Multiagent Control of Airplane Wing Stability with “Feathers” under the Flexural Torsional Flutter
Abstract
:1. Introduction
2. Related Work
3. Dynamics Equations of Wing with “Feathers”
- b is the wing section chord;
- h is the transverse deflection of the section ;
- is the angle of attack in the section of the undeformed wing;
- is the wing twist angle, which is considered positive if it increases the angle of attack in the section;
- are the wing stiffness in bending and torsion, respectively;
- and are the linear force and moment relative to the stiffness axis acting on the wing.
- m is the linear mass of the wing;
- is the linear mass moment of inertia of the wing relative to its stiffness axis;
- and are the linear aerodynamic force of the wing and the linear moment of the aerodynamic force relative to the stiffness axis, due to wing vibrations.
- and are the linear aerodynamic force and moment relative to the stiffness axis respectively, arising due to wing oscillations in the neutral position of the “feathers”;
- and are the linear aerodynamic force and moment created by changing the orientation of the “feathers”.
- ; is the wing lift coefficient;
- is considered constant along the span;
- is the instant value of the angle of attack when the wing moves;
- is the value of the angle of attack at which ;
- is the air density.
- E and are SC’s of the wing section in the static position and during vibrations, respectively; and is the deflection of the wing;
- X-axis corresponds to the main stream speed;
- Y-axis is perpendicular to X-axis and to the axis of rigidity of the undeformed wing;
- are wing profiles (considered to be sufficiently thin) in a static position (before vibrations);
- are wing profiles during oscillations;
- are the front and back edges of the “feather” 1 in the neutral position on the corresponding profiles;
- are the front and back edges of the “feather” 2 (analog of the aileron) in the neutral position on relevant profiles;
- and are the trailing edges of the “feathers” 1 and 2, respectively, after their deviations.
- and are the distances from the leading and trailing edges of the “feather” 1 to the leading edge of the wing;
- and are similar parameters for the “feather” 2;
- is the angle of twisting of the wing near the point ;
- is the angle of deviation of the “feather” 1 from the neutral position;
- is the angle of deviation of the “feather” 2 from the neutral position;
4. Control Synthesis with the Speed-Gradient Method
5. Multiagent Control
- are coordinates of the point to which the i-th “feather” is attached;
- and are deflection and angle of twisting of the wing at the location of the i-th “feather” (deviations from the curve (2));
- is the subset of “feathers” exchanging information with the i-th “feather”;
- is a non-negative weighting coefficient denoting the significance of information passing from the i-th “feather” to the j-th. Here, we assume that and ; here if the i-th and j-th “feathers” are not connected;
- is the adjacency matrix of the network.
5.1. Non-Multiagent Control Synthesis
5.2. Multiagent Control Synthesis
6. Simulation
- time step ;
- number of time instants ;
- airspeed ;
- air density .
- linear mass of the wing ;
- wing section chord ;
- wing (half-span) length ;
- from (5) the derivative of the wing lift coefficient with respect to , ;
- wing stiffness in bending ;
- wing stiffness in torsion ;
- distance between stiffness centers and gravity centers in the wing cross section assumed to be constant and equal ;
- wing cross-section is assumed to be elliptical with height ;
- linear mass moment of inertia of the wing relative to its stiffness axis
- number of “feathers” ;
- “feather” coordinates in Z axis ;
- “feather” coordinates in X axis according to formulas (9) ;
- the coordinates of “feathers” trailing edges in X axis equal ;
- the coordinates of “feathers” joint in X axis ;
- the distance from the leading edge of the wing to the SC section .
7. Conclusions and Outlook
- a mathematical model of the bending-torsional vibrations of an airplane wing with controlled “feathers” on its surface was constructed;
- three different conditions of the control problem corresponding to distinct subject functions were considered;
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
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Shalymov, D.; Granichin, O.; Ivanskiy, Y.; Volkovich, Z. Multiagent Control of Airplane Wing Stability with “Feathers” under the Flexural Torsional Flutter. Mathematics 2022, 10, 236. https://doi.org/10.3390/math10020236
Shalymov D, Granichin O, Ivanskiy Y, Volkovich Z. Multiagent Control of Airplane Wing Stability with “Feathers” under the Flexural Torsional Flutter. Mathematics. 2022; 10(2):236. https://doi.org/10.3390/math10020236
Chicago/Turabian StyleShalymov, Dmitry, Oleg Granichin, Yury Ivanskiy, and Zeev Volkovich. 2022. "Multiagent Control of Airplane Wing Stability with “Feathers” under the Flexural Torsional Flutter" Mathematics 10, no. 2: 236. https://doi.org/10.3390/math10020236
APA StyleShalymov, D., Granichin, O., Ivanskiy, Y., & Volkovich, Z. (2022). Multiagent Control of Airplane Wing Stability with “Feathers” under the Flexural Torsional Flutter. Mathematics, 10(2), 236. https://doi.org/10.3390/math10020236