A Multi-Objective Point Response Prediction Method for Vertical Tail Buffeting Based on Elastic Scaling Transformation

Abstract

Aircraft with a twin vertical tail and leading-edge extension configuration may experience vertical tail buffeting during high-angle-of-attack maneuvering flight. This issue can lead to structural fatigue damage in the vertical tail, shortening its service life and increasing maintenance costs, ultimately compromising flight safety. Therefore, accurate prediction of buffeting loads and responses is essential during design. In the preliminary stage, wind tunnel testing is the primary means to obtain dynamic data such as fluctuating pressure and acceleration response, which can be transformed to full-scale conditions through similitude principles. However, the elastic scaling model used in buffeting tests is usually established for a specific flight condition. When the flow velocity or objective flight condition changes, the similitude relationship becomes invalid, limiting the applicability of test results and preventing full-envelope strength verification. To overcome this limitation, this study proposes a multi-objective point response prediction method for vertical tail buffeting. The method enables the prediction of full-scale responses at multiple objective flight conditions using wind tunnel data that do not strictly satisfy similitude criteria. A complete aircraft vertical tail buffet (rigid/elastic) hybrid model was developed for testing, and an Adjusted Model incorporating elastic scaling transformation was established. The proposed method was validated through experiments, demonstrating improved test data utilization and prediction accuracy across multiple-objective flight conditions.

1. Introduction

Modern high-maneuverability aircraft commonly adopt a twin vertical tail configuration to balance superior maneuverability and stealth performance. When combined with a leading-edge extension design, this configuration can provide additional vortex lift, further enhancing the aircraft’s aerodynamic performance. Typical examples include advanced fighter aircraft such as the F/A-18, F-22, and F-35 from the United States, and the J-20 from China. However, during high-angle-of-attack maneuvering flight, the leading-edge vortices shed from the wing, leading-edge extension, and canard tend to break down before reaching the vertical tails, exposing the tail structures directly to high-energy turbulent wakes. The unsteady turbulent excitation induced by the breakdown of these vortices often contains frequency components that overlap with certain natural frequencies of the vertical tail structure. Under the combined action of aerodynamic loads and structural characteristics, this coupling may cause forced vibration of the vertical tail, manifested as large-amplitude vertical tail buffeting [1,2].

The occurrence of vertical tail buffeting not only accelerates structural fatigue damage and significantly shortens the service life of the vertical tail, but also adversely affects the aircraft’s operational availability and mission capability. Furthermore, the high-frequency vibration induced by buffeting can degrade pilot handling comfort, interfere with critical operations such as weapon aiming and control, and, in severe cases, may even induce flight accidents, posing a threat to flight safety [3]. During high-angle-of-attack maneuvers, these leading-edge vortices propagating aft can also induce buffeting on the horizontal tail [4,5], thereby affecting aircraft handling qualities. While this phenomenon is acknowledged, the present study focuses on the more prevalent and severe issue of vertical tail buffeting.

The buffeting phenomenon in aircraft can generally be categorized into two main types: lift-type buffeting and non-lift-type buffeting [6]. The vertical tail buffeting investigated in this study belongs to the lift-type category. This type also includes the transonic buffeting of wings that occurs during transonic flight, which is induced by the interaction between shock waves and boundary layers [7]. Since the operational flight envelope of military fighter aircraft extends far beyond the onset boundary of vertical tail buffeting, predicting and analyzing the buffeting response and loads is of greater importance than merely identifying the buffeting boundary. Research on this topic has primarily been conducted through numerical simulations and wind tunnel tests. The vertical tail buffeting encountered during the development of the U.S. F/A-18 aircraft is the most representative example [8,9,10,11], where extensive experimental and numerical investigations were carried out by the designers. With the advancement of computer technology and numerical methods, researchers have performed numerous studies using Computational Fluid Dynamics (CFD) to simulate vertical tail buffeting loads and responses. Meng Dehong et al. combined the Reynolds-Averaged Navier–Stokes (RANS) and Large Eddy Simulation (LES) methods, employing a hybrid approach to simulate the fluctuating pressure characteristics of a fighter model undergoing vertical tail buffeting [12]. Elmekawy et al. investigated the prediction accuracy of nonlinear turbulence models for vertical tail buffeting and verified their advantages in capturing complex vortex structures [13]. Bing Han et al. proposed a CFD/CSD coupling framework based on Radial Basis Functions (RBF) to analyze flow features and resonance phenomena during buffeting [14]. Delgado-Gutiérrez et al. developed an efficient CFD simulation framework based on the Single-Step Simplified Lattice Boltzmann Method (SSLBM) for evaluating the buffeting loads of delta wings with twin vertical tails at high angles of attack [15].

Using wind-tunnel tests to obtain vertical-tail buffeting responses and fluctuating pressures is an effective experimental approach. According to classical aeroelastic theory, buffeting is defined as a forced vibration [16]. While the analysis of buffeting must account for the fluid-structure coupling effect, it was widely held in early academic research that this coupling is weaker than that associated with flutter [17]. Early transonic buffeting studies employed rigid wing models and treated the problem in a decoupled manner; with further investigation, however, researchers recognized that structural elasticity significantly influences buffeting behavior and that including elastic effects yields results different from those of decoupled approaches [18]. Similarly, early studies of vertical-tail buffeting treated the problem as a weakly coupled aeroelastic phenomenon by modeling the fuselage, wings, and vertical tails as rigid bodies [1,19]. This decoupled strategy has certain engineering applicability but neglects the influence of elastic deformation on the fluctuating pressure field. Wind-tunnel models used to evaluate vertical-tail buffeting mitigation measures have often been designed without enforcing dynamic similitude [20,21]; such models are effective for validating mitigation concepts but cannot be used to predict buffeting responses. Subsequently, a dynamically similar elastic vertical-tail model was developed and combined with a rigid tail model to form a complete aircraft vertical tail buffet (rigid/elastic) hybrid model [8,22], enabling simultaneous measurement of fluctuating pressure and dynamic response. In this configuration, one vertical tail is a geometrically similar rigid model that preserves the principal flow-field features and provides fluctuating-pressure measurements, while the other is a dynamically similar elastic model that preserves the effect of elastic deformation on buffeting loads and maintains the fluid–structure coupling. The buffeting responses measured on the elastic model can be similitude-transformed to full scale; however, because the elastic model is designed according to scale-similitude principles, it corresponds to a single objective flight condition and cannot be used to predict responses for flight conditions outside that design point.

In summary, the “rigid/elastic hybrid model” represents a novel experimental approach. The geometrically similar rigid model measures the buffeting fluctuating pressure induced by vortex breakdown, while the dynamically similar elastic model accounts for aeroelastic effects to measure acceleration responses. Test data can be converted to full-scale aircraft values based on similitude relations, enabling the prediction of buffeting responses at aircraft scale. However, the similitude relations for the elastic model are satisfied only at its specific design point, preventing response prediction for flight conditions outside this design state. In actual aircraft operation, numerous flight conditions within the flight envelope can induce vertical tail buffeting. Consequently, relying solely on tests with an elastic model makes it difficult to effectively predict buffeting responses across the flight envelope. This one-to-one correspondence between an elastic scaling model and its design flight condition limits the utilization efficiency of wind tunnel test data. Designing separate elastic scaling models for multiple flight conditions would substantially increase wind tunnel testing costs and duration. Therefore, response prediction based on elastic scaling transformation is currently limited to individual target points, and the efficiency of utilizing wind tunnel test data for vertical tail buffeting studies requires further improvement.

Faced with this challenge of low test data utilization efficiency and the inability to perform multi-point state assessment and prediction, related research has been conducted in other engineering fields. In the realm of static aeroelastic analysis, Sun Yuchen et al. combined CFD data to perform elastic correction on wind tunnel test results, thereby extending their applicability [23]. In structural health monitoring, Jasper R et al. utilized limited test data to update simulation models for structural damage localization, achieving the goal of extrapolating test data to predict multiple states [24]. These efforts demonstrate a research approach that utilizes limited test data to predict structural behavior and correct structural parameters.

