Flutter Optimization of Large Swept-Back Tri-Wing Flight Vehicles

Abstract

The aerodynamic configuration of large swept-back tri-wings is generally adopted for hypersonic vehicles, but the structural stiffness of the ailerons is weak, which may lead to damage due to the flutter behavior. In the initial stage of structural design, studying the flutter characteristics of tri-wing flight vehicles is necessary and can provide the stiffness index of the tri-wing structural design. To assess the flutter characteristics of tri-wing flight vehicles efficiently, a rapid modeling technique of the finite element method was used in this paper. For the structural scheme of large swept-back tri-wing flight vehicles, a structural dynamic model was modeled using the rapid modeling technique, the unsteady aerodynamic was computed using the double-lattice method, and the flutter characteristics were analyzed using the P-K method. Variable parametric studies were conducted to evaluate the effects of the stiffness of the aileron skin, the stiffness of the control mechanism, and the mass distribution of the aileron on the flutter characteristics of large swept-back tri-wing flight vehicles. The results showed that the key flutter coupling modes of such vehicles are symmetric and anti-symmetric combinations of aileron rotation and torsion. Additionally, optimizing the control mechanism stiffness and mass distribution of the aileron could improve the flutter boundary, which can be helpful in the structural design of such vehicles. The flutter optimization technique effectively improved the flutter boundary, significantly enlarged the flight envelope, and accurately provided the stiffness index for the structural design of large swept-back tri-wing flight vehicles.

1. Introduction

Aeroelastic instabilities are some of the critical issues affecting the reliability and safety of military and commercial aircraft [1,2,3]. The tri-wing configuration is usually adopted for large swept-back hypersonic flight vehicles [4,5], such as the American X-37B, the Soviet Union’s Buran, the British Hotol, and the German Sanger [6,7,8]. The aileron structure of such flight vehicles is thin, which may lead to the flutter behavior. Analyzing the flutter characteristics of a tri-wing flight vehicle is a necessary part of its structural stiffness assessment. Many scholars and experts have studied the flutter of tri-wing flight vehicles. Sziroczak [9] provided an overview of the current technical issues and challenges associated with the design of hypersonic vehicles and took a multi-disciplinary approach to the assessment of these vehicles. Vedeneev and Nesterov [10] investigated the impact of the nonequilibrium reacting airflow on the flutter boundary; the results obtained could be useful in the design of lightweight and robust flight vehicles. Decreasing the structural mass and increasing structural stiffness are considered to be crucial to the structural design; therefore, flutter optimization can be used to obtain find the optimal solution and to deal with the flutter problem. Ricciardi [11] presented a novel approach to the solutions of multi-phase, multi-objective optimal control problems, and the approach was applied to the solution of three test cases of increasing complexity: an atmospheric re-entry problem, an ascent and abort trajectory scenario, and a three-objective system and trajectory optimization problem for tri-wing flight vehicles. Dai et al. [12] presented an improved initial sizing method for the conceptual design of airbreathing hypersonic aircraft. The initial sizing results from the method were reasonable and satisfactory for the conceptual design of airbreathing hypersonic aircraft. Yuan et al. [13] developed a coupled computational fluid dynamics and computational structural dynamics capability for wing flutter that is applicable to tri-wing configurations. Scholars have also focused on and studied the following key technical issues faced by tri-wing flight vehicles in terms of aeroelasticity: the wide-speed range flight adaptability problem, the nonlinear aeroelastic vibro-acoustic problem, analysis and evaluation methods for handling stability characteristics, and optimization methods [14,15,16,17,18]. Furthermore, many scholars have studied optimization methods for flutter analysis. According to the current survey, some scholars have studied multi-objective optimization methods to analyse the flutter of wing structures and composite wings. Sleesongsom et al. [19,20] aimed to reduce the solution complexity in the multi-objective reliability-based design optimization (MORBDO) of an aircraft wing structure, which is a symmetrical part of the aircraft structure. They proposed the multi-objective reliability-based partial topology optimization of a composite aircraft wing using a fuzzy-based metaheuristic (MRBPTOFBMH) approach. Some scholars have studied the optimization procedures under nonlinear flutter constraints. Jonsson et al. [21,22] developed a framework for integrating a geometrically nonlinear flutter constraint, which considers in-flight deflections, into a high-fidelity gradient-based structural optimization. Zhang et al. [23] developed an optimization procedure involving nonlinear aeroelastic effects for galloping control based on the quasi-steady aeroelastic force model, and for flutter control based on a nonlinear unsteady model. Others have studied the design optimization applications under the flutter and post-flutter constraints. Jonsson et al. [24] presented flutter and post-flutter constraints for aircraft design optimization applications. These considerations and challenges are broadly applicable to the optimization of engineering systems, including stability and post-critical dynamic constraints. However, these scholars only focused on long straight wings, which are simple, and they did not involve complex layout fight. Few scholars have studied the flutter optimization problem of the aileron of large swept-back tri-wing flight vehicles.

