Enhancing Sustainability in Marine Structures: Nonlinear Energy Sink for Vibration Control of Eccentrically Stiffened Functionally Graded Panels Under Hydrodynamic Loads

Abstract

The research examines the impact of nonlinear energy sinks (NES) on the reduction in the nonlinear vibratory responses of eccentrically stiffened functionally graded (ESFG) panels exposed to hydrodynamic loads. To simulate real marine environments, hydrodynamic forces, such as lift and drag that change with velocity, have already been determined experimentally using Matveev’s equations for a particular ship. The material composition of both the panel and the stiffeners varies across their thickness. The stiffeners are modeled using Lekhnitskii’s smeared stiffener approach. Additionally, analytical approaches implement the classical shell theory (CST) with considerations for geometric nonlinearity, along with the Galerkin method for calculations. The P-T method is subsequently employed to determine the nonlinear vibratory behavior of ESFG panels. In this method, the piecewise constant argument is used jointly with the Taylor series expansion, which is why it is named the P-T method. The findings reveal that NES can effectively dissipate vibrational energy, contributing to the extended service life of marine structures while reducing the need for frequent maintenance. This study supports sustainability objectives by increasing energy efficiency, lessening structural fatigue, and improving the overall environmental impact of marine vessels and infrastructure.

1. Introduction

Functionally graded (FG) panels are widely used in a variety of engineering applications, including aircraft, marine, fusion energy, and combustion engines. These panels play important roles in a variety of structural designs, including complex engineering frameworks such as ship structures, submarines, offshore platforms, satellites, and airplanes. In industries that demand high-strength, lightweight structures, ESFG panels are particularly advantageous. NES can be utilized effectively to dissipate vibrational energy in such applications. NES absorbers represent a specific type of vibration absorbers that mitigate undesired vibrations by transferring energy through nonlinear mechanisms. Therefore, their passive design and ability to attenuate high-amplitude vibrations make NES widely valuable across multiple industries. NES play a crucial role in marine structures, particularly in enhancing durability and operational longevity under dynamic hydrodynamic conditions. These devices are pivotal in mitigating the high-energy vibrations that marine structures frequently encounter, thus preventing structural failures and reducing maintenance needs. In this regard, NES can play a crucial role in marine structures, particularly in enhancing durability and operational longevity under hydrodynamic loads. These devices are pivotal in mitigating the high-energy vibrations that marine structures frequently encounter, thus preventing structural failures and reducing maintenance needs.

A significant body of research investigates the dynamic behavior of FG plates and shells to understand structural performance under vibrational loads [1,2,3,4]. In parallel, numerous studies examine the unique vibrational responses of FG panels specifically [5,6,7]. Also, for vibration suppression of these structures, several works have focused on vibration control in FG shells [8,9,10,11] and FG panels [12,13,14].

Numerous studies have concentrated on enhancing structural strength through the incorporation of stiffening elements to boost load-bearing capabilities [15,16,17,18]. At the same time, research has also examined the dynamic and vibrational responses of diverse types of stiffened shells [19,20,21,22]. Additionally, studies have been conducted on the vibration behavior of stiffened panels composed of FG materials under various excitations [23,24,25]. Other works have specifically explored vibration reduction strategies for stiffened FG shells in various configurations [26,27]. In [26], the damped vibration behaviors of stiffened FG (SFG) cylindrical shells embedded within an elastic foundation and optimized using genetic algorithms were addressed. In [27], the nonlinear active vibration control of SFG cylindrical shells exposed to periodic excitation was examined.

In spite of the extensive application of NES absorbers in various industries, previous research has not thoroughly examined their impact on reducing the vibrational behaviors of structures. In this regard, some researchers have utilized these absorbers to suppress vibrations [28,29,30]; meanwhile, others have examined the vibrational behaviors of FG shells equipped with the absorbers [31,32]. It should be explained that the theoretical analysis in [31] is utilized to investigate the vibration behaviors of FG graphene platelet-reinforced composite cylindrical shells with periodically embedded dynamic vibration absorbers. In [32], the study focuses on the impact of the new linear vibration absorber on reducing the vibration of SFG cylindrical shells.

The existing literature has examined the vibrational behaviors of stiffened panels composed of FG materials. However, no study has yet focused on the vibration control of ESFG panels under hydrodynamic loads using a NES absorber. Investigating the vibrational behavior of these structures with an NES absorber is crucial for enhancing sustainability in engineering, particularly in marine structures. In this regard, it should be explained that the primary challenge addressed in this research is the efficient management of vibratory responses in marine structures exposed to dynamic hydrodynamic pressures. These pressures, which vary significantly with marine conditions, can lead to premature material failure and increased maintenance costs. NES offers a promising solution by dissipating vibrational energy effectively, thus enhancing the structural integrity and sustainability of marine infrastructure. This study, therefore, investigates the impact of an NES absorber—comprising linear and nonlinear springs and a damper—on reducing the nonlinear vibrations of ESFG panels subjected to hydrodynamic loads. The material properties of both the panel and the stiffeners are continuously graded through their thickness. Hydrodynamic forces, including lift and drag that vary with velocity, were experimentally determined using Matveev’s equations to accurately simulate real marine conditions for a specific vessel, enabling the calculation of hydrodynamic forces under varying operational conditions, as detailed in Reference [33]. By utilizing these experimentally validated parameters, the present study leverages robust foundational data to assess the effectiveness of NES in realistic marine conditions. Stiffeners are modeled according to Lekhnitskii’s smeared stiffener approach. Analytical methods involve CST with geometric nonlinearity, combined with the Galerkin method for calculations. The integration of CST and the Galerkin method allows for a robust analysis of the dynamic behavior of ESFG panels. By starting from the precise physical assumptions dictated by CST and discretizing via the Galerkin method, we can simulate and predict the nonlinear dynamic response of the panels under various loading conditions. This approach not only ensures the accuracy of our model but also enhances the reliability of the outcomes pertinent to practical applications in marine structures where these types of panels are employed. Subsequently, the P-T method is applied to evaluate the nonlinear vibratory behavior of ESFG panels. This research offers valuable insights for those in engineering and academia who seek to improve sustainability in marine and other structural applications.