To address the limitations of wind tunnel testing for buffeting response prediction, this paper proposes a multi-objective points response prediction method. Based on similitude theory and dynamic equations, the method constructs an Adjusted Model via elastic scaling transformation. It clarifies how to achieve effective conversion of wind tunnel test data to full-scale aircraft conditions when the target flight state does not satisfy the similitude criteria, thereby enabling response prediction for multiple target flight states. First, the general workflow of conducting buffeting wind tunnel tests using a full-configuration vertical tail “rigid/elastic” hybrid model is briefly described, and the similitude principles for converting wind tunnel test results to full-scale aircraft conditions are derived. Then, the concept of multi-objective points response prediction and the methodology for constructing the elastic scaling transformation-based Adjusted Model are introduced. The design and manufacturing of the full-configuration vertical tail “rigid/elastic” hybrid buffeting wind tunnel test model were completed, followed by ground vibration tests, wind tunnel tests, and finite element simulation analysis. Six target flight states inconsistent with the model’s design state were selected. The acceleration response was predicted using the Adjusted Model constructed based on the multi-objective points prediction method. The results indicate that the acceleration responses obtained via the multi-objective points prediction method show better agreement with the reference values.

2. Prediction Method of Buffeting Response Based on Wind Tunnel Test

The aerodynamic loads acting on the vertical tail during buffeting exhibit strong nonlinear characteristics, making theoretical analysis highly challenging. Simulating vortex-breakdown-dominated buffet flow fields using CFD methods poses significant challenges. On one hand, high-fidelity CFD methods demand substantial computational resources. On the other hand, due to the highly turbulent and high-frequency nature of buffet fluctuating pressures, their coupling with the structure represents a complex aeroelastic problem, making accurate simulation and prediction difficult [25]. Therefore, studies on vertical tail buffeting at home and abroad mostly rely on flight tests and wind tunnel experiments. Compared with flight tests, wind tunnel tests offer advantages such as lower cost and reduced risk. Moreover, when properly designed, wind tunnel test data show good consistency with the loads and responses of full-scale aircraft [26]. Consequently, from an engineering application perspective, during the early design stage of an aircraft, vertical tail buffeting wind tunnel tests using scaled models can be conducted to obtain dynamic data such as surface acceleration responses and fluctuating pressures, providing key references for subsequent structural design and safety assessment.

The full-configuration vertical tail buffeting model differs from conventional aerodynamic pressure measurement or flutter test models. It requires a combined model consisting of a rigid vertical tail model (left side) and an elastically similar vertical tail model (right side) for testing, i.e., the full-configuration vertical tail “rigid/elastic” hybrid buffeting wind tunnel test model [8,22]. The test principle is illustrated in Figure 1. Based on this model, both the acceleration response and fluctuating pressure during vertical tail buffeting can be measured simultaneously. The rigid vertical tail model is instrumented with pairs of fluctuating pressure sensors distributed over its surface to obtain the spatial distribution of pressure fluctuations acting on the tail during buffeting. The elastic vertical tail model, designed according to the dynamic similitude ratio, is equipped with acceleration sensors to measure the corresponding acceleration responses. For details regarding full-configuration vertical tail “rigid/elastic” hybrid buffeting wind tunnel test model used in this study, please refer to Section 5.

During the model design process, it is first necessary to verify theoretically whether the excitation frequency associated with vortex breakdown and shedding, measured under the same wind-tunnel flow conditions, is consistent with the structural response frequencies of the elastic vertical tail. On this basis, a similitude relationship is established to convert wind-tunnel measurements to full-scale aircraft conditions. Taking into account the dynamic similitude requirements of the elastic vertical tail model, a complete aircraft vertical tail buffet (rigid/elastic) hybrid model is then designed and tested. Finally, using the established similitude relationship, the fluctuating pressure and acceleration response data at full scale are obtained, providing the dynamic load inputs required for buffeting intensity analysis of the aircraft vertical tail structure. The workflow of a complete aircraft vertical tail (rigid/elastic) hybrid model buffeting wind-tunnel test is illustrated in Figure 2.

According to the workflow shown in Figure 2, the procedure begins by determining the scaling and similitude relations for the wind-tunnel model from the test-design constraints (including wind-tunnel dimensions and flow-velocity range) and the representative objective flight conditions for which buffeting intensity must be verified. Based on these similitude relations, the complete aircraft vertical tail buffet (rigid/elastic) hybrid model is then designed. The rigid vertical tail model only needs to satisfy geometric similitude and is instrumented with fluctuating-pressure sensors on its surface to capture the pressure distribution during buffeting. The elastic vertical tail model must satisfy dynamic-similitude requirements and is used to measure buffeting-induced acceleration responses and dynamic internal loads. Prior to wind-tunnel testing, ground vibration tests are conducted on the elastic model to verify that its dynamic characteristics meet the prescribed similitude criteria.

For the complete aircraft vertical tail (rigid/elastic) hybrid model buffeting wind-tunnel tests, the study focuses on the buffeting characteristics of the vertical tail. Measured wind-tunnel data include dynamic responses, internal loads, and fluctuating pressures. The dynamic response refers to acceleration, which is obtained from accelerometers mounted on the elastic vertical tail model. During testing, the variation of the buffeting acceleration response with angle of attack and Mach number can be observed directly; further analysis enables identification of the buffeting onset boundary, and the acceleration response data can be used to assess the effectiveness of buffeting-mitigation measures. Internal loads denote bending moments and torsional moments, which are measured by strain gauges applied at the root of the elastic vertical tail model. The acceleration responses and internal loads measured in the model tests can be scaling-transformed to full-scale aircraft conditions according to the design objective-point scaling ratios. Fluctuating pressure data are acquired by paired pressure sensors distributed over the surface of the rigid vertical tail model. Analysis of these data yields the spectral and statistical characteristics of the pressure fluctuations induced by vortex breakdown on the tail surface, as well as the spatial distribution of the fluctuating-pressure loads. Subsequent processing produces time-domain fluctuating-pressure histories, which are then converted to full-scale values to serve as input dynamic loads for the vertical tail structural fatigue and strength assessments.

Finally, by applying the scaling transformation, the measured data can be converted to full-scale aircraft values for buffeting intensity verification of the vertical tail. Wind-tunnel results can guide the stiffness design of the tail surface and its supporting structure so that their natural frequencies avoid the dominant excitation frequencies of the broken-off vortical wake during high-angle-of-attack flight, thereby reducing buffeting responses and improving structural fatigue resistance to meet dynamic strength requirements.

3. Scaling Transformation Relations of the (Rigid/Elastic) Hybrid Model

Wind-tunnel test results can be converted to full-scale aircraft values according to similitude relations for dynamic strength design. Below, the scaling ratios between the vertical-tail buffeting wind-tunnel model and the full-scale aircraft are derived. Define the similitude ratio $k_X$ as the ratio of a model parameter $X_m$ to the corresponding full-scale (prototype) parameter $X_a$. To convert dynamic test results obtained in the wind tunnel to full-scale flight conditions, it is necessary to ensure dynamic compatibility between the excitation frequency of the flow field following vortex breakdown and the structural response frequencies of the elastic vertical tail. A review of relevant domestic and international literature indicates that, to achieve dynamic similitude, the transformation criterion commonly used is the equality of the Strouhal number (St, i.e., reduced frequency) between the model and the prototype aircraft [27]. The St is an important dimensionless parameter that characterizes flow unsteadiness and vortex-shedding behavior. The St number represents the ratio between the time required for the fluid to pass a characteristic length and the structural vibration period. It can also be interpreted as the ratio of the structural characteristic length to the wavelength of the vortex disturbance, reflecting the time-varying characteristics of the flow and indicating the significance of unsteady aerodynamic effects. Ensuring equality of the St number between the model and the full-scale prototype guarantees consistency in the dynamic characteristics of flow separation and vortex shedding; that is, it ensures dynamic similarity in the vortex-induced excitation.

$${\eta }_m={\displaystyle \frac{{\omega }_m{L}_m}{V_m}}={\displaystyle \frac{{\omega }_a{L}_a}{V_a}}={\eta }_a$$

(1)

$$k_f=k_{\omega }={\displaystyle \frac{{\omega }_m}{{\omega }_a}}={\displaystyle \frac{V_m{L}_a}{V_a{L}_m}}={\displaystyle \frac{k_V}{k_L}}$$

(2)

In Equation (1), $\eta$ denotes the reduced frequency, $\omega$ the angular frequency, $L$ the characteristic length, and $V$ the airflow velocity.

