10 01-01 team cheese-final report_lunar landing gear

MECH 460 | QUEEN’S UNIVERSITY, KINGSTON ONTARIO, CANADA

Lunar Landing Gear Group 27 | Team Cheese

Nabeil Alazzam – [email protected]

Ben Banks – [email protected]

James Burke – [email protected]

Ian Cameron – [email protected]

As part of a student project with Queen’s University, this report contains a documented approach to the

conceptual design of a system to dampen the shock impulse of an unmanned vehicle landing on the

surface of the moon. The project has been adapted from an open task provided by Team FREDNET, a

registered competitor for the Google Lunar X Prize.

http://teamcheese.com

12/7/2009

http://teamcheese.com/

Lunar Landing Gear

Group 27 | Team Cheese

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“Ask not what open source can do for you,

but what you can do for open source.”

MECH 460 – Fall 2009


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Special Thanks

Dr. Ronald Anderson

Ryan Weed

Chau Ngai

Sean Casey

Team FREDNET

Dr. Kevin Deluzio

Professor Alan Ableson

Andy Bryson

Eric Benson

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EXECUTIVE SUMMARY The Google Lunar X PRIZE is a thirty million dollar international competition to safely land a robot on

the surface of the Moon, which will travel five hundred meters over the lunar surface and send images

and data back to Earth.

Team Cheese is a group of four mechanical engineering undergraduates at Queen’s University. They

have worked with Team FREDNET, which is a registered official team competing for the Google Lunar X

Prize, over the semester. Team FREDNET is a special competitor as it consists of an international group

of developers, engineers, scientists and other members dedicated to the open source philosophy.

Previous missions to the moon and to mars as well as various energy absorption systems were

investigated which helped to design and gain knowledge on landing mechanisms. Design constraints

such as weight restrictions, absorption methods, operation, reliability and cost were identified. These

constraints helped each analysis on force absorption, strut design, and foot pad design.

There are two design elements that TC analyzed; the energy absorption system and the structural

support system.

The energy absorption system chosen incorporated plastically deforming material. The energy

absorption system would be contained in the secondary struts and these struts would be able to house

crush cartridges .12 m long with cross-sectional diameters of .01905 m. All the acceleration values fall

under the 10 G acceleration constraint, while the energy absorbing structure reduces the penetration

into regolith and the acceleration beyond the results of obtained for a rigid structure if touchdown

occurs on pure regolith, with properties similar to those suggested by data from the Apollo missions.

More accurate mission constraints would have to be put forward for more accurate accelerations

and force values. The error propagated though with unrealistic amounts due to suggested uncertainties

of up to 30%. As design work progresses, the theoretical modeling of the touchdown scenario will

improve. These are all issues that can be investigated during prototyping and testing.



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Through the use of Matlab optimization, along with graphical analysis, key relationships between

design variables were examined. Key relationships were evaluated to determine dimensions. 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 3.25cm. The angle between the primary strut and the vertical (theta) was found to

be 20o, and the angle between the secondary strut 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.

The structural system also included foot pads for each strut. They needed to provide stability and

support for each leg while making contact. The foot pads contact area helped reduce the force

transferred to the legs and minimized the penetration into the regolith. These two ensured the

acceleration felt by the landing module was minimal.

The physical areas analyzed included the overall shape including the joint, the area, and the

thickness of the pads. The overall shape incorporated a cup shaped pad with a ball joint welded to the

bottom. To reduce weight, both the ball joint and pad were designed to be made of titanium. The area

was determined to be 0.071m2 which resulted in a penetration of 8cm and a force of 14KN felt by the

landing module. An acceleration of 7 G’s would be experienced by the landing module which was much

less than the required 10G Team FREDNET restriction. Using mechanics of materials knowledge, and

applying methods of superposition, the thickness of each pad was found to be 1.76cm. This was found

to be the maximum the thickness had to be.

With the help of Team FREDNET, Professor Anderson, other faculty members, strong design analysis

processes, and available resources, Team Cheese was able to design a preliminary conceptual lunar

landing gear system. The design incorporated key aspects of energy absorption, structural design, and

foot pad design. A landing gear system was developed to the design constraints proposed by Team

FREDNET. Further analysis and prototyping is needed to finalize the design, make it ready for

manufacturing, and finally incorporate it into Team FREDNET’s final Google X Prize landing module

design.

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Table of Contents

EXECUTIVE SUMMARY................................................................................................................................. iii

LIST OF FIGURES......................................................................................................................................... ix

1 Introduction .................................................................................................................................. 1

1.1 Google Lunar X Prize ................................................................................................................. 1

1.2 Team FREDNET .......................................................................................................................... 1

1.3 Team Cheese ............................................................................................................................. 1

2 Project Description ........................................................................................................................ 2

2.1 Constraints ................................................................................................................................ 2

2.1.1 Successful Design .............................................................................................................. 2

2.2 Scope ......................................................................................................................................... 4

3 Research ........................................................................................................................................ 4

3.1 Background ............................................................................................................................... 4

3.2 Energy Absorption Techniques ................................................................................................. 4

3.3 Past Missions ............................................................................................................................. 8

3.3.1 Apollo ................................................................................................................................ 8



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3.3.2 Surveyor .......................................................................................................................... 13

3.3.3 Mars ................................................................................................................................ 14

3.4 Lunar Conditions ..................................................................................................................... 15

4 Design .......................................................................................................................................... 16

4.1 Evaluative criteria ................................................................................................................... 16

4.2 Design Methodology ............................................................................................................... 18

4.2.1 Structural System Design Candidates ............................................................................. 19

4.2.2 Energy absorption Design Candidates ............................................................................ 23

4.3 Design Selection ...................................................................................................................... 26

4.3.1 Structural System ............................................................................................................ 26

4.3.2 Energy Absorption System .............................................................................................. 27

4.3.3 Strengths ......................................................................................................................... 27

4.3.4 Weaknesses..................................................................................................................... 28

4.3.5 Combined Design ............................................................................................................ 28

5 Design Refinement ...................................................................................................................... 28

5.1 Impact design .......................................................................................................................... 29

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5.2 Lever Design ............................................................................................................................ 30

6 Design Analysis ............................................................................................................................ 32

6.1 Structural System .................................................................................................................... 32

6.1.1 Optimization design and setup ....................................................................................... 33

6.1.2 Optimization analysis ...................................................................................................... 36

6.1.3 Determining optimal wall thickness and diameters ....................................................... 38

6.1.4 Determining optimal position for secondary strut interface .......................................... 41

6.1.5 Optimization Program ..................................................................................................... 45

6.2 Energy Absorption................................................................................................................... 46

6.2.1 System Energy Modeling................................................................................................. 49

6.3 Landing Pad Design ................................................................................................................. 55

6.3.1 Landing Pad Shape .......................................................................................................... 55

6.3.2 Area of landing pads ....................................................................................................... 56

6.3.3 Thickness of Pads ............................................................................................................ 56

6.4 Ball Joint .................................................................................................................................. 58

6.5 Pad Connection ....................................................................................................................... 60



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6.5.1 Other Considerations ...................................................................................................... 60

6.6 Thermal Considerations .......................................................................................................... 61

7 Design Results ............................................................................................................................. 63

8 Prototyping ................................................................................................................................. 65

8.1 Materials ................................................................................................................................. 65

8.2 Manufacturing......................................................................................................................... 66

8.3 Cost ......................................................................................................................................... 66

9 Recommendations ...................................................................................................................... 66

10 Conclusion ................................................................................................................................... 67

APPENDIX A ............................................................................................................................................. 74

APPENDIX B ............................................................................................................................................. 75

APPENDIX C ............................................................................................................................................. 84

APPENDIX D ............................................................................................................................................. 86

APPENDIX E ............................................................................................................................................. 89

APPENDIX F ............................................................................................................................................. 91

APPENDIX G ............................................................................................................................................. 96

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LIST OF FIGURES Figure 1 - Opened extrusion profile with APM aluminum foam–polymer hybrid filling(12) ................. 6

Figure 2 - Open cell structure with regular geometry (13) .................................................................... 6

Figure 3: Piston connected to suspension strut pushes the hydraulic fluid which compresses the

Nitrogen. ....................................................................................................................................................... 7

Figure 4: Stowed and deployed positions of landing gear for the Apollo 11 landing module(15) ........ 9

Figure 5: Detailed design of landing gear for Apollo 11 lunar landing module(15) ............................. 10

Figure 6: Landing-gear primary strut honeycomb structure.(15) ........................................................ 11

Figure 7 - Landing-gear secondary strut honeycomb structure.(15) ................................................... 11

Figure 8 - Secondary Strut depicting compression and tension stroke(16) ......................................... 12

Figure 9 - Photograph of Aluminum honeycomb shock-absorbing cartridges from full-scale landing-

gear struts before and after impact crush(16) ........................................................................................... 12

Figure 10 - Surveyor 1 Landing Gear Dimensions (18) ......................................................................... 13

Figure 11 - Operation of Surveyor Landing Gear a) During Descent b) Initial Contact and Compressed

Shock Absorber c) Rest Position (18) .......................................................................................................... 14

Figure 12 – Early Structural Lunar Landing Module Concept (3) ......................................................... 18

Figure 13 –QFD Chart for Structural Strut Systems .............................................................................. 23

Figure 14 - QFD Chart for Energy absorption Systems ........................................................................ 26

Figure 15 - Impact Design ..................................................................................................................... 29

Figure 16 - High stress location, impact design .................................................................................... 30

Figure 17 - Lever design ........................................................................................................................ 31

Figure 18 - Variability in pod leg position during touchdown .............................................................. 32

Figure 19: Free body diagram for primary strut ................................................................................... 34



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Figure 20: Normal Stress vs. Theta for Primary Strut ........................................................................... 37

Figure 21: Shear Stress vs. Theta for Primary Strut .............................................................................. 38

Figure 22: Outside Diameter vs. Theta for Primary Strut ..................................................................... 39

Figure 23: Thickness vs. theta for primary strut ................................................................................... 40

Figure 24: Weight vs. Theta for varying inside diameters .................................................................... 41

Figure 25: Weight vs. theta for varying lengths of a. ........................................................................... 42

Figure 26: Wall thickness of secondary struts for varying lengths of a. ............................................... 43

Figure 27: Plot of 3D Secondary strut length vs. theta ........................................................................ 44

Figure 28: Plot of alpha vs. theta .......................................................................................................... 45

Figure 29 - Column of regolith per strut pad modeled with a spring stiffness .................................... 49

Figure 30 - Complete plastic deformation and densification of crush cartridge ................................. 51

Figure 31 - Simplified cross sectional view of half a lunar pad ............................................................ 57

Figure 32 - Cross-sectional area of foot pad ........................................................................................ 58

Figure 33 - steering link ball joint (Ningbo Pair Industrial Co.) ............................................................. 59

Figure 34 - Final landing pad design ..................................................................................................... 61

Figure 35 – Conceptual thermally isolated spherical joint ................................................................... 62

Figure 36 - Three dimensional rendering of entire lunar landing gear assembly (note: dimensions are

not to scale, this rendering is only for an approximate visual representation) ......................................... 63

Figure 37 – Three dimensional rendering of entire landing assembly (note: dimensions are not to

scale, this rendering is only for an approximate visual representation) .................................................... 65

Figure 38 - Rigid Landing Structure Energy Analysis ............................................................................ 84

Figure 39 - Energy Analysis with an Absorbing Structure .................................................................... 85

Figure 40 - Mass conservation of crush cartridge ................................................................................ 85

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Figure 41 - Simplified Foot Pad ............................................................................................................. 86

Figure 42 - Cross sectional Area of Foot pad ........................................................................................ 86

Figure 43 - Simplified model of landing mechanism ............................................................................ 91

Figure 44 - Spring system modeled at impact ...................................................................................... 92

Figure 45 - Inclined crush spring model ............................................................................................... 93

Figure 46 - Vertical crush equivalent spring system ............................................................................. 93

Figure 47 - Combined vertical spring system ....................................................................................... 94

Figure 48 - Equivalent spring system .................................................................................................... 94



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LIST OF TABLES

Table 1 - Lunar Regolith Properties (20) ............................................................................................... 15

Table 2 – Evaluative Design Criteria ..................................................................................................... 16

Table 3 – Advantages and disadvantages of vertical static strut structural system design ................. 20

Table 4 - Advantages and disadvantages of vertical dynamic strut structural system design ............. 20

Table 5 - Advantages and disadvantages of angled static strut structural system design ................... 21

Table 6 - Advantages and disadvantages of angled dynamic strut structural system design .............. 22

Table 7 – Advantages and disadvantages of hydro-pneumatic suspension system ............................ 23

Table 8 - Advantages and disadvantages of plastic deformation structure system ............................ 24

Table 9 - Advantages and disadvantages of shock absorber system ................................................... 25

Table 10 - Advantages and disadvantages of hydro-pneumatic suspension in combination with

plastic deformation structure system ......................................................................................................... 25

Table 11 - Landing Characteristics of Lunar Module ............................................................................ 53

Table 12 - Effect of mass and free fall height error for a system with energy absorption .................. 53

Table 13 - Summary of structural dimensions ..................................................................................... 64

Table 14 - Operation Variables ............................................................................................................. 74

Table 15 - Design Variables .................................................................................................................. 74

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1 Introduction MECH 460 is one of three possible capstone project courses for Mechanical and Materials

Engineering students at Queen’s University. The course aims to prepare graduating students for the

engineering workforce by working on real-world team design problems as proposed by industry based

clients. Students will participate as part of an engineering design team to complete a significant

engineering design project while working in a professional manner within the constraints of limited time

and budgets. (1)

1.1 Google Lunar X Prize

The Google Lunar X PRIZE (GLXP) is a thirty million dollar international competition to safely land a

robot on the surface of the Moon, which will travel five hundred meters over the lunar surface and send

images and data back to the Earth. (2)

1.2 Team FREDNET

Team FREDNET (TF(X)) is a registered official team competing for (GLXP). TF(X) is a special

competitor as it consists of an international group of developers, engineers, scientists and other

members dedicated to the open source philosophy. Their goal, besides winning the GLXP, is to bring the

same successful approach used in developing major software systems to bear on the problems

associated with space exploration and research. (3)

1.3 Team Cheese

Team Cheese (TC) is a group of four mechanical engineering undergraduates at Queen’s University

taking part in the MECH 460 Project Course. TC is cooperating with TF(X)’s open effort to design and

build the systems necessary to land a functional rover on the moon. TC has adapted the task of

designing a conceptual landing system that will absorb the impact loading on an unmanned lunar

landing craft during touchdown.