2. Materials and Methodology

2.1. ESFG Panels with NES

Figure 1 demonstrates the configuration of the ESFG panels with NES exposed to hydrodynamic loads (hydrodynamic drag ($q_1$) and lift ($q_2$)). The characteristics of ESFG panels, including thickness ($h$), length ($a$), width ($b$), and radii of curvatures ($R$), and the dimensions of the stiffeners, such as $d_i$, $S_i$, and $h_i$ ($i=r,s$) are, respectively, width, spacing and thickness. In this regard, the subscripts $s$ and $r$ correspond to the stringer and ring stiffeners, respectively. Additionally, the NES with linear ($k_l$) and nonlinear ($k_{nl}$) springs, damping coefficient ($C$), and mass ($M$) is positioned at $x=d.$ Also, in this study, the material composition of both the panel and the stiffeners is FG, as illustrated in Figure 2.

Material properties of the FG panel and stiffeners, including mass density ($\rho$) and Young’s modulus ($E$), are determined by considering the material volume fraction.

• FG panels:

$$P_p=P_c+P_{mc} {\left({\displaystyle \frac{2z+h}{2h}}\right)}^{N_p} ;-{\displaystyle \frac{h}{2}}\le z\le {\displaystyle \frac{h}{2}}.$$

(1a)

• FG stiffeners:

$$\begin{gathered} P_s=P_m+P_{cm} {\left({\displaystyle \frac{2z-h}{2h_s}}\right)}^{N_s} ;{\displaystyle \frac{h}{2}}\le z\le {\displaystyle \frac{h}{2}}+h_s; \\ {P}_r=P_m+P_{cm} {\left({\displaystyle \frac{2z-h}{2h_r}}\right)}^{N_r} ;{\displaystyle \frac{h}{2}}\le z\le {\displaystyle \frac{h}{2}}+h_r, \end{gathered}$$

(1b)

where $P_{cm}=P_c-P_m$, $P_{mc}=P_m-P_c$, $P=E,\rho$, and $N_i\left(i=p, s,r\right)\ge 0$ represent the material power-law index for panels, stringer and ring. The subscripts $s$, $p$, $c$, and $m$ denote the stiffeners, panels, ceramic, and metal, respectively.

2.2. Governing Equations

In this research, the CST is utilized to derive the nonlinear governing equations necessary for investigating the nonlinear dynamic behaviors. The strain components and the nonlinear strain-displacement relations are formulated based on the von-Kármán nonlinear geometric assumptions [34]:

$$\begin{gathered} {\epsilon }_x={\epsilon }_x^0-z{ \chi }_x; {\epsilon }_y={\epsilon }_y^0-z{ \chi }_y; {\gamma }_{xy}={\gamma }_{xy}^0-2z{ \chi }_{xy} ; \\ { \chi }_x=w_{,xx}; { \chi }_y=w_{,yy}; {\chi }_{xy}=w_{,xy}; \\ {\epsilon }_x^0=u_{,x}+{\displaystyle \frac{1}{2}}{w}_{,x}^2; {\epsilon }_y^0=v_{,y}-{\displaystyle \frac{w}{R}}+{\displaystyle \frac{1}{2}}{w}_{,y}^2; {\gamma }_{xy}^0=u_{,y}+v_{,x}+w_{,x}{w}_{,y}. \end{gathered}$$

(2)

The compatibility equation for ESFG panels utilizing Equation (2) is presented as follows [33,34]:

$${\epsilon }_{x,yy}^0+{\epsilon }_{y,xx}^0-{\gamma }_{xy,xy}^0=w_{,xy}^2-{\displaystyle \frac{1}{R}}{w}_{,xx}-w_{,xx}{w}_{,yy},$$

(3)

in which ${\gamma }_{xy}^0$ is a shear strain and ${\epsilon }_y^0$, ${\epsilon }_x^0$ are normal strains. Also, $w$ is the displacement component in the $z$-direction. Also, the following equations describe the stress–strain relationships of FG panels and stiffeners:

• FG panels:

$$\left\{\begin{array}{c}{\sigma }_x^p\\ {\sigma }_y^p\\ {\tau }_{xy}^p\end{array}\right\}=\left[\begin{array}{ccc}{\displaystyle \frac{E_p}{1-{\nu }^2}}& {\displaystyle \frac{{\nu E}_p}{1-{\nu }^2}}& 0\\ {\displaystyle \frac{{\nu E}_p}{1-{\nu }^2}}& {\displaystyle \frac{E_p}{1-{\nu }^2}}& 0\\ 0& 0& {\displaystyle \frac{E_p}{2\left(1+\nu \right)}}\end{array}\right]\left\{\begin{array}{c}{\epsilon }_x\\ {\epsilon }_y\\ {\gamma }_{xy}\end{array}\right\}.$$

(4a)

• FG stiffeners:

$$\left\{\begin{array}{c}{\sigma }_x^s\\ {\sigma }_y^r\end{array}\right\}=\left[\begin{array}{cc}{E}_s& 0\\ 0& E_r\end{array}\right]\left\{\begin{array}{c}{\epsilon }_x\\ {\epsilon }_y\end{array}\right\},$$

(4b)

where ${\tau }_{xy},$ ${\sigma }_x$, and ${\sigma }_y$ are the shear and normal stresses. Considering the contribution of stiffeners through the smeared stiffeners technique while neglecting their twisting effect, and integrating the stress–strain relationships along with their corresponding moments across the panel’s thickness, the force and moment resultants for ESFG panels can be formulated [35] as follows:

• Resultant forces:

$$\left\{\begin{array}{c}{N}_x\\ N_y\\ N_{xy}\end{array}\right\}=\left[\begin{array}{ccc}{A}_{11}& A_{12}& 0\\ A_{12}& A_{22}& 0\\ 0& 0& A_{66}\end{array}\right]\left\{\begin{array}{c}{\epsilon }_x^0\\ {\epsilon }_y^0\\ {\gamma }_{xy}^0\end{array}\right\}-\left[\begin{array}{ccc}{B}_{11}& B_{12}& 0\\ B_{12}& B_{22}& 0\\ 0& 0& 2B_{66}\end{array}\right]\left\{\begin{array}{c}{\chi }_x\\ {\chi }_y\\ {\chi }_{xy}\end{array}\right\}.$$

