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UASE 501 Worksheet 7.2
Worksheet: UAS Launch and Recovery
UAS 501: Introduction to Unmanned Aircraft Design
Embry-Riddle Aeronautical University
This worksheet reviews and clarify some of the equations as a supplement to Chapter 11 of Gundlach’s textbook Designing Unmanned Aircraft Systems. First, an overview of general equations used within the text are presented. The worksheet then addresses key equations for several launch and recovery approaches discussed within the text.
Exercises are distributed throughout the worksheet provide an opportunity to apply the calculations to realistic problems.
This worksheet is not a replacement for the textbook. The textbook provides alternative approaches, equations, and key insights to help guide your analysis and better understand the benefits and constraints of each approach.
The equations presented can be applied to both SI and English units. Please remember that weight is a force. In British Imperial units, a pound is a unit of force (represented as lb or lbf). In SI units, newtons are the unit for force. With respect to energy, the SI unit is joules, and the British Imperial unit is foot-pound force (equal to 1.3558 J).
Within this worksheet, the units specified are give such that the SI unit is on the left and the British Imperial unit is on the right.
The following are some general equations that have broad applicability across different launch and recovery systems directly or with some adaptation.
The Impulse-Momentum Equation defines the relationship between an impulse, i.e. change in force over time, and change momentum. For constant force over the period of time, the relationship is represented as:
F⋅Δt=WTOg⋅|ΔV|
Where,
This equation is valuable because it defines a relationship between the force associated with the launch or recovery event and parameters of that event including the weight of the aircraft at the initiation of the event and the change in velocity magnitude over the duration of the event.
Note: the equation above includes WTO
for takeoff weight, but this relationship can be generalized to recovery as well using recovery weight instead.
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Exercise:
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The chapter frequently utilizes the following equation to specify the length of a motion or stroke such as the distance a vehicle must move to accelerate using a rail launcher, or the length of travel down a runway for a conventional takeoff.
L=|V|22a
(m
or ft
)
Where,
Like the stroke length for acceleration discussed above, the equation to calculate the stroke length for deceleration is the negation of the previous equation. The negative ensures that we have a positive length despite the negative value for acceleration as the vehicle decelerates to a stop.
L=-|V|22a
(m or ft)
Where,
At launch, energy is imparted onto the vehicle for it to transition from an at rest state to a launched state. The following equation defines the relationship between energy imparted from common sources to the sum of kinetic energy, potential energy, and consumption of stored energy.
ΔELauncher+ΔEPropulsion+ΔELosses=WTO⋅ΔV22⋅g+Δh+ΔEStored
Where,
Dependent upon the launcher used, one or multiple of the ΔE
components may not be applicable. For instance, a conventional takeoff does not utilize a launcher; therefore, ΔELauncher=0
. As another example, consider a rail launch system where the UA’s propulsion system is not active until some post-release velocity is attained, which would make ΔEPropulsion=0
.
Additionally, during preliminary design where energy losses are unknown, ΔLosses
is often neglected.
Finally, if stored energy is not consumed during the launch event, ΔEStored=0
.
ERecovery=WRecovery2⋅g⋅|ΔV|2+WRecovery⋅Δh
Where,
When a vehicle is launched or recovered, the wing loading should be considered to ensure that the safety margin for peak loads is not exceeded.
The following equation estimates the wing loading in terms of the lift applied to the wing surface divided by the wing’s surface area. Manipulating the Lift equation presented in the aerodynamics and airframe design worksheet, the wing load can be solved as:
L/S=0.5⋅ρ⋅V2⋅CL
(N/m2
or lb/ft2
)
Where,
Please note that this is not a precise measurement across various portions of the wing surface. Rather, it provides an estimate across the entirety of the wing.
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Exercise:
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The energy imparted during launch is provided by the propulsion system and not by a launch mechanism. Therefore, for the energy equation (see Section 1), ΔELauncher
is set to zero (0).
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Exercise:
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For recovery of a UAS using conventional landing gear, the energy equation can be rewritten as to account for energy expended for braking:
ΔEPropulsion+ΔELosses+ ΔEBraking=WTO⋅-VApproach22⋅g+Δh+ΔEStored
Where,
For rail launch systems, the UAS is pushed along a rail and released with a sufficient force that it can attain a desired launch velocity after release. Rocket, pneumatic, spring, and other mechanisms can be used to impart energy for the launch event.