In Equation (2), $k_f$, $k_{\omega }$, $k_V$ and $k_L$ are the scale ratios of frequency, angular frequency, velocity and length between the model and the full-scale aircraft, respectively. $k_f$ denotes the similitude relation between the modal natural frequencies of the elastic vertical tail model and those of the full-scale vertical tail. The elastic model must satisfy dynamic similitude; analogous to flutter-model design, starting from the aeroelastic equation of motion of the model [6], the scaling ratios between the model and the full-scale prototype are determined based on the principles of dimensional analysis. To ensure that the model and the prototype share the same equation of motion, the condition $k_K=k_{\omega }^2{k}_m=k_g{k}_K=k_{\rho }{k}_V^2{k}_L{k}_A$ must be satisfied. Here, $k_K$, $k_m$, $k_g$, $k_{\rho }$ and $k_A$ represent the scaling ratios for generalized stiffness, generalized mass, damping, density, and aerodynamic influence coefficients, respectively, between the model and the prototype [28].

At this point, it is necessary that $k_g=1$. The model and the prototype must share identical structural damping characteristics. When the model is fabricated using conventional metallic materials, it is generally accepted in engineering practice that the structural damping of the model and the prototype can be considered the same. This is an idealized assumption. If consistency in the damping ratio cannot be guaranteed, the model should be designed with a damping ratio lower than that of the prototype to obtain more conservative test results. Furthermore, if their Mach number $Ma$, Reynolds number $Re$, specific heat ratio $\gamma$, gravity parameter ${\displaystyle \frac{V}{gL}}$, and reduced frequency ${\displaystyle \frac{\omega L}{V}}$ are all equal, and they have similar aerodynamic shapes and vibration modes, then $k_A=1$. Using the three fundamental scaling ratios—length ratio $k_L$, density ratio $k_{\rho }$, and velocity ratio $k_V$—combined with the analysis above, the derived scaling ratios can be obtained as follows:

$$k_K=k_{\omega }^2{k}_m=k_{\rho }{k}_V^2{k}_L$$

(3)

It can be further derived that the mass ratio and stiffness ratio are as follows:

$$k_m={\displaystyle \frac{k_{\rho }{k}_V^2{k}_L}{k_{\omega }^2}}=k_{\rho }{k}_L^3$$

(4)

$$k_K=k_{\rho }{k}_V^2{k}_L$$

(5)

In practical wind tunnel testing, achieving equality of the Reynolds number is a difficult condition to fulfill, as the test Reynolds number is typically lower than the actual flight Reynolds number. Relevant studies suggest that for aeroelastic model wind tunnel testing, Reynolds number effects can be neglected when the Reynolds number exceeds 4 × 10⁵ [29]. $k_A=1$ is based on the assumptions of small perturbations and linear aerodynamics, whereas the buffet flow field is highly nonlinear. This equation is primarily used for establishing similitude relations, while the nonlinear characteristics of the excitation are accounted for by the actual fluctuating pressures measured on the rigid model. It should be noted that, due to constraints in test conditions such as wind tunnel size and velocity range, it is not possible to ensure that all flow similarity criteria are satisfied. For the vertical tail buffeting problem induced by vortex shedding and breakdown investigated in this paper, the primary design requirement for the wind tunnel model is to ensure equality of the Strouhal number between the model and the full-scale prototype. Other flow similarity criteria, such as the Reynolds number mentioned earlier, cannot be fully satisfied. Consequently, the model designed in this study is a ‘not-perfect similar’ model that satisfies only a subset of the flow similarity criteria. This introduces certain errors, referred to as scale effects, in the subsequent analysis of the model test data and similitude relations. Therefore, the scale ratio between the model and the prototype should not be too small, in order to minimize the impact of such errors.

By maintaining consistency of the reduced frequency between the model and the full-scale aircraft vertical tail, dynamic similitude in vortex-induced vibration is ensured, thus enabling the conversion of wind-tunnel results to actual flight conditions. When the unsteady aerodynamic excitation frequency measured by the rigid vertical tail model in the wind-tunnel flow is matched with the structural response frequencies of the elastic model, the scaling relations for other physical quantities between the model and the full-scale aircraft can be further derived.

Based on the three fundamental scaling ratios of the elastic model and using the principles of dimensional analysis, the scaling relation for the acceleration dynamic response is derived as follows:

$$K_{a\mathrm{R}\mathrm{M}\mathrm{S}}=K_L{K}_f^2={\displaystyle \frac{K_V^2}{K_L}}$$

(6)

$$K_{a\mathrm{P}\mathrm{S}\mathrm{D}}=K_L^2{K}_f^3={\displaystyle \frac{K_V^3}{K_L}}$$

(7)

where $K_{a\mathrm{R}\mathrm{M}\mathrm{S}}$ is the similitude scaling factor for the root-mean-square (RMS) of the acceleration response, and $K_{a\mathrm{P}\mathrm{S}\mathrm{D}}$ is the scaling factor for the power spectral density (PSD) of the acceleration response. The responses measured on the elastic model at the wind-tunnel design point can be directly converted to the objective flight conditions using the above scaling relations, thereby providing guidance for high-angle-of-attack flight testing of the aircraft.

Conversion relations for the fluctuating pressure measured on the rigid vertical-tail model to full-scale aircraft values can also be derived.

$$K_{C\mathrm{R}\mathrm{M}\mathrm{S}}=K_{\rho }{K}_V^2$$

(8)

$$K_{C\mathrm{P}\mathrm{S}\mathrm{D}}=K_{\rho }^2{K}_V^3{K}_L$$

(9)

Here, $K_{C\mathrm{R}\mathrm{M}\mathrm{S}}$ denotes the scaling factor for the RMS of the fluctuating pressure, and $K_{C\mathrm{P}\mathrm{S}\mathrm{D}}$ denotes the scaling factor for the fluctuating-pressure PSD. The fluctuating pressures measured on the rigid vertical-tail model can be similitude-transformed to the objective flight conditions and used as full-scale dynamic load inputs for vertical-tail structural strength and fatigue analyses.

4. Multi-Objective Point Response Prediction Method for Vertical Tail Buffeting

4.1. Construction of the Adjusted Model Based on Elastic Scaling Transformation

During wind-tunnel testing, the elastic vertical-tail model satisfies the following matrix-form equation of dynamics.

$$\left[M\right]\left\{\ddot{u}\left(t\right)\right\}+\left[C\right]\left\{\dot{u}\left(t\right)\right\}+\left[K\right]\left\{u\left(t\right)\right\}=\left\{P\left(t\right)\right\}$$

(10)

In Equation (10), $\left[M\right]$ is the mass matrix, $\left[C\right]$ the damping matrix, $\left[K\right]$ the stiffness matrix, $\left\{P\left(t\right)\right\}$ the buffeting fluctuating-pressure vector, $\left\{\ddot{u}\left(t\right)\right\}$, $\left\{\dot{u}\left(t\right)\right\}$, and $\left\{u\left(t\right)\right\}$ denote the acceleration, velocity, and displacement vectors in physical coordinates respectively. By transforming the physical coordinates to modal coordinates using $\left\{u\left(t\right)\right\}=\left[\varphi \right]\left\{X\left(t\right)\right\}$, the matrix-form equation of motion in modal coordinates is obtained:

$$\left[\tilde{M}\right]\left\{\ddot{X}\left(t\right)\right\}+\left[\tilde{C}\right]\left\{\dot{X}\left(t\right)\right\}+\left[\tilde{K}\right]\left\{X\left(t\right)\right\}=\left\{F\left(t\right)\right\}$$

(11)

Here, $\left[\varphi \right]$ represents the structural mode shape matrix normalized by its maximum value. From Equation (11), the buffeting response in modal coordinates can be obtained. In that equation, $\left[\tilde{M}\right]={\left[\varphi \right]}^T\left[M\right]\left[\varphi \right]$ is the generalized mass matrix, $\left[\tilde{C}\right]={\left[\varphi \right]}^T\left[C\right]\left[\varphi \right]$ is the generalized damping matrix, and $\left[\tilde{K}\right]={\left[\varphi \right]}^T\left[K\right]\left[\varphi \right]$ is the generalized stiffness matrix, $\left\{F\left(t\right)\right\}={\left[\varphi \right]}^T\left\{P\left(t\right)\right\}$ is the generalized buffeting fluctuating-pressure vector expressed in modal coordinates, and $\left\{\ddot{X}\left(t\right)\right\}$, $\left\{\dot{X}\left(t\right)\right\}$, and $\left\{X\left(t\right)\right\}$ denote the acceleration, velocity and displacement vectors in modal coordinates, respectively.

