Nonlinear Dynamic Stability Analysis of Ground Effect Vehicles in Waves Using Poincaré–Lindstedt Perturbation Method

Abstract

In this study, we present an analytical tool that can be used to predict the nonlinear dynamic response of ground effect vehicles (GEVs) advancing through sinusoidal head-sea waves. GEVs exhibit a unique instability phenomenon known as porpoising, which is an oscillatory motion along the heave and pitch axes that can cause serious structural damage. The heaving and pitching equations of motion are presented in the form of coupled, forced, and nonlinear Duffing-type equations with cubic nonlinearity. The analytical model developed in this study leverages the Poincaré–Lindstedt perturbation method to express the amplitude and frequency of motion in terms of all physical parameters. The accuracy and reliability of the proposed model were validated through computational fluid dynamics (CFD) simulations based on incompressible unsteady Reynolds-averaged Navier–Stokes (RANS) equations. The results show a strong agreement between the analytical tool and the CFD simulations, with minor discrepancies due to assumptions inherent in the $2D$ simulations, particularly the assumption that seawater only passes beneath the hull, resulting in increased buoyancy forces and reduced damping. This study offers a novel and practical method for predicting the dynamic stability of GEVs under realistic sea conditions, potentially enhancing safety and operational efficiency by mitigating the risks associated with porpoising.

1. Introduction

With the increasing number of passengers and the need for more freight transport around the world, logistics and transport companies are pressured to increase their number of flights. However, this has led to a major increase in global air traffic, and the aviation industry is under pressure to reduce noise and emissions. One of the potential solutions to this issue is to build airports away from populated areas in offshore locations. This means moving the takeoff and landing paths over water. However, the cost of land reclamation and the need for new terminal buildings and pathways to be constructed are very high. An alternative to expensive infrastructure would be the use of a hybrid configuration that can reach high speeds while carrying a heavy payload. The introduction of high-speed planing hulls has led to the development of a hybrid configuration known as the ground effect vehicle (GEV), which is fast, agile, and capable of covering a wide range of applications because it combines the characteristics of airplanes and hydroplanes. The configuration of the GEV starts as a ship, but with the advantage of using the aerodynamic phenomenon known as the ground effect to fly just above the surface of the water.

When a craft flies next to the surface of the water or ground, it is influenced by the surface effect of aero-hydrodynamics, which leads to a high lift-to-drag ratio. The deceleration of air trapped between the ground and the wing surface causes a significant increase in the pressure on the under-surface of the wing [1]. Thus, the ground effect (GE) can be defined as the increase in the lift-to-drag ratio of a lifting surface flying at a small relative distance from the underlying surface [2]. This technology was most notably used in the development of the “Ekranoplan” designed in the Soviet Union at the beginning of the 1960s [1,2,3].

GEVs have a unique instability phenomenon called porpoising, which, according to Faltinsen [4], can be defined as a periodic, bounded vertical motion that a craft might exhibit at certain speeds, as illustrated in Figure 1. This phenomenon can be seen as an oscillatory motion in the heave and pitch axes and can cause severe damage to the structure of the craft [5]. In some cases, if the hull leaves the water and returns with a negative trim angle, the craft will submarine [6]. Porpoising can be fatal, even in calm water, because the loss of longitudinal stability can cause self-induced heave and pitch oscillations (porpoising) and the submergence of the bow area [7].

It is of great importance to study the response of the coupled motions of an object when it is subjected to an external periodic force. Such a setup finds applications in resonance vibrations, similar to what is seen when a GEV is accelerating to take off. This could lead to minimizing unwanted resonance and avoiding chaos [8]. In addition, the exact analytical solutions of the nonlinear, coupled, and forced equations of motion are very helpful for performing comprehensive analyses. These solutions can be used to capture the main nonlinear phenomena, such as nonlinear resonance and chaos, that characterize these systems with respect to the linear ones. External periodic stresses on floating objects can induce instability, particularly if they coincide with the object’s or its systems’ resonance frequency. This can exacerbate vibrations and result in structural damage or system failure. For vessels, these forces may damage hydraulic systems, limiting performance and potentially causing structural failures. Understanding and managing these impacts is critical to maintaining the stability and lifetime of floating objects and their systems [9]. In contrast to the numerical solutions, which can be useful for investigating a specific case, the analytical solution can be helpful for conducting a detailed parametric analysis aimed at understanding the whole system’s behavior. This is needed in the pre-design of any mechanical structure that interacts with a moving fluid [10]. Hence, understanding the role of coupling and nonlinearity on system behavior could enhance the dynamic stability of GEVs and provide insight into key design parameters. In fact, predicting the takeoff performance by combining aerodynamic and hydrodynamic calculations is a major challenge in GEV design [11]. The structural design of a sea-craft requires the evaluation of its motions and wave-induced pressure distributions over the hull in wavy sea conditions. Thus, a nonlinear, time-dependent seakeeping solution must simulate, with sufficient accuracy and efficiency, the seakeeping behavior of a sea-craft in wave trains that may be several hours long in duration [12]. With this goal in mind, the development of a nonlinear periodic solution was initiated.