Compared with classical wing flutter (the coupling mode of wing bending and torsion), the critical flutter velocity of aileron flutter is smaller. The reason is that the natural frequency of the wing torsion mode is higher than that of the aileron rotation and aileron torsion, which is less likely to appear. Therefore, studying aileron flutter is necessary and useful. In this paper, a rapid modeling technique of the finite element method was adopted to build a finite element model of a large swept-back tri-wing flight vehicle [25]. The rapid modeling technique was applied to the Automatic Meshing Tool developed by our team. The technique requires the use of commercial software to extract geometrical surfaces and lines. These geometrical surfaces and lines are topologically processed to ensure that one edge is shared by the connecting surfaces or lines. The material and property are defined by a text file. The Automatic Meshing Tool is run to obtain the Quad or Tria elements of the surfaces and the Bar element of the lines. The material does not contain the density term. The masses of the structure and non-structure are modeled so that they are concentrated at the grid points via CONM2 entries and connected to the primary structure by a multipoint constraint (RBE3). The grid nodes of CONM2 are used as the reference grid points (dependent nodes), and the grid nodes of the primary structures, which are the cross-nodes of the ribs, skin, and walls, are used as the independent nodes. This technique can contribute to parametric analysis and fleetly iterating models. Fast iterative models are also an important part of flutter optimization.

The natural frequency and mode shape of a tri-wing flight vehicle were calculated using the Lanczos method. The aeroelastic coupling method employs the Infinite Plate Spline (IPS), which is an improvement on the two-dimensional interpolation method and is very suitable for flat structures. The IPS method was proposed by Harder and Desmarais [26]. The flutter characteristics were analyzed using the P-K method [27,28]. According to the results, the main influencing factors of tri-wing flight include the stiffness of the aileron surface, the mass distribution of the aileron, and the stiffness of the control mechanism. Then, the influences of different parameters and flight conditions on the flutter boundary were studied. The flutter characteristics of the aileron were investigated based on the computational results of tri-wing flight vehicles. Finally, the flutter optimization was developed in this paper. The optimization variables are the structural parameters, the optimization target is the flutter boundary, and the optimization constraint is the structural mass.

2. Flutter Analysis

The structural stiffness of tri-wing flight ailerons is weak. In the whole flight envelope, it is often difficult for the structural design of tri-wing aircraft to meet the requirements of the flutter boundary. Flutter reflects the vibration instability of an aircraft in an airflow, so the structural dynamic characteristics, namely the natural frequency and mode shape, are the most essential factors in flutter analysis. In this study, a rapid modeling technique of the finite element method was adopted to build a finite element model. The mesh types of the fuselage were Quad and Tria, and the number of elements was 11,806. The mesh types of the main plane were Quad and Tria, and the number of elements was 6562. The mesh types of the aileron were Quad, Tria, and Bar, and the number of elements was 944 [25]. For the tri-wing flight vehicle (Figure 1), the Lanczos method was used to solve the eigenvalues of the structural dynamic equations of the complex and large structure, so as to obtain the natural frequencies and mode shapes. According to the initial flutter analysis, the flutter is related to the first bend of the fuselage (BOF), the symmetric rotation of the aileron (SROA), the anti-symmetric rotation of the aileron (ASROA), the symmetric torsion of the aileron (STOA), the anti-symmetric torsion of the aileron (ASTOA), and the second bend of the fuselage. Therefore, only 6 modes were used in the flutter analysis.