(5a)

• Resultant moments:

$$\left\{\begin{array}{c}{M}_x\\ M_y\\ M_{xy}\end{array}\right\}=\left[\begin{array}{ccc}{B}_{11}& B_{12}& 0\\ B_{12}& B_{22}& 0\\ 0& 0& B_{66}\end{array}\right]\left\{\begin{array}{c}{\epsilon }_x^0\\ {\epsilon }_y^0\\ {\gamma }_{xy}^0\end{array}\right\}-\left[\begin{array}{ccc}{D}_{11}& D_{12}& 0\\ D_{12}& D_{22}& 0\\ 0& 0& 2D_{66}\end{array}\right]\left\{\begin{array}{c}{\chi }_x\\ {\chi }_y\\ {\chi }_{xy}\end{array}\right\},$$

(5b)

where

$$\begin{gathered} A_{11}={\displaystyle \frac{E_1}{1-{\nu }^2}}+{\displaystyle \frac{E_{1s}{d}_s}{S_s}}; A_{12}={\displaystyle \frac{E_1\nu }{1-{\nu }^2}}; A_{22}={\displaystyle \frac{E_1}{1-{\nu }^2}}+{\displaystyle \frac{E_{1r}{d}_r}{S_r}}; A_{66}={\displaystyle \frac{E_1}{2\left(1+\nu \right)}} ; \\ {B}_{11}={\displaystyle \frac{E_2}{1-{\nu }^2}}+{\displaystyle \frac{E_{2s}{d}_s}{S_s}}; B_{12}={\displaystyle \frac{E_2\nu }{1-{\nu }^2}}; B_{22}={\displaystyle \frac{E_2}{1-{\nu }^2}}+{\displaystyle \frac{E_{2r}{d}_r}{S_r}}; B_{66}={\displaystyle \frac{E_2}{2\left(1+\nu \right)}} ; \\ {D}_{11}={\displaystyle \frac{E_3}{1-{\nu }^2}}+{\displaystyle \frac{E_{3s}{d}_s}{S_s}}; D_{22}={\displaystyle \frac{E_3}{1-{\nu }^2}}+{\displaystyle \frac{E_{3r}{d}_r}{S_r}}; D_{12}={\displaystyle \frac{E_3\nu }{1-{\nu }^2}}; D_{66}={\displaystyle \frac{E_3}{2\left(1+\nu \right)}}, \end{gathered}$$

(6)

in which

$$\begin{gathered} E_1={\int }_{-{\displaystyle \frac{h}{2}}}^{{\displaystyle \frac{h}{2}}}{E}_{p}dz; E_2={\int }_{-{\displaystyle \frac{h}{2}}}^{{\displaystyle \frac{h}{2}}}{E}_{p}zdz; E_3={\int }_{-{\displaystyle \frac{h}{2}}}^{{\displaystyle \frac{h}{2}}}{E}_p{z}^{2}dz; \\ {E}_{1i}={\int }_{-{\displaystyle \frac{h}{2}}}^{{\displaystyle \frac{h}{2}}+h_i}{E}_{i}dz; E_{2i}={\int }_{-{\displaystyle \frac{h}{2}}}^{{\displaystyle \frac{h}{2}}+h_i}{E}_{i}zdz; E_{3i}={\int }_{-{\displaystyle \frac{h}{2}}}^{{\displaystyle \frac{h}{2}}+h_i}{E}_i{z}^{2}dz; i=s,r. \end{gathered}$$

(7)

The nonlinear equilibrium equations governing ESFG panels, derived from CST, are presented as follows [33,35]:

$${N_x}_{,x}+{N_{xy}}_{,y}=0;$$

(8a)

$${N_{xy}}_{,x}+{N_y}_{,y}=0;$$

(8b)

$$\begin{gathered} {M_x}_{,xx}+2{M_{xy}}_{,xy}+{M_y}_{,yy}+N_x{w}_{,xx}+2N_{xy}{w}_{,xy}+N_y\left(w_{,yy}+{\displaystyle \frac{1}{R}}\right)+q_1+q_2= \\ {\rho }_1{w}_{,tt}, \end{gathered}$$

(8c)

in which $M_x$, $M_y$, and $M_{xy}$ represent the bending and twisting moments, while $N_x$, $N_y$, and $N_{xy}$ denote the in-plane normal forces and shear forces, respectively. Also, ${\rho }_1$ is the integrated mass density, which can be determined as follows:

$${\rho }_1=\left({\rho }_c+{\displaystyle \frac{{\rho }_{mc}}{N_p+1}}\right) h+\left({\rho }_m+{\displaystyle \frac{{\rho }_{cm}}{N_s+1}}\right) {\displaystyle \frac{d_s{h}_s}{S_s}}+\left({\rho }_m+{\displaystyle \frac{{\rho }_{cm}}{N_r+1}}\right) {\displaystyle \frac{d_r{h}_r}{S_r}}.$$

(9)

In this study, an NES is employed to mitigate the vibrations of the ESFG panels. Accordingly, based on Equation (8), the governing equation that incorporates both the dynamics of the ESFG panel and the NES is formulated as follows:

$${N_x}_{,x}+{N_{xy}}_{,y}=0;$$

(10a)

$${N_{xy}}_{,x}+{N_y}_{,y}=0;$$

(10b)

$$\begin{gathered} \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{M_x}_{,xx}+2{M_{xy}}_{,xy}+{M_y}_{,yy}+N_x{w}_{,xx}+2N_{xy}{w}_{,xy}+N_y\left(w_{,yy}+{\displaystyle \frac{1}{R}}\right)+q_1+q_2 \\ +\left[c\left(w_{,t}-s_{,t}\right)-k_l\left(s-w\right)-k_{nl}{\left(s-w\right)}^3\right]{\delta }_k\left(x-d,y-R\pi /4\right)={\rho }_1{w}_{,tt}; \end{gathered}$$

