Modal Density Evaluation of a Fluid-Loaded Free-Damping Stiffened Plate

Abstract

An analytical method is developed to evaluate the modal density of a fluid-loaded stiffened plate with a damping layer. The effects of the damping layer, ribs, and fluid load on the structure’s equivalent bending rigidity and surface density are analyzed. The vibration equation is obtained by applying the Hamilton principle, and the modal density is calculated by counting modes in the specific band. The modal density calculation method for both ribbed-type plates and uniform-type plates is verified through numerical simulation. The increase in the number of ribs has made the rib-off frequency at which the effect of the ribs can be neglected become higher, since the wavelength needs to be shorter when the ribbed plate can be treated as a uniform-type plate. The introduction of the damping layer has slightly increased the modal density compared to the uniform plate. In contrast, the introduction of fluid load has dramatically increased the modal density of the corresponding base plate in the low-frequency domain, and the effect of the fluid load can be ignored in the high-frequency domain.

1. Introduction

Based on the characteristics of the noise source, underwater noise from ships is typically classified into mechanical noise, propeller noise, and hydrodynamic noise. For ordinary electro-mechanical propulsion marine vessels, mechanical noise increases slowly or remains essentially unchanged with increasing speed, while hydrodynamic noise intensity often increases rapidly above ten or more knots. Generally, at low speeds, hydrodynamic noise is usually a secondary noise source, while mechanical noise, mainly caused by reciprocating engine operation, is the primary noise source of the underwater noise.

The vibration and noise emitted by ships and underwater vehicles may affect the comfort and efficiency of the crews. Moreover, ship noise pollution may have an adverse effect on the living environment of ocean creatures. Therefore, the precise prediction of ship noise at the design stage is of great value in realizing the reduction of vibration and noise from ships. Ships are primarily composed of plate-type structures, such as stiffened plates and damping plates, and the fluid-loaded boundary condition cannot be ignored for ship parts submerged in water. Therefore, research on the vibration characteristics of a fluid-loaded, free-damping, stiffened plate is of great importance.

The first step in understanding the vibration response of a structure is to investigate its free vibration and characteristic frequency. Recently, Guo et al. [1] used the two-dimensional spectral Tchebyshev (2D-ST) technique to solve the free vibration problem of a concentric stiffened rectangular plate under arbitrary boundary conditions. Wang et al. [2] employed a Nitsche-based non-uniform rational B-splines (NURBS) isogeometric analysis (IGA) method to study the free vibration of a stiffened plate with holes, in which the plate with stiffeners was modeled using the Reissner–Mindlin theory and the Timoshenko beam theory. Moreover, Zhang et al. [3] established and validated an analytical model of the vibroacoustic response and sound transmission characteristics of a ribbed plate coupled with a cavity. Peng et al. [4] and Zhou et al. [5] both studied the free and forced vibration of a composite stiffened plate. In their studies, Peng et al. applied the mixed analytical–numerical method (MA-NM) to investigate the vibration characteristics of a sandwich plate. At the same time, Zhou et al. presented a meshfree method that combined the first-order shear deformation (FSDT) theory for static and dynamic analyses of composite laminated stiffened plates. Moein et al. [6] examined the free vibration characteristics of coupled nested conical shells (CNCSs) made of porous composite materials. In this work, the rule of mixtures is employed to ascertain the equivalent material properties of the hybrid materials, while the Chamis approach is utilized for three-phase materials. The first-order shear deformation theory (FSDT) and Donnell’s theory are utilized to model the conical shells. On this basis, they took the thermal conditions into consideration in their recent work [7].

Except for the structural form of the stiffened plate itself, the treatment method of the boundary condition has also attracted the attention of researchers. Most research is limited to classical boundary conditions, such as simple support or clamping, whereas elastic boundary conditions also exist in engineering applications, which have significant research value. Du et al. [8] proposed a theoretical model of the stiffened plate with multiple dynamic vibration absorbers under different boundary constraints, in which the boundary conditions and the connection relationship between the stiffened plate and dynamic vibration absorbers were implemented through virtual boundary spring stiffness and the introduction of auxiliary sine functions in the displacement function. Zhao et al. [9] employed a Fourier series combined with auxiliary functions to satisfy the arbitrary boundary conditions. In the process of coupling the stiffeners and the plate, Mercer et al. [10] applied the transfer matrix method to predict the natural frequencies and normal modes of a row of skin-stringer panels, allowing for non-uniform stringer spacing and different characteristics of individual stringers. Zhang et al. [11] have simplified the ribs as forces and moments applied to plates and presented a unified integral transform solution for analyzing the free and forced vibration of ribbed rectangular plates with arbitrary classical boundary conditions. However, Xu et al. [12] consider the rigid coupling between the ribs and plate to be inappropriate, as they believe that the stiffeners are often spot-welded or fixed to a plate through screws and rivets. Thus, the effects of elasticity connection should be taken into consideration.

