1
A MATHEMATICAL MODEL FOR STUDYING THE
REACTION OF AN AIRCRAFT ENGINE BLADE TO A BIRD
STRIKE
Dilshod Eshmuradov Elmuradovich
1
, Seungjik Lee
2
, Behzod Zhumamuratov Akramjonovich
3
,
Temurmalik Elmuradov Dilshod_o`g`li
4
1
Head of the Department of "Energy Supply Systems" of the Tashkent University of Information Technologies,
Candidate of Technical Sciences, Associate Professor, Academician of the Academy of Sciences of Turon
2
Ph.D,
Professor of the Department of “Artificial Intelligence” of the Tashkent University of Information Technologies
3
doctoral student of the Department of "Metrology, technical regulation, standardization and certification" of Tashkent State
Technical University named after Islam Karimov
4
Senior Lecturer of Tashkent State Technical University named after Islam Karimov
https://doi.org/10.5281/zenodo.10471056
Abstract.
The method of studying the reaction of an aircraft engine blade to a bird strike is considered. A model of the
contact interaction of a soft div with the blade of an aircraft engine has been developed. By comparing the
results and mathematical modeling with the results of the experiment, the efficiency and operability of the
proposed model and method for studying the bird resistance of aircraft engine blades are proved.
Keywords:
aircraft engine, blade, bird, impact, stability.
INTRODUCTION
Cases of birds getting into an aircraft engine raise
a number of questions related to the reliability of
aircraft equipment and flight safety. The annual cost
of bird strikes to commercial aircraft worldwide is
estimated at US $1.3 billion. Considering that in the
coming years the likelihood of solving the problem of
eliminating cases of birds entering an engine during
operation is very low, one of the effective ways to
reduce the negative impact of birds and improve
operational quality and flight safety is to create
engines that are resistant to damage resulting from
these collisions. In birds, the durability of aircraft
engines is assessed using mathematical modeling
methods and experimentally. One of the reasons for
the high cost of developing an aircraft engine is the
fact that the design process involves the need for
expensive full-scale testing. One of the effective ways
to reduce the cost of engine development is to reduce
the number of full-scale tests and partially replace
them with numerical experiments. In addition to
reducing cost, the use of a computational experiment
can reduce development time by 3 times and improve
the quality of the finished product. Therefore, the
development of numerical models of contact
interaction between a bird (soft div) and an aircraft
engine blade (obstacle) with the aim of introducing
them into the practice of designing bird-resistant
blades is an urgent scientific and technical problem.
The goal of the work is to develop a mathematical
model of the interaction of a soft div with an aircraft
engine blade and a method for studying its response to
impact.
The process of a soft div colliding with an
obstacle is a complex physical and mechanical
process, with its inherent features and certain
methodological difficulties associated with its
modeling. The problem of a collision of a soft div
with an obstacle is a non-stationary, spatial, contact
problem of continuum mechanics. Figure 1 shows a
block diagram of the computational and experimental
method for studying the mechanical processes of
contact of a soft div with an engine blade.
2
Figure 1 - Computational and experimental method for studying the mechanical processes of contact
of a soft div with an engine blade
The practical implementation of the method under
consideration involves a consistent transition from a
real phenomenon to an idealized representation. It has
the form of continuous media in order to obtain
qualitative and quantitative results from numerical
simulation of a real phenomenon. The physical model
describes the phenomenon of water hammer [1], which
accompanies the collision process. There are four
stages of water hammer: 1) active (initial), which is
associated with the propagation of the shock wave; 2)
the stage of pressure decline, which is accompanied by
the propagation of a rarefaction wave; 3) stage of
steady flow and 4) termination of the process. A
mathematical model, represented by a system of
partial differential equations, describes the mechanical
motion and thermo mechanical state of deformable
bodies. Together with geometric and physical
relationships, as well as limit, initial and contact
conditions, the equations of the mathematical model
constitute a general initial-boundary value problem.
The construction of a numerical model involves
the transition from differential equations (strong form)
to the integral equation of motion (weak form) through
the variational principle of virtual works. Within the
framework of this stage, the following problems were
solved: 1) the method of sampling the soft div and
shoulder blade was chosen; 2) the effect of the
sampling step on the accuracy of the resulting solution
was investigated; 3) the influence of the shape of the
soft div on the pressure distribution was
investigated; 4) models of continuous media are
selected; 5) a contact interaction model is selected and
described; 6) the solution method is selected.
In order to verify the numerical model of contact
interaction of the soft div with the blade and
substantiate the reliability of the obtained results, the
results of numerical modeling were compared with the
results of a full-scale experiment. Comparison criteria
were
developed:
qualitative
and
quantitative
comparison using integral indicators and by the
distribution of physical parameters.