(10c)

$$M{s}_{,tt}+\left[k_l\left(s-w\right)+c\left(s_{,t}-w_{,t}\right)+k_{nl}{\left(s-w\right)}^3\right]{\delta }_k\left(x-d,y-R\pi /4\right)=0,$$

(10d)

in which $s$ represent the displacement component of NES, the subscript $t$ denotes time, and ${\delta }_k$ represents the Dirac delta function. Regarding Equation (10a) and (10b), the stress function ($\psi$) is also expressed as follows:

$$N_x={\psi }_{,yy}; N_y={\psi }_{,xx}; N_{xy}=-{\psi }_{,xy}.$$

(11)

Based on the studies by Bich et al. [35] and by using the resultant force and moment (Equation (10a) and (10b)), and according to Equations (2), (3), (10c) and (10d), the system’s governing equations are derived as follows:

$$\begin{gathered} {\mathsf{\Sigma }}_{11}{\psi }_{,xxxx}+\left({\mathsf{\Sigma }}_{66}-2{\mathsf{\Sigma }}_{12}\right){\psi }_{,xxyy}+{\mathsf{\Sigma }}_{22}{\psi }_{,yyyy}+{\mathsf{\Sigma }}_{21}^{*}{w}_{,xxxx} \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} +\left({\mathsf{\Sigma }}_{11}^{*}+{\mathsf{\Sigma }}_{22}^{*}-2{\mathsf{\Sigma }}_{66}^{*}\right)w_{,xxyy}+{\mathsf{\Sigma }}_{12}^{*}{w}_{,yyyy}+{\displaystyle \frac{1}{R}}{w}_{,xx}+\left[w_{,xy}^2-w_{,xx}{w}_{,yy}\right]=0; \end{gathered}$$

(12a)

$$\begin{gathered} {\rho }_1{w}_{,tt}+{\mathsf{\Sigma }}_{11}^{**}{w}_{,xxxx}+\left({\mathsf{\Sigma }}_{12}^{**}+{\mathsf{\Sigma }}_{21}^{**}+4{\mathsf{\Sigma }}_{66}^{**}\right)w_{,xxyy}+{\mathsf{\Sigma }}_{22}^{**}{w}_{,yyyy}-{\mathsf{\Sigma }}_{21}^{*}{\psi }_{,xxxx} \\ -\left({\mathsf{\Sigma }}_{11}^{*}+{\mathsf{\Sigma }}_{22}^{*}-2{\mathsf{\Sigma }}_{66}^{*}\right){\psi }_{,xxyy}-{\mathsf{\Sigma }}_{12}^{*}{\psi }_{,yyyy}-{\displaystyle \frac{{\psi }_{,xx}}{R}}-{\psi }_{,yy}{w}_{,xx}+2{\psi }_{,xy}{w}_{,xy} \\ \hspace{1em}\hspace{1em} -{\psi }_{,xx}{w}_{,yy}+\left[c\left(w_{,t}-s_{,t}\right)-k_l\left(s-w\right)-k_{nl}{\left(s-w\right)}^3\right]{\delta }_k\left(x-d,y-{\displaystyle \frac{R\pi }{4}}\right); \end{gathered}$$

(12b)

$$M{s}_{,tt}+\left[k_l\left(s-w\right)+c\left(s_{,t}-w_{,t}\right)+k_{nl}{\left(s-w\right)}^3\right]{\delta }_k\left(x-d,y-R\pi /4\right)=0,$$

(12c)

in which the coefficients ${\mathsf{\Sigma }}_{ij}$, ${\mathsf{\Sigma }}_{ij}^{*}$, ${\mathsf{\Sigma }}_{ij}^{**}$ are as follows:

$$\begin{gathered} {\mathsf{\Sigma }}_{11}={\displaystyle \frac{1}{\Delta }}{A}_{11}; {\mathsf{\Sigma }}_{22}={\displaystyle \frac{1}{\Delta }}{A}_{22}; {\mathsf{\Sigma }}_{12}={\displaystyle \frac{A_{12}}{\Delta }}; {\mathsf{\Sigma }}_{66}={\displaystyle \frac{1}{A_{66}}}; \Delta =A_{11}{A}_{22}-A_{12}^2; \\ {\mathsf{\Sigma }}_{11}^{*}={\mathsf{\Sigma }}_{22}{B}_{11}-{\mathsf{\Sigma }}_{12}{B}_{12}; {\mathsf{\Sigma }}_{22}^{*}={\mathsf{\Sigma }}_{11}{B}_{22}-{\mathsf{\Sigma }}_{12}{B}_{12}; {\mathsf{\Sigma }}_{12}^{*}={\mathsf{\Sigma }}_{22}{B}_{12}-{\mathsf{\Sigma }}_{12}{B}_{22}; \\ {\mathsf{\Sigma }}_{21}^{*}={\mathsf{\Sigma }}_{11}{B}_{12}-{\mathsf{\Sigma }}_{12}{B}_{11}; {\mathsf{\Sigma }}_{66}^{*}={\displaystyle \frac{B_{66}}{A_{66}}}; {\mathsf{\Sigma }}_{11}^{**}=D_{11}-B_{11}{\mathsf{\Sigma }}_{11}^{*}-B_{12}{\mathsf{\Sigma }}_{21}^{*}; \\ {\mathsf{\Sigma }}_{22}^{**}=D_{22}-B_{12}{\mathsf{\Sigma }}_{12}^{*}-B_{22}{\mathsf{\Sigma }}_{22}^{*}; {\mathsf{\Sigma }}_{12}^{**}=D_{12}-B_{11}{\mathsf{\Sigma }}_{12}^{*}-B_{12}{\mathsf{\Sigma }}_{22}^{*}; \\ {\mathsf{\Sigma }}_{21}^{**}=D_{12}-B_{12}{\mathsf{\Sigma }}_{11}^{*}-B_{22}{\mathsf{\Sigma }}_{21}^{*}; {\mathsf{\Sigma }}_{66}^{**}=D_{66}-B_{66}{\mathsf{\Sigma }}_{66}^{*}. \end{gathered}$$

(13)

2.3. Dynamic Galerkin Approach

ESFG panels with simply supported edges and NES are subjected to hydrodynamic loads. The corresponding boundary conditions are specified as follows:

$$\begin{gathered} w=0,{ M}_x=0{, N}_{xy}=0, at x=0; a; \\ w=0,{ M}_y=0{, N}_{xy}=0, at y=0; b. \end{gathered}$$

