Portal:Lunar Lander/Landing Gear/LLG-TC
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This should represent the bulk of the report. Apologies for any incoherence or formatting issues (still learning wiki). These will get fixed eventually. In the meantime, please refer to the [Team Cheese Blog] for the full report and complimentary files. --Ian C 07:54, 23 December 2009 (UTC)
Conceptual lunar landing gear design by Team Cheese (TC).
Contents |
Evaluative Criteria
The following criteria guide the design effort to account for the design constraints.
| Criteria | Description |
|---|---|
| Mass | The mass of a landing system must be minimized in order to minimize the cost of launch. |
| Volume | The volume of landing system components must be minimized to reduce volume occupancy in the launch payload and to maximize cargo space available within the landing module itself. |
| Force Attenuation | The properties of the energy absorption system must reasonably limit the maximum acceleration that develops on the landing module. |
| Stability | The intended final orientation of the landing module must be able to withstand moments and anticipated external forces acting on its structure. |
| Clearance | The landing system must provide adequate clearance within specifications over a range of possible surface conditions. |
| Simplicity | Ease of manufacture, operation and implementation in the landing module. |
| Reliability | Confidence in successful operation, unobtrusive integration and no interference with landing module subsystems. |
| Design Time | TC’s design efforts were restricted by time and academic deadlines. |
| Cost | Important factor considering an open effort (cost of launch and manufacture). This criterion was omitted in design comparison as it would influence in the weighting of every other criteria. |
Design Methodology
Assumed mission operation specifications direct preliminary design of the landing system.
- The static launch payload is cylindrical in geometry (2 meter diameter and 3.3 meter in height).
- The landing module will occupy 70% to 75% of this volume.
- The mass is assumed to be 190kg ± 30%.
- The landing module center of mass will be assumed to be the center of its simplified volume.
- The impact force developed must be less than a 10G static load.
- The landing structure must withstand and limit the force of landing.
- The thrust engine shuts down 1 m above lunar surface.
- The thrust engine will only reduce the impact velocity, it will not remove it. The landing gear operates in close proximity to the thrust engine. The material will have to resist large temperature variations.
- Level support of the landing module with some degree of stability on a flat surface or an incline is ideal.
- The surface area of the landing module in contact with lunar regolith must spread the weight out appropriately and prevent slip to a reasonable degree. Uneven impact may affect the levelness of the final orientation.
- A clearance of 0.3m ± 30% is required.
- Clearance is needed to restrict contact of the landing module and fuel systems with the lunar surface. It might be beneficial for rover deployment while mitigating the potential for minor surface protrusions to affect mission performance.
- The readiness of the landing system is required by the time the landing module is in lunar orbit.
- Proper remote or programmed deployment and activation of the landing system while in lunar orbit is essential if the system involves an inactive or stowed configuration.
As the landing module comes to its final rest position after touchdown, components of the landing system may be sacrificial. Successful performance of the landing system is geared towards rover deployment and landing an undamaged base station in a position of rest.
While the balloon mechanism used to land Spirit and Opportunity on Mars is interesting and thought provoking, the atmosphere on the moon is so thin that a parachute could not be used to slow descent let alone stabilize a thrust platform. This removes the possibility of using an inflatable balloon shell as there would be no cheap and readily available alternative to slow or orient the landing module before impact (heat from onboard thrusters would likely damage the shell). Without negative acceleration as the landing module approaches the surface of the moon it could possibly crash land or bounce away at escape velocity depending on its impact angle.
The use of electromagnetic flux to resist the motion due to impact between the landing gear and the landing module was omitted in the design comparison; a large power source would make for a high mass and low feasibility. The pod leg concept offers the most versatility; it is a modular design that has a low mass and volume. The approach has been used successfully in a number of past missions like the Apollo and Surveyor missions. Conceptual design will look for an optimal pod leg design.
The design was split into two sections for analysis to help identify the best combination of structural support and energy absorption systems. Individual analysis of the two different systems allows for a more comprehensive look at each component and will provide a stronger final solution to design. Four designs were chosen for the structural support analysis, and four designs were chosen for the energy absorption system analysis.
The four different structural support designs being analyzed consist of a combination of static and dynamic struts with vertical or angled orientations. Static struts will be rigid bodies protruding from the landing module while dynamic struts will have a system that allows them to extend from a collapsed state to a deployed state and that will lock them in this position for landing. A three strut configuration will be assumed for this process as the majority of research done has shown that this is lightest configuration while still having a lot of stability.(14) Angled struts will also have two support beams in a triangle configuration providing extra strength to stop them from deflecting upwards upon landing.
