Project Bellerophon 32 4.0 Detailed Design 4.1 200g Payload 4.1.1 Vehicle Overview The launch vehicle carrying the 200 g payload (Fig. 4.1.1.1) hitches a ride on a balloon up to an altitude of 30 km where the first of three stages is ignited. At 30 km, the rocket launches in a vertical orientation from a gondola that is attached to the balloon. Once the rocket finishes burning the propellant in all three stages, the designed orbit perigee is 486 km. When random uncertainties in vehicle performance characteristics are included in the design (Monte Carlo analysis), the launch vehicle achieves an average perigee of 437 km. Fig. 4.1.1.1: Launch vehicle stack up – 200g payload. Author: Amanda Briden
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The equation of motion for a cart on a track can be · Web viewAn additional feature of the nose cone is a blunted tip made of titanium, which is a heat resistant material. The nose
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Project Bellerophon 32
4.0 Detailed Design
4.1 200g Payload
4.1.1 Vehicle Overview
The launch vehicle carrying the 200 g payload (Fig. 4.1.1.1) hitches a ride on a balloon up to an
altitude of 30 km where the first of three stages is ignited. At 30 km, the rocket launches in a
vertical orientation from a gondola that is attached to the balloon. Once the rocket finishes
burning the propellant in all three stages, the designed orbit perigee is 486 km. When random
uncertainties in vehicle performance characteristics are included in the design (Monte Carlo
analysis), the launch vehicle achieves an average perigee of 437 km.
Fig. 4.1.1.1: Launch vehicle stack up – 200g payload.(Daniel Chua)
Author: Amanda Briden
Project Bellerophon 33
4.1.1.1 Launch System Breakdown
4.1.1.1.1 Gondola and Balloon Components
Providing support to the launch vehicle and guidance at take-off, the gondola is an all aluminum
structure. To support the launch vehicle there are three equally spaced, horizontally oriented,
rings that attach to the launch vehicle’s outer structure (Fig. 4.1.1.1.1.1). Also positioned
horizontally are a square frame (at the bottom of the gondola) and flange (at the top of the
gondola). Connecting these rings and frame are four equally spaced, vertically oriented launch
rails that guide the launch vehicle off the gondola at ignition.
Variable Value UnitsVacuum Specific Impulse 352.30 sChamber Pressure 2,068,000 PaMass Flow Rate 10.69 kg/sPropellant Mass 1,462.00 kgEngine Mass 96.94 kgThrust (vac) 34,045.3 NBurn Time 136.8 sExit Area 0.543 m2
Exit Pressure 2,821.67 Pa
A conical nozzle was chosen because of the solid particles of propellant that will be coming out
of the combustion chamber. The combustion process does not necessarily combust the fuel 100%
and these particles can deteriorate a nozzle if it is let’s say Bell shaped. Some of our early MAT
codes had values based off of a 12° conical nozzle and that is one of the reasons we decided on
this cone angle for the final design. Also having a smaller cone angle reduces the divergence loss
at the exit of the nozzle. A picture of the nozzle can be seen below in Fig. 4.2.4.1.2.
Author: Stephan Shurn
Project Bellerophon 52
Figure 4.2.4.1.2: Our 12° conical nozzle
The second stage of the launch vehicle uses a solid rocket motor, with hydroxyl-terminated
polybutadiene/ ammonium perchlorate/ aluminum (HTPB/AP/AL) as the propellant. The nozzle
is a 12º conical nozzle with LITVC attached. The LITVC has the same configuration as the first
stage, with the exception of the H2O2. Since there is no H2O2 already present due to the solid
motor, a pressurized tank is added in a curved configuration sitting beneath the solid motor. The
tank wraps around the nozzle and is pressurized with gaseous nitrogen so that the H2O2 can flow
into the lines for injection. The specifics of the propulsion system can be seen in Table 4.1.4.1.2.
Variable Value UnitsVacuum Specific Impulse 309.3 sChamber Pressure 6,000,000 PaMass Flow Rate 2.728 kg/sPropellant Mass 566.64 kgEngine Mass 51.53 kgThrust (vac) 8,782.6 NBurn Time 207.7 sExit Area 0.040 m2
Exit Pressure 11,453.660 Pa
Author: Stephan Shurn
Project Bellerophon 53
The third stage of the launch vehicle uses a solid rocket motor, with hydroxyl-terminated
polybutadiene/ ammonium perchlorate/ aluminum (HTPB/AP/AL) as the propellant. The nozzle
is a 12º conical shape. The specifics of the propulsion system can be seen in Table 4.1.4.1.3.
Variable Value UnitsVacuum Specific Impulse 309.3 sChamber Pressure 6,000,000 PaMass Flow Rate 0.194 kg/sPropellant Mass 37.26 kgEngine Mass 8.40 kgThrust (vac) 625.0 NBurn Time 191.9 sExit Area 0.003 m2
Exit Pressure 11,453.660 Pa
Project Bellerophon 54
4.1.4.2 Aerothermal
In our aerodynamic analysis, we use linear perturbation theory to determine the aerodynamic
loading on the launch vehicle. Linear perturbation theory is the method in which the pressure
over the top and bottom surfaces of the launch vehicle is integrated to solve for the normal and
axial force coefficients acting on the launch vehicle. It is valid in the subsonic and supersonic
regimes, but falls apart in the transonic regime. For this reason, we have ignored the
aerodynamic outputs in the transonic regime and only pay attention to the outputs in the subsonic
and supersonic regimes. By integrating the change in pressure around the launch vehicle we are
able to solve for bending and pitching moments, drag coefficient, axial forces, normal forces,
shear forces, and the center of pressure location. All of these aerodynamic moments, coefficients,
and forces are based on the final geometry of the launch vehicle as well as the Mach number,
angle of attack, and time spent in the atmosphere.
Mach number, variation in angle of attack, use of LITVC, stage separation, as well as wind gusts
all have a large impact on the aerodynamic loadings of the launch vehicle. As the launch vehicle
makes its way through the atmosphere, the change in density also has a significant effect on the
impact of these forces and moments. The results for the variation of bending moment and
pitching moment with respect to Mach number at zero degree angle of attack can be found in
Figs. 4.1.4.2.1 and 4.1.4.2.2 respectively. Once the launch vehicle reaches a speed of Mach 4.7,
it exits the atmosphere. At this point, the first stage has still not separated; therefore, moments
are shown as they act on the entire launch vehicle.
Author: Jayme Zott
Project Bellerophon 55
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
-3500
-3000
-2500
-2000
-1500
-1000
-500
0
Mach
Bend
ing M
omen
t (Nm
)
Fig. 4.1.4.2.1: Variation of bending moment with respect to Mach number at zero angle of attack. 200g.
(Alex Woods, Jayme Zott)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
200
400
600
800
1000
1200
Mach
Pitc
hing
Mom
ent (
Nm)
Fig. 4.1.4.2.2: Variation of pitching moment with respect to Mach number at zero angle of attack. 200 g.