Most studies conducted to date on the dynamic stability of seaplanes have been experimental or numerical in nature. For instance, Guo et al. [11] used the volume-of-fluid (VOF) approach to numerically simulate the air–water flow field to study the resistance characteristics and attitude of a seaplane at various takeoff speeds. They concluded that a seaplane’s trim angle should be operated in an ideal trim range, typically between 4° and 6°, to minimize hydrodynamic resistance. Song et al. [13] conducted a numerical study to understand the characteristics of a seaplane’s takeoff performance. The study only focused on the suction force, pressure, free surface, and motion of the seaplane model to understand the characteristics of the planing motion of the seaplane when moving through waves of different velocities, wavelengths, and heights. A two-dimensional numerical investigation was performed using the VOF approach proposed by Xie et al. [14]. The latter used a gas–liquid two-phase model to simulate complex fluid fluxes, with the air–water interface captured using the VOF technique. Their findings elucidated the mechanism of complicated flows under the influence of the bottom cabin and inclined postures. After conducting a comprehensive review of the academic literature on seaplane hydrodynamics, Tavakoli et al. [15] concluded that more work needs to be conducted on mathematical models, particularly those that simulate maneuvering motions, to fully account for the challenges of operating in a real-world maritime environment. Zhou et al. [16] stated that the majority of the present research on amphibious aircraft planing in waves is still concentrated on experiments and numerical simulations in calm water or under low-wave conditions. According to Wang et al. [17], it is no longer difficult to forecast the porpoising conditions for a specific sea-craft, experimentally or numerically. The challenge is finding out how to solve the porpoising problem, which is achievable with an understanding of the nonlinear behavior of sea-crafts accelerating on a wavy sea surface. Zheng et al. [18] used a modified two-phase flow solver in OpenFOAM to simulate the porpoising phenomenon of a large amphibious aircraft with four turbines during takeoff. They discovered that heaving and pitching oscillations are primarily determined by the hydrodynamic force, with the aerodynamic force playing an auxiliary role. A numerical study by Zha et al. [19] investigated the water-landing behavior of seaplanes and concluded that careful control of seaplane motion is essential to prevent potential damage from overwashing events. After thoroughly reviewing the methods used in seaplane hydrodynamic design, Morabito [20] stated that, because the majority of the research in this field dates back to the 1940s, studying the hydrostatic and hydrodynamic challenges associated with seaplane design is currently an important endeavor. Hwang et al. [21] investigated the longitudinal static stability of Ground Effect Vehicles (GEVs) in calm waters. In contrast, the current study focuses on the nonlinear dynamic stability of GEVs under sinusoidal head sea waves, which includes dynamic fluid–structure interactions such as porpoising and nonlinear oscillations. This work provides a more thorough understanding of GEV stability by addressing real-world issues in dynamic marine situations, taking into account both environmental and structural factors that static stability models do not account for. Collu et al. [22] developed a mathematical model to analyze stability in steady conditions, with a focus on how aerodynamic surfaces reduce pitching moments and improve stability, particularly at higher speeds. In contrast, the current study looks at the nonlinear dynamic stability of GEVs under sinusoidal head-sea waves, focusing on complicated fluid–structure interactions. While Collu et al. [22] focus on stability in calm, steady environments, this study investigates stability in more dynamic, real-world settings with wave-induced effects. Using CFD simulations and perturbation approaches, the current study provides a more comprehensive investigation of dynamic stability in marine environments, broadening the theoretical framework to account for nonlinear effects and dynamic problems. Huang et al. [23] analyze the effect of wave height and wavelength on water load during seaplane alighting, with an emphasis on how changing conditions affect the structural safety and seakeeping performance of amphibious aircraft during rough-water interactions. In contrast, the current study adds dynamic fluid–structure interactions like porpoising and nonlinear oscillations, providing a more complete knowledge of how environmental influences and structural attributes influence seaplane stability in operational settings. This study extends beyond static or localized behavior, offering a broader perspective on dynamic stability and addressing the real-world issues that seaplanes encounter when operating in dynamic marine situations. In contrast, the work of Lu et al. [24] focuses on hydrodynamic load reduction during landings using V-hydrofoils under various wave conditions; this present study extends the understanding of GEV stability by investigating its nonlinear dynamic behavior in sinusoidal head-sea waves. The work by Lu et al. [24], which focused on how wave parameters affect the load reduction during landing, is extended to consider the long-term dynamic stability of GEVs in waves, including nonlinear fluid–structure interactions such as porpoising and nonlinear oscillations. Chen et al. [25] study the hydrodynamic loads and landing performance of amphibious aircraft in waves, whereas this research focuses on the nonlinear dynamic stability of Ground Effect Vehicles (GEVs) in sinusoidal head-sea waves. Although Chen et al. [25] focused on the impact phase during water landings, in this case, it is shown more generally, as the long-term dynamic stability of a system, including porpoising and much more complicated fluid–structure interaction oscillations. Hence, a wider picture of how GEVs behave in realistic wave conditions is offered.

This study aimed to present an analytical tool that can be used to predict the nonlinear motion of GEVs advancing through head-sea waves. The analytical tool is a system of coupled, nonlinear heaving and pitching equations of motion, which are assumed to be externally excited by a sinusoidal head-sea wave. These equations are presented in the form of a system of coupled Duffing equations with cubic nonlinearity. The focus is on heave and pitch motions because, at a speed-to-length ratio of less than four (takeoff and landing speeds), the hydrodynamic coefficients associated with motions in other directions are neglected [26]. Hence, the dynamic stability of GEVs during takeoff and landing was examined from heaving and pitching motions. The analytical tool was validated using computational fluid dynamics (CFD) simulations performed on Ansys Fluent, using the VOF method, as the flow was two-phase. We used CFD simulations to validate the results, given their accuracy and ability to model complex phenomena, such as nonlinear dynamics, with a higher level of detail compared to traditional analytical methods. Although CFD results cannot be considered in a strict sense as directly experimental, they do provide a reliable numerical approximation derived from well-known principles of fluid dynamics. The primary objective at this stage is to confirm that our mathematical model aligns with the expected physical behavior. Analytical data for aero-hydrodynamics and flight mechanics have a direct impact on safety and reliability and can assist in enhancing both. This provides an opportunity to investigate the nature of the nonlinearity derived directly from motion physics. This should provide a deeper understanding of the nonlinear phenomenon associated with motion across waves, by explaining the effect of the nonlinear and coupling coefficients on the magnitude of the external force/moment. This study is the first of its kind to use the coupled nonlinear Duffing equation to describe the heaving and pitching motions of GEVs advancing through head-sea waves. A closed-form analytical solution to the equations of motion, describing the dynamic response of the GEV hull, is also presented in this study.