From the derivation in Section 3, the acceleration responses measured on the elastic scaling model at the design condition can be directly converted to the objective flight condition according to the similitude relations given in Equations (6) and (7). Suppose the initial design condition corresponds to an objective flight condition with flight altitude $H_0$, flight speed $V_0$, length ratio $k_{L_0}$, velocity ratio $k_{V_0}$, and density ratio $k_{{\rho }_0}$. The resulting scaling relations between the model and the full-scale aircraft are then derived as follows:

$$K_{f_0}={\displaystyle \frac{K_{V_0}}{K_{L_0}}}$$

(12)

$$K_{m_0}=K_{{\rho }_0}{K}_{L_0}^3$$

(13)

$$K_{a\mathrm{R}\mathrm{M}{\mathrm{S}}_0}=K_{L_0}{K}_{f_0}^2={\displaystyle \frac{K_{V_0}^2}{K_{L_0}}}$$

(14)

The elastic vertical-tail model at the initial design condition satisfies the following matrix-form equation of motion in modal coordinates:

$$\left[{\tilde{M}}_0\right]\left\{{\ddot{X}}_0\left(t\right)\right\}+\left[{\tilde{C}}_0\right]\left\{{\dot{X}}_0\left(t\right)\right\}+\left[{\tilde{K}}_0\right]\left\{X_0\left(t\right)\right\}=\left\{F_0\left(t\right)\right\}$$

(15)

The meanings of the parameters are the same as in Equation (11), and the subscript “0” denotes the initial design state. Matrices $\left[{\tilde{M}}_0\right]$ and $\left[{\tilde{K}}_0\right]$ are obtained from the finite-element model, while matrix $\left[{\tilde{C}}_0\right]$ must account for aerodynamic damping. Based on the time-domain acceleration responses measured in the wind-tunnel tests, damping ratios are derived by applying a fast Fourier transform (FFT) to the data and using the half-power bandwidth method [30].

Under stable flight conditions (where Mach number and angle of attack remain constant), the dynamic loads and responses induced by vertical tail buffeting can be treated as a stationary random process. The stationarity test for the data measured in the wind tunnel tests of this study is detailed in Section 5.3. The RMS values of the acceleration response can therefore be adopted as the metric for quantifying the buffeting response. When the test flow velocity changes or when dynamic responses at other objective flight conditions are required, the conversion rules between the model and the full-scale aircraft derived from the initial similitude relation cease to be valid. In such cases, the dynamic response data measured in the wind tunnel cannot be directly similitude-transformed, and the applicability of the similitude criteria must be reassessed. The present study is conducted in this context: when the wind-tunnel test condition remains fixed but the objective flight condition under analysis varies—so that the similitude relations given by Equations (6) and (7) no longer hold—how can the full-scale acceleration response be predicted from wind-tunnel data obtained on the initially designed model?

If one intends to continue using the elastic-similitude approach to convert wind-tunnel measurements to full-scale values for a new objective flight condition, the model must satisfy the similitude relations between that objective state and the wind-tunnel test state—including key scale factors such as the frequency scale, mass scale, and so on. These scale factors must be adjusted according to the flow-field characteristics and the structural dynamic parameters at the new objective condition to ensure the accuracy and validity of the scaling transformation. Assume the flight altitude and flight speed of the new objective condition are $H_n$ and $V_n$, respectively, and that the model length scale, velocity scale, and density scale that satisfy the new similitude relations are $k_{L_n}$, $k_{V_n}$, and $k_{{\rho }_n}$. The resulting scaling ratios between the model and the full-scale aircraft are then:

$$k_{f_n}={\displaystyle \frac{k_{V_n}}{k_{L_n}}}$$

(16)

$$k_{m_n}=k_{{\rho }_n}{k}_{L_n}^3$$

(17)

$$k_{a\mathrm{R}\mathrm{M}{\mathrm{S}}_n}=k_{L_n}{k}_{f_n}^2={\displaystyle \frac{k_{V_n}^2}{k_{L_n}}}$$

(18)

Under an unchanged test condition, the matrix-form equation of motion for the elastic vertical-tail model that satisfies the above similitude relations is expressed as follows. This elastic model is hereafter defined as the Adjusted Model.

$$\left[{\tilde{M}}_n\right]\left\{{\ddot{X}}_n\left(t\right)\right\}+\left[{\tilde{C}}_n\right]\left\{{\dot{X}}_n\left(t\right)\right\}+\left[{\tilde{K}}_n\right]\left\{X_n\left(t\right)\right\}=\left\{F_0\left(t\right)\right\}$$

(19)

Equation (19) describes the dynamic behavior of the Adjusted Model at the initial design wind-tunnel test condition. The meanings of the equation parameters are the same as in Equation (11), and the subscript n denotes that the Adjusted Model corresponds to the new objective flight condition. With the wind-tunnel test condition (flow velocity) held constant, the generalized force vector $F_0$ is identical to that in Equation (15). Because the wind-tunnel data from the initially designed model are still used, the model length scale remains unchanged (i.e., $k_{L_0}=k_{L_n}$), whereas the similitude ratios for other physical quantities (such as frequency and mass) must be redefined according to the new objective flight condition. Under these circumstances, the generalized mass, generalized stiffness, and modal frequencies of the Adjusted Model and the initial-design model should satisfy the following relations:

$$\left[{\tilde{M}}_n\right]={\displaystyle \frac{k_{{\rho }_n}}{k_{{\rho }_0}}}\left[{\tilde{M}}_0\right]$$

(20)

$$\left[{\tilde{K}}_n\right]={\displaystyle \frac{k_{{\rho }_n}{k}_{V_n}^2}{k_{{\rho }_0}{k}_{V_0}^2}}\left[{\tilde{K}}_0\right]$$

(21)

$$f_n={\displaystyle \frac{k_{f_n}}{k_{f_0}}}{f}_0={\displaystyle \frac{k_{V_n}}{k_{V_0}}}{f}_0$$

(22)

The Adjusted Model is the model configured to satisfy the scaling transformation for the new objective flight condition. The dynamic parameters of the Adjusted Model are obtained from Equations (20)–(22). According to the similitude relations between the elastic model and the prototype derived in Section 3, the model and the prototype must possess identical structural damping characteristics, i.e., $k_g=1$. Consequently, the structural damping of the Adjusted Model remains consistent with that of the initial design model. It is important to note that the damping acting on the vertical tail during buffeting consists of both structural damping and aerodynamic damping. The total damping ratio can be identified from the acceleration response signals obtained in the wind tunnel tests. Based on the similitude relations derived in Section 3, and assuming a series of conditions are met, $k_A=1$ can be obtained, meaning the generalized aerodynamic coefficients of the model and the prototype are equal. The aerodynamic damping effect of the model is related to the test conditions, such as angle of attack and Mach number [29]. Therefore, under the same test conditions, the aerodynamic damping for Adjusted Models satisfying different similitude relations and for the initial design model is identical. Hence, the generalized damping matrix of the Adjusted Model remains consistent with that of the initial design model, i.e., $\left[{\tilde{C}}_n\right]=\left[{\tilde{C}}_0\right]$.