(Alex Woods, Jayme Zott)
The moments presented in Figs. 4.1.4.2.1 and 4.1.4.2.2 correlate well with the magnitude of
moments expected for a launch vehicle of our size and shape. It is important for us to determine
Author: Jayme Zott
Project Bellerophon 56
these moments because the structures group uses them to determine appropriate materials and
thicknesses for the final launch vehicle design.
The results for the variation of normal, axial, and shear forces with respect to Mach number at a
zero degree angle of attack can be found in Figs. 4.1.4.2.3, 4.1.4.2.4, and 4.1.4.2.5 respectively.
The normal and axial forces are important for the D&C group’s analysis. D&C uses the normal
and axial forces acting on the launch vehicle to help determine the amount of LITVC needed for
control at any given moment in time. The shear force is important for the structures group’s
analysis. Structures uses the shear force acting on the vehicle to help determine appropriate
materials and thicknesses for the final launch vehicles design.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
50
100
150
200
250
300
Mach
Norm
al Fo
rce
(N)
Fig. 4.1.4.2.3: Variation of normal force with respect to Mach number at zero angle of attack. 200 g.
(Alex Woods, Jayme Zott)
Author: Jayme Zott
Project Bellerophon 57
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
100
200
300
400
500
600
700
800
900
1000
Mach
Axial
Forc
e (N
)
Fig. 4.1.4.2.4: Variation of axial force with respect to Mach number at zero angle of attack. 200 g.
(Alex Woods, Jayme Zott)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
-200
-180
-160
-140
-120
-100
-80
-60
-40
-20
0
Mach
Shea
r For
ce (N
)
Fig. 4.1.4.2.5: Variation of shear force with respect to Mach number at zero angle of attack. 200 g.
(Alex Woods, Jayme Zott)
The variation of CD with Mach number at a constant zero angle of attack is shown in figure
4.1.4.2.6. Because the diameter of the 200g launch vehicle is quite large, the coefficient of drag
CD is also quite large.
Author: Jayme Zott
Project Bellerophon 58
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.2
0.4
0.6
0.8
1
1.2
1.4
Mach
Cd
Fig. 4.1.4.2.6: Impact of Mach number on CD at zero angle of attack. 200 g.
(Alex Woods, Jayme Zott)
As previously mentioned, we use the linear perturbation theory to determine all aerodynamic
forces, coefficients, and moments, including CD. This method requires complete knowledge of the
launch vehicle geometry before any aerodynamic forces, coefficients or moments can be
determined. This causes a problem because the trajectory analysis requires use of CD long before
the final geometry is determined. Because the CD variation shown in Fig. 4.1.4.2.6 is determined
after the final launch vehicle geometry has been designed, it cannot be used in the trajectory
analysis. Instead, we use a CD trend based on historical data for the trajectory analysis.1,2 While
this historical CD trend is not based on our own geometry, it is based on successful launch
vehicles with geometries similar to our final design. The CD based on historical data at zero angle
of attack is shown in the Fig. 4.1.4.2.7
Author: Jayme Zott
Project Bellerophon 59
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Mach
Cd
Fig. 4.1.4.2.7: Impact of Mach number on CD at zero angle of attack based on historical data. 200 g.
(Jayme Zott)
Given additional time, we could complete a better trajectory analysis by including the correct CD
based on the linear perturbation theory into the trajectory code. If we created an intermediate file
between the initial propulsion sizing output and the trajectory input, a more accurate CD value
could be used within the trajectory code. Fig. 4.1.4.2.8 shows the error caused by the using the
CD trend based on historical geometries, rather than the CD determined directly from our own
Fig. 4.1.4.2.8: Comparison of CD based on historical data and CD based on dimensional analysis (linear perturbation
theory). 200 g.
(Jayme Zott)
Table 4.1.4.2.1 Summary of Maximum Aerodynamic Loading 200 g.
Aerodynamic Load Subsonic Supersonic Bending Moment [Nm] -1850.7 -1133.1Pitching Moment [Nm] 626.6 383.6Normal Force [N] 146.6 89.8Axial Force [N]Shear Force [N]Center of Pressure [% length]Coefficient of Drag CD
Dynamic Pressure [Pa]CD % error [%]
486.9 -99.4 38.3 1.38
54
298.1 -60.8 38.3 0.85 279 17
(Jayme Zott)
References
1Sutton, George P., and Oscar Biblarz. Rocket Propulsion Elements. New York: John Wiley & Sons, Inc., 2001.
2The Martin Company, “The Vanguard Satellite Launching Vehicle”, Engineering Report No. 11022, April 1960.
Author: Jayme Zott
Project Bellerophon 61
4.1.4.3 Structures
The structural components of the launch vehicle for the 200 g payload design include a gondola
structure encasing a three-stage launch vehicle. We decided to create the gondola from
aluminum bars of 0.04 m thickness. Aluminum was chosen because of its light weight, low cost,
strength, low weight, and ease of manufacturing.
Fig. 4.1.4.3.1 Gondola frame for rocket element of launch vehicle of a 200 g payload.
(Sarah Shoemaker)
For our 200 g payload design the frame of the gondola extends 3.3460 m in height and has a
square base with sides of 0.8760 m. Ring diameters are 1.3015 m. We then have a total mass of
the gondola to 177.188 kg. This frame can withstand the approximately 55 kPa of pressure
exerted on it by our launch vehicle.
The first stage of the rocket is 7.1478 m in length and 1.3015 m in diameter. It is comprised of
an engine, oxidizer tank, fuel tank, pressurant tank, inter-tank structure, and avionics equipment.
The tanks are all made from aluminum and are of various sizes and thicknesses. An inter-stage
“skirt” connects the first and second stages of the rocket. It maintains a 10° slope from the lower
stage to the upper stage. It is also made of aluminum and is reinforced with support stringers and
ring supports. The first inter-stage structure is 1.7789 m in length. We take our upper and lower
Author: Molly Kane
Project Bellerophon 62
diameters from the diameters of the first and second stages. For this design, the maximum
diameter is 1.3015 m and the minimum diameter is 0.6741 m.
The second stage of the rocket consists of an engine, fuel tank, and avionics equipment. This
stage has a diameter of 0.6741 m and a length of 2.5594 m. The fuel tank is also made of
aluminum. The alloy Al-7075 makes all of our tanks because of its historical use and very high
strength to weight ratio. An inter-stage structure connects the second and third stages of the
rocket. This second inter-stage skirt is 1.1400 m in height with diameters ranging from 0.6741
m, maximum, to 0.2721 m, minimum, with a constant slope of 10°.
The third and final stage of the rocket, itself, reaches 0.8945 m in height and is 0.2721 m in
diameter. It is composed of an engine, fuel tank, and avionics equipment. We choose the tank to
be constructed from aluminum. The payload sits atop the thirst stage, protected by a titanium-
tipped nose cone. The nose cone is a simple cone with a blunted nose, and extends 0.3375 m.
The end is made of titanium to protect from heat, and the remainder is made of aluminum.