2. Mathematical Formulation

Coupled Duffing oscillators were previously investigated analytically in [8,10,27,28,29]. In all previous works, the attention has been focused on the route to chaos and the frequency response. No attempts have been made to obtain an exact, closed-form analytical solution representing the heaving and pitching responses of a GEV advancing through head-sea waves, which was the aim of this work. The chaos theory is one of the most significant achievements in nonlinear science. The Duffing equation is associated with mathematical chaotic behavior. This chaotic behavior exists in many natural systems, such as weather and climate systems [30]. It also occurs spontaneously in systems with artificial components, such as road traffic. Chaos theory has been applied in several other disciplines, including meteorology, sociology, and environmental sciences. Chaos is defined as a periodic long-term behavior in a deterministic system that exhibits a sensitive dependence on the initial conditions [31]. Three properties must exist in a dynamical system for it to be classified as chaotic:

• It must exhibit periodic long-term behavior, meaning that the solution of the system settles into an irregular pattern as $t\to \infty$. The solution either does not repeat or oscillates periodically.
• It is sensitive to the initial conditions. This means that any small change in the initial condition can change the trajectory, which may result in significantly different long-term behavior.
• It must be “deterministic”, which means that the irregular behavior of the system is due to the nonlinearity of the system, rather than external forces.

Thus, Duffing oscillators find applications in chaos theory, which is a field of study in mathematics that investigates the behavior of dynamical systems that are highly sensitive to the initial conditions. Small differences in the initial conditions (such as rounding errors in the numerical computation) yield widely diverging outcomes for such dynamical systems, rendering long-term predictions generally impossible [31]. There are several analytical methods that can be used to solve the Duffing equation in its various forms, such as the method of multiple scales [32], the Bogolubov–Mitropolski method [33], He’s energy balance method [34], the global error minimization method [35], the variational iteration method [36], the Jacobi elliptic functions method [37], the homotopy perturbation method [38], and the Poincaré–Lindstedt method [39].

Several physical problems faced by engineers today demonstrate certain fundamental features, such as nonlinearities, that prevent exact analytical solutions. Thus, perturbation methods are used to obtain information regarding the solutions of nonlinear equations. The solution is presented in the form of convergent power series expansions with respect to a small dimensionless parameter that is of the same order of nonlinearity and amplitude of motion [40]. The Poincaré–Lindstedt method is used to uniformly approximate periodic solutions to ordinary differential equations. This method eliminates the secular terms arising from the straightforward application of the regular perturbation theory to weakly nonlinear equations [41]. Secular terms are those that grow without bounds. These terms have a singularity point at which a given mathematical object is not defined [41].

The Poincaré–Lindstedt method was adopted in this research to find a closed-form solution to the coupled equations of motion because it is used to obtain periodic solutions to nonlinear differential equations. It describes the period of the unstable motion of GEVs such that porpoising can be predicted analytically. The use of this technique leads to an expression of the desired solution in terms of a formal power series with a small parameter ($\epsilon$), known as a perturbation series, that quantifies the deviation from the exact solvable problem [42,43,44].

2.1. Heave and Pitch Equations of Motion

The equations of motion for a GEV advancing at a constant forward velocity with an arbitrary heading in regular sinusoidal sea waves are presented in this Section. The performance of the GEV is presented using a six-DOF (degrees of freedom) system. Considering that the responses are linear and harmonic, the six linear equations of motion can be written using the subscript notation as follows [4]:

$$\sum _{k=1}^6\left[\left(M_{jk}+A_{jk}\right){\ddot{\mathrm{ƞ}}}_k+B_{jk}{\dot{\mathrm{ƞ}}}_k+C_{jk}{\mathrm{ƞ}}_k\right]=F_j{e}^{i\omega t},$$

(1)

where

• $j=1-6.$
• $m_{jk}$ is the component of the generalized mass matrix of the craft in the $j^{th}$ direction owing to the $k^{th}$ motion.
• $A_{jk}$ is the added-mass coefficient in the $j^{th}$ direction due to the $k^{th}$ motion.
• $B_{jk}$ is the damping coefficient in the $j^{th}$ direction due to the $k^{th}$ motion.
• $C_{jk}$ is the hydrostatic restoring force coefficient in the $j^{th}$ direction due to the $k^{th}$ motion.
• $F_j$ is the complex amplitude of the exciting forces and moments in the $j^{th}$ direction.

Heaving and pitching due to moving through head-sea waves are oscillatory motions along the vertical and sagittal axes, as illustrated in Figure 2. Heaving can be defined as the translational (linear) motion in the vertical direction, while pitching is the rotational (angular) motion of the center of gravity of the aircraft [45].

The heaving and pitching system of GEV motion behaves like a two-degrees-of-freedom spring-mass system. According to Faltinsen [4], this assumption is clear when a craft model is given heave or pitch displacements from its equilibrium position; it rapidly oscillates several times before it comes to rest. Therefore, the resulting equations for the heave and pitch motions of the GEV can be expressed as follows:

$$\left(m+A_{33}\right){\ddot{ƞ}}_3+A_{35}{\ddot{ƞ}}_5+B_{33}{\dot{\mathrm{ƞ}}}_3+B_{35}{\dot{\mathrm{ƞ}}}_5+C_{33}{\mathrm{ƞ}}_3+C_{35}{\mathrm{ƞ}}_5=F\left(t\right),$$

(2)

$$A_{53}{\ddot{\mathrm{ƞ}}}_3+{(A}_{55}+I_{55}){\ddot{\mathrm{ƞ}}}_5+B_{53}{\dot{\mathrm{ƞ}}}_3+B_{55}{\dot{\mathrm{ƞ}}}_5+C_{53}{\mathrm{ƞ}}_3+C_{55}{\mathrm{ƞ}}_5=M\left(t\right).$$