The four different energy absorption systems consist of fluid dampening systems, plastic deformation structures and elastic dampening systems. The plastic deformation structure system is considered to be housed entirely in the strut. Other systems may be housed in the strut or designed as an interface between the strut and the landing module.
Structural System Candidates
Design 1 - Static Vertical Strut
| Advantages | Disadvantages |
|---|---|
| Simple ( very easy to model, easy to test) | Does not support significant lateral loading |
| Strong vertical support (materials are usually strongest in compression) | Due to lack of lateral support stronger materials will have to be used |
| Uses little volume in payload | Stability is limited in instances of lateral loading |
Design 2 - Dynamic Vertical Strut
| Advantages | Disadvantages |
|---|---|
| Reasonably Simple Design ( very easy to model after extension of struts, easy to test) | Does not support significant lateral loading |
| Strong vertical support (materials are usually strongest in compression) | Due to lack of lateral support stronger materials will have to be used |
| Uses little volume in payload | Lateral loading capability is reduced even further with this design due to locking mechanism being weakest link |
| Reliability is reduced due to added complexity of design | |
| Stability is limited in instances of lateral loading
Dynamic mechanism adds extra weight |
Design 3 - Static Angled Strut
| Advantages | Disadvantages |
|---|---|
| Reasonably Simple Design | Harder to model than vertical strut designs |
| Less materials and size are required due to strength | Uses the most volume in payload of all four designs |
| Stability exceeds that of vertical strut designs | More complicated to model and test |
| Most reliable design |
Design 4 - Dynamic Angled Strut
| Advantages | Disadvantages |
|---|---|
| Uses the least volume in payload of all four designs | Most complicated design (testing, modeling, and designing) |
| Stability exceeds that of vertical strut designs | Locking mechanism will reduce overall strength as it will be the weakest link |
| Less materials and size are required due to strength | Reliability is reduced slightly due to added complexity of design |
Energy Absoprtion System Candidates
Design 1 - Hydor-pneumatic
| Advantages | Disadvantages |
|---|---|
| No oscillation of system | Takes up the most volume |
| Provides variable resistance to applied load | Most complex design |
| Clearance below structure after landing is invariable and controllable | Fluid is susceptible to extreme temperature variances |
| One of the least reliable designs due to the number of components with potential to fail |
Design 2 - Plastically Deforming
| Advantages | Disadvantages |
|---|---|
| Provides one of the best energy absorption density ratios | Requires a large amount of volume |
| Requires the least mass | Has the largest variability in clearance after landing |
| The simplest design (testing, modeling and design time) | Less reliable than other designs due to variable nature of collapsing structures |
| Has a variable clearance after landing since amount of plastic deformation that will occur is unknown |
Design 3 - Shock Absorbing
| Advantages | Disadvantages |
|---|---|
| Requires the least volume as it is a compact design | Has more mass due to the requirement of working fluid as well as surrounding spring |
| Clearance below structure after landing is invariable and controllable | Fluid is susceptible to extreme temperature fluctuations |
| Will require the least design time as there are a large amount of resources on the subject available | Oscillation will occur in the spring till the system is fully dampened by the fluid. |
| Is the most reliable system |
Design 4 - Hydro-pneumatic & Plastically deforming
| Advantages | Disadvantages |
|---|---|
| Fairly Reliable design as there will be less | Complicated design requiring integration of both systems |
| No oscillations since there is no elastic absorption of force. | Will be very heavy due to the added weight of the working fluids. |
| Requires less material then most other designs as each system is specialized for certain ranges | Fluid is susceptible to extreme temperature fluctuations |
| Will provide the best energy absorption as each system supports the other | Variability in clearance after landing due to unknown amount of plastic deformation occurring |
Design Selection
Based on the analysis preformed, and after a review of the design constraints, the candidate design for an analysis is the angled static strut structural system with the plastic deformation structure energy absorption design. The combination of these two systems will provide the most stable, reliable and strongest support, while using the least volume and mass and providing the most energy absorption.
The primary strengths of the combined design will be its simplicity, stability and energy absorption ratio. The plastic deformation of the absorption material will allow large forces to be absorbed while reducing the force transfer to the lunar module by dissipating the kinetic energy of impact through the buckling of the honeycomb. The static strut design will allow for energy absorption system to be housed within the module reducing the overall volume of the craft and the angle of the struts will help to support a degree of horizontal loading. The combined design is simple with few moving parts increasing the probability of a successful landing which is important in the design. This design seems easy to integrate with the current conceptual design of the module’s structure without significant change. The weakness in the design will be in the increased base radius that the angled static struts will require leading to an increased volume that the landing gear will take up in the launch payload. The mass required for the primary and secondary struts will also be a weakness however integration of the landing gear into the structural aspects of the module may account for this.
