Composite Boom Design Final Report

Composite Boom Design Final Report Team 1

Matthew Gaudioso, John Emoto-Tisdale, Armin Moosazadeh, Jeff Kandel, Stephen Potter

Section 1: Summary

Design Spec Current Spec Achieved

δ1 1 in 0.85 in

δ2 1 in 0.16 in

δ3 1 in 0.60 in

Frequency S 1, S2, S3 > 32 hz 110.94 hz, 110.94 hz, 42.85 hz

Geometry (length) 50 ft 50 ft

Geometry (Maximum envelope) 13 inches diameter 13 inches diameter

Maximum Weight 150 lb 149.4 lb

Withstand maximum moment 550,000 lb-in 550,570 lb-in

Thermal deflection stowed NA 3.0738 in

Thermal deflection released NA -0.0102 in

For beam A, CFRP, epoxy matrix (isotropic) was used with 40% +/- 45 deg plies and 60% 0 deg plies. For

beam B, boron carbide was used.

The strategy for picking the beams was as follows:

Beam A: look for an optimization of low density and high shear modulus

Beam B: look for an optimization of low density and high elastic modulus

A hollow cylinder was used for both beams. Beam A used the maximum envelope, whereas beam B

used a slightly smaller outer diameter.

Torsional springs were going to be added to decrease δ3, but the trade-off for decreased δ3 did not

outweigh the negative consequence of increased weight as shown in Appendix C.

Characteristic Value

L_A 300 in

L_B 300 in

E_zA 17.08*10^6 psi E_zB 68.5*10^6 psi R_Ao 6.5 in R_Ai 6.357 in R_Bo 6.25 in R_Bi 6.194 in G_A 6.84*10^6 psi rho1 .054 lb/in^3 rho2 .085 lb/in^3 F_tu1 130.34*10^3 psi F_tu2 81.2*10^3 psi F_cu1 98.6*10^3 psi F_cu2 825*10^3 psi alpha1 2.22e-6 alpha2 1.89e-6

Section 2: Introduction

The design problem was to design an arm boom which could be able to withstand a deflection

requirement in three different configurations, and satisfy a frequency requirement, geometry

requirement, weight requirement, and be able to withstand a bending moment. In order to meet the

challenges, different cross-sections were considered and different materials were considered. The

design specifications that were required are shown in table 2.1.

Design Spec

δ1 1 in

δ2 1 in

δ3 1 in

Frequency S 1, S2, S3 > 32 hz

Geometry (length) 50 ft

Geometry (Maximum envelope) 13 inches diameter

Maximum Weight 150 lb

Withstand maximum moment 550,000 lb-in

Thermal deflection stowed NA

Thermal deflection released NA

Table 2.1 Design Requirements

The three different configurations that have a deflection requirement of 1 in maximum given the force

of 25 lb in the given directions are shown in Figure 2.1:

Figure 2.1a Configuration 1

Figure 2.1b Configuration 2

Figure 2.1c Configuration 3

The cross sections that were considered were the hollow cylinder, the I-beam, and the hollow

square cross section. The cross section that was predicted to be able to meet the design specifications

best was the hollow cylinder. The cross sections can be seen in Figure 2.2.

Figure 2.2 Cross-Sections

The materials that were considered were aluminum, titanium, graphite epoxy with multiple ply

combinations, CFRP epoxy matrix, and boron carbide. Titanium was originally thought to be a better

material than aluminum because titanium has a greater elastic modulus, shear modulus, and tensile

strength. One downside of titanium is that it has a greater density than aluminum which meant that the

weight requirement would be more difficult to meet for titanium. Since titanium has higher values for

the elastic modulus, shear modulus, and tensile strength, the deflection requirements will be easier to

meet for a smaller thickness than that of aluminum, and thus it was thought that the weight

requirement could still have been met with titanium along with the other specifications despite the

greater density than aluminum.

Composites were also considered in order to meet the design specifications. Graphite Epoxy

was the first composite considered. By varying the plies, different elastic modulus and shear modulus

values could be obtained. The flexibility to achieve different elastic and shear modulus values is an

advantage of composites over isotropic materials. In the first part of the arm boom, a low density and

high shear modulus was desired in order to the meet the deflection requirement and a high shear

modulus is needed to be high to stop the rotation of the second part of the arm boom in the 3rd

deflection case. A low density and high elastic modulus was desired for the second part of the beam so

as to have a small deflection for the second part of the arm boom.

Other composites were also considered. CFRP epoxy matrix was used in the first part of the

beam, which met the requirement of low density and high shear modulus. Boron Carbide was used in

the second part of the beam, which met the requirement of low density and high elastic modulus.