(3)

Subscripts 3 and 5 represent the heave and pitch motions, respectively. In addition, $u$ is the heave translational displacement, and $v$ is the pitch angular displacement. $F\left(t\right)$ is the external excitation force and $M\left(t\right)$ is the external excitation moment. Both quantities were assumed to be caused by sinusoidal waves. Hence, the excitation forces and moments can be expressed as follows:

$$F\left(t\right)=F\cos \left(\omega t\right),M\left(t\right)=M\cos (\omega t+\phi ).$$

(4)

Both the time-dependent sinusoidal force and moment functions oscillate at an external excitation frequency $\omega$. Here, $t$ denotes time, $F$ represents the amplitude of the force, $M$ is the amplitude of the moment, and $\phi$ is the phase angle, indicating a phase shift in the moment relative to the force.

The hydrostatic restoring force coefficient (buoyancy), hydrodynamic coefficients (added mass and damping), and external forces and moments (amplitudes and frequency) acting on a planing hull were calculated using strip theory. This theory utilizes the integration of the hydrodynamic pressure acting on the wetted surface of the planing hull to obtain the forces and moments. The calculation of the coefficients is discussed in detail in Section 4.1.

The general equations of the heave and pitch motions of GEVs were reduced to Duffing oscillator equations by assuming that the restoring forces behave according to Hooke’s law but have a nonlinear nature (cubic nonlinearity). The system is excited by a sinusoidal head-sea wave, and the damping effect is negligible because the hydrodynamic coefficients associated with damping are much smaller than the added mass and restoring force coefficients. This is because the damping coefficients are functions of frequency and speed, whereas the added mass and restoring force coefficients are functions of the GEV mass [46]. In addition, the pitch’s angular acceleration was neglected in the heave equation, and the heave translational acceleration was neglected in the pitch equation. Structural deformation was neglected in this study for two main reasons: first, the primary focus is on dynamic stability as the load shifts from hydrostatic to hydrodynamic, where structural deformations have minimal impact on the dynamic response; and second, for the scale of motions considered, the influence of structural deformation is negligible, particularly in small-amplitude oscillations where fluid forces dominate the dynamics. A mathematical model was developed to predict porpoising when the craft is under maximum buoyancy support; hence, the hydrostatic restoring force is dominant. Taking this into consideration, this two-degrees-of-freedom system can be modeled using two coupled second-order nonlinear differential equations with the following forms:

$$\ddot{u}+{\omega }_0^{2}u+a_{1}v+a_2{u}^3=F_1\cos \left(\omega t\right),$$

(5)

$$\ddot{v}+{\omega }_0^{2}v+b_{1}u+b_2{v}^3=M_1\cos \left(\omega t\right)+M_2\sin \left(\omega t\right),$$

(6)

where

• $a_1$ is the pitch coupling term coefficient.
• $a_2$ is the heave nonlinear term coefficient.
• $b_1$ is the heave coupling term coefficient.
• $b_2$ is the pitch nonlinear term coefficient.
• $F_1$ is the heave amplitude.
• $M_1$ and $M_2$ are the pitch amplitudes.
• ${\omega }_0$ is the natural frequency.
• $\omega$ is the external excitation frequency.

Equations (5) and (6) are harmonically excited Duffing oscillator equations. They can be solved using the Poincaré–Lindstedt method by introducing a new, small, non-dimensional parameter ($\epsilon$) into the equations. This was achieved by assuming that the nonlinearity, coupling, and excitation force/moment are small relative to the harmonic motion and that they appear in the same order in each equation. In addition, ($\epsilon$) can be used as a measure of the strength of the nonlinearity. Hence, a small parameter ($\epsilon$) was introduced, and Equations (5) and (6) can be rewritten as follows:

$$\ddot{u}+{\omega }_0^{2}u+{\epsilon a}_{1}v+{\epsilon a}_2{u}^3={\epsilon F}_1\cos \left(\omega t\right),$$

(7)

$$\ddot{v}+{\omega }_0^{2}v+{\epsilon b}_{1}u+{\epsilon b}_2{v}^3={\epsilon M}_1\cos \left(\omega t\right)+{\epsilon M}_2\mathrm{s}\mathrm{i}\mathrm{n}\left(\omega t\right).$$

(8)

2.2. Amplitude Equations

We next turned to the question of determining an expression for the amplitude and frequency of porpoising oscillations in terms of the nonlinearity, coupling, and external forcing parameters. The frequency of the system $\omega$ can be explicitly presented in the differential equation by introducing the following transformation:

$$\tau ={\displaystyle \frac{\omega }{n{\omega }_0}}t.$$

(9)

The frequency of the forcing term was initially assumed to be unknown, but near a multiple of the natural frequency, ${\omega }_0$. To account for this, $n$ was introduced to define the order of resonance, allowing the solution to cover various cases (i.e., different orders of resonance). By applying the chain rule, the equations were transformed as follows:

$${\omega }^2{u}^{″}+{\omega }_0^{2}u+{\epsilon a}_{1}v+{\epsilon a}_2{u}^3={\epsilon F}_1\cos \left(\tau \right),$$

(10)

$${\omega }^2{v}^{″}+{\omega }_0^{2}v+{\epsilon b}_{1}u+{\epsilon b}_2{v}^3={\epsilon M}_1\cos \left(\tau \right)+{\epsilon M}_2\mathrm{s}\mathrm{i}\mathrm{n}\left(\tau \right),$$