Strucutural System
The structural system will consist of three primary angled struts extending in an equilateral manner from the landing module. These struts will contain a structural geometry or a material that will plastically deform to absorb impact load. Two secondary struts will cantilever with the primary legs in a similar fashion to Apollo 11 design. The angled struts will be fixed in a static fashion to the lunar module and extend outwards increasing the overall radius of the module.
Strengths
The strength of the angled static struts lies in its increase of system stability and resistance to variable loading conditions. The use of angled struts over vertical struts allows for support of some horizontal loading in the case of a slightly non uniform vertical landing. This means that if the landing module comes in at a slight angle, or lands on a slight incline the struts will still be able to support the landing force without failing. The angled aspect of the struts also helps to reduce rotation of the module upon landing by expanding the base radius. This design is fairly simple to model, and the lack of moving parts will reduce the probability of a part failing during operation of the landing gear which increases the probability of a successful mission.
Weaknesses
The primary weakness of an angled static strut design is the increase in the amount of space within the launch payload that the module will occupy. Without the ability to retract the struts increased base radius caused by the extruding struts will need to be accommodated. The angled strut design will also require more mass than the vertical strut design due to the need for secondary struts supporting vertical forces to compensate for the difference in vertical strength between the angled strut design and the vertical strut design.
Energy Absorption System
The energy absorption system will consist of sacrificial materials contained in each of the primary angled static struts that will plastically deform.
Strengths
The strength of the plastic deformation materials lies in the amount of force it can absorb relative to its weight. The impact force is absorbed through the landing pads and transferred to the primary strut where the impact materials absorb the kinetic energy as they buckle. This design will also not provide any oscillation as the process does not consist of any elastic damping of force and all the force is dispersed through buckling or crush of the materials. The structure is also fairly simple and does not consist of many moving parts meaning there is less probability that something will go wrong causing failure of the landing gear system which is important as reliability of the landing gear is a strong concern for the project.
Weaknesses
The weakness of the plastic deformation structure is the large amount of volume it takes up. The strut diameter will need to be larger to incorporate the absorption material within it and the length of the strut will need to be long enough to allow for buckling of the absorption material while still providing the clearance required of the lunar module. There may be variability in the clearance between the lunar module and the lunar surface with the use of this type of energy absorption system as the variability in the force of landing will affect the degree of buckling of the absorption material.
Design Refinement
Prior work justified three angled pod legs fixed in a static manner such that no deployment would be necessary for use. Each primary leg is cantilevered by two secondary struts (projecting outwards and upwards) in order to stabilize the primary strut against non-axial forces. In this regard, the selected impact design was refined into a lever design.
Impact Design
The impact design consists of the angled primary strut housing the energy absorption system below its connection to two rigid secondary struts. The primary strut mates with the landing pad cylinder so that the impact at touchdown accelerates the landing pad strut into the energy absorption system as in Figure 15. The joints for the impact design can be pinned or fixed. Apart from the energy absorption, the structure is rigid.
This design is capable of absorbing vertical forces. The energy absorption component is located bellow the secondary strut connection. Note that during touchdown, the landing pad will be accelerated at an angle whereby its final position is less extended; the landing pad will essentially drag inwards across the surface until the landing module has reached its rest position ultimately reducing the base diameter. This dynamic constraint will create bending where the landing pad cylinder mates with the landing structure cylinder as in Figure 16.
Further the combination of three pod legs with the impact design will not adequately address horizontal forces that may develop in non-ideal touchdown scenarios. These forces will induce further bending forces in the high stress location depicted in Figure 16. The final clearance of the impact design will depend upon the deflection of the energy absorption component easily resolved into a vertical component, and the compression of the lunar regolith.
Lever Design
Due to the shortcomings of the impact design it evolved into a new design. The lever design makes use of pin joints, universal, spherical or any other type imparting a larger range of motion to the landing structure, to allow the primary strut to act as a lever. The energy absorption system is contained within each secondary strut and used to restrict the lever motion of the primary strut as in Figure 17. With this in mind, the whole structure will be dynamic until it is rigid in its final rest position under load. The dynamic action of the design however will only be present during touchdown and wil not impinge on the ability of the design to be stowed in a constrained payload.
While any axial force developed on the primary strut will be transmitted to the landing module, this situation is unlikely due to the ability for the landing pad to slip outwards along the regolith extending the base diameter of the landing module. The lever design is better suited to accommodate both vertical and non-vertical forces that may develop during touchdown. Each leg regardless of it touchdown position relative to the acting forces will be able to dampen both horizontal and vertical force. This design makes the landing structure more versatile and capable of dealing with a variety of touchdown scenarios. However, its final rest position is variable depending on what forces develop during touchdown and the angle at which the secondary struts are connected; an incline at rest could develop due to non-symmetric crush of the energy absorption cartridges and the angle of the pod legs is then likely to change as in Figure 18, a top perspective.