Torsion springs were also considered to decrease the deflection due to rotation. Figure 2.3

shows the torsion spring.

Figure 2.3 Torsion Spring

Section 4: Technical Development

The manipulator arm booms, which are a part of the mechanical arm assembly for the Remote

Manipulator System (RMS), have a set of design specifications to adhere to. During launch the arm

assembly must be firmly attached to the Shuttle Orbiter by retention latches. These latches are released

in orbit so that the arms can be deployed for duty and are re-latched after use before re-entry and

landing. Both the weight and the overall arm stiffness are important design considerations for the RMS.

Stiffness is critical from the standpoint of control system stability and operator acceptability. Thus,

based on the weight, stiffness, and geometrical requirements given, there is a great deal of technical

background that the group had to concentrate on in order to be successful. In addition, with each of the

3 configurations and design specifications, there were key assumptions that needed to be addressed to

develop the fundamental equations and data to base the designs on.

The first configuration (Case 1) was the straight, fully extended arm that was exposed to a

bending moment from the load acting upon the tip of the arm. The way we solved the deflection of this

case was with the use of Castigliano’s Theorem. This energy method may be unusual, but it is a powerful

and surprisingly simple approach to deflection analysis. Castigliano’s Theorem states that when forces

act on elastic systems subject to small displacements, the displacement corresponding to any force, in

the direction of the force, is equal to the partial derivative of the total strain energy with respect to that

force. Even though this theorem can approach many cases, as evident with expressions (equations)

relating to tension and compression, torsion, and bending, Case 1 only had to use the equation for

bending. The expression for bending is the partial of the total strain energy with respect to the moment,

as seen below.

∫

(

)

The assumption made for solving the displacement of Case 1 is that the arm is exposed purely to a

bending moment. In reality, the arm is also exposed to shear. However, when referring to Shigley’s

Mechanical Engineering Design book (page 165) and solving for the total energy of the arm, the error

introduced when shear is neglected is less than 1 percent. Thus, the key assumption is that the shear

effects the displacement in such a minute level that we approximate it to be zero. Furthermore, with the

calculations of Castigliano’s Theorem, we achieved the displacement in Case 1 to be:

Purely to simplify the equation, if LA = LB = L we would get:

The second configuration (Case 2) was the arm forming right angle, load in-plane. Similar to the previous

case, Case 2 was exposed to a bending moment from the load acting upon the tip of the arm. The way

we solved the deflection of this case was with the use of superposition. The arm booms were broken

down into 2 separate arms. The displacement for the second half of the arm was modeled as a simple

cantilever, and the moment that was formed from the second arm was superimposed into the first arm.

The assumption made in this case was the small angle approximation, where sinθ = θ. This angle θ

relates to the movement in the second arm from the tip load. Since it such a small displacement

(maximum 1 inch for a 50 foot boom), we can assume the sine of θ is essentially θ. Thus, the equation

for the total displacement of Case 2 is:

Purely to simplify the equation, if LA = LB = L we would get:

The third configuration (Case 3) was the arm forming right angle, load normal to plane. As with the

other two cases, Case 3 had a bending moment, but was also exposed to torsion due to the load acting

normal to the plane. Consequently, the problem of determining the displacement was broken down into

three segments: displacement in the first half of the arm due to bending, displacement in the second

half of the arm due to bending, and displacement in the first half of the arm due to torsion. Due to this

array of calculations, the method of superposition was applied once again in this case. In solving for the

twist angle in the displacement due to torsion, an assumption was made that was utilized in the

previous case. This assumption is indeed the small angle approximation, where sin(θ) = θ. Thus, the

equation for the total displacement of Case 3 is:

Simplifying this equation, where LA = LB = L and J

, we get:

The relative equations of Young’s Modulus and Moment of Inertia for arms A and B in all 3 cases can be

seen in Appendix C.

The geometrical constraints for the project involved a 50 foot arm boom with a center joint (elbow)

located at 25 feet from the end support point (shoulder). Hence, our calculations broke down the 50

foot boom into two 25 foot arms. The assumption made with this center joint is that it is built-in for

locked condition. Similarly, the elbow joint can be assumed to be rigid, for all arm positions, in the

locked condition. Finally, the maximum envelope along the boom was not to exceed 13 inches diameter.

Moving on to the next design specification, there was a maximum weight allowed, which played a

crucial role in the development of the technical capabilities. The arm boom was not to exceed 150

pounds, based on the following material options: Aluminum, Titanium, and Fiber Reinforced Composite.