The inert mass breakdowns of components and stages are summarized in the following table.
Table 4.1.4.3.1: Inert Mass Breakdown of Three-Stage Rocket, 200 g Payload
Stage 1 Stage 2 Stage 3 Totals Units
Fuel Tank 68.05 47.98 3.15 119.19 kg
Ox Tank 56.03 0.00 0.00 56.03 kg
Pressure Tank 12.66 0.00 0.00 12.66 kg
Engine 96.94 51.53 8.40 156.87 kg
Pressure Addition 58.99 0.00 0.00 58.99 kg
Avionics 3.96 23.82 3.96 31.74 kg
Inter-tank 0.31 0.00 0.00 0.31 kg
Skirt/Nose Cone 11.07 3.40 1.75 16.22 kg
Total 308.02 126.73 17.26 452.01 kg
To determine the length of the first stage we take into consideration the lengths of the fist stage
nozzle, engine, fuel tank, oxidizer tank, pressurant tank, inter-tank structure, and first inter-stage
Author: Molly Kane
Project Bellerophon 63
skirt lengths. For the second stage we do not take the nozzle length into consideration because it
rests inside of the inter-stage skirt. We do include the second inter-stage skirt length in the
dimensions of the second stage. The third stage, similarly, does not include the nozzle length,
but has the addition of the nose cone height. This brings the total height of our rocket to 10.6017
m. The following table shows the specifications of each part and each stage.
Table 4.1.4.3.2: Dimensions of the Three-Stage Rocket, 200 g Payload
Stage 1 Stage 2 Stage 3 Totals Units
Nozzle Length 1.7041 0.4645 0.1239 2.2925 m
Engine Length 0.4260 0.1161 0.0310 0.5731 m
Fuel Tank Length 1.3015 1.3033 0.5260 3.1308 m
Fuel Tank Thickness 0.0037 0.0055 0.0022 n/a m
Ox Tank Length 1.3318 0.0000 0.0000 1.3318 m
Ox Tank Thickness 0.0037 0.0000 0.0000 n/a m
Inter-tank Thickness 0.0020 0.0020 0.0020 n/a m
Press. Tank Diam. 0.6055 0.0000 0.0000 0.0000 m
Press. Tank Thick. 0.0039 0.0000 0.0000 n/a m
Nose Cone Length 0.0000 0.0000 0.3375 0.3375 m
Diameter 1.3015 0.6741 0.2721 n/a m
Length of Stage 7.1478 2.5594 0.8945 10.6017 m
The connecting inter-stage skirts are designed to meet the requirements of the main parts of the
launch vehicle. The slope of 10º was chosen because the closer to vertical they are, the more
weight they can withstand. However, they still must provide the transition between the two
different diameters. Choosing this low-angled slope gave us more efficient stringers with less
material needed for each stringer. Ultimately, this reduced the cost of the stringers. The inter-
stage skirt between stages one and two requires a minimum load bearing of 42.380 kN. The
inter-stage skirt between stages two and three calls for a minimum load bearing capability of
2.986 kN. The skirts are reinforced with stringers and ring supports, detailed as follows.
Author: Molly Kane
Project Bellerophon 64
Table 4.1.4.3.3: Details of the Interstage Structure, 200 g PayloadStage 1-2 Stage 2-3 Units
The gondola is connected to a spherical balloon, filled with helium, made of polyethylene
plastic. During flight, the gondola carrying the launch vehicle is suspended below the balloon.
We assume that the balloon pops right before the launch vehicle passes through it. As the
balloon rises, the gas expands and the balloon is sized to hold the gas at an altitude of up to 30
km. The battery, that powers the communications with the range safety officer on the ground, is
Author: Amanda Briden
Project Bellerophon 69
attached to the flanges of the gondola. Neither the balloon or gondola are reused. Fig.
4.2.1.1.1.2 puts the size of these components with respect to the launch vehicle into perspective.
Fig. 4.2.1.1.1.2: Size comparison of the gondola, launch vehicle, and balloon – 1kg payload.(CJ Hiu, Sarah Shoemaker)
4.2.1.1.2 First Stage
Fig. 4.2.1.1.2.1 is an exploded view of the launch vehicle. A reference table, summarizing the
sizing and propulsion information for each stage, is also provided. Please refer back to it while
reading the descriptions of each stage.
Author: Amanda Briden
Project Bellerophon 70
Fig. 4.2.1.1.2.1: Exploded view of launch vehicle stack up and and parameter summary – 1kg payload.(Stephen Bluestone, Amanda Briden, Nicole Bryan, CJ Hiu, Molly Kane, William Ling , Sarah Shoemaker)
Author: Amanda Briden
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Project Bellerophon 71
A hybrid first stage with a hydroxy-terminated polybutadiene (HTPB) solid fuel and hydrogen
peroxide (H2O2) liquid oxidizer pairing is pressurized with gaseous nitrogen and provides a thrust
of 21 kN. Part of the first stage propellant is tapped off to support the liquid injection thrust
vector control (LITVC), which is used to steer the rocket. Made out of light-weight space-grade
aluminum, the structure can withstand a maximum acceleration of 2.86 Gs. The first stage is
70.48% of the launch vehicle’s gross liftoff mass (GLOM) and the length of this stage is 5.81 m.
Fig.4.2.1.1.2.2 is a dimensional drawing of the first stage.
Fig. 4.2.1.1.2.2: Dimensional drawing of the first stage – 1kg payload.(Jesii Doyle)
4.2.1.1.3 Second Stage
The second stage is an ammonium perchlorate (AP), aluminum (Al), and HTPB solid motor with
an extra tank of H2O2 to provide fuel for the LITVC. The H2O2 is again pressurized with gaseous
nitrogen. This stage imparts a thrust of 6.1 kN. Able to withstand a maximum acceleration of
Author: Amanda Briden
Project Bellerophon 72
3.67 Gs, the second stage is made of space-grade aluminum. A cone truncated in the mid-section
is used to connect one stage diameter to the next such that there are no gaps in the structure; this
is called a skirt. The most significant part of the avionics package is located on the interior of the
skirt connecting the second and third stages. The avionics package located in the skirt includes a
battery, telecom, central processing unit (CPU), and CPU equipment. These features increase
the avionics mass from the first by a factor of 5, for a total avionics mass on the second stage of
30 kg. The second stage is 25.97% of the launch vehicle’s GLOM. This stage is
2.44 m long and a dimensional drawing is shown in Fig. 4.2.1.1.3.1.
Fig. 4.2.1.1.3.1: Dimensional drawing of the second stage – 1kg payload.(Jesii Doyle)
Author: Amanda Briden
Project Bellerophon 73
4.2.1.1.4 Third Stage
Since the avionics is jettisoned along with the second stage at the end of its burn, we spin the
third stage of the launch vehicle to maintain stability. The propellant type and structural material
are identical to the second stage. The third stage is 3.55% of the launch vehicle’s GLOM. Stage
three is 1.12 m in length and a dimensional drawing follows in Fig. 4.2.1.1.4.1.
