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Statement and general technique for solving
problem of oscillation of a hereditarily
deformable aircraft
Botir Usmonov1, Quvvatali Rakhimov2, Akhror Akhmedov2*
1Tashkent Chemical-Technological Institute, Tashkent, Uzbekistan
2Fergana state university, Fergana, Uzbekistan
Abstract. This article provides a general statement and a technique for
solving the problem of oscillations of a hereditarily deformable aircraft in a
gas flow with a finite number of degrees of freedom. Using the Lagrange
equations and the variational principle of the hereditary theory of
viscoelasticity, the equations of motion of the problem under consideration
are derived. The generalized forces acting on the aircraft in the subsonic
flight mode are determined according to the stationarity hypothesis. As a
result, closed interconnected weakly singular integro-differential equations
are obtained that describe the mathematical model of the problem with a
finite number of degrees of freedom. General schemes for the numerical
solution of these equations are outlined. As an example, the flexural-
torsional-aileron flutter of the transient process of a hereditarily deformable
wing with a finite freedom number is considered.
1 Introduction
An algorithm for the numerical solution of a system of linear and nonlinear weakly singular
integro-differential equations is constructed by eliminating weakly singular singularities of
integral and integro-differential equations.
The problem of flexural-torsional-aileron flutter of a hereditarily deformable aircraft
wing with finite degrees of freedom is considered. During the research, a special approach
to solving this problem was developed. According to this approach, the problem's solution
ultimately reduces to solving a system of homogeneous linear algebraic equations with
complex coefficients. From the conditions for the existence of non-trivial solutions of this
system, transcendental algebraic equations are obtained, the solution of which can be
obtained numerically using applied computer programs.
*Corresponding author: axmedov1981@gmail.com
E3S Web of Conferences 365, 05007 (2023) https://doi.org/10.1051/e3sconf/202336505007
CONMECHYDRO - 2022
© The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative
Commons Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/).
2 General statement of the problem. Numerical solution
technique
A hereditarily deformable aircraft moving in a gas flow is a complex mechanical system
with infinite degrees of freedom, continuously exchanging mechanical and thermal energy
with the environment. The description of the motion of such a system in the most general
form seems unlikely. Therefore, in most problems, the described motions of mechanical
systems have to be considered as a structure consisting of a finite number of deformable
and rigid elements, the relative mobility of which is limited by holonomic constraints.
Thus, in this case, along with external loads, the equations of motion will also include
unknown reaction forces of constraints. These forces can be eliminated from the equation
of motion using the principle of possible displacements. As a result, we obtain a system of
equations describing the motion of a non-free material system with holonomic constraints -
the Lagrange equation of the second kind
−
=0, =1,2… (1)
where,=−−,, -k is the kinetic and potential energy, Wis the work of
external forces, ,are the generalized coordinates and its time derivative.
The kinetic energy of a mechanical system with holonomic constraints is represented in
the form of a homogeneous quadratic function of generalized velocities
=
∑
, (2)
Potential energy, according to the variational principle of the hereditary theory of
viscoelasticity [2], can be expressed in the form of a homogeneous quadratic function of
generalized coordinates
=
∑−2∗
, (3)
where, сis inertial coefficients, is stiffness coefficients,
=∑
(4)
where is the generalized force,
∗()=(−)
- case of transient process,
∗()= (−)
∞
- for steady process
(−)is the core of heredity having a weakly singular feature of the Abel type, i.e.,
(−)=()(−);>0,>0,0<<1
E3S Web of Conferences 365, 05007 (2023) https://doi.org/10.1051/e3sconf/202336505007
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Substituting (2) - (4) into (1), we obtain a system of ordinary weakly singular integro-
differential equations of the second order of the form:
−
+
=, =1, (5)
Methods for solving such weakly singular integro-differential equations both for the
transient process and for the steady process are well described in [1,3].
In particular, for a flexural-torsional-aileron flutter of a hereditarily deformable wing,
expression (2)–(4) takes the form [1, 4]
=
+
+
+++ (6)
=
[−2∗]+
[−2∗]+
[−2∗] (7)
=++ (8)
Integro-differential equations (5) for this problem take the form
−
=; =1
,3
(9)
where =−,
If the external forces are aerodynamic, then according to the stationarity
hypothesis, for ,,we have [5,6]
=−+
++1
+1
=−+
++
+1
=−+
+++
(10)
where coefficients ,, (=1
,5
) are known constants [1,2]; ρ is air density; tis
wing chord; Vis flight speed.
Substituting (10) into (9), we obtain a system of weakly singular integro-differential
equations for determining the generalized coordinates q1=h, q2=α,q3=s, which in matrix
form can be written as
+(1−∗)++=0 (11)
with initial conditions
(0)=; (0)= (12)
that
=
;=
;= 00
0
0
00
=; =; =;V is flight speed;
E3S Web of Conferences 365, 05007 (2023) https://doi.org/10.1051/e3sconf/202336505007
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=
;=
The system of weakly singular integro-differential equations (11) together with the
initial conditions (12) is described by a discrete mathematical model in the general
formulation of the problem of the flexural-torsional-aileron flutter of a hereditarily
deformable wing in a gas flow. This system is quite general. In particular, it can be obtained
from it: a) if q3=0is the flexural-torsional flutter of a hereditarily deformable wing: b) if q2
=0– flexural-aileron flutter and finally c) if q1=0 – torsion -aileron flutter.
