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Loading and Stress Analysis for Hypersonic Flight
Conditions of Falcon Launch VII
Adam L. Comer1
USAF Academy, Colorado Springs, CO, 80840
In an effort to reach new altitudes, the Falcon Launch VII program at the Air Force
Academy selected a two-stage rocket featuring passive separation and a purely kinetic
second stage. This design and its predicted hypersonic speeds presented new design and
analysis challenges for the program. With limited simulation capabilities and hypersonic
flow knowledge, the team determined the expected aerodynamic loads and center of
pressure. Combined with an estimated center of gravity, pitching mass moment of inertia,
simulation data, and common aerospace assumptions, the aerodynamic data were used to
ascertain the overall rocket dynamics in a worst-case situation. Cut free body diagrams and
more detailed aerodynamic data for sections of this vehicle were used to determine the loads
and stresses throughout the rocket. A brief description of how testing was applied to reduce
the risk posed by the uncertainties of this analysis is also included.
Nomenclature
α
α’
A
ax
ay_cg
ay_cg_cut
CNbody
CNfin
cg
cp
D
dbh
dbolt
dcomp
dcp
dcp_bb
dcp_bf
dcp_db
dcp_df
dedge
FN
Fx
Fy
I
Iz_cg
Iz_cg_cut
Lcut
Lbb
Ldart
1
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
body angle of attack
fin angle of attack
cross sectional area at a section
axial acceleration of the rocket
acceleration of the center of gravity of the entire rocket in the y-direction
acceleration of a cut section of the rocket in the y-direction
normal force coefficient for a body
normal force coefficient for a fin
center of gravity
center of pressure
force of drag on rocket
diameter of a bolt hole
diameter of a fastener
diameter of a component
axial distance between the total normal force cp and rocket cg
axial distance between the booster body normal force cp and rocket cg
axial distance between the booster fin normal force cp and rocket cg
axial distance between the dart body normal force cp and rocket cg
axial distance between the dart fin normal force cp and rocket cg
distance from the center of a fastener to the edge of a joined component
axial normal force at a cross section
force in the x-direction
force in the y-direction
moment of inertia at a cross section
mass moment of inertia of the rocket about its cg
mass moment of inertia of a cut section of the rocket about the rocket cg
length of a cut section
length of booster body
exposed length of dart when the rocket is fully assembled
FLVII Mechanical Team Chief of Design and Analysis, Department of Astronautics, P.O. Box 1131 USAF
Academy, CO 80841, Student.
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American Institute of Aeronautics and Astronautics
l/d
m
mcut
Mz
Mz_cg
n
Nbody
Nfin
Nbb
Nbf
Ndb
Ndf
NTotal
Peq
Q
σ
σe
σM
σmin
r
Sfin
Sref
τ
τbolt
τto
t
T
V
xcg
xcp
xcp_bb
xcp_bf
xcp_db
xcp_df
xcut
y
ωdot
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
length of body over body diameter
mass of entire rocket
mass of cut
moment about the z axis
moment about the z axis at the rocket center of gravity
number of bolts at a bolted joint
normal force on a body
normal force on a fin
normal force on the booster body
normal force on the booster fins
normal force on the dart body
normal force on the dart fins
total normal force on the rocket
equivalent normal force at a cross section
dynamic pressure
normal stress
equivalent normal stress
normal stress due to bending moment
normal stress at the minimum cross sectional area of a bolt pattern
distance from the axis of a rocket to the interface between two joints
fin planform area
reference area for fin normal force calculations
shear stress at a cross section
shear stress along the cross section of a bolt
shear stress along the shear tear out area of a bolted joint
thickness of a component
thrust force
shear force at the cut end of a section of the rocket
location of the center of gravity of the entire rocket measured from the nose of the dart
location of the center of pressure of the entire rocket’s normal force measured from the dart’s nose
location of the center of pressure of booster body normal force measured from the nose of the dart
location of the center of pressure of booster fin force measured from the nose of the dart
location of the center of pressure of dart body force measured from the nose of the dart
location of the center of pressure of dart fin normal force measured from the nose of the dart
distance between cut of the section and the overall cg of the rocket
maximum possible distance from the neutral axis for a cross section
angular acceleration of the rocket
I. Introduction
W
ITH the major goal of reaching the edge of space, the Falcon Launch rocket program chose a two-stage
rocket design with a low drag, passively separated upper stage. Simulation data indicated that a short, high
thrust burn would be optimal for the program
requirements. This type of boost phase
resulted in the highest acceleration and
velocity values predicted in program history,
and to exacerbate the difficulty of the flight
environment and design process, the high
Figure 1. CAD rendering of rocket.
