 150
(b) Simulated vertical force.
Figure B.8 : Three-dimensional plots of integrated forces vs. slip with constant hw/rw.
at neighborhood of sw=0.3, and thus this proves a steady state can be achieved pertaining to the translational velocity. Moreover, Figure B.10 plots the change of DPw with respect to sw at c4=0.015,0.03,0.045. The critical state that DPwalways indicates less than zero with larger c4 is confirmed by this result.
Prediction of Tractive Limitations
The tractive limitations are given by a function of the sinkage, especially the sinkage ratio.
Figure B.12 plots minimal conditions of hw/rwwith change of the wheel radius rw, which
satis-B.3 Parametric Analysis based on Terramechanics Model
Figure B.9 : Traveling results regarding slip and steady sinkage [92].
0 0.2 0.4 0.6 0.8 1
−20
−15
−10
−5 0 5 10
Slip Drawbar Pull, DP w [N]
c4 = 0.015 c4 = 0.03 c4 = 0.045
Figure B.10 : Drawbar pull vs. slip with various c4.
Drawbar Pull
Slip 1
0
much slip sinkage less slip sinkage
Figure B.11 : Description of slip-traction characteristics.
0 0.1 0.2 0.3 0.4 0.5
0 0.1 0.2 0.3 0.4 0.5 0.6
Wheel Radius, r
w [m]
Minimal h w / r w
K = 1.0 [cm]
K = 2.5 [cm]
K = 4.0 [cm]
(a) Effect of K.
0 10 20 30 40 50
0 0.1 0.2 0.3 0.4 0.5 0.6
rw / K Minimal h w / r w
K = 1.0 [cm]
K = 2.5 [cm]
K = 4.0 [cm]
(b) Effect of rw/K.
Figure B.12 : Minimal hw/rwsatisfying DPw≤0 with various rwtargeting dry sand.
fies negative drawbar pull DPw≤0, where targeted terrain is assumed to be dry sand. Seen from Figure B.12(a), it can be confirmed that the minimal hw/rwbasically increases with an increase
B.3 Parametric Analysis based on Terramechanics Model
Table B.2 : Nominal parameters of lunar soil in simulation analyses [81, 84].
Soil Parameter Symbol Value Unit
Internal Friction Angle φ 35 deg
Cohesion Stress C 170 Pa
Pressure-Sinkage Modulus for Internal Friction Angle kφ 814.4 kN/mn+2 Pressure-Sinkage Modulus for Cohesion Stress kc 1379 N/mn+1
Deformation Modulus K 0.0178 m
Pressure-Sinkage Ratio n 1.0
-Cohesion for determining the Relative Position c1 0.4 -of Maximum Radial Stress
Cohesion for determining the Relative Position c2 0.15 -of Maximum Radial Stress
0 0.1 0.2 0.3 0.4 0.5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Wheel Radius, r
w [m]
Minimal h w / r w
φ = 35 [deg], K = 1.02 [cm]
φ = 35 [deg], K = 1.78 [cm]
φ = 35 [deg], K = 2.54 [cm]
(a) Effect of K.
0 0.1 0.2 0.3 0.4 0.5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Wheel Radius, r
w [m]
Minimal h w / r w
φ = 35 [deg], K = 1.78 [cm]
φ = 31 [deg], K = 1.78 [cm]
φ = 39 [deg], K = 1.78 [cm]
(b) Effect ofφ.
Figure B.13 : Minimal hw/rwsatisfying DPw≤0 with various rwtargeting lunar soil.
sand. In the meantime, this advantage is saturated beyond a certain rw(e.g. the minimal hw/rw
becomes an almost same value at rw >0.5m in Figure B.12(a)). With normalizing rw/K, the characteristics of the minimal hw/rwbecome marked as shown in Figure B.12(b).
Furthermore, simulations targeting lunar soil [81] are demonstrated to investigate the limita-tions. The nominal property of the lunar soil is shown in Table B.2. Figure B.13 depicts the simulation results with the lunar soil. In accordance with Figure B.13(a), similar tendencies of the limitations with the dry sand can be shown. On the other hand, the additional simulation results graphed in Figure B.13(b) indicate the effect ofφ on the limitations. It is concluded that higher frictional soil poses a better tractive performance. Consequently, the theoretical predic-tion of the wheel’s limitapredic-tions in the steady state is achieved by the sinkage condipredic-tion.