Study on the Dynamic Magnification Effect of Structure Stiffness Based on the Gust Coupling Analysis of Civil Aircraft

Abstract

Regarding the dynamic magnification effect of structure stiffness on the gust analysis of civil aircraft, the following three methods are presented: rigid modes analysis, secondary processing based on elastic modes, and analysis with enlarged stiffness. These methods provide consistent gust load and address the challenge of extracting internal gust loads of rigid aircraft. The coupling resonant effects of the inertial force, the aerodynamic force, and the gust-induced aerodynamic force at different frequencies are examined. The response of flexible aircraft is nonlinearly related to frequency. It exhibits a significant increase in the inertial force and the aerodynamic force at higher frequencies, while a quasi-rigid response at very low frequencies shows the importance of sufficient analysis time. In addition, compared with rigid aircraft, flexible aircraft experiences a delay in the occurrence of extreme gust loads with the delay interval proportional to the frequency. The maximum gust load of flexible aircraft under a certain range of frequencies exceeds that of rigid aircraft, although this is not necessarily the case at the specific frequency. The dynamic magnification factor is 1.25 for the model in this study, which is almost constant and reaches its maximum value together with the gust loads when the frequency coincides with the frequency of the first bending mode.

1. Introduction

Gust and turbulence loads are critical scenarios in the flight load requirements outlined by civil aviation regulations. The distribution of gust loads across the entire aircraft must be assessed to evaluate structural strength and safety from the preliminary design stage to the civil aircraft design certification stage. Gust load limits are a crucial criterion in aircraft design.

According to civil aviation regulations [1,2,3], the evaluation of limit gust loads must be conducted through dynamic analysis within the aircraft’s flight envelope, accounting for unsteady aerodynamic characteristics and all critical structural degrees of freedom, including rigid body motions. The stiffness of the aircraft is a significant factor influencing gust loads. With the increasing size of modern civil aircraft and the extensive use of composite materials, the airframe’s flexibility has risen, amplifying the impact of inertial force and aerodynamic force in gust analysis. This change alters the relationship of inertial force, aerodynamic force, and gust-induced aerodynamic force, resulting in different gust loads of flexible aircraft compared with rigid aircraft analyses, and the difference depends on the aircraft’s stiffness.

Regarding the importance of aircraft stiffness on gust load, Castrichini et al. [4] investigated the effect of exploiting the folding wingtips in flight as a device to reduce dynamic gust loads with the introduction of a passive nonlinear negative stiffness hinge spring. It was found that significant reductions in the dynamic loads are possible with different structure stiffnesses. This study focuses mainly on the local stiffness difference, so the influence of wing stiffness on gust load will not be discussed. Ricci and Scotti [5] performed the analytical activity to design a gust alleviation system on an unconventional flexible three-surface aircraft using the rigid modes and the elastic modes of the aircraft to show the gust alleviation effect with control law. Stiffness in this paper is the factor considered by the control law, and the gust load difference between flexible aircraft and rigid aircraft is not discussed. Guo et al. [6] developed a methodology for calculating the flight dynamic characteristics and gust response of free flexible aircraft in a coupled numerical tool based on Computational Fluid Dynamics (CFD) and Computational Structure Dynamics (CSD). The coupled way is not suitable in engineering, and the relationship between the gust load of rigid aircraft and flexible aircraft is not presented in this paper. Su and Cesnik [7] investigated the dynamic response of highly flexible wings. Their structural model is a strain-based finite element framework. The aerodynamic model is based on the 2D finite inflow theory and includes stall and unsteady effects. The effect of stiffness is not included in this paper, and the method is not suitable in engineering. Gov and Karpel [8] developed a novel model integration using a modal formulation with a 3D nonlinear structural dynamic model for the effective study of very flexible structures under gust load. The presented modal formulation links linear segments with nonlinear coupling terms to perform large geometric nonlinear analysis. This method is also not suitable in engineering, and the stiffness effects were not discussed.

In the context of considering gust loads in aircraft design with composite materials, Wang et al. [9] developed an aeroelastic optimization framework in 2022 for the preliminary design of variable stiffness composite wing structures. This framework minimizes wing mass by optimizing lamination parameters, laminate thickness, and wing jig twist distribution during aeroelastic tailoring. The load conditions considered include the limit gust load case, underscoring the importance of stiffness design for composite wings and its close relationship with gust loads. This study focuses on gust load as a restrictive load condition for wing stiffness design and will not delve into the composition of gust loads or the effects of stiffness on gust load.

The study of the structure stiffness’ impact on gust load is also crucial for developing the civil aircraft fatigue test load spectrum. In 1972, Boeing’s fatigue program explicitly highlighted the need to consider the dynamic magnification effect of gust loads when determining equivalent gust loads in the fatigue test environment [10]. The gust analysis with rigid and flexible degrees of freedom was still important for the fatigue analysis in Boeing in 2024 [11]. The same practice is also recognized by the Chinese aviation industry. Min et al. [12] emphasized that in the fatigue analysis of civil aircraft, particularly for flight profiles, including cruise, the dynamic magnification factor of gust loads for flexible aircraft relative to rigid aircraft should be accounted for. They provided a method for calculating this dynamic magnification factor to guide the compilation of fatigue load spectra. Recently, Kemper [13] introduced two different frequency-based levels of simplification for the future codification or engineering design work. The level 2 simplification in this paper is denoted as the damage equivalence factor approach, considering the influence of wind characteristics and structural dynamic properties on fatigue life. Rajpal et al. [14] developed a numerical design methodology for optimizing composite wings subject to gust using static and dynamic experiments to assess the effect of fatigue on the aeroelastic performance of the wing and validate the analytical fatigue model. In this paper, the fatigue process resulted in degradation of the wing stiffness, leading to the change in the aeroelastic response of the wing, and a current knockdown factor was produced for a lighter wing compared with the traditional knockdown factor considering the degradation in stiffness over its design life. Simon et al. [15] developed a method for the continuous estimation of structural aircraft loads from recorded flight data for fatigue analysis. The responses of different gust zones under different gust gradients were used to solve the inverse problem. In conclusion, examining the differing response processes of rigid and flexible aircraft is essential for effective fatigue load spectrum compilation and fatigue analysis.

