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The T-tail configuration, while offering advantages for large transport aircraft, is susceptible to peculiar aerodynamic phenomena such as deep stall and flutter, necessitating high-fidelity dynamic scaling for wind tunnel testing. In order to address the issue of similarity in the dynamic characteristics of scaled T-tail models, we propose a comprehensive optimization design method for dynamic scaled equivalent models of T-tail structures with rear fuselages. The development of an elastic-scaled model is accomplished through the integration of the least squares method with a genetic sensitivity hybrid algorithm. In this framework, the objective function is defined as minimizing a weighted sum of the frequency errors and the modal shape discrepancies ( $1 -$ Modal Assurance Criterion) for the first five modes, subject to lower and upper bound constraints on the design variables (e.g., beam cross-sectional dimensions). The findings indicate that the application of finite element modelling in conjunction with multi-objective optimization results in the scaled model that closely aligns with the dynamic characteristics of the actual aircraft structure. Specifically, the frequency error of the optimized model is maintained below 2%, while the modal confidence level exceeds 95%. A ground vibration test (GVT) was conducted on a fabricated scaled model, with all frequency errors below 3%, successfully validating the optimization approach. This GVT-validated high-fidelity model establishes a reliable foundation for subsequent wind tunnel tests, such as flutter and buffet experiments, the results of which are vital for validating the full-scale aircraft’s aeroelastic model and informing critical flight safety assessments. The T-tail elastic model design methodology presented in this study serves as a valuable reference for the analysis of T-tail characteristics and the design of wind tunnel models. Furthermore, it provides insights applicable to multidisciplinary optimisation and the design of wind tunnel models for other similar elastic scaled-down configurations.

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1. Introduction

A T-tail is a type of aircraft tail configuration, in which the horizontal stabiliser (consisting of the tailplane and elevator) is mounted on top of the vertical stabiliser [1]. This configuration enables the tailplane to remain unaffected by the exhaust gases of the engine [2], thus facilitating its wider adoption on aircraft equipped with rear-mounted engines. Seaplanes and amphibious aircraft also typically employ T-tail configurations in order to maintain the horizontal stabiliser as far as possible from the water surface, thereby reducing internal vibration and noise. T-tails are also employed in the fabrication of long-endurance drones that are characterised by stringent quality requirements [3]. The T-tail is a special aerodynamic structure. When the angle of attack is minimal, this configuration enables the horizontal stabiliser to circumvent the impact of wing flow, thereby enhancing control efficiency. Furthermore, the configuration of the layout facilitates the integration of openings in the rear fuselage, thereby offering a distinct advantage in terms of facilitating cargo transportation. Consequently, numerous military transport aircraft designs adopt the T-tail configuration.

However, the T-tail configuration results in complex aerodynamic characteristics due to its high horizontal tail layout. These characteristics pose a threat to flight safety, particularly in the form of flutter [4,5], buffet and deep stall issues [6,7]. This is due to the fact that the horizontal stabiliser possesses a substantial mass and is affixed to the uppermost extremity of the vertical stabiliser. Consequently, even negligible movements of the vertical stabiliser can result in substantial in-plane and out-of-plane movements of the horizontal stabiliser. Conversely, the weight of the horizontal stabiliser is located at the top of the vertical stabiliser, resulting in the imposition of significant loads on the vertical stabiliser [8]. Junction flows also generally occur at the junctions of horizontal and vertical tail parts, the flow in the corner region oscillates and flaps the connecting components, which may threaten aircraft safety [9]. Consequently, research conducted on T-tail holds considerable importance in the context of the development of large transport aircraft.

Nonetheless, due to the extreme operating conditions faced by large transport aircraft (load, temperature), it is impossible to conduct full-scale aircraft experiments. Notwithstanding the feasibility of conducting prototype testing, the experiments are costly, time-consuming and difficult to control [10]. Consequently, the necessity arises to design a scaled-down equivalent model that exhibits structural dynamics characteristics analogous to those of the aircraft [11]. Thereafter, wind tunnel tests are to be conducted within a wind tunnel environment, thereby simulating real aerodynamic load conditions.

With regard to wind tunnel tests on T-tails, relevant scholars have conducted wind tunnel tests at low speeds, high speeds, and high angles of attack [12]. Farokhi et al. [13,14] conducted research on asymmetric buffering loads on horizontal stabilisers in large-scale separated flows using rigid models. In 2015, Fichera and Ricci [15] developed a control-chain-free-floating T-tail aerodynamic elastic wind tunnel model. However, the horizontal tail of the T-tail was represented solely by its main structural component, the main spar. The aerodynamic shape of the horizontal tail was not utilised to reproduce its aerodynamic contribution. In 2018, Cunningham et al. [16,17] at NASA Langley Research Center conducted preliminary simulations of a general T-tail transport aircraft configuration using a T-tail transport aircraft model. The primary objective of this experimental simulation was to assess the fidelity requirements of the aerodynamic model. However, the model solely simulated the mass and aerodynamic shape characteristics of the aircraft, while the stiffness simulation was omitted. In 2023, Mamou and Broughton [18] utilised a rigid mould to assess the aerodynamic support interference sources of a half-scale model of an aircraft with a T-tail installed in a wind tunnel. Pusztai et al. [19,20] conducted wind tunnel tests using a carbon fibre-based universal T-tail transport aircraft model to study the dynamic characteristics of T-tail aircraft after stalling. The majority of extant literature focuses on researching the characteristics of T-tails, or else it neglects to consider the simulation of real stiffness by the model. The paucity of available information on the simulation of real aircraft frequencies and vibration modes is a matter of concern.

While optimization methods have been widely applied in fields such as structural health monitoring (SHM) for damage detection using natural frequency changes [21,22,23,24,25], the design of dynamically scaled aeroelastic models for wind tunnel testing presents a distinct set of challenges. Unlike SHM applications, which often utilize optimization as an inverse problem to identify structural damage based on measured data, the present study tackles a forward design problem. The objective here is to synthesize a physical structure whose inherent dynamic characteristics (both natural frequencies and mode shapes) precisely match pre-defined targets derived from scaling laws. This necessitates a multi-objective optimization approach that simultaneously constrains frequency errors and modal shape fidelity (quantified via the Modal Assurance Criterion), all under stringent manufacturing and mass distribution constraints. This requirement for high-fidelity dynamic similitude is paramount for reliably predicting aeroelastic phenomena such as flutter and buffet, which are critically sensitive to both frequency and mode shape.