PHYSICAL MODEL
The physical model was built with the following
assumptions in mind:
1)
the soft div is a cylinder having a length to
diameter ratio of 2;
2)
soft div material is considered
homogeneous;
3)
the strength of the soft div is small
compared to the strength of the blade, and it is
neglected;
4)
given the assumption specified in claim
3, there is no rebound of the soft div;
5)
forces of
viscous damping in the material and friction forces on
the contact surface are neglected;
6)
flow in the
material beyond the shock wave front is one-
dimensional, adiabatic and irreversible.
Figure 2 shows the four phases of the impact. The
first phase-active (Fig. 2a) is characterized by a sharp
increase in pressure due to sharp braking of particles
in the zone of contact of the soft div with the obstacle
and is associated with the propagation of the shock
wave in the direction opposite to movement. The
active phase of the impact is described using two
parameters: Hugoniot pressure and pressure build-up
time. The Hugoniot pressure [1] is defined according
to the expression (1):
𝑝
𝐻
= 𝑝
2
− 𝑝
1
= 𝜌
1
𝑣
𝑠
𝑣
0
(1)
3
𝑝
1
and
𝑝
2
- pressure before and behind the shock
wave front;
𝑣
- the rate of propagation of the shock
wave in the medium;
𝑣
0
- impact speed.
The second phase-propagation of the vacuum wave
(Fig. 2 b) is associated with the propagation of the
vacuum wave from the free surface of the soft div to
the center due to the formation of a zone of high
pressure gradients. This in turn causes the free surface
of the soft div to move radially relative to the
obstacle. When the vacuum wave reaches the center of
the soft div (point c, Fig. 2 b), a decrease in pressure
is observed. The law of distribution of pressure (2)
along the radius of the cylindrical volume is
determined by the ratio [1]:
𝑝
𝑟
= 𝑝
H
𝑒
−𝑘𝑟
𝑅(𝑡)
(2)
k-
constant
;
an
r-
radius vector determining the
location of the point at which the pressure is measured
;
R (t)
is the maximum contact radius at time
t.
Figure 2 - Impact phases
In the third phase of steady-state flow (Fig. 2 v) -
there is a decrease in radial pressures in the soft div
and the occurrence of tangential stresses. Since the
strength of the soft div under the action of tangential
stresses is low, it spreads over the surface of the
obstacle. At this stage, stationary pressure and velocity
fields occur in the soft div. Braking pressure [1] at
the central point "c" is estimated using expression (3):
𝑝
𝑠
=
1
2
𝜌
0
𝑣
0
2
(3)
𝜌
0
- density of soft div material at zero porosity.
In the fourth phase, as the upper free surface of the
soft div approaches the obstacle, the speed of its
movement decreases and the pressure increases. The
pressure field is non-stationary and reaches the
maximum value at the braking point, with subsequent
reduction to atmospheric value as it moves away from
the center. As the free surface of the soft div is in this
pressure field, an instantaneous pressure drop occurs
and the flow process stops (Fig. 2 g).
The duration of the impact process [1] can be
estimated using expression (4):
𝑡
𝐷
= 𝐿 𝑣
0
⁄
(4)
L - is the length of the soft div.
MATHEMATICAL MODEL
The system of equations describing the motion and
thermo mechanical state of deformable solid media is
recorded in the actual configuration, and their
differentiation and integration is carried out according
to Euler coordinates.
𝜌𝑉 = 𝜌
0
, 𝑋 ∈ 𝑉
𝑇
∪ 𝑉
𝑏
(5)
𝜌
and
𝜌
0
- density of the medium at the current and
initial
time,
respectively;
𝑉 = 𝐽 = det(𝐹)
- relative volume;
𝑉
𝑇
- part of the
space of a given volume occupied by an obstacle;
𝑉
𝑏
-
part of the space of a given volume occupied by a soft
div.
𝜌
𝑑𝑣
𝑑𝑡
= 𝑑𝑖𝑣𝜎, 𝑋 ∈ 𝑉
𝑇
∪ 𝑉
𝑏
(6)
𝜌
𝑑𝑒
𝑑𝑡
𝑉𝑠
𝑖𝑗
𝜀
𝑖𝑗
− (ρ + q)𝑉,̇ 𝑋 ∈ 𝑉
𝑇
∪ 𝑉
𝑏
(7)
v
- velocity vector;
𝑑𝑒
𝑑𝑡
- acceleration vector;
𝑑𝑖𝑣𝜎 = ∇ ∙ 𝜎
- voltage tensor divergence;
4
∇=
𝜕(… )
𝜕𝑥
𝑖 +
𝜕(… )
𝜕𝑦
𝑗 +
𝜕(… )
𝜕𝑧
𝑘
- Hamilton operator
(Nabl operator);
σ
- Cauchy stress tensor;
е
- specific
internal energy; ε - strain velocity tensor;
р
- pressure;
q
- volume viscosity;
𝑠
𝑖𝑗
- components of the voltage
deviator.