Fig. 4.2.1.1.4.1: Dimensional drawing of the third stage – 1kg payload.(Jesii Doyle)
4.2.1.1.5 Nose Cone Component
The nose cone protecting the top of the launch vehicle from extreme heating is made of
aluminum and titanium. An additional feature of the nose cone is a blunted tip made of titanium,
which is a heat resistant material. The nose cone is jettisoned once the vehicle reaches an
altitude of 90 km (out of the Earth’s atmosphere). The nose cone jettison occurs prior to the
separation of the first stage.
Author: Amanda Briden
Project Bellerophon 74
4.2.1.2 Mission Requirements Verification
What are the chances that we reach an orbit with a periapsis of at least 300 km?
There is a 99.99% chance that our launch vehicle reaches a periapsis of 300km. After 10,000
Monte Carlo simulations launch vehicle only fails once (Fig. 4.2.1.2.1). We therefore meet the
mission requirement of 99.86% success rate, considering only non-catastrophic failures. An
average perigee, shown as the peak of the histogram in Fig. 4.2.1.2.1, of 368 km is achieved.
280 300 320 340 360 380 400 4200
100
200
300
400
Periapsis altitude(km)
num
ber o
f cas
es
Fig. 4.2.1.2.1: 1kg periapsis altitude histogram with std = 15.774 km and mean = 367.727 km.(Alfred Lynam)
What are the chances of a failure that results in complete loss of mission?
Accurately predicting the mission success rate, including failures that result in complete loss of
mission, is difficult to do without built and tested hardware. Therefore, we turn to the historical
success rates of the Ariane IV, Ariane V, and Pegasus, to predict ours. We use the success
pattern of Pegasus as it is the only vehicle is air-launched. We predict a 93.84% success rate,
which includes catastrophic failures and is achieved after 24 launches.
Author: Amanda Briden
Project Bellerophon 75
4.2.1.3 Mission Timeline - A Launch in the Life of the 1kg Payload Launch Vehicle
T- 1:35:34 to launch
The entire launch system begins its 1 hour and 35 minute ascent to its launch altitude of 30 km.
On average, the system drifts 120 km before reaching the launch altitude. Prior to ignition, a
range safety officer on the ground checks the status of the launch system and has the authority to
proceed with or abort the launch. Fig. 4.2.1.3.1 is a visual representation of the stages of flight
described in the timeline.
T+ 00:00:00 to launch – We are go for launch!
If all systems are go, the first stage is ignited and the launch vehicle is guided off the gondola via
four launch rails. We assume that the balloon pops as the launch vehicle passes through it.
Throughout the course of the burn, the position of the launch vehicle is determined at every
instant by the control system which follows a near optimal steering law. During the first stage
the launch vehicle climbs out of the atmosphere and jettisons the nose cone.
T+ 00:02:20 First Stage Burn-out
Approximately two thirds of the way through the first stage burn, the launch vehicle begins a
pitch over maneuver. This initial maneuver is of the same form of that used in the Apollo
program. After burning for 140.8 s and climbing to an altitude of 87.4 km, the first stage
separates.
T+ 00:05:20 Second Stage Burn-out
During this phase, the launch vehicle continues to pitch over to burn off velocity in the radial
direction. At orbit insertion, radial velocity needs to be zero in order for a circular orbit to be
achieved. With the burn duration of 179.2 s and burn out altitude of 257 km, the second stage
separates, jettisoning the bulk of the system’s avionics.
T+ 00:08:35 Third Stage Burn-out – We’re in orbit!
For the duration of the third stage burn, the launch vehicle uses spin stabilization to maintain its
orientation and does not require avionics control or LITVC. This means that the vehicle’s
Author: Amanda Briden
Project Bellerophon 76
orientation from the end of the second stage burn through the third is maintained. After a 195.3 s
third stage burn time, the launch vehicle ends its ascent and enters an orbit with a perigee of 368
The trajectory given is the best case that coincides with the vehicle designed by the team. The
final decision to use a high altitude balloon to launch from proved to be beneficial. A balloon
launching configuration reduces the amount of drag from the atmosphere which in turn reduces
the ΔV necessary to get into Low Earth Orbit (LEO). Table 4.2.2.1 shows the results of the most
pertinent orbit parameters and other data that describes the final orbit and trajectory the vehicle is
inserted into.
Table 4.2.2.1 Orbit Parameters and Other Important Results
Variable Value UnitsPeriapsis* 406 kmApoapsis* 481 kmEccentricity 0.0054 --Inclination 28.5 degSemi-Major Axis 6,819 kmPeriod 5,604 sFootnotes: *Altitudes are from the surface of the Earth.
Some special notes about Table 4.2.2.1 need to be stated. The mission requirement is to insert
the launch vehicle into a 300 km orbit. Table 4.2.2.1 shows the periapsis of the orbit is well
above the requirement. We choose a trajectory that allows for errors that might propagate
throughout the flight that lowers the resulting periapsis. An important characteristic is the
nominal trajectory is very circular with an eccentricity of 0.0054. Finally, a specific value for
the inclination is not requested of the team; therefore the inclination is not of great importance to
the resulting orbit.
Noted in Table 4.2.2.2 is the ΔV budget necessary for the trajectory. The parameter ΔV total is the
amount of ΔV the launch vehicle needs to deliver to obtain the stated orbit. We used the ΔV total
to size the vehicle.
Author: Allen Guzik
Project Bellerophon 78
Table 4.2.2.2 ΔV Breakdown
Variable Value Units Percent of TotalΔVtotal 9,379 m/s --ΔVdrag 6 m/s 0.043ΔVgravity 2057 m/s 21.745ΔVEarth assist 411 m/s 4.394ΔVleo 7727 m/s 82.606
Figure 4.2.2.1 shows a plot of the resulting trajectory and orbit for the 1 kg payload.
Figure 4.2.2.1: Nominal trajectory and orbit for the 1 kg payload.(Allen Guzik)
Besides the resulting ΔV the trajectory predicts, the other important parameter other we require
is the steering law coefficients. For D&C analysis we need these values to match the nominal
ascending path the trajectory analysis calculates. These steering coefficients are found by
optimizing the ending orientation of the vehicle. The orientation is defined by three angles, Ψ1,
Ψ2, and Ψ3, where they represent and define the orientation of the vehicle at the end of the first,
second, and third stages respectively. Figure 4.2.2.2 depicts how Ψ1, Ψ2, and Ψ3 define the
orientation of the vehicle during the flight. Table 4.2.2.3 shows the steering angles defined for
the nominal trajectory.
Author: Allen Guzik
Project Bellerophon 79
Figure 4.2.2.2: Ψ steering law angle orientation definition.(Amanda Briden)
Table 4.2.2.3 Angles from the Steering Law
Variable Value UnitsEnd of 1st stage 30 degEnd of 2nd stage -10 degEnd of 3rd stage -10 deg
From these steering angles the linear tangent steering law coefficients can be defined. Equation
4.2.2.1 defines the linear tangent steering law Trajectory uses.
