The WEB structure is designed using plate theory, and the suspension, mast, and arm are designed using Bernoulli-Euler beam theory.
3.4.8.1 Structural calculations – suspension system
The design procedure for the suspension system is as follows:
1. Assume rocker/bogie suspension geometry of the form used by Sojourner and MER, and shown in Figure 3.4.1.
2. Choose equations relating L2 and L1 to L. Assume the following relations, which produce relative dimensions similar to those shown in Figure 3.4.1.
L2= 0.4L
L1 = L −
(
L2 / 2 + d)
(3.4.7)3. The main body of the rover is attached at approximately the center point of the beam labeled L1, and that beam is attached to the center point of the beam labeled L2. Both these beams can be approximated as pin supported at each end, with a point load at the center. The following equation describes the deflection, δ, of a pin-supported beam with a center point load.
Other quantities in the equation are moment of inertia I, factor of safety fs,, force N, length of the beam Lx, and modulus of elasticity E. The factor of safety is set to a default value of five.
NL3 x
I = fs (3.4.8)
48Eδ
4. Given a maximum allowable deflection, a known force, and known material properties, the moment of inertia of the suspension beam can be calculated. For the beams of length L1, the force is
N1 = mg (3.4.9)
2
5. Because the attachment between the mobility system and the WEB occurs at the midpoint of the beams of length L1, the force on the beams of length L2 is one half that of the force on the beams of length L1.
N 2 =mg (3.4.10)
4
6. Given the inertia of each beam, and assuming a square cross-section and known ratios of beam height to width and wall thickness to width, the beam dimensions can be determined by solving the following set of equations for the beam width b.
I =12 1 bh3 − 12 1
(
b − t)(
h − t)
3(3.4.11)
= 12 1
(
3bth2 -3bht 2+bt 3+th3-3h2 t2+3ht3-t4)
The ratios currently assumed are h=b and t=b/2 (i.e. a solid bar, though the algorithm has been tested with smaller values as well), as these relationships applied to a MER-sized rover returns a suspension system that appears similar to that of MER.
7. The mass of the differential (the part of the mobility system that is located inside the WEB) is assumed to be 4.4 kg. When information on the MER or MSL suspension system becomes available, this number can be replaced with a higher-fidelity estimate.
8. The mass of each wheel is determined by the following equation, where D is the wheel diameter, b is the wheel width, and T is the thickness of the wheel.
π D T2
mwheel = ρ + πDbT (3.4.12)
4
9. The mass of each suspension system beam is determined by multiplying the cross-sectional area of the beam by its length, and then multiplying by a factor of two to account for unmodeled components such as standup motors, hinges, latches, etc.
10. The mass of each drive and steering motor is assumed to be linearly related to the mass of the rover. When motor mass data for MER and MSL become available, the fidelity of this model can be improved.
mmotor =0.05 mrover (3.4.13)
100
11. The total mass of the mobility system is the sum of the four beam masses (two of length L1 and two of length L2), six wheel masses, ten motor masses (six drive and four steering
motors) and the mass of the differential.
12.
3.4.8.2 Structural calculations – mast
The mast design algorithm is a low-fidelity placeholder. The design procedure for the mast is as follows:
1. Determine the mass of the instruments at the top of the mast.
2. Using the Bernoulli-Euler relation for a simply-supported beam, and given a max deflection δ, the moment of inertia, I, of the mast can be determined. The force N, is equal to the gravitational acceleration on Mars times the mass of the instruments at the end of the mast, and E is the modulus of elasticity. The factor of safety, fs, is set to 5.
NL3
I = fs (3.4.14)
Eδ
3. Assume a square cross-section mast. Given a ratio of wall thickness to width equal to 0.5, the dimensions of the mast are determined by
I =12 1 bh3 − 12 1
(
b − t)(
h − t)
3(3.4.15)
= 12 1
(
3bth2 -3bht +bt 2 3+th3-3h2 t2+3ht3-t4)
4. The total mass of the mast is equal to the mass of the beam, which can be determined from the dimensions and the material properties, plus the mass of the motor used to raise the mast, which is assumed to be 1 kg.
3.4.8.3 Structural calculations - arm
The arm design algorithm is a low-fidelity placeholder. The design procedure for the arm is as follows:
1. Determine the mass of the instruments and tools at the end of the arm.
2. Select the number of elbow-type joints. The default number of elbow joints is one.
3. Using the Bernoulli-Euler relation for a simply-supported beam, and given a max deflection, δ, the moment of inertia I of the outermost arm segment can be determined. The force N is equal to the gravitational acceleration on Mars times the mass of the instruments at the end of the arm, and E is the modulus of elasticity. The factor of safety, fs, is set to 5.