(14)

In this regard, according to boundary conditions Equation (14), the deflection of panels is as follows:

$$w=W\left(t\right)\sin {\mathrm{\alpha }}_{m}x\sin {\mathrm{\alpha }}_{n}y.$$

(15)

Here, ${\mathrm{\alpha }}_n={\displaystyle \frac{n\mathsf{\pi }}{b}}$ and ${\mathrm{\alpha }}_m={\displaystyle \frac{m\mathsf{\pi }}{a}}$, where $n$ and $m$ represent the respective number of half-waves along the $y$- and $x$-axis. Substituting Equation (15) to Equation (12a) and solving the resultant equation, $\psi$ obtained as follows:

$$\psi ={\psi }_1\sin {\mathrm{\alpha }}_{m}x\sin {\mathrm{\alpha }}_{n}y+{\psi }_2\cos 2{\mathrm{\alpha }}_{m}x+{\psi }_3\cos 2{\mathrm{\alpha }}_{n}y,$$

(16)

in which

$$\begin{gathered} {\psi }_1={\displaystyle \frac{{\alpha }_n^2}{32{\mathsf{\Sigma }}_{11}{\alpha }_m^2}}{W}^2, {\psi }_2={\displaystyle \frac{{\alpha }_m^2}{32{\mathsf{\Sigma }}_{22}{\beta }_n^2}}{W}^2; \\ {\psi }_3={\displaystyle \frac{{\mathsf{\Sigma }}_{21}^{*}{\alpha }_m^4+\left({\mathsf{\Sigma }}_{11}^{*}+{\mathsf{\Sigma }}_{22}^{*}-2{\mathsf{\Sigma }}_{33}^{*}\right){\alpha }_m^2{\beta }_n^2+{\mathsf{\Sigma }}_{12}^{*}{\beta }_n^4-{\alpha }_m^2/R}{{\mathsf{\Sigma }}_{11}{\alpha }_m^4+\left({\mathsf{\Sigma }}_{33}-2{\mathsf{\Sigma }}_{12}\right){\alpha }_m^2{\beta }_n^2+{\mathsf{\Sigma }}_{22}{\beta }_n^4}}W. \end{gathered}$$

(17)

Equations (15) and (16) are substituted into Equation (12b) and (12c), and the Galerkin method is subsequently applied to yield the following results in the presented form:

$$\begin{gathered} \ddot{W}+a_1\dot{W}+a_2\dot{S}+a_{3}W+a_4{k}_{l}W+a_5{k}_{l}S+a_6{k_{nl}W}^3+a_7{k_{nl}S}^3+ \\ {a}_8{k}_{nl}{W}^{2}S+a_9{k}_{nl}{S}^{2}W+a_{10}{W}^3+a_{11}{W}^2+a_{12}{q}_1+a_{13}{q}_2=0; \end{gathered}$$

(18a)

$$\begin{gathered} \ddot{S}+b_1\dot{S}+b_2\dot{W}+b_3{k}_{l}S+b_4{k}_{l}W+b_5{k}_{nl}{S}^3+b_6{k}_{nl}{W}^3+b_7{k}_{nl}{S}^{2}W+ \\ {b}_8{k}_{nl}{W}^{2}S=0, \end{gathered}$$

(18b)

where $a_i$ and $b_j$ $\left(i=\mathrm{1,2},\dots 13;j=\mathrm{1,2},\dots ,8\right)$ are presented in Appendix A. It should be explained that the Galerkin method was chosen for its efficacy in reducing complex problems into simpler, solvable forms, crucial for handling the nonlinear dynamics observed in ESFG panels with NES.

2.4. Free Vibration Analysis

To perform the linear and free vibration analysis of the ESFG panels, the NES and hydrodynamic loads in Equation (18a) are neglected, resulting in:

$$\ddot{W}+a_{3}W=0.$$

(19)

Thus, according to Equation (19), the natural frequency (NF) of the ESFG panels is as follows:

$${\omega }_{mn}=\sqrt{a_3}.$$

(20)

2.5. P-T Method

The P-T method is employed to numerically solve Equation (18) to obtain the nonlinear vibration responses. Notably, the P-T technique ensures higher accuracy and reliability than many other numerical methods, as evidenced in previous studies [36]. The P-T method, which is based on the principles of Taylor series expansion and employs a piecewise constant strategy, is utilized here. Accordingly, an alternative representation of Equation (18) is provided below:

$$\ddot{W}+2{\mu }_1\dot{W}+{\omega }_1^{2}W={\xi }_1\left(W,\mathrm{S},\dot{S},t\right);$$

(21a)

$$\ddot{S}+2{\mu }_2\dot{S}+{\omega }_2^{2}S={\xi }_2\left(\mathrm{S},W,\dot{W},t\right),$$

(21b)

in which

$$\begin{gathered} {\mu }_1=a_1/2; {\mu }_2=b_1/2; {\omega }_1^2=a_3+a_4{k}_l; {\omega }_2^2=b_3{k}_l; \\ {\xi }_1\left(W,\mathrm{S},\dot{S},t\right)=-\left[a_2\dot{S}+a_5{k}_{l}S+a_6{k_{nl}W}^3+a_7{k_{nl}S}^3+a_8{k}_{nl}{W}^{2}S+ \\ {a}_9{k}_{nl}{S}^{2}W+a_{10}{W}^3+a_{11}{W}^2+a_{12}{q}_1+a_{13}{q}_2\right]; \\ {\xi }_2\left(\mathrm{S},W,\dot{W},t\right)=-\left[b_2\dot{W}+b_4{k}_{l}W+b_5{k}_{nl}{S}^3+b_6{k}_{nl}{W}^3+b_7{k}_{nl}{S}^{2}W+b_8{k}_{nl}{W}^{2}S\right]. \end{gathered}$$

(22)