Table 1. Mode parameters.

Mode Shape	Frequency/Hz
1st bending of fuselage (1st BOF)	22.72
Symmetric rotation of aileron (SROA)	28.22
Anti-symmetric rotation of aileron (ASROA)	28.51
Symmetric torsion of aileron (STOA)	39.40
Anti-symmetric torsion of aileron (ASTOA)	44.01
2nd bending of fuselage (2nd BOF)	58.05

shows the frequencies of the first bend of the fuselage, the symmetric rotation of the aileron, the anti-symmetric rotation of the aileron, the symmetric torsion of the aileron, the anti-symmetric torsion of the aileron, and the second bend of the fuselage.

During flutter analysis, it is necessary to reasonably select the key mode shapes, so as to obtain an accurate judgment of the flutter modes [29]. There can be one or more combinations of different mode shapes from the frequency spectrum of the aircraft, which are the most likely relation to the lower critical flutter velocity [30]. For the multi-order mode shapes of aircraft, only the first six order modes (Table 1) are the key modes in the flutter analysis because the flutter velocities are calculated using multi-order mode combinations and the first six modes are enough to study the flutter characteristics. Finally, two types of flutter coupling modes are compared to analyze the flutter characteristics.

Six order mode shapes (Table 1) were selected for flutter analysis. According to the flutter results (

Table 2. The flutter results.

Mode Shapes Combination	Flutter Coupling Modes	Flutter Speed (m/s)	Flutter Frequency (Hz)
Six modes	See Table 1	225	39.11
		289	43.60
Symmetric modes	SROA/28.22 Hz and STOA/39.40 Hz	227	39.11
Anti-symmetric modes	ASROA/28.51 Hz and ASTOA/44.01 Hz	288	43.59

), two types of flutter could exist. The first corresponds to a flutter velocity of 225 m/s and a flutter frequency of 39.11 Hz; the second corresponds to a flutter velocity of 289 m/s and a flutter frequency of 43.60 Hz. From Figure 2, the damping of the STOA and ASTOA changes from negative to positive; moreover, the frequencies of SROA and ASROA increase gradually with the velocity, which means that SROA, ASROA, STOA, and ASTOA are the main flutter modes. According to the classical flutter condition, this flutter mechanism type is one of the classical flutter mechanisms. Because the flutter mode consists of two or more modes, in order to further judge the flutter mode, the mode selection method was adopted to analyze the flight flutter behavior.

To study the relationship between the two types of flutter behavior, the symmetric vibration (the frequencies of the SROA and STOA are 28.22 Hz and 39.40 Hz) and the anti-symmetric vibration (the frequencies of the ASROA and ASTOA are 28.51 Hz and 44.01 Hz) of the tri-wing aircrafts were selected to calculate the flutter velocity. The symmetric vibration types of the aileron (SROA and STOA) were selected for the flutter analysis. According to the results (Figure 3) of the symmetric rotation and symmetric torsion, the flutter velocity was 227 m/s and the flutter frequency was 39.11 Hz. Comparing to the results of six order mode shapes, the relative error was 0.9%. Thus, it can be concluded that the first flutter mode is the symmetric combination of the SROA and STOA mode shapes.