When the aircraft encounters gust excitation, the nose enters the gust first, followed by the empennage, resulting in a phase difference in gust excitation across different components. The structure vibrates under the excitation of gust-induced aerodynamic force. The gust load is the interaction result of inertial force, aerodynamic force, and gust-induced aerodynamic force. It is important to note that, for both elastic and rigid aircraft, the external forces under gust excitation must always include inertial force, aerodynamic force, and gust aerodynamic force. The inertial force and the aerodynamic force are caused by the free movements and vibration of the aircraft under gust excitation.

Additionally, the aerodynamic force and gust-induced aerodynamic force are unsteady forces that can be simulated in both time and frequency domains. The methods for analyzing unsteady aerodynamic forces can be categorized into the strip method, panel method, and Computational Fluid Dynamics (CFD) method [16]. The strip method, which originated in the 1930s [17], is known for its simplicity and fast computational speed for the unsteady aerodynamic force of two-dimensional incompressible fluid flow. Shams et al. [18] established a nonlinear aeroelastic response model for slender wings based on the Wagner function in 2008, and Masrour et al. [19] conducted an unsteady response analysis of flexible aircraft with nonlinear structures using the Wagner function in 2023. However, the strip method has limitations in calculating the unsteady aerodynamic forces of the three-dimensional compressible fluid flow on the complex surfaces of the actual aircraft. The panel method is effective and simple for engineering, dividing the aircraft surface into a large number of boxes with fundamental solutions, i.e., source, eddy, and doublet [20]. The doublet-lattice method [21] and the Unsteady Vortex-Lattice Method (UVLM) [22] are the commonly used panel methods. The doublet-lattice method, which was first developed by Albano and Rodden [23] and then enhanced by Rodden [24,25,26], is also implemented in the aerodynamic force of gust analysis in the business software MSC.Nastran and Zaero. The UVLM is a time-domain method used for the unsteady aerodynamic force, capable of coupling analysis with the structure geometric nonlinear analysis [27,28], and the UVLM has not been widely used in thousands of gust analysis cases in engineering.

Numerous researchers have employed CFD to perform fluid–structure coupling calculations of gust loads [29,30]. However, the CFD method is time-consuming and not practical for analyzing the thousands of gust cases encountered in engineering. As a result, these methods have not been widely adopted in engineering applications. Conversely, the doublet-lattice method applied to plane elements is widely used in engineering due to its efficiency in calculating unsteady aerodynamic forces in the frequency domain [31,32,33]. After obtaining the aerodynamic forces using the aforementioned methods, gust loads can be determined through fluid–structure coupling analysis in both the time domain and frequency domain, where the aircraft structure stiffness influences the inertial force and the aerodynamic force in gust load analysis.

Research on gust load analysis for rigid aircraft began as early as 1988. Hoblit [34] provided a function for the gust response of rigid aircraft, focusing solely on the heaving motion. This approach, derived from solving the differential equation of motion with the quasi-steady and unsteady treatments of the aerodynamic forces, has proven useful in engineering applications. However, due to the omission of pitch motion, Hoblit’s method lacks sufficient accuracy in simulating unsteady aerodynamic force and unsteady gust-induced aerodynamic force. Even when replacing the aerodynamic calculation method with CFD, it is essential to account for the coupling effects of aerodynamic forces and structural dynamics response on gust load calculations. In other words, regardless of the accuracy of the aerodynamic calculations, it is critical to include sufficient degrees of freedom in gust load analysis. For rigid aircraft, this means incorporating all free modes, while for flexible aircraft, elastic modes must be superimposed onto the rigid modes.

As noted earlier, gust response in engineering is typically analyzed in frequency-domain methods using the doublet-lattice method for aerodynamic forces. Regarding gust load establishment, Karpel et al. [35] utilized the modal approach in 1995 to analyze flexible, dynamic loads in response to impulsive excitation. This approach involves constructing first-order, time-domain equations of motion in generalized coordinates, both with and without considering unsteady aerodynamic effects. The dynamic loads associated with structural responses are expressed using the mode displacement (MD) method and the summation-of-forces (SOF) method. The key advantage of the SOF method is its ability to distinguish between different types of external forces and accumulate these forces, providing better convergence speed compared to the MD method. The phase and magnitude relationships of the three types of forces determine the gust load, and this method is less influenced by local load excitations. On the other hand, the MD method focuses on establishing a specific stiffness matrix and structural deformation, which is not constrained by the type of dynamic response analysis. However, the MD method requires a higher number of modes for accurate results, exhibits slower convergence speed, and is more susceptible to local excitation effects. Despite these limitations, the MD method remains applicable for gust load analysis.

When using the MD method to extract internal loads for gust analysis, the process involves multiplying the element stiffness matrix by the relative deformation of the element. This method is well suited for stick models with beam elements under gust excitation. However, it is not applicable for the gust analysis of rigid aircraft to deduce the internal load based on the deformation of the structure, as the structural elements in such aircraft are rigid and lack relative deformation. Consequently, the gust load cannot be determined using the MD method in this case. Therefore, the SOF method remains more practical for analyzing gust loads in rigid aircraft.

However, the SOF method is not always preferred. Firstly, when the structural force transmission path is complex, it can be challenging to determine how to accumulate external forces using the SOF method. Secondly, the SOF method typically requires considerably more computational time and storage space compared to the MD method. Lastly, the MD method offers advantages in obtaining specific section loads for structures, with localized stiffness defined at particular sections.

To address the difficulties associated with the MD method in gust load analysis, this paper presents the following three solution approaches: (1) rigid modes analysis, (2) secondary processing based on elastic modes, and (3) analysis with enlarged stiffness. All these three methods yield consistent results and effectively resolve the issue of extracting internal loads for rigid aircraft. These approaches can be extended to other gust response calculations that do not use the doublet-lattice method for aerodynamic forces calculation and other dynamic response calculations where the aerodynamic forces are absent.

