Mod´elisation du comportement dynamique d’une plaque ´elastique en interaction avec un ´ecoulement de fluide

Abstract

As part of this thesis, a finite element method-based approach is proposed to study the aeroelastic instabilities of rectangular flat plates under the effect of supersonic flow. The mathematical model of the structure is developed using a combination of the finite element method and Sanders’ shell theory. The components of the membrane displacement of the plate element are modeled by two-dimensional polynomials, and the bending displacement component is modeled by the exact solution of the motion equation. The aerodynamic load induced by the supersonic flow is modeled by the first-order linearized piston theory, including the effect of the flow direction. The mass, stiffness, and damping matrices are constructed by exact analytical integration. The limits of flutter are obtained by solving the resulting system of equations. During our numerical investigations, we mainly addressed the effect of different types of boundary conditions, aspect ratio, and air flow orientation. After observing the remarkable nu- merical properties of the proposed formulation for homogeneous plates, we considered extending it to the case of functionally graded material (FGM) plates. This new for- mulation, based on the concept of the neutral plane, was evaluated in cases of free and aeroelastic vibrations. The results were then compared to those of other scientific lite- rature. The developed approach distinguishes itself by significant advantages in terms of accuracy and convergence.

As part of this thesis, a finite element method-based approach is proposed to study the aeroelastic instabilities of rectangular flat plates under the effect of supersonic flow. The mathematical model of the structure is developed using a combination of the finite element method and Sanders’ shell theory. The components of the membrane displacement of the plate element are modeled by two-dimensional polynomials, and the bending displacement component is modeled by the exact solution of the motion equation. The aerodynamic load induced by the supersonic flow is modeled by the first-order linearized piston theory, including the effect of the flow direction. The mass, stiffness, and damping matrices are constructed by exact analytical integration. The limits of flutter are obtained by solving the resulting system of equations. During our numerical investigations, we mainly addressed the effect of different types of boundary conditions, aspect ratio, and air flow orientation. After observing the remarkable nu- merical properties of the proposed formulation for homogeneous plates, we considered extending it to the case of functionally graded material (FGM) plates. This new for- mulation, based on the concept of the neutral plane, was evaluated in cases of free and aeroelastic vibrations. The results were then compared to those of other scientific lite- rature. The developed approach distinguishes itself by significant advantages in terms of accuracy and convergence.