The anti-symmetric mode shapes of the aileron (ASROA and ASTOA) were selected for flutter analysis. According to the results (Figure 4) of the anti-symmetric rotation and anti-symmetric torsion, the flutter velocity was 289 m/s and the flutter frequency was 43.59 Hz. Comparing the results of the six modes, the relative error was 0.3%. So, it can be concluded that the second flutter mode is the anti-symmetric combination of the ASROA and ASTOA mode shapes.

By comparing the flutter results of the symmetric vibration and anti-symmetric vibration, we can conclude that the flutter modes of large swept-back tri-wing flight vehicles are mainly the symmetric mode (SROA and STOA) and anti-symmetric mode (ASROA and ASTOA). The critical flutter velocity of the anti-symmetric vibration is higher than that of the symmetric vibration.

3. Parametric Analysis

3.1. Variable Stiffness of Aileron Skin

For the tri-wing aircraft, the flutter characteristics of different parameters were studied by analyzing of various structural parameters. In this subsection, the effect of the aileron stiffness on the flutter characteristics was studied by changing the aileron skin thickness of the tri-wing flight vehicle. The aileron skin thickness was altered by changing the thickness item of PSHELL.

According to the results of flutter under different aileron stiffnesses (Figure 5), the frequencies of the symmetric rotation and anti-symmetric rotation rise gradually with the increase in skin thickness, and the symmetric torsion and anti-symmetric torsion of the aileron change slowly. By increasing the aileron skin stiffness, the velocities of both the symmetric and anti-symmetric flutter modes increase, and the symmetric flutter mode velocity increases more significantly than the anti-symmetric flutter mode velocity. Therefore, the increase in aileron stiffness has an impact on the symmetric flutter mode. Obviously, increasing the aileron stiffness can improve the flutter speed, but it will increase the mass of the aileron simultaneously.

3.2. Variable Stiffness of Control Mechanism

In this subsection, the effects of the flutter velocity on the control mechanism stiffness were studied by changing the stiffness of the control mechanism. The stiffness of the aileron control mechanism has an obvious effect on the frequency of the symmetric and anti-symmetric rotation and torsion modes. Firstly, the variable regularity of the symmetric and anti-symmetric rotation and torsion frequencies of the aileron, along with the stiffness of the aileron control mechanism, was studied. As shown in Figure 6a, with the variable stiffness in the control mechanism, the frequencies of the symmetric and anti-symmetric rotations are obvious, while the frequencies of the symmetric and anti-symmetric torsion are not obvious. Furthermore, the anti-symmetric rotation and torsion frequencies are always higher than the symmetric vibration. As the control mechanism stiffness increases, the ratio of the rotation frequency to torsion frequency also changes, which affects the flutter coupling modes, thus manifesting as a difference in the critical flutter velocity.

3.3. Variable Mass of Aileron

The effect of the mass on the flutter characteristics was studied by changing the mass coefficient of the aileron. According to the common principles of the structural dynamics, increasing the mass can reduce the structural frequency. As shown in Figure 7, with the increase in the coefficient of the aileron, the frequency of the symmetric and anti-symmetric torsion frequency decreases slowly, and the flutter velocity decreases quickly and then increases slowly. Therefore, in an appropriate situation, the mass of the aileron can help to improve the flutter velocity, and reducing the aileron mass is better for improving the flutter velocity than increasing the mass.