Regarding the effects of gust dynamic magnification factors in civil aircraft fatigue analysis, this paper establishes a model of a typical single wing with a high aspect ratio; analyzes the resonant coupling effects of inertial force, aerodynamic force, and gust-induced aerodynamic force using the doublet-lattice method; and determines the dynamic gust load amplification factor of flexible aircraft relative to rigid aircraft. The results indicate the following:

• At very low gust excitation frequencies, the response of flexible aircraft closely resembles that of rigid aircraft, exhibiting a quasi-rigid response. As the gust excitation frequency increases, the inertial force and the aerodynamic force of flexible aircraft significantly surpass those of rigid aircraft.
• The three forces act in succession. The flexible aircraft experience a delay in the occurrence of extreme gust loads compared to rigid aircraft, with the delay time interval proportional to the gust excitation frequency.
• The gust load of flexible aircraft is more affected by gust excitation frequency than rigid aircraft. Both loads change nonlinearly under different gust analysis frequencies with quite low values under low gust gradients than the ones under high gradients generally.
• The maximum gust load of flexible aircraft under a certain range of gust analysis frequencies exceeds that of rigid aircraft, although this does not necessarily occur at the specific gust frequency.
• The dynamic magnification factor is almost constant throughout the wing with a notable increase at the wing tip, which depends on gust excitation frequency and reaches its maximum value when the gust excitation frequency coincides with the first bending mode frequency, where the gust loads for both elastic and rigid aircraft also peak.

The study examines the coupling resonant effects of inertial force, aerodynamic force, and gust-induced aerodynamic force at different frequencies, providing a reference for the dynamic magnification effects of aircraft structure stiffness. The difficulty of the MD method in gust load analysis is resolved using the above methods. These three methods exhibit the suitable approaches and underlying principles of the gust magnification factor considering aircraft stiffness in engineering, with the relationship between the gust load of elastic aircraft and rigid aircraft revealed under different gust excitation frequencies. The above methods could be extended to gust analysis with other aerodynamic forces calculation methods and other dynamic response calculations without aerodynamic forces.

2. Methods

2.1. Gust Load Analysis Method for Flexible Aircraft

When the nonlinear characteristics of the structure are not pronounced, gust load analysis can be conducted using frequency-domain calculations. In the modal coordinate system, and without considering the deflection of the control surfaces, the aeroelastic analysis equation in the frequency domain for an open-loop flexible aircraft can be established as follows:

$$\left(-M_{hh}{\omega }^2+i{C}_{hh}\omega +\left(1+ig\right)K_{hh}-{\displaystyle \frac{1}{2}}\rho V^2{Q}_{hh}\left(m,k\right)\right)q=P\left(\omega \right),$$

(1)

When the control surface deflection is considered, the closed-loop analysis equation is established as follows:

$$\left(-M_{hh}{\omega }^2+i{C}_{hh}\omega +\left(1+ig\right)K_{hh}-{\displaystyle \frac{1}{2}}\rho V^2{Q}_{hh}\left(m,k\right)\right)q=\left(M_{hc}{\omega }^2+{\displaystyle \frac{1}{2}}\rho V^2{Q}_{hc}(m,k)\right)\delta +P\left(\omega \right),$$

(2)

In Equation (2), $M_{hh}$ is the generalized mass matrix; $M_{hc}$ is the generalized rudder coupled inertial mass matrix; ${\mathit{C}}_{hh}$ is a generalized damping matrix; ${\mathit{K}}_{hh}$ is the generalized stiffness matrix; $m$ is the Mach number; $V$ is the speed of flight; $k$ is the reduction frequency where $k=\omega c/2V$; $c$ is the length of mean aerodynamic chord; $Q_{hh}(m,k)$ is the generalized aerodynamic coefficient matrix; $Q_{hc}(m,k)$ is the generalized aerodynamic influence coefficient matrix of control surface; $\omega$ is the circular frequency; $g$ is structural damping; $\rho$ is the atmospheric density; $q$ is the modal amplitude vector; and $\delta$ is the generalized coordinate corresponding to the deflection of the control surface, where the closed-loop relationship with q is established through the transfer function $\mathrm{T}$, i.e., $\delta =\mathrm{T}·q$. $P(\omega )$ is the gust-induced aerodynamic force converted to the modal coordinate system.

According to the civil aircraft regulation discrete gust design criteria, the shape of the discrete gust is defined by a 1-COS function, as shown in Figure 1.

Figure 1 illustrates the discrete gust velocity $w(t)$ with a gust velocity amplitude of 1 m/s and a gust gradient of 52.00 m. The gradient determines the gust excitation frequency. At the same flight speed, a larger gradient corresponds to a lower gust excitation frequency. While the 1-COS gust velocity shape is a standard form, actual gust excitation can take various forms. The frequency-domain gust excitation used in continuous gust analysis can also be converted into time-domain gust excitation [36,37]. The gust excitation velocity remains consistent at the same longitudinal position, but there is a phase difference in gust excitation between different longitudinal positions.

The Fourier transform is applied to the vertical time-domain gust excitation shown in Figure 1 to obtain the frequency-domain representation. Using the doublet-lattice method, which is commonly employed in engineering, the aerodynamic surface model for gust analysis is established. The excitation function $P(\omega )$ in the modal coordinate system is then expressed as follows:

$$P\left(\omega \right)={\displaystyle \frac{\rho V^2\max (g(t))}{2V\Delta f}}{(q_{kh})}^{\text{'}}\cdot WSKJF\cdot A_{jjest}(\omega )\cdot W_j{e}^{-i\omega (X_j-X_0)/V}\cdot g(\omega ),$$

(3)

where $\Delta f$ is the frequency calculation interval; $q_{kh}$ is the vector of the modal amplitude vector on the aerodynamic interpolation point $k$ set; $WSKJF$ is the integral matrix of the force and moment on each element; $A_{jjest}$ is the aerodynamic coefficient matrix of the current frequency $\omega$, which can be obtained by the interpolation of $A_{jj}$ at the given frequency as several decreasing frequencies are usually given in gust analysis for interpolation. $X_j$ and $X_0$ are the longitudinal positions of the specific excitation point and the fixed gust reference point, respectively. $W_j$ is the dot product of the unit vector in the normal direction of the aerodynamic panel element and the unit vector in the gust excitation direction; $g(\omega )$ is the Fourier transform of gust velocity in the frequency domain. $P(\omega )$ is the gust-induced aerodynamic force converted to the modal coordinate system. The excitation force $P(\omega )$ is the distribution value over the frequency interval $\Delta f$, taking into account the phase difference caused by the distance difference between $X_j$ and $X_0$.