As the theory, methods and applications of structural finite element analysis continue to evolve, the demand for accurate and reliable mathematical models that accurately reflect structural characteristics has increased. This has made modelling issues increasingly critical. The utilisation of a meticulously formulated structural dynamics model is imperative for a range of applications, including response calculations, load estimation, and stability analysis. However, conventional modelling methods frequently depend on the expertise and discernment of designers, necessitating repeated trial-and-error adjustments to attain satisfactory outcomes. Advances in structural optimisation design technology have led to the development of methods that employ optimisation techniques in the design of structural dynamics models [26]. The integration of these methods has the potential to enhance the efficiency and precision of the design process, thereby reducing the necessity for manual intervention.

In the context of structural optimisation of T-tail, Lv et al. [27] employed genetic sensitivity algorithms and dynamic simplified models to perform structural vibration optimisation on T-tail. This approach proved to be highly efficient in terms of calculation. However, it should be noted that the analysis was conducted exclusively on the structural natural frequency, with the structural mode shape being overlooked. Li and Sun [28] used Nastran software to perform structural optimization on composite T-tail fins with the aim of optimizing T-tail mass. Cai and Sun [29] also used Nastran software to optimize composite T-tail fins with the aim of improving their static strength, stiffness, and stability. The investigation further encompassed the exploration of the impact of composite layer orientation on the ultimate outcomes of the optimisation process. Ji et al. [30] employed a multi-level optimisation method to enhance the T-tail structure, with the objective of optimising static strength, stability, and flutter. Research in this field to date has focused primarily on quality and structural strength. However, there are few publications that comprehensively consider the natural frequency and vibration modes of T-tail.

The present paper puts forward a comprehensive optimization method combining the least squares and genetic sensitivity hybrid algorithm. By establishing a detailed dynamic finite element model of the T-tail with simulated aft fuselage stiffness, the natural frequencies and mode shapes of the target model are constrained, thereby completing the optimization design of the model. The optimization goal is formulated as a multi-objective problem, aiming to minimize a weighted-sum function that aggregates the relative frequency errors and the modal shape discrepancies ( $1 -$ MAC) for the first five structural modes. This optimization is subject to boundary constraints on the design variables, which include the cross-sectional dimensions (e.g., width, height) of the beam elements representing the primary structure.

It is worth emphasizing that frequency optimization underpins the design and qualification of virtually all major aerospace components, from airframes to propulsion systems. Accurate dynamic similitude is fundamental not only for predicting and avoiding aeroelastic instabilities, such as flutter, in the airframe [31] but also for ensuring that structural modes do not couple with critical engine rotor frequencies, a scenario that could precipitate high-cycle fatigue failures in the engine subsystem. Genetic Algorithms and other optimization techniques are consequently indispensable tools for achieving these critical dynamic design objectives across the discipline [32,33]. The central role of frequency correlation in validating high-fidelity aeroelastic models has been further highlighted in recent model updating methodologies [34].

By investigating the influence of quantified fuselage stiffness on modal frequencies, the feasibility of the single-beam simulated rear fuselage stiffness design for the T-tail elastic model was verified. To validate the methods described in this paper, a scaled model was fabricated, and ground vibration test (GVT) were conducted to verify the feasibility and effectiveness of the optimization methods and model design proposed in this paper.

2. Comprehensive Optimization Theory of Least Squares-Genetic Sensitivity Hybrid Algorithm

It is crucial to clarify that the target dynamic characteristics (frequencies and mode shapes) for the optimization are derived from a high-fidelity, full-scale finite element model of the original T-tail structure. The values from this reference model are scaled down according to the similarity laws detailed in Section 3.1 to establish the specific optimization targets $(λ_{n, aim}, {φ_{n}})$ for the scaled equivalent model. The objective of the optimization is thus to make the dynamics of the scaled finite element model match these scaled targets. Experimental data from a GVT is used subsequently in Section 5 to validate the physical realization of the optimized model, not to define the optimization targets herein.

The present chapter provides a comprehensive explanation of the mathematical methods employed in the comprehensive optimization theory of the least squares-genetic sensitivity hybrid algorithm. These include structural dynamics sensitivity analysis, least squares methods and mathematical models of optimization methods. The structural optimisation process of this method is illustrated in the Figure 1.

2.1. Hybrid Optimization Strategy

The comprehensive optimization framework integrates a genetic algorithm (GA) with a gradient-based least squares (LS) method. This hybridization is designed to leverage the global search capability of the GA and the local convergence efficiency of the least squares approach. The process, illustrated in Figure 2, follows these stages:

Initialization and Global Search: A GA population is initialized with random design variables within their bounds. The GA evolves this population over generations via selection, crossover, and mutation operators to explore the design space globally.

Transition: The GA phase continues until its convergence criterion is met. The best individual from the final GA population is selected as the initial point for the local search.

Local Refinement: The least squares method, utilizing the sensitivity derivatives from the Nelson method (Section 2.2), performs a local optimization starting from the GA-provided point. This step efficiently fine-tunes the design variables to achieve high precision on the objective function.

