Professor Kedward TA: Zi Yie ME 154 Team 1: Matthew Gaudioso, Jeff Kandel, Armin Moosazadeh, Stephen Potter, John Emoto-Tisdale

Y X Z

Ea, Ia

Eb, Ib

La

Lb

For our design, we considered these three cross-sections. The hollow cylinder cross-section was used for Beam A in all configurations.

Ri

All three cross-sections were constrained so that the diagonal is 13 inches max.

t h

hi

Circle Square

Ro=6.5in

b t

I-beam

D

Deflection Stiffness: Case 1 ◦ The result for the deflection was found using a finite element method and was also found using Castigliano’s Theorem.

Deflection Stiffness: Case 1

Castigliano’s Theorem Energy method for deflection analysis Very useful when forces act on elastic systems subject to small displacements bending

ď ˝

Deflection Stiffness: Case 2

â—Ś Using theory of superposition and using a finite element method

Deflection Stiffness: Case 3 ◦ Using theory of superposition

Bending Strength

◦ Bending Moment at which the structure will fail given the material’s failure stress Fu. ◦ For composite materials the lowest failure stress was always the compressive failure stress.

Hollow cylinder

I-Beam

Hollow Square

Thermal Deflection ◦ V is the maximum deflection due to thermal expansion. ◦ The section with the higher deflection due to thermal expansion is shown on the tables on the later slides. T1

T2

Stowed

Released

ď ˝

Stowed Frequency

Minimum Flexural 32 Hz Frequency Boundary Conditions Pinned ends on beam

ď ˝

Weight â—Ś Section A for all situations uses the hollow cylinder geometry to reduce torsion.

đ?‘Š = đ?œŒđ?‘‰ Hollow cylinder

Hollow Square

I-Beam

Hollow I-Beam h=b=9.19in Cylinder Ro=6.5in hi=9.08in Ri=6.44in t=.056in

Hollow Square D=9.19in T=.07in

Restriction

Deflection 1 (in)

3.4727

3.9228

3.6642

1 in max

Deflection 2 (in)

0.7597

1.2098

0.9511

1 in max

6.9865

4.5657

1 in max

Deflection 3 (in)

4.3743

Failure Bending Moment (in-lb)

2.82*105

2.82*105

2.82*105

550,000 in-lb Minimum due to dynamic launch conditions

Applied Bending Moment due to Temperature gradient (in-lb)

1.96*105

1.36*105

1.92*105

Must be less than failure bending moment above

Weight(lb)

142.7

146.7

146.06

150 lb max

No geometry passed all conditions with Aluminum. Many conditions failed by too large of a margin to consider Aluminum a viable choice. We considered Titanium next.

Hollow Cylinder Ro=6.5in Ri=6.46in

I-Beam

Restriction

hi=9.125in t=.033in

Hollow Square D=9.19in T=.04in

Deflection 1 (in)

3.3803

3.8419

3.6119

1 in max

Deflection 2 (in)

0.7395

1.2010

0.9710

1 in max

Deflection 3 (in)

4.1873

6.8301

4.4189

1 in max

5.84 *105

5.84*105

550,000 in-lb min due to launch conditions

Failure Bending Moment (in-lb)

5.84 *105

h=b=9.19in

Applied Bending Moment due to Temperature gradient (in-lb)

7.86 *104

5.3 *104

7.2 * 104

Must be less than failure bending moment above

Weight (lb)

147.3656

149.3139

145.2912

150 lb max

ď ˝

Stowed Frequency: Hollow Cylinder Material

Natural Frequency

Passes 32 Hz requirement?

Aluminum

9.01 Hz

No

Titanium

9.05 Hz

No

Aluminum and titanium failed the vibrations test as well. So, we moved on to composites.

No basic geometry can allow Titanium to satisfy our requirements. Part still failed by too large of a margin to consider Titanium a viable design choice.

Isotropic materials are a poor choice.

Composites are now considered.

Hollow Cylinder Ro=6.5in Ri=6.43in

I-Beam

Maximum Spec

hi=9.12in t=.036in

Hollow Square D=9in T=.081in

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

Failure Bending Moment (in-lb)

5.943*105 2.295*105

5.534*105 550,550 in-lb compression

Bending Moment due to Temperature gradient (in-lb)

5.525*104 6.55*104

5.592*104

550,550 in-lb

Weight (lb)

149.27

149.47

150 lb

compression

h=b=9.19in

Compression

149.1

Of the cross-sections tested, the hollow circular cross-section proved to be the best for all requirements Changes in Length

◦ Adding Length to Beam A and removing from beam B reduces deflection due to torsion by decreasing the lever arm that is deflected.

Changes in Thickness

◦ Adding thickness to beam A decreases all deflection at the risk of adding more weight.

Choosing more 45⁰ Plies to limit deflection ◦ A higher Shear Modulus can decrease the deflection due to torsion in case 3, but since it corresponds to a lower Young’s modulus the bending deflection in all cases will increase

Choosing Plies to increase Ftu & Fcu

◦ Decreasing the % 45 plies results in higher values of Ftu & Fcu. The higher the values, the better it can withstand maximum applied bending moment

Addition of Torsional springs at elbow may decrease deflection due to torsion for the case 3 out of plane load. Other composite materials were next considered.

Materials from CES Software Database • 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 Young’s 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.

Final Design Parameters Characteristic

Value

Length Beam A

300 in

Length Beam B

300 in

Youngâ€™s Modulus Beam A

17.08*10^6 psi

Youngâ€™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

Stowed Frequency: Hollow Cylinder Different weight and dimensions (inner/outer radii) in both members of the beam Cannot treat it as beam with uniformly distributed mass Solution: Add third pin in middle and treat it as 2 beams

Beam A

Beam B

Material

Natural Frequency

Passes 32 Hz requirement?

CFRP Epoxy Matrix (40% 45o, 60% 0o)

27.74 Hz (Beam A)

No

Boron Carbide

42.85 Hz (Beam B)

Yes

Stowed Frequency: Hollow Cylinder 3 pins (both ends and middle) did not pass minimum requirement of 32 Hz for both beams Beam A (left beam) failed Solution: Add fourth pin in midpoint of Beam A and treat it as 3 beams Beam A1

Beam A2

Beam B

Material

Natural Frequency

Passes 32 Hz requirement?

CFRP Epoxy Matrix (40% 45o, 60% 0o)

110.95 Hz (Beam A1) 110.95 Hz (Beam A2)

Yes Yes

Boron Carbide

42.85 Hz (Beam B)

Yes

Design Spec vs. Current Spec Achieved 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

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

110.94 hz, 110.94 hz, 42.85 hz

Other materials can be tested for more desirable properties that allow the implementation of a safety factor in the design.

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 fatigue strength and fracture toughness could be studied to ensure a more robust design.

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