The impact will increase the base diameter as the primary strut levers outward, increasing the final stability of the landing module. The sweep of the primary legs will be to the extent of the resolved deflection lengths of the crush material and will not be extreme so instability is not likely. An extreme tilt of the landing module is also not likely for the same reasons.
Design Analysis
Analysis was done into three main areas. The structural design analysis looked at determining optimal angles, lengths and the thickness of the primary and secondary struts. The footpad design analysis looked at determining the optimal shape and thickness of the footpad and determining the optimal attachment mechanism of the footpad to the primary strut. Energy absorption design analysis looked at removing the most acceleration from the system within the limits of the structural design. These sections had to be designed with the primary design constraints taken into account.
Structural Optimization
The matlab script used for optimization can be foudn in the appendices of the full report on the [Team Cheese Blog]
Note: The matlab scrip is configurable for a similar lever design and can accept different material property and constraint constants.
Optimization Setup
The first method discussed for optimizing the structural design was to create a program that optimized all the design variables on its own. This program would be designed in conjunction with the impact design outlined in the design selection section. The struts would be designed as a rigid body with known variables such as material properties input into the program, and inequality constraints defined to optimize the structure. This method however had some key problems. The structure would be statically indeterminate and would require a Finite Element Analysis program to determine the optimal design. Allowing the program to optimize the variables on its own would hide the cost of changing variables. One design could be optimal, but the cost of halving weight might be a minimal increase in angle above the set boundary and this cost would not be shown. The design is not feasible within the given constraints however, changes could be made to the design to allow this slight increase, but this case could not be analyzed since this information would not be outputted from the optimization.
With the design selection changed to the lever design, the method chosen for analysis of the structural design was also changed. Instead of creating a program to optimize the structure, a program was created to output graphs of key design variables. In this way trends between variables could be analyzed to come up with a favorable design. This new method would allow easy comparisons between variables, and allow slight changes to the design with large effects on highly valued weightings to be identified.
To begin the analysis, free body diagrams were created for the primary and secondary struts. Figure 19 shows the free body diagram for the primary strut with area 1 being the interface between the footpad and the primary strut, area 2 being the interface between the secondary strut and the primary strut and area 3 being the interface between the lunar module and the primary strut. Key forces were determined for each of the free body diagrams and statics methods were used to find the unknown forces. Firstly, the forces in the x and y directions were found and set equal to zero. These equations can be seen in Equation 6 2 and Equation 6 3. The second moments of area about the x-y axis for the primary and secondary struts were then determined using the equation for a hollow cylinder. The sum of the moments about point 1 were found and set equal to zero and this can be seen in Equation 6 1.
Using these force and moment balance equations, equations for the unknown design variables were built. These equations could be used to build trend graphs of variables. The equation for the force from the secondary struts (P) was built first; this can be seen in Equation 6 4. Next, the equations for the unknown reaction forces acting at point 3 in Figure 19 were built based on Equation 6 4. These can be seen in Equation 6 5 and Equation 6 6
Using these forces, the next task was to optimize the angle between the strut and the vertical at point 3. The angle at point 3 was chosen instead of at point 1 as it allowed for easier programming of the optimization loops. The angle between the secondary struts and the primary strut also had to be optimized along with the distance between point 2 and point 3. Along with this, optimization of the diameter of the hollow cylinders was found to minimize mass by maximizing stress within a safety factor below the yield strength.
To build the graphs, design constraints had to be defined. The first constraint was the minimum angle that the legs could be set to. If the angle was too small, the first problem is that the pads could not slip outwards meaning the energy absorption would not be used. This slip is a measure of the angle of the struts combined with the friction of the lunar regolith. The friction coefficient of lunar regolith μ was found to be 0.3 (22), and the normal force N acting up the struts is 4079N. Equation 6 7 is constructed from the force balance for friction at the foot pad, using this equation the minimum angle θ can be found by rearranging as seen in Equation 6 8.
The minimum angle therefore is 16.7 degrees. This angle however, is also important for another design constraint. A slight tilt in the landing module could be experienced if the landing is not perfect. If the legs are angled too vertically, this slight tilt could cause all the forces to be focused straight up a single strut and could cause failure. In this way the struts need to be angled enough that the tilt in the landing module would be too extreme a case to be considered. In this regard however 16.7o is a satisfactory angle.