2.2. Structural Dynamics Sensitivity Analysis

The sensitivity analysis of eigenvectors is calculated using the Nelson method. The eigenvalue equation can be expressed as follows:

(1) $([K] - λ_{n} [M]) {φ_{n}} = 0$

In Equation (1), $λ_{n}$ and ${φ_{n}}$ represent the nth-order eigenvalues and eigenvectors, respectively. [ $K$ ] denotes the structural generalized stiffness matrix, and $[M]$ signifies the structural generalized mass matrix. Taking the partial derivative with respect to the ith design variable $θ$ yields

(2) $([K] - λ_{n} [M]) \frac{\partial {φ_{n}}}{\partial θ_{i}} + (\frac{\partial [K]}{\partial θ_{i}} - λ_{n} \frac{\partial [M]}{\partial θ_{i}}) {φ_{n}} = \frac{\partial λ_{n}}{\partial θ_{i}} [M] {φ_{n}}$

The aforementioned equation can be expressed as follows:

(3) $([K] - λ_{n} [M]) \frac{\partial {φ_{n}}}{\partial θ_{i}} = - (\frac{\partial [K]}{\partial θ_{i}} - λ_{n} \frac{\partial [M]}{\partial θ_{i}} - \frac{\partial λ_{n}}{\partial θ_{i}} [M]) {φ_{n}}$

Given that $([K] - λ_{n} [M])$ item is singular, it is not possible to use this formula directly to calculate the sensitivity of the eigenvector. In order to proceed, it is necessary to reduce the matrix scale by one order of magnitude. It is the contention of the mass regularization method that

(4) $\{\begin{matrix} {φ_{n}}^{T} [M] {φ_{n}} = 1 \\ {φ_{n}}^{T} [K] {φ_{n}} = λ_{n} \end{matrix}$

In Equation (4), T is defined as the transpose of the matrix. The sensitivity formula for frequency to structural parameters is as follows:

(5) $\frac{\partial λ_{n}}{\partial θ_{i}} = {φ_{n}}^{T} (\frac{\partial [K]}{\partial θ_{i}} - λ_{n} \frac{\partial [M]}{\partial θ_{i}}) {φ_{n}}$

2.3. Mathematical Expression of the Least Squares Method

Suppose there is a set of data $(x_{1}, y_{1}), (x_{2}, y_{2}), \dots, (x_{i}, y_{i})$ , then find a function $f (x)$ to fit these data. The goal of the least squares method is to minimize the sum of squared errors, i.e.,

(6) $L = \sum_{i = 1}^{m} {(y_{i} - f (x_{i}))}^{2}$

In the context of linear regression problems, under the assumption that $f (x) = a_{0} + a_{1} x$ , the objective function can be expressed as follows:

(7) $L = \sum_{i = 1}^{m} {(y_{i} - a_{0} - a_{1} x_{i})}^{2}$

The optimal values of $a_{0}$ and $a_{1}$ can be obtained by solving for the partial derivatives of L with respect to $a_{0}$ and $a_{1}$ , and subsequently setting L equal to zero. The final linear regression equation is as follows:

(8) $y = a_{0} + a_{1} x$

In Equation (8):

(9) $a_{1} = \frac{n \sum_{i = 1}^{m} x_{i} y_{i} - \sum_{i = 1}^{m} x_{i} \sum_{i = 1}^{m} y_{i}}{n \sum_{i = 1}^{m} x_{i}^{2} - {(\sum_{i = 1}^{m} x_{i})}^{2}}$

(10) $a_{0} = \frac{\sum_{i = 1}^{m} y_{i} - a_{1} \sum_{i = 1}^{m} x_{i}}{n}$

2.4. Mathematical Formulation of the Optimization Problem

The residual vector is introduced and the least squares method is derived through the norm. It is hypothesised that a linear equation system $A \hat{x} = b$ is in existence, where A is an $m \times n$ matrix, $\hat{x}$ is an $n \times 1$ unknown vector, and $b$ is an $m \times 1$ vector. The objective of the least squares method is to ascertain a $\hat{x}$ such that the error between $A x$ and $b$ is minimised, i.e.,

(11) $min_{\hat{x}} {∥ A \hat{x} - b ∥}_{2}^{2}$

among them,

{∥ \cdot ∥}_{2}

indicates

L_{2}

norm. The objective function of the least squares method can be expressed as follows:

(12) ${\hat{x}}_{a i m} = arg min_{\hat{x}} {∥ A \hat{x} - b ∥}_{2}^{2} = arg min_{\hat{x}} \sum_{i = 1}^{m} {(\sum_{j = 1}^{n} A_{i j} x_{j} - b_{i})}^{2}$

The residual vector is defined by constraining the modal shape term:

(13) $r = 1 - MAC (ϕ_{n}, φ_{n})$

The target nth-order modal vector is represented by $ϕ_{n}$ , whilst the optimization model n th-order modal vector is denoted by $φ_{n}$ . Modal Assurance Criterion is represented by MAC.

(14) $MAC = \frac{{(ϕ_{n}^{T} φ_{n})}^{2}}{(ϕ_{n}^{T} ϕ_{n}) (φ_{n}^{T} φ_{n})}$

The objective function of the method described in this paper is

(15) $f = {∥1 - MAC (ϕ_{n}, φ_{n})∥}_{2}^{2} = {[1 - \frac{(ϕ_{n}^{T} φ_{n})}{(ϕ_{n}^{T} ϕ_{n}) (φ_{n}^{T} φ_{n})}]}^{2}$

(16) $\bar{φ_{n}} = arg min_{φ_{n}} {∥ 1 - MAC (φ_{n}, ϕ_{n}) ∥}_{2}^{2} = arg min_{φ_{n}} {[1 - \frac{{(ϕ_{n}^{T} φ_{n})}^{2}}{(ϕ_{n}^{T} ϕ_{n}) (φ_{n}^{T} φ_{n})}]}^{2}$

$\bar{φ_{n}}$ is the n th-order target mode. And the gradient of $φ_{n}$ is

(17) $\nabla f (φ_{n}) = - 2 (1 - \frac{(ϕ_{n}^{T} φ_{n})}{(ϕ_{n}^{T} ϕ_{n}) (φ_{n}^{T} φ_{n})}) \cdot \frac{(ϕ_{n}^{T} φ_{n})}{∥ ϕ_{n} ∥_{2}^{2} {∥ φ_{n} ∥}_{2}^{4}} (∥ φ_{n} ∥_{2}^{2} ϕ_{n} - (ϕ_{n}^{T} φ_{n}) φ_{n})$

The structural dynamic optimization problem for the T-tail scaled model is formally stated as a constrained minimization problem. The objective is to find the set of design variables that minimizes the discrepancy between the dynamic characteristics of the scaled model and the target values, subject to manufacturing and physical constraints.

