IEEE - Aerospace and Electronic Systems - December 2022 - 21

Panigrahi et al.
Figure 8.
Propeller deflection in aerial and aquatic conditions. (a) Propeller deflection in aerial operation. (b) Propeller deflection in aquatic operation.
(c) Variation ofpropeller deflection ofAPC 12--4-in due to material properties.
propellers, as can be seen in 12-in propellers in comparison
to 4-in propellers, as shown in Figure 7. Hence, the given
framework can be used to accurately predict changes in the
thrust and torque profiles in any fluid medium for a given
change in diameter, pitch, and operational RPM.
The preliminary screening of the entire propeller database
can be performed by comparing the absolute aerial
thrust obtained from the BEMT methodology, with that
required for the takeoffofthe multimedium vehicle, which
is user-defined. The required absolute aerial thrust, which
is used for screening is also illustrated as a horizontal plane
in Figure 7(a) and (c). As a result, the set ofpropellers suitable
for the specific vehicle can be chosen after considering
the minimum thrust required for aerial takeoff, which is
essentially derived from the initial design specifications of
the multimedium vehicle. For example, considering the
minimum thrust required for aerial takeoff is 4.9 N (for a 2
kg multimedium vehicle with four thrusters in Quad-copter
X0 configuration), the entire range of propellers from 7-in
to 12-in can be considered as a plausible choice as the generated
thrust meets the takeoffrequirement.
ANALYSIS OF PROPELLER DEFLECTION
Standard approaches for evaluating the propeller tip
deflection include modeling the propeller as a rigid body
hinged to the hub or constrained to each other by torsional
DECEMBER 2022
springs [29]. Analytical relationships can also be obtained
by performing the static load analysis in computational
finite element platforms like STAR CCMþ, ANSYS, or
Solidworks. However, here we implemented an alternative
strategy by using the Euler Bernoulli theorem [30].
Propellers moving in any fluid environment can be
modeled as a rigid body cantilevered at the root and suffering
maximum deflection at the tip. Considering we
have the force moment profile acting on each section of
the propeller, as discussed earlier in the section " Blade
Elemental Momentum Theory, " the propeller can be
modeled as a cantilever beam following the differential
equation [31]:
d2z
dx2 ¼
Mb
EI
(12)
where z is the deflection of a propeller section at a radial
distance of x from the root, Mb is the bending moment
acting on the propeller blade, E is the elastic modulus,
and I is the propeller's moment of inertia about the spinning
axis. Solving the above equation for the given
boundary conditions (at x ¼ 0; z ¼ 0, and dz
dx ¼ 0at x ¼
0Þ provides the deflection of the blade at any cross section
along the propeller span length, which is computed
for both aerial and underwater conditions as shown in
Figure 8. Propellers of different materials can also be
analyzed for their deflection in an aquatic medium, as
illustrated in Figure 8(c). Here Figure 8(a) and (b) represent
the deflection of plastic propellers of varying
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IEEE - Aerospace and Electronic Systems - December 2022

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