System of equations (5-7) is supplemented with
kinematic (8) and geometric (9), (10) relations.
𝑑𝑢
𝑑𝑡
= 𝑣, 𝑋 ∈ 𝑉
𝑇
∪ 𝑉
𝑏
(8)
и
- vector of movements.
𝜀
𝑖𝑗
=
1
2
(
𝜕𝑢
𝑖
𝜕𝑥
𝑗
+
𝜕𝑢
𝑗
𝜕𝑥
𝑖
+
𝜕𝑢
𝑘
𝜕𝑥
𝑖
𝜕𝑢
𝑘
𝜕𝑥
𝑗
) ;
(9)
𝜀̇
𝑖𝑗
=
1
2
(
𝜕𝑣
𝑖
𝜕𝑥
𝑗
+
𝜕𝑣
𝑗
𝜕𝑥
𝑖
)
(10)
Physical relations (11-13) describe peculiarities of
behavior of deformable media manifested in the form
of deformation resistance.
In the case of elastic-plastic behavior of the
interference material, the components of the stress
tensor are as follows:
𝑝 = 𝐾(
1
𝑉
− 1);
(11)
𝑠
𝑖𝑗
∇
+ 2𝐺Λ̇s
𝑖𝑗
= 2𝐺(𝜀̇
𝑖𝑗
− 𝜀̇
𝑘
𝑔
𝑖𝑗
);
(12)
К
- bulk compression modulus;
s
𝑖𝑗
- derivative of
voltage deviator; G- rigidity modulus;
λ
- scalar
parameter;
𝑔
𝑖𝑗
- metric tensor components.
To build a numerical method for solving a system
of defining equations of a mathematical model, the
variation principle of virtual works is used.
For the sampling of the soft div, the Smoothed
Particle Hydrodynamics was used, which uses the
Lagrangian approach to describe the movement of the
continuous medium [2, p. 637-642; 3; 4]. The solid
medium is represented by a discrete set of mobile
particles that allow arbitrary connectivity with each
other. Each of the particles is an interpolation point at
which the media properties are set. The particle is
defined by the spatial coordinates
xi(t)
and the mass
mt(t).
The properties of the particle are determined by
the smoothing length
(h)
using the kernel function
(W)
.
The particle property
A
at an arbitrary point
r
is
determined by summing the corresponding values of
all particles within two smoothing lengths:
𝐴(𝑟
𝑖
) =
∑
𝑚
𝑗
𝑁
𝑗=1
𝐴
𝑗
𝜌
𝑗
𝑊((𝑟
𝑖
− 𝑟
𝑗
), ℎ);
𝑚
𝑗
- mass of the j
th
particle;
𝐴
𝑗
- value of parameter
А of the j
th
particle;
𝜌
𝑗
- density of the j
th
particle;
r
-
coordinate;
h
- smoothing length;
W
- weight function
or core;
N
- number of adjacent to j
th
particles.
The kernel function is defined by the anti-aliasing
function in
θ (x):
𝑊(𝑥, ℎ) =
1
ℎ(𝑥)
𝑑
𝜃(𝑥);
d
- parameter that determines the dimension of the
space,
𝑥 = 𝑟 ℎ
⁄
𝜃(𝑥) =
1
𝜋ℎ
3
{
1 −
3
2
(
𝑟
ℎ
)
2
+
3
4
(
𝑟
ℎ
)
3
; 0 ≤
𝑟
ℎ
≤ 1
1
4
(2 −
𝑟
ℎ
)
3
; 1 ≤
𝑟
ℎ
≤ 2
0;
𝑟
ℎ
> 2 }
;
After sampling, the main equations of the
mathematical model take the form (13-14):
𝜌
𝑖
= ∑
𝑚
𝑗
𝑊
𝑖𝑗
𝑁
𝑗=1
;
(13)
𝑑𝑣
𝑖
𝛼
𝑑𝑡
= ∑
𝑚
𝑗
(
𝜎
𝑖
𝛼𝛽
𝜌
𝑖
2
+
𝜎
𝑗
𝛼𝛽
𝜌
𝑗
2
)∇W
𝑖𝑗
𝑁
𝑗=1
;
(14)
To sample the equation of motion in time, a
modification of the method of central differences was
used, which is implemented in the form of an explicit
scheme of the 2
nd
order with variable time increments
[2, c. 501]. To find the solution to equation (21), the
processing time is divided into nrs time intervals or
steps in
∆t
time, where
n
=
1...nTs
. Vector of nodal
accelerations (15) on the
n
th
time layer is determined
as a result of the rotation of the mass matrix:
𝛼
𝑛
= 𝑀
−1
(𝑓
𝑖𝑛𝑡
(𝑢
𝑛
, 𝑡
𝑛
) + 𝐻
𝑛
);
(15)
The finite difference expression for determining
the velocity vector on a half-integer time layer is (16):
𝑣
𝑛+
1
2
= 𝑣
𝑛−
1
2
+ ∆𝑡
𝑛
𝛼
𝑛
;
(16)
The finite difference expression for determining
the node motion vector on the following time layer
tn
+ 1
is (17):
𝑢
𝑛+1
= 𝑢
𝑛
+ ∆𝑡
𝑛+
1
2
𝑣
𝑛+
1
2
;
(17)
The updated position of the nodes is obtained by
adding to the initial position vector the values of nodal
movements calculated on the following time layer
(18):
𝑥
𝑛+1
= 𝑥
0
+ 𝑢
𝑛+1
.