The optimization problem is mathematically defined as follows:

(18) $min_{θ} f (θ) = α \sum_{n = 1}^{5} |\frac{λ_{n} (θ) - λ_{n, aim}}{λ_{n, aim}}| + β \sum_{n = 1}^{5} {(1 - MAC (ϕ_{n} (θ), φ_{n}))}^{2}$

subject to:

(19) $θ_{i}^{l} \leq θ_{i} \leq θ_{i}^{u} for i = 1, 2, \dots, k,$

where:

$θ$ is the vector of k design variables (here, $k = 200$ ), representing the cross-sectional dimensions (e.g., width b, height h, web thickness $t_{w}$ , flange thickness $t_{f}$ ) of the beam elements in the primary structure.

$f (θ)$ is the scalar objective function to be minimized.

$λ_{n} (θ)$ and $ϕ_{n} (θ)$ are the n-th natural frequency and mode shape of the finite element model, respectively, which are functions of the design variables $θ$ .

$λ_{n, aim}$ and $φ_{n}$ are the target frequency and mode shape for the n-th mode, derived from the scaling laws in Section 3.1.

$MAC (ϕ_{n} (θ), φ_{n})$ is the Modal Assurance Criterion between the computed and target mode shapes, as defined in Equation (14).

$α$ and $β$ are weighting coefficients that balance the contributions of frequency and modal shape errors. In this study, $α = 1$ and $β = 1$ were employed to assign equal importance to both terms.

$θ_{i}^{l}$ and $θ_{i}^{u}$ are the lower and upper bounds for the i-th design variable, ensuring manufacturability and physical feasibility.

Function Evaluation Process: The evaluation of the objective function $f (θ)$ for a given candidate design $θ$ involves the following steps:

Finite Element Analysis: An eigenvalue analysis is performed on the detailed T-tail FE model (described in Section 3) with the current set of design variables $θ$ to compute the first five natural frequencies $λ_{n} (θ)$ and corresponding mass-normalized mode shapes $ϕ_{n} (θ)$ .

Error Calculation: The relative frequency error for each mode is calculated as $|(λ_{n} (θ) - λ_{n, aim}) / λ_{n, aim}|$ . The modal shape discrepancy is quantified as $1 - MAC (ϕ_{n} (θ), φ_{n})$ , where the MAC value is computed according to Equation (14).

Objective Function Aggregation: The weighted sum of the frequency errors and modal shape discrepancies across the first five modes is computed, yielding the value of the objective function $f (θ)$ .

The hybrid optimization strategy integrates a GA for global exploration with a gradient-based LS method for local refinement. This hybrid approach is designed to leverage the global search capability of the GA, thereby mitigating the susceptibility of pure gradient methods to local minima, while the subsequent LS step accelerates convergence to a high-precision solution. The algorithm proceeds as follows: The GA initializes with a population of 200 individuals, evolving over generations through tournament selection, simulated binary crossover, and polynomial mutation. The fitness of each individual is evaluated based on the objective function defined in Equation (18). After each GA generation, the top 10% of individuals (elites) are selected for local intensification. For each elite, a local search is conducted using the LS method, which leverages the sensitivity derivatives (computed via the Nelson method in Equations (16) and (17) to solve a local approximation problem Equations (11) and (12), efficiently driving the solution towards a nearby local optimum. The population is then updated, and this iterative process of global exploration and local refinement continues until the change in the best fitness value remains below a tolerance of $1 \times 10^{- 6}$ for 10 consecutive generations.

The optimization problem formulated herein underscores a fundamental distinction from typical tasks in the Structural Health Monitoring (SHM) domain [21,22,23,24,25]. In SHM, optimization often serves as an inverse problem for damage identification, where algorithms are designed to be sensitive to minimal frequency shifts or mode shape changes, and the MAC may be used primarily as a validation metric. Conversely, our forward design formulation treats frequency matching and MAC value maximization as dual, equally critical, and tightly constrained objectives within the optimization itself. This rigorous requirement stems from the criticality of aeroelastic scaling: even minor deviations in mode shapes can significantly alter the phase and energy transfer between structural deformation and unsteady aerodynamic loads, leading to non-conservative or inaccurate predictions of flutter and buffet boundaries. The present hybrid algorithm is thus developed to navigate this high-dimensional, non-linear design space effectively, ensuring that the final scaled model is not merely frequency-equivalent but also mode shape-equivalent to the full-scale aircraft structure—a prerequisite for reliable wind tunnel testing.

2.5. Algorithm Selection Rationale

The selection of the hybrid GA-LS algorithm was deliberate, grounded in the specific challenges posed by the structural dynamics optimization of the T-tail scaled model. The design problem is characterized by a high-dimensional, non-linear, and constrained search space, where the objective function (evaluated via computationally expensive FE analysis) is likely non-convex and multi-modal.

Under these conditions, pure gradient-based methods (e.g., Sequential Quadratic Programming, MMA, or BFGS) are highly efficient for local refinement but exhibit a strong dependence on the initial guess and are prone to convergence to suboptimal local minima. Conversely, population-based metaheuristics like Genetic Algorithms are particularly suited for such problems due to their global exploration capability, robustness, and ability to handle complex design spaces without requiring gradient information [32,33]. However, a pure GA often suffers from slow convergence and may lack the precision required for high-fidelity matching of dynamic characteristics.

The hybrid GA-LS strategy was therefore employed to synergize the global search strength of the GA with the local convergence efficiency of a gradient-based method. This approach is a well-established and effective paradigm in aerospace structural optimization for navigating complex design spaces and achieving high-precision solutions [26,27,28,29,30,32,33,34]. The primary objective of this study was the reliable design and experimental validation of a physical scaled model, not a broad benchmarking of optimization algorithms. The choice of the GA-LS hybrid is justified by its proven track record in handling similar engineering design problems and, most importantly, by the conclusive experimental validation provided by the GVT, which confirmed the model’s high dynamic fidelity (Section 5).

3. Finite Element Modeling of T-Tail Equivalence Model with Reduced Scale

3.1. Model Design

The optimization methodology developed in this study is a general framework for designing dynamically-scaled models. To demonstrate this framework, a representative set of dynamic characteristics—namely, the natural frequencies and mode shapes of a T-tail configuration—is required as the foundational input or target for the scaling process. It is crucial to note that this input data serves as an exemplar; in practical industrial applications, this data would be sourced from a high-fidelity full-scale aircraft FE model or experimental tests.