3.4. Variable Flight Altitude and Mach Number

The upper boundary of the flight envelope is limited by the flutter boundary. Therefore, it was necessary to study the flutter boundary of the tri-wing aircraft under different flight conditions by changing the altitude and Mach number. Figure 8 shows the flutter boundary of the tri-wing aircraft at different Mach numbers, which shows a flutter risk point at Mach 0.8. Therefore, the structural design of tri-wing aircraft should focus on the flutter characteristics at Mach 0.8 under flight conditions. During structural design, the flutter behavior often involves the whole flight envelope. In order to intuitively analyze the influence of different aileron control mechanism stiffnesses on the flutter characteristics, the problem was switched to the flutter dynamic pressure of tri-wing aircraft, and the influence regularity on the flutter boundary was studied. By confirming the sea level height and changing the flight Mach number and the stiffness of the aileron control mechanism, the flutter regularity of the ailerons of tri-wing flight vehicles under the same flight condition (flight altitude and Mach number) is obtained. In the subsonic range, the flutter boundary decreases when the stiffness of the aileron control mechanism increases from 0 to 1 × 10⁷ N·rad/mm, and the flutter boundary increases when the stiffness of the aileron control mechanism increases sequentially. In the supersonic range, the trend is basically consistent with that of the subsonic velocity, but the trend is more intense. Overall, increasing the stiffness of the aileron control mechanism can contribute to improving the flutter boundary of the tri-wing aircraft.

During the structural design of an aircraft, it is necessary to determine whether the structural parameters have obvious flutter behavior based on the flutter boundary. As shown in Figure 8, when the flutter boundary is above the flight envelope ①, the tri-wing aircraft has no flutter behavior in the whole flight envelope; when the flutter boundary is in contact with the flight envelope ②, the tri-wing flight vehicle can be in a critical condition of flutter behavior; and when the flutter boundary intersects the flight envelope ③, the tri-wing aircraft has flutter problems in the full flight envelope, which means that the structural design scheme needs to be considered, and the flutter boundary must be improved by changing the control mechanism stiffness, the aileron stiffness, and the aileron mass. Of course, this study is based on the theoretical results, and the actual results also need to be verified using the flutter tunnel test.

Based on the results of the structural parametric analysis, the conclusion is that increasing the stiffness of the control mechanism and the mass of the aileron helps to improve the flutter velocity. To contrast the effects of the mass of aileron and the control mechanism stiffness on the flutter characteristics of the flight, two structural parameters were changed simultaneously. As shown in Figure 9, when the mass of the aileron is fixed and the control mechanism stiffness of the aileron increases, the flutter velocity first decreases and then increases. When the stiffness of aileron control mechanism is fixed and the mass of the aileron increases, the flutter velocity continually increases. As shown in Figure 10, the effect of the control mechanism stiffness on the flutter speed is significantly higher than that of the aileron (the bar chart on the right side of the symmetry line is significantly higher than on the left side). Therefore, increasing the stiffness aileron control mechanism is better than increasing the quality of the ailerons.

4. Flutter Optimization

By researching the flutter characteristics of large swept-back tri-wing flight vehicles and the influence regularity of the structural parameters, the flutter characteristics and factors can be mastered. To determine a reasonable structural improvement scheme, a flutter optimization technique for tri-wing aircraft was developed. The optimization design variables are the control mechanism stiffness (1.0 × 10⁷ ≤ k_c ≤ 5.0 × 10⁷ N·rad/mm), the thickness of the aileron skin (${1\le t}_a\le 10 \mathrm{m}\mathrm{m}$), and the mass of the ailerons (${0\le m}_a\le 0.01 \mathrm{t}$). The flutter boundary ($\mathit{Q}$) is the constraint function. The Efficient Global Optimization (EGO) algorithm is adopted for the optimizer. The Efficient Global Optimization (EGO) algorithm is a hybrid optimization algorithm in which an interpolating response surface model is built in every iteration and new simulation points are added based on the result of an optimization that is performed on the response surface model. The response surface model that is chosen for the EGO algorithm is the Kriging model [31].

Response surface modeling (RSM) is used to show the sensitivity of different variables to the flutter velocity. RSM is a collection of mathematical and statistical techniques that are useful for the modeling and analysis of problems in which a response of interest is influenced by several variables. By using the Design of Experiments (DOE) methods, which is a systematic approach to obtaining the maximum amount of information out of various types of experiments while minimizing the number of experiments, combined with response surface modeling, the response between variables of the points set up in the design can be predicted efficiently. RSM methodology allows for further processing of the DOE results. The use of 3D graphing of the RSM for flutter optimization is possible, representing an efficient approach. Figure 11, Figure 12 and Figure 13 show the sensitivity of the control mechanism stiffness, the stiffness of the aileron, and the mass at mass_3 (Figure 14) to the flutter characteristics of the tri-wing aircraft. The mass at mass_3 is more sensitive to the flutter velocity than the manipulation stiffness and the stiffness of the aileron.