ϕ=tan−1
( at+b ) Eq. 4.2.2.1
The D&C analysis uses these coefficients to control the launching vehicle. Table 4.2.2.4 shows
the coefficients used for our launching scenario. Figure 4.2.2.3 shows a close up, view of the
ascending trajectory.
Author: Allen Guzik
Project Bellerophon 80
Table 4.2.2.4 Coefficients for Steering Law
Variable Value Unitsa1 - 1.9900e-1 --b1 2.8636e1 --a2 - 4.0000e-3 --b2 1.1700e0 --a3 8.6720e-20 --b3 - 1.1760e-1 --Footnotes: Values are coefficients so no units.
Figure 4.2.2.3: Close up view of the ascending trajectory for the 1kg launch configuration.(Allen Guzik)
In conclusion, we are very pleased with the resulting nominal trajectory of the 1 kg launch
configuration. Our periapsis is above the required 300km, and the orbit is very circular. The
trajectory also allows for error to be tolerable and still meet the required orbit.
Author: Allen Guzik
Project Bellerophon 81
4.2.3 Controlled Trajectory
We are not able to exactly match the designed trajectory due to many factors. The trajectory
group models the launch vehicle as a point mass to determine the nominal orbit. To arrive at the
controlled trajectory the D&C group models the launch vehicle as a rigid body. Also, the
Trajectory group’s steering law includes sharp corners which are not physically possible. To
keep the launch vehicle under control those corners have to be smoothed out. These factors
combine to make the controlled trajectory differ from the nominal one. At orbit insertion, the
launch vehicle is at a lower altitude which leads to a more eccentric orbit which is illustrated in
the following figures.
Fig. 4.2.3.1: Close up view of launch trajectory; designed orbit (red), and actual controlled orbit (yellow)
(Mike Walker, Alfred Lynam, and Adam Waite)
Author: Albert Chaney
Project Bellerophon 82
Fig. 4.2.3.2: Designed orbit (red), and actual controlled orbit (yellow)
(Mike Walker, Alfred Lynam, and Adam Waite)
Below is a table of the orbital parameters for the orbit we achieve. The value a is the semi-major
axis, e is the eccentricity, i is the inclination, Ω is the right ascension of the ascending node, and
Fig. 4.2.3.3: Ground Track of the controlled portion of the launch
(Mike Walker, Alfred Lynam, and Adam Waite)
Figure 4.2.3.3 is a ground track for the controlled portion of the launch. Ground tracks are
important in the design of ground tracking stations and range safety concerns. The ground track
is vital in the mission planning of the launch.
Author: Albert Chaney
Project Bellerophon 84
4.2.4 Subsystem Details
4.2.4.1 Propulsion
The propellants we selected for the 1 kilogram payload launch vehicle were a hybrid first stage
and a solid second and third stage. Our selection process involved the use of an optimization
code which gave us the best results for a 1 kilogram payload launch vehicle. The code gave us a
propulsion system described in the following section.
Our launch vehicles first stage consists of a hybrid fuel rocket motor. This fuel consists of
hydrogen peroxide as the oxidizer and hydroxyl terminated polybutadiene (HTPB) as the solid
propellant. The hydrogen peroxide is first catalyzed and then fed through the grain of the solid
fuel where it combusts and travels through the nozzle. The nozzle is a 12° conical nozzle with
LITVC attached. The LITVC system is composed of four valves that allow H2O2 to be injected
into the nozzle at a 90º angle to the centerline of the nozzle. A schematic of the LITVC can be
seen below in Figure 4.2.4.1.1.
Figure 4.2.4.1.1: LITVC and Nozzle Configuration
Authors: Ricky Hinton, Stephan Shurn
Project Bellerophon 85
In Figure 4.2.4.1.1, the nozzle is shaded grey and all LITVC components are highlighted in
orange. The pipes are run from the H2O2 tank that is used for the hybrid motor, and then is
distributed to each valve. The valves are connected to the controller which relays a signal for a
certain valve to open and allow pressurized H2O2 to be injected into the main flow in the nozzle,
which produces a side thrust. This side thrust allows for control of the launch vehicle during its
ascent. There is only one engine for this stage. The specific values for the first stage can be seen
below in Table 4.2.4.1.1.
Table 4.2.4.1.1 1 kg Payload Stage 1 Propulsion Specifics
Variable Value UnitsVacuum Specific Impulse 352.3 sChamber Pressure 2,068,000 PaMass Flow Rate 6.730 kg/sPropellant Mass 947.9 kgEngine Mass 72.62 kgThrust (vac) 21,435.5 NBurn Time 140.8 sExit Area 0.342 m2
Exit Pressure 2,821.167 Pa
A conical nozzle was chosen because of the solid particles of propellant that will be coming out
of the combustion chamber. The combustion process does not necessarily combust the fuel 100%
and these particles can deteriorate a nozzle if it is let’s say Bell shaped. Some of our early MAT
codes had values based off of a 12° conical nozzle and that is one of the reasons we decided on
this cone angle for the final design. Also having a smaller cone angle reduces the divergence loss
at the exit of the nozzle. A picture of the nozzle can be seen below in Fig. 4.2.4.1.2.
Authors: Ricky Hinton, Stephan Shurn
Project Bellerophon 86
Figure 4.2.4.1.2: Our 12° conical nozzle
For our second stage we chose a solid rocket propellant. The compound for this propellant is
Hydroxyl-terminated Polybutadiene/ Ammonium Perchlorate/ Aluminum (HTPB/AP/AL). The
nozzle once again is a 12° conical nozzle due to the solid propellant. The LITVC system is
attached to the nozzle. The LITVC has the same configuration as the first stage, with the
exception of the H2O2. Since there is no H2O2 already present due to the solid motor, a
pressurized tank is added in a curved configuration sitting beneath the solid motor. The tank
wraps around the nozzle and is pressurized with gaseous nitrogen so that the H2O2 can flow into
the lines for injection. There is again only one engine for this stage. Table 4.2.4.1.2 below shows
the specifics for this stage.
Table 4.2.4.1.2 1 kg Payload Stage 2 Propulsion Specifics
Variable Value UnitsVacuum Specific Impulse 309.3 sChamber Pressure 6,000,000 PaMass Flow Rate 1.880 kg/sPropellant Mass 336.92 kgEngine Mass 36.44 kgThrust (vac) 6,052.4 NBurn Time 179.2 sExit Area 0.028 m2
Exit Pressure 11,453.660 Pa
Authors: Ricky Hinton, Stephan Shurn
Project Bellerophon 87
The third and final stage for this launch vehicle consists of a solid propellant motor. The
propellant for this stage once again is Hydroxyl-terminated Polybutadiene/ Ammonium
Perchlorate/ Aluminum (HTPB/AP/AL). The nozzle is once again a 12° conical nozzle but does
not have LITVC control for this stage. There is a single engine for stage three. The specifics can
be seen in Table 4.2.4.1.3 below for stage three of this one kilogram launch vehicle.