ﻓﻲ ﺇﻁﺎﺭ ﻫﺫﻩ ﺍﻷﻁﺭﻭﺣﺔ، ﻳﺗﻡ ﺍﻗﺗﺭﺍﺡ ﻁﺭﻳﻘﺔ ﺗﺳﺗﻧﺩ ﺇﻟﻰ ﺍﻟﻌﻧﺎﺻﺭ ﺍﻟﻣﻧﺗﻬﻳﺔ ﻟﺩﺭﺍﺳﺔ ﻋﺩﻡ ﺍﻻﺳﺗﻘﺭﺍﺭ ﺍﻟﻬﻭﺍﺋﻲ ﺍﻟﻣﺭﻥ ﻟﻠﻭﺡ ﻣﺳﺗﻁﻳﻝ ﺗﺣﺕ ﺗﺄﺛﻳﺭ ﺗﺩﻓﻖ ﻓﻭﻕ ﺻﻭﺗﻲ ﻓﻲ ﺍﺗﺟﺎﻫﺎﺕ ﻣﺧﺗﻠﻔﺔ ﻭﻣﻭﺍﺯﻱ ﻟﻠﺳﻁﺢ ﺍﻟﻌﻠﻭﻱ. ﺗﻡ ﺗﻁﻭﻳﺭ ﺍﻟﻧﻣﺫﺟﺔ ﺍﻟﺭﻳﺎﺿﻳﺔ ﻟﻠﻬﻳﻛﻝ ﺑﺎﺳﺗﺧﺩﺍﻡ ﻣﺯﻳﺞ ﻣﻥ ﻁﺭﻳﻘﺔ ﺍﻟﻌﻧﺎﺻﺭ ﺍﻟﻣﺣﺩﻭﺩﺓ ﻭﻧﻅﺭﻳﺔ ﺍﻟﻘﺷﺭﺓ ﻟﺳﺎﻧﺩﺭ. ﻳﺗﻡ ﺗﻣﺛﻳﻝ ﻣﻛﻭﻧﺎﺕ ﺍﻻﻧﺣﺭﺍﻑ ﺍﻟﻐﺷﺎﺋﻲ ﻟﻠﻭﺡ ﺑﺎﺳﺗﺧﺩﺍﻡ ﻣﺗﻌﺩﺩﺍﺕ ﺍﻟﺣﺩﻭﺩ ﺍﻟﺛﻧﺎﺋﻳﺔ، ﻓﻲ ﺣﻳﻥ ﻳﺗﻡ ﺗﻌﻭﻳﺽ ﺍﻻﻧﺣﺭﺍﻑ ﺍﻟﺟﺎﻧﺑﻲ ﺑﻧﺎءﺍ ﻋﻠﻰ ﺍﻟﺣﻝ ﺍﻟﺩﻗﻳﻖ ﻟﻣﻌﺎﺩﻟﺔ ﺍﻟﺣﺭﻛﺔ. ﻳﺗﻡ ﺗﻣﺛﻳﻝ ﺍﻟﺣﻣﻝ ﺍﻟﻬﻭﺍﺋﻲ ﺍﻟﻧﺎﺗﺞ ﻋﻥ ﺍﻟﺗﺩﻓﻖ ﺍﻟﻔﻭﻕ ﺻﻭﺗﻲ ﺑﺎﺳﺗﺧﺩﺍﻡ ﻧﻅﺭﻳﺔ ﺍﻟﻣﻛﺑﺱ ﺍﻟﺧﻁﻲ ﻣﻥ ﺍﻟﺩﺭﺟﺔ ﺍﻷﻭﻟﻰ، ﺑﻣﺎ ﻓﻲ ﺫﻟﻙ ﺗﺄﺛﻳﺭ ﺍﺗﺟﺎﻩ ﺍﻟﺗﺩﻓﻖ. ﻳﺗﻡ ﺑﻧﺎء ﻣﺻﻔﻭﻓﺎﺕ ﺍﻟﻛﺗﻠﺔ ﻭﺍﻟﺻﻼﺑﺔ ﻭﺍﻟﺗﺧﻣﻳﺩ ﺑﺎﺳﺗﺧﺩﺍﻡ ﺍﻟﺗﻛﺎﻣﻝ ﺍﻟﺗﺣﻠﻳﻠﻲ ﺍﻟﺩﻗﻳﻖ. ﻳﺗﻡ ﺍﻟﺣﺻﻭﻝ ﻋﻠﻰ ﺣﺩﻭﺩ ﺍﻟﺗﺫﺑﺫﺏ ﻋﻥ ﻁﺭﻳﻖ ﺣﻝ ﻧﻅﺎﻡ ﺍﻟﻣﻌﺎﺩﻻﺕ ﺍﻟﺣﺎﻛﻣﺔ. ﺃﻭﻻً، ﺗﻡ ﺍﺳﺗﻛﺷﺎﻑ ﺗﺄﺛﻳﺭﺍﺕ ﺍﻟﺣﺩﻭﺩ ﺍﻟﻌﺎﺩﻳﺔ ﻭﻏﻳﺭ ﺍﻟﻌﺎﺩﻳﺔ، ﻭﻧﺳﺑﺔ ﺍﻟﺟﺎﻧﺏ، ﻭﺍﺗﺟﺎﻩ ﺍﻟﺗﺩﻓﻖ. ﺛﻡ ﺑﻌﺩ ﺫﻟﻙ ﺗﻡ ﺍﺳﺗﺧﺩﺍﻡ ﺍﻟﺻﻳﻐﺔ ﺍﻟﻣﻘﺗﺭﺣﺔ ﻟﻠﺗﻧﺑﺅ ﺑﺎﻻﻫﺗﺯﺍﺯﺍﺕ ﺍﻟﺣﺭﺓ ﻭﻋﺩﻡ ﺍﻻﺳﺗﻘﺭﺍﺭ ﺍﻟﻬﻭﺍﺋﻲ ﻟﻠﻭﺣﺎﺕ ﺍﻟﻣﺳﺗﻁﻳﻠﺔ ﺫﺍﺕ ﺍﻟﺗﺩﺭﺝ ﺍﻟﻭﻅﻳﻔﻲ. ﻗﻣﻧﺎ ﺑﻣﻘﺎﺭﻧﺔ ﺍﻟﻧﺗﺎﺋﺞ ﺍﻟﻣﺳﺗﻣﺩﺓ ﻣﻥ ﻫﺫﻩ ﺍﻟﺻﻳﻐﺔ ﻣﻊ ﺗﻠﻙ ﺍﻟﻣﻧﺷﻭﺭﺓ ﻓﻲ ﺍﻷﺑﺣﺎﺙ ﺍﻷﺧﺭﻯ ﺍﻟﻣﻧﺷﻭﺭﺓ ﻓﻲ ﺍﻷﺩﺑﻳﺎﺕ.