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2 Project Description The landing gear is a crucial structural component of a lunar landing craft. The system has to absorb

the kinetic energy associated with touchdown in a controlled manner and support the static load of the

landing module in an upright position allowing the deployment of a rover. TF(X) is seeking conceptual

designs to address these issues. (4) TC is pursuing the design of a structural system that will absorb the

energy associated with landing and provide the correct orientation to the landing craft on the lunar

surface.

2.1 Constraints

The nature of the design task will impart constraints on a proposed system. A conceptual design will

have to account for:

• The absorption of an expected impact force and the support and upright orientation of the

landing module within an appropriate safety factor.

• Forces to be reacted against post-landing (i.e. rover deployment, robotic component

operation).

• Appropriate and safe stowing into a launch payload.

• Reliable activation or deployment while in lunar orbit prior to landing if stowing is only

possible with the system in an inactive or stowed configuration.

• Appropriate operation of the system while not conflicting with the function of other landing

module subsystems.

• The expected horizontal and vertical motion and orientation of the landing module during

touchdown.

• Conditions and terrain of the lunar surface.

2.1.1 Successful Design

Considering the constraints, assumptions were implemented to direct the successful design of a

landing system.

1. The static launch payload is cylindrical in geometry (2 meter diameter and 3.3 meter in

height).

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• The landing module will occupy 70% to 75% of this volume.

2. 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.

3. The impact force developed must be less than a 10G static load.

• The landing structure must withstand and limit the force of landing.

4. 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.

5. 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.

6. 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.

7. 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 a consequence of the mission specifications, many components of the landing system can be

sacrificial; the landing module comes to its final rest position after touchdown. Successful performance

of the landing system is geared towards rover deployment and landing an undamaged base station in a

position of rest.



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2.2 Scope

The scope of this project consists of conceiving and justifying a lunar module landing system that will

function within the constraints described. Aspects of the landing gear such as storage, deployment and

the process and mechanics of the absorption of forces created upon impact on the moon will be

analyzed.

3 Research

TC has had assistance from the helpful library staff at Queen’s University. Initial research snowballed

off of the prior work of the TF(X) community documented by Anders Feders in the TF(X) Wiki. TC has

studied the background components or systems of proposed landing mechanisms and several energy

absorption techniques to complement work performed in previous missions. Lunar conditions were

investigated to determine their effect on the conceptual design.

3.1 Background

For the proposed SELENE project, the Japanese Aerospace Exploration Agency will be landing

unmanned probes onto the lunar surface. Simulation of the landing of a 520 kg module with four legs

from a freefall height (engine shutdown) of 3 m at lunar gravity predicted a peak acceleration of 293.3

m/s2 for a maximum impact force of 38.3 kN acting on each leg.(5)

Research into an optimal pod leg configuration suggested that a three-legged structure only

required a slightly larger landing gear diameter than a four-legged structure for a given probability of

stable landing. (6) The smaller leg configuration would better accommodate the mass constraint of the

design.

3.2 Energy Absorption Techniques

An energy absorption system can dissipate the kinetic energy of impact load by employing a damped

elastic structure (i.e. a piston structure), a controlled fluid damper (i.e. an airbag or restricted fluid flow

chamber), electromagnetic magnetic resistance on moving components (i.e. resistive magnetic flux) or

the property of certain materials or structures to plastically deform (i.e. crushing certain materials or

buckling structural components). A combination of several of these techniques may synergize the

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energy absorption capabilities of a whole system. A benefit within the scope of the mission is that the

energy absorption system may be sacrificial; the landing module will not move after arriving at its final

resting location.

The structural benefit of aluminum honeycomb or foam is observed as a reduction in density with

restricted loss of energy absorption capabilities compared to a solid structural member. An increase in

energy absorption is observed after filling a structural element with aluminum foam.(7) The yield stress

of aluminum honeycomb is dependent on the direction of loading. (8) (9) Under dynamic compressive

loading, the escape of trapped air within the structure is suspected for strain-rate dependence. (9) In

design applications, the direction of compression may be controlled within the design. A combined

orientation of honeycomb cells may benefit the overall structure and control the manner of material

buckling.

Concerning the axial crushing of honeycomb material structures, a dynamic (as opposed to quasi-

static) loading increases the crash strength or force transmission by up to 74%. The preservation of

energy absorption capabilities is achieved by using a smaller cell volume. (10) Alternative materials may

provide greater strength also.

Structural foam can be used to remove energy from a system. Work can be done on the structural

foam in terms of the crush force and the distance by which the structure plastically deforms.(11)

Material bonding to the containing structure upon injection is a design concern regarding the use of

structurally internal aluminum foam if it is designed to fail. Closed cell polymer composite aluminum

foam (Figure 1) will not suffer the same shortcoming and will also fill whatever the shape the cells are

poured into while providing high yield strength. (12)



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Figure 1 - Opened extrusion profile with APM aluminum foam–polymer hybrid filling(12)

Open cell structures (Figure 2) employ manufactured imparted geometry to reduce the mass of the

structure and increase the crushability of the substance. The deformation of these structures exhibit

brittle behaviours and the relative stiffness of the structure is dependent on its size. (13)

Figure 2 - Open cell structure with regular geometry (13)

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A proprietary hydro-pneumatic damping system uses nitrogen as a resistive medium in a suspension

system. The nitrogen is approximately six times more flexible than conventional steel, making it very

elastic.

A pump pressurizes the hydraulic fluid. This work can also be used in another system. Since the

hydraulic fluid is pressurized by the car, this system features ride height control. The pressure sinks in

this system consist of one sphere per wheel. These spheres are hollow metal balls that have a hole on

the bottom, with a rubber membrane (usually a flexible desmopan) filling the opening. The top of the

sphere is filled with pressurized nitrogen (up to 75 bar), while the bottom of the sphere is connected to

the hydraulic fluid system of the car (up to 180 bar). This can be seen in Figure 3 below.

Figure 3: Piston connected to suspension strut pushes the hydraulic fluid which compresses the Nitrogen.

The suspension system works by have a rod from the wheel push the hydraulic fluid into the sphere

compressing the nitrogen. The hydraulic fluid flows through a two way leaf valve in and out of the

sphere dampening the system. This type of dampening system is one of the most efficient and simplest.

The hydraulic fluid that is used by Citroën is a mineral oil, similar to automatic transmission fluid.



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Mineral oil is used because it is not hygroscopic, which means it does not absorb water. This prevents

bubbles forming in the hydraulic fluid and corrosion occurring.

Advantages:

• It is a progressive spring-rate suspension since the more it is compressed, the harder it becomes.

As the piston is pushed in, as the volume of the nitrogen halves, the pressure doubles. This

relationship produces a suspension system with an infinite number of elastic rates. The further it is

pushed in the harder it is to further go in. The main benefit is the fact that you can have a very

soft suspension at the beginning yet not bottom out like a soft steel suspension.

• This system adjusts to the weight of the car as passengers load and unload. The hydraulic fluid

can be pressurized and depressurized respectively as the weight increases and decreases. This

allows the car to maintain a constant ride height.

• Ride height is manually adjustable

• Compact suspension design.

• Maintenance is relatively easy, as it doesn't require any specific tools.

Disadvantages:

• Specifically trained mechanic may be required to service certain parts increasing cost of owning

this suspension system.

• Expensive to replace if poorly maintained

• If failure occurs, ride height will drop and the braking power will be reduced. It takes a large failure

before braking power is reduced however.

3.3 Past Missions

There have been thirteen successful missions to the moon involving the landing of manned and

unmanned crafts. (14) The landing mechanism of the Mars Rover landings was also investigated.

3.3.1 Apollo

The NASA Apollo 11 mission was the first manned mission to the lunar surface. The design of the

lunar module initially started out as a static five-legged, inverted-tripod-type landing gear attached to a

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cylindrical descent stage.(15) A six-legged landing gear configuration was found to have a lower stability

to weight ratio then the five-legged design, and the four-legged configuration was found to require too

large a diameter for deployment and retraction for stowage. The design of the module was later

changed to a cruciform-type structure and with this change the landing gear was changed to a four

legged structure. (15)The original design of an inverted-tripod-type landing gear was reviewed at this

point and it was found that the use of a cantilever type design provided a smaller overall diameter of the

landing module and due to the availability of stowage space on the rocket this design was chosen.

(15)The final design can be seen below in Figure 4 and Figure 6Figure 5. Figure 4 shows a detailed design

of the stowed and deployed positions of the Apollo 11 landing gear. The truss design can be seen as the

secondary struts are connected approximately half way up the primary strut. In the inverted-tripod

design the secondary struts were connected to the foot of the primary strut in the landing pad area.

Figure 4: Stowed and deployed positions of landing gear for the Apollo 11 landing module(15)

Figure 5 shows a detailed design of the Apollo 11 landing gear components. The secondary struts

can be seen along with a detailed view of the supporting deployment truss designed to provide

structural support while allowing for stowing the landing gear before launch.(15)



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Figure 5: Detailed design of landing gear for Apollo 11 lunar landing module(15)

The energy absorption system for the Apollo 11 Lunar Lander was a collapsible aluminum

honeycomb structure contained in the primary and secondary struts. The design consisted of an inner

cylinder that fitted into an upper outer cylinder to provide compression stroking at touchdown. (15)The

inner cylinder of the main strut was connected the footpad through the use of a ball-joint fitting. Figure

6 shows the cartridge interface for the primary struts.

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Figure 6: Landing-gear primary strut honeycomb structure.(15)

For the secondary struts the same method was chosen however the inner cylinder was attached to

the primary strut and the outer cylinder was attached to the deployment truss through the use of a ball-

joint fitting.(15) Figure 7 shows the cartridge interface for the secondary struts.

Figure 7 - Landing-gear secondary strut honeycomb structure.(15)

In testing, non-vertical components of impact force (due to geometric orientation) on the secondary

struts were addressed in compression and tension stroke damping. This configuration can be seen in

Figure 8. (16)



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Figure 8 - Secondary Strut depicting compression and tension stroke(16)

Figure 9 shows images of the aluminum honeycomb shock-absorbing cartridges from the Apollo 11

Lander before and after impact crush testing.

Figure 9 - Photograph of Aluminum honeycomb shock-absorbing cartridges from full-scale landing-gear struts before

and after impact crush(16)

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3.3.2 Surveyor

The surveyor program consisted of seven unmanned lunar landing spacecraft of similar design that

were sent to the moon. Five of the seven spacecraft successfully landed on the lunar surface. (17)

Surveyor 1 landed with a vertical velocity of about 3.6 m/s and a lateral velocity of about 0.3 m/s.