(11)

where the prime symbol indicates the derivative with respect to the re-scaled time variable $\tau$. An unknown frequency of the system appeared in the differential equations. Hence, approximate solutions were obtained for both equations in the form of power series expansions in terms of $\epsilon$. In other words, the following were introduced:

$$u=u_0\left(\tau \right)+\epsilon u_1\left(\tau \right)+{\epsilon }^2{u}_2\left(\tau \right)+O\left({\epsilon }^3\right),$$

(12)

$$v=v_0\left(\tau \right)+\epsilon v_1\left(\tau \right)+{\epsilon }^2{v}_2\left(\tau \right)+O\left({\epsilon }^3\right),$$

(13)

$$\omega =n{\omega }_0+\epsilon {\omega }_1+{\epsilon }^2{\omega }_2+O\left({\epsilon }^3\right).$$

(14)

The first term in the expansion represents the linear solution of the equation, also referred to as the leading-order term. In this study, the leading-order frequency, ${\omega }_0$, was set to unity, representing the natural frequency of the system. Substituting Equations (12)–(14) into Equations (10) and (11) yielded the following:

$${[{\omega }_0+\epsilon {\omega }_1+{\epsilon }^2{\omega }_2]}^2\left[u_0^{″}+\epsilon u_1^{″}+{\epsilon }^2{u}_2^{″}\right]+{\omega }_0^2\left[u_0+\epsilon u_1+{\epsilon }^2{u}_2\right]+\epsilon a_1\left[v_0+\epsilon v_1+{\epsilon }^2{v}_2\right]+{\epsilon a_2[u_0+\epsilon u_1+{\epsilon }^2{u}_2]}^3=\epsilon F_1\cos \left(\tau \right),$$

(15)

$${[{\omega }_0+\epsilon {\omega }_1+{\epsilon }^2{\omega }_2]}^2\left[v_0^{″}+\epsilon v_1^{″}+{\epsilon }^2{v}_2^{″}\right]+{\omega }_0^2\left[v_0+\epsilon v_1+{\epsilon }^2{v}_2\right]+\epsilon a_1\left[u_0+\epsilon u_1+{\epsilon }^2{u}_2\right]+{\epsilon a_2[v_0+\epsilon v_1+{\epsilon }^2{v}_2]}^3=\epsilon M_1\cos \left(\tau \right)+\epsilon M_2\sin \left(\tau \right).$$

(16)

Separating the terms of the same order and eliminating the terms of order $O\left({\epsilon }^3\right)$ yielded the following equations:

• Heaving terms:

  $$u_0^{″}+u_0=0,$$

  (17)

  $${\omega }_0^2\left[u_1^{″}+u_1\right]+2{\omega }_0{\omega }_1{u}_0^{″}+a_2{u}_0^3=F_1\mathit{cos}\left(\tau \right),$$

  (18)

  $${\omega }_0^2\left[u_2^{″}+u_2\right]+2{\omega }_0{\omega }_1{u}_1^{″}+2{\omega }_0{\omega }_2{u}_0^{″}+{\omega }_1^2{u}_0^{″}+3a_2{u}_0^2{u}_1=0.$$

  (19)
• Pitching terms:

  $$v_0^{″}+v_0=0,$$

  (20)

  $${\omega }_0^2\left[v_1^{″}+v_1\right]+2{\omega }_0{\omega }_1{v}_0^{″}+a_2{v}_0^3=M_1\cos \left(\tau \right)+M_2\mathrm{s}\mathrm{i}\mathrm{n}\left(\tau \right),$$

  (21)

  $${\omega }_0^2\left[v_2^{″}+v_2\right]+2{\omega }_0{\omega }_1{v}_1^{″}+2{\omega }_0{\omega }_2{v}_0^{″}+{\omega }_1^2{v}_0^{″}+3a_2{v}_0^2{v}_1=0.$$

  (22)

The leading-order solutions of the two equations were assumed to have the following form:

$$u_0=Acos\left({\omega }_0\tau \right),v_0=Bcos\left({\omega }_0\tau \right)+Csin\left({\omega }_0\tau \right).$$

(23)

The corrections to the linear solution were determined in the course of the analysis by requiring the expansion of $u$ and $v$ to be uniform for all $\tau$. However, the particular solution of the first-order terms of $u$ and $v$ contains secular terms that make the expansion non-uniform. To achieve uniform expansion, the secular terms in $u_1$, $v_1$, $u_2$, and $v_2$ must be eliminated. To this end, the coefficients of the secular terms were set to zero to find expressions for the first-order frequency ${\omega }_1$. Thus, only the inhomogeneous terms in Equations (18), (19), (21), and (22) governing $u_1$, $v_1$, $u_2$, and $v_2$ were inspected. This resulted in the expansions being uniform for all first- and second-order solutions because secular terms did not appear in them. Removing all secular terms up to the first order required satisfying the following system of equations:

$$a_{1}B-F_1=0,$$

(24)

$$3a_2{A}^3+4a_{1}C-8A{\omega }_0{\omega }_1=0,$$

(25)

$$3b_2{B}^3+3b_2{C}^{2}B-8B{\omega }_0{\omega }_1-4M_1=0,$$

(26)

$$4A{b}_1+3b_2{C}^3+3b_2{B}^{2}C-8C{\omega }_0{\omega }_1-4M_2=0,$$

(27)

while $A$ satisfied the following cubic polynomial:

$$3F_1\left(b_1^2{b}_2{F}_1^2-a_1^2{a}_2{M}_1^2\right)A^3-6b_1{b}_2{F}_1^3{M}_2{A}^2+\left(4a_1^2{b}_1{F}_1^2{M}_1-4a_1^3{M}_1^3+3b_2{F}_1^3{M}_1^2+3b_2{F}_1^3{M}_2^2\right)A-\left(4a_1^2{F}_1^2{M}_1{M}_2\right)=0.$$

(28)