velocities were predicted for relatively low
altitudes. Such extreme flight conditions and a completely new design required new loading and stress analysis tools
for the program to design and dimension the rocket to meet the program’s factor of safety of 1.5. Relying on the
flight heritage of previous one-stage designs and implementing the same analysis conservatisms of previous years
were simply not an option.
The final design is shown in Figure 1. The assembled rocket has a total length of 139 in, consisting of 45 in of
exposed dart length (total length of 51 in) and 94 in of booster length. The propellant case of the booster is bolted to
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American Institute of Aeronautics and Astronautics
an avionics bay which is connected to the interstage. A boat tail of approximately 6 inches in length fits inside the
interstage of the booster, providing an interface between the booster and dart with no fasteners.
II. Defining the Flight Environment and Worst-Case Loads
Using Missile Datcom simulations with three degrees of freedom, the team determined that the maximum
dynamic pressure would be experienced at the end of the boost phase. The specific condition features a Mach
number of 5.53 and an altitude of approximately 15,000 ft. Furthermore, the assumed angle of attack was 2 deg in
order to obtain an estimate of normal force on the rocket.2
A. Thrust
Based on static fire testing and grain design programs, the team obtained a thrust versus time curve for the burn.
This curve featured a peak thrust value of 6700 lbf with a sharp drop in thrust at the end of the burn. To ensure
adequate conservatism, the team assumed that the thrust will be at its maximum value when the rocket is subjected
to the previously described worst-case flight conditions.
B. Normal Force
Given the small angle of attack expected for a ballistic trajectory, except at the low-velocity condition of apogee,
the team assumed that lift and normal force were approximately equal. The team applied equations for the
coefficient of normal force available in Ref. 1, and each body and each set of fins were treated separately.
Specifically, a normal force was calculated for the dart body, the dart fins, the booster body, and the booster fins.
The following equation was applied for the calculation of the coefficient of normal force for all fins:
(1)
where CNfin is the coefficient of normal force for a fin, α’ is the fin angle of attack, M is the Mach number, Sfin is the
fin’s planform area, and Sref is a fin reference area constant provided by Tactical Missile Design.1 For bodies, the
following coefficient of normal force equation was applied:
(2)
where CNbody is the coefficient of normal force of the body, α is the body angle of attack, and l/d is the ratio of body
length to body diameter.1 These coefficients were multiplied by the product of dynamic pressure (Q) and reference
area (Sref) for fins or cross sectional area (A) for bodies to determine the normal forces (N).
(3)
(4)
Applying these equations to the two bodies and two sets
of fins resulted in the normal forces summarized in Table 1.
Each set of fins features four fins, and to simplify the
analysis, it is assumed that the fins are oriented so that only
two fins produce normal forces and the other two fins are
located on the top and bottom of the body so that any
normal forces on these fins is not significantly influenced by
the angle of attack. The difference between the two booster
fin normal forces is due to a 0.2 deg fin cant used to induce
spin stabilization.
2
Component/Normal Force
Booster Fin 1
Booster Fin 2
Booster Body
Dart Fin 1
Dart Fin 2
Dart Body
TOTAL
Magnitude (lbf)
294
237
700
34
34
70
1369
Table 1. Summary of normal forces for worst case
conditions.
This angle of attack value is based on conventions for rockets with ballistic trajectories according to private
communications with Professor Bill Saylor (Schriever Chair and Visiting Professor at the U.S. Air Force Academy)
and Dr. Russ Cummings (Professor specializing in applied aerodynamics and computational fluid dynamics at the
U.S. Air Force Academy).
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American Institute of Aeronautics and Astronautics
C. Drag
Due to the complexities of calculating drag, the maximum axial acceleration and thrust values available from
Missile Datcom simulations and propulsion analyses were combined with component mass estimates from
SolidWorks CAD software to perform a conservative analysis that neglects drag to determine internal normal force.
