1 Introduction

Longitudinally moving continuum systems have attracted much interest due to their wide application in modern industry. Examples of longitudinally moving continuum systems include paper webs during production, extrusion processes, deployment of appendages in space, and continuous hot-dip galvanizing process. To better understand the vibration behavior of such systems, proper modeling and dynamic analysis of the systems are essential. There are two formulations for longitudinally moving continuums, namely, the one-dimensional model and the two-dimensional model.

Beams, belts, and strings are the typical one-dimensional model and widely used in existing references. Nonlinear dynamics and bifurcations for simply supported moving beams were examined by Pellicano and Vestroni^[1], who considered the dynamics of the system subjected to an axial transport of mass. By using the Maxwell and Kelvin-Voigt models, Marynowski^[2] studied the instability and vibration behaviors of an travelling viscoelastic beam. A literature review of work on various aspects of the dynamics of longitudinally moving strings in vacuum has been given by Chen^[3]. By using the method of averaging, Yang and Chen^[4] examined the instability of a longitudinally translating beam with a time-dependent speed. An and Su^[5] calculated the vibration response of a longitudinally traveling beam, where the Timoshenko model and integral transform method were adopted. Yan et al.^[6] analyzed the nonlinear vibration of a longitudinally translating Timoshenko beam. The time-dependent speed and viscoelasticity were considered in the study. Zhang and Song^[7] and Zhang et al.^[8] investigated high-dimensional dynamics, bifurcation, and chaos of a viscoelastic travelling belt. The Galerkin method and perturbation method were utilized to solve differential equations of motion. The parametric vibration and stability of an longitudinally accelerating string guided by a partial nonlinear elastic foundation were analytically investigated by Ghayesh^[9]. Yang et al.^[10] developed a discretized model for the mode analysis of longitudinal moving continuum, which showed good agreement with continuous model. Chen and Ding^[11] and Chen and Wang^[12] investigated the transverse response and stability of a longitudinally moving viscoelastic beam with constant and time-dependent speed, respectively. By using the Galerkin method, Ding and Chen^[13] studied numerically the natural frequencies of planar vibration of longitudinally moving beams in the supercritical ranges. The convergence of Galerkin method on dynamic response of beams on a nonlinear foundation was discussed by Ding et al.^[14]. Chang et al.^[15] considered a travelling Rayleigh beam and delved into its vibrational characteristics and instability. They used the finite element method and Hamilton's principle to derive the governing equation. In the subcritical speed regime, Ghayesh et al.^[16] studied the nonlinear vibration characteristics of an longitudinally accelerating beam. Considering a travelling steel strip, Li et al.^[17] investigated the dynamic responses of the steel strip and then vibration control of the system was carried out. By using the Fourier differential quadrature method, Zhang et al.^[18] probed into the transverse nonlinear vibrations of an axially accelerating viscoelastic beam. The nonlinear dynamics of a deploying-and-retreating wing was studied by Zhang et al.^[19]. The wing was treated as a cantilever beam travelling along the axial direction in the paper. Recently, Ding and Zu^[20] examined steady state responses of an axially moving viscoelastic beam in pulley-belt systems with a one-way clutch together with belt-bending stiffness. Considering longitudinal motion coupled with transversal motion, Yang and Zhang^[21] investigated nonlinear vibrations of an longitudinally moving beam. More recently, a study on longitudinally moving beams with time-varying length was carried out^[22], in which the invariants and energetics of the time-varying system were discussed in detail by utilizing the method of assumed-mode.