The generalized frequency-domain excitation value $P(\omega )$ of discrete gust can be obtained by Equation (3). This value can also be output in matrix $PDFX$ by adding the DMAP statement in MSC. Nastran, i.e., adding the following statement to the execution control segment. $PDFX$ and $P(\omega )$ are the same matrix.

• COMPILE FREQRS SOUIN=MSCSOU LIST
• ALTER ‘GUST  CASES‘
• OUTPUT4 PDFX,,,,//0/37/99//9
• ENDALTER

The number 37 in the above contents is the FORTRAN unit number on which the $PDFX$ matrix will be written into. Other matrices in Equation (3) can be output using the DMAP statement [38,39] in MSC.Nastran. The aerodynamic coefficient matrix $A_{jj}$ is output by MSC. Nastran must be inverted before use. The inverted matrix $A_{jj}^{-1}$ corresponds directly to the matrix $A_{jjest}$ in Equation (3), preserving its intended meaning and application within the context of the equation.

In the frequency-domain analysis, the initial step is to obtain the frequency-domain response for the primary vibration of each mode. For the open-loop gust response, the response of h-the mode at a given frequency $\omega$ is represented by Equation (4).

$${\xi }_h\left(\omega \right)={\displaystyle \frac{P(\omega )}{-2{\omega }^2{M}_{hh}+K_{hh}-\frac{1}{2}\rho V^2{Q}_{hh}}},$$

(4)

In Equation (4), both ${\mathit{M}}_{hh}$ and ${\mathit{K}}_{hh}$ are diagonal matrices, while $Q_{hh}$ is full matrix. There is a coupling effect between different modes in $Q_{hh}$.

After obtaining the frequency-domain response for all modes, the next step involves calculating the response over a range of frequencies starting from 0 Hz. This is carried out by dividing the frequency range into $N$ equal intervals of $\Delta f$. Using $\Delta t$ as the uniform output time, the displacement and acceleration functions for all stations are then calculated, as shown in Equations (5) and (6).

$$u\left(t\right)={\displaystyle \frac{1}{\mathsf{\pi }}}{\displaystyle {\int }_0^{N·2\mathsf{\pi }\Delta f}q{\xi }_h(\omega )} e^{i\omega t}d\omega ,$$

(5)

$$a\left(t\right)=-{\displaystyle \frac{1}{\mathsf{\pi }}}{\displaystyle {\int }_0^{N·2\mathsf{\pi }\Delta f}q{\xi }_h(\omega )} {\omega }^2{e}^{i\omega t}d\omega ,$$

(6)

In Equation (5), the integration is performed over a frequency range with 0 Hz and $2\mathsf{\pi }N\Delta f$ as the starting and ending points, respectively. These boundary points should be included in the integration, but only half of their response values ensure accurate calculation.

As the fundamental responses, displacement and acceleration represent the dynamic behavior of the structure at every moment. In the context of gust analysis, the cumulative load on the structure can be determined using two methods. The first method involves obtaining the element load by multiplying the element stiffness matrix by the displacement difference matrix between elements. This relationship is expressed in Equation (7).

$$F_1\left(t\right)=K_{ele}\Delta u,$$

(7)

Using the stick model with beam elements, the function for calculating the vertical bending moment and shear force with the element stiffness matrix $K_{ele}$ is given as follows:

$$K_{ele}=\left[\begin{array}{cccc}12& 6L& -12& 6L\\ 6L& 4L^2& -6L& 2L^2\\ -12& -6L& 12& -6L\\ 6L& 2L^2& -6L& 4L^2\end{array}\right]\cdot {\displaystyle \frac{EI}{L^3}},$$

(8)

where $L$ is the length of the element; $EI$ is the vertical bending stiffness of the element; and the displacement difference matrix $\Delta u$ between the elements takes the vertical displacement and deflection angle of the adjacent elements in turn. The principle is straightforward and will not be elaborated further. The symbols used in Equation (7) correspond to the actual coordinate system employed in the analysis.

Most of the aircraft structure can be established as beam elements including the fuselage, the wing, and the empennage. The advantage of using Equation (7) is its convenience and speed, allowing for the calculation of section loads when the element stiffness matrix is known. However, it has the drawback of slow convergence, and the accuracy of the results depends on the number and shape of the modes. When the number of modes is insufficient, the section load obtained may exhibit significant deviation.

Gust loads can be determined by accumulating the contributions from inertial force, aerodynamic force due to structural movement, and aerodynamic force induced by gust to address this issue. The frequency-domain response functions for these three types of forces are given by Equations (9)–(11).

$$I_h(\omega )=-{\omega }^2{M}_{aa}q{\xi }_h(\omega ),$$

(9)

$$Q_h(\omega )={\displaystyle \frac{\rho V^2}{2}}\cdot {(GDAKT)}^{\text{'}}\cdot \mathrm{WSKJF}\cdot A_{jhest}(\omega )\cdot {\xi }_h(\omega ),$$

(10)

$$Q_g(\omega )={\displaystyle \frac{\rho V^2\max (g(t))}{2V\Delta f}}{(GDAKT)}^{\text{'}}\cdot \mathrm{WSKJF}\cdot A_{jjest}(\omega )\cdot W_j\cdot e^{-i\omega (X_j-X_0)/V}\cdot g(\omega ),$$

(11)

where ${\mathit{M}}_{aa}$ is the mass matrix on the analytical freedom set, i.e., a-set; $\mathit{GDAKT}$ is a transformation matrix from the a-set of structural analysis degrees of freedom to the k-set of aerodynamic interpolation degrees of freedom. The $A_{jhest}$ matrix can be obtained using $A_{jh}$ matrix interpolation at different frequencies. The relationship between $A_{jh}$ and $A_{jj}$ is as shown in Equation (12).

$$A_{jh}(k)=A_{jj}(k)\cdot ({(D1JK)}^{\text{'}}+ik{(D2JK)}^{\text{'}})\cdot q_{kh},$$

(12)

where $D1JK$ and $D2JK$ are the real and imaginary parts of the differential matrix of downwash generated by aerodynamic surface elements due to structural deformation.