$A$ represents the amplitude of the external force, which is the amplitude of the sea wave. The nonlinear coefficients $a_2$ and $b_2$ were obtained from the polynomial, assuming in the first approximation that both coefficients are equal. The other terms were obtained from the following equations:

$$a_1={\displaystyle \frac{C_{35}}{m+A_{33}}},$$

(29)

$$b_1={\displaystyle \frac{C_{53}}{A_{55}+I_{55}}},$$

(30)

$$F_1={\displaystyle \frac{F_a}{m+A_{33}}},$$

(31)

$$M_1={\displaystyle \frac{M_a}{A_{55}+I_{55}}},$$

(32)

$$M_2={\displaystyle \frac{M_b}{A_{55}+I_{55}}},$$

(33)

$${\omega }_0=\sqrt{{\displaystyle \frac{C_{33}}{m+A_{33}}}}=\sqrt{{\displaystyle \frac{C_{55}}{I_{55}+A_{55}}}.}$$

(34)

Hence, the amplitude equations obtained are as follows:

• Heave:

  $$u\left(\mathrm{t}\right)=\mathrm{A}\mathrm{s}\mathrm{i}\mathrm{n}\left(\mathsf{\omega }\mathrm{t}\right)-\mathsf{\epsilon }\left[{\displaystyle \frac{a_2{A}^3}{32{\omega }_0^2}}\cos \left(3\omega t\right)\right].$$

  (35)
• Pitch:

  $$v\left(t\right)={\displaystyle \frac{F_1}{a_1}}\cos \left(\omega t\right)+{\displaystyle \frac{F_1(M_2-A{b}_1)}{a_1{M}_1}}\sin \left(\omega t\right)+\epsilon \left[{\displaystyle \frac{b_2{F}_1^3(A{b}_1-M_2)(M_2^2-2A{b}_1{M}_2+A^2{b}_1^2-3M_1^2)}{32a_1^3{M}_1^3{\omega }_0^2}}\sin \left(3\omega t\right)-{\displaystyle \frac{b_2{F}_1^3\left(3M_2^2-6A{b}_1{M}_2+3A^2{b}_1^2-M_1^2\right)}{32a_1^3{M}_1^2{\omega }_0^2}}\mathrm{c}\mathrm{o}\mathrm{s}(3\omega t)\right].$$

  (36)
• Frequency of oscillations:

  $$\omega ={\omega }_0+\epsilon \left[{\displaystyle \frac{3a_2{A}^3{M}_1-4A{b}_1{F}_1+4F_1{M}_2}{8A{M}_1{\omega }_0}}\right].$$

  (37)

The time-dependent solutions for Equations (35)–(37) were visualized using MATLAB R2022b. To verify the accuracy of the analytical solution, MATLAB’s ODE45 solver was used to numerically solve Equations (7) and (8), and the numerical results were found to strongly match the analytical solution, confirming its accuracy.

3. CFD Validation

3.1. The Viscous Model

The computational fluid dynamics (CFD) solver used was Ansys Fluent, which is based on the incompressible unsteady Reynolds-averaged Navier–Stokes equations (RANSEs) [47]. This solver decomposes the solution variables of the exact Navier–Stokes equations into mean (time-averaged) and fluctuating components. The turbulence model used was the two-equation $k-\omega$ shear stress transport (SST) model. This model includes two additional transport equations to represent the turbulent properties of flow in order to account for historical effects, such as the convection and diffusion of turbulent energy [48]. The transport variable $k$ determines the energy in turbulence, whereas $\omega$ determines the scale of the turbulence. It is widely used to simulate aerodynamic flow for aeronautical applications and is known for its ability to handle a variety of turbulent flows, such as rapid length-scale changes [49].

3.2. The Computational Domain

3.2.1. Geometry

The two geometries used in this research were the $2D$ hulls presented in [45] and [50], which are later referred to as model 1 and model 2, respectively. The two models differ in geometry, weight, kinematics, and shape, demonstrating that our analytical tool can predict performance across various hull designs without limitations. As the aim was to investigate the dynamic stability during takeoff and landing, only the hull of the GEV was considered in the simulations. During takeoff and landing, the hydrodynamic stability prediction is of great importance in the design of GEVs, as it provides information about the water drag resistance and porpoising [7]. As shown in Figure 3, the computational domain is divided into three zones: the moving zone, the re-meshing zone, and the stationary zone. This is because the area around the hull of the GEV must oscillate according to the dynamic response of the hull to sea waves. The re-meshing zone does not oscillate; however, the mesh in this zone responds to the motion of the moving zone and constantly reconstructs the elements of the mesh. This is performed to reduce the computational time, because only the re-meshing and moving zones have a high-definition mesh.

The distance from the free surface (seawater level) to the top of the domain was assumed to be equal to the hull length. The depth of the domain was also equal to the hull length. Moreover, the distance from the center of gravity of the hull to the outlet was assumed to be four times the length of the hull. Nevertheless, the distance from the center of gravity (CG) of the hull to the inlet was twice the length of the hull. This agrees with the International Towing Tank Conference (ITTC)’s practical guidelines for ship CFD applications [51]. The body-fixed reference frame axes were chosen to coincide with the principal axes of inertia, and the origin was taken coincident with the center of gravity of the hull.

3.2.2. Mesh

As presented in Figure 4, a hybrid mesh was generated that combined structured (hexahedral) elements in the stationary zone and unstructured (tetrahedral) elements in the re-meshing and moving zones. This was conducted to reduce the computational time. The stationary zone does not necessitate any need for a high-definition mesh, as it is just a freestream region. The mesh density was evaluated for seven different boundary-layer edge sizes and studied against the coefficient of the drag of the hull. The optimal edge size of the hull was found to be 0.0082 ft. This is the point at which the solution starts to converge, in which the difference in the coefficient of drag produced by any smaller edge size is negligible. In addition, this edge size produces approximately 165,000 mesh elements, which can be simulated using a high-specification PC within an acceptable time. In unsteady flow simulations, using an appropriate dimensionless wall distance, $y+$, is crucial for accurately resolving the boundary layer around the hull [52]. To ensure that $y+$ is less than one, the boundary layer was refined by setting the first-layer height to $6\times {10}^3\times L$, where $L$ is the length of the hull. This guarantees that there are at least 45 layers surrounding the boundary layer regions.