However, those were the baseline materials that needed to be considered and it was allowable to work

beyond those options. To determine the weight of the arm boom, the relationship between density and

volume was utilized.

The volume was dependent on whether the geometry was a hollow cylinder, hollow square, or an I-

beam. The equations for these relative volumes can be found in Appendix C.

In order to have the designed RMS be successful, there was a design specification relating to a maximum

applied bending moment. This maximum bending moment is developed under the dynamic conditions

during launch. The design had to withstand an ultimate bending moment of 550,000 lb-in. at all

positions away from the supports to survive these conditions. The relative bending moment at which

the structure will fail is based upon the relative material’s failure stress Fu. In addition, since there were

3 different material options to work with, there were different moments of inertia. As an example, the

equation for the bending moment at which a hollowed-cylinder structure will fail is seen below. The

other equations can be seen in Appendix C.

Once the mentioned design specifications were adhered to, the arm boom had to comply with a

minimum flexural frequency in a straight stowed configuration. The design specification was known that

the flexural frequency had to be greater than 32 Hz to provide acceptable dynamic response during

launch vibration environment. The boom was to be supported with a minimum number of retention

latches. The assumption made in the calculated frequency was that the latches provided a simple

support condition. Thus, we started with 2 latches (1 on each end) and solved for the frequency. If that

value did not adhere to the 32 Hz minimum, then an addition latch would be added to increase the

frequency, and so on until the requirement was met. The equation for the stowed frequency was known

to be (as mentioned by Professor Kedward):

The boundary condition for this model is that the supports of the beam are pinned (not clamped

because that is difficult to achieve on flexible vehicle).

Since the RMS would be in extreme heat conditions, the design had to withstand a maximum thermal

deflection. Since half of the arm may be facing the sun while the other will be facing away from the sun,

the temperature difference ΔT could have been as extreme as 400 degrees F. (+200 on exposed surface,

-200 on hidden surface). This was evaluated under 2 conditions: when the arm its stowed (folded at

elbow such that wrist is next to shoulder, pinned at both ends) and when it is released (fully extended

arm). For the stowed case, the maximum deflection occurs in the middle of the two pinned edges, and

the equation for the maximum deflection is:

Where is , is the thermal deflection, L is the length of the arm in its respective

position, and h is the height from top to bottom.

For the released case, the maximum thermal deflection occurs at the tip. The maximum deflection is

characterized as:

When it comes to the construction of the arm boom, there is a designated manufacturing process that

needs to be taken account for. Based upon the material, the manufacturing process differs. In our

situation, the design involved Carbon-Fiber Reinforced Polymer. In addition, the process can vary based

upon the part being created, the required finish (outside gloss) and the number of parts being created.

The choice of ply matrix can also affect the process because the properties of the composite will be

influenced. As an instructional tool to understand its construction, a brief manufacturing flow chart is

shown below for the carbon reinforced composite.

Carbon Fiber Cloth

Layering sheets of Carbon Fiber Cloth into a mold in

shape of final product

Fill mold with epoxy and then heat or air-cure

Vacuum Bagging to harden the part (remove air bubbles)

Resin is applied to the fabric in a vacuum mold

Final Product

Section 4: Results, Conclusions, and Recommendations

To query results for different arm boom materials and geometries, the characterizing equations listed in

the previous section were coded as a Matlab function such that the user need only enter the material

properties and geometrical parameters of the design at hand. Execution of said program returns the

performance values of the particular arm boom design with respect to each requirement—see Appendix

C for code used to generate the following results. As mentioned previously, the first design materials

considered were those with isotropic properties, namely aluminum and titanium. Table 4.1 depicts the

performance values yielded given an arm boom composed of aluminum, with the weight constrained

such that it did not exceed the maximum weight requirement of 150 pounds. The cross-section for the

“shoulder to elbow” member was kept circular due to the torsional stiffness requirement, while three

different cross-section geometries were considered for the “elbow to wrist” member — hollow circular,

I-beam, and hollow square — as seen in the following results tables. Another design approach would

have been to determine the minimum boom weight required to meet all performance requirements,

but using a benchmark weight was the technique invoked to generate all of the following results.