A damping plate can be considered the simplest form of composite plate for its double-layer construction. Commonly used theories [13,14,15] include equivalent single-layer (ESL) theory, Zig-Zag theory, layer-wise (LW) theory, and three-dimensional elasticity theory. The ESL theory contains classical shell (Kirchhoff) theory, first-order shear deformation (Reissner–Mindlin) theory, and higher-order shear deformation theory [16,17]. The classical shell theory is suitable for thin plates, and the other two theories are ideal for thick shells. In fact, most of the plate-type structures in the ship can be considered thin plates. Thus, the ESL theory is usually accurate enough for vibration investigation of ship structures.

Although the vibration of stiffened plates has been extensively studied for decades, most reported investigations have focused on the investigation method of various boundary conditions and coupling patterns between stiffeners and plates. Depending on the plate’s thickness and the required calculation accuracy, multiple theories of the damping plate’s vibration are applied. The research mentioned above primarily focused on the accuracy of the characteristic frequency and response of the ribbed and damping plate itself. This significantly limits the applicability of these methods in engineering practice, for instance, in predicting the noise and vibration of large structures, such as ships. Statistical energy analysis (SEA) is suitable for solving dynamic problems of large, complex systems coupled with mechanical, acoustic, and other subsystems. The precise vibration characteristic and subsystem response are less critical under the statistical framework. Of fundamental importance is the modal density, which represents the number of resonant modes of the subsystem within a frequency band. The modal density is the basis for predicting vibration and noise radiation in large structures, such as ships and underwater vehicles.

Efforts have been made to evaluate the modal density of plate-type ribbed structures. Mikulas [18] has derived the dynamic equilibrium equations from energy principles for eccentrically stiffened flat plates, and the modal density is calculated by counting modes in a specific band. Abderrazak et al. [19,20] investigated the vibroacoustic behavior of stiffened metallic or composite panels under airborne and structure-borne excitations by the modal expansion technique. Dickow et al. [21] established a model of a ribbed plate by applying Fourier sine modes, and the coupling between the plate and ribs is incorporated using Hamilton’s principle. Except for plate-type structures, the modal density calculation methods of unstiffened [22,23,24] and the stiffened cylindrical shell [25,26,27,28,29,30] also attract much attention.

In previous investigations, most researchers have focused on studying plates with ribs, plates with damping layers, or plates with fluid load. In their studies, only a single factor was considered, including ribs, a damping layer, or fluid load. However, all factors should be considered for precisely estimating the ships’ or underwater vehicles’ noise and vibration prediction. The modal density is a crucial parameter in predicting medium and high-frequency vibration and noise. Thus, research on the modal density evaluation of a fluid-loaded, free-damping, stiffened plate is of great importance.

In the present investigation, an estimation method for the modal density of a fluid-loaded, free-damping, stiffened plate is presented and validated using numerical calculation methods. In addition, the effects of the damping layer, ribs, and fluid load on the structure’s modal density were also investigated to better understand the modal density characteristics of fluid-loaded, free-damping, stiffened structures.

2. Theoretical Estimation Model of Modal Density

2.1. Basic Structure Model

From the definition of the modal density, it can be shown that the evaluation of the modal density is strongly related to the solution of the structure’s vibration equation. The modal density can be acquired from the structure’s modal frequencies. The modal frequency is directly determined by the structure’s stiffness and mass, which can be affected by the addition of reinforced ribs, damping layers, and fluid load. For the reasons mentioned above, one of the focuses in this section is to explore the influence of the attachment structures on the overall stiffness and surface density.

The considered model of a fluid-loaded, free-damping, stiffened plate is shown in Figure 1.

As shown in Figure 1, the uniform base steel plate is reinforced by orthogonally arranged ribs, while the damping layer is attached to the opposite side of the base plate. The semi-infinite water domain is in contact with the damping layer, which introduces vibroacoustic coupling between the structure and fluid.

In the following sections, the vibrational modeling method for attaching a damping layer, ribs, and fluid load is sequentially given.

2.2. Modeling of the Attached Damping Layer

The damping plate can be regarded as a double-layer structure, and a neutral plane (where there is no in-plane deformation) exists between the top and bottom surfaces of the double-layer plate. By designating the xOy plane to coincide with the neutral plane of the base layer, the coordinate of the neutral plane of the damping layer in the z-direction can be defined as $z_0$. The schematic of the damping plate is shown in Figure 2, where the thicknesses of the damping layer and the base layer are h_p and h, respectively.

The coordinate $z_0$ can be obtained by letting the integration of stress in the thickness direction be zero:

$$\left({\displaystyle {\int }_{-\frac{h}{2}}^{\frac{h}{2}}{\sigma }_x}dz+{\displaystyle {\int }_{\frac{h}{2}}^{\frac{h}{2}+h_p}{\sigma }_{xp}dz}\right)+\left({\displaystyle {\int }_{-\frac{h}{2}}^{\frac{h}{2}}{\sigma }_y}dz+{\displaystyle {\int }_{\frac{h}{2}}^{\frac{h}{2}+h_p}{\sigma }_{yp}dz}\right)=0$$

(1)

By solving Equation (1), the expression of z₀ can be acquired by

$$z_0={\displaystyle \frac{Q_p\left[h_p\left(h+h_p\right)\right]}{2\left(Q+Q_p\right)}}$$

(2)

where $h$ and $h_p$ are the thicknesses of the base layer and the damping layer, respectively. The expression of $Q$ and $Q_p$ are $Q=E/\left(1-\mu \right)$, $Q_p=E_p/\left(1-{\mu }_p\right)$, respectively. The terms $E$, $\mu$, $E_p$, ${\mu }_p$ refer to Young’s modulus and Poisson’s ratio of the base layer and damping layer, respectively.