The next design constraint was the maximum clearance between the lunar module and the lunar surface. This constraint in combination with angle and the maximum radial distance allowed for the struts would give us the length of the struts. The clearance was built up of a combination of factors. The first is the initial clearance constraint given by Team FREDNET of 30cm ± 30%. Secondly, the maximum penetration into the lunar regolith of 8 cm needs to be included, and finally the amount of crush in the energy absorption system has to be taken into account to ensure the initial clearance. Combining these factors the maximum clearance was found to be 70cm. The maximum radial distance from the central strut had to be calculated as well. Given the design constraints of the payload, and including the footpad diameter, the maximum distance the struts could extend was found to be 75cm. knowing this height, the maximum angle that the struts can extend to is 47 degrees.
Program operation
The optimization program is used to determine stress plots and using that weight, diameter, and angle plots for structural designs.
First the diameter variables are set for the primary and secondary struts. The length from the lunar module along the primary strut to the secondary strut interface is then defined as “a”. The level of accuracy for the loops is set. The offsets in the x and y direction for the pin locations on the lunar module for the secondary struts in relation to the primary struts are then determined. Once this is done, the material properties are defined such as the yield strength, shear strength, and density. At this point variables are defined to determine which plots will be outputted.
The max angle within the constraints is calculated and stress vs. theta for angles between 0 and max angle are found using a defined function. This function outputs normal stress, shear stress, length of the main strut, length of the secondary struts, the force P and the angle alpha. This function works by first defining the second moment of area for the primary strut as a hollow cylinder. Secondly, the length for each theta being iterated for (from 0 to the maximum angle) is calculated as the loop iterates. The alpha for this length and theta angle is calculated using the sin and cosine laws. Forces are determined using equations outlined in the structural analysis section above. Shear stress for the main strut is determined and then bending stress on the main strut is determined. The bending stress on the main strut is found by calculating the bending moment at a length along the primary strut (determined by the level of accuracy for loops defined earlier) and determining if this bending stress is larger than previous bending stresses encountered so far in the loop. Once the loop has finished, the maximum bending stress in the strut will be stored and this will be added to the maximum axial stress to give the maximum stress in the beam, which is one of the variables outputted back to the main program.
The main program takes all the variables outputted by the stress analysis function and organizes them. It then checks which plot was called for before the stress equations and plots it given the variables provided by the function.
Optimization Analysis
The first graph created was normal stress in the primary strut vs. theta which can be seen in Figure 20. This graph gives a clear indication of how the normal forces in the beam (not including shear) change as the legs are angled outwards for various diameters of the primary strut. The stress increases in the strut as the angle increases. This means that according to the stress distribution, a minimum angle would be preferred. It can also be seen that as diameter increases, stress decreases due to the fact that as the diameter increases and the bending moment experienced by the strut decreases.
In Figure 21 the shear stresses experienced in the primary strut with respect to theta can be seen. As theta increases the shear stresses can be seen to increase. This is because as the angle increases, the components of force in directions not used for bending moment, turns into shear stress. The shear stress chart suggests that a minimum angle would be optimal in order to reduce the shear stress and thus reduce the diameter required for the strut.
Strut Thicknesses and Diameters
In Figure 22 the minimum main strut outside diameter for given inside diameters vs. theta is shown. This plot will help to find the minimum wall thickness acceptable for the primary struts. The resolution for these plots is not as good as other plots due to problems in computing power. For higher resolution plots the iteration time increased exponentially due to the nature of the loops. For further refinement of the plots a more precise method for iterating the diameters would be required, and better computational power would be needed.
Figure 23 is another representation of Figure 22. Figure 23 shows double the wall thickness of the primary strut vs. theta for various inside diameters. This plot further shows the compromises in wall thickness when increasing inside diameter to decrease weight. Through analysis of Figure 23 and Figure 22 an inside diameter of 3cm was chosen which would give a wall thickness of 2.5mm.
Figure 24 shows the total weight of the structure given various angles of theta for various inside diameters of the primary strut. As the inside diameter of the primary strut increases the total weight decreases as expected however the wall thickness also decreases. The total weight of the structure does decrease as the inside diameter increases however the percentage change decreases as the diameter increases. This means that the gain in total weight reduced becomes smaller and smaller for increased diameters.
Secondary Strut Interface
Figure 25 shows the total weight of the structure for varying lengths of a based on the angle of theta. As shown the total weight decreases as the length of “a” increases, however as in Figure 24 the percentage decrease in total weight decreases as a increases.
Figure 26 shows wall thicknesses for the secondary struts for varying lengths of “a” vs. theta. As the length of “a” increases the wall thickness for varying degrees of theta decreases. The inside diameter of the secondary struts was found to be 1.905cm to allow storage of the aluminum foam, and using this figure the outside diameter was found to be 2.35cm.