A full discussion of this approach is presented in the “Determining Internal Loads” section.
III. Defining Rocket Dynamics
In order to determine the internal loads at various locations throughout the rocket and develop stress calculations,
one must determine the overall dynamics of the rocket and obtain critical component data. Mass estimates for every
component were obtained using SolidWorks, and once the rocket was integrated, this CAD software estimated the
center of gravity (cg) of the entire rocket as 66 in from the nose of the rocket. Using these masses and modeling each
component as a basic shape, the team determined the pitching mass moment of inertia of each component and of the
entire rocket about the overall cg, which was calculated as 141,500 lbm-in2. This mass moment of inertia calculation
was checked with the value calculated by SolidWorks. The next steps of developing the overall dynamics were the
involved processes of determining the centers of pressure and applying the equations of motion.
A. Determining Centers of Pressure
By inputting the aforementioned worst-case conditions and rocket geometry into Missile Datcom, the team’s
simulation expert was able to determine that the location of the center of pressure of the entire rocket (x cp) is
approximately 86 in from the nose of the dart at an angle of attack of 2 deg. In order to obtain enough information to
calculate internal loads, the team also needed to assign locations of action to each individual normal force
calculated. The centers of pressure for the fins were approximated as acting at half the chord length according to
common supersonic convention.2 For the two bodies, the dart and booster, the team isolated each one in Missile
Datcom and simulated the worst-case conditions. The program’s output for each body provided an approximation of
the location of the cp on each body. To ensure convergence between the empirical curves used by Missile Datcom
and the normal forces calculated using Tactical Missile Design, a weighted average of the six normal forces of Table
1 that accounted for the individual cp locations was used to approximate overall cp as follows:
(5)
where Nbf1 is the normal force on booster fin 1, Nbf2 is the normal force on booster fin 2, Nbb is the normal force on
the booster body, Ndf1 is the normal force on the dart fin 1, Ndf2 is the normal force on dart fin 2, Ndb is the normal
force on the dart body, NTotal is the total normal force, xcp is the location of the overall rocket center of pressure
measured from the nose of the dart, and all the x values are the distances from the nose of the dart to the center of
pressure of the associated normal force. Using this method, the overall center of pressure was calculated as 87 in
from the nose of the dart, which is reasonably close to the overall value predicted by Missile Datcom. Such a
similarity provided confidence in the relative magnitudes of the normal forces calculated. The calculated xcp value of
87 in was used to determine overall dynamics since its incorporation of data from both Missile Datcom and Tactical
Missile Design provides more consistency in the analysis.
B. Applying the Equations of Motion
With all the necessary cg, cp, and lift
values, the team was ready to apply equations
of motion to obtain more information about
the rocket’s dynamics. Although the
simulation featuring three degrees of freedom
provided some of the needed dynamic
information,
specifically
the
axial
acceleration, the team still needed to
determine the acceleration normal to the body
and the angular acceleration about the
pitching axis for the worst-case conditions
before analyzing cut free body diagrams. The
dynamics in the yaw direction and about the
roll axis were neglected since the forces
Figure 2. Free body diagram of rocket.
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American Institute of Aeronautics and Astronautics
generating these dynamics were considered negligible compared to the axial and pitching forces. In short, the added
complexity of a three dimensional analysis was not warranted by these relatively insignificant effects.
Figure 2 provides the free body diagram under consideration with the forces located at their approximate
locations. By applying Newton’s second law, the team developed Eqn. 6 below:
(6)
where Fy is a force in the y-direction, Ndf is the sum of the two normal forces on the dart fins, N bf is the sum of the
forces on the booster fins, m is the mass of the entire rocket at the worst-case condition, and ay_cg is the acceleration
of the rocket cg in the y-direction. When solved for ay_cg, this equation results in an acceleration of 18.3 g’s. By
summing the moments on the rocket, one obtains Eqn. 7:
(7)
where Mz is a moment about the z-axis, dcp_db is the longitudinal distance between the dart body’s cp and the rocket
cg, dcp_df is the longitudinal distance between the dart fins’ cp and the rocket cg, d cp_bb is the longitudinal distance
between the booster body’s cp and the rocket cg, dcp_bf is the longitudinal distance between the booster fins’ cp and
the rocket cg, Iz_cg is the mass moment of inertia about the z-axis, and ωdot is the angular acceleration of the rocket. 3
When solved for ωdot, this equation results in an angular acceleration of -79.9 rad/s2.
