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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.

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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

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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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