Figure 27 shows a plot of the length of the secondary struts vs. theta for various lengths of “a”. Combining analysis of this figure with Figure 25 and Figure 26 an optimum value of “a” of 0.4m was determined. This value of a sacrifices a low percentage change of total weight reduction while ensuring a manageable wall thickness and providing enough length for energy absorption systems to be incorporated into the secondary strut.
Figure 28 shows a plot of alpha vs. theta for varying degrees of theta. This simply organizes the data given in previous plots to provide a better understanding of the relationships between the two angles.
Considering weight is the primary design variable for the structural design we can see that the optimal angle for theta is our minimum angle shown by Figure 24 and Figure 25. Since the minimum angle allowing for frictional forces to spread the pads outwards radially is 17 degrees, giving this and a safety factor the optimum angle was determined to be 20 degrees. For this angle the angle of alpha is 50o given by Figure 28. The minimum inside diameter is 3cm from Figure 22, Figure 23 and Figure 24.
Energy Absorption
An internal cylindrical column of aluminum foam was selected to absorb the impact energy during touchdown as its isotropic properties simplify its integration in the design, and it would not store the energy for release as a spring would. The preferable landing was assumed to have no bounce. The properties of the crush cartridge to be housed in the secondary or primary struts depend on the relative density of the structure and the material selected according to manufacturer specifications. With an optimal structural design, the energy absorption system will be integrated in order to minimize the acceleration experienced on the landing module within the design constraints as best as possible. It is important to note that the compression of the lunar regolith will act in unison with the energy absorption system contained in the landing gear.
The following assumptions validate the energy analysis performed to obtain a final acceleration value developed by the landing module.
- Energy is conserved during touchdown.
- This allows the system to be simply modeled as a spring system. The potential energy of the landing module transfers completely to kinetic energy at impact. It is assumed that there are no heat or sound losses. Note that the final energy of the system will be negative as the landing module is assumed to come to rest below the datum based on the deflection of the landing volume. The energy is stored but no longer usable within the system after the regolith and the crush are plastically deformed.
- The structural components of the pod legs are rigid.
- The structural elements of the landing gear will not buckle, bend or shear. All the energy absorption occurs within the crush and the regolith. This assumption is later justified by designing the structural elements to withstand bending and shear stresses appropriately. Further, the landing pads do not bend, and transmit all the landing force to the leg.
- The thrusters shut off at a predetermined free fall height. The velocity of the landing module at this height is zero.
- This allows for a simple energy balance to be preformed.
- During touchdown, the upright orientation of the landing module is normal to the horizontal surface of the moon. Further, there are no moments acting on the landing module at touchdown.
- This ensures that all pod legs impact at the same time. The spring systems corresponding to each leg thus act in unison and the landing module has no pitch, yaw, or roll that might increase the force experienced in individual pod legs in a non-symmetric manner. Realistically, the landing may have a tilt to the horizon. Appropriate safety factors in the design can be assumed to account for minor deviation.
- The lunar impact of the landing pads generate only compressive forces on the lunar regolith. Further, there are no horizontal components of the landing module’s impact velocity.
- In order to model the touchdown mechanism as a simple spring system, only regolith compression will be assumed. Horizontal components of landing velocity will introduce regolith shear during touchdown and exhibit unaccounted bending and shear forces in the structural design. Optimizing the design variables in a MatLab script can account for the bending and shear stresses developed within the structure however these forces will depend on the effective stiffness of the regolith which will overlook the complex analysis necessary to determine the surface properties if shear is present. Both structural designs will inherently introduce shear as they will either drag the landing pad outward (lever design) or inward (impact design), but this effect is neglected in the analysis.
- During the compression that occurs, the properties of lunar regolith are constant.
- The density of surface lunar regolith will likely be smaller than the density of regolith at depth. This will vary the effective stiffness of regolith as it deflects under compression. An appropriate value representative of a conservative estimate is justified for a holistic approach to the model.
- During compression, the touchdown contact area is constant.
- This models the lunar soil as a column of material that effectively acts like a spring based on its load bearing strength. This also means that the Landing volume material will exhibit the same effective stiffness throughout compression.
- The structural composition of the lunar soil at the impact location is pure regolith for the total deflection realized by compression.
- For the simplified spring system to be justified, the effective stiffness of the regolith cannot be subject to variations in composition of the lunar regolith. This analysis assumes an appropriate landing site is located in order to reduce the risk of landing on a purely rigid surface. In reality, a variation of composition will exist but appropriate safety factors may sufficiently account for these variations.
- The landing module touches down in a linearly inelastic manner.