IV. Determining Internal Loads
With these overall dynamics and component data (i.e. mass moments of inertia and masses), the rocket can be
cut into sections and the equations of motion can be applied to determine the internal loads at nearly any location
along the rocket. The only location at which the team considered the dart and the booster as separate bodies is at the
booster interstage to dart boat tail interface. At this location, the team used overall dynamics to determine the
dynamics of each body separately and solved for the bearing loads between the two bodies by treating those loads as
external loads on each of the two bodies. Treating the bearing loads as external forces, the team was then able to
better approximate the internal loads within this region of interface.
A. Major Assumptions
When solving for internal loads, the team decided to ignore the torque throughout the rocket created by the
booster fin cant. Since this cant is so small
and the spin rate so low compared to the
pitch angular acceleration, this assumption
seems reasonable. Additionally, any forces
creating yaw or resulting from dynamics in
the yaw direction are ignored due to the
dominant magnitude of the normal forces
and pitching dynamics. Moreover, the
rocket is assumed to have no angular
velocity about the pitching axis because the
2 deg angle of attack is assumed to be a
maximum.
B. Applying Dynamics to a Cut Section
Figure 3 shows an example of a cut free
body diagram used to demonstrate the
process of solving for internal loads. This
Figure 3. Cut free body diagram of booster.
particular cut is at the bolt pattern
connecting the avionics bay to the booster
body. The normal force is determined by summing the forces in the x-direction, resulting in the following:
–
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American Institute of Aeronautics and Astronautics
(8)
where T is thrust, FN is the normal force at the cut, D is the drag on the section, mcut is the mass of the section, and ax
is the axial acceleration. After solving for FN, one notes that ignoring drag is a conservative assumption that will
result in a higher internal normal force than the cross section actually experiences.3
To determine the internal shear force, one must first determine the acceleration of the cut section in the ydirection. An application of rigid body kinematics and the team’s assumptions results in Eqn. 9 below:
(9)
where ay_cg_cut is the magnitude of the acceleration of the cut section in the y-direction, ay_cg is the magnitude of the
acceleration of the entire rocket in the y-direction, and xcg_cut is the distance from the overall rocket cg to the cut
section’s cg.3 Other problems that arise with a cut body are how much of the normal force calculated for the entire
body acts on the section and determining the location at which this fraction of the normal force acts on the section.
To address these complications, the team assumed that the ratio of the length of the cut to the total length of the
body is equal to the ratio of the normal force on the section to the normal force on the entire body. Furthermore, the
team assumed that the ratio of the distance between the body’s cp and the body’s leading edge to the length of the
body is equal to that of any cut section. Thus, with the acceleration value and these assumptions, the team applied
Newton’s second law to obtain the equation below:
(10)
where V is the shear force at the cut, Lcut is the length of the section, and Lbb is the total length of the booster body,
including the avionics bay and interstage. By solving for V, one obtains the internal shear, which is also critical to
solving for the internal moment at the cross section. To find the internal moment with the previously calculated
component mass moments of inertia, one must sum the moments on the section about the overall rocket cg to obtain
the following:
(11)
where Mz_cg is a moment about the z-axis of the cg, xcut is the distance between the cut and the rocket cg, xcp_bb is the
distance between the cp of the entire booster
Location
FN (lbf) V (lbf) Mz (lbf-in)
and the tip of the dart’s nose, Ldart is the
Interstage
to
Avionics
Bay
4320
-98.9
20000
length of the dart, Lbb is the length of the
Connection
booster body, Lcut is the length of the cut
Avionics Bay to Propellant 4790
-131
24400
section, and Iz_cg_cut is the mass moment of
Case
Connection
inertia of the section about the rocket’s cg.
Table 2. Summary of internal loads at critical booster locations.
When solved for Mz, Eqn. 11 provides the
internal moment. Example values are provided for two critical locations in the booster in Table 2.