Table 4.1: Aluminum (E = 10.15 Msi; Gxy= 3.77 Msi; Fu = 35.97 ksi; ρ = .0975 lb/in3; α = 12.3 µin/in/F°

Hollow Cylinder Ro=6.5in Ri=6.44in

I-Beam h=b=9.19in hi=9.08in t=.056in

Hollow Square D=9.19in T=.07in

Restriction

Deflection 1 (in) 3.4727 3.9228 3.6642 1 in max



Failure Bending Moment (in-lb)

2.82*105 2.82*10

5 2.82*10

5 550,000 in-lb Minimum

due to dynamic launch conditions

Applied Bending Moment due to Temperature

gradient (in-lb)

1.96*105 1.36*10

5 1.92*10

5 Must be less than failure

bending moment above

Weight(lb) 142.7 146.7 146.06 150 lb max

As indicated by the above table, none of the geometry considerations producing a passing arm

boom design with aluminum. The performance of an aluminum arm boom was too poor to

consider more geometries, such as a tapered beam. Arm boom design was next attempted

with titanium, another isotropic material.

Table 4.2: Titanium (E= 16.82 Msi Gxy= 6.38 Msi Fu = 120 Ksi ρ= .163 lb/in3; α= 4.8 μin/in/F°)



Hollow Square D=9.19in T=.04in

Restriction





5.84 *105 5.84 *10

5 5.84*10

5 550,000 in-lb min due to

launch conditions

Applied Bending Moment due to Temperature

gradient (in-lb)

7.86 *104 5.3 *10

4 7.2 * 10

4 Must be less than failure

bending moment above

Weight (lb) 147.3656 149.3139 145.2912 150 lb max

Results for titanium designs were similar to those for aluminum. Neither material could meet

the deflection requirements with a total boom weight of less than 150 pounds. This is due to

the fact that, although titanium boasts a higher Elastic Young’s Modulus, its Young’s Modulus to

density ratio is nearly the same as that for aluminum — E/ρ = 104.1 and 103.2 Mega inches for

aluminum and titanium, respectively. Thus, the strength provided for 150 pounds was nearly

equal for both isotropic materials. In fact, most isotropic materials have Young’s Modulus to

density ratios roughly equal to those for aluminum and titanium. Therefore, standard isotropic

materials are not a viable choice for an arm boom design that will meet all requirements.

Further confirmation that isotropic materials would not suffice is provided by the following

vibrations analysis of the stowed configuration.

Table 4.3: Vibration Results for Isotropic Arm Boom Designs

Material Natural Frequency Passes 32 Hz requirement?

Aluminum 9.01 Hz No

Titanium 9.05 Hz No

Although no isotropic design did meet all requirements, the geometric trend was clear —

hollow circular cross-sections yield the best results, followed closely by the hollow square and

lastly the I-beam. Given all of the results for isotropic arm boom design, the intuitive conclusion

was to consider composites. Composite materials fostered more confidence designing because

they posses much higher Young’s Modulus to density ratios, which help with lowering

deflection and increasing the natural frequency of the boom.

The following table lists the results using the graphite epoxy carpet plots provided to the

ME156A class.

Table 4.4: Graphite Epoxy (7.5% 90° 40% 45° 52.5% 0° Plies E=16.7Msi Gxy=3Msi Ftu = 75ksi Fcu = 65ksi ρ=.06lb/in3 α=.000001μin/in/degF)



Hollow Square D=9in

T=.081in

Maximum Spec

Deflection 1 (in) 1.0834 1.4173 1.1756 1 in

Deflection 2 (in) 0.2646 0.7265 0.3722 1 in

Deflection 3 (in) 2.2262 3.9934 2.3570 1 in


5.943*105

compression 2.295*10

5

Compression 5.534*10

5

compression 550,550 in-lb

Bending Moment due to Temperature gradient (in-lb)

5.525*104 6.55*10

4 5.592*10

4 550,550 in-lb

Weight (lb) 149.27 149.1 149.47 150 lb

The superiority of the hollow circular cross-section was confirmed with the above results.

While usage of the given graphite epoxy did not yield passing performances, the respective

performances were much better than those for the isotropic materials considered, as expected.

Results and analysis demonstrated that adding length to the base beam and removing length

from the outer beam reduced deflection, as did adding thickness to the inner arm from the

outer arm. Choosing more 45° plies results in a higher Shear Modulus, which decreased the

deflection due to torsion. However, that corresponds to a lower Young’s Modulus, which

resulted in more deflection due to bending. There was no combination of plies for the given

epoxy that could meet both the torsional and the bending deflection at the same time. Thus,

the addition of torsion springs was next considered, as were other composite materials found

with the CES software. Since carpet plots are not provided for those composites provided by

the CES software, the carpet plot for the composite given to the ME156A class was used to

interpolate a carpet plot for the composite used from CES.