If the Adjusted Model’s buffeting response were to be computed with commercial finite-element software, a finite-element mesh satisfying the similitude relations for the new objective flight condition would have to be generated first, substantially increasing the FE modelling workload. In this study, a custom MATLAB R2020a dynamic response solver was developed. Based on the elastic model at the initial design state, the mass matrix, stiffness matrix, and the modal shape matrix normalized by its maximum value were extracted from the initial FE model. By adjusting the generalized mass and frequency characteristics of the model according to different target flight states, the construction of the Adjusted Model was achieved. The generalized force vector was constructed based on the fluctuating pressures measured in the wind tunnel tests on the rigid tail model. Since the mass and stiffness of the Adjusted Model are obtained by globally scaling the initial design model according to Equations (20) and (21), ensuring that the frequency scaling ratio satisfies Equation (22), it can be assumed that the maximum-value-normalized mode shapes remain unchanged, i.e., $\left[{\varphi }_n\right]=\left[{\varphi }_0\right]$. When solving responses with this program, modal damping is defined as $\left[\tilde{C}\right]$, and Equation (19) can be cast in a decoupled form so that the dynamic response of each mode can be computed independently. For the i-th mode of the Adjusted Model, the equation of motion is:

$${\tilde{M}}_i{\ddot{X}}^i\left(t\right)+{\tilde{C}}_i{\dot{X}}^i\left(t\right)+{\tilde{K}}_i{X}^i\left(t\right)=F^i\left(t\right)$$

(23)

In Equation (23), ${\tilde{M}}_i$, ${\tilde{C}}_i$ and ${\tilde{K}}_i$ denote the generalized mass, generalized damping and generalized stiffness of the i-th mode of the Adjusted Model, respectively; ${\ddot{X}}^i\left(t\right)$, ${\dot{X}}^i\left(t\right)$ and $X^i\left(t\right)$ represent the acceleration, velocity and displacement responses of the i-th modal coordinate.

The external force and displacement are taken as the averages at the three time instants $t_{k-1}$, $t_k$ and $t_{k+1}$. The acceleration and velocity are computed using the Central Difference Method. Further rearrangement yields [30]:

$$\left\{{\displaystyle \frac{{\tilde{M}}_i}{\Delta t^2}}+{\displaystyle \frac{{\tilde{C}}_i}{2\Delta t}}+{\displaystyle \frac{{\tilde{K}}_i}{3}}\right\}{X}_{k+1}^i=\left\{{\displaystyle \frac{2{\tilde{M}}_i}{\Delta t^2}}-{\displaystyle \frac{{\tilde{K}}_i}{3}}\right\}{X}_k^i+\left\{-{\displaystyle \frac{{\tilde{M}}_i}{\Delta t^2}}+{\displaystyle \frac{{\tilde{C}}_i}{2\Delta t}}-{\displaystyle \frac{{\tilde{K}}_i}{3}}\right\}{X}_{k-1}^i+{\displaystyle \frac{1}{3}}\left(F_{k+1}^i+F_k^i+F_{k-1}^i\right)$$

(24)

By sequentially solving the decoupled single-degree-of-freedom equations for all modal coordinates and finally transforming the modal responses back to physical coordinates via $\left\{u\left(t\right)\right\}=\left[\varphi \right]\left\{X\left(t\right)\right\}$, the buffeting response solution of the Adjusted Model is completed.

4.2. Predicted Response Workflow for Multiple Objective Points of Buffeting

Ideally, if the flight altitude or speed of an objective point deviates from the initial design state, the wind-tunnel model would automatically adjust its mass and stiffness distribution to satisfy the similitude relations of the new objective condition. However, since the wind-tunnel model is only compliant with similitude at the initial design state, it cannot adapt to new objectives. Therefore, when the objective point’s flight condition changes while the test condition remains unchanged, and one still wishes to convert the dynamic response data from wind-tunnel to full-scale aircraft conditions, one must construct an Adjusted Model following Section 4.1. The construction procedure is illustrated in Figure 3 below.

As illustrated in Figure 3, if an objective flight condition coincides with the initial design state, the dynamic response data from wind-tunnel testing can be directly transformed via the existing similitude relation. However, when the objective flight condition deviates from the design state, the original similitude relation no longer applies. In such cases, a new similitude ratio must be determined based on the objective flight condition and the test condition, and an Adjusted Model satisfying the new similitude relation must be constructed to enable prediction of the dynamic response for the current objective condition.

When the objective flight condition changes, constructing the Adjusted Model by re-establishing the similitude ratios allows for scaling transformation, forming the basis for multi-objective points response prediction. The elastic-similitude-based multi-objective points response prediction method is as follows:

• Design and fabricate an elastic vertical-tail model that satisfies scaling transformation for the initial objective flight condition; generate its finite element (FEM) mesh and perform modal simulation to obtain the modal shape vectors, generalized masses, and natural frequencies for the modes of interest.
• Determine the mass and frequency scaling ratios required for scaling transformation under the new objective flight condition; adjust the generalized masses and frequencies of the initial design model accordingly to construct the Adjusted Model (see the flowchart in Figure 3).
• Process the fluctuating pressure data measured on the rigid tail model in the wind-tunnel test; partition and apply these pressure loads to the Adjusted Model. Using the dynamic response solver (MATLAB), compute the acceleration response at the wing-tip position, denoted as $A_{cal}^D$.
• Use the FEM model of the elastic vertical tail from Step 1, apply the same partitioned fluctuating pressure loads, and perform a transient-response simulation in Nastran software to obtain the acceleration response at the wing-tip, denoted as $A_{cal}^L$.
• Process the acceleration response data measured on the elastic vertical tail model in the wind tunnel. Since this model was designed to satisfy the similitude relation for the initial objective condition, its measured wing-tip acceleration RMS is denoted as $A_{test}^L$.
• Let the Adjusted Model’s acceleration RMS at the current test condition be $A_{test}^D$. Then enforce the relation $A_{test}^D=A_{test}^L\times {\displaystyle \frac{A_{cal}^D}{A_{cal}^L}}$; this equation implies that the error factors between the initial design model and the Adjusted Model are consistent, so their computed results and the test data should scale proportionally.

Here, $A_{cal}^D$, $A_{cal}^L$, $A_{test}^L$, and $A_{test}^D$ are all RMS values of acceleration responses. Superscript “L” (local) refers to the model under the original design state; superscript “D” (debug) denotes the Adjusted Model after mass and stiffness modifications to satisfy similitude under the new objective flight condition. The subscript “test” denotes experimental data, and “cal” denotes calculated values.

The workflow of the multi-objective points response prediction method is illustrated in Figure 4.

The response $A_{test}^D$ obtained through this method can be directly similitude-transformed—according to the scaling ratios corresponding to the new objective flight condition—to predict the full-scale aircraft acceleration response. In the flowchart, the dashed line from the Adjusted Model to $A_{test}^D$ indicates that the corresponding experiment is not actually performed: rather, $A_{test}^D$ is derived equivalently through simulation data and wind-tunnel data of the initial design model. By following this method, even when the objective flight condition changes, the dynamic response data at the new objective point can still be predicted based on the initial design model, thereby achieving multi-objective points response prediction.

5. Design and Testing of the (Rigid/Elastic) Hybrid Model

5.1. Design of the Buffeting Wind Tunnel Test Model

In order to investigate the dynamic aerodynamic loads and structural responses of the vertical tail under vortex breakdown excitation, and to validate the reliability of the proposed multi-objective points response prediction method, a complete aircraft vertical tail buffet (rigid/elastic) hybrid model was designed and manufactured to enable measurements of fluctuating pressure and dynamic response in realistic complex flow fields.

As shown in Figure 5, the fuselage, wings, and horizontal tail are all constructed as rigid structures to faithfully reproduce the primary flow-field features. The vertical tail employs a (rigid/elastic) combined design: one side is rigid, primarily used to measure the fluctuating pressures on the surface during buffeting and thereby derive the dynamic aerodynamic loads; the other side is designed as an elastic structure according to dynamic similitude principles, to acquire the dynamic response of the vertical tail under buffeting excitation [22].