C. Dart and Booster Interface
The interface between the dart and booster creates a
further complication in the analysis (see Fig. 4). To model
this situation dynamically, all the mating surfaces
between the boat tail and the interstage were replaced
with bearing loads, and the loads perpendicular to the axis
of the rocket were oriented so that a force couple is
created to provide the moment transferred between the
booster and the dart. Worst-case conditions and the
associated normal forces were assumed. Isolating either
stage, the booster or the dart, and applying the approach
outlined above by treating the booster-dart interaction
3
Figure 4. Cross sectional view of interstage.
This realization was made by Captain Matt Snyder, Instructor of Engineering Mechanics at the U.S. Air Force
Academy.
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American Institute of Aeronautics and Astronautics
forces as external forces on the body, one can apply Newton’s second law in the x-direction to solve for the normal
force on the end of the boat tail. To find the two bearing forces acting perpendicular to the axis of the rocket, the
team applied the sum of the forces in the y-direction and the sum of moments about the z-axis to develop a system
of equations to solve for each of these two bearing loads. With these bearing loads calculated and treated as external
loads, the team was then able to use each body separately to perform analyses on cross sections in the interface
region.
V. Stress Calculations
In calculating the stress at various cross sections throughout the rocket, the team used different approaches for
component cross sections, bolted joints, and welded joints. These approaches are broadly outlined below.
A. Simple Cross Sections
With the three dominant loads at the cross section determined, shear force, moment, and normal load, the team
was then able to calculate the area and moment of inertia of the cross section to find the shear stress, normal stress
due to bending moment, and normal stress due to axial load as outlined below in Eqs. 12:
(12)
where τ is the shear stress, A is the cross sectional area, σM is the normal stress due to an internal moment, y is the
maximum possible distance from the neutral axis (the axis of the rocket in this case), I is the moment of inertia of
the cross section, and σFN is the normal stress due to the axial load. 4 To combine these stresses, the normal stresses
are first summed to form a general σ value, and then the maximum-distortion-energy theory is applied to obtain the
equivalent stress (σe) of Eqn. 13, which can be compared directly to the yield strength of the component’s material.
(13)4
B. Bolted Joints
For bolted joints, four major failure modes were considered: bolt shear, bolt bearing, failure across minimum
cross section, and shear tear out. For these joints, shear force was largely ignored and Eq. 14 was used to obtain an
equivalent axial load from the calculated axial load due to thrust opposed by drag (FN) and the bending moment at
the cross section:
(14)
where r is the distance from the axis of the rocket to the mating surface of the two components.4 This equivalent
load is divided by the number of bolts to approximate the load on each bolt. To approximate the shear stress in the
bolt, the following equation was used:
(15)
where τbolt is the shear stress in the bolt, n is the number of bolts in the pattern, and Abolt is the cross sectional area of
the bolt.5 Comparing this value to the bolt’s shear strength provided the factor of safety for the bolt shear failure
mode. For bolt bearing, Eq. 16 was used to calculate the appropriate stress:
(16)
where σb is the bearing stress, t is the thickness of either of the joined components, and d bolt is the diameter of the
bolt hole.5 This stress is calculated for both of the components joined by the pattern and compared to the appropriate
component’s tensile yield strength. For failure across the cross section of minimum area, Eq. 17 was applied:
4
This equation is based on private communication with Mr. Pat O’Grady of Instar Engineering.
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American Institute of Aeronautics and Astronautics
(17)
where dcomp is the average diameter of the component with thickness t and dbh is the diameter of the bolt hole.5 These
stress values were compared to the yield strength of the appropriate component’s material to determine factor of
safety. Finally, for shear tear out of the bolts through either component’s material, Eq. 18 was applied:
(18)
where τto is the shear tear out stress in the designated component and d edge is the distance from the center of the bolt
hole to the nearest edge of the component. By comparing these stress values to the shear yield strengths of the
appropriate component materials, a final factor of safety was developed for bolted joints.
C. Welded Joints
For welded joints, the same approach delineated for simple cross sections was applied with all areas and
moments of inertia based on the thickness of the throat section of the weld. For much of the analysis, the effects of
the shear force were neglected due to the dominance of the stress due to bending moment and axial loads.