After much iteration with CES materials, the final materials/ply configurations chosen were:

Table 4.5: Final Design Materials

Arm Boom Member Material Source

Shoulder to Elbow CFRP, epoxy matrix 40% ±45° and 60% 0° plies

CES Software

Elbow to Wrist Boron Carbide CES Software

The strategy used to choose the material for the shoulder to elbow member was to optimize

the shear modulus to density ratio. The strategy for the elbow to wrist material was to

optimize the young’s modulus to density ratio. The interpolated carpet plot used to determine

the ply configuration to use for the CFRP epoxy is shown below:

The hollow circular cross-section was chosen for both members due to the fact that it

performed best for all previous material choices. Torsional springs were added to decrease the

torsional deflection, but the trade-off for decreased deflection did not outweigh the negative

consequence of increased weight, so they were not included in the final design. The following

are the final design parameters:

Characteristic Value

Length Beam A = Length Beam B 300 in

Young’s Modulus Beam A 17.08*10^6 psi

Young’s Modulus Beam B 68.5*10^6 psi

Outer Radius Beam A 6.5 in

Inner Radius Beam B 6.357 in

Inner Radius Beam A 6.25 in

Inner Radius Beam B 6.194 in

Shear Modulus Beam A 6.84*10^6 psi

Density Beam A .054 lb/in^3

Density Beam B .085 lb/in^3

Ultimate Tensile Strength Beam A 130.34*10^3 psi

Ultimate Tensile Strength Beam B 81.2*10^3 psi

Ultimate Compressive Strength Beam A 98.6*10^3 psi

Ultimate Compressive Strength Beam B 825*10^3 psi

Coefficient of Thermal Expansion Beam A 2.22e-6

Coefficient of Thermal Expansion Beam B 1.89e-6

The vibration design process for the final design is shown below. Because each member had

different weight and dimensions, the arm boom could not be treated as a beam with uniformly

distributed mass. Thus a third pin was added in the middle to treat it as two beams.


CFRP Epoxy Matrix (40% 45

o, 60% 0

o)

Boron Carbide

27.74 Hz (Beam A) 42.85 Hz (Beam B)

No Yes

Table 4.6: Vibration Results, 3 pins

Using three pins did not pass the minimum requirement of 32 Hz natural frequency for both

beams. The solution was to add a fourth pin in the midpoint of Beam A and treat the arm boom

as 3 beams.


CFRP Epoxy Matrix (40% 45

o, 60% 0

o)

Boron Carbide

110.95 Hz (Beam A1) 110.95 Hz (Beam A2) 42.85 Hz (Beam B)

Yes Yes Yes

Table 4.7: Vibration Results, 4 pins

As shown above, four pins yielded natural frequencies that exceed the minimum requirement.

The following is the table of all of the achieved performances versus the requirements.

Design Spec Current Spec Achieved

δ1 1 in 0.85 in

δ2 1 in 0.16 in

δ3 1 in 0.60 in

Frequency

Section 1, Section 2, Section 3

> 32 hz 110.94 hz, 110.94 hz, 42.85 hz

Geometry (length) 50 ft 50 ft

Geometry (Maximum envelope) 13 inches diameter 13 inches diameter

Maximum Weight 150 lb 149.4 lb

Withstand maximum moment 550,000 lb-in 550,570 lb-in

Thermal deflection stowed NA 3.07 in

Thermal deflection released NA -.0102 in

Table 4.8: Performance Achievements vs. Requirements

All requirements were achieved.

Future improvements could include testing of other materials for more desirable properties

that allow the implementation of a larger safety factor. More ply orientations can be tested to

meet more desirable material properties as well. Geometrical considerations such as tapering

could be explored to enhance performance. Mechanical behavior such as fracture toughness

for the materials chosen could be studied to ensure a more robust design.

Appendix A: Sources

Material Sources

Instructor provided carpet plots; Source: Original Version of Advanced Composite Design Guide

Ashby, Mike. CES EduPack. Granta Design, 2011. Computer software.

Formulas and Equations

Budynas, Richard G., J. Keith. Nisbett, and Joseph Edward. Shigley. Shigley's Mechanical Engineering

Design. New York: McGraw-Hill Higher Education, 2010. Print.

Appendix B: Project Team with Assigned Responsibilities

Matthew Gaudioso: Team Leader, deflection and thermal analysis, matlab coding

John Emoto-Tisdale: Deflection analysis, geometrical design, matlab coding

Armin Moosazadeh: Vibrational analysis and design, geometrical design

Jeff Kandel: Deflection and thermal analysis, material research and design

Stephen Potter: Vibrational analysis and design, material research and design

Appendix C: Drawing Package

See attached hand calculations and Matlab code

Appendix D: Presentation Package

See attached presentation

Composite Boom Design Final Report

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