The elastic vertical-tail model was fabricated as a composite of a metal core plate and contouring foam, as shown in Figure 6. The core plate is made from 2 mm-thick 7075-T6 aluminum alloy and serves as the primary load-bearing element, providing the overall structural stiffness required by the model. A PMI-70 foam skin is bonded to the metal core for fairing and contouring, ensuring good aerodynamic geometric similitude between the model and the full-scale vertical tail.

Take H = 8 km and Ma = 0.85 as the initial design objective flight condition for the elastic model, with a wind-tunnel test velocity V_m = 40 m/s. The full-configuration vertical tail “rigid/elastic” hybrid buffeting wind tunnel test model investigated in this study references the aerodynamic shape of an advanced fighter aircraft, particularly for the fighter aircraft F/A-18 with a leading-edge extension and twin vertical tail configuration, its documented occurrences of vertical tail buffeting are referenced for the design of the wind tunnel test model in this study [8]. The design process incorporates key findings from F/A-18 research, including: the angle-of-attack, altitude, and velocity ranges in which buffeting occurs, as well as the specific modes, along with their frequencies and mode shapes, that contribute to the buffeting response. But this design process does not represent a specific aircraft type. Consequently, fixed inputs such as mass and frequency are not predefined. The model design comprehensively considers factors including test wind tunnel dimensions, model structural strength, and sensor installation. The three fundamental scaling ratios for the elastic model were selected as shown in Equations (25)–(27). Based on these fundamental ratios, the finite element model of the elastic vertical tail was subjected to overall scaling of its dimensions, mass, and stiffness. This process yielded a full-scale finite element model for subsequent computational validation.

$$K_{\rho }={\displaystyle \frac{{\rho }_m}{{\rho }_a}}=2.33$$

(25)

$$K_V={\displaystyle \frac{V_m}{V_a}}=0.1527$$

(26)

$$K_L={\displaystyle \frac{L_m}{L_a}}=0.2$$

(27)

Based on the three fundamental scaling ratios, the frequency and mass scaling factors can be derived from Equations (2) and (4) as follows:

$$K_f={\displaystyle \frac{K_V}{K_L}}=0.7637$$

(28)

$$K_m=K_{\rho }{K}_L^3=0.01864$$

(29)

The rigid vertical tail was manufactured from high-strength 30CrMnSiA steel (see Figure 7) and consists of two main parts: the wing structure and a cover plate. Sixteen pairs (32 in total) of pressure taps are symmetrically arranged along four chordwise lines on both sides of the fin surface (see Figure 8), and fluctuating-pressure sensors are mounted at these locations to measure the spatial distribution of pressure fluctuations on the tail surface.

5.2. Ground Vibration Test of the Elastic Vertical-Tail Model

As shown in Figure 9, a finite-element (FE) model of the elastic vertical-tail was developed and subjected to modal analysis; the FE model was then used for subsequent transient-response simulations.

The modal simulation results of the elastic vertical-tail model are listed below in Figure 10. This study primarily focuses on the first four natural modes.

The Ground Vibration Test (GVT) of the elastic vertical tail model was conducted using the impact hammer testing method. A single accelerometer was used, while an impact hammer was moved to successively excite measurement points. The acceleration response signals were acquired to obtain Frequency Response Functions (FRFs), from which modal parameters were then fitted. The test procedure is illustrated in Figure 11. During the GVT, the elastic vertical tail model was individually fixed to a ground test rig, which provided a relatively ideal clamped boundary condition.

In the wind tunnel configuration, the elastic vertical tail model is mounted on the rigid fuselage, which is connected to the support structure (including the angle-of-attack adjustment mechanism). Prior to the start of each wind tunnel test run, the elastic vertical tail model is impacted to acquire acceleration response signals. This is to verify whether the frequencies of its primary modes (first bending and first torsion) are consistent with the results from the GVT.

Table 1. Comparison between GVT and simulation results of the elastic vertical tail.

Modal	Simulation Results/Hz	GVT Results/Hz	Theoretical Results/Hz	Relative Error
1st Bending	15.83	15.72	15.80	−0.51%
1st Torsional	28.71	29.50	28.60	3.15%
2nd Bending	50.75	54.09	53.10	1.86%
2nd Torsional	73.25	78.69	77.10	2.07%

below lists the frequencies of the first four natural modes of the elastic vertical-tail model, together with a relative-error analysis. The simulated frequencies were obtained from finite-element calculations, the experimental frequencies are from the GVT results, and the theoretical frequencies correspond to the design objective frequencies that satisfy the frequency scaling ratio.

A comparative analysis of the experimental and theoretical frequencies indicates that the model’s natural frequencies are in good agreement with the design objective frequencies, with frequency errors maintained within acceptable limits. This demonstrates that the elastic vertical-tail model possesses satisfactory dynamic similitude. Furthermore, a Modal Assurance Criterion (MAC) analysis was performed on the experimental and simulated mode shapes. The resulting MAC matrix is shown in Figure 12. The diagonal elements of the MAC matrix are 0.98, 0.99, 0.97, and 0.90, respectively, indicating good agreement between the experimental and simulated mode shapes.

In summary, the test results satisfy the similitude criteria for aeroelastic scaling. Therefore, they serve as a valid foundation for subsequent wind tunnel testing and demonstrate the feasibility of conducting dynamic response measurements and data scaling transformations.

5.3. Wind-Tunnel Buffeting Test of the Model

The buffeting wind tunnel test of the full-configuration vertical tail (rigid/elastic) hybrid model was conducted in the FL-12 wind tunnel at the China Aerodynamics Research and Development Center. This facility is a single-return, closed-test-section low-speed wind tunnel with a test section size of 4 m × 3 m × 8 m. The Reynolds number was 1.81 × 10⁶. During the test, transition strips were applied on the front sections of the rigid fuselage and wings to further increase the effective Reynolds number. Following the scaling ratios designed for the elastic model, the test section flow velocity was set at 40 m/s, with a total pressure of 90,981 Pa and a dynamic pressure of 928 Pa. To investigate buffeting characteristics at high angles of attack, the wind tunnel test commenced at an angle of attack of 20°, with subsequent test points set every 3°, resulting in a total of 15 test conditions, as detailed in

Table 2. Wind tunnel test condition.

No.	Wind Speed/(m/s)	Angle of Attack/(°)
1	40	20
2	40	23
3	40	26
4	40	29
5	40	32
6	40	35
7	40	38
8	40	41
9	40	44
10	40	47
11	40	50
12	40	53
13	40	56
14	40	59
15	40	62

.

The wind tunnel test of the model is shown in Figure 13, where the main strut at the belly and the tail strut at the rear fuselage are used in combination to adjust the angle of attack.

During the test, sampling begins once the wind-tunnel velocity and the model’s angle-of-attack stabilize. Each test condition is sampled for 20 s at a rate of 6400 Hz. Shown in Figure 14 below are the time-domain signals of the fluctuating pressure at selected measurement points on the rigid vertical tail model at an angle of attack of 20°. A stationary random process test was performed on the fluctuating pressure signals. The Mean and RMS values were calculated for data segments of 5 s, 10 s, 15 s, and 20 s in length. A comparison of this data is presented in

Table 3. The statistical values for fluctuating pressure (points 1, 5 and 9, AoA = 20°).

Data Segments	Mean Point 1/Pa	RMS Point 1/Pa	Mean Point 2/Pa	RMS Point 2/Pa	Mean Point 3/Pa	RMS Point 3/Pa
5 s	2.21	191.01	−2.95	132.71	−0.09	52.86
10 s	−2.84	194.85	−0.81	131.69	0.54	52.51
15 s	−1.71	192.63	0.07	131.68	−0.98	52.53
20 s	−0.01	196.02	0.04	131.93	0.05	52.55

. It can be observed that the statistical values for the same measurement point across different time segments are essentially consistent.

The autocorrelation functions for the corresponding measurement points are plotted, as shown in Figure 15. The curves rapidly decay to zero, confirming that the buffeting fluctuating pressure data constitute a stationary random process.

Shown in Figure 16 below are the time-domain acceleration signals from the wingtip of the elastic vertical tail model at an angle of attack of 20°. Tests confirm that this acceleration signal also conforms to a stationary random process.