The time-domain responses corresponding to Equation (9), Equation (10), and Equation (11) are given by Equation (13), Equation (14), and Equation (15).

$$I_h\left(t\right)=-{\displaystyle \frac{1}{\mathsf{\pi }}}{M}_{aa}{\displaystyle {\int }_0^{N·2\mathsf{\pi }\Delta f}q{\xi }_h(\omega )} {\omega }^2{e}^{i\omega t}d\omega ,$$

(13)

$$Q_h\left(t\right)={\displaystyle \frac{1}{\mathsf{\pi }}}{\displaystyle {\int }_0^{N·2\mathsf{\pi }\Delta f}{Q}_h(\omega ) }{e}^{i\omega t}d\omega ,$$

(14)

$$Q_g\left(t\right)={\displaystyle \frac{1}{\mathsf{\pi }}}{\displaystyle {\int }_0^{N·2\mathsf{\pi }\Delta f}{Q}_g(\omega ) }{e}^{i\omega t}d\omega ,$$

(15)

In Equations (13)–(15), the integration is performed over the frequency range from 0 Hz to $2\mathsf{\pi }N\Delta f$, with half of the response values of these boundary points to ensure accurate integration.

To determine the cumulative load at a specific station caused by gust excitation, the section load can be obtained by summing the three forces described by Equations (13)–(15), according to the cumulative relationship of node freedom to the section load in the a-set of Figure 2. The external cumulative forces are represented by Equation (16).

$$F_2\left(t\right)={\displaystyle \sum _1^{N_a}\left(I_h^a(t)+Q_h^a(t)+Q_g^a(t)\right)},$$

(16)

where $N_a$ is the number of degrees of freedom of nodes in a-set included when accumulated to a specific station. The symbols used in Equation (16) correspond to the actual coordinate system employed in the analysis. As illustrated in Figure 2, to determine the gust load at the cumulative station, the degrees of freedom of all structural nodes from the wing tip station to the cumulative station should be considered. By accumulating the three forces, the gust section load for the wing can be obtained. The symbols used in Equation (16) correspond to the actual coordinate system employed in the analysis. The same principle applies to handling gust loads for the fuselage and empennage. In Equation (16), $I^a(t)$, $Q_h^a(t)$, and $Q_g^a(t)$ are, respectively, the inertial force, the aerodynamic force caused by structural movement, and the gust-induced aerodynamic force in the time domain corresponding to the degree of freedom a in the a-set.

2.2. Gust Load Analysis Method of Rigid Aircraft

The movement of the aircraft structure is induced by gust excitation, and the resulting acceleration corresponds to the inertial force. For vertical gust excitation, the distinction between rigid and flexible aircraft lies in how the inertial force and aerodynamic force, resulting from structural movement, differ between the two models.

Given that $Q_{hh}$ is a full array in Equation (4) and the vibrations of different modes are coupled, calculating the gust load for a rigid aircraft cannot be accomplished by merely considering the primary vibration responses of the first six rigid body modes after obtaining the primary vibration responses from the elastic structural modes. The following three methods are proposed for calculating the gust load of rigid aircraft to address this issue.

The matrix required in Equation (16) is extracted by using the DMAP statement of MSC.Nastran. For other aeroelastic analysis software or programming processes, the approach remains consistent. Method 1 involves using only the first six rigid body modes to obtain the gust load. This approach can be achieved by setting up a gust analysis using rigid body modes in MSC. Nastran and applying the MONITOR card to obtain the cumulative section load. However, this method does not provide the internal load of the element, as there is no deformation in the structural elements of a rigid aircraft.

Method 2, described as a secondary process based on elastic modes calculation, involves the following steps,

Increase Stiffness: Since the structure stiffness of a rigid aircraft is theoretically infinite compared to the structure stiffness of a flexible aircraft after analyzing the gust load for the flexible aircraft, the stiffness of the flexible aircraft is assumed to be uniformly increased by a large factor, denoted as $\mathrm{MAX}$, using the modal mass normalization method.

Frequency Adjustment: If the flexible aircraft modes are calculated up to a natural frequency of Fmax, then the modes for the rigid aircraft should be calculated up to a frequency of $\sqrt{\mathrm{MAX}}\cdot F_{\max }$.

Modal Consistency: The structural distribution stiffness remains unchanged, so the modal amplitude vector $\mathit{q}$ for both the elastic and rigid structure models remain consistent.

Gust Load Calculation: Equation (7) is used to determine the gust section load using the element stiffness matrix. Equation (16) is applied to compute the external force accumulation when the rigid aircraft encounters gusts.

The gust load calculated by this method is generally consistent with the gust load obtained by considering only the first six rigid body modes as described in Method 1. The reasons are analyzed as follows: after the stiffness of the flexible aircraft is uniformly increased by a factor of $\mathrm{MAX}$, this process is equivalent to multiplying ${\mathit{K}}_{hh}$ by $\mathrm{MAX}$, while maintaining modal mass normalization with ${\mathit{M}}_{hh}$ as the unit diagonal matrix. Consequently, the main vibration response of the mode described by Equation (4) transforms into Equation (17).

$${\xi }_h\left(\omega \right)={\displaystyle \frac{P(\omega )}{-2{\omega }^2{M}_{hh}+\mathrm{MAX}\cdot K_{hh}-\frac{1}{2}\rho V^2{Q}_{hh}}},$$

(17)

For the generalized stiffness matrix ${\mathit{K}}_{hh}$, the values of the first 6 orders of ${\mathit{K}}_{hh}$ are 0, while in the actual calculation, the frequency should be a real number close to 0.