- In reality the landing module will likely bounce because the composition of the landing volume will not entirely consist of lunar regolith. The rigidity of the landing structure or the landing volume, beyond what lee way the crush and load bearing properties of the lunar regolith provide, may also induce reverberations that create bounce.
- The crush cartridge is completely plastically deforms.
- It is assumed that the crush force develops on all the crush cartridges. All energy in the process is converted to work.
System Energy
A maximum acceleration was determined in two cases by modeling the energy of a rigid landing structure touching down on pure regolith and an energy absorbing landing structure touching down on pure regolith. The system energy components, the lunar regolith and the energy absorption mechanism represent the main components of simple energy balance.
Compression of Lunar Regolith
Subjected to pure compression during impact, the landing area is assumed to be constant which allows the compression of the landing volume to be modeled with effective spring stiffness, as in Figure 29 based off of the compressibility of lunar regolith.
The effective stiffness (k1) of the landing volume depends on the load bearing strength (β) in terms of regolith compressibility [Pa/m] and the total landing area (Aimpact) provided by the landing pads.
Equation 6 9 k_1=βA_impact
Crush Plastic deformation
The total work done by the plastic deformation of all crush elements represents absorption of the touchdown kinetic energy.
Equation 6 10 E_crush=F_crush δ×6
Ecrush is the total energy absorbed by all six crush cartridges housed in the secondary struts. This energy is removed from the system. Fcrush is the force necessary to induce plastic deformation and acts axially to the secondary strut. This force acts over the distance δ, the total deflection of the crush cartridge.
Process
The crush force is determined by evaluating the crush strength (σcrush) of the crush cartridge. This is based off of the relation in Equation 6 11 (11).
Equation 6 11 (11) σ_crush=0.58σ_solid*ρ_rel^(3⁄2)
The crush strength relation is an approximate estimate provided by ERG Aerospace using the yield strength of the parent material (σsolid) and the relative density (ρrel) the foam is manufactured to. The achievable manufacturing range of relative density varies between 4 % and 10%. (11)
Equation 6 12 F_crush=σ_crush*A_crush
The crush force (Fcrush) is obtained in relation to the crush strength and the the cross-sectional area of the crush cartridge (Acrush). This area is limited by internal dimensions of the secondary struts.
Assume the touchdown impact develops the necessary crush force.
Assume complete plastic deformation as in Figure 30 (densification of the crush cartridge until ρrel is 1).
To determine the deflection (δ) of the crush cartridge, the total amount of solid mass contained in the foam (mcrush) is found.
Equation 6 13 m_crush=A_crush *l*ρ_solid *ρ_rel
The initial length of the crush cartridge (l) is constrained by the dimensions of the secondary struts. The density of the parent material (ρsolid) is weighted by the relative density of the foam structure.
A conservation of mass is applied in Equation 6 14 to solve for the deflection of the crush in Equation 6 15. χ refers to the length of the densified crush material.
Equation 6 14 x=m_crush/(A_x ρ_solid ρ_(rel=1) )
Equation 6 15 δ=l-x
A conservation of energy in Equation 6 16 is applied to solve for deflection of the lunar regolith (Δ1) in Equation 6 17 and thus the maximum vertical force in Equation 6 18. Using the maximum force, the maximum acceleration can be calculated with Equation 6 19. In the case of the rigid landing structure, the energy term of the crush cartridge representing energy removal from the system due to plastic deformation is neglected.
The mass of the landing module (m) is constrained as is the free fall height (h) by operating conditions. gmoon refers to the gravitational force of the moon. It is assumed that the residual velocity of the landing module at the free fall height (v0) is zero. The maximum vertical force component (Fmax) acting on the landing module is determined. Fmax is resolved into its component acting axially along each secondary strut. This force must be greater than the crush force to validate the assumed induced plastic deformation.
If this assumption is not validated, the cross sectional area of the crush cartridge and the relative density of its material can be reduced, the landing pad area can be increased within its bounding conditions, or the operating free fall height can be increased to obtain a lesser crush force or a greater impact force.
Results
The excel sheet used to calculate the energy and force equations is found in the report on the [Team Cheese Blog]
Using the regolith compressibility data from the Apollo mission and within the design constraints, the energy analysis proves that the acceleration developed on the landing module can be limited below a 10G static load. The landing pad diameter is constrained to the space where thrusters do not fire. An appropriate pad diameter of 0.3 meters was selected as the resulting impact area minimized the extent to which the landing module would sink into the regolith while developing a reasonably low impact force. The secondary struts are able to house crush cartridges 0.12 m long with cross-sectional diameters of 0.01905 meters. Table 11 compares the accelerations developed by the rigid landing structure and the energy absorbing landing structure during touchdown on pure lunar regolith. All the values fall under the 10 G acceleration constraint, while the energy absorbing structure reduces sink into regolith and the acceleration beyond the results of the rigid structure.