The PSD curves corresponding to the time-domain fluctuating pressure signals are obtained as shown in Figure 17. Analysis of the fluctuating pressure PSD at the three measurement points indicates that the energy is predominantly concentrated in the low-frequency band below approximately 150 Hz.

Based on the acceleration time-domain data, a PSD analysis was conducted to obtain the frequency-domain characteristics of the acceleration signals, as shown in Figure 18. The acceleration energy is concentrated below 250 Hz, with peaks occurring near the natural frequencies of the elastic model. Influenced by aeroelastic effects (aerodynamic stiffness and damping), these peaks do not exactly coincide with the natural frequencies. The two peaks below 100 Hz correspond to the first and third modes measured in the GVT test.

6. Application of the Multi-Objective Point Response Prediction Method

6.1. Numerical Method Validation

As described in Section 4.2, the response $A_{cal}^D$ must be obtained using the custom MATLAB transient-response solver, so it is essential to verify the solver’s computational accuracy and reliability. To this end, a cantilevered rectangular plate model is selected as a benchmark case. A random fluctuating load is applied to its surface, and the problem is solved by both the MATLAB transient solver and a commercial finite-element software. The acceleration time-domain responses at the same node are extracted from both solution methods.

Both methods use identical damping parameters, time integration step sizes, and output intervals. The resulting acceleration time histories are plotted together, and their response trends and numerical errors are compared to validate the accuracy and applicability of the custom program under the present research conditions, as illustrated in Figure 19.

The time-domain data computed by the two methods were subjected to PSD analysis, and the resulting PSD curves were plotted together, as shown in Figure 20.

The comparison indicates that the in-house MATLAB transient-response solver is highly consistent with the commercial finite-element software in the time domain: the response traces match well in both amplitude and phase, with numerical discrepancies kept within acceptable bounds. The PSD comparison further shows a strong overall overlap between the two methods, with only minor differences in the identification of dominant frequencies and peak amplitudes. These results validate the high accuracy and stability of the custom solver for transient dynamic problems. In summary, the self-developed program demonstrates sufficient reliability for use in the subsequent multi-objective points response prediction and dynamic-response computations.

6.2. Comparison of Predicted Responses

Determining the damping required for response solving is necessary. While only structural damping can be obtained from ground resonance tests, the vertical tail buffeting problem must also account for aerodynamic damping effects. In this study, the equivalent damping (incorporating aerodynamic damping effects) was identified by performing a Fast Fourier Transform (FFT) on the acceleration response time-domain data measured in the wind tunnel and applying the half-power bandwidth method. Figure 21 shows the amplitude-frequency curve. Let $A_0$ denote the peak amplitude and $f$ the corresponding frequency; let $f_1$ and $f_2$ be the frequencies at which the amplitude corresponds to ${\displaystyle \frac{\sqrt{2}}{2}}{A}_0$. Then, considering aerodynamic damping effects, the equivalent damping ratio satisfies the following relation [30]:

$$\zeta ={\displaystyle \frac{f_2-f_1}{2f}}$$

(30)

Based on the method described above, the damping corresponding to each peak frequency was sequentially determined according to the peak distribution in the FFT spectrum of the buffeting acceleration response. These values were then used to configure the settings for both the finite element software and the MATLAB dynamic response solver. According to the PSD curves of the fluctuating pressure and acceleration obtained from the wind tunnel tests, the frequency band of interest for the dynamic response solution was determined to be within 250 Hz. Within this band, the acceleration spectrum exhibited four distinct peaks. The equivalent damping was calculated at these peak frequencies and used as the damping ratio for the respective local frequency bands. The results are presented in

Table 4. Equivalent damping by frequency band.

Frequency Band/Hz	Damping/%
0~35	1.14
35~86	7.51
86~165	7.06
165~250	2.63

below:

As can be seen from Table 4, the influence of aerodynamic damping effects varies across different frequency bands and does not follow a simple positive or negative correlation. The effect is most significant within the frequency bands of the second and third peak frequencies (35 Hz to 165 Hz), where the equivalent damping ratio reaches over 7%. Analysis in conjunction with Figure 16 indicates that this is because the fluctuating pressure energy induced by leading-edge vortex breakdown is concentrated within this frequency band.

Based on Equation (30), the damping ratio is calculated, and the derived value is used to configure the solver parameters for both the finite-element simulation and the MATLAB transient-response program. The operational flight envelope of fighter aircraft extends far beyond the onset boundary of buffeting. Combining insights from studies on F-18 vertical tail buffeting with the wind tunnel test conditions of the model in this paper, six typical target flight conditions within the flight envelope were selected, as shown in

Table 5. Flight condition information.

Flight Condition	$\mathit{H}$/km	$\mathit{M}\mathit{a}$	$\mathit{A}\mathit{o}\mathit{A}$/°
1	4	0.3	26
2	4	0.5	26
3	6	0.3	23
4	6	0.5	23
5	8	0.3	20
6	8	0.5	20

, to evaluate the acceleration dynamic response at the wingtip position when vertical tail buffeting occurs. Since none of the selected objective conditions coincide with the initial design state of the wind-tunnel model, data cannot be directly converted using the elastic scaling principle. Therefore, the Adjusted Model must be constructed using the method described in Section 4.1 in order to predict dynamic responses for non-design objective flight conditions.

According to the six selected objective flight conditions and the fluctuating-pressure similitude relations given in Section 3, the fluctuating-pressure data measured on the rigid vertical-tail model in the wind tunnel were similitude-transformed to full-scale aircraft values and applied as input loads to the full-scale finite-element model. Transient response analysis in the FE software was then performed to obtain the RMS of the acceleration response at the vertical-tail tip; this result serves as the reference (control group) response for subsequent relative-error comparisons to validate the multi-objective points prediction method.

On the one hand, using the acceleration scaling factor for each selected objective condition, the acceleration RMS $A_{test}^L$ directly measured in the wind-tunnel on the initial design model was similitude-transformed to full scale to yield the unadjusted acceleration response. Using the multi-objective points prediction procedure described above, an Adjusted Model was constructed and its acceleration response RMS $A_{test}^D$ was computed; $A_{test}^D$ was then converted to the corresponding full-scale value for the objective condition, denoted as the adjusted acceleration response. To assess the effectiveness of the multi-objective points prediction method, the unadjusted and adjusted acceleration responses were compared with the baseline reference values and subjected to relative-error analysis. The comparison results are presented in

Table 6. Comparison of acceleration response errors before and after adjustment.

Flight Condition	Reference Response/g	Unadjusted Response/g	Relative Error	Adjusted Response/g	Relative Error
1	33.92	25.54	−24.69%	31.99	−5.68%
2	88.60	70.95	−19.92%	89.72	1.26%
3	22.11	20.48	−7.38%	20.88	−5.58%
4	72.82	56.89	−21.87%	75.38	3.52%
5	9.26	10.44	12.75%	8.43	−8.98%
6	42.49	29.01	−31.72%	37.73	−11.20%

.

The results listed in the table are all RMS values of the acceleration response at the vertical-tail tip. The baseline reference data (control group) were obtained by finite-element simulation: the fluctuating pressures measured on the rigid model in the wind tunnel were scaled to full-scale values and applied regionally to the full-scale FE model. The pre-adjustment acceleration responses are based on the initial design model, while the post-adjustment responses are obtained from the constructed Adjusted Model. The relative errors of the pre- and post-adjustment responses with respect to the baseline reference are plotted together as a scatter plot in Figure 22.

The comparison results indicate that, relative to directly applying scaling transformation using the original-design model, the Adjusted Model constructed by the multi-objective points response prediction method yields markedly smaller relative errors between the RMS acceleration responses and the baseline reference values. For the six selected representative objective flight conditions, the acceleration response errors obtained with the multi-objective method are generally concentrated within ±10%, demonstrating good stability and consistency. By contrast, acceleration responses obtained by direct similitude conversion of the initial-design model exhibit substantially larger errors with pronounced scatter; for some objective conditions the relative error exceeds 30%.