After multiplying ${\mathit{K}}_{hh}$ by $\mathrm{MAX}$, the values of the first six orders of ${\mathit{K}}_{hh}$ remain either zero or very close to zero. However, despite this multiplication of ${\mathit{K}}_{hh}$, the influence of elastic modes on rigid modes cannot be directly eliminated. The effect of this influence is dependent on the shape of the mode. Specific values and examples illustrating this impact are provided in a later subsection.

The ${\mathit{K}}_{hh}$ for the elastic modes is non-zero. For a civil aircraft, where the first wing bending frequency is approximately 2 Hz, the corresponding modal stiffness value in ${\mathit{K}}_{hh}$ is 16π2 (the value is 157.9), and the modal mass value in ${\mathit{M}}_{hh}$ is 1, so ${\mathit{K}}_{hh}$ is the dominant term. When multiplied by a larger factor $\mathrm{MAX}$, $\mathrm{MAX}\cdot K_{hh}$ becomes the dominant term in the denominator of Equation (17), even though ${\mathit{Q}}_{hh}$ remains a full matrix. As a result, after multiplying ${\mathit{K}}_{hh}$ by $\mathrm{MAX}$, the response of the elastic modes at each frequency point depends primarily on ${\mathit{K}}_{hh}$, and the value ${\xi }_h\left(\omega \right)$ approaches zero.

After the above operations, the gust load in Equation (7) is given in the following form:

$$F_1\left(t\right)=\mathrm{MAX}\cdot K_{ele}\Delta u,$$

(18)

Considering the displacement time-domain response integration process described in Equation (5), after multiplying ${\mathit{K}}_{hh}$ by $\mathrm{MAX}$, the displacement between elements $\Delta u_{\max }$ is given by Equation (19), and the section load $F_1\left(t\right)$, based on $\Delta u_{\max }$, is shown in Equation (20).

$$\Delta u_{\max }=\Delta \left({\displaystyle \frac{1}{\mathsf{\pi }}}{\displaystyle {\int }_0^{N·2\mathsf{\pi }\Delta f}\frac{q\cdot P(\omega )\cdot e^{i\omega t}}{-2{\omega }^2{M}_{hh}+\mathrm{MAX}\cdot K_{hh}-\frac{1}{2}\rho V^2{Q}_{hh}}}\mathrm{d}\omega \right),$$

(19)

$$F_1\left(t\right)=\mathrm{MAX}\cdot K_{ele}\Delta u_{\max },$$

(20)

When $\mathrm{MAX}\cdot K_{hh}$ dominates the denominator in Equation (20), ignoring the influence of $-2{\omega }^2{\mathit{M}}_{hh}$ and $\rho V^2{\mathit{Q}}_{hh}/2$, Equation (20) can be simplified as

$$F_1\left(t\right)=\mathrm{MAX}\cdot K_{ele}\Delta \left({\displaystyle \frac{1}{\mathsf{\pi }}}{\displaystyle {\int }_0^{N·2\mathsf{\pi }\Delta f}\frac{q\cdot P(\omega )\cdot e^{i\omega t}}{\mathrm{MAX}\cdot K_{hh}}}\mathrm{d}\omega \right),$$

(21)

After the numerator and denominator eliminate $\mathrm{MAX}$, Equation (21) is the static response result of the structure under gust excitation $P(\omega )$.

This process can also be derived using Equation (16). When $\mathrm{MAX}\cdot K_{hh}$ in Equation (17) dominates and the vibration response corresponding to the elastic mode ${\xi }_h\left(\omega \right)$ approaches zero, then the inertial force and aerodynamic force in Equation (16) include only the rigid modes. The gust-induced aerodynamic force is unaffected, allowing the gust section load $F_2\left(t\right)$ to be obtained by considering only the rigid modes.

Method 1, Method 2, and Method 3 are performed by programming in the same method as the frequency-domain method of [41], which is widely used in engineering. The only difference is there are only rigid modes in Method 1 and an amplification treatment of the stiffness matrix in Method 2 and Method 3 to address the difficulties associated with the MD method in gust load analysis in engineering.

Additionally, Method 2 requires the establishment of structural model matrices and additional post-processing, which may not be suitable for engineering software applications. Instead, Method 3 can be adopted, which follows a similar principle to Method 2: performing gust analysis after increasing the stiffness of the structural model and obtaining the gust section load through either the internal load calculation from Equation (7) or the section load accumulation process from Equation (16). Accordingly, the internal loads are obtained from the CBAR element in the stick model using beam elements in MSC. Nastran. The accumulated section load output is produced using the MONITOR card in MSC. Nastran. The key difference is that in Method 3, the modal amplitude vector $q$ must be recalculated, whereas in Method 2, this recalculation is not necessary. Method 3 is generally easier to implement in engineering applications compared to Method 2, although Method 2 provides the underlying principle for Method 3.

3. Results and Discussion

3.1. The Gust Analysis Model

A single-wing model of civil aircraft for gust calculation is established. The coupling effects of inertial force, aerodynamic force, and aerodynamic force induced by vertical discrete gust excitation, considering both elastic and rigid modes, are analyzed using the three methods described in Section 2.2.

Figure 3 depicts the single-wing structural model, which has a length of 16.24 m and an aspect ratio of 9. The structure is simplified as an elastic beam positioned along the rigid axis of the component and is simulated using 29 CBAR beam elements with bending stiffness to exhibit the bending and shear effect in the vertical and lateral direction and the torsional stiffness to exhibit the torsional effect. The structure model consists of discrete masses and moments of inertia, including the structure mass distribution, the system mass distribution, and the fuel mass distribution. Considering the wing model is a half model, the total mass of the model is 78 t, with a balance weight at the wing root, to indicate the actual center of gravity of the whole aircraft model. Figure 4 illustrates the aerodynamic model of the single wing, simplified to consist of aerodynamic surface elements. The unsteady aerodynamic force is computed using the doublet-lattice method, and the aerodynamic force is applied to the structure through spline function interpolation. The rigid body modes include the heaving and pitching modes of the wing, while the elastic modes are calculated up to 70 Hz, incorporating a total of 23 modes. The mode of the whole aircraft is affected by the mass distribution and stiffness distribution of fuselage, wing, and empennage, which affect the gust load of the wing consequently, while the mode of the wing plays the most important role in the gust load of the wing. Considering the inertial effects of the fuselage and the empennage, the balance weight at the wing root is used to account for the fuselage and the empennage mass.