For the purpose of this demonstration, the vibration modes and frequencies of a baseline T-tail design, referred to as the “original model,” are used as the optimization targets for the structural dynamics characteristics. Based on the original model of a T-tail, its vibration mode, and frequency are used as the optimization targets for the structural dynamics characteristics. Scaled parameters derived from actual wind tunnel test conditions are used to design a scaled model of the T-tail dynamics. The primary vibration modes under consideration encompass bending of Vertical tail (VT), bending of rear fuselage, 1st unsymmetrical bending of Horizontal tail (HT), 1st symmetrical bending of HT, 2nd unsymmetrical bending of HT. The first five orders frequencies of the original model of a T-tail are displayed in Table 1:

The first five orders mode shape diagram of the T-tail optimised model is illustrated in Figure 3, where the black wire frame represents the original model, and the color wire frame represents the modal shape diagram.

The following definition sets out to delineate the fundamental scale of the reduced similarity theory: The scale ratio is to be defined in relation to the dimensions of the wind tunnel:

(20) $K_{L} = 1 / 7$

The density ratio is to be defined on the basis of the air density at the cruise flight altitude and the gas density in the wind tunnel:

(21) $K_{ρ} = 3.03$

The dynamic pressure ratio is to be defined in relation to the actual speed of the aircraft and the flow velocity of wind tunnel test:

(22) $K_{q} = 0.125$

The induction scale is as follows: The quality ratio is:

(23) $K_{M} = K_{ρ} \times K_{L}^{3} = 0.00884$

The frequency ratio is defined as follows:

(24) $K_{f} = \frac{\sqrt{\frac{K_{q}}{K_{ρ}}}}{K_{L}} = 1.42$

In accordance with the aforementioned formula, the target modes of the optimization model are enumerated in Table 2:

As shown in the Figure 4, the physical model of the T-tail is installed on the rear fuselage section of the full-scale model. The fuselage is much stiffer than the rear fuselage beam of the T-tail, which constrains the T-tail in the wind tunnel. Meanwhile, the wing is an elastic model that can accurately simulate the flow field of a real aircraft in the wind tunnel, laying the foundation for subsequent full-scale wind tunnel tests.

The primary structural framework of the T-tail scaled model design incorporates a beam frame and box section skin structure. The primary structural component of T-tail is an aluminium alloy beam frame, which provides stiffness. The external shape of the T-tail is formed using aerospace plywood, and glass fibre composite materials for the skin. The rudder and elevator are connected to the vertical stabilizer and horizontal stabilizer using variable-angle plates. The canopy is composed of foam and 3D-printed photopolymer resin. In order to enhance the model’s structural strength, the leading and trailing edges of the wooden frame box sections are filled with lightweight wood as reinforcing structures. Finally, the rear fuselage beam is composed of channel-shaped aluminium beams, which can more effectively simulate the effect of rear fuselage stiffness on the T-tail modal behaviour. The internal structure of the T-tail scaling model is shown in Figure 5.

Create a detailed T-tail finite element model and allocate mass according to the target distribution to ensure mass similarity. Use this model for initial optimization. The boundary conditions consist of fixed constraints at the front end of the rear fuselage beam that constrain six degrees of freedom. The initial optimization model is shown in Figure 6.

In order to facilitate parameter optimization of the model, one-dimensional elements were employed for the rear fuselage beam and T-tail beam frame, while surface elements were utilised for the wooden frame and skin. Solid elements were employed for the leading and trailing edges, which were made of lightweight wood, and the foam canopy. The element distribution of the detailed dynamic finite element model of the T-tail is shown in Table 3.

3.2. Optimize Partitioning

There are 200 elements in the total T-tail structure optimization. The areas of optimisation encompass the VT spring plate, the HT spring plate, the VT stabiliser beam frame, the HT stabiliser beam frame, the rudder skeleton, the elevator skeleton, and the rear fuselage beam. It is evident that each beam section attribute is defined as an optimisation area. Each optimized partitions is shown in the Figure 7.

The design variables for the left and right sides of the HT stabilizer and elevator are congruent, meaning that only one side of the horizontal tailplane is involved in the optimisation design. Furthermore, the variable-angle connection plate parameters are not included in the optimisation. The design variables for the optimization model are shown in Table 4.

3.3. Physical Realization of the Optimized Model

To bridge the gap between the optimized finite element model and the manufacturable physical structure, a direct translation process was employed, ensuring equivalence in both stiffness and mass.

Stiffness Equivalence: The cross-sectional dimensions of the 1D beam elements (e.g., width, height), which served as the primary design variables in the optimization, were directly used to define the geometry of the corresponding aluminum alloy beam frames in the computer-aided design (CAD) model and subsequent machining drawings. The stiffness properties (bending and torsional) of these physical beams are thus equivalent to their numerical counterparts. For sub-structures modelled with a combination of elements, the optimized parameters define the effective stiffness of the assembly, which is realized by fabricating the individual components (eams, skins) to their specified dimensions and assembling them as per the design.

Mass Equivalence: The total mass of the scaled model was controlled to meet the target scaled mass ( $K_{M}$ ). The structural mass was inherently achieved by using the specified materials (aluminum alloy, aerospace plywood, glass fiber skin) manufactured to the optimized geometries. To precisely match the target mass distribution and center of gravity position, the non-structural mass defined by concentrated mass elements (CONM2) in the FE model was physically implemented by attaching precisely manufactured tungsten alloy balance weights at the designated locations during the final assembly stage.

4. Results and Discussion

4.1. Model Optimization Result

Optimize the model based on the first five orders of frequency and MAC values. The first five orders frequencies and mode shape vectors are used as optimization objectives and constraints, and 99 element parameters are selected, for a total of 200 optimization parameters. The goal during optimization was to minimize the sum of f for the first five modes of the T-tail. The analysis results indicate that convergence occurs after the 43rd iteration.

The specific values of the key design variables before and after optimization are summarized in Table 5. These values represent a subset of the 200 total optimization parameters, selected from components identified as having a major influence on the T-tail’s dynamic response, such as the rear fuselage beam and the root sections of the stabilizers. The final dimensions provided in this table were used to generate the technical drawings for manufacturing the physical scaled model.