Figure 10 shows dimensions and structure of the landing gear used for the Surveyor spacecraft. Figure

11 depicts the operation of the landing gear. A portion of the impact load was absorbed by crushable

aluminum foam attached on the underside of the landing module and on the footpads of the three pod

legs. The remainder of the landing energy was damped by an elastic shock absorber (11 cm stroke) on

each of three primary pod legs. The spacecraft frame was designed to come to a rest position parallel to

the lunar surface. (18)

Figure 10 - Surveyor 1 Landing Gear Dimensions (18)



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Figure 11 - Operation of Surveyor Landing Gear a) During Descent b) Initial Contact and Compressed Shock Absorber c)

Rest Position (18)

Strain gauges on the shock absorbers indicated the loads of impact and the bounce that some of the

Surveyor crafts experienced. Surveyor 1 rebounded clear of the surface for approximately 500 meters

after initial touchdown before coming to rest. This corresponds to a bounce height of approximately 6

cm. (18)

Surveyor 1 had a mass of 292 kg. The static load needed to support the spacecraft on its three pads

was approximately 3kPa. The initial impact loading of landing acting on each pod leg was about 2.2 kN.

The second impact generated loads of approximately one-quarter initial loads. The footpads exerted

approximately 40 to 70 kPa on the lunar surface during landing. (18)

As a shock absorber was used, the landing module had an oscillation before coming to rest after the

second touchdown.(18)

3.3.3 Mars

As of 2008, there have been fifteen missions involving landing on mars. Of these fifteen, five have

been successful. Device malfunctions have resulted in many of these failures. Three of the last five mars

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missions have used an air bag absorption system for landing. One specific Lander was researched, called

the Mars Pathfinder. [1]

The mars pathfinder relied on air bags to absorb the force of impact on the surface of mars. This

enabled the payload to bounce off the surface without harming the inner components. The air bag

system was made up of 24 inner connected spheres that inflated within half a second right before

impact. The spheres made up a pyramid shape 17 feet tall and 17 feet in diameter, with the payload at

the center of the pyramid. This landing gear system was able to absorb a 20-27 G’s of acceleration and

was able to accomplish this at temperatures as low as -80oC. [3]

The main purpose of this type of landing gear was to eliminate the contamination of the soil and

rock samples that were picked up. It also was a great way of absorbing impact. This sort of design was

able to take much more acceleration compared to other designs used. The Mars Pathfinder was tested

to resist 75 G’s of acceleration. [2]

However this type of landing gear also had its drawbacks. The final resting place of the payload was

never known and such things as slopes and rocks could easily veer it off course. After the initial contact

opportunity, the module bounced 24km away from the intended landing site. The airbags also weighed

a lot. Being made of the same material as space suits, this increased the cost dramatically. (22)

3.4 Lunar Conditions

The moon has a negligible atmosphere. The lunar acceleration is approximately 1/6th terrestrial

acceleration. Lunar surface temperature ranges from 40K to 396K. During the day, the temperature

average is 380K and at night the temperature average is 120 K. (19)

Table 1 - Lunar Regolith Properties (20)

Bulk Density [g/cm3] Lunar Regolith Load Bearing

Strength [N/cm2]

1.15 0.02-0.04

1.9 30-100



Page | 16

Data from surveyor 3 suggest regolith has a bulk density of 1.6 g/cm3 at 2.5 cm depth.(20) The

Apollo 11 mission had a probe that stuck into the lunar surface. Data from this mission suggests that the

lunar regolith has a load bearing strength of 2-3psi/inch penetration.(15)

From a study of rock sizes on the crest of the North Ray crater rim, approximately 20% of surface

protrusions greater than 20 cm in size. This value falls to 10% in the outer eject area of the crater.

(21)The design of a landing system can assume that a clearance of 30 cm will account for all surface

protrusions and that the landing surface will be relatively flat whether angled or level.

4 Design Dr. Tim Bryant suggested rewording the design challenge so as not to unintentionally restrict design

possibilities to a narrow scope. The original title of the project, the conceptual design of lunar landing

gear, implied an inherent mechanical aspect to the design of a landing mechanism.

Designating the problem as the requirement of a system to dampen the shock impulse of a lunar

landing allowed TC to identify the underlying design conflict with greater ease. The optimal design will

absorb the most energy in a simple and cost effective manner. It will have the lowest mass and the

smallest volume within the constraints of the problem; we are aiming to maximize the ratio of energy

absorption to density. This will minimize launch cost and maximize available room in the launch payload

and lunar landing module itself.

4.1 Evaluative criteria

The evaluative criteria relate to the assumptions integrated into the design effort. Each criterion is

intended to guide design efforts relating to a part of the system.

Table 2 – Evaluative Design Criteria

Design 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

Lunar Landing Gear


Page | 17

minimized to reduce volume occupancy in the launch

payload and to maximize cargo space available within the

landing module itself

Energy absorption The energy absorption properties of the damping system

must be maximized.

Material Temperature Resistance Structural material must perform within specifications

regardless of large temperature variation. (Lunar surface,

retrorocket fire, )

Support The landing system must provide a reasonable degree of

level orientation after landing and minimize potential slip

and sink in its rest position.

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 a

landing module.

Reliability Confidence in successful operation, unobtrusive

integration and no interference with landing module

subsystems.

Design Time TC’s design efforts are restricted by time and academic

deadlines. Availability of research pertaining to proposed

design system and design time will play a factor in

choosing a certain design.

Cost Important factor considering an open effort. Cost of

launch and manufacture. This criterion was kept in mind

but omitted in design comparison as it would influence in

the weighting of every other criteria.



Page | 18

4.2 Design Methodology

Several landing mechanisms were explored. TC will seek to optimize a simple and affordable

design, one that an open community can hopefully develop. Ideally the designed system will also easily

integrate with the structural concept of the landing module Figure 12 that has previously been

developed.

Figure 12 – Early Structural Lunar Landing Module Concept (3)

To achieve a soft landing on the surface of the moon, the landing module will have to decelerate

to a reasonable velocity before hitting the moon.

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.

Lunar Landing Gear


Page | 19

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.

4.2.1 Structural System Design Candidates

Design 1 – Vertical Static Strut

The vertical static strut is the simplest of the four designs being considered. It consists of 3 struts

that are oriented vertically from the base of the landing module. Due to the orientation of the struts

and the potential for lateral loading stronger material will be required. During landing there is a

potential for the landing module to have a horizontal velocity which could put a significant amount of

lateral stress on the strut. However, the lateral stress can also occur during static loading after the

landing if the weight distribution is unbalanced, which may occur when the rover disembarks the

landing module.



Page | 20

Table 3 – Advantages and disadvantages of vertical static strut structural system design

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 – Vertical Dynamic Strut

The vertical dynamic strut is more complex than its static counterpart. It is very similar to the static

design as it consists of three struts oriented vertically from the base of the landing module. However, to

decrease volume used in the payload, a mechanism is used to extend the struts from a collapsed

configuration. Once extended the struts lock into place and act in the same manner as the vertical static

struts. This mechanism would be designed very similar to the Apollo 11 lunar landing gear locking

mechanism. (14)

Table 4 - Advantages and disadvantages of vertical dynamic strut structural system design


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

Lunar Landing Gear


Page | 21

Design 3 – Angled Static Strut

This concept is more complicated to design and analyze than the vertical static, however it provides

the best method of support. The reason to this is the angled design accommodates lateral loading and

as a result material and size requirements decrease significantly. The angled struts also provide greater

stability not only because of their capability to handle lateral loading but also because the base area is

increased. The increase in base area provides a greater resistance to any induced moments that would

cause the module to tip over.

Table 5 - Advantages and disadvantages of angled static strut structural system design


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 – Angled Dynamic Strut

This concept is the most complicated of all four designs. This type of strut provides all the benefits of

angled static struts with the benefit of reducing the volume used in the payload. Once extended the

struts lock into place and act in the same manner as the vertical static struts. This mechanism would be

designed very similar to the Apollo 11 lunar landing gear locking mechanism. (14)



Page | 22

Table 6 - Advantages and disadvantages of angled dynamic strut structural system design


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

4.2.1.1 Decision Matrix - Structural Systems

A Quality Function Deployment chart, Figure 13, was used to determine the optimal design to

proceed with for the structural section of the landing gear. The requirements were weighted based on

their effect on the evaluative criteria. Mass was considered the most important factor as it not only

significantly affects the cost of launch but the client highlighted the importance of a light weight design.

Since the primary function of the strut design is to support the landing module the structural integrity of

the unit was weighted heavily. Reliability was also one of the most important factors as there is no room

for failure. Material strength is also considered as it significantly affects the cost of the final strut

design. Stability is also important as the landing module may also be used as the antenna to

communicate with Earth. Simplicity is important to increase reliability, decrease cost and most

importantly to increase TC’s probability of success. Design time was also taken into consideration, since

the tight timeline quality, reliability and the probability of success.

Lunar Landing Gear


Page | 23

Figure 13 –QFD Chart for Structural Strut Systems

4.2.2 Energy absorption Design Candidates

Design 1 – Hydro-pneumatic Suspension

The Hydro-pneumatic suspension system is one of the most complex systems being considered. The

design being considered this time is an altered version of the Hydro-pneumatic suspension system used

on cars. A piston pushes on a liquid fluid which alternately pushes on a compressed gas. This system

provides energy absorption however it does not result in oscillation as it has an internal dampening

system.

Table 7 – Advantages and disadvantages of hydro-pneumatic suspension system


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

Weights

Volume

Weight

Force on In

terna

l

compo

nents

Thermal

Conducti

vity

Tensil

e Stre

ngth

Force D

ispert

ion too

lunar su

rface

(N)

Height (m

)

# movin

g par

ts

Time N

eeded

(res

earch

,

Implem

entat

ion

Probab

ility o

f Succ

ess

(%)

Design #1

Design #2

Design #3

Design #4

Mass 19.8 9 9 3 1 1 9 9 3 3.1 3.0 2.8 2.6

Volume 11.0 3 1 1 3 3 1 3 3 3.1 3.5 2.9 3.1

Material 13.2 3 9 1 9 9 1 1 3 3 3 3.0 2.8 3.5 3.1

Stability 8.8 3 3 9 2.5 2.2 4.0 3.6

Structural 17.6 9 9 3 3 3.0 2.7 4.0 3.5

Simplicty 7.7 3 1 1 1 1 1 3 9 3 9 4.0 2.7 3.8 2.5

Design Time 4.4 1 1 1 1 1 9 9 9 4.0 2.9 3.6 2.6

Reliablity 17.6 1 3 3 3 9 1 1 9 9 9 2.9 2.6 4.0 3.6100.0 281.2 255.5 322.4 285.3



Page | 24

Design 2 – Plastic Deformation Structure

The Plastic Deformation Structure system is the simplest design being considered as it does not rely

on various interacting components. The deformation structure will be compartmentalized allowing for

changes to easily be made in future design refinements. No oscillation will occur in this system as the

energy is not being absorbed elastically but rather dissipated through the destruction of material.

Table 8 - Advantages and disadvantages of plastic deformation structure system


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 absorber

The Shock Absorber system is a more complex system then the plastic deformation structure

however the design process should be simpler due to the abundance of resources available for

reference. This type of system is used on everything from cars to bikes and provides the most energy

absorption for its volume. The required mass is high due to the need for working fluid as well as spring

material, and a small amount of oscillation will occur since some energy is being absorbed elastically.

Lunar Landing Gear


Page | 25

Table 9 - Advantages and disadvantages of shock absorber system


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


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 Suspension in combination with Plastic Deformation structure This system is the most complicated of all four designs and will require the most design time. The

combination of systems allows for each one to be specialized for a certain range of the energy

absorption allowing for a smoother landing. This specialization also allows the individual components to

be smaller than if they were used on their own.

Table 10 - Advantages and disadvantages of hydro-pneumatic suspension in combination with plastic deformation

structure system


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


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



Page | 26

4.2.2.1 Decision Matrix - Energy absorption

Figure 14 - QFD Chart for Energy absorption Systems

4.3 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.

4.3.1 Structural 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.

4.3.1.1 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

Weights

Volume

Weight

Force on In

ternal

componen

ts

Thermal

Conductivit

y

Tensil

e Stre

ngth

Force D

ispert

ion too

lunar su

rface

(N)

Height (m

)

# movin

g parts

Time N

eeded

(rese

arch,

Implem

entat

ion

Probab

ility o

f Succ

ess

(%)

Design #1

Design #2

Design #3

Design #4

Force Absorption 16.7 1 9 9 1 3 9 1 9 9 3.0 3.8 3.4 4.0Mass 17.5 9 9 3 1 1 9 9 3 2.0 3.5 1.8 1.5

Volume 10.0 3 1 1 3 3 1 3 3 1.2 1.5 2.2 1.8Material 10.0 3 9 1 9 9 1 1 3 3 3 2.2 3.5 2.4 2.8Clearance 8.3 3 9 3 3 4.0 1.5 4.0 2.6Simplicty 8.3 3 1 1 1 1 1 3 9 3 9 1.6 3.2 2.4 2.2

Design Time 12.5 1 1 1 1 1 9 9 9 3.0 3.0 3.5 2.5Reliablity 16.7 1 3 3 3 9 1 1 9 9 9 2.6 2.6 3.5 3.0

100 246.50 294.58 289.58 260.17

Lunar Landing Gear


Page | 27

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.