Table 1. The natural frequencies.

Order	Frequency/Hz
1	2.41
2	5.23
3	6.33
4	6.77
5	13.27

lists the first five natural frequencies of the single-wing model, with the first wing-bending mode occurring at 2.41 Hz. The shape of this first bending mode is shown in Figure 5.

3.2. The Gust Analysis Cases

For the model described in Section 3.1, the flight condition with maximum load is selected, with a flight altitude of 7500 m. The wing bending moment is positive when the wing is bending upward. The vertical discrete gust speed is set to 1 m/s upward in a 1-COS shape, and gust gradients ranging from 9.10 m to 106.70 m are considered. The gust gradients and excitation frequencies for various gust cases are summarized in

Table 2. The different gradients of gust cases.

Case No.	Gust Gradient/m	Gust Excitation Frequency/Hz
1	9.10	14.00
2	10.65	11.90
3	13.30	9.50
4	17.80	7.10
5	26.70	4.80
6	52.00	2.40
7	80.00	1.60
8	106.70	1.20

. When the gust gradient is 52.00 m, the excitation frequency is close to the first wing-bending mode.

The Frobenius norm of the full matrix ${\mathit{Q}}_{hh}$ for the elastic wing, as presented in Section 3.1, over the frequency range from 0 Hz to 30 Hz is shown in

Table 3. The different Frobenius norms of ${\mathit{Q}}_{hh}$ at different frequencies.

No.	Frequency/Hz	The Frobenius Norm
1	0.00	2.4725 × 105
2	5.00	2.5450 × 105
3	10.00	2.9166 × 105
4	15.00	3.4855 × 105
5	20.00	4.2629 × 105
6	25.00	5.2787 × 105
7	30.00	6.5573 × 105

. The Frobenius norm of ${\mathit{K}}_{hh}$ is 8.3301 × 10⁵, which is equivalent to the values given in Table 3. It is evident that after multiplying ${\mathit{K}}_{hh}$ by a large factor $\mathrm{MAX}$, $\mathrm{MAX}\cdot K_{hh}$ dominates the denominator in Equation (17) for the external force accumulation method, thereby effectively eliminating the influence of ${\mathit{Q}}_{hh}$. Furthermore, the frequency range from 0 Hz to 30 Hz generally covers the range used for gust analysis in engineering.

3.3. The Gust Load of Rigid Aircraft in the Three Methods

The gust loads for rigid aircraft, calculated using the three methods outlined in Section 2.2, are presented. The wing-bending moment for the gust gradient of 52.00 m, obtained via the external force accumulation SOF method, is shown in Figure 6. The results from all three methods are identical, and hence, the paper does not differentiate between them further.

3.4. The Effect of the Structure Stiffness on Gust Load

The wing-bending moment for both elastic and rigid aircraft under the gust gradient of 52.00 m was obtained using the following two methods: the internal load method (Equation (7)), labeled as MD; and the external force accumulation method (Equation (16)), labeled as SOF. The coinciding results, illustrated in Figure 7 with the loads of 6 sections using the MD method and the loads of 28 sections using the SOF method, show that the outcomes from Equation (7) and Equation (16) are consistent. This confirms that both load-handling methods are equivalent for both elastic and rigid aircraft, and thus, they are not further distinguished in this paper.

The gust load depicted in Figure 7 represents the combined effect of inertial force, aerodynamic force caused by structural movement, and aerodynamic force excited by the gust. With regard to the response process of three typical gust gradients—9.10 m, 52.00 m, and 106.70 m—the response curves for the three types of forces (inertial force, aerodynamic force, and gust-induced aerodynamic force), which are the components of the wing root-bending moment are shown in Figure 8, Figure 9, and Figure 10, respectively, to show the different external force response of flexible aircraft and rigid aircraft under different gust excitation frequencies.

In Figure 8, Figure 9 and Figure 10, the gust excitation aerodynamic force remains unaffected by the stiffness of the aircraft structure, with its peak value mainly determined by the excitation frequency (in other words, the gust gradient). Under conditions of very low gust excitation frequencies, the response of flexible aircraft is essentially similar to that of rigid aircraft, exhibiting quasi-static characteristics. Conversely, under conditions of high gust excitation frequency, the structure movement response of flexible aircraft is much larger than that of rigid aircraft. In other words, as the gust excitation frequency increases, the inertial force and the aerodynamic force in flexible aircraft significantly surpass those in rigid aircraft.

The dynamic magnification effect is similar to the result of [19], where the heave acceleration of flexible aircraft in unsteady analysis is larger than that of the rigid aircraft under 1-COS gust excitation. In addition, the rigid and flexible aircraft trim simulations in [42] are performed using the validated code at the same flight conditions and gust velocity to show that both the angle of attack and the elevator deflection of flexible aircraft are larger than those of rigid aircraft because of the deformation of the aircraft.

The aerodynamic force and gust-induced aerodynamic force phenomenon is similar to reference [14], where the 10% reduction in the lift curve slope in both the static wind tunnel test and the 15% reduction in the maximum life coefficient in the dynamic wind tunnel test under different gust frequency are accounted for the degradation of stiffness in fatigued wings compared with pristine wings.

Initially, during the gust response, the aerodynamic force is predominantly influenced by the gust. Once the gust has passed, the inertial force and aerodynamic force become the primary contributor. To accurately capture the peak load value, the calculation time must be long enough to adequately account for the succession of the three forces. Additionally, the occurrence of the gust peak load for flexible aircraft is delayed compared to that for rigid aircraft, with the time interval of this delay being proportional to the gust excitation frequency. In other words, the peak load for flexible aircraft occurs later than for rigid aircraft.