Figure 8 shows the first five orders of the mode shape diagram of the optimized model, with blue representing the finite element model and yellow denoting the modal shape diagram.

The detailed simulation model, modal frequencies and error values relative to the target frequencies are displayed in Table 6.

As demonstrated in Table 5, the optimised model exhibits a frequency error of less than 2%. Furthermore, as shown in Figure 9, the MAC values are all greater than 95%. Thereby substantiating the efficacy of the comprehensive algorithm, which successfully balances global search and local accuracy.

In the subsequent section, the quantitative impact of the rear fuselage stiffness parameters will be analysed in order to reveal the underlying mechanism of the optimisation results.

4.2. The Influence of Rear Fuselage Stiffness on the Modal Frequency of the T-Tail

Following a thorough analysis, the impact of the fuselage stiffness of the trough-shaped single-beam structure on the T-tail frequency mode shape was examined, incorporating lateral bending stiffness, vertical bending stiffness, and torsional stiffness.

4.2.1. Derivation of Moment of Inertia $I_{X}$

The height of the web is designated by h, and its thickness is measured in units of $t_{ω}$ . The moment of inertia of the web of the rear fuselage channel beam about the X-axis is:

(25) $I_{X, web} = \frac{t_{w} h^{3}}{12}$

the width of the wing edge is designated by b, and the thickness is denoted by

t_{f}

. The moment of inertia of each wing edge around the X-axis includes its own term and the shifted term.

(26) $I_{X, flange} = \frac{b t_{f}^{3}}{12} + b t_{f} {(\frac{h}{2})}^{2}$

The total moment of inertia is the sum of the web and the two flanges:

(27) $I_{X} = \frac{t_{w} h^{3}}{12} + \frac{b_{f} h^{2}}{2} + \frac{b^{3} t_{f}}{6}$

As demonstrated in Equation (27), $I_{X} \propto h^{3}$ , It has been demonstrated that an increase in the height h of the web can lead to a substantial enhancement in the transverse bending stiffness.

4.2.2. Derivation of Moment of Inertia $I_{Y}$

The moment of inertia of the web of the rear fuselage channel beam about the Y-axis is:

(28) $I_{Y, web} = \frac{h t_{w}^{3}}{12}$

The moment of inertia of each wing edge around the Y-axis includes its own term and the shifted term.

(29) $I_{Y, flange} = \frac{t_{f} b^{3}}{12} + b t_{f} {(\frac{b}{2})}^{2} = \frac{t_{f} b^{3}}{12} + \frac{t_{f} b^{3}}{4} = \frac{t_{f} b^{3}}{3}$

Total moment of inertia is:

(30) $I_{Y} = \frac{h t_{w}^{3}}{12} + \frac{2 t_{f} b^{3}}{3}$

As demonstrated in Equation (30), $I_{Y} \propto b^{3}$ , It has been demonstrated that an augmentation in the width of the flange b can result in a substantial enhancement in vertical bending stiffness.

4.2.3. Derivation of Saint-Venant’s Torsion Constant J

The J approximation of the open thin-walled section is derived as the sum of the contributions of each plate, where the web contribution is defined as follows:

(31) $J_{web} = \frac{1}{3} h_{w} l^{3}$

The collective contribution of the two sides of the wing is as follows:

(32) $J_{flange} = 2 \cdot \frac{1}{3} b t_{f}^{3}$

The total torsion constant is:

(33) $J = \frac{1}{3} (h t_{w}^{3} + 2 b t_{f}^{3})$

As demonstrated in Equation (33), $J \propto t^{3}$ , Channel steel has been shown to possess low torsional stiffness. Increasing the wall thickness or switching to a closed section has been demonstrated to enhance torsional stiffness.

To verify the above formula, this paper calculates the first five order frequencies of the optimized model under different h conditions, as shown in Figure 10a. As can be seen in the figure, the first five frequencies of the optimized model increase slowly as h increases.

As shown in Figure 10b that the vertical bending stiffness $I_{Y} \propto b^{3}$ is significantly enhanced by an increase in the flange width b of the channel beam. The alteration of b from 15 mm to 45 mm leads to an augmentation in the bending modal frequency of the rear fuselage from 6.52 Hz (original model) to $8.7$ Hz (target model), signifying a 33.4% rise. It is noteworthy that when $b >$ 45 mm, the frequency increase trend slows down, indicating that the stiffness gain has a marginal effect.

Figure 11 show the first five orders frequencies of the optimized model when only the $t_{w}$ and $t_{f}$ property values of the channel beam cross-section are changed. For both symmetric and asymmetric bending of HT, the frequency changes of the asymmetric bending mode are more pronounced. When the $t_{w}$ and $t_{f}$ values are small, the frequency of the asymmetric bending mode is lower than that of the symmetric bending mode. However, once these values surpass a certain threshold, the frequency of the asymmetric bending mode surpasses that of the symmetric bending mode.

In summary, when the lateral bending stiffness is significantly increased, there is little effect on the first five orders of mode of the T-tail. However, a substantial increase in the vertical bending stiffness results in a discernible effect on the bending of the rear fuselage. It is evident that as the vertical bending stiffness increases, the bending mode frequency of the rear fuselage increases significantly. It is evident that as the value attains a specified point, the rate of increase in the upward trend becomes more gradual. It is evident that the frequency of the symmetrical bending mode of the HT undergoes a more pronounced alteration in response to the symmetric and antisymmetric bending. When the vertical bending stiffness value is minimal, the frequency of the symmetrical mode of the HT is reduced relative to that of the antisymmetric mode. When the vertical bending stiffness reaches a certain value, the frequency of the symmetrical bending of the HT will exceed that of the antisymmetric mode. It has been demonstrated that an increase in torsional stiffness, whether symmetric or asymmetric, results in a more pronounced frequency change in the asymmetric bending mode. When the torsional stiffness value is minimal, the frequency of the asymmetric bending mode is reduced relative to that of the symmetric bending mode. It has been established that, upon attaining a specific value of torsional stiffness, the frequency of the asymmetric bending mode surpasses that of the symmetric bending mode.