The vibration equation of the damping plate can be obtained by applying the Hamilton principle, and can be given as

$$\left(D_e{\nabla }^4-m_e{\omega }^2\right)w=0$$

(3)

where $D_e$ and $m_e$ are the equivalent bending rigidity and the equivalent surface density, respectively. They can be expressed as

$$\left\{\begin{array}{l}{D}_e=D+D_0+D_p\\ m_e=\rho h+{\rho }_p{h}_p\end{array}\right.$$

(4)

where $\rho$ and ${\rho }_p$ are the densities of the base layer and damping layer, respectively; $D$ is the bending rigidity of the base plate. The expressions of $D_0$ and $D_p$ are given as

$$\left\{\begin{array}{l}{D}_0={\displaystyle \frac{Eh{z}_0^2}{1-{\mu }^2}}\\ D_p={\displaystyle \frac{E_p}{3\left(1-{\mu }_p^2\right)}}\left[h_p^3+3h_p^2\left({\displaystyle \frac{h}{2}}-z_0\right)+3h_p{\left({\displaystyle \frac{h}{2}}-z_0\right)}^2\right]\end{array}\right.$$

(5)

It can be concluded from Equation (4) that the damping layer affects both the bending rigidity and the surface density.

2.3. Modeling of the Attached Fluid Load

The second step is to introduce the fluid loading. When the damping plate is attached to the fluid loading, the vibration equation in the time domain can be expressed as:

$$D_e{\nabla }^{4}w+m_e{\displaystyle \frac{{\partial }^{2}w}{\partial t^2}}=-p{|}_{z=0}$$

(6)

where $D_e$, $m_e$, and $p$ are the equivalent bending rigidity, equivalent surface density, and acoustic pressure, respectively. The acoustic pressure satisfies the wave equation,

$${\nabla }^{2}p-{\displaystyle \frac{1}{{c_0}^2}}{\displaystyle \frac{{\partial }^{2}p}{\partial t^2}}=0$$

(7)

The general solution of the displacement of the damping plate and acoustic pressure can be expressed as

$$\begin{array}{l}w=W{\mathrm{e}}^{-\mathrm{j}{k}_{\mathrm{x}}x}{\mathrm{e}}^{-\mathrm{j}{k}_{y}y}{e}^{j\omega t}\\ p=P{\mathrm{e}}^{-\mathrm{j}{k}_{\mathrm{x}}x}{\mathrm{e}}^{-\mathrm{j}{k}_{y}y}{\mathrm{e}}^{-\mathrm{j}{k}_{z}z}{e}^{j\omega t}\end{array}$$

(8)

where $W$ and $P$ are the displacement magnitude of the damping plate and the acoustic pressure. The vibration equation of the damping plate in the frequency domain can be obtained by substituting Equation (8) into Equation (6),

$$D_e{k}_p{}^{4}W-{\omega }^2{m}_{e}W=-P$$

(9)

where $k_p$ refers to the flexural wavenumber of the damping plate. Continuous boundary conditions of the structure and the fluid domain should be introduced to solve the vibration equation, which can be expressed as

$$P={\displaystyle \frac{j{\omega }^2{\rho }_{0}W}{k_z}}$$

(10)

where ${\rho }_0$ is the density of the fluid, and $k_z$ is the acoustic wave number in the $+z$ direction. The relationship between $k_z$ and $k_p$ can be expressed as

$$k_z=\pm \sqrt{k^2-k_p{}^2}$$

(11)

where $k$ is the acoustic wavenumber. As $k>k_p$, $k_z$ is a real number, and an acoustic wave is radiated by the damping plate. While $k<k_p$, $k_z$ is an imaginary number, and no acoustic radiation exists. In this case, the fluid load can be regarded as added mass. Here, we only consider the situation of $k<k_p$, and $k_z$ can be expressed as

$$k_z=-j\sqrt{k_P^2-k^2}$$

(12)

The negative sign in Equation (12) means the fluid load is coupled with the damping plate in the $+z$ direction. The vibration characteristic equation can be acquired by substituting Equation (10) and Equation (12) into Equation (9), which can be given as

$$D_e{k}_P^4-m_e^{\prime }{\omega }^2=0$$

(13)

where $m_e^{\prime }$ is the equivalent surface density of the damping plate under fluid loading, which can be given as

$$m_e^{\prime }=m_e+{\displaystyle \frac{{\rho }_0}{\sqrt{k_p^2-k^2}}}$$

(14)