4.3.1.2 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.

4.3.2 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.

4.3.3 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.



Page | 28

4.3.4 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.

4.3.5 Combined Design

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.

5 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.

Lunar Landing Gear


Page | 29

5.1 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.

Figure 15 - Impact Design

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.



Page | 30

Figure 16 - High stress location, impact design

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.

5.2 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.

Lunar Landing Gear


Page | 31

Figure 17 - Lever design

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.



Page | 32

Figure 18 - Variability in pod leg position during touchdown

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.

6 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.

6.1 Structural System

The structural design incorporated key aspects in determining how the landing pads and struts

would be oriented and located. Methods of statics and solid mechanics along with optimization

techniques were used to determine optimal angles, lengths and diameters of the primary strut and

secondary struts.

Lunar Landing Gear


Page | 33

6.1.1 Optimization design and 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.



Page | 34

Figure 19: Free body diagram for primary strut

Equation 6-1

�𝑀𝑀1 = 𝑃𝑃 sin𝛼𝛼 (𝐿𝐿 − 𝑎𝑎) + 𝐿𝐿𝑉𝑉 sin𝜃𝜃 × 𝐿𝐿 − 𝐿𝐿𝐻𝐻 cos𝜃𝜃 × 𝐿𝐿 = 0

Equation 6-2

�𝐹𝐹𝑥𝑥 = 𝐺𝐺𝐻𝐻 − 𝐿𝐿𝐻𝐻 − 𝑃𝑃 sin(𝜃𝜃 − 𝛼𝛼) = 0--

Equation 6-3

�𝐹𝐹𝑦𝑦 = 𝐺𝐺𝑉𝑉 − 𝐿𝐿𝑉𝑉 − 𝑃𝑃 cos(𝜃𝜃 − 𝛼𝛼) = 0--

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

G

x

G

y

L

y

L

x

P

X

Y

1

2

3 𝜃𝜃

𝑎𝑎

Lunar Landing Gear


Page | 35

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.

Equation 6-4

𝑃𝑃 = 𝐺𝐺𝐻𝐻 cos𝜃𝜃 × 𝐿𝐿 − 𝐺𝐺𝑉𝑉 sin𝜃𝜃 × 𝐿𝐿

sin𝛼𝛼 (𝐿𝐿 − 𝑎𝑎) + sin(𝜃𝜃 − 𝛼𝛼) cos𝜃𝜃 × 𝐿𝐿 − cos(𝜃𝜃 − 𝛼𝛼) sin𝜃𝜃 × 𝐿𝐿

Equation 6-5

𝐿𝐿𝑉𝑉 = 𝐺𝐺𝑉𝑉 − 𝑃𝑃 cos(𝜃𝜃 − 𝛼𝛼)

Equation 6-6

𝐿𝐿𝐻𝐻 = 𝐺𝐺𝐻𝐻 − 𝑃𝑃 sin(𝜃𝜃 − 𝛼𝛼)

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 𝜇𝜇 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 𝜃𝜃 can be

found by rearranging as seen in Equation 6-8.

Equation 6-7

𝜇𝜇(𝐹𝐹 × cos𝜃𝜃) = 𝐹𝐹 × sin𝜃𝜃



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Equation 6-8

𝜃𝜃 = tan−1 𝜇𝜇

The minimum angle therefore is 16.7o. 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 47o.

6.1.2 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.

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Figure 20: Normal Stress vs. Theta for Primary Strut

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.



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Figure 21: Shear Stress vs. Theta for Primary Strut

6.1.3 Determining optimal wall thickness 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.

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Figure 22: Outside Diameter vs. Theta for Primary Strut

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.



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Figure 23: Thickness vs. theta for primary strut

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.

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Figure 24: Weight vs. Theta for varying inside diameters

6.1.4 Determining optimal position for 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.



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Figure 25: Weight vs. theta for varying lengths of a.

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.

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Figure 26: Wall thickness of secondary struts for varying lengths of a.

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.



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Figure 27: Plot of 3D Secondary strut length vs. theta

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.

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Figure 28: Plot of alpha vs. theta

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.

6.1.5 Optimization Program

The optimization program is used to determine stress plots and using that weight, diameter, and

angle plots for structural designs. The program runs in the following way and the code can be found in

Appendix B.

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



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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.

6.2 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

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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.

1. 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.

2. 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.

3. 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.

4. 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.



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5. 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.

6. 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.

7. 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.

8. 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.

9. The landing module touches down in a linearly inelastic manner.

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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.

10. 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.

6.2.1 System Energy Modeling

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,

can be integrated into a simple energy balance.

6.2.1.1 Lunar Regolith under compression

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.

Figure 29 - Column of regolith per strut pad modeled with a spring stiffness



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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

𝑘𝑘1 = 𝛽𝛽𝐴𝐴𝑖𝑖𝑖𝑖𝑖𝑖𝑎𝑎𝑖𝑖𝑖𝑖

6.2.1.2 Crush Cartridge Plastic Deformation

The total work done by the plastic deformation of all crush elements represents absorption of the

touchdown kinetic energy.

Equation 6-10

𝐸𝐸𝑖𝑖𝑐𝑐𝑐𝑐𝑐𝑐 ℎ = 𝐹𝐹𝑖𝑖𝑐𝑐𝑐𝑐𝑐𝑐 ℎ𝛿𝛿 × 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.

6.2.1.3 Calculation 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 .

Equation 6-11 (11)

𝜎𝜎𝑖𝑖𝑐𝑐𝑐𝑐𝑐𝑐 ℎ = 0.58𝜎𝜎𝑐𝑐𝑠𝑠𝑠𝑠𝑖𝑖𝑠𝑠 𝜌𝜌𝑐𝑐𝑟𝑟𝑠𝑠3

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

𝐹𝐹𝑖𝑖𝑐𝑐𝑐𝑐𝑐𝑐 ℎ = 𝜎𝜎𝑖𝑖𝑐𝑐𝑐𝑐𝑐𝑐 ℎ𝐴𝐴𝑖𝑖𝑐𝑐𝑐𝑐𝑐𝑐 ℎ

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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).

Figure 30 - Complete plastic deformation and densification of crush cartridge

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

𝑖𝑖𝑖𝑖𝑐𝑐𝑐𝑐𝑐𝑐 ℎ = 𝐴𝐴𝑖𝑖𝑐𝑐𝑐𝑐𝑐𝑐 ℎ 𝑠𝑠𝜌𝜌𝑐𝑐𝑠𝑠𝑠𝑠𝑖𝑖𝑠𝑠 𝜌𝜌𝑐𝑐𝑟𝑟𝑠𝑠

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

𝑥𝑥 =𝑖𝑖𝑖𝑖𝑐𝑐𝑐𝑐𝑐𝑐 ℎ

𝐴𝐴𝑥𝑥𝜌𝜌𝑐𝑐𝑠𝑠𝑠𝑠𝑖𝑖𝑠𝑠 𝜌𝜌𝑐𝑐𝑟𝑟𝑠𝑠=1

Equation 6-15

𝛿𝛿 = 𝑠𝑠 − 𝑥𝑥

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,



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the energy term of the crush cartridge representing energy removal from the system due to plastic

deformation is neglected.

Equation 6-16

12𝑘𝑘1∆1

2 + 𝐸𝐸𝑖𝑖𝑐𝑐𝑐𝑐𝑐𝑐 ℎ = 𝑖𝑖𝑔𝑔𝑖𝑖𝑠𝑠𝑠𝑠𝑚𝑚 ℎ +12𝑖𝑖𝑣𝑣0

2

Equation 6-17

∆1= �2(𝑖𝑖𝑔𝑔ℎ − 𝐸𝐸𝑖𝑖𝑐𝑐𝑐𝑐𝑐𝑐 ℎ)

𝑘𝑘1

Equation 6-18

𝐹𝐹𝑖𝑖𝑎𝑎𝑥𝑥 = 𝑘𝑘1∆1

Equation 6-19

𝑎𝑎𝑖𝑖𝑎𝑎𝑥𝑥 =𝐹𝐹𝑖𝑖𝑎𝑎𝑥𝑥𝑖𝑖

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.

6.2.1.4 Results

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

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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.

Table 11 - Landing Characteristics of Lunar Module

Δ1 ± Δ1 Fmax ± Fmax amax ± amax

[m] ± [m] [N] ± [N] [m/s2] ±[m/s2] G’s ± G’s Worst Case G

Rigid Structure

0.065 0.015 9354 2883 49.2 21.2 5.02 2.16 7.18

Energy Absorbing Structure

0.023 0.040 3346 5809 17.6 31.0 1.80 3.17 4.96

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.

Table 12 - Effect of mass and free fall height error for a system with energy absorption

G’s ± G’s Percentage error 30% error mass and

free fall height 1.80 3.17 176%

10% error mass and free fall height

1.80 1.37 76%

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



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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. Several of the

key error equations are depicted in Appendix E.

6.2.1.5 Concluding Notes

Originally, an effective spring stiffness for the aluminium crush was sought. This analysis is flawed as

the aluminium crush would plastically deform and not behave in a linearly elastic manner. A

conservative approximation could be obtained by appropriately varying the elastic modulus of segments

of the cartridge during crush with an increasing relative density. Each segment represented an effective

spring stiffness which combined with the entire system. This analysis was dismissed as the design was

refined; the crush cartridge became subject to a varying angle and the effective spring stiffness would

change not only with the segmented crush, but also with the angle variation. This analysis would have

been complex, overly conservative and was developed from a flawed assumption. Appendix depicts the

system energy modeling using a spring stiffness as an energy absorber in the struts should TF(X) choose

to continue the design in a different manner.

A representative from Erg Aerospace, a company that manufactures energy absorbing foams,

suggested assuming a constant acceleration of a force to ultimately determine the distance and relative

density of the crush to reduce the velocity of the landing module to zero without surpassing the

acceleration limit. However, this method would constrain the design of the structural elements to the

required dimensions of the energy absorption system as depicted and thus was not pursued due to the

constrained volume of the payload.

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.

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6.3 Landing Pad Design

The landing pads were designed considering the load bearing strength of the regolith. Since the load

bearing strength of lunar soil is much less than that of the earth, the pad size needed to be relatively

larger to ensure penetration was minimal while keeping the restricting the force transmission to ensure

that the acceleration experienced by the lunar module was within constraints. The regolith was modeled

as a spring, meaning that the larger the pad diameter, the larger the force each strut would experience

and the larger the acceleration acting on the landing module.

In the design and analysis of the landing pads, titanium was decided to be used. Titanium is a very

strong material, can be welded and is readily available. Titanium however may not be the best for

prototyping because of the expense and the challenge in machining it. Aluminum could be used in this

case because it is cheap and strong. More research needs to be conducted into the best materials to use

for the design give other aspects of the Lunar Lander project.

6.3.1 Landing Pad Shape

The landing pad shape resembled the pads used in the Apollo mission to the moon. Apollo 11’s pads

had a flat bottom and curved sides and were large in area which was decreased the penetration into the

regolith. The pads needed to take this into consideration and the design that was made to ensure

minimal penetration.

The energy absorption analysis assumed a compression of lunar soil against a flat contact area of the

pads. The design of a flat bottom and curved sides was pursued. Consequently, the data obtained was a

reasonable estimation concerning the area of contact with the pads and regolith.

The curved sides increased the area of contact as well as prevented shocks from rocks and other

material from making contact with the struts. Therefore, the only force felt by each strut would be

coming up directly from the base. This ensured there would be no horizontal forces contacting the

struts, only the force exerted by the ball joint directly up the primary strut.

The curved sides were also needed for the possibility of horizontal velocity experienced on landing.

If there was some horizontal velocity, the pads would need to have some curvature so they would be



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able to slide. Like ski’s, the pads would need to slide over the regolith. This would ensure the pads

would not get stuck or caught causing torques that could result in failure in a strut.

6.3.2 Area of landing pads

The ideal area was found by comparing the amount of penetration into the regolith and the

resultant force. The smaller the area, the more penetration would result. This would lead to a smaller

force because the regolith acts like a spring and would decrease the deceleration on impact. Although,

there are limits on the amount of penetration desired; there is no guarantee that the regolith properties

would be constant for a deep penetration. A larger pad area would decrease this penetration, increase

the force exerted, and increase the amount of acceleration. Restrictions from TF(X) such as payload

area limits, landing module thruster location and weight limitations also had to be considered.

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.

6.3.3 Thickness of Pads

Using a material mechanics analysis, the thickness of the titanium pads was determined. Since there

would be no need for energy absorption within the pads, the pads were designed for rigidity. Separate

analysis was needed for both the bottom of the pad as well as the sides.