All gust cases with varying gradients listed in Table 2 were calculated. The peak values of the gust response were taken as the gust loads for the wing, as shown in Figure 11 and Figure 12, for both elastic and rigid aircraft.

As illustrated in Figure 11 and Figure 12, the gust load of flexible aircraft is more affected by increasing gust excitation frequency, in other words, decreasing gust gradients, than that of rigid aircraft. The gust loads of flexible and rigid aircraft do not vary linearly with the gradient, and they are quite lower under low gust gradients than the ones under high gradients generally. The maximum gust load in the wing root of flexible aircraft with different gust gradients is 298.8% of the minimum value, with a ratio of 129.8% for rigid aircraft. The maximum gust load of flexible aircraft under a certain range of gust analysis frequencies exceeds that of rigid aircraft, although the maximum gust load of flexible aircraft with the specific gust gradient does not necessarily exceed that of rigid aircraft. When the gust excitation frequency approaches the frequency of the first wing-bending mode of the structure, specifically with the gust gradient of 52.00 m (the excitation frequency of 2.40 Hz), the gust load on the flexible aircraft reaches its maximum value. Conversely, the gust load on the rigid aircraft peaks under the gust gradient of 52.00 m, with the second largest value under the gust gradient of 80 m. Accordingly, the gust gradient of 52.00 m is important for both flexible aircraft and rigid aircraft. At the preliminary stage of aircraft design, the gust load of aircraft with uncertain stiffness should be analyzed within a certain range of stiffness variation to avoid deviation.

All gust section loads with varied gust gradients are compared between flexible and rigid aircraft. The proportional relationships are exhibited as the dynamic magnification factor, which is the value of the bending moment of flexible aircraft divided by that of the rigid aircraft depicted in Figure 13, Figure 14 and Figure 15.

In Figure 13, Figure 14 and Figure 15, the proportional relationships between the gust loads of flexible and rigid aircraft indicate that the dynamic magnification factor remains relatively stable throughout the wing under different gust excitation frequencies. The dynamic magnification factor is almost constant through the wing. However, there is a notable increase in the dynamic magnification factor at the wing tip. The average ratio, excluding the wing tip region, is calculated for various gust excitation frequencies and presented in Figure 16.

As shown in Figure 16, by analyzing the dynamic magnification factor, the coupling resonant characteristic in gust analysis is obtained. The dynamic magnification factor peaks at 1.25 when the gust excitation frequency of the flexible aircraft matches the frequency of the first wing-bending mode. At other excitation frequencies, the dynamic magnification factor can vary, being either greater than or less than 1, depending on the gust excitation frequency under the specific stiffness characteristic, mass characteristic, and the unsteady characteristic of the aircraft structure.

Civil aircraft exhibit numerous structural modes in engineering, with distinct mode shapes and mode frequencies of the wings, fuselage, and empennage. Gust gradients should be adjusted on a large scale accordingly to excite enough modes to ensure that the dynamic magnification factor analysis remains conservative for various components and robust for the entire aircraft structure.

In this paper, the analysis of open-loop gust loads for civil aircraft is presented. With regard to civil aircraft equipped with fly-by-wire control systems, closed-loop gust analysis can be performed using the methods outlined herein. In such analyses, the gust dynamic magnification factor of flexible aircraft will simultaneously account for the effects of structure stiffness and the control laws, providing a comprehensive evaluation of the system’s response.

4. Conclusions

• For the gust load analysis of rigid aircraft, three methods—rigid modes analysis, secondary processing based on elastic modes, and analysis with enlarged stiffness—yield the same results using the mode displacement (MD) method and the summation-of-forces (SOF) method. These methods address the difficulty of extracting internal loads from rigid aircraft where deformation is absent.
• The gust excitation aerodynamic force peak value is mainly determined by the excitation frequency. At very low gust excitation frequencies, the response of flexible aircraft is essentially the same as that of rigid aircraft, exhibiting a quasi-static response. As the gust excitation frequency increases, the inertial force and the aerodynamic force in flexible aircraft significantly surpass those in rigid aircraft.
• The gust-induced aerodynamic force, the inertial force, and the aerodynamic force act in succession. To accurately capture the peak load value, the calculation time must be long enough to adequately account for the succession of the three forces. Under gust excitation, flexible aircraft experience a delay in the peak gust moment compared to rigid aircraft, with the time interval of this delay being proportional to the gust excitation frequency.
• The gust load of flexible aircraft is more affected by increasing the gust excitation frequency, in other words, decreasing the gust gradient, than that of rigid aircraft. The gust loads of flexible and rigid aircraft do not vary linearly with the gradient, and they are quite lower under low gust gradients than the ones under high gradients generally. The maximum gust load of flexible aircraft under a certain range of gust analysis frequencies exceeds that of rigid aircraft, although the maximum gust load on flexible aircraft under a specific gust gradient does not necessarily exceed that of rigid aircraft.
• The coupling resonant characteristic in gust analysis is obtained by analyzing the dynamic magnification factor, which is not necessarily greater than 1. The dynamic magnification factor, i.e., the ratio of the gust load of flexible aircraft to the gust load of rigid aircraft, is almost constant through the wing, with a notable increase at the wing tip, and depends on the specific gust excitation frequency under the specific stiffness characteristic, mass characteristic, and the unsteady characteristic of the aircraft structure. Both the gust load of flexible and rigid aircraft, as well as the dynamic magnification factor, reach their maximum values when the gust excitation frequency matches the frequency of the first wing-bending mode. With regard to other aircraft components, enough gust gradients should be included to excite enough modes to ensure the conservative estimation of the gust dynamic magnification factor of flexible aircraft for the entire aircraft structure.

The methods presented in this paper are applicable to both open-loop and closed-loop gust load analyses for civil aircraft and could be extended to gust analysis with other aerodynamic forces calculation methods and other dynamic response calculations without aerodynamic forces. The gust dynamic magnification factor of flexible aircraft provides a comprehensive evaluation of the system’s response considering the effects of structure stiffness and the control laws in the case of specific aerodynamic characteristics.