Section 5.3 has already verified that the vertical tail buffeting acceleration response constitutes a stationary random process; thus, the RMS value can effectively characterize the statistical properties of the acceleration signal. Furthermore, the probability distribution curves of the acceleration response signals from the control group, the pre-adjustment case, and the post-adjustment case were plotted together for probability distribution analysis. Taking Condition 1 as an example, as shown in Figure 23, it can be observed that the probability distribution of the post-adjustment acceleration response signal aligns more closely with that of the control group, whereas the probability distribution of the pre-adjustment acceleration signal shows a greater deviation from the control group. The same trend is observed for the other conditions, which are not individually plotted here.

Based on the above work, PSD analysis of the acceleration time histories was performed to verify whether the frequency-domain characteristics of the responses predicted by the multi-objective point response prediction method conform to expectations. The fluctuating-pressure data measured in the wind tunnel were converted to full-scale values using Equation (8) and applied, in partitioned form, to the full-size finite-element model; the resulting acceleration responses were computed and subjected to PSD analysis to serve as the reference data (control group). The acceleration responses measured on the wind-tunnel initial-design model were analyzed in the frequency domain and converted to full scale according to Equation (7); this result is denoted the pre-adjustment PSD. The acceleration responses obtained from the Adjusted Model constructed via the multi-objective point prediction method were likewise PSD-analysed and transformed to full scale; this result is denoted the post-adjustment PSD. The pre-adjustment PSD, post-adjustment PSD and baseline PSD were plotted together and compared for error analysis as shown in Figure 24, Figure 25, Figure 26, Figure 27, Figure 28 and Figure 29.

Comparison of PSD curves across the six flight conditions shows that the post-adjustment PSD (red) exhibits substantially better overlap with the reference (control group) PSD (green) than the pre-adjustment PSD (blue). The dominant spectral peaks of the post-adjustment and baseline spectra are closely aligned; the relative errors of the primary dominant frequency are listed in

Table 7. Relative error of dominant PSD frequencies (primary) between the control group and the post-adjustment response.

Flight Condition	Reference Frequency/Hz	Adjusted Frequency/Hz	Relative Error
1	20.94	20.96	0.10%
2	21.10	21.12	0.09%
3	20.89	20.91	0.10%
4	20.57	20.96	1.90%
5	20.80	21.03	1.11%
6	20.80	20.83	0.14%

and are within 2% for all six cases.

Compared to the primary dominant frequency, the relative error for the second dominant frequency between the two is larger, with a maximum error approaching 7%, as shown in

Table 8. Relative error of dominant PSD frequencies (second) between the control group and the post-adjustment response.

Flight Condition	Reference Frequency/Hz	Adjusted Frequency/Hz	Relative Error
1	69.38	66.46	−4.21%
2	68.98	66.54	−3.54%
3	69.30	66.93	−3.42%
4	69.62	64.87	−6.82%
5	69.79	67.01	−3.98%
6	69.32	66.63	−3.88%

. This discrepancy is likely related to modal frequency errors in the elastic model, given that the frequency error for its 1st mode is approximately 0.5%, while for the 2nd mode it reaches about 3%. Furthermore, the concentration of fluctuating pressure energy in that frequency band may also have an influence.

In addition to the peak frequencies, a certain pattern is observed when comparing the peak responses of the adjusted PSD with those of the control group. For the first dominant frequency, the peak response of the adjusted PSD is consistently lower than that of the control group PSD. For the second dominant frequency, with the exception of Condition 4, the pattern of peak response magnitudes is consistent with that of the first dominant frequency. The underlying reason for this is attributed to errors in the damping identification of the wind tunnel model.

The two principal dominant frequencies correspond respectively to the first bending and the first torsion modes, which are the two modal contributions with the largest influence on vertical-tail buffeting response. By contrast, the pre-adjustment PSD (blue) shows large deviations in dominant frequencies relative to the baseline (green) and therefore does not correctly represent the true structural natural frequencies; the spectral overlap between these two curves is poor. In summary, the acceleration responses obtained from the Adjusted Model via the multi-objective point prediction method exhibit the expected frequency-domain characteristics and show markedly improved agreement with the baseline. Regarding the PSD analysis, this paper focuses on the discussion of the dominant frequencies and peak responses. In conjunction with the assumptions adopted in Section 4.1 for constructing the Adjusted Model, that the equivalent damping and the maximum-value-normalized mode shapes of the Adjusted Model remain consistent with those of the initial design model—it is acknowledged that these assumptions introduce unavoidable errors. Nonetheless, the analysis effectively demonstrates the advantages of the response prediction based on the Adjusted Model.

A comprehensive analysis of the above comparisons indicates that directly converting the wind-tunnel results of the initial-design model to full-scale aircraft values yields large errors, because the initial-design model satisfies dynamic similarity only at its design point—its mass and stiffness characteristics correspond to a single objective flight condition. When the test condition or the objective flight condition changes, the relevant scaling factors (e.g., mass ratio, frequency ratio) also change, so the initial model no longer satisfies the new similitude relations and the wind-tunnel data cannot be directly used to predict responses at non-design objective states. The multi-objective points prediction method proposed here remedies this by constructing an Adjusted Model that meets the similitude relations for each new objective condition, thereby reducing transformation error and ensuring improved prediction accuracy.

7. Conclusions

This study investigates the vertical tail buffeting problem encountered by aircraft with a twin vertical tail configuration during high-angle-of-attack maneuvering flight. To address the limitation of elastic model scaling transformation being dependent on a specific design point, a multi-objective point response prediction method for vertical tail buffeting is proposed. This method tackles the engineering bottleneck where direct scaling transformation is impossible when the test conditions and the target flight state do not satisfy the required similitude relations. The research is based on a full-configuration vertical tail “rigid/elastic” hybrid buffeting wind tunnel test model. Through wind tunnel testing, finite element simulation, and a custom-developed MATLAB solver, the accuracy, stability, and engineering applicability of the multi-objective point response prediction method were systematically validated. The main conclusions are as follows:

1. For six typical non-design states within the ranges of flight altitude 4–8 km, Mach number 0.3–0.5, and angle of attack 20–26°, the relative error between the predicted RMS values of the vertical tail wingtip acceleration response (using the proposed multi-point buffeting response prediction method) and the control group was essentially controlled within 10%. The probability density distribution curves showed good agreement with those of the control group. Frequency-domain analysis indicated that the relative error of the dominant frequencies in the acceleration response PSD obtained from the Adjusted Model, compared to the control group, was within 7%. The PSD curves overlapped well, and their frequency-domain characteristics met expectations. The proposed multi-objective points response prediction method demonstrates good stability and reliability, meeting the accuracy requirements for engineering applications.

2. The proposed multi-point buffeting response prediction method is applicable when the test condition remains unchanged, and the target flight state to be analyzed/predicted deviates from the design point. The construction of the Adjusted Model applied the following key assumptions:

(a) The aerodynamic damping effects of the elastic model can be considered consistent under identical test conditions;

(b) The structural damping of the full-scale aircraft and the model scale are approximately equal;

(c) The model obtained by globally scaling mass and stiffness maintains the same maximum-value-normalized mode shapes as before scaling.

Among these, damping identification relies on signals from wind tunnel tests and may contain certain errors. Furthermore, frequency and mode shape errors of higher-order modes can cumulatively affect response prediction accuracy. Additionally, the fluctuating pressure used in this study originated from wind tunnel measurements with sensors placed at a limited number of discrete points, and the tests were conducted in a low-speed wind tunnel. This setup cannot fully characterize the strong nonlinearity and stochastic nature of transonic buffet excitation.

3. This study establishes a buffeting response prediction workflow that does not rely on CFD simulation but is instead based on testing and dynamic correction. By constructing an “Adjusted Model” to modify generalized mass and frequency characteristics, the method enables the extrapolation of test data from a single elastic model to multiple target flight states. The method is not only suitable for acceleration response prediction but can also be extended to the prediction of other dynamic loads such as displacement, bending moment, and torque, demonstrating strong engineering value. Furthermore, it avoids the need to repeatedly design and manufacture elastic models for each target state, significantly improves the utilization efficiency of wind tunnel test data, and reduces development costs and cycle time.