3.2.3. The Numerical Domain

The volume of fluid (VOF) method was selected in the multiphase flow tab. The computational domain was defined as a two-phase flow domain with air as the primary fluid and water as the secondary fluid. The open-channel flow and open-channel wave BC (boundary conditions) should be enabled to define the head-sea wave. The numerical beach should be enabled in the stationary zone to account for the wave damping at the outlet and reduce the computational time. This suppresses wave reflection at the outlet. A dynamic mesh was used in this investigation. This allowed the hull to respond to sea waves in the form of motions with only two degrees of freedom. The motion was restricted by the use of a user-defined function (UDF). The UDF was also used to record the heave and pitch motions and define the mass and mass moment of inertia of the hull. Smoothing and re-meshing were enabled to allow the mesh to reconstruct the moving cells.

4. Results

4.1. Hydrodynamic Coefficient Calculation

The determination of the coefficients, exciting force, and moments of Equations (1) and (2) is a major problem in the analytical study of motion. To simplify this problem, the boat was divided into transverse strips or segments [53]. Then, the added mass, damping, restoring force, external excitation force, and moment coefficients were calculated using two-dimensional hydrodynamic theory [45]. This method is known as the “strip theory”. It has been widely used to calculate the hydrodynamic coefficients of ships, boats, planing hulls, and sea-crafts. The objective of this theory is to predict the motion of floating bodies in the heave and pitch directions when the oscillation frequency is known [45]. Over the past century, several strip theory methods have been formulated, such as those of Korvin-Kroukovsky [50]; Salvesen, Tuck, and Faltinsen [54]; and Lewis [55]. The key differences between them are in the number of segments the hull is divided into, and in the assumptions made when calculating the sectional added mass, sectional damping, sectional restoring force, and moment coefficients.

Strip theory is a very efficient tool owing to its ability to analytically determine the coefficients in the coupled heave and pitch equations of motion under any sea wave or speed condition. This facilitates the improvement of the ship design so that the performance can be enhanced in severe weather or sea conditions. In this Section, the strip theory method proposed by Bhattacharyya [45] was applied to obtain the values of the coefficients of the coupled heave and pitch equations so that further analytical investigations could be carried out. To be compatible with the objectives of this research, a few assumptions were made in the mathematical formulation of the strip theory, such as the assumptions that the sea waves are periodic and linear (regular sea waves/sinusoidal); the forces and moments generated by the wind and propellers can be neglected; the sea-craft is unrestrained, is rigid, and has a slender shape; the sea-craft is symmetrical in the x–z plane (transverse axis); the GEV is heading into the waves in a direction transverse to its peak line (i.e., the heading angle ($\mu$) is 180°); the vertical motion is composed of coupled pitching and heaving motions only; and the wavelength is equal to the hull length.

With these assumptions in mind, the hull was divided into a few segments or strips that were rigidly connected to each other, and the flow around it was considered two-dimensional in nature. These considerations allowed the problem to be reduced from three to two dimensions, with each segment treated as part of an infinite cylinder with two-dimensional flow around it. This implies that there is no interaction between the flows at adjacent segments [45]. The response of each segment was then calculated separately, and the total response of the hull was determined by integrating the component reactions of all the segments over the total length of the planing hull. In summary, strip theory can be performed using the following steps:

• The hull is divided into four sections to provide a representation of the underwater hull shape.
• The sectional added mass, damping, and restoring force coefficients are then determined by considering the hull as a series of transverse segments.
• Subsequently, the hydrodynamic coefficients of the heave and pitch equations are calculated by performing pressure integration along the length of the hull.
• Finally, the excitation forces and moments are calculated using Newton’s second law based on the given sea wave characteristics and hull shape.

4.2. Geometry and Conditions

The geometric properties of the two hull models used in this study are listed in

Table 1. Geometrical properties of hulls.

Parameter	Model 1	Model 2
Overall length (ft)	19.2	5
Beam length (ft)	2.592	0.666
Draft (ft)	1.144	0.081
Weight (lb)	2838	33.2
Radius of gyration (ft)	4.588	1.25
Moment of inertia (lb.sec2.ft)	1780.2	1.611

. The center of gravity (CG) of the hull was assumed to coincide with the center of mass and center of rotation. Strip theory can now be applied to determine the coefficients of the heave and pitch equations. In order to study the stability during takeoffs and landings, investigations were carried out on two Froude numbers ($F_n$); 0.1 and 0.2. According to Fossen [56], this is the takeoff or landing regime where the hull is supported by hydrodynamic forces. This is the regime of maximum hydrodynamic resistance, where a porpoising prediction is of great importance. The Froude number ($F_n$) is a non-dimensional quantity used in hydrodynamics to study the effect of gravity on fluid motion [56]. The incident wave amplitude was considered to be 0.2 ft and 0.1 ft in the simulations performed for models 1 and 2, respectively.