It can be concluded from Equation (14) that considering fluid loading is equivalent to introducing a frequency-dependent additional mass and has no effect on the bending rigidity. The relationship between $k_p$ and $\omega$ can be obtained by substituting Equation (14) into Equation (13), which can be given as

$$D_e{k}_P^4-m_e{\omega }^2={\displaystyle \frac{{\rho }_0{\omega }^2}{\sqrt{k_p^2-k^2}}}$$

(15)

By squaring both sides of the equation and moving the term $k_p^2-k^2$ to the left of the equation, Equation (15) can be rewritten as

$$\left(D_e{}^2{k}_P^8-2D_e{k}_P^4{m}_e{\omega }^2+m_e{}^2{\omega }^4\right)\left(k_p^2-k^2\right)-{\rho }_0{}^2{\omega }^4=0$$

(16)

Equation (16) can be transformed into a polynomial form,

$$D_e^2{k}_p^{10}-{\displaystyle \frac{D_e^2}{c_0{}^2}}{k}_p^8{\omega }^2-2D_e{m}_e{k}_p^6{\omega }^2+2{\displaystyle \frac{D_e}{c_0{}^2}}{m}_e{k}_p^4{\omega }^4+m_e^2{k}_p^2{\omega }^4-{\displaystyle \frac{1}{c_0{}^2}}{m}_e^2{\omega }^6-{\omega }^4{\rho }_0^2=0$$

(17)

Thus, from Equation (17), the flexural wavenumber $k_p$ of the fluid-loaded damping plate can be determined at a given circular frequency $\omega$. Then, the added mass introduced by the fluid load can be obtained using Equation (14).

2.4. Modeling of the Reinforced Ribs

The third step is to introduce the reinforced ribs. The effects of ribs on equivalent surface density $m_e$ and equivalent bending rigidity in the x-direction and y-direction can be determined by comparing the flexural wavelength (${\lambda }_B$) and the spacing of ribs in two directions $\left(S_x,S_y\right)$, as presented in the following four cases.

For case 1, where ${\lambda }_B>\max \left(S_x,S_y\right)$, the mass and stiffness of the ribs can be smeared on the base plate, and the equivalent surface density can be expressed as $m_e^{″}=m_e^{\prime }+{\rho }_x{\displaystyle \frac{A_x}{S_x}}+{\rho }_y{\displaystyle \frac{A_y}{S_y}}$. The corresponding equivalent bending rigidity in the x-direction and y-direction can be described as $D_x=D_e+{\displaystyle \frac{E_x{I}_x}{S_x}}+{\displaystyle \frac{G_x{J}_x}{S_x}}$ and $D_y=D_e+{\displaystyle \frac{E_y{I}_y}{S_y}}+{\displaystyle \frac{G_y{J}_y}{S_y}}$, respectively.

For case 2, where $S_x<{\lambda }_B<S_y$, the mass and stiffness can be smeared on the base plate only in the x-direction. Thus, $m_e^{″}=m_e^{\prime }+{\rho }_x{\displaystyle \frac{A_x}{S_x}}$, $D_x=D_e+{\displaystyle \frac{E_x{I}_x}{S_x}}+{\displaystyle \frac{G_x{J}_x}{S_x}}$, and $D_y=D_e$.

For case 3, where $S_y<{\lambda }_B<S_x$, the mass and stiffness can be smeared on the base plate only in the y-direction. Thus, $m_e^{″}=m_e^{\prime }+{\rho }_y{\displaystyle \frac{A_y}{S_y}}$, $D_x=D_e$, and $D_y=D_e+{\displaystyle \frac{E_y{I}_y}{S_y}}+{\displaystyle \frac{G_y{J}_y}{S_y}}$.

For case 4, where ${\lambda }_B<\min \left(S_x,S_y\right)$, the effects of ribs on the stiffness and surface density of the base plate are ignored, indicating that $m_e^{″}=m_e^{\prime }$, $D_x=D_e$, and $D_y=D_e$.

In all the cases mentioned above, $A_x$, $E_x$, $G_x$, and $J_x$ represent the cross-sectional area, Young’s modulus, shear modulus, and torsion constant of the ribs, respectively. When the subscript is ‘y’, it denotes ribs in the y direction. The terms $D_e$ and $m_e^{\prime }$ are the equivalent bending rigidity and the equivalent surface density of the fluid-loaded damping plate, respectively. Thus, the expressions for the equivalent parameters can be found in Equations (4), (5), and (14), which will not be repeated here.