Table 4.2.4.1.2 1 kg Payload Stage 3 Propulsion Specifics
Variable Value UnitsVacuum Specific Impulse 309.3 sChamber Pressure 6,000,000 PaMass Flow Rate 0.231 kg/sPropellant Mass 45.09 kgEngine Mass 9.53 kgThrust (vac) 743.4 NBurn Time 195.3 sExit Area 0.003 m2
Exit Pressure 11,453.660 Pa
Authors: Ricky Hinton, Stephan Shurn
Project Bellerophon 88
4.2.4.2 Aerothermal
In our aerodynamic analysis, we use linear perturbation theory to determine the aerodynamic
loading on the launch vehicle. Linear perturbation theory is the method in which the pressure
over the top and bottom surfaces of the launch vehicle is integrated to solve for the normal and
axial force coefficients acting on the launch vehicle. It is valid in the subsonic and supersonic
regimes, but falls apart in the transonic regime. For this reason, we have ignored the
aerodynamic outputs in the transonic regime and only pay attention to the outputs in the subsonic
and supersonic regimes. By integrating the change in pressure around the launch vehicle we are
able to solve for bending and pitching moments, drag coefficient, axial forces, normal forces,
shear forces, and the center of pressure location. All of these aerodynamic moments, coefficients,
and forces are based on the final geometry of the launch vehicle as well as the Mach number,
angle of attack, and time spent in the atmosphere.
Mach number, variation in angle of attack, use of LITVC, stage separation, as well as wind gusts
all have a large impact on the aerodynamic loadings of the launch vehicle. As the launch vehicle
makes its way through the atmosphere, the change in density also has a significant effect on the
impact of these forces and moments. The results for the variation of bending moment and
pitching moment with respect to Mach number at zero degree angle of attack can be found in
Figs. 4.2.4.2.1 and 4.2.4.2.2 respectively. Once the launch vehicle reaches a speed of Mach 4.5,
it exits the atmosphere. At this point, the first stage has still not separated; therefore, moments
are shown as they act on the entire launch vehicle.
Author: Jayme Zott
Project Bellerophon 89
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
-900
-800
-700
-600
-500
-400
-300
-200
-100
0
Mach
Bend
ing M
omen
t (Nm
)
Fig. 4.2.4.2.1: Variation of bending moment with respect to Mach number at zero angle of attack. 1 Kg.
(Alex Woods, Jayme Zott)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
50
100
150
200
250
300
350
400
450
Mach
Pitc
hing
Mom
ent (
Nm)
Fig. 4.2.4.2.2: Variation of pitching moment with respect to Mach number at zero angle of attack. 1 Kg.
(Alex Woods, Jayme Zott)
The moments presented in Figs. 4.2.4.2.1 and 4.2.4.2.2 correlate well with the magnitude of
moments expected for a launch vehicle of our size and shape. It is important for us to determine
these moments because the structures group uses them to determine appropriate materials and
thicknesses for the final launch vehicle design.
Author: Jayme Zott
Project Bellerophon 90
The results for the variation of normal, axial, and shear forces with respect to Mach number at a
zero degree angle of attack can be found in Figs. 4.2.4.2.3, 4.2.4.2.4, and 4.2.4.2.5 respectively.
The normal and axial forces are important for the D&C group’s analysis. D&C uses the normal
and axial forces acting on the launch vehicle to help determine the amount of LITVC needed for
control at any given moment in time. The shear force is important for the structures group’s
analysis. Structures uses the shear force acting on the vehicle to help determine appropriate
materials and thicknesses for the final launch vehicles design.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
-25
-5
15
35
55
75
95
115
Mach
Norm
al Fo
rce
(N)
Fig. 4.2.4.2.3: Variation of normal force with respect to Mach number at zero angle of attack. 1 Kg.
(Alex Woods, Jayme Zott)
Author: Jayme Zott
Project Bellerophon 91
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
50
100
150
200
250
300
350
400
Mach
Axial
Forc
e (N
)
Fig. 4.2.4.2.4: Variation of axial force with respect to Mach number at zero angle of attack. 1 Kg.
(Alex Woods, Jayme Zott)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
-50
-45
-40
-35
-30
-25
-20
-15
-10
-5
0
Mach
Shea
r For
ce (N
)
Fig. 4.2.4.2.5: Variation of shear force with respect to Mach number at zero angle of attack. 1 Kg.
(Alex Woods, Jayme Zott)
Author: Jayme Zott
Project Bellerophon 92
The variation of CD with Mach number at a constant zero angle of attack is shown in Fig.
4.2.4.2.6. Because the diameter of the 1 kg launch vehicle is quite large, the coefficient of drag
CD is also quite large.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Mach
Cd
Fig. 4.1.4.2.6: Impact of Mach number on CD at zero angle of attack. 1 Kg.
(Alex Woods, Jayme Zott)
As previously mentioned, we use the linear perturbation theory to determine all aerodynamic
forces, coefficients, and moments, including CD. This method requires complete knowledge of the
launch vehicle geometry before any aerodynamic forces, coefficients or moments can be
determined. This causes a problem because the trajectory analysis requires use of CD long before
the final geometry is determined. Because the CD variation shown in Fig. 4.2.4.2.6 is determined
after the final launch vehicle geometry has been designed, it cannot be used in the trajectory
analysis. Instead, we use a CD trend based on historical data for the trajectory analysis.1,2 While
this historical CD trend is not based on our own geometry, it is based on successful launch
vehicles with geometries similar to our final design. The CD based on historical data at zero angle
of attack is shown in the Fig. 4.2.4.2.7
Author: Jayme Zott
Project Bellerophon 93
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Mach
Cd
Fig. 4.2.4.2.7: Impact of Mach number on CD at zero angle of attack based on historical data. 1 Kg.
(Jayme Zott)
Given additional time, we could complete a better trajectory analysis by including the correct CD
based on the linear perturbation theory into the trajectory code. If we created an intermediate file
between the initial propulsion sizing output and the trajectory input, a more accurate CD value
could be used within the trajectory code. Fig. 4.1.4.2.8 shows the error caused by the using the
CD trend based on historical geometries, rather than the CD determined directly from our own
geometry.
Author: Jayme Zott
Project Bellerophon 94
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Cd (historical)
Cd (dimensional)
Mach
Cd
Fig. 4.2.4.2.8: Comparison of CD based on historical data and CD based on dimensional analysis (linear perturbation theory). 1 Kg.
(Alex Woods, Jayme Zott)
Table 4.1.4.2.1 Summary of Maximum Aerodynamic Loading 1 Kg.
Aerodynamic Load Subsonic Supersonic Bending Moment [Nm] -773.0 -388.7Pitching Moment [Nm] 357.8 179.9Normal Force [N] 94.2 47.3Axial Force [N]Shear Force [N]Center of Pressure [% length]Coefficient of Drag CD
Dynamic Pressure [Pa]CD % error [%]
335.7 -43.0 40.0 1.44
63
168.9 -21.6 40.0 0.81 240 21
(Jayme Zott)
References:1Sutton, George P., and Oscar Biblarz. Rocket Propulsion Elements. New York: John Wiley & Sons, Inc., 2001.