The bottom of the pad was designed to be flat and circular with a ball joint welded centrally. As only

force transfer was analyzed, the thickness of the pad needed to provide rigidity for the most extreme

cases experienced when landing. It is assumed that the surface would be relatively flat and the ideal

case would be landing on pure regolith. If the pad landed on a rock however, the pad would have to be

able to resist the concentrated force at any point along the pad without deforming.

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. A

simplified view of the landing pad can be seen below, in Figure 31, showing the force of a rock being

applied at a point of the pad where the resulting moment would be greatest. Figure 31 shows half of a

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horizontal cut out of the pad. The sides of the curve were also neglected in this analysis because they

would not contribute to a lot of the resistivity when a force was applied.

Figure 31 - Simplified cross sectional view of half a lunar pad

The superposition equation is depicted by Equation 6-20.

Equation 6-20

𝜈𝜈𝑖𝑖𝑎𝑎𝑥𝑥 = +𝐹𝐹𝐿𝐿3

3𝐸𝐸𝐸𝐸

𝝂𝝂max was the maximum deflection that would be seen by the landing pads. It was assumed that a

reasonable deflection of 0.5 cm could be seen in the worst case scenario.

F was the force exerted by a rock or obstruction and was found to be 4 KN per landing pad.

E was the modulus of elasticity of tI-6Al- 4v annealed titanium and was found to be 113.8 GPa (23)

L was 0.125m and was calculated knowing the geometry of the beam.

The calculations can be found in Appendix B.

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

Weld Ball Joint

𝝂𝝂max



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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.

The landing pad shape, area and thickness combined to produce Figure 32 below.

Figure 32 - Cross-sectional area of foot pad

6.4 Ball Joint

A ball joint would be used to connect the landing pad to each leg. A ball joint was chosen because of

its strong ability to handle and transmit forces from the landing pads when landing in different

orientations, while reducing unwanted torque to the upper legs. The ball joint could easily

accommodate any touchdown angle that the pad may experience due to landing on rocks or slopes.

Regardless of the orientation of each leg, the force would be concentrated through the joint up the leg,

resulting in very little torque. If the pad were rigidly connected to each leg, large moments would be

seen at each connection in the upper leg. A design similar to a steering link ball joint could be used in

this case. The ball joint is wide on the bottom and connects to a lubricated ball with the ability to rotate.

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This ball is connected to a shaft which is then threaded. An example of a possible ball joint can be seen

below in Figure 33.

Figure 33 - steering link ball joint (Ningbo Pair Industrial Co.)

Specifically, the ball joint must have the ability to:

1. Transfer a maximum impact force of 6KN through it. This has to be done without deflection, or

any material failure. This is also assuming a safety factor of 1.4.

2. Have at least 45 degree’s of freedom assuming the worst case scenarios of landing. If slopes or

rocks are contacted when landing, the ball joint must be able to rotate in order to handle the

angled forces applied.

3. Rotate easily and hang vertically before landing. The ball joints must be loose to ensure that the

landing pads are hanging loosely and are parallel with the surface. This will ensure that no

failures will occur including the possibility of the pads flipping before landing.

4. Resist cold welding. The ball joint will have to deal with cold welding when in space. Therefore,

the design must incorporate a solution to stop this from occurring. To counter this, the ball joint

could have a thermal separator to ensure there isn’t metal to metal contact. The ball and socket

is the only area in which there needs to be a substantial analysis for the footpad. A thermal

lubricant could be used or, coating the ball and socket with a fiberglass shell would ensure no

cold welding will occur. This is further pursued in thermal considerations.



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The weight is one of the most important parts of the pad therefore the ball joint would be made out

of titanium. Andy Bryson at the university machine shop discussed involving the ball joint and the

design.(24) Research was done using various resources including websites, catalogues, and Kingston

companies. Most of the ball joints found were made out of steel for other applications or were too

small. The larger sizes capable of withstanding the forces used in industry and were made of steel, not

titanium. More research is needed to determine if there is a ball joint on the market that would work.

Otherwise, a custom design would be needed for this specific purpose.

6.5 Pad Connection

The best way to connect them would be by welding. Since the bottom of the landing pad is flat and

the bottom of the joint is assumed to be flat, they could be welded together around the circumference

of the ball joint base. This would then be heat treated to reinforce the titanium weld and provide the

strength needed so no failures would result.

The cup would be made in two parts. First, there is the base to which the ball joint would be welded.

Then, there would be the angled sides. The angled sides would be welded on after the bottom had been

welded to the ball joint. Therefore the outer rim of the base would be welded to the bottom edge of the

side piece. After welding the two pieces together, the whole pad would be heat treated. To complete

the pads, the exterior edges on the bottom formed from the weld would be machined to ensure a

smooth and continuous finish.

The connection of the ball joint to each strut was also examined. Possible ball joints that could be

used and all joints analyzed seemed to have their ends threaded for connection to other parts. This

technique could be used in the design. It provides strength and direct contact with the struts. If

sufficient threading is used, it would ensure failure of the ball joint shaft. This also ensures the force

experienced on contact will be transferred directly through each strut.

6.5.1 Other Considerations

The orientation of the ball joint when it is connected to the landing pad is a concern. Ideally, the ball

joint should not be welded vertically onto the landing pad. The overall degree of freedom would have to

be much greater to take into account if the landing module landed on a rock. The ball joint would have

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to be able to incorporate the initial strut angle as well as the additional angle a rock might cause. This

could result in a final angle of more than 80 degrees which is not ideal. To solve this problem, the ball

joint should be integrated to the landing pad on an angle equal to the angle of the strut. This would

ensure that the ball joint is at 0 degrees with respect to the axis of the strut. Therefore, the ball joint

would have freedom both ways which is best if it made contact with a rock or obstruction.

The final design can be seen below in Figure 34.

Figure 34 - Final landing pad design

6.6 Thermal Considerations

The landing structure will be subject to thermal energy transfer. During transit, radiation from the

sun may adversely affect any movable parts. While a constraint clearance will insulate the landing

module against radiation from lunar regolith, the entire landing structure will be exposed to this

radiation. It may be beneficial to wrap the landing gear structure with a foil to increase its reflectivity.

Thermal conductivity isolation will be necessary to limit the thermal energy transfer between the

lunar surface and the landing module, through the landing structure. This can be accomplished

strategically by thermally isolating the joints in the landing structure. The connection between the



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landing pads and the primary strut is to be fitted with a spherical joint. The mating parts of this joint

could fabricated with a large tolerance and outfitted with a smooth fiberglass coating to reduce heat

transfer through these key parts. Further aero gel powder could be contained within the joint and used

as a dry lubricant between the mating fiberglass layers.

Figure 35 – Conceptual thermally isolated spherical joint

The structure will also be subject to thermal expansion. In conjunction with temperature variation,

the materials will expand and contract typically at different rates which may fatigue the material or

induce structural weaknesses. Further, the rapid contraction of mating components may induce cold-

welding. The risk of non-functional configurations due to such thermal occurrences justifies a deeper

thermal analysis into construction materials and the manufacturing tolerances which are not adequately

addressed in this design report.

Also it is important to note that the material properties used to determine the crush strength of the

energy absorbing foam and the dimensions of the landing structure were tabulated values at 298 Kelvin.

These properties will likely change and alter the analysis depending on the operating temperature of the

landing mechanism.

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7 Design Results The final design results are described below. Figure 36 below shows an approximate visual

representation of the lunar landing gear assembly. The detailed part dimensions are shown below in

Table 13 as well as part drawings in Appendix G.

Figure 36 - Three dimensional rendering of entire lunar landing gear assembly (note: dimensions are not to scale, this

rendering is only for an approximate visual representation)

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,



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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.

Table 13 - Summary of structural dimensions

Variable Value Primary Strut ID 3 cm Primary Strut OD 3.5 cm

Primary Strut Length 75 cm Secondary Strut ID 1.905 cm Secondary Strut OD 2.35 cm

Secondary Strut Length 50.8 cm Alpha 50o Theta 20o

A 40cm

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.

The final three dimensional assembly of the landing module is shown below in Figure 37. The

dimensions are not to scale, as this rendering is only for an approximate visual representation.

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Figure 37 – Three dimensional rendering of entire landing assembly (note: dimensions are not to scale, this rendering is only

for an approximate visual representation)

8 Prototyping

Due to the large error from uncertainty in operation variables, prototyping and testing of a scaled

model will be the only reliable method of obtaining accurate data. Due to the symmetry involved in the

proposed conceptual landing gear, it may be possible to build only a third of the structure to scale and

test for the accelerations that develop. Progressive loading can test for the structural limits of the

design.

8.1 Materials



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For prototyping, material selection will be an important factor. Titanium will be too expensive to use

in prototyping so a cheaper material will need to be used and calculations done to convert the

prototype to the full scale model. Materials such as 6061-T6 aluminum would be optimal for the

prototyping as they are lightweight, strong, and relatively cheap. Fabrication of parts using these

materials would also be cheap as they are easy to work with, and are common enough that many shops

would be able to machine them.

8.2 Manufacturing

The final construction of the landing gear structure will ideally make use of a homogenous material

for ease of integration. This will also mitigate the effect of non-uniform thermal expansion. The

machining of titanium will likely be hard and expensive. If it is possible to use prefabricated parts of

dimensions that will satisfy the constraints as developed structurally and conceptually in this report, the

final design will mitigate manufacturing costs. Dimensioning should also consider the feasibility of its

manufacture.

8.3 Cost

The cost of design implementation is related to the mass and manufacture of the chosen design.

The launch cost decreases dramatically with the decrease of structural mass and associated fuel mass. A

2002 report put launch costs to low earth orbit in a range around 20000 $(USD)/kg, but depending on

the launch vehicle, the cost can be as high as 50000$(USD)/kg. (25) Launch vehicle technology has not

changed significantly from 2002 so these figures are reasonable. Therefore, material choice is extremely

important, weight of fasteners is important, and minimizing the thickness and mass of structural

components. Further analysis will be needed to reduce the weight in areas that were not considered;

fastening the struts to the landing module, and the exact thickness of the connecting plates to the base

of the landing module will have to be determined.

9 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

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system down into smaller and more manageable 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.

10 Conclusion

The design analysis was broken into three main components. These were energy absorption, landing

pad design, and a structural system.

The energy absorption system chosen incorporated plastically deforming material. The energy

absorption system would be contained in the secondary struts and these struts would be able to house

crush cartridges. All the acceleration values fall under the 10 G acceleration constraint, while the energy

absorbing structure reduces the penetration into regolith and the acceleration beyond the results of

obtained for a rigid structure assuming touchdown occurs on pure regolith with properties similar to

those suggested by data from the Apollo missions.

Large uncertainty in operation variables means that either prototyping or improved mission

variables are required to obtain more accurate data. The error propagated though with unrealistic

amounts due to suggested uncertainties of up to 30%. As future design work progresses, the theoretical

modeling of the touchdown scenario will improve. Also, these are all issues that can be investigated

during prototyping and testing. Prototyping should be performed on a scalable model with cheap

material that is easy to machine. The final landing gear structure should consist of a homogenous

material to simply integration and to account for thermal effects. Dimensioning should consider the

feasibility of manufacture.



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Landings pads provided stability and support for each leg during contact. The landing pads contact

area balanced the force transferred to the legs and minimized the penetration into the regolith. This

ensured the acceleration developed by the landing module was minimal.

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 most important aspect of cost is the weight. It costs $20 000.00 per kilogram to launch

something into space. Manufacturing the design with an expensive material is justified as launch costs

are prohibitive. As a result of cost, material is the one of the most important aspects in the overall

design of the module. An optimal design with reduced material mass in the struts and the landings pads

provides a foundation for further design work.

Future thermal design work would benefit the project and the design of the landing structure.

Future work could break the system into smaller more manageable design tasks. The primary strut

could be outfitted with its own energy absorption mechanism to reduce the risk of axial force

transmission. The design will also have to integrate a feasible method to connect to the landing module.

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REFERENCES

1. Sellens, Dr. Rick. Mech 460: Team Design Project Conceive and Design. Mechanical and Materials

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8. Yasuki, Tsuyoshi and Mikutsu, Satoshi. A Study on Yielding funtion of Aluminium Honeycomb.

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24. Bryson, Andy. Machinist. [interv.] James Burke. 10 19, 2009.