2.5. Modal Density Evaluation

Since the equivalent bending rigidity and the equivalent surface density are determined, the last step is to calculate the modal frequency. The vibration equation of ribbed plates can be obtained by applying the principle of minimum potential energy ($\delta U=0$). Therefore, the modal frequency of the system can be calculated by [18]:

$${\omega }_{mn}={\pi }^2\sqrt{{\displaystyle \frac{1}{m_e^{″}}}\left(B_F+B_M\right)}$$

(18)

where $B_F$ and $B_M$ are the structure’s bending rigidity and membrane rigidity, respectively. They can be expressed as

$$B_F={\displaystyle \frac{D_x{m}^4}{L_x{}^4}}+{\displaystyle \frac{2D_e{m}^2{n}^2}{L_x{}^2{L}_y{}^2}}+{\displaystyle \frac{D_y{n}^4}{L_y{}^2}}$$

(19)

$$B_M={\displaystyle \frac{12\left(1-{\mu }^2\right)D_e{m}^4}{L_x{}^4}}{\displaystyle \frac{\overline{S}{\left(1+{\beta }^2\right)}^2{\left(\frac{h_x}{h_t}\right)}^2+\overline{R}{\beta }^4{\left(1+{\beta }^2\right)}^2{\left(\frac{h_y}{h_t}\right)}^2+\overline{S}\overline{R}C}{{\left(1+{\beta }^2\right)}^2+2{\beta }^2\left(1+\mu \right)\left(\overline{S}+\overline{R}\right)+\left(1-{\mu }^2\right)\left(\overline{S}+{\beta }^4\overline{R}+2{\beta }^2\overline{S}\overline{R}\left(1+\mu \right)\right)}}$$

(20)

where $\beta ={\displaystyle \frac{n{L}_x}{m{L}_y}}$, $\overline{S}={\displaystyle \frac{E_x{A}_x}{E{h}_t{S}_x}}$, $\overline{R}={\displaystyle \frac{E_y{A}_y}{E{h}_t{S}_y}}$ are nondimensional parameters; $h_x$ and $h_y$ are the distances from the middle surface of the plate to the centroid of ribs in the x-direction and y-direction, respectively; $h_t$ is the total thickness of the damping plate; $D_e$ is the bending rigidity of the fluid-loaded damping plate, and $C$ is the coupling term of ribs. The term $C$ can be given by

$$C={\beta }^2\left[{\beta }^2\left(1-{\mu }^2\right)+2\left(1+\mu \right)\right]{\left({\displaystyle \frac{h_x}{h_t}}\right)}^2+2{\beta }^4{\left(1+\mu \right)}^2\left({\displaystyle \frac{h_x}{h_t}}\right)\left({\displaystyle \frac{h_y}{h_t}}\right)+{\beta }^4\left[1-{\mu }^2+2{\beta }^2\left(1+\mu \right)\right]{\left({\displaystyle \frac{h_y}{h_t}}\right)}^2$$

(21)

In Equation (20), the membrane rigidity $B_M$ is present because of eccentric stiffening. It is closely related to the characteristic of ribs, which can be neglected when $h_x$ and $h_y$ are relatively small compared to $h_t$. While $B_M$ must be considered for the situation that the height of the ribs can not be ignored.

The number of modal frequencies that lie in the frequency range of interest will give the mode count $N_F\left(\omega \right)$, and the modal density can be given by

$$n_F\left(f_c\right)={\displaystyle \frac{N}{f_H-f_L}}$$

(22)

where $f_L$, $f_c$, and $f_H$ are the lower, center, and upper frequencies of the frequency band of interest, respectively.

3. Results and Discussion

3.1. Illustrative Example and Validation

The modal density of the fluid-loaded, free-damping, stiffened plate can be obtained by applying the theoretical model presented in Section 2. The length and width of the base layer and damping layer are 4.7124 m and 2 m, respectively. The thicknesses of the base layer and damping layer are 0.008 m and 0.02 m, respectively. The width and height of the ribs’ section are 0.016 m and 0.04 m, respectively. The spacings between ribs in the x and y directions are 0.4712 m and 0.016 m, respectively. The materials of the base layer and ribs are both steel, while the damping layer is made of rubber, with the Young’s modulus of 1 × 10⁸ Pa, density of 1100 kg/m³, and Poisson’s ratio of 0.495. The geometric and material parameters of the fluid-loaded free-damping stiffened plate are listed in

Table 1. Geometric and material parameters of the fluid-loaded free-damping stiffened plate.

	Quantity	Value	Unit
Base layer	Length (Lx)	4.7124	m
	Width (Ly)	2.0000	m
	Thickness (h)	0.0080	m
	Young’s modulus (E)	2.1 × 1011	Pa
	Density (ρ)	7800	kg/m3
	Poisson’s ratio (μ)	0.3125	/
Ribs	Width (br)	0.0160	m
	Height (hr)	0.0400	m
	Young’s modulus (Er)	2.1 × 1011	Pa
	Density (ρr)	7800	kg/m3
	Poisson’s ratio (μr)	0.3125	/
	Spacing between ribs in the x direction (Sx)	0.4712	m
	Spacing between ribs in the y direction (Sy)	0.0160	m
Damping layer	Young’s modulus (Ep)	1 × 108	Pa
	Density (ρp)	1100	kg/m3
	Poisson’s ratio (μp)	0.495	/
Fluid	Density (ρ0)	1000	kg/m3
	Sound velocity (c)	1500	m/s

.