According to the optimum results (Figure 14), it was found that changing the mass at the trailing edge of the aileron has a significant effect on the flutter boundary, which increased significantly along with the increase in mass. Figure 14 shows that changes in the flutter dynamic pressure with the optimization design variables under different Mach numbers. By changing the Mach number, the optimization design variables have different effects on the flutter dynamic pressure, which causes discontinuity in the different fold lines. The maximum flutter boundary represents the optimum result, which is at the mass_3 of the trailing edge. At the optimal position (the highest subsonic domain of the flutter boundary), the control mechanism stiffness is 1 × 10⁷ N·rad/mm and the mass coefficient is calculated as 1.5. According to the results of the variable parameters, it can be seen that the control mechanism stiffness and mass coefficient reach an ideal state, but the flutter boundary is greatly improved. From analyzing the data, the mass of the aileron trailing edge is higher than the other parameters. Therefore, the optimal effect of the mass of the aileron trailing edge on the flutter boundary needs to be determined.

The masses of three points (mass_1, mass_2, and mass_3) were changed at the trailing edge to calculate the flutter velocity. Figure 15 shows that increasing the mass at the trailing edge of the aileron contributes to increasing the flutter velocity, and that the mass closest to the root of the posterior edge (mass_3) of the wing has a greater effect on the flutter velocity.

Through the flutter optimization results, it can be concluded that the mass of mass_3 (at the trailing edge of the aileron) and the stiffness of the aileron control mechanism have the most obvious influence on flutter speed, while the thickness of the aileron skin has less influence than the former two.

5. Results and Discussion

Aiming at the flutter problem faced by large swept-back tri-wing flight vehicles, a parametric rapid finite element model of a tri-wing aircraft was built. The flutter characteristics of the tri-wing aircraft were analyzed, and the flutter coupling mode shapes of the aircraft were obtained. The flutter regularity for various aileron stiffnesses, various control mechanism stiffnesses, and various aileron mass distributions was studied. The following three results were obtained.

(1)

For the tri-wing aircraft, there are generally two flutter coupling modes: one is the symmetric combination of the SROA and STOA mode shapes; the other is the antisymmetric combination of the ASROA and ASTOA mode shapes.

(2)

According to the parametric analysis, the control mechanism stiffness and mass distribution of the aileron have more obvious effects on the flutter velocity than the structural stiffness of the aileron. Additionally, in the whole flight envelope, the flutter risk point is Mach 0.8.

(3)

The flutter optimization results show that increasing the mass of the aileron trailing edge could be conducive to improving the flutter boundary of large swept-back tri-wing aircraft.

Based on these results, the flutter coupling modes, a method for improving the flutter velocity, and the flutter risk Mach number have been clearly determined, which is useful for the structural design of large swept-back tri-wing flight vehicles.

6. Conclusions

In the present study, flutter optimization procedures were adopted to analyze the flutter behavior of large swept-back tri-wing flight vehicles. According to the results, three conclusions were obtained accordingly, as follows.

(1)

The flutter mechanism type is one of the classical flutter mechanisms, and the flutter behavior can easily occur in the aileron part. The critical flutter velocity of the symmetric flutter mode shapes is generally larger than that the anti-symmetric flutter mode shapes.

(2)

The structural design of large swept-back tri-wing flight vehicles should be focused on the control mechanism stiffness and the mass distribution of the aileron. It is recommended to appropriately increase the control mechanism stiffness and mass at the trailing edge of the aileron.

(3)

The flutter optimization process can be used in the structural design process for aircraft with complex layouts, such as large swept-back tri-wing flight vehicles.