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APPENDIX A

Table 14 - Operation Variables

Table 15 - Design Variables

m Landing Module Mass [kg] 190 30 57h Height of Engine Cutoff [m] 1 30 0.3

gl Lunar Gravity [m/s2] 1.6 0 0

vh Velocity at Engine Cutoff [m/s] 0 0 0

vi Velocity at Impact [m/s] 1.79 - 0.45

ge Earth Gravity [m/s2] 9.8 0 0

# pod legs

Number of Pod Legs - 3 - -

# crush cartidges

Number of energy absorption units - 6 - -

β Lunar Regolith Load Bearing Strength [N/m2/m] 678618 20 135723.6

θ Primary Leg strut Angle [˚]

Fmax Maximum Impact Force [N]

amax Maximum Static Acceleration [m/s2]

Δ1 Regolith Penetration [m]

E6061T6 Young's Modulus of Crush Material [gPa]

σ6061T6 Yeild Strength of Aluminium [mPa]

ρ6061T6 Density of Aluminium [kg/m3]

Dpad Pad Diameter [m]

Apad Pad Area [m2]

Aimpact Projected Impact Area [m2]

Dcrush Crush Diameter [m]

Acrush Crush Area [m2]

lcrush Length fo Crush Cartridge [m]

σcrush Crush Strength [mPa]

Fcrush Crush force of the aluminium foam [N]

δ Crush Deformation Length along strut [m]

ρrel Relative density of Foam [kg/m3]

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APPENDIX B MAIN MATLAB SCRIPT

clear; % Variable List % ============= Coef_Fric = 0.3; % kinetic friction on surface of moon MOutsideDiameter = .05; MInsideDiameter = 0.03; SOutsideDiameter = .0381; SInsideDiameter = 0.01905; Clearance = .7; ExDistance = .75; a=.4; LevAcur = 100; % Level of accuracy variable controls number of iterations

that are run for each variation of the strut xoffset = 0.3; yoffset = 0.3; %Metal Properties: Titanium Ti-6AL-4V Annealed yieldStrength = 880 * 10^6; %in Pa shearStrength = 550 * 10^6; %in Pa density = 4.43 * 1000000 / 1000; %4.43 is in grams per cubic centimeter % Plotting Variables % ================== A=0; %Normal Stress vs Theta B=0; %Shear Stress vs Theta C=0; %Main Diameter vs Theta D=0; %Secondary Diameter vs Theta E=0; %Alpha vs Theta F=0; %Weight vs Theta G=0; %P vs Theta H=0; %main strut length vs Theta I=1; %Plot of 3D Sec Strut Length vs Theta J=0; %Plot of Main Strut Weight vs Theta K=0; %Plot of Sec Strut Weight vs Theta M=1; %Plot of Main Strut Wall Thickness vs Theta



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N=1; %Plot of Sec Strut Wall Thickness vs Theta % Force acting on strut at ground % =============================== Gv = 4079; %Vertical force in N %HF = 1500; Gh = -Coef_Fric * Gv; Gm = 0; % Max Angle Strut can make with body % ================================== MaxAngle = (pi/2) - atan(Clearance / ExDistance ); % Counter Variables for plotting cx=1; % Theta & Stress Only % =============================== for theta= 0.122:(pi/360):MaxAngle [x(cx),ystress(cx),yshear(cx),P(cx),alpharad(cx),mL(cx),sL(cx)]=

StressVsTheta(theta,Clearance,Gv,Gh,xoffset,a,MOutsideDiameter,MInsideDiameter);

alpha(cx)=(alpharad(cx)/pi)*180; % Counting cx=cx +1; end cx2=1; % Diameter vs. Theta & Weight vs. Theta % =============================== for theta= 0.122:(pi/360): MaxAngle pmODiameter = 0.06; mODiameter = pmODiameter *2; % Main Strut Diameter Calculation while abs(mODiameter - pmODiameter) > 0.00001

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mODiameter = pmODiameter; [xD,ystressD,yshearD,Ps,alpharad2,mLD(cx2),sLD(cx2)]=

StressVsTheta(theta,Clearance,Gv,Gh,xoffset,a,mODiameter,MInsideDiameter); if ystressD > (0.7 * yieldStrength) || yshearD > (0.7 *

shearStrength) pmODiameter = mODiameter + 0.0031243; elseif ystressD < (0.6 * yieldStrength) || yshearD > (0.67 *

shearStrength) pmODiameter = mODiameter - 0.0015453; end rsLD(cx2)= sqrt(sLD(cx2)^2 + yoffset^2); end OMOutsideDiameter(cx2)= mODiameter; WallThickness(cx2) = (OMOutsideDiameter(cx2) - MInsideDiameter)/2; % Secondary Strut Diameter Calculation SArea = (abs(Ps)/2) / (0.01* yieldStrength); psODiameter = sqrt(4*(SArea + (1/4*pi*((SInsideDiameter)^2)))/pi); % while abs(sODiameter - psODiameter) > 0.00001 % % sODiameter = psODiameter; % % SArea = (1/4*pi*((sODiameter)^2)) -

(1/4*pi*((SInsideDiameter)^2)); % Ax = Ps / SArea; % % if abs(Ax) > (0.6 * yieldStrength) % % psODiameter = sODiameter + 0.008; % % elseif abs(Ax) < (0.5 * yieldStrength) % % psODiameter = sODiameter - 0.004; %



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% end % % end OSOutsideDiameter(cx2)= psODiameter; WallThicknessS(cx2) = (OSOutsideDiameter(cx2) - SInsideDiameter)/2; WeightS(cx2)= density * 2*((1/4*pi*((OSOutsideDiameter(cx2)^2)) -

(1/4*pi*((SInsideDiameter)^2)))*rsLD(cx2)); WeightM(cx2)= density * ((1/4*pi*((OMOutsideDiameter(cx2)^2)) -

(1/4*pi*((MInsideDiameter)^2)))*mLD(cx2)); Weight(cx2)= density * ( 2*((1/4*pi*((OSOutsideDiameter(cx2)^2)) -

(1/4*pi*((SInsideDiameter)^2)))*rsLD(cx2)) + ((1/4*pi*((OMOutsideDiameter(cx2)^2)) - (1/4*pi*((MInsideDiameter)^2)))*mLD(cx2)) );

% Plotting Variables xod = theta*180/pi; % Counting cx2 = cx2 +1; end % Output % ====== hold on if A==1 xlabel('\Theta - Degrees') ylabel('Max Axial Stress (Pa)') title('Plot of Max Axial Stress vs \Theta') plot(x,ystress,'-g') elseif B==1; xlabel('\Theta - Degrees') ylabel('Max Shear Stress (Pa)') title('Plot of Max Shear Stress vs \Theta') plot(x,yshear,'-b')

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elseif C==1 xlabel('\Theta - Degrees') ylabel('Main Strut Diameter (m)') title('Plot of Main Strut Diameter vs \Theta') plot(x,OMOutsideDiameter,'-k') elseif D==1 xlabel('\Theta - Degrees') ylabel('Secondary Strut Diameter (m)') title('Plot of Secondary Strut Diameters vs \Theta') plot(x,OSOutsideDiameter,'-g') elseif E==1 xlabel('\Theta - Degrees') ylabel('\Alpha - Degrees') title('Plot of Alpha vs \Theta') plot(x,alpha) elseif F==1 xlabel('\Theta - Degrees') ylabel('Weight (kg)') title('Plot of Total Weight vs \Theta') plot(x,Weight,'-k') elseif G==1 xlabel('\Theta - Degrees') ylabel('Force P (N)') title('Plot of Force P vs \Theta') plot(x,P) elseif H==1 xlabel('\Theta - Degrees') ylabel('Length (m)') title('Plot of Main Strut Length vs \Theta') plot(x,mLD) elseif I==1 xlabel('\Theta - Degrees') ylabel('Length (m)') title('Plot of 3D Sec Strut Length vs \Theta') plot(x,rsLD,'-k') elseif J==1 xlabel('\Theta - Degrees') ylabel('Weight (kg)') title('Plot of Main Strut Weight vs \Theta') plot(x,WeightM) elseif K==1 xlabel('\Theta - Degrees') ylabel('Weight (kg)')



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title('Plot of Sec Strut Weight vs \Theta') plot(x,WeightS) elseif M==1 xlabel('\Theta - Degrees') ylabel('Wall Thickness (m)') title('Plot of main Strut Wall thickness vs \Theta') plot(x,WallThickness,'-k') elseif N==1 xlabel('\Theta - Degrees') ylabel('Wall Thickness (m)') title('Plot of main Strut Wall thickness vs \Theta') plot(x,WallThicknessS,'-k') end hold off

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MATLAB STRESSVSTHETA FUNCTION

function [x,ystress,yshear,P,alpha,mL,sL] = StressVsTheta(theta,Clearance,Gv,Gh,xoffset,a,MOutsideDiameter,MInsideDiameter)

% Variable List % ============= LevAcur = 100; % Level of accuracy variable controls number of iterations

that are run for each variation of the strut % Counter Variables for plotting cx=1; % Moment of Inertia Calculation % ============================= Ixy_main = (1/4*pi*((MOutsideDiameter/2)^4)) -

(1/4*pi*((MInsideDiameter/2)^4)); %Ixy_sec = (1/4*pi*((SOutsideDiameter/2)^4)) -

(1/4*pi*((SInsideDiameter/2)^4)); MaxStress(cx) = 0; % Max stress determined in this iteration of the strut L = Clearance / cos(theta); % Determining alpha sL = sqrt( xoffset^2 + (a)^2 - (2*(a)*xoffset*cos((pi/2) - theta))); alpha = asin(xoffset*( (sin((pi/2)-theta))/sL)); % Determine forces acting on main strut P(cx) = ( Gh*L*cos(theta) - Gv*L*sin(theta)) / ( (L-a)*sin(alpha) +

L*sin(theta - alpha)*cos(theta) - L*sin(theta)*cos(theta-alpha) ); Lv(cx) = Gv - P(cx)*cos(theta - alpha); Lh(cx) = Gh - P(cx)*sin(theta-alpha);



Page | 82

% Shear Stress on main strut % ========================== MArea = (1/4*pi*((MOutsideDiameter)^2)) - (1/4*pi*((MInsideDiameter)^2)); % Angle1 = tan(Gh/Gv); %Angle1 is angle between combined force acting

from ground and main strut % % GF = sqrt(Gv^2 +Gh^2); % % Gs = sin(theta-Angle1)* GF; MaxShear(cx)= 1.5 * abs(Gv*sin(theta) - Gh*cos(theta)) / MArea; % for L = 0: Length/LevAcur: Length % % % % Bending Moment % % M(i) = Fy * L; % % BS(i) = (M(i)* (OutsideDiameter/2))/Ixy; % % if BS(i) > MaxStress % MaxStress = BS(i); % end % % i=i+1; % % end % Bending stress on main strut % ======================= i=1; %Angle2 = tan(Lv(cx)/Lh(cx)); %Angle2 is angle between combined force

acting from lunar module and main strut %LF = sqrt(Lv(cx)^2 +Lh(cx)^2); %Ls = sin((pi/2)-theta -Angle2)* LF; Ps = sin(alpha)* P(cx); for Length = 0:L/LevAcur : L

Lunar Landing Gear


Page | 83

% Bending Moment if Length <= (L-a) M(i)= (Gv*sin(theta))*Length - (Gh*cos(theta))*Length

- Ps*((L-a)- Length) - Lv(cx)*sin(theta)*(L-Length) + Lh(cx)*cos(theta)*(L-Length);

else M(i)= (Gv*sin(theta))*Length - (Gh*cos(theta))*Length -

Ps*(Length - (L-a)) - Lv(cx)*sin(theta)*(L-Length) + Lh(cx)*cos(theta)*(L-Length);

end BS(i) = (M(i)* (MOutsideDiameter/2))/Ixy_main; if BS(i) > MaxStress(cx) MaxStress(cx) = BS(i); end i=i+1; end % Axial Stress on strut % ======================= Ax(cx) = (Gv*cos(theta) + Gh*sin(theta)) / MArea; % Max Stress on strut % =================== MaxStress(cx) = MaxStress(cx) + Ax(cx); % Weighted Score % ============== %WeightedScore = (Weight_Stress * (MaxStress / AbsMaxStress)) +

(Weight_Cost * ( x = theta*180/pi; ystress = MaxStress(cx); yshear = MaxShear(cx); P = P(cx); mL=L;



Page | 84

APPENDIX C

Figure 38 - Rigid Landing Structure Energy Analysis

k1

Min

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e

Min

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e

Dpad Apad Aimpact B*Aimpact Error: B*Aimpact Δ Error: Δ Fmax Error: Fmax amax Error: amax Ge's Error: Ge's Worst Case Ge