Upon reaching the end of the rail, the system has attained is release velocity. This velocity is different from its launch velocity, which is the desired velocity for which the vehicle begins its climb-out. The following equation can be used to determine the release velocity
VRelease=|ΔV|2+2⋅g⋅Δh-ΔhRelease
(m/s or ft/s)
Where,
The rail is comprised of an acceleration section and a deceleration section (if a shuttle is used). The total length is the sum of the length of these two sections. Each length can be derived utilizing the stroke length equations for acceleration deceleration as specified above.
LRail=LAccel+LDecel
(m or ft)
When a pneumatic launch mechanism is used, a pneumatic piston drives a shuttle along the rail.
The force delivered by the piston is equal to the:
FPiston=ΔP⋅Π4⋅DPiston2
(N
or lb
)
Where,
The force delivered by the shuttle to launch the vehicle can be calculated as follows:
FShuttle=WTO+WShuttle⋅ag+sinθlauncher
(N
or lb
)
Where,
The piston stroke length corresponds to the mathematical relationship for acceleration stroke length. That equation can be used to derive unknowns if the stroke length is already known. Given the ratio of piston length to acceleration length, the following relationship between the Force delivered by the shuttle given the force delivered by the piston exists.
FShuttle=FPiston⋅LPistonLAccel
Where,
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Exercise:
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For a rocket launch, a variable rate of Thrust (N or lb) propels the rocket toward a desired launch velocity. The total impulse is described in the textbook in equation 11.41 as the integral of thrust over time through the duration of the launch event.
The total impulse of the rocket launch relates to the change in momentum as described in the following equation.
ITot=WTOg⋅ΔV
(N⋅s
or lb ⋅s
)
Where,
From the above equation, you can solve for the change in velocity resulting from the rocket launch if the total impulse is known.
One advantage of a rocket-based launch mechanism is it can achieve a zero-length takeoff distance. To achieve this goal, the vertical component of the thrust vector must exceed the takeoff weight of the vehicle and the takeoff weight of the rocket. This relationship is expressed as follows:
TRocket⋅sinθRocket≥WTO+WRocket
Where
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Exercise:
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The tension-line launch technique supports the launch of small unmanned aircraft systems.
The textbook presents the general equation for the transfer of energy from the tensioned line to the vehicle in equation 11.48 as the integration of the tension of the cable as a function of its stretch distance from its starting point.
This relationship simplifies for spring and elastic chord tension lines utilizing the spring constant, k
. This equation is given as:
ΔELine=12⋅k?⋅ΔX2
Where,
Using the above equation for ΔELine
, additional unknowns can be derived using the impulse-momentum equation for launch events.
Within the textbook, a horizontal arresting cable is described as a type of arresting cable used for recovery. In the ideal case, the aircraft snags the arresting cable such that the tension applied symmetrically between the two sides of the arresting cable as the UA continues to move forward in the x-axis.
The forces and distance traveled to arrest the vehicle can be calculated as a step-wise analysis updating the distance traveled and the force of the cable in the x-axis utilizing the following equations.
Fx,Cable=-2⋅TCable⋅sinθCable
(N
or lb
)
Where
θCable=tan-1ΔXAV1/2 ⋅ LCable
(degrees)
Where
a=Fx,Cable ⋅gWRecovery
(m/s2
or ft/s2
)
Where,
ΔXAV,i=Vi-1⋅ΔT+12ai⋅ΔT2
Utilizing the above equations, the stop distance and stop time can be estimated. Additionally, the loads on the vehicle can be estimated by calculating Fx,Cable.
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Exercise:
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Parachute recovery permits both gliding and non-gliding descent profiles. This section briefly highlights the calculations required for a non-gliding UAS descending at the chute’s terminal velocity, VT
.
The terminal velocity is attained when the drag produced by the chute equals the recovery weight of the aircraft.
WRecovery=DChute=12⋅ρVT2⋅CD,Chute⋅SChute
(N
or lb
)
Where,
The surface area of a round chute relative to its diameter is equal to:
SChute=π4⋅DChute2
Where,
Utilizing the above equations, the diameter of the parachute can be calculated as:
DChute=8⋅WRecoveryπ⋅ρ⋅VT2⋅CD,Chute
Where,
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Exercises:
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Section 11.17 describes impact attenuation if the vehicle were permitted to impact the ground as part of its arresting technique. If the vehicle is designed with crumple zones, the impulse-momentum equation can be described as:
FZ⋅LStroke=WRecovery2⋅g⋅Vz,Impact2
Where,
From this equation, the level of force of the impact, FZ
, along the z-axis can be derived based upon the length of the vehicle’s crumple zone, LStroke
along the z-axis and the vehicle’s velocity along the z-axis, Vz , Impact
.
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Exercise:
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