G. A. Ankilov, P. A. Velmisov, A. S. Zharkova

Ulyanovsk State Technical University (Ulyanovsk, Russian Federation)

Abstract. This paper examines the mathematical modeling of ventilation systems consisting of deformable air ducts through which an air flow is supplied. Using constructed three- dimensional mathematical model described by a system of partial differential equations, the paper investigates dynamic stability of the elastic wall of an air duct where some gas flows. The Lyapunov dynamic stability criterion is used to study the mechanical system’s stability. To study stability in problems of aerohydroelasticity in compressible and incompressible medium models, Lyapunov-type functionals are constructed for deduced systems of differential equations. By studying these functionals stability conditions are obtained. They ensure that the functional is positive and its time derivative is negative. For a compressible medium model, the dependence between the longitudinal force compressing the plate and the air flow velocity is constructed for specific parameters of the mechanical system. Using the plot constructed, a comparison of the stability conditions for compressible and incompressible medium models is made. It is shown that the medium compressibility has negative effect on the stability of the deformable wall of the air duct and leads to decrease of the stability region.

Key Words: partial differential equations, dynamic stability, aerohydroelasticity, compressible and incompressible medium, air duct, elastic plate, Lyapunov-type functionals

For citation: G. A. Ankilov, P. A. Velmisov, A. S. Zharkova. Study of influence of flow compressibility on dynamic stability of elastic wall of air duct. Zhurnal Srednevolzhskogo matematicheskogo obshchestva. 27:4(2025), 488–499. DOI: https://doi.org/10.15507/2079-6900.27.202504.488-499

Information about the authors:

Grigory A. Ankilov, Postgraduate Student, Department of Higher Mathematics, Ulyanovsk State Technical University (32 Severny Venets St., Ulyanovsk 430027, Russia), ORCID: http://orcid.org/0009-0006-6180-0652, ankilov1996@mail.ru

Petr A. Velmisov, Dr. Sci. (Phys. and Math.), Professor, Department of Higher Mathematics, Ulyanovsk State Technical University (32 Severny Venets St., Ulyanovsk 430027, Russia), ORCID: http://orcid.org/0000-0001-7825-7015, velmisov@ulstu.ru

Alina S. Zharkova, Postgraduate Student, Department of Higher Mathematics, Ulyanovsk State Technical University (32 Severny Venets St., Ulyanovsk 430027, Russia), ORCID: http://orcid.org/0009-0004-8126-3589, mon16blan@yandex.ru