[m] [m2] [m2] [N/m] ±[N/m] [m] ±[m] [N] ±[N] [m/s2] ±[m/s2] [unit] ±[unit] [unit]0.1 0.008 0.024 15990 3198 0.195 0.024 3118 731 16.4 4.2 1.67 0.43 2.100.11 0.010 0.029 19347 3869 0.177 0.022 3430 804 18.1 4.6 1.84 0.47 2.310.12 0.011 0.034 23025 4605 0.162 0.020 3742 877 19.7 5.0 2.01 0.51 2.520.13 0.013 0.040 27022 5404 0.150 0.018 4053 951 21.3 5.4 2.18 0.55 2.730.14 0.015 0.046 31340 6268 0.139 0.017 4365 1024 23.0 5.9 2.34 0.60 2.940.15 0.018 0.053 35977 7195 0.130 0.016 4677 1097 24.6 6.3 2.51 0.64 3.150.16 0.020 0.060 40933 8187 0.122 0.015 4989 1170 26.3 6.7 2.68 0.68 3.360.17 0.023 0.068 46210 9242 0.115 0.014 5301 1243 27.9 7.1 2.85 0.73 3.570.18 0.025 0.076 51806 10361 0.108 0.013 5612 1316 29.5 7.5 3.01 0.77 3.780.19 0.028 0.085 57722 11544 0.103 0.013 5924 1389 31.2 7.9 3.18 0.81 3.990.2 0.031 0.094 63958 12792 0.097 0.012 6236 1462 32.8 8.4 3.35 0.85 4.200.21 0.035 0.104 70514 14103 0.093 0.011 6548 1536 34.5 8.8 3.52 0.90 4.410.22 0.038 0.114 77389 15478 0.089 0.011 6860 1609 36.1 9.2 3.68 0.94 4.620.23 0.042 0.125 84585 16917 0.085 0.010 7171 1682 37.7 9.6 3.85 0.98 4.830.24 0.045 0.136 92100 18420 0.081 0.010 7483 1755 39.4 10.0 4.02 1.02 5.040.25 0.049 0.147 99935 19987 0.078 0.010 7795 1828 41.0 10.5 4.19 1.07 5.250.26 0.053 0.159 108089 21618 0.075 0.009 8107 1901 42.7 10.9 4.35 1.11 5.460.27 0.057 0.172 116564 23313 0.072 0.009 8418 1974 44.3 11.3 4.52 1.15 5.670.28 0.062 0.185 125358 25072 0.070 0.009 8730 2047 45.9 11.7 4.69 1.20 5.880.29 0.066 0.198 134472 26894 0.067 0.008 9042 2121 47.6 12.1 4.86 1.24 6.09

0.3 0.071 0.212 143906 28781 0.065 0.008 9354 2194 49.2 12.6 5.02 1.28 6.300.31 0.075 0.226 153660 30732 0.063 0.008 9666 2267 50.9 13.0 5.19 1.32 6.510.32 0.080 0.241 163733 32747 0.061 0.007 9977 2340 52.5 13.4 5.36 1.37 6.720.33 0.086 0.257 174126 34825 0.059 0.007 10289 2413 54.2 13.8 5.53 1.41 6.930.34 0.091 0.272 184839 36968 0.057 0.007 10601 2486 55.8 14.2 5.69 1.45 7.140.35 0.096 0.289 195872 39174 0.056 0.007 10913 2559 57.4 14.6 5.86 1.49 7.360.36 0.102 0.305 207225 41445 0.054 0.007 11225 2632 59.1 15.1 6.03 1.54 7.570.37 0.108 0.323 218897 43779 0.053 0.006 11536 2706 60.7 15.5 6.20 1.58 7.780.38 0.113 0.340 230889 46178 0.051 0.006 11848 2779 62.4 15.9 6.36 1.62 7.990.39 0.119 0.358 243201 48640 0.050 0.006 12160 2852 64.0 16.3 6.53 1.66 8.200.4 0.126 0.377 255833 51167 0.049 0.006 12472 2925 65.6 16.7 6.70 1.71 8.410.41 0.132 0.396 268785 53757 0.048 0.006 12784 2998 67.3 17.2 6.87 1.75 8.620.42 0.139 0.416 282056 56411 0.046 0.006 13095 3071 68.9 17.6 7.03 1.79 8.830.43 0.145 0.436 295647 59129 0.045 0.006 13407 3144 70.6 18.0 7.20 1.84 9.040.44 0.152 0.456 309558 61912 0.044 0.005 13719 3217 72.2 18.4 7.37 1.88 9.250.45 0.159 0.477 323789 64758 0.043 0.005 14031 3291 73.8 18.8 7.54 1.92 9.460.46 0.166 0.499 338339 67668 0.042 0.005 14343 3364 75.5 19.2 7.70 1.96 9.670.47 0.173 0.520 353209 70642 0.041 0.005 14654 3437 77.1 19.7 7.87 2.01 9.880.48 0.181 0.543 368399 73680 0.041 0.005 14966 3510 78.8 20.1 8.04 2.05 10.09 Threshold0.49 0.189 0.566 383909 76782 0.040 0.005 15278 3583 80.4 20.5 8.21 2.09 10.300.5 0.196 0.589 399739 79948 0.039 0.005 15590 3656 82.1 20.9 8.37 2.13 10.51

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Lunar Landing Gear


Page | 85

Figure 39 - Energy Analysis with an Absorbing Structure

Figure 40 - Mass conservation of crush cartridge

k1

Min

imiz

e

Min

imiz

e

Dpad Apad Aimpact B*Aimpact Error: B*Aimpact Δ Error: Δ Fmax Error: Fmax amax Error: amax Ge's Error: Ge's Worst Case Ge

[m] [m2] [m2] [N/m] ±[N/m] [m] ±[m] [N] ±[N] [m/s2] ±[m/s2] [unit] ±[unit] [unit]0.1 0.008 0.024 15990 3198 0.070 0.051 1115 843 5.9 4.5 0.60 0.46 1.060.11 0.010 0.029 19347 3869 0.063 0.046 1227 927 6.5 4.9 0.66 0.50 1.160.12 0.011 0.034 23025 4605 0.058 0.042 1339 1012 7.0 5.4 0.72 0.55 1.270.13 0.013 0.040 27022 5404 0.054 0.039 1450 1096 7.6 5.8 0.78 0.59 1.370.14 0.015 0.046 31340 6268 0.050 0.036 1562 1180 8.2 6.3 0.84 0.64 1.480.15 0.018 0.053 35977 7195 0.047 0.034 1673 1265 8.8 6.7 0.90 0.69 1.580.16 0.020 0.060 40933 8187 0.044 0.032 1785 1349 9.4 7.2 0.96 0.73 1.690.17 0.023 0.068 46210 9242 0.041 0.030 1896 1433 10.0 7.6 1.02 0.78 1.790.18 0.025 0.076 51806 10361 0.039 0.028 2008 1517 10.6 8.1 1.08 0.82 1.900.19 0.028 0.085 57722 11544 0.037 0.027 2119 1602 11.2 8.5 1.14 0.87 2.010.2 0.031 0.094 63958 12792 0.035 0.025 2231 1686 11.7 9.0 1.20 0.91 2.110.21 0.035 0.104 70514 14103 0.033 0.024 2342 1770 12.3 9.4 1.26 0.96 2.220.22 0.038 0.114 77389 15478 0.032 0.023 2454 1855 12.9 9.8 1.32 1.00 2.320.23 0.042 0.125 84585 16917 0.030 0.022 2565 1939 13.5 10.3 1.38 1.05 2.430.24 0.045 0.136 92100 18420 0.029 0.021 2677 2023 14.1 10.7 1.44 1.10 2.530.25 0.049 0.147 99935 19987 0.028 0.020 2789 2108 14.7 11.2 1.50 1.14 2.640.26 0.053 0.159 108089 21618 0.027 0.020 2900 2192 15.3 11.6 1.56 1.19 2.740.27 0.057 0.172 116564 23313 0.026 0.019 3012 2276 15.9 12.1 1.62 1.23 2.850.28 0.062 0.185 125358 25072 0.025 0.018 3123 2361 16.4 12.5 1.68 1.28 2.960.29 0.066 0.198 134472 26894 0.024 0.018 3235 2445 17.0 13.0 1.74 1.32 3.06

0.3 0.071 0.212 143906 28781 0.023 0.017 3346 2529 17.6 13.4 1.80 1.37 3.170.31 0.075 0.226 153660 30732 0.023 0.016 3458 2613 18.2 13.9 1.86 1.42 3.270.32 0.080 0.241 163733 32747 0.022 0.016 3569 2698 18.8 14.3 1.92 1.46 3.380.33 0.086 0.257 174126 34825 0.021 0.015 3681 2782 19.4 14.8 1.98 1.51 3.480.34 0.091 0.272 184839 36968 0.021 0.015 3792 2866 20.0 15.2 2.04 1.55 3.590.35 0.096 0.289 195872 39174 0.020 0.015 3904 2951 20.5 15.7 2.10 1.60 3.700.36 0.102 0.305 207225 41445 0.019 0.014 4016 3035 21.1 16.1 2.16 1.64 3.800.37 0.108 0.323 218897 43779 0.019 0.014 4127 3119 21.7 16.6 2.22 1.69 3.910.38 0.113 0.340 230889 46178 0.018 0.013 4239 3204 22.3 17.0 2.28 1.74 4.010.39 0.119 0.358 243201 48640 0.018 0.013 4350 3288 22.9 17.5 2.34 1.78 4.120.4 0.126 0.377 255833 51167 0.017 0.013 4462 3372 23.5 17.9 2.40 1.83 4.220.41 0.132 0.396 268785 53757 0.017 0.012 4573 3456 24.1 18.4 2.46 1.87 4.330.42 0.139 0.416 282056 56411 0.017 0.012 4685 3541 24.7 18.8 2.52 1.92 4.430.43 0.145 0.436 295647 59129 0.016 0.012 4796 3625 25.2 19.2 2.58 1.96 4.540.44 0.152 0.456 309558 61912 0.016 0.012 4908 3709 25.8 19.7 2.64 2.01 4.650.45 0.159 0.477 323789 64758 0.016 0.011 5019 3794 26.4 20.1 2.70 2.06 4.750.46 0.166 0.499 338339 67668 0.015 0.011 5131 3878 27.0 20.6 2.76 2.10 4.860.47 0.173 0.520 353209 70642 0.015 0.011 5243 3962 27.6 21.0 2.82 2.15 4.960.48 0.181 0.543 368399 73680 0.015 0.011 5354 4047 28.2 21.5 2.88 2.19 5.070.49 0.189 0.566 383909 76782 0.014 0.010 5466 4131 28.8 21.9 2.94 2.24 5.170.5 0.196 0.589 399739 79948 0.014 0.010 5577 4215 29.4 22.4 3.00 2.28 5.28

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rea)

# crush cartridges 6

[kg/m3] ± [kg/m3] [m] ± [m] [N] ±[N]

ρsolid 2700 13.5 Dcrush 0.01905 0.001 Fcrush 383.5269 40.26529

[Pa] ± [Pa] [m2] ± [m2] [J] ±[J]

σsolid 290000000 0 Acrush 0.000285023 2.99E-05 Ecrush 265.0938 36.16161

[Pa] ± [Pa] [m] ± [m]

σfoam 1345600 0 l 0.12 0.01

[kg] ± [kg]

ρrel 0.04 mcrush 0.003693898 0.000495

ρrel1 1 [m] ± [m]

δ 0.1152 0.010033



Page | 86

APPENDIX D Landing Pad Calculations:

A simplified version of the pad can be seen below in Figure 41.

Figure 41 - Simplified Foot Pad

First, the moment of inertia needed to be simplified, to solve for the thickness t. A cross sectional

view of the area can be seen below in Figure 42.

Figure 42 - Cross sectional Area of Foot pad

t is the thickness of the pad throughout. A was determined by assuming the contact area of a rock.

We assumed that a pointed, jagged rock could make contact with the pad area of 1cm2. This would

result in 1 cm sides. Therefore, assuming the worst case, with the rock being rather pointed, A was

assumed to be 1 cm.

The moment of inertia of a rectangular section used can be seen below in Equation 0-1.

Equation 0-1

𝐸𝐸 =1

12𝐴𝐴𝑖𝑖3

𝝂𝝂max

A

t

Lunar Landing Gear


Page | 87

𝐸𝐸 =1

120.01𝑖𝑖3

𝐸𝐸 = (8.333𝑥𝑥10−4)𝑖𝑖3

Using methods of superposition, the resulting moment of inertia I, could be found. Equation 0-2

below relates the total deflection with material properties and lengths. This was used to determine the

moment of Inertia.

Equation 0-2

𝜈𝜈𝑖𝑖𝑎𝑎𝑥𝑥 = +𝑃𝑃𝐿𝐿3

3𝐸𝐸𝐸𝐸

P= 4079 N (force per foot pad)

L = 0.125 m. This was found by first knowing the whole length was 30cm. The ball joint is assumed

to be around 5 cm in diameter. Since we analyzed half of the pad, the length had to be divided

by two. It is assumed that the area under the ball joint is fixed and would not bend. Therefore

the length under the ball joint was taken away as well.

E = 113.8 GPa titanium tI-6Al- 4v annealed.

𝝂𝝂max= assumed to be, in worst case scenario, 0.005m.