The modal density calculated by the theoretical model is validated by the finite element method (FEM) below 1 kHz, which is shown in Figure 3a. Six meshes are needed in one wavelength to ensure the accuracy of the simulation. Considering the computer’s performance, the maximum frequency is set to 1 kHz. The center frequency of a 1/3-octave band is selected as a reference value when searching for the characteristic frequency. The results calculated by the present model are also compared with the simulation results from the SEA software VA One below 100 kHz.

As the FEM method considers most of the structure’s modeling details, it has the highest evaluation accuracy. However, it costs a significant amount of computing resources. Thus, it is unsuitable to evaluate large-scale structures in the high or middle frequency range. In this research, the calculation frequency of the FEM model is below 1 kHz to help assess the calculation accuracy of the present theoretical model. As shown in Figure 3a, from 10 Hz to 1 kHz, all the modal density results estimated by VA One, FEM, and the present theoretical model have a certain consistency in the variation trend with frequency. Specifically, compared with the FEM model, the theoretical model has a great consistency below 31.5 Hz and above 125 Hz, and has a macroscopic discrepancy between 40 Hz and 100 Hz. This discrepancy may be caused by the simplification in modeling the reinforced ribs for improving computational efficiency. Compared with the FEM model, the VA One model has a great consistency below 80 Hz, while it exhibits a rather severe fluctuation from 100 Hz to 1 kHz. In this frequency range, the present theoretical model performs with better evaluation accuracy.

Restricted by computing resources, the modal density results of the FEM model above 1 kHz are not calculated. The corresponding results from 1 kHz to 100 kHz by both the VA One method and the present theoretical method are given in Figure 3b for comparison purposes. As shown in the figure, the model density curve obtained by VA One still exhibits severe fluctuations, which fall short of meeting engineering expectations. Since the calculating process of VA One is not open source, it is not applicable to analyze the fluctuation phenomenon of SEA. However, the theoretical result tends to be gentler with the increase in frequency, and the value of the theoretical results is close to the peak value of the curve given by VA One. In the high-frequency domain, the wavelength of the ribbed plate is shorter than the spacing between ribs; therefore, the influence of the ribs on the base plate can be neglected (can be found in Section 2.4, case 4). Since the modal density of a plate is a constant, the modal density of the ribbed plate should be in convergence with the corresponding plate, and a smooth trend in the high-frequency domain is expected, which is shown in the results by the theoretical model.

The modal overlap factor is a dimensionless quantity that gives an indication of the number of modes that contribute to the response when a subsystem is excited at a single discrete frequency. It provides an indication of when a statistical description of the subsystem’s dynamics is appropriate. The expression of the modal overlap factor is given as follows,

$$M_e={\displaystyle \frac{\pi }{2}}{\omega }_c\eta n$$

(23)

where $\eta$ is the damping loss factor, $n$ is the modal density, ${\omega }_c$ is the center frequency of the band. In this case, the value of the damping loss factor is $\eta =0.01$. The results of the modal overlap factor of the fluid-loaded free-damping stiffened plate calculated by the theoretical model and VA One are shown in Figure 4.

When the modal overlap factor is larger than four, the response is no longer dominated by distinct modal resonances and is typically a relatively smooth function of frequency, which is the definition of the Schroeder frequency. It can be concluded from Figure 4 that the Schroeder frequency is approximately 1 kHz; therefore, the modal density curve may probably show a smooth trend above 1 kHz, which is consistent with the result of the theoretical model shown in Figure 3.

3.2. Effect of Ribs on Modal Density

The effects of ribs on the modal density of the fluid-loaded, free-damping, stiffened plate are studied to further understand the modal density properties of this type of plate.

The ribs will introduce increased equivalent surface density and bending rigidity, as mentioned in Section 2. This is determined by the magnitude relationship between the wavelength and the spacing of the ribs in two directions. In the low-frequency domain, the wavelength is larger than the spacing of the ribs in both directions, and all the ribs are smeared on the plate. In the middle-frequency domain, the wavelength is only larger than the spacing of the ribs in one of the x or y directions. Therefore, the ribs along the corresponding direction can be smeared on the plate, while the ribs along the other direction can be neglected. Similarly, the effects of ribs can be ignored in the high-frequency domain since the wavelength is shorter than the spacing of the ribs in both directions.

Unlike the plate reinforced by ribs, the wavenumber domain integral method is used to calculate the modal density of the uniform plate. It can be concluded from Section 2 that the introduction of a damping layer has changed the equivalent bending rigidity and surface density of the original uniform plate in certain expressions. However, the complexity has restricted the application of the wavenumber domain integral method in acquiring equivalent parameters of the ribbed plate. When no ribs exist, the wavenumber domain integral method is introduced as follows.

The uniform plates, damping plates, fluid-loaded plates, and fluid-loaded damping plates are called uniform-type plates in this article, whose modal density can be calculated by the dispersion curve, which can be acquired by the vibration equation that

$$\left(D_e{\nabla }^4-m_e{\omega }^2\right)w=0$$

(24)

where $D_e$ and $m_e$ are equivalent bending rigidity and equivalent surface density, respectively. The expressions for the equivalent parameters can be found in Equations (4) and (14).