VI. Integrating Thermal Considerations
Unfortunately, the constraints on time and capabilities required the team to complete the loading analysis and
design prior to obtaining meaningful and reasonable thermal data. In the absence of reliable data, the team chose
materials with superior thermal properties or used protective insulators, such as cork, to protect the components
experiencing the most severe flow conditions. With more reliable data, future programs could use calculated
temperatures and short-term temperature data for materials to determine their strength under the expected thermal
conditions. The emphasis on short-term data is essential since many materials, such as tempered aluminums,
experience a less severe drop in material strength when exposure time is short. 5
VII. Limited Testing for a Low-Cost Program
With a tight schedule and limited financial means and machining capabilities, the team was unable to conduct
much of the proposed testing for the program. However, loading testing was conducted on the interstage due to the
critical nature of the component’s function and because of a departure from the proposed design to simplify the
machining of the component. During testing, the students discovered that one of the shortcomings of the described
analysis is the failure to predict deflection of the components. This error was discovered after deflection in the
interstage led to an interference fit between the prototype boat tail and interstage. Thus, the design was modified to
account for potential deflection and to stiffen the bearing portion of the interstage. Bending moment testing was also
conducted to determine the accuracy of the bolted joint analysis. A preliminary analysis of the data revealed a failure
to account for the deflection resulting from the settling of the bolts within their holes. Further analysis is being
conducted to understand the results of testing and the effects of testing fixture flexing on the applied conditions.
VIII. Conclusion
In conclusion, ambitious program requirements led the team to a new design with extreme flight conditions.
Such conditions have made a more detailed analysis with fewer overly conservative assumptions a must. The first
step in the team’s approach was to determine the worst-case loading condition through simulations. With this
condition established, the team determined the aerodynamic loads and their points of application to determine the
entire rocket’s dynamics. By determining the contributions of each major component, including the booster body,
booster fins, dart body, and dart fins, to the normal loads and the centers of pressure for those loads, the team was
able to apply two dimensional equations of motion to cut sections of the rocket to determine internal loads. These
loads were the basis of the stress calculations. Future improvements should incorporate reliable thermal data to
obtain more accurate factors of safety and use more testing to address the limitations of this analysis.
5
This insight about material properties was gained through private discussion with Dr. Ralph Bush (Engineering
Mechanics associate professor at the U.S. Air Force Academy).
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Acknowledgments
The contributions of many people made this paper and analysis possible. First, my instructors, including Major
Tyler Hague, Captain Luke Sauter, Captain Matt Snyder, and Professor Bill Saylor, provided guidance on the
analysis and helped me work through many of the more difficult facets of the analysis. Additionally, Instar engineer
Mr. Pat O’Grady was a major help with many of the more technical challenges, especially bolted joint analysis.
Finally, the analysis was certainly a group effort with everyone on the mechanical team doing part of the analysis.
These individuals included Cadets Ed Bae, Mary Barrett, Andrea Berger, Ken Bull, Trevor Johnson, Preston Moon,
Ozzie Ortiz, and Cedric Palacci. Special thanks go to Ken Bull for providing critical component data through
SolidWorks CAD software, Tony O’Shea for providing the vital simulation data from Missile Datcom, and Cedric
Palacci for his aerodynamic analysis.
References
1
Freeman, E. L., Tactical Missile Design, 2nd ed., AIAA Education Series, Reston, VA, 2006, pp. 44-46.
2
Brandt, S. A., Stiles, R. J., Bertin, J. J., and Whitford, R., Introduction to Aeronautics: A Design Perspective, 2nd ed., AIAA
Education Series, Reston, VA, 2004, Chap. 4.
3
Bedford, A., and Fowler, W., Engineering Mechanics: Dynamics, 4th ed., Pearson-Prentice Hall, Upper Saddle River, NJ,
2005, pp. 109, 312, 365.
4
Juvinall, R.C., and Marshek, K. M., Fundamentals of Machine Component Design, 4th ed., John Wiley & Sons, Hoboken,
NJ, 2006, pp. 117,120, 123,
5
Bush, R. W., “Fastener Failure Modes,” United States Air Force Academy, Colorado. 24 Jan. 2008.
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American Institute of Aeronautics and Astronautics