5. Model Production and Ground Vibration Test

5.1. Translation of Optimization Results to Physical Manufacture

The process of transforming the optimized finite element model into a high-fidelity physical structure was executed through a meticulous engineering workflow. The procedure encompassed the following key stages:

CAD Modeling and Drawing Generation: The optimized values of the design variables, particularly the cross-sectional dimensions of the beam elements (e.g., the values listed in Table 5), served as the direct input for constructing a detailed 3D Computer-Aided Design (CAD) model. This digital prototype ensured the geometric accuracy of all components, including the complex interfaces. From the CAD model, standard 2D machining drawings with full dimensions and tolerances were generated for each part to guide the manufacturing.

Material Selection and Component Fabrication: All components were manufactured using the materials specified in the design phase. The primary aluminum alloy beam frames were machined via Computer Numerical Control (CNC) milling to achieve the precise cross-sectional dimensions obtained from the optimization. The skin and wooden ribs were cut using a laser cutter for dimensional fidelity.

Precision Assembly and Mass Control: The components were assembled into the complete T-tail structure using aerospace-grade adhesives and mechanical fasteners. To accurately replicate the mass distribution defined in the FE model, the non-structural mass represented by concentrated mass elements (CONM2) was physically implemented by bonding precisely manufactured and weighed tungsten alloy balance weights at the predetermined nodal locations on the structure. The total mass and center of gravity of the final assembled model were verified against the target values to ensure dynamic similarity.

This rigorous process ensured that the manufactured physical model was a faithful realization of the optimized design, thereby providing a valid basis for the subsequent GVT.

5.2. Ground Vibration Test Setup and Results

According to the optimized model, the T-tail structure data is modified and the processing drawings are finished. Then, the physical model is processed. The GVT was performed using the moving hammer method, with a total of 100 beam joints selected as signal acquisition points. The GVT sampling points are the intersection points of the VT stabilizer, rudder, HT stabilizer and elevator spar frame ribs. As shown in Figure 12, the excitation was provided by an impact hammer, while the structural response was captured by a network of high-sensitivity ICP accelerometers. These accelerometers feature high sensitivity and a wide frequency response range, enabling accurate capture of vibration responses across different frequencies and providing high-quality signal input for modal parameter identification. The accelerometers were mounted at 100 pre-defined measurement points, primarily located at the intersections of the beam frame ribs. The data acquisition system recorded the time-domain signals, which were subsequently processed to obtain Frequency Response Functions (FRFs) and finally the modal parameters using a polyreference frequency domain algorithm.

Figure 13 illustrates the first five orders of the mode shape diagram of the T-tail physical model, where blue signifies the structural model and yellow denotes the mode shape. The frequencies of each order of the physical model and the frequency error values compared with the optimized model are shown in Table 7. All of the errors were within 3%, previous studies have shown that this error range can be determined that the physical model has high fidelity [35,36], proving the optimization method’s feasibility in physical production and paving the way for subsequent wind tunnel testing.

6. Conclusions

In this paper, a reasonable optimization scheme is designed by using a comprehensive optimization method combining the least squares and genetic sensitivity hybrid algorithm, and the model structure of the target frequency and vibration mode is optimized. Finally, the frequency and vibration mode optimization effect of the T-tail is good, and the comprehensive performance is excellent. This method can provide ideas for multidisciplinary optimization and wind tunnel model design of other similar structures.

The present study demonstrates that the stiffness of the rear fuselage exerts a significant influence on the modal characteristics of the T-tail. In the process of simulating the stiffness of the rear fuselage through the utilisation of a channel-shaped single-beam structure, it becomes evident that the symmetric and asymmetric bending modes of the horizontal tail can be decoupled, thereby enabling the design of the frequency to a certain extent. The vertical bending stiffness of the rear fuselage exerts a substantial influence on the frequency of the symmetric bending mode of the T-tail horizontal tail. Concurrently, the torsional stiffness of the vertical tail rear fuselage exerts a substantial influence on the frequency of the T-tail horizontal tail asymmetric bending mode. This includes the elastic model design of the T-tail wing for the rear fuselage, which can provide a reference for subsequent T-tail characteristic analysis and wind tunnel model design.

It is important to discuss the role of structural damping within the context of this study. The optimization methodology presented herein focused exclusively on achieving high-fidelity similitude in natural frequencies and mode shapes, which are the primary drivers for determining aeroelastic instability boundaries such as flutter speed. Structural damping ratios were not included as targets in the optimization process. This approach is consistent with standard practice in the design of dynamically-scaled wind tunnel models, as damping is a complex, system-level property influenced by material hysteresis, joint mechanics, and energy dissipation mechanisms that are notoriously difficult to scale predictively [37]. While damping critically influences the response amplitude and the precise damping level can shift the flutter speed, its accurate scaling is often less critical for initial flutter clearance testing than the correct replication of frequency and mode shape distributions. The damping characteristics of the physical scaled model were, however, identified during the GVT (Section 5). These experimentally obtained damping values will be essential for correlating and refining future aeroelastic predictions from wind tunnel tests, such as buffet and flutter experiments, ensuring a more comprehensive validation of the full-scale aircraft’s aeroelastic model.

In this study, we focused on optimizing the natural frequencies and mode shapes of the scaled T-tail model to achieve dynamic similitude, with all frequency errors below 3% and MAC values exceeding 95%. However, it is important to note that the modal masses and generalized mass participation factors were not explicitly optimized, as the mass distribution was primarily controlled through balance weights to meet the target scaled mass. While this approach ensured overall mass equivalence, future work should incorporate detailed modal mass analysis to further enhance dynamic similarity, particularly for non-similar models in wind tunnel response correction.

Finally, while the hybrid GA-LS algorithm proved highly effective in achieving the dynamic similitude targets validated by physical testing, this study primarily focused on its application within an integrated design-for-testing workflow. Future work could valuably explore a comparative performance analysis of this optimizer against other advanced gradient-based and metaheuristic algorithms to further generalize the findings.