There is a large error associated with the energy analysis. This is a result of the broad mission variables; mass and the free fall height have associated errors of 30%. This error is further inflated in error propagation regarding an integrated energy absorbing component in the structural system. As seen in Table 12, by reducing the uncertainty on the operation variables of only mass and free fall height the percentage error is drastically reduced.
The large error on the force value creates difficulty in verifying that the axial crush forces of the crush cartridges have been reached. It becomes evident that as design progresses concerning the touchdown characteristics of the landing module, more accurate accelerations and force transfer values can be deduced. It still remains the case that constraint breaching accelerations will not likely develop, provided the regolith has similar properties to the data that the Apollo missions suggest.
Notes
The analysis presented employs checking the assumption that the structurally constrained crush cartridge would completely plastically deform reducing the energy of the system as best it could. This approach is justified as a completely rigid landing structure can be shown to develop less than 10G’s of acceleration on the landing module if touchdown occurs on pure regolith with properties similar to those suggested by data from the Apollo missions. More accurate mission constraints will have to be put forward for more accurate acceleration and force values. This will also allow confirmation of the critical assumption that the cartridge crush force has been reached to allow plastic deformation. These are all issues that can be investigated during prototyping and testing.
Landing Pad
The final diameter of the footpad was determined to be 0.3m and the resulting area was 0.071m2. This area produced a force of around 12.25 KN experienced by the landing module and a worst care resultant acceleration of 7 G’s, which is below the 10G threshold.
Using deflection techniques and assuming a maximum deflection for the pads, we were able to find a thickness that provided rigidity but also minimized material. Specifically, using methods of superposition, the moment of inertia was found and the resulting thickness could be calculated.
Using this analysis, the resulting thickness was found to be 1.78 cm. This was found making reasonable assumptions and taking into account the worst case scenario of landing.
The thickness found was a high estimate. It did not take into account that the landing pads were circular. However, it proves that a beam with a cross sectional area that takes into account the obstruction contact area will be rigid. The amount of deflection in a circular pad would be much less than seen here and would reduce the thickness needed for each landing pad. Further mechanics analysis and testing will be needed to find the minimum thickness needed for the pads based on prototyping.
Results
Figure 36 below shows an approximate visual representation of the lunar landing gear assembly. The detailed part dimensions are shown below in Table 13.
The structural system was analyzed to determine optimal angles, thicknesses and lengths. Through the use of Matlab optimization, along with graphical analysis, key relationships between design variables were examined. The final inside diameter of the primary strut was found to be 3cm, and the final outside diameter of the primary strut was found to be 3.5cm. The angle between the primary strut and the vertical (theta) was found to be 20o, and the angle between the secondary struts and the primary strut (alpha) was found to be 50O. The length between the primary strut interface to the lunar module, and the secondary strut interface to the primary strut (a) was found to be 0.4mm. The inside diameter of the secondary strut was found to be 1.905cm and the outside diameter was found to be 2.35cm.
With a rigid structure and a total pad area that fit within dimension constraints, the threshold acceleration was not reached. Further reduction to the acceleration figure was achieved with a crush cartridge with a relative density of 0.04, a length of 0.12 m and a cross sectional diameter of 0.01905 m. The complete deformation of the crush cartridges remove 265 joules of energy from the system, though high error made it impossible to check whether the crush strength had been reached. An optimal area was found to be 0.071m2 for each foot pad. This reduced penetration and reduced the acceleration experienced by the landing module. A maximum thickness of 1.78cm was found for each pad ensuring they would be rigid. A ball joint was used to connect each pad with the struts to ensure force was directed upwards and so different pad orientations could result without failure.
Recommendations
The project outlines a scalable methodology to determine the structural dimensions of the landing module and key dimensions of the energy absorption system. Future work would ideally break the system down into smaller parts such as the design of the landing pad, the design of the spherical connection joints, and the method of breaking the structure into parts that can be manufactured. The primary strut could further be designed to house a separate energy absorbing mechanism to reduce the risk axial force transmission. This would further reduce the acceleration developed if the holistic energy absorbing system is designed to plastically deform. The design will also have to integrate a feasible method of bolting or connecting to the landing module. If manufactured parts that are within specifications a readily available, a top-down design approach may be worthwhile to justify lower manufacturing costs. However, the unique operating conditions impose limits that may not be possible to work around. Ultimately this project work provides a foundation from which to further design efforts.
References
Until the information is properly referenced on the wiki, please refer to the bibilography of the report on the Team Cheese Blog crossreferenced with the numerical entry on the page.