To derive an approximate or numerical solution for Equation (21) over the entire time domain using the Taylor series expansion and piecewise linear arguments, it is necessary to linearize the governing equation. This step is crucial for obtaining an analytical expression within the limited time intervals considered. By applying the P-T method, the functions ${\xi }_i$ ($i=1, 2$) are expanded as a Taylor series to the required level of accuracy within the $i$th time interval, defined by ${\displaystyle \frac{\left[Nt\right]}{N}}\le t<{\displaystyle \frac{\left[Nt\right]+1}{N}}$. The resulting series expansion is given by the following equations:

$$\begin{gathered} \ddot{W}+2{\mu }_1\dot{W}+{\omega }_1^{2}W={{\xi }_1}_{\left[Nt\right]/N}+{{\xi }_1^{\text{'}}}_{\left[Nt\right]/N}\left(t-{\displaystyle \frac{\left[Nt\right]}{N}}\right) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\displaystyle \frac{1}{2!}}{{\xi }_1^{\text{'}\text{'}}}_{\left[Nt\right]/N}{\left(t-{\displaystyle \frac{\left[Nt\right]}{N}}\right)}^2+{\displaystyle \frac{1}{3!}}{{\xi }_1^{\text{'}\text{'}\text{'}}}_{\left[Nt\right]/N}{\left(t-{\displaystyle \frac{\left[Nt\right]}{N}}\right)}^3+\cdots ; \end{gathered}$$

(23a)

$$\begin{gathered} \ddot{S}+2{\mu }_2\dot{S}+{\omega }_2^{2}S={{\xi }_2}_{\left[Nt\right]/N}+{{\xi }_2^{\text{'}}}_{\left[Nt\right]/N}\left(t-{\displaystyle \frac{\left[Nt\right]}{N}}\right); \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} +{\displaystyle \frac{1}{2!}}{{\xi }_2^{\text{'}\text{'}}}_{\left[Nt\right]/N}{\left(t-{\displaystyle \frac{\left[Nt\right]}{N}}\right)}^2+{\displaystyle \frac{1}{3!}}{{\xi }_2^{\text{'}\text{'}\text{'}}}_{\left[Nt\right]/N}{\left(t-{\displaystyle \frac{\left[Nt\right]}{N}}\right)}^3+\dots . \end{gathered}$$

(23b)

To solve the equations using the 4th-order P-T method, the first four terms of Equation (23) for ${\xi }_i$ ($i=1, 2$) are employed. Following this, the corresponding numerical values for the interval ${\displaystyle \frac{\left[Nt\right]}{N}}\le t<{\displaystyle \frac{\left[Nt\right]+1}{N}}$ can be determined through the outlined steps below:

$$\begin{gathered} \hspace{1em}\mathrm{W}\left(t\right)=e^{-{\mu }_1\left(t-{\displaystyle \frac{\left[Nt\right]}{N}}\right)}\left\{{\mathrm{B}}_{11}\cos \left[{\mathsf{\Gamma }}_1\left(t-{\displaystyle \frac{\left[Nt\right]}{N}}\right)\right]+{\mathrm{B}}_{12}\sin \left[{\mathsf{\Gamma }}_1\left(t-{\displaystyle \frac{\left[Nt\right]}{N}}\right)\right]\right\}+{\mathsf{\Pi }}_{11}+ \\ {\mathsf{\Pi }}_{12}\left(t-{\displaystyle \frac{\left[Nt\right]}{N}}\right)+{\mathsf{\Pi }}_{13}{\left(t-{\displaystyle \frac{\left[Nt\right]}{N}}\right)}^2+{\mathsf{\Pi }}_{14}{\left(t-{\displaystyle \frac{\left[Nt\right]}{N}}\right)}^3; \end{gathered}$$

(24a)

$$\begin{gathered} \hspace{1em}\mathrm{S}\left(t\right)=e^{-{\mu }_2\left(t-{\displaystyle \frac{\left[Nt\right]}{N}}\right)}\left\{{\mathrm{B}}_{21}\cos \left[{\mathsf{\Gamma }}_2\left(t-{\displaystyle \frac{\left[Nt\right]}{N}}\right)\right]+{\mathrm{B}}_{22}\sin \left[{\mathsf{\Gamma }}_2\left(t-{\displaystyle \frac{\left[Nt\right]}{N}}\right)\right]\right\}+{\mathsf{\Pi }}_{21}+ \\ {\mathsf{\Pi }}_{22}\left(t-{\displaystyle \frac{\left[Nt\right]}{N}}\right)+{\mathsf{\Pi }}_{23}{\left(t-{\displaystyle \frac{\left[Nt\right]}{N}}\right)}^2+{\mathsf{\Pi }}_{24}{\left(t-{\displaystyle \frac{\left[Nt\right]}{N}}\right)}^3, \end{gathered}$$

(24b)

in which

$$\begin{gathered} {\mathsf{\Gamma }}_i=\sqrt{{\omega }_i^2-{\mu }_i^2}; {\mathsf{\Pi }}_{i4}={\displaystyle \frac{1}{6{\omega }_i^2}}{{\xi }_i^{\text{'}\text{'}\text{'}}}_{\left[Nt\right]/N}; {\mathsf{\Pi }}_{i3}={\displaystyle \frac{1}{{\omega }_i^2}}\left({\displaystyle \frac{1}{2}}{{\xi }_i^{\text{'}\text{'}}}_{\left[Nt\right]/N}-6{\mu }_i{\mathsf{\Pi }}_{i4}\right); \\ {\mathsf{\Pi }}_{i2}={\displaystyle \frac{1}{{\omega }_i^2}}\left[{\displaystyle \frac{1}{2}}{{\xi }_i^{\text{'}}}_{\left[Nt\right]/N}-4{\mu }_i{\mathsf{\Pi }}_{i3}-6{\mathsf{\Pi }}_{i4}\right]; {\mathsf{\Pi }}_{i1}={\displaystyle \frac{1}{{\omega }_{mn}^2}}\left[{{\xi }_1}_{\left[Nt\right]/N}-2{\mu }_i{\mathsf{\Pi }}_{i2}-2{\mathsf{\Pi }}_{i3}\right]; \\ {\mathrm{B}}_{11}=W-{\mathsf{\Pi }}_{11}; {\mathrm{B}}_{12}={\displaystyle \frac{1}{{\mathsf{\Gamma }}_1}}\left(\dot{W}-{\mu }_1{\mathrm{B}}_{11}-{\mathsf{\Pi }}_{12}\right); \\ {\mathrm{B}}_{21}=S-{\mathsf{\Pi }}_{21}; {\mathrm{B}}_{22}={\displaystyle \frac{1}{{\mathsf{\Gamma }}_2}}\left(\dot{S}-{\mu }_2{\mathrm{B}}_{21}-{\mathsf{\Pi }}_{22}\right). \end{gathered}$$