I = fs NL3 (3.4.16) Eδ
4. Assume a square cross-section arm. Given a ratio of wall thickness to width equal to 0.5, the dimensions of the mast are determined by
I =12 1 bh3 − 12 1
(
b − t)(
h − t)
3(3.4.17)
= 12 1
(
3bth2 -3bht2+bt 3+th3-3h2 t2+3ht3-t4)
5. The mass of the arm segment can be determined from the dimensions and the material properties.
6. For the next outermost arm segment, the tip mass is the mass of the instruments and tools plus the mass of the outermost arm segment. The same equations are used to calculate the
dimensions. This process repeats for all arm segments, where at each step the tip mass is the sum of the instrument and tool mass and the masses of all previously calculated arm segments.
7. The total mass of the arm is the sum of the masses of the arm segments, plus the masses of each of the arm motors, which are currently assumed to be zero.
3.4.8.4 Structural calculations – plate bending
As explained in the assumptions section, the plates are designed to meet a maximum deflection
requirement. This condition determines the plate’s flexural rigidity and thickness. This method applies to both uniform and sandwich plates, but the equations are different. First for a rectangular uniform plate with built-in edges uniformly loaded, the maximum deflection is given by [ST-59]:
∆
Max= C q a
4D
(3.4.18)Where C depends on the plate’s aspect ratio (see table 35 of [ST-59]), q is the load per unit area, a is the length of the small side of the plate and D is the flexural rigidity. Given ∆Max, D is then known and the following equation is used to find the plate thickness, t (see Equation 3 of [ST-59]).
D
unif= Et
(3.4.19)12(1 − ν
2)
Where E and ν are the Young modulus and the Poisson coefficient of the plate respectively and t is the thickness. If the plate is a sandwich structure, the core thickness is first estimated and then the maximum deflection condition is applied to find the skin thickness. The core thickness is determined by following a weight-minimization expression [HEX].
t
cw
cβ qa
2=
(3.4.20)2w F
s sWhere wc and ws are the densities of the core and skin respectively, Fs is the allowable facing stress and β depends on the aspect ratio and is conservatively set to 0.12. The relation between the maximum deflection and the flexural rigidity then is represented by [HEX]:
16qa
4∆
Max=
6C
1 (3.4.21)π D
Where C1 depends on the aspect ratio but is conservatively set to 2.
Hence, the given maximum deflection determines flexural rigidity, which is related to the plate’s total thickness t by the following equation [HEX].
1
Core
Where tc is the core thickness and t the total plate thickness
Finally the skin thickness is just half of the difference between the total and core thicknesses. Knowing the plate’s thicknesses, dimensions and material densities, the mass is easily deduced.
3.4.8.5 Structural calculations – plate buckling
The same kind of method is used for the design of a plate under compression. The plate’s thickness is sized so that the actual compressive load is less than the critical load given in [ST-40] section 64.
E
3 3E
D
sandwich= t − t −
(3.4.22)12(1 − ν
2)
cE
skin
4
π
2 2 3 3 2
NCrit = Da + +
4 b4 a2b2
3 a (3.4.23)
NCrit ≥ fs N Plate
Where a is the plate side in the direction of the compression, b is the other side, NWall is the actual compressive load and D is the flexural rigidity as formerly defined. The second equation represents the design condition where fs is a factor of safety. Hence, the combined equations give a value for D and consequently for the thickness.
3.4.8.6 Structural calculations – plate tension
This time the plate is designed to resist a tension load. The plate must be thick enough so that the actual tensile strength acting on it is less than the ultimate tensile strength.
t = f
sT
plate(3.4.24)
bT
ultWhere t is the designed thickness, Tplate is the tension load, Tult is the material ultimate stress and b is the length of the plate perpendicular to the tension direction.
3.4.8.7 Thermal calculations
Thermal transfer occurs by three primary methods: conduction, convection, and radiation. All three of these heat transfer methods are included in the WEB thermal model.
Heat transfer through a flat object (such as a wall) by conduction is proportional to the difference
between the surface temperatures at each side of the object, and can be modeled by (Eq. 3.4.25), where Q is the heat transfer rate, L is the thickness of the material, A is the area of the material, and the temperature-independent conduction coefficient k depends on the properties of the material.
Q = kA L
(
T − T0)
(3.4.25)Heat transfer by convection is proportional to the difference between the surface temperature of the object and the ambient temperature. Convective heat transfer can be modeled by the following
equation, where Q is the heat transfer rate, A is the area of the material, and the convection coefficient h depends on environmental factors such as wind speed and the density and chemical composition of the convecting medium.
Q =hA
(
T − T0)
(3.4.26)Heat transfer by radiation depends on the difference between the fourth powers of the surface
temperature of the object and the ambient temperature. The material may have different emission and absorption properties, which are characterized by the emissivity coefficient ε and the absorptivity coefficient α. Radiative heat transfer can be modeled by the following equation, where Q is the heat transfer rate, σ is the Stephan-Boltzmann constant, and A is the area of the material.
Q =σA