𝜈𝜈𝑖𝑖𝑎𝑎𝑥𝑥 = +𝑃𝑃𝐿𝐿3

3𝐸𝐸𝐸𝐸

𝐸𝐸 = +𝑃𝑃𝐿𝐿3

3𝐸𝐸𝜈𝜈𝑖𝑖𝑎𝑎𝑥𝑥

𝐸𝐸 = +(4079)0.1253

3(113.8𝑥𝑥109)(0.005)

𝐸𝐸 = 4.667𝑥𝑥10−9

Therefore, from Equation 0-1 above,



Page | 88

𝐸𝐸 = (8.333𝑥𝑥10−4)𝑖𝑖3

𝑖𝑖3 =𝐸𝐸

(8.333𝑥𝑥10−4)

𝑖𝑖3 =4.667𝑥𝑥10−9

8.333𝑥𝑥10−4

𝑖𝑖3 = 5.603𝑥𝑥10−6

𝑖𝑖 = 1.78 cm

Using a simple beam analysis and a concentrated force acting as the rock, the moment of inertia was

found. Using this moment of inertia, the thickness needed to withstand this force was found. The

thickness needed for the bottom of the pad was found to be 1.78 cm.

Lunar Landing Gear


Page | 89

APPENDIX E The error from the presented energy analysis was propagated based on the design variable

uncertainty by the method of root of the sums squared. Below is an example of some of the important

error that is propagated through the excel spread sheet in terms of the energy absorption analysis.

From Equation 6-19, the error on amax:

𝑟𝑟𝑎𝑎𝑖𝑖𝑎𝑎𝑥𝑥 = ��𝜕𝜕𝐹𝐹𝑖𝑖𝑎𝑎𝑥𝑥𝜕𝜕𝑎𝑎𝑖𝑖𝑎𝑎𝑥𝑥

𝑟𝑟𝐹𝐹𝑖𝑖𝑎𝑎𝑥𝑥 �2

+ �𝜕𝜕𝑖𝑖

𝜕𝜕𝑎𝑎𝑖𝑖𝑎𝑎𝑥𝑥𝑟𝑟𝑖𝑖�

2

From Equation 6-18, the error on Fmax:

𝑟𝑟𝐹𝐹𝑖𝑖𝑎𝑎𝑥𝑥 = ��𝜕𝜕∆1

𝜕𝜕𝐹𝐹𝑖𝑖𝑎𝑎𝑥𝑥𝑟𝑟∆1�

2

+ �𝜕𝜕𝑘𝑘1

𝜕𝜕𝐹𝐹𝑖𝑖𝑎𝑎𝑥𝑥𝑟𝑟𝑘𝑘1�

2

From Equation 6-9, the error on k1:

𝑟𝑟𝑘𝑘1 = ��𝜕𝜕𝛽𝛽𝜕𝜕𝑘𝑘1

𝑟𝑟𝛽𝛽�2

+ �𝜕𝜕𝐴𝐴𝑖𝑖𝑖𝑖𝑖𝑖𝑎𝑎𝑖𝑖𝑖𝑖𝜕𝜕𝑘𝑘1

𝑟𝑟𝐴𝐴𝑖𝑖𝑖𝑖𝑖𝑖𝑎𝑎𝑖𝑖𝑖𝑖 �2

From Equation 6-17, the error on Δ1:

𝑟𝑟∆1 = ��𝜕𝜕𝑖𝑖𝜕𝜕∆1

𝑟𝑟𝑖𝑖�2

+ �𝜕𝜕𝑔𝑔𝜕𝜕∆1

𝑟𝑟𝑔𝑔�2

+ �𝜕𝜕ℎ𝜕𝜕∆1

𝑟𝑟ℎ�2

+ �𝜕𝜕𝐸𝐸𝑖𝑖𝑐𝑐𝑐𝑐𝑐𝑐 ℎ𝜕𝜕∆1

𝑟𝑟𝐸𝐸𝑖𝑖𝑐𝑐𝑐𝑐𝑐𝑐 ℎ �2

+ �𝜕𝜕𝑘𝑘1

𝜕𝜕∆1𝑟𝑟𝑘𝑘1�

2

As an example of the partial derivatives from Equation 6-17:

𝜕𝜕𝑖𝑖𝜕𝜕∆1

= (𝑔𝑔ℎ)�𝑘𝑘1−0.5�[2(𝑖𝑖𝑔𝑔ℎ − 𝐸𝐸𝑖𝑖𝑐𝑐𝑐𝑐𝑐𝑐 ℎ)]−0.5



Page | 90

𝜕𝜕𝑔𝑔𝜕𝜕∆1

= (𝑖𝑖ℎ)�𝑘𝑘1−0.5�[2(𝑖𝑖𝑔𝑔ℎ − 𝐸𝐸𝑖𝑖𝑐𝑐𝑐𝑐𝑐𝑐ℎ)]−0.5

𝜕𝜕ℎ𝜕𝜕∆1

= (𝑔𝑔𝑖𝑖)�𝑘𝑘1−0.5�[2(𝑖𝑖𝑔𝑔ℎ − 𝐸𝐸𝑖𝑖𝑐𝑐𝑐𝑐𝑐𝑐 ℎ)]−0.5

𝜕𝜕𝐸𝐸𝑖𝑖𝑐𝑐𝑐𝑐𝑐𝑐 ℎ𝜕𝜕∆1

= −�𝑘𝑘1−0.5�[2(𝑖𝑖𝑔𝑔ℎ − 𝐸𝐸𝑖𝑖𝑐𝑐𝑐𝑐𝑐𝑐 ℎ)]−0.5

𝜕𝜕𝑘𝑘1

𝜕𝜕∆1= −

12 �𝑘𝑘1

−1.5�[2(𝑖𝑖𝑔𝑔ℎ − 𝐸𝐸𝑖𝑖𝑐𝑐𝑐𝑐𝑐𝑐 ℎ)]0.5

Lunar Landing Gear


Page | 91

APPENDIX F The following analysis models the system energy with an energy absorber housed in the struts that

can be depicted with a spring stiffness should TF(X) pursue an alternate design. It builds of off the same

assumptions found in the energy absorption analysis presented. This analysis is also preformed on a per

leg analysis to integrate further flexibility in future design of a landing system. The mass of the module

(mtot) borne by each leg will be defined as m.

Equation 0-1

𝑖𝑖 =𝑖𝑖𝑖𝑖𝑠𝑠𝑖𝑖

(# 𝑠𝑠𝑜𝑜 𝑖𝑖𝑠𝑠𝑠𝑠 𝑠𝑠𝑟𝑟𝑔𝑔𝑐𝑐)

The force obtained from this analysis will reflect a force experienced per leg.

For a static inclined strut with contained crush, the touchdown mechanism can be expressed as the

loading of a combined spring system.

The initial system state is modeled as depicted in Figure 43. The landing module will have a potential

energy depending on h, the height of the landing module at engine shutdown quantified with the lunar

surface as the datum.

Figure 43 - Simplified model of landing mechanism



Page | 92

The impact collision is assumed to behave in a linearly inelastic manner. Modeled as a spring system,

the crush and the lunar surface will experience a deformation upon touchdown. Combining the springs

systems at touchdown, the maximum acceleration developed on the landing module (amax) during

touchdown will be expressed by the maximum vertical force (Fmax) acting on the component mass (m) of

the lunar module borne by each pod leg, depicted in Figure 44.

Equation 0-2

𝑎𝑎𝑖𝑖𝑎𝑎𝑥𝑥 =𝐹𝐹𝑖𝑖𝑎𝑎𝑥𝑥𝑖𝑖

Figure 44 - Spring system modeled at impact

To obtain Fmax, the spring system must be combined to relate the total deflection of an equivalent

system to the energy that is conserved.

The inclined crush of the landing module spring system in Figure 45 must be expressed in terms of a

vertical equivalent as in Figure 46 before combining it with the lunar surface spring system.

Lunar Landing Gear


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Figure 45 - Inclined crush spring model

Figure 46 - Vertical crush equivalent spring system

The vertical deflection (∆2) is expressed as ratio of the inclined deflection (∆crush) to the incline of the

pod leg (θ).

Equation 0-3

∆𝑖𝑖𝑐𝑐𝑐𝑐𝑐𝑐 ℎ= ∆2cos(𝜃𝜃)

The force acting through the strut is first expressed in terms of the spring deflection and then can be

resolved into its vertical component.

Equation 0-4

𝐹𝐹𝑐𝑐𝑖𝑖𝑐𝑐𝑐𝑐𝑖𝑖 = 𝑘𝑘𝑖𝑖𝑐𝑐𝑐𝑐𝑐𝑐 ℎ∆crush

Equation 0-5

𝐹𝐹𝑖𝑖𝑎𝑎𝑥𝑥 = 𝐹𝐹𝑐𝑐𝑖𝑖𝑐𝑐𝑐𝑐𝑖𝑖 cos(𝜃𝜃)

Combining the result of Equation 0-3 and Equation 0-4 into Equation 0-5 will express Fmax in terms of

the vertical deflection, kcrush and θ.

Equation 0-6

𝐹𝐹𝑖𝑖𝑎𝑎𝑥𝑥 = 𝑘𝑘𝑖𝑖𝑐𝑐𝑐𝑐𝑐𝑐 ℎ𝑖𝑖𝑠𝑠𝑐𝑐2(𝜃𝜃)∆2

Comparing Equation 0-6 to Fmax in Figure 46 expressed in terms of its vertical spring deflection yields

a value for the equivalent vertical stiffness (k2) of the system.

Equation 0-7



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𝑘𝑘2 = 𝑘𝑘𝑖𝑖𝑐𝑐𝑐𝑐𝑐𝑐 ℎ𝑖𝑖𝑠𝑠𝑐𝑐2(𝜃𝜃)

An equivalent spring system Figure 47 can now be expressed from the combination of the vertical

equivalent landing module system and the system exhibited by the regolith Figure 48.

Figure 47 - Combined vertical spring system

Figure 48 - Equivalent spring system

The equivalent stiffness of the system (keq) can now be expressed in terms of k1 and k2.

Equation 0-8

𝑘𝑘𝑟𝑟𝑒𝑒 =𝑘𝑘1𝑘𝑘2

𝑘𝑘1 + 𝑘𝑘2

The total displacement of the landing module during impact (∆tot) can be expressed into the

deflection of the regolith (∆1) and of the vertical deflection of the crush (∆2).

Equation 0-9

∆𝑖𝑖𝑠𝑠𝑖𝑖 = ∆1 + ∆2

An energy balance to the equivalent system can now be applied knowing that the component mass

per leg of the landing module (m) will drop from a height h at lunar gravity (gmoon) while assuming no

losses.

Equation 0-10

Lunar Landing Gear


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12𝑘𝑘𝑟𝑟𝑒𝑒∆𝑖𝑖𝑠𝑠𝑖𝑖 2= 𝑖𝑖𝑔𝑔𝑖𝑖𝑠𝑠𝑠𝑠𝑚𝑚 (ℎ + ∆𝑖𝑖𝑠𝑠𝑖𝑖 )

Applying the quadratic equation to solve for the positive root yields the total deflection of the spring

system.

Equation 0-11

∆𝑖𝑖𝑠𝑠𝑖𝑖=𝑖𝑖𝑔𝑔𝑖𝑖𝑠𝑠𝑠𝑠𝑚𝑚𝑘𝑘𝑟𝑟𝑒𝑒

+��𝑖𝑖𝑔𝑔𝑖𝑖𝑠𝑠𝑠𝑠𝑚𝑚𝑘𝑘𝑟𝑟𝑒𝑒

�2

+2𝑖𝑖𝑔𝑔𝑖𝑖𝑠𝑠𝑠𝑠𝑚𝑚 ℎ

𝑘𝑘𝑟𝑟𝑒𝑒

A force balance on the combined spring system, previously shown in Figure 48, expresses

equivalence between the spring systems.

Equation 0-12

𝐹𝐹𝑖𝑖𝑎𝑎𝑥𝑥 = 𝑘𝑘1∆1= 𝑘𝑘2∆2

With Equation 0-11 and Equation 0-12, ∆2 or ∆1 can be expressed in terms of ∆tot and the ratio of the

spring stiffness’s.

Equation 0-13

∆1=∆𝑖𝑖𝑠𝑠𝑖𝑖

�1 + 𝑘𝑘1𝑘𝑘2�

Equation 0-14

∆2=∆𝑖𝑖𝑠𝑠𝑖𝑖

�1 + 𝑘𝑘2𝑘𝑘1�

With either Equation 0-13 or Equation 0-14 and Equation 0-12, Fmax can be calculated. Subsequently

with Equation 0-4, the maximum acceleration experienced by the landing module can be determined.



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APPENDIX G Attached are the drawings for some of the key parts of this lunar landing assembly. More design

time was focused on the analysis. The compiled assembly drawing is attached as a Solidworks file.

10 01-01 team cheese-final report_lunar landing gear

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