2The Martin Company, “The Vanguard Satellite Launching Vehicle”, Engineering Report No. 11022, April 1960.
Author: Jayme Zott
Project Bellerophon 95
4.2.4.3 Structures
Our launch vehicle, designed to deliver 1kg of payload into low earth orbit, consists of a three-
stage rocket lifted to an altitude with a balloon and gondola. Bars of aluminum, 0.04 m thick,
provide the necessary strength required for the launch vehicle. This gondola is able to with stand
the approximately 31 kPa of pressure from launch.
Fig. 4.2.4.3.1 Gondola frame for rocket element of launch vehicle of a 1 kg payload.
(Sarah Shoemaker)
The gondola is an aluminum frame which extends 3.849 m in height. We made the base is
square in shape with sides of 1.0 m and support ring diameters are 1.1264 m. The total mass of
our gondola is then 227.114 kg.
The first stage of our rocket contains an engine, oxidizer tank, fuel tank, pressurant tank, inter-
tank structure, and avionics equipment. This stage is 6.0803 m in length and 1.1264 m in
diameter. We make all tanks from aluminum and they are of various sizes and thicknesses. The
Al-7075 alloy that we employ is proven historically and also has a very high strength to weight
ratio. The second stage is made up of an engine, fuel tank, and avionics equipment. It extends
1.9776 m in length with a constant diameter of 0.5669 m. As with the first stage, the fuel tank is
made from aluminum. The third stage is 0.8540 m long and has a diameter of 1.0554 m. This is
Author: Molly Kane
Project Bellerophon 96
the final stage of the rocket element for the launch vehicle. Above the third stage is a nose cone,
0.3596 m long and made of titanium. It protects the 1 kg payload that sits over the third stage.
Inert masses of the components vary between stages as follows.
Table 4.2.4.3.1: Inert Mass Breakdown of Three-Stage Rocket, 1 kg Payload
Variable Value UnitsVacuum Specific Impulse 352.3 sChamber Pressure 2,068,000 PaMass Flow Rate 23.571 kg/sPropellant Mass 4122.85 kgEngine Mass 193.49 kgThrust (vac) 75073.2 NBurn Time 174.9 sExit Area 1,198 m2
Exit Pressure 2,821.167 Pa
A conical nozzle was chosen because of the solid particles of propellant that will be coming out
of the combustion chamber. The combustion process does not necessarily combust the fuel 100%
and these particles can deteriorate a nozzle if it is let’s say Bell shaped. Some of our early MAT
codes had values based off of a 12° conical nozzle and that is one of the reasons we decided on
this cone angle for the final design. Also having a smaller cone angle reduces the divergence loss
at the exit of the nozzle. A picture of the nozzle can be seen below in Fig. 4.3.4.1.1.
Authors: Stephan Shurn, Ricky Hinton, Jerald A. Balta
Project Bellerophon 120
Figure 4.3.4.1.1: Our 12° conical nozzle
The second stage of the launch vehicle uses a solid rocket motor, with hydroxyl-terminated
polybutadiene/ ammonium perchlorate/ aluminum (HTPB/AP/AL) as the propellant. The nozzle
is a 12º conical nozzle with LITVC attached. The LITVC has the same configuration as the first
stage, with the exception of the H2O2. Since there is no H2O2 already present due to the solid
motor, a pressurized tank is added in a curved configuration sitting beneath the solid motor. The
tank wraps around the nozzle and is pressurized with gaseous nitrogen so that the H2O2 can flow
into the lines for injection. The specifics of the propulsion system can be seen in Table 4.3.4.1.2.
Variable Value UnitsVacuum Specific Impulse 309.3 sChamber Pressure 6,000,000 PaMass Flow Rate 4.739 kg/sPropellant Mass 1009.33 kgEngine Mass 75.72 kgThrust (vac) 15256.9 NBurn Time 213.0 sExit Area 0.070 m2
Exit Pressure 11,453.660 Pa
Authors: Stephan Shurn, Ricky Hinton, Jerald A. Balta
Project Bellerophon 121
The third stage of the launch vehicle uses a solid rocket motor, with hydroxyl-terminated
polybutadiene/ ammonium perchlorate/ aluminum (HTPB/AP/AL) as the propellant. The nozzle
is a 12º conical shape. The specifics of the propulsion system can be seen in Table 4.3.4.1.3.
Variable Value UnitsVacuum Specific Impulse 309.3 sChamber Pressure 6,000,000 PaMass Flow Rate 0.215 kg/sPropellant Mass 38.37 kgEngine Mass 8.56 kgThrust (vac) 692.4 NBurn Time 178.4 sExit Area 0.003 m2
Exit Pressure 11,453.660 Pa
Authors: Stephan Shurn, Ricky Hinton, Jerald A. Balta
Project Bellerophon 122
4.3.4.2 Aerothermal
In our aerodynamic analysis, we use linear perturbation theory to determine the aerodynamic
loading on the launch vehicle. Linear perturbation theory is the method in which the pressure
over the top and bottom surfaces of the launch vehicle is integrated to solve for the normal and
axial force coefficients acting on the launch vehicle. It is valid in the subsonic and supersonic
regimes, but falls apart in the transonic regime. For this reason, we have ignored the
aerodynamic outputs in the transonic regime and only pay attention to the outputs in the subsonic
and supersonic regimes. By integrating the change in pressure around the launch vehicle we are
able to solve for bending and pitching moments, drag coefficient, axial forces, normal forces,
shear forces, and the center of pressure location. All of these aerodynamic moments, coefficients,
and forces are based on the final geometry of the launch vehicle as well as the Mach number,
angle of attack, and time spent in the atmosphere.
Mach number, variation in angle of attack, use of LITVC, stage separation, as well as wind gusts
all have a large impact on the aerodynamic loadings of the launch vehicle. As the launch vehicle
makes its way through the atmosphere, the change in density also has a significant effect on the
impact of these forces and moments. The results for the variation of bending moment and
pitching moment with respect to Mach number at zero degree angle of attack can be found in
Figs. 4.3.4.2.1 and 4.3.4.2.2 respectively. Once the launch vehicle reaches a speed of Mach 4.4,
it exits the atmosphere. At this point, the first stage has still not separated; therefore, moments
are shown as they act on the entire launch vehicle.
Author: Jayme Zott
Project Bellerophon 123
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
-10000
-9000
-8000
-7000
-6000
-5000
-4000
-3000
-2000
-1000
0
Mach
Bend
ing M
omen
t (Nm
)
Fig. 4.3.4.2.1: Variation of bending moment with respect to Mach number at zero angle of attack. 5 Kg.
(Alex Woods, Jayme Zott)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
200
400
600
800
1000
1200
1400
1600
1800
2000
Mach
Pitc
hing
Mom
ent (
Nm)
Fig. 4.3.4.2.2: Variation of pitching moment with respect to Mach number at zero angle of attack. 5 Kg.