(25)

3. Numerical Results

In this section, first, the result obtained is validated with previous research, and then the impact of stiffeners and the NES absorber on the vibration reduction in FG panels exposed to hydrodynamic loads is investigated. In this contest, it should be explained that while the core of the present research is based on simulations rather than experimental trials, it is crucial to understand that the simulation parameters have been meticulously crafted to closely mimic real-world conditions. Hydrodynamic forces, such as lift and drag, which are critical to our models, were determined using established experimental data from Matveev’s equations, adapted specifically for a ship of known dimensions as detailed in Reference [33]. This approach allows harnessing the reliability of real-world data while exploiting the control and versatility of simulation, providing a solid foundation for analyzing the effectiveness of NES in marine structures under dynamic hydrodynamic loads.

3.1. Validation of the Present Study

To validate the present outcomes, in

Table 1. Comparison of the dimensionless NF ($h/a=0.1; a/b=1; E_c=38\times {10}^{10} \mathrm{N}/{\mathrm{m}}^2; E_m=7\times {10}^{10} \mathrm{N}/{\mathrm{m}}^2; {\rho }_c=3800 \mathrm{k}\mathrm{g}/{\mathrm{m}}^3; {\rho }_m=2796 \mathrm{k}\mathrm{g}/{\mathrm{m}}^3$).

$$\mathit{a}/\mathit{R}$$	$${\mathit{N}}_{\mathit{p}}$$	Present	Alijani et al. [37]	Matsunaga [38]	Chorfi and Houmat [39]
FG plate
0	1	0.0456	0.0456	0.0430	0.0442
	0.5	0.0506	0.0506	0.0492	0.0490
FG cylindrical panel
0.5	1	0.0501	0.0501	0.0485	0.0490
	0.5	0.0553	0.0553	0.0535	0.0540

, the dimensionless NF (${\overline{\omega }}_{mn}={\omega }_{mn}h\sqrt{{\displaystyle \frac{{\rho }_c}{E_c}}}$) obtained in this study are compared with the results reported by Alijani et al. [37], Matsunaga [38], and Houmat [39]. Similarly, in

Table 2. Comparison of the NF (Hz) for panels ($b=0.2286; a=0.2794;\nu =0.33; E=7.2\times {10}^{10}; \rho =2800 \mathrm{k}\mathrm{g}/{\mathrm{m}}^3$).

$$\mathit{R}\left(\mathit{m}\right)$$	$$\mathit{h}\left(\mathit{m}\mathit{m}\right)$$	$$\left(\mathit{m},\mathit{n}\right)$$	Present	Soedel [40]	Shen and Xiang [41]	Nath [42]	Chern and Chao [43]	Bardell and Mead [44]
2.4384	1.2190	(1,1)	162.2	163.7	163.2	163.9	161.4	163.1
1.2192	0.7112	(1,2)	182.0	182.1	181.7	182.2	180.8	181.6

, the NF of the panels are compared with findings from Soedel [40], Shen and Xiang [41], Nath [42], Chern and Chao [43], and Bardell and Mead [44]. Furthermore, Figure 3 illustrates the comparison of the obtained NFs with the results from Szilard [45] and Troitsky [46]. These comparisons demonstrate a strong level of agreement.

3.2. Vibration Behaviors of ESFG Panels

In this sub-section, it is decided to investigate the impact of stiffeners and the NES absorber on the vibration reduction in FG panels exposed to hydrodynamic loads. It should be noted that the NES absorber is placed at the center of the panel. The central placement of the NES absorber on the ESFG panel, under hydrodynamic loads, is strategically advantageous as it leverages the symmetrical properties of the panel and maximizes the damper’s ability to absorb and dissipate vibrational energy. This arrangement significantly reduces the maximum deflection and enhances the structural resilience and service life of the panel in marine environments. As previously mentioned, to simulate real marine environments, hydrodynamic forces such as lift and drag, which vary with velocity, have been determined experimentally using Matveev’s equations for a specific ship (width of 3.4 m, length of 16.4 m and occupied water volume 12,000 kg) [47], as listed in

Table 3. Hydrodynamic lift and drag loads at various velocities [47].

$\mathit{V}\left(\mathbf{m}/\mathbf{s}\right)$	Drag (N)	Lift (N)
12	4416.1	129,529
10.9	5130.6	121,090
10.6	5243.2	117,836
10.4	6165.6	114,204
10.2	5380.6	112,730
10	5550.8	110,537

. Additionally, the structural dimensions and material parameters used to obtain the results are as follows:

$${\rho }_m=2702 \mathrm{Kg}/{\mathrm{m}}^3; {\rho }_c=3800\mathrm{Kg}/{\mathrm{m}}^3; E_m=70 \mathrm{GPa};E_c=380 \mathrm{GPa}.$$

(26)

Also, it should be explained that the Poisson’s ratio for metal and ceramic is assumed to be equal (i.e., ${\nu }_c={\nu }_m=0.3$).

$$\begin{gathered} a=0.75 \mathrm{m}; a/b=1;h=0.002 \mathrm{m}; R/h=250; m=1; n=7; \\ {N}_i=1 \left(i=p,s,r\right); \\ {h}_s=d_s=0.002 \mathrm{m}; \mathrm{N}\mathrm{u}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{r} \mathrm{o}\mathrm{f} \mathrm{s}\mathrm{t}\mathrm{i}\mathrm{f}\mathrm{f}\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{s}=30;V=10.9 \mathrm{m}/\mathrm{s}; \\ M=0.5 kg; d=a/2; C=0.2 \mathrm{N}\mathrm{s}/\mathrm{m}; k_l=300 \mathrm{N}/\mathrm{m}; k_{nl}=5000 \mathrm{N}/{\mathrm{m}}^5. \end{gathered}$$

(27)

Figure 4 illustrates the effect of hydrodynamic lift and drag loads at a velocity of 10.9 m/s on FG panels that lack stiffeners and NES absorbers. The results indicate that the inclusion of hydrodynamic loads leads to a sudden increase in vibration amplitude.