4.3. Discussion of Results

The analytical and CFD simulation results of heaving and pitching for model 1 are shown in Figure 5 and Figure 6, respectively. The CFD solution predicted the motion to oscillate with an unsteady frequency, whereas the analytical solution only produced a constant frequency of oscillations. This is because, in $2D$ simulations, the water is restricted to only flow underneath the hull, whereas in reality, a significant amount of water is diverted around the sides of the hull. This leads to the motion response of the hull being greater than expected, as less energy is dissipated from the hull to the water. This is owing to the lower damping rate experienced by two-dimensional hulls compared to three-dimensional ones. This explains the high heaving amplitude of motion and the lower pitch amplitude produced in the CFD solution. The lower pitch amplitude can be explained by the fact that pitching is restricted because it is a $2D$ hull, and the buoyancy force is greater when the center of buoyancy changes in response to the increase in pitching. There is an inconsiderable discrepancy between the results of the two methods in the first $5s$ of Figure 5 and Figure 6. This is because the location of the free surface was defined based on a length-to-draft ratio of 10. However, this depends on the weight, location of the CG, and speed of the hull, and it takes a few seconds to stabilize. When the Froude number $F_n$ was increased to 0.2, the motion amplitude decreased, while the oscillation frequency increased. The predicted amplitude matched the CFD results very accurately, although the pitching frequency from the CFD solution showed a slight increase over time. This can be attributed to the increased motion response of the $2D$ hull, as previously mentioned.

The findings for model 2 are shown in Figure 7 and Figure 8. Owing to the significantly higher frequency of oscillations in this model compared with the previous model, the data are only displayed for $10s$. This is because of its much lower overall length and weight. The results of this model demonstrate that the heaving and pitching amplitudes obtained from both methods are in good agreement. Because the center of buoyancy traveled a shorter distance compared to model 1, the pitching amplitudes obtained from both methods showed good agreement. In addition, because the hull, in this case, was much lighter, it stabilized quickly after the simulation began. As a result, the amplitude and frequency predictions were closely aligned. When the speed was increased, the results became more stable, and the oscillation frequency from the CFD solution remained nearly constant. Although the heaving amplitude from the CFD solution was slightly higher than the analytically predicted value, the pitch amplitude and frequency were well matched.

The observed phase shifts in the oscillations result from variations in the initial conditions applied to each case. Although the system exhibits a harmonic-like behavior overall, different initial zero-state conditions of the hull were deliberately set to assess the system’s response under different starting conditions, leading to the observed phase shifts.

The short-period response observed in the CFD solution corresponds to high-frequency oscillations, which align with the primary dynamic mode of the system as predicted by the Poincaré–Lindstedt perturbation method. These oscillations reflect the hull’s primary heaving and pitching motions in response to sinusoidal wave forces. The long-period response, on the other hand, is caused by low-frequency modes that result from fluid–structure interactions. These interactions are more thoroughly captured in the CFD model than in the analytical solution. Specifically, the long-period oscillations are due to the structural flexibility of the hull, higher-order fluid-structure coupling, and wave-induced damping effects. The CFD model accounts for complex phenomena like hydrodynamic forces, aerodynamic damping, and wave–structure interactions, which can produce slower oscillations. Additionally, the long-period oscillations are linked to resonant coupling between the hull and the wave frequencies, potentially leading to slower drift motions or subharmonic resonances. These effects, which are better represented in the CFD model, reflect the more detailed interactions between the hull and the waves that the analytical model cannot capture.

To evaluate the accuracy of our mathematical model’s predictions, we used Root Mean Square Error (RMSE), a metric that quantifies the average magnitude of the differences between the predicted and actual values. The RMSE values presented in Figure 9 highlight the accuracy of the model’s predictions when compared to the CFD results. The figure illustrates the discrepancy between the numerical solutions obtained from the CFD simulations and the analytical model. As shown, the RMSE for each case indicates how well the model predicted the amplitudes and frequencies of heave and pitch motions, with lower RMSE values signifying better agreement. The calculated RMSE values range from 0.003 to 0.02, suggesting that the model’s predictions are in close alignment with the CFD results. These findings demonstrate that the analytical model effectively captures the dynamic behavior of the system, with only minor deviations observed.

4.4. Porpoising

In this study, porpoising is observed as a coupled oscillatory motion between the heave (vertical displacement) and pitch (rotational motion about the longitudinal axis) of the GEV. The phenomenon is characterized by large amplitude oscillations in pitching, which occur with a short-period cycle, and is coupled with an increase in the amplitude of the heaving motion. These pitching oscillations exhibit high frequency and substantial amplitude, indicative of nonlinear dynamic behavior in the system. The system’s response exhibits significant energy transfer between heave and pitch motions, which is characteristic of porpoising. The coupling between the heave and pitch motions results from the complex interaction of the hull’s hydrodynamic forces with its dynamic characteristics, particularly under wave-induced excitation. This coupled motion, involving both high-frequency and large-amplitude oscillations, leads to dynamic instability in the GEV, resulting in substantial variations in both vertical displacement (heaving) and rotational motion (pitching), which define the porpoising phenomenon.

5. Conclusions

This study presents a novel analytical tool for predicting the nonlinear dynamic stability of GEVs in head-sea waves, focusing on coupled heave and pitch motions. Using the Poincaré–Lindstedt perturbation method, we derived a closed-form solution to the nonlinear equations of motion, validated through CFD simulations, which showed strong agreement with minor discrepancies. The model effectively predicts porpoising behavior, characterized by large-amplitude, high-frequency oscillations in pitching, coupled with increased heaving motion. This dynamic instability, resulting from the interaction of the hull’s hydrodynamic forces and its dynamic characteristics, leads to oscillations in both heave and pitch motions, defining the porpoising phenomenon.

Future directions include applying nonlinear wave theories, such as the Cnoidal [57,58,59] and nonlinear Stokes wave theories [60,61], to study GEV motion under irregular sea waves [62,63], which represent nonharmonic oscillations. Integrating hydrodynamic coefficients into nonlinear strip theory will capture three-dimensional effects, and an asymptotic approach will explore additional degrees of freedom, such as rolling. Experimental validation will be pursued to confirm the model’s accuracy. These advancements, along with the integration of the mathematical model into a Digital Twin Lab, will enable real-time simulations and continuous data to optimize GEV performance.