The dispersion relationship can be obtained from Equation (24) with the following expression:

$${\omega }^2={\displaystyle \frac{D_e}{m_e}}{\left(k_x^2+k_y^2\right)}^2={\displaystyle \frac{D_e}{m_e}}{k}_B{}^4$$

(25)

where $k_B$ is the flexural wave number, while $k_x=m\pi /L_x$ and $k_y=n\pi /L_y$ are components of $k_B$ in the x and y direction, respectively. The dispersion curve obtained by Equation (25) is shown in Figure 5.

In Figure 5, the shadow region contains all modes below the frequency corresponding to the flexural wavenumber $k_B$, whose area can be calculated by a double integral. Furthermore, the mode number of the flexural wave in the frequency band can be expressed as:

$$N\left(k_B\right)={\displaystyle \frac{A}{\Delta k_x\Delta k_y}}={\displaystyle \frac{\pi k_B{}^2}{4}}\cdot {\displaystyle \frac{L_x{L}_y}{{\pi }^2}}$$

(26)

where A is the area of the shadow region. Modal density is then calculated as the mode number in the unit frequency band, which can be given as:

$$n\left(k_B\right)={\displaystyle \frac{\mathrm{d}N\left(k_B\right)}{\mathrm{d}{k}_B}}={\displaystyle \frac{L_x{L}_y{k}_B}{2\pi }}$$

(27)

$$n\left(\omega \right)=n\left(k_B\right){\displaystyle \frac{\mathrm{d}{k}_B}{\mathrm{d}\omega }}={\displaystyle \frac{L_x{L}_y}{4\pi }}\sqrt{{\displaystyle \frac{m_e}{D_e}}}$$

(28)

where $n\left(k_B\right)$ and $n\left(\omega \right)$ are the modal densities in the wavenumber and frequency domains, respectively. The modal density of the fluid-loaded free-damping plate, as predicted by the theoretical model, is verified in Figure 6.

Figure 6 illustrates excellent consistency between the theoretical model and VA One in calculating the modal density of the fluid-loaded free-damping plate. Therefore, the correctness of the theoretical model in estimating the modal density of a fluid-loaded damping plate without ribs is verified. All the comparison results in the following content are produced using the present theoretical method.

The effect of the ribs on the fluid-loaded damping plate is shown in Figure 7.

Figure 7 shows that the effect of ribs is mainly in the low-frequency domain. The introduction of the ribs has increased the modal frequency and made the modes more distinct in specific frequency bands in the low-frequency domain, resulting in a reduction in modal density. In the high-frequency domain, because the wavelength is pretty small compared to the spacing between the ribs in the x and y directions, the effect of the ribs can be ignored. The number of ribs also affects the modal density of the fluid-loaded free-damping ribbed plate, as shown in Figure 8.

Let the frequency at which the modal density curve meets its maximum value and remains constant be $f_0$. The modal density of the fluid-loaded damping plate tends to be the same as that of the uniform plate as $f>f_0$. It can be concluded from Figure 8 that $f_0$ is positively correlated with the number of ribs, which means the corresponding $f_0$ increases when the number of ribs increases. As mentioned in Section 2, the effect of ribs can be neglected when the wavelength of the corresponding modal frequency is shorter than the spacing between ribs in both directions. The rib-off frequency $f_0$ is the dividing point in the frequency domain that divides the ribbed plate and the uniform plate, which means the ribbed plate can be treated as a uniform-type plate when $f>f_0$. Therefore, when the number of ribs increases, the wavelength needs to be shorter when the ribbed plate can be treated as a uniform-type plate, which means the rib-off frequency $f_0$ needs to be higher.

The characteristic of the calculation method for modal density determines the computing cost. The wavenumber domain integral method for uniform-type plates requires fewer computing resources than the discrete mode counting method for ribbed plates. Therefore, we can apply the wavenumber domain integral method when the calculation frequency is higher than $f_0$. In conclusion, determining $f_0$ can help calculate modal density in the high-frequency domain.

3.3. Effects of the Damping Layer and Fluid Load on Modal Density

The dispersion curves of the uniform-type plate acquired by Equation (25) at 5 kHz are compared in Figure 9. The parameters of the damping layer and fluid load can be found in Table 1.

Figure 9 shows that the dispersion curve of the uniform plate varies when different boundary conditions are applied. The radius of the 1/4 circle is the flexural wave number, which is only related to the ratio of $D_e$ to $m_e$, as shown in Equation (25). By letting $\alpha =m_e/D_e$, the values of $\alpha$, $m_e$, and $D_e$ are listed in

Table 2. Equivalent bending rigidity, surface density, and their ratio of uniform-type plate (f = 5 kHz).

