Projects Using CES EduPack - MAELabs UCSD

Substituting (3) into (5) and then (4) into the new equation gives:
L
2
L
2
∆T
δ = = α
(6)
2R
2 h
where δ is the tip deflection (in), and α is the thermal expansion coefficient (°F -1 ).
The mass of the beam is:
m = ρ A L
(7)
The bending frequency can be found in Appendix A.12 of Ref 1 as:
3.
52 EI
f = (8)
2π
mL
3
Substituting (7) into (8) gives:
3.
52 EI
f = (9)
2π
4
ρAL
Using the definition of shape factor and substituting gives:
I I φ A
φ 4π
B
B = ⇒ =
(10)
A
2 A 4π
3.
52 EφB
A
f = (11)
2π
ρ 4πL
4
where f is the natural frequency (Hz), E is Young’s Modulus (psi), and I is the
bending moment of inertia (in 4 ) and m is the mass of the beam (lbm), φB is the
bending shape factor and A is the beam cross sectional area (in 2 ). Note that for
this formula to work you must account for the difference between lbf and lbm
using the conversion 1 lbf = 386.4 lbm-in/s 2 or 1 lbf = 32.2 lbm-ft/s 2 .
Consideration of shape factors
In addition to selecting the appropriate material for this application, you are to
also consider what the appropriate shape for this application might be. See
Reference 1, Chapter 11 and especially Table 11.3 for a description of shape
factors. Make your materials selection based on solid cross-sections and then
optimize the performance of your antenna by considering various shapes.
Given these design criteria, use the CES EduPack software, Level 3, to narrow the
list of possible materials.
References
1. “Materials Selection in Mechanical Design”, 3 rd Edition, by M.F. Ashby,
Butterworth Heinemann, Oxford 2005.
This Project is courtesy of Dr. Tom Dragone at Orbital Science Corporation and
Professor Kevin Hemker of the Engineering Department, Johns Hopkins
University.
© Granta Design, March 2009 39