(18)
THE RESULTS OF THE
RESEARCH
Using the method considered in the work and the
constructed mathematical model of the contact
interaction of a soft div with a blade of an aircraft
engine, numerical studies of the reaction of a titanium
alloy blade to the impact of a soft div of different
weights, at different speeds and at different angles
were carried out. Figure 3 shows the pattern of
deformation of the soft div and blade during the first
250 μs for the case of an oblique impact of an 82.6 g
soft div at a speed of 302.1 m/s at an angle of 36.4 °
to a cantilevered titanium alloy blade, which has the
following dimensions: length 311.2 mm, width 88.9
mm and thickness 4.27 mm. The point of impact of the
soft div on the blade is located at a distance of 70%
of the blade span.
5
Figure 3 - Deformation pattern of soft div and blade in case of oblique impact
The obtained result makes it possible to
analyze the trajectories of movement of particles of the
soft div and to estimate the size and nature of
probable damage to the blade. In the case of an oblique
impact of a soft div on the blade, the soft div is
divides into two parts, one of which interacts with the
surface of the blade, and the other moves in the initial
direction.
Figure 4 - Graph of the change in the dynamic deflection
of the blade in the final section at frontal impact
Figure 4 shows a graph of the change in the
dynamic deflection of the blade in the final section in
the event of a frontal impact of a soft div weighing
100.5 g at a speed of 177.4 m/s along a cantilevered
titanium alloy blade. Figure 5 shows the result of
comparing the bending stresses in the root section of
the blade for frontal and oblique impacts. The
influence of the speed and angle of collision on the
distribution of stresses in the root section of the blade
was investigated.
Figure 5 - Impact of impact process parameters on the
distribution of normal stresses in the root section of the
blade
When analyzing the distribution of normal stresses
at the root intersection of the blade for both impact
cases, the following should be noted: In terms of the
probability of damage, the case of frontal impact is
more dangerous than the case of oblique impact.
Although in the case of oblique impact, the speed of
the soft div is higher than in the case of frontal
impact, the stress level for this case is lower than the
corresponding stress level in the case of frontal impact.
This indicates a more significant impact of the
collision angle on the stress level than the speed. For
both cases, the stress level exceeds the yield strength,
as evidenced by the development of plastic
deformations in the root section of the blade.
CONCLUSION
1. A hybrid model of the contact interaction of a
soft div with an aircraft engine blade has been
developed, which combines two sampling methods:
the finite element method for a blade and the mesh-
free method of smoothed particles for a soft div.
2. The use of a grid-free method of smoothed
particles to sample a soft div eliminated problems of
numerical instability and expanded the field of
6
modeling and investigation of mechanical processes
accompanying impact.
3. Qualitative coordination of the results of
numerical modeling with the results of a full-scale
experiment, indicating the operability of the proposed
model and the possibility of using it as an alternative
to full-scale tests, was obtained. This, in turn,
simplifies, accelerates, and reduces the material cost
of designing new bird-resistant blades.
4. The use of first-order shell elements in a
numerical model with a single integration point for
sampling the blade reduces the computational cost
compared to elements that use a complete integration
scheme, and this in turn increases the computational
efficiency of the model.
5. The model allows you to analyze the possible
consequences of a soft div impact on the blade,
assess the size and type of probable damage, as well as
obtain a distribution of parameters characterizing the
thermo mechanical state of the blade in time, as well
as in volume.
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URL:
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