Type of Plate	${\mathit{m}}_{\mathit{e}}$ (kg·m−2)	${\mathit{D}}_{\mathit{e}}$ (N·m)	$\mathit{\alpha }$ (kg·N−1·m−3)
Uniform plate	62.4	9.85 × 103	6.3 × 10−3
Damping plate	84.4	1.05 × 104	8.1 × 10−3
Fluid-loaded plate	87.5	9.85 × 103	8.9 × 10−3
Fluid-loaded damping plate	109.8	1.05 × 104	10.5 × 10−3

.

The wavenumber $k_B$ and the modal density $n$ are both positively correlated with α. The wavenumber $k_B$ is proportional to ${\alpha }^{0.25}$, while the modal density $n$ is proportional to ${\alpha }^{0.5}$. The effects of the damping layer and surrounding fluid on the modal density of uniform-type plates are illustrated in Figure 10.

As shown in Figure 10, introducing the damping layer slightly increased the modal density compared to the uniform plate, and the curve exhibits no frequency dependence. However, the inclusion of the surrounding fluid has dramatically increased the modal density of the corresponding base plate (i.e., the fluid-loaded uniform plate corresponds to the uniform plate) in the low-frequency domain. At the same time, the two curves tend to be identical in the high-frequency domain. This is because the frequency-dependent additional mass caused by the surrounding fluid has significantly reduced the modal frequency of structures in the low-frequency domain. Consequently, some modes have entered the frequency band that previously had no modes or only a few modes, resulting in an increased modal density. The additional mass can be ignored in the high-frequency domain, because the relationship between the wavenumber of the plate $k_p$ and the acoustic wavenumber $k$ is $k>k_p$. In this case, the plate can generate acoustic radiation, and the fluid load will not be considered as added mass.

The frequency-dependent additional mass can be obtained by Equation (14), and the expression of the effective density of the water-loaded plate is

$${\rho }_e={\rho }_s+{\displaystyle \frac{{\rho }_0}{h}}{\displaystyle \frac{1}{\sqrt{k_P^2-k^2}}}$$

(29)

where ${\rho }_0$ is the density of the surrounding fluid medium. For the water-loaded plate, ${\rho }_0=1000 \mathrm{kg}/{\mathrm{m}}^3$; for the air-loaded plate, ${\rho }_0=1.21 \mathrm{kg}/{\mathrm{m}}^3$.The normalized effective density is defined as ${\rho }_{\mathrm{ne}}={\rho }_{\mathrm{e}}/{\rho }_{\mathrm{s}}$. Therefore, ${\rho }_{ne}$ of the water-loaded plate and the air-loaded plate as a function of frequency are given in Figure 11.

It can be seen from Figure 11 that the ${\rho }_{\mathrm{ne}}$ of the water-loaded plate is larger than that of the air-loaded plate. However, as the frequency increases, the two circumstances gradually converge to the unit value, which is the normalized effective density of the air-loaded plate. For the air-loaded situation, ${\rho }_0=1.21 \mathrm{kg}/{\mathrm{m}}^3$, the added mass can be neglected, and ${\rho }_{ne}$ is close to unity. For the water-loaded situation, ${\rho }_0=1000 \mathrm{kg}/{\mathrm{m}}^3$, the added mass should be considered. With the increase in the frequency, the term $\sqrt{k_P^2-k^2}$ shows an increasing trend, which causes a decreasing trend of ${\rho }_{ne}$. When $\sqrt{k_P^2-k^2}$ approachs infinity, the normalized effective density ${\rho }_{ne}$ will converge to unity. Therefore, the effect of fluid load on the vibration characteristics of the plate can be ignored beyond 10 kHz, and the resulting calculation error is negligible.

4. Conclusions

A theoretical model for the modal density of a fluid-loaded, free-damping, stiffened plate was established based on the Hamilton principle and the fluid-structure coupling boundary condition. It can be concluded from the vibration equation of a fluid-loaded, free-damping, stiffened plate that the introduction of ribs and a damping layer leads to an increase in equivalent bending rigidity and surface density. At the same time, the fluid load is treated as a frequency-dependent additional mass in the low-frequency domain, and this additional mass can be ignored in the high-frequency domain. The accuracy of the theoretical model is verified by FEM in the low-frequency domain and VA One in the high-frequency domain, respectively.

Moreover, the introduction of ribs has increased the modal frequency, making the modes more distinct in specific frequency bands within the low-frequency domain and resulting in a reduction in modal density. The increase in the number of ribs has made the rib-off frequency, at which the effect of the ribs can be neglected, higher, as the wavelength needs to be shorter when the ribbed plate can be treated as a uniform plate. The modal density of the uniform-type plate can be calculated using the wavenumber domain integral method when the ribs are absent. The effect of the damping layer and the fluid load is analyzed based on the theoretical model. In conclusion, the introduction of the damping layer has resulted in a slight increase in modal density compared to the uniform plate. In contrast, the introduction of the fluid load has significantly increased the modal density of the corresponding base plate in the low-frequency domain, and the effect of the fluid load can be neglected in the high-frequency domain.

The theoretical model of the fluid-loaded, free-damping stiffened plate enables the prediction of vibration and sound radiation in the high-frequency domain for structures such as ships and underwater vehicles using the SEA method, which has excellent engineering application prospects.