(Alex Woods, Jayme Zott)
The moments presented in Figs. 4.3.4.2.1 and 4.3.4.2.2 correlate well with the magnitude of
moments expected for a launch vehicle of our size and shape. It is important for us to determine
these moments because the structures group uses them to determine appropriate materials and
thicknesses for the final launch vehicle design.
Author: Jayme Zott
Project Bellerophon 124
The results for the variation of normal, axial, and shear forces with respect to Mach number at a
zero degree angle of attack can be found in Figs. 4.3.4.2.3, 4.3.4.2.4, and 4.3.4.2.5 respectively.
The normal and axial forces are important for the D&C group’s analysis. D&C uses the normal
and axial forces acting on the launch vehicle to help determine the amount of LITVC needed for
control at any given moment in time. The shear force is important for the structures group’s
analysis. Structures uses the shear force acting on the vehicle to help determine appropriate
materials and thicknesses for the final launch vehicles design.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
50
100
150
200
250
300
350
Mach
Norm
al Fo
rce
(N)
Fig. 4.3.4.2.3: Variation of normal force with respect to Mach number at zero angle of attack. 5 Kg.
(Alex Woods, Jayme Zott)
Author: Jayme Zott
Project Bellerophon 125
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
200
400
600
800
1000
1200
Mach
Axial
Forc
e (N
)
Fig. 4.3.4.2.4: Variation of axial force with respect to Mach number at zero angle of attack. 5 Kg.
(Alex Woods, Jayme Zott)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
-600
-500
-400
-300
-200
-100
0
Mach
Shea
r For
ce (N
)
Fig. 4.3.4.2.5: Variation of shear force with respect to Mach number at zero angle of attack. 5 Kg.
(Alex Woods, Jayme Zott)
Author: Jayme Zott
Project Bellerophon 126
The variation of CD with Mach number at a constant zero angle of attack is shown in Fig.
4.3.4.2.6. Because the diameter of the 1 Kg launch vehicle is quite large, the coefficient of drag
CD is also quite large.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Mach
Cd
Fig. 4.3.4.2.6: Impact of Mach number on CD at zero angle of attack. 5 Kg.
(Alex Woods, Jayme Zott)
As previously mentioned, we use the linear perturbation theory to determine all aerodynamic
forces, coefficients, and moments, including CD. This method requires complete knowledge of the
launch vehicle geometry before any aerodynamic forces, coefficients or moments can be
determined. This causes a problem because the trajectory analysis requires use of CD long before
the final geometry is determined. Because the CD variation shown in Fig. 4.3.4.2.6 is determined
after the final launch vehicle geometry has been designed, it cannot be used in the trajectory
analysis. Instead, we use a CD trend based on historical data for the trajectory analysis.1,2 While
this historical CD trend is not based on our own geometry, it is based on successful launch
vehicles with geometries similar to our final design. The CD based on historical data at zero angle
of attack is shown in the Fig. 4.3.4.2.7.
Author: Jayme Zott
Project Bellerophon 127
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Mach
Cd
Fig. 4.3.4.2.7: Impact of Mach number on CD at zero angle of attack based on historical data. 5 Kg.
(Jayme Zott)
Given additional time, we could complete a better trajectory analysis by including the correct CD
based on the linear perturbation theory into the trajectory code. If we created an intermediate file
between the initial propulsion sizing output and the trajectory input, a more accurate CD value
could be used within the trajectory code. Fig. 4.1.4.2.8 shows the error caused by the using the
CD trend based on historical geometries, rather than the CD determined directly from our own
geometry.
Author: Jayme Zott
Project Bellerophon 128
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Cd (dimensional)Cd (historical)
Mach
Cd
Fig. 4.3.4.2.8: Comparison of CD based on historical data and CD based on dimensional analysis (linear perturbation theory). 5 Kg.
(Alex Woods, Jayme Zott)
Table 4.3.4.2.1 Summary of Maximum Aerodynamic Loading 5 Kg.
Aerodynamic Load Subsonic Supersonic Bending Moment [Nm] -6505.0 -3346.0Pitching Moment [Nm] 1246.0 640.9Normal Force [N] 215.9 111.0Axial Force [N]Shear Force [N]Center of Pressure [% length]Coefficient of Drag CD
Dynamic Pressure [Pa]CD % error [%]
760.0 -407.4 38.2 1.31
62.5
390.9 -209.6 38.2 0.78 225 19.5
(Jayme Zott)
References:1Sutton, George P., and Oscar Biblarz. Rocket Propulsion Elements. New York: John Wiley & Sons, Inc., 2001.
2The Martin Company, “The Vanguard Satellite Launching Vehicle”, Engineering Report No. 11022, April 1960.
Author: Jayme Zott
Project Bellerophon 129
4.3.4.3 Structures
The launch vehicle for the 5 kg payload design is comprised of a three-stage rocket carried by a
gondola and balloon. The gondola is made of aluminum bars, 0.04 m thick.
Fig. 4.3.4.3.1 Gondola frame for rocket element of launch vehicle of a 5 kg payload.
(Sarah Shoemaker)
We mad the frame 5.133 m tall with a square base of sides 1.38 m long. The ring supports have
a diameter of 1.8386 m. The mass of the gondola add up to 338.32 kg. The material properties
and dimensions give the structure the ability to withstand the approximately 36 kPa of pressure it
experiences during launch.
The first stage of our rocket is 10.6004 m in length and 1.8386 m in diameter. An engine,
oxidizer tank, fuel tank, pressurant tank, inter-tank structure, and avionics equipment make up
this first stage. All of our tanks are created from aluminum, but have different sizes and
thicknesses. An inter-stage “skirt” connects the first and second stages of the rocket. It has a
constant 10° slope from the lower stage to the upper stage. We also made this of aluminum and
it is reinforced with aluminum support stringers and rings. Between the first and second stages is
a 2.8965 m inter-stage structure. Its maximum diameter is that of the third stage 1.8386 m, and
the minimum diameter is the diameter of the second stage 0.8172 m.
Author: Molly Kane
Project Bellerophon 130
Only an engine, fuel tank, and avionics equipment make up the second stage of the rocket. This
stage has a diameter of 0.8172 m and a length of 3.2709 m. The second stage fuel tank is also
made from aluminum. The inter-stage structure connects the second and third stages of the
rocket, having a constant slope of 10° and vertical length of 1.5380 m.
The third stage is 0.9046 m in length and is 0.2748 m in diameter. It is also composed of an
engine, fuel tank, and avionics equipment. The fuel tank is constructed from aluminum. The 5
kg payload sits above the third stage and is protected by the nose cone. The nose cone is 0.3408
m in length and has a mass of 1.7927 kg.
The inert mass breakdowns of components and stages are summarized in the following table.
Table 4.3.4.3.1: Inert Mass Breakdown of Three-Stage Rocket, 5 kg PayloadStage 1 Stage 2 Stage 3 Totals Units