Author Contributions

Conceptualization, Z.C. and W.Q.; methodology, Z.C.; software, Z.C.; validation, X.A., R.Y. and W.Q.; formal analysis, Z.C.; investigation, Z.C.; resources, R.Y. and W.Q.; data curation, Z.C.; writing—original draft preparation, Z.C.; writing—review and editing, X.A. and W.F.; visualization, Z.C.; supervision, W.Q.; project administration, W.Q.; funding acquisition, W.Q. All authors have read and agreed to the published version of the manuscript.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Data is contained within the article.

Conflicts of Interest

The authors declare no conflicts of interest.

Abbreviations

The following abbreviations are used in this manuscript:

GVT	Ground vibration test
GA	Genetic algorithm
LS	Least squares
VT	Vertical tail
HT	Horizontal tail
MAC	Modal assurance criterion

Footnotes

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Figures and Tables

Figure 1 Optimization flowchart of least squares-genetic sensitivity hybrid algorithm for comprehensive optimization theory.

Figure 2 Flowchart of the hybrid least squares-genetic algorithm optimization process.

Figure 3 Mode shape diagrams of original model: (a) Bending of VT is 3.3 Hz. (b) Bending of rear fuselage is 6.52 Hz. (c) 1st unsymmetrical bending of HT is 7.6 Hz. (d) 1st symmetrical bending of HT is 7.92 Hz. (e) 2nd unsymmetrical bending of HT is 22.44 Hz.

Figure 4 Whole aircraft model.

Figure 5 Internal structure of the T-tail scaled model.

Figure 6 Initial optimization model of T-tail.

Figure 7 Optimized partitions.

Figure 8 Mode shape diagrams of optimized model: (a) Bending of VT is 4.705 Hz. (b) Bending of rear fuselage is 8.568 Hz. (c) 1st unsymmetrical bending of HT is 10.87 Hz. (d) 1st symmetrical bending of HT is 11.341 Hz. (e) 2nd unsymmetrical bending of HT is 31.69 Hz.

Figure 9 The MAC values of the optimized modal vibration mode and the original vibration mode.

Figure 10 (a) The first five order frequencies of the optimized model under different h conditions. (b) The first five order frequencies of the optimized model under different b conditions.

Figure 11 (a) The first five order frequencies of the optimized model under different $t_{w}$ conditions. (b) The first five order frequencies of the optimized model under different $t_{f}$ conditions.

Figure 12 Ground Vibration Test setup for the T-tail scaled model.

Figure 13 Mode shape diagrams of physical model: (a) Bending of VT is 4.823 Hz. (b) Bending of rear fuselage is 8.329 Hz. (c) 1st unsymmetrical bending of HT is 10.841 Hz. (d) 1st symmetrical bending of HT is 11.627 Hz. (e) 2nd unsymmetrical bending of HT is 32.149 Hz.

Table 1

The mode of the original model.

Order	Mode Name	Original Frequency (Hz)
1	Bending of VT	3.3
2	Bending of rear fuselage	6.52
3	1st unsymmetrical bending of HT	7.6
4	1st symmetrical bending of HT	7.92
5	2nd unsymmetrical bending of HT	22.44

Table 2

Target mode of optimization model.

Order	Mode Name	Target Frequency (Hz)
1	Bending of VT	4.73
2	Bending of rear fuselage	8.7
3	1st unsymmetrical bending of HT	10.76
4	1st symmetrical bending of HT	11.25
5	2nd unsymmetrical bending of HT	31.86

Table 3

Detailed finite element model element list.

Element Type	Element Name	Quantity
Surface elements	CQUAD4	143,698
Surface elements	CTRIA3	868
Solid elements	CTETRA	203,874
	CHEXA	17,852
	CPENTA	5192
Rigid elements	RBE2	75
Concentrated mass elements	CONM2	23
Nodes		200,536

Table 4

The design variables for the optimization model.

Variable Type	Optimized Partitions	Quantity
Width	VT stabilizer	28
	HT stabilizer	24
	Rudder	20
	Elevator	22
	VT spring plate	2
	HT spring plate	2
	Rear fuselage beam	1
Height	VT stabilizer	28
	HT stabilizer	24
	Rudder	20
	Elevator	22
	VT spring plate	2
	HT spring plate	2
	Rear fuselage beam	1
Web thickness of channel beam	Rear fuselage beam	1
Flange thickness of channel beam	Rear fuselage beam	1

Table 5

Optimized values of key design variables.

Component	Variable Description	Initial Value (mm)	Optimized Value (mm)
Rear Fuselage Beam	Channel Height $(h)$	45	80
	Channel Flange Width $(b)$	100	140
	Web Thickness $(t_{w})$	14	10
	Flange Thickness $(t_{f})$	14	10
VT stabilizer	Root Beam Width	15	13
VT stabilizer	Root Beam Height	50	43.5
HT Stabilizer	Root Beam Width (Port Side)	30	35.8
HT Stabilizer	Root Beam Height (Port Side)	20	26.8
VT Spring Plate	Beam Width	50	60
VT Spring Plate	Beam Height	30	36.6
HT Spring Plate	Front Beam Width	40	46
	Front Beam Height	8	11.2
	Rear Beam Width	40	45.6
	Rear Beam Height	8	11

Table 6

The mode of the optimized model.

Orders	Modal Name	Target Frequency (Hz)	Optimize Frequency (Hz)	Frequency Error
1	Bending of VT	4.73	4.705	−0.52%
2	Bending of rear fuselage	8.7	8.568	−1.51%
3	1st unsymmetrical bending of HT	10.76	10.87	1.02%
4	1st symmetrical bending of HT	11.25	11.341	0.81%
5	2nd unsymmetrical bending of HT	31.86	31.69	−0.53%

Table 7

The mode of the physical model.

Modal Order	Modal Name	Optimize Frequency (Hz)	Physical Frequency (Hz)	Frequency Error
1	Bending of VT	4.705	4.823	2.5%
2	Bending of rear fuselage	8.568	8.329	−1.51%
3	1st unsymmetrical bending of HT	10.87	10.841	−0.26%
4	1st symmetrical bending of HT	11.341	11.627	2.5%
5	2nd unsymmetrical bending of HT	31.69	32.149	1.4%

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Design and Optimization Method for Scaled Equivalent Model of T-Tail Configuration Structural Dynamics Simulating Fuselage Stiffness

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