The impact of FG stiffeners and the NES absorber on reducing the vibration of FG panels under hydrodynamic lift and drag loads at a velocity of 10.9 m/s is illustrated in Figure 5. As shown, although the FG stiffeners significantly reduce the amplitude of nonlinear vibrations of the FG panels, the NES absorber has an even greater effect on reducing the vibration amplitude than the FG stiffeners. It should be explained that the increased vibration amplitude in FG panels, as depicted in this figure, results from dynamic hydrodynamic forces, material heterogeneity due to functional grading and stiffening, and geometric nonlinearity. These factors interact to produce complex vibrational responses under varying operational conditions.

Figure 6 illustrates how the NES absorber reduces the vibration of ESFG panels under hydrodynamic lift and drag loads at a velocity of 10.9 m/s. This figure demonstrates how the vibration amplitude can be significantly decreased for the FG panels by taking into account both the NES and FG stiffeners. Thus, using the NES absorber for the ESFG panels can effectively decrease the vibrational energy and consequently leads to increasing the service life of marine structures.

In Figure 7, the effect of the NES absorber on various positions of the ESFG panels on the reducing the vibration of the system is demonstrated. The results show that when the NES absorber is placed near the center of the panel, the vibration amplitude decreases significantly. In fact, the maximum deflection of the ESFG panels under hydrodynamic lift and drag loads at a velocity of 10.9 m/s is substantially reduced by positioning the NES absorber at the center of the panels.

The impact of the material power-law index for panels, stringer and ring ($N=N_p=N_s=N_r$) on the reduction in the vibration of ESFG panels is illustrated in Figure 8. According to this figure, when $N=1$, the vibration amplitude decreases significantly. In fact, when the material properties of both the stiffeners and panels are FG, the maximum deflection of the ESFG panels under hydrodynamic lift and drag loads at a velocity of 10.9 m/s is substantially reduced. Additionally, the outcomes show that when the material properties of the panel and stiffeners approach those of ceramic and metal, respectively, the vibration amplitude is lower than when the material properties of the panel and stiffeners approach those of metal and ceramic, respectively.

The influence of the number of the stiffeners on the reduction in the vibration of ESFG panels is demonstrated in Figure 9. The results reveal that increasing the number of stiffeners leads to decreasing the vibration amplitude of ESFG panels. Consequently, increasing the number of stiffeners effectively decreases vibrational energy.

The effects of the NES absorber on the reduction in the vibration of ESFG panels under hydrodynamic lift and drag loads for various velocities are revealed in Figure 10. The outcomes illustrate that the vibration amplitude can be significantly decreased for the ESFG panels under hydrodynamic lift and drag loads for various velocities by taking into account the NES absorber. Therefore, the present findings show that NES can effectively dissipate vibrational energy of ESFG panels, contributing to the extended service life of marine structures while reducing the need for frequent maintenance.

4. Conclusions

The influence of the NES on the reduction in the nonlinear vibrational behavior of ESFG panels exposed to hydrodynamic lift and drag loads is investigated. Using Lekhnitskii’s smeared stiffener approach, the stiffeners are modeled. To derive the nonlinear governing equations, the CST and Galerkin method are implemented. Finally, the P-T method is employed to determine the nonlinear vibratory behavior of ESFG panels with NES. The findings of this study not only underscore the effectiveness of NES in mitigating vibrational impacts on marine structures but also open avenues for their implementation in real-world marine environments. By integrating NES into the design of marine panels, engineers can significantly enhance the durability and operational lifespan of these structures. Furthermore, this research paves the way for future studies to explore alternative configurations and materials for NES, potentially leading to even more robust and adaptable vibration control solutions. The implications for marine sustainability are profound, offering potential reductions in maintenance costs and improvements in energy efficiency. Below is a summary of the primary findings:

• The FG stiffeners significantly reduce the amplitude of nonlinear vibrations of the FG panels; however, the NES absorber has an even greater effect on reducing the vibration amplitude than the FG stiffeners.
• Using the NES absorber for the ESFG panels under hydrodynamic lift and drag loads for various velocities can effectively decrease the vibrational energy and consequently leads to increasing the service life of marine structures.
• The maximum deflection of the ESFG panels under hydrodynamic lift and drag loads is substantially reduced by positioning the NES absorber at the center of the panels.
• When the material properties of both the stiffeners and panels are FG, the maximum deflection of the ESFG panels is substantially reduced. Additionally, when the material properties of the panel and stiffeners approach those of ceramic and metal, respectively, the vibration amplitude is lower than when the material properties of the panel and stiffeners approach those of metal and ceramic, respectively.
• Increasing the number of stiffeners effectively decreases vibrational energy.

Therefore, the findings of this study show that NES can effectively dissipate the vibrational energy of FG panels, contributing to an extended service life for marine structures and reducing the need for frequent maintenance, especially when both FG stiffeners and NES are used together. Additionally, NES can present a pivotal advancement in marine structural design, offering a passive, yet effective, solution to managing the intense vibratory environments experienced by marine structures. The capability of NES to absorb and dissipate high-energy vibrations significantly enhances the durability and service life of marine infrastructures subjected to dynamic marine loads.

Building on these findings, it would be valuable for future research to explore how NES could be integrated with smart monitoring systems, enabling dynamic adjustments to fluctuating environmental conditions. Furthermore, experimental designs that simulate more complex or severe marine conditions could deepen our understanding of NES’s operational boundaries and capabilities. Breakthroughs in materials science could enhance the efficiency and durability of NES systems, possibly by incorporating new materials with superior energy absorption properties. Practically, NES application could transform the safety and longevity of large vessels, especially in high-energy areas. Adopting this technology could markedly improve energy efficiency and structural resilience. Thus, utilizing NES technology could significantly reduce maintenance costs.