Then the problem becomes a classical flutter problem with two degrees of freedom:
∗=0we obtain the problem of the flexural-torsional-aileron flutter of an ideally elastic
wing together with the above-mentioned particular cases.
The exact solution of the system of weakly singular integro-differential equations (11)
presents significant mathematical difficulties. Therefore, an approximate solution,
according to the method of eliminating weakly singular singularities of integral and integro-
differential equations [3,7], is found from the following linear systems of recurrent
algebraic equations:
(+)=(+)+−
−∑(−)
+−
∑
−∑, =1,2...
(13)
Where
С==
2;=;=;=1,−1;=()
2;=();
=()
2[(+1)−(−1)];=1,−1;=()
2[−(−1)]
Determining the critical flight speed at which the flutter phenomena of aircraft begins is
one of the most important tasks of aeroelasticity. The solution is to study the oscillatory
instability (flutter) of the unperturbed motion of the aircraft using the developed
computational algorithm (13) and a special algorithm for finding critical speeds кр based on
a computational experiment for given geometric and mechanical parameters. According to
this technique, the loss of dynamic stability is determined from the conditions for the
existence of undamped amplitudes (critical time, critical speed) [1].
Thus, the use of the proposed mathematical models and the numerical solution
algorithm (13) makes it possible to investigate the problem of vibration of a wing with an
aileron made of hereditarily deformable materials in an air flow in subsonic flight modes.
Suppose the nature of the natural vibrations of a structure is known. In that case, it is
possible to judge its inherent internal properties that manifest themselves under the action
of external disturbances. On fig. 1. bending, torsional, and aileron oscillations are shown on
an elastic setting.
E3S Web of Conferences 365, 05007 (2023) https://doi.org/10.1051/e3sconf/202336505007
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Fig. 1. =320,=0.25,=0.05,=0.1
On fig. 2. bending, torsional, and aileron vibrations are shown in a viscous-elastic
setting.
Fig. 2. =320,=0.25,=0.05,=0.0
3 Conclusions
1. Computational experiments have shown that a slight decrease in the singularity
parameter or a slight increase in the viscosity parameter leads to a significant decrease in
the critical flutter speed. Accounting for aerodynamic damping in the viscoelastic case
leads to an increase in the critical flutter velocity. This can be explained by the fact that in
the viscoelastic case, the damping terms of the aerodynamic forces turn out to be a
destabilizing factor that causes oscillatory instability of hereditarily deformable systems.
2. It has been found that the damping rate of free flexural-torsional vibrations essentially
depends on the rheological parameters of the construction material. A decrease in the
E3S Web of Conferences 365, 05007 (2023) https://doi.org/10.1051/e3sconf/202336505007
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singularity parameter leads to an increase in the coefficient of internal energy absorption of
the system. Thus free oscillations of the system in practice will disappear after a certain
period, which provides a new opportunity to optimize the damping properties of the
material of vibrating structures used in aerospace technology.
3. It is determined that at low flight speeds, natural damped oscillations occur near the
equilibrium position, then with an increase in flight speed, the oscillatory process occurs
slightly above the equilibrium position with slowly increasing amplitudes.
4. It has been established that the excess of the critical flutter velocity in the ideal elastic
case does not mean the immediate destruction of the structure, the destruction occurs only
after a certain period and is of a fatigue nature. To find the expected service life of a
structure, it is necessary to determine the amplitude of its oscillations in the flutter region,
taking into account the structure material's nonlinear and hereditarily deformable
properties.
References
1. Badalov F.B. Ganikhanov Sh.F. Vibrations of hereditary - deformable structural
elements of aircraft. Tashkent. TGAI.2002. 230s.
2. Rzhanitsyn A.R. Creep theory. M. Nauka 1968. 416s
3. Matyash V.I. Flutter of an elastic-viscous plate. Mechanics of polymers. 1971. No. 6.
p. 1077-1083
4. Usmonov B.Sh. Oscillations of a hereditarily deformable wing with an aileron.
Problems of mechanics. Tashkent. 2006. No. 4. p.19-22
5. Keldysh M.V., Grossman E.P., Parkhomovsky Ya.M. Vibrations of the wing with
aileron. Proceedings of TsAGI 1937. no . 337. p.1-98.
6. Keldysh M.V. Selected works. Mechanics. M. science. 1985.348s.
7. Usmonov, B., Rakhimov, Q., Akhmedov, A. The study of the influence of the gamma
function on the flutter velocity. International Conference on Information Science and
Communications Technologies: Applications, Trends and Opportunities, ICISCT 2019.
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