Plates and shells are the widely used two-dimensional model. Based on the Mindlin-Reissner plate theory, Wang^[23] developed a mixed finite element formulation for a moving orthotropic thin plate. Zhou and Wang^[24] calculated natural frequencies of an longitudinally moving viscoelastic plate by using the differential quadrature method. Hu and Zhang^[25] studied the parametric vibration and stability of a longitudinally accelerating plate. The electromagnetic force in the magnetic field was considered in the study. Considering a thin viscoelastic plate travelling in the axial direction at a constant velocity, Hatami et al.^[26] used the finite strip method to obtain the natural frequencies of the system. The instability of longitudinally travelling elastic plates was analytically investigated by Banichuk et al.^[27]. The elastic plate was assumed to move at a constant velocity and transverse vibration was concentrated particularly. Tang and Chen^[28] studied the free vibration of a longitudinally moving rectangular plate, where four different boundary conditions were considered. Considering a circular cylindrical shell that travels along the longitudinal direction, Wang et al.^[29] introduced an improved formulation based on the Donnell nonlinear shallow shell theory to describe vibrational characteristics of the shell. They examined particularly the 1:1:1:1 internal resonance phenomenon of the structure.

All the work cited above considers moving continuums without interactions with fluid. In applications such as shipping engineering, marine engineering, and continuous hot-dip galvanizing process, however, longitudinally moving continuums work in dense fluids. Specially, in the continuous hot-dip galvanizing process, as shown in Fig. 1, the moving plate between touch rolls and stabilizing rolls always exhibits significant vibration during operation, which worsens the local dipping environment and results in vibration striation defect for the galvanized plate. Therefore, it is important to understand the dynamic behavior of longitudinally moving continuums coupled with dense fluid. There are limited studies on the dynamics of longitudinally moving continuums submerged in the dense fluid. Wang and Ni^[30] investigated the vibration and stability of an longitudinally moving beam completely immersed in liquid and constrained by simple supports with torsion springs. The differential quadrature method was used to obtain natural frequencies of the system in their study. The nonlinear dynamic instability of an axially translating unidirectional plate was presented by Wang et al.^[31], where the effects of the surrounding fluid and aerodynamic excitation were taken into account. Ni et al.^[32] numerically analyzed the natural frequencies of a cantilever beam attached to an longitudinally moving base in a liquid. The Galerkin approach was used, and the effects of moving speed of the base and several other system parameters on the dynamics and stability of the beam were discussed. Wang et al.^[33-34] examined the free vibrations and stability of longitudinally travelling rectangular plates fully and partially coupled with finite incompressible liquid. Using the multiple scale method, Wang et al.^[35] performed the internal resonance analysis for longitudinally moving plates immersed in a finite fluid domain. More recently, Wang and Zu^[36-37] discussed the instability of longitudinally accelerating viscous plates in liquid and the nonlinear steady-state responses of longitudinally traveling functionally graded material plates in contact with liquid.

In this paper, the vibration of a longitudinally travelling plate immersed in an infinite liquid domain is studied. Effects of rotation, compressibility, and viscidity of the liquid are neglected and the velocity potential is used to describe the liquid velocity. The classical thin plate theory is utilized to derive mechanical energies of the traveling plate. An exponential function is introduced to depict the dynamic deformation of the moving plate. The vibrational characteristics are solved using the Rayleigh-Ritz method. Furthermore, the convergence study is performed, and it shows a quick convergence speed for the immersed moving plate. In addition, the parametric study is carried out to demonstrate effects of key system parameters on vibrational characteristics of the immersed moving plate. To extend the study, the added virtual mass incremental (AVMI) factor method is also adopted for the vibration analysis of the immersed moving plate.

2 Methodology 2.1 Problem statement

The problem raised here comes from the continuous hot-dip galvanizing process, as shown in Fig. 1. Our purpose is to investigate the vibration of the steel strip between touch rolls and stabilizing rolls because the steel strip always exhibits significant vibration when it works. The modeling part is considered as a longitudinally moving rectangular plate with the length b, the width a, the thickness δ, the density ρ_p, Young's modulus E, and Poisson's ratio υ, which is immersed in a dense fluid with the mass density ρ_f, as shown in Fig. 2. The liquid domain is finite in the x- and y-directions, but is infinite in the z-direction. The width and height of the liquid domain are d and h, respectively. The material of plate considered here is homogeneous and isotropic. In this study, we adopt the Cartesian coordinate system (O, x, y, z) to depict the liquid motion and (O', x', y', z') to depict the vibration of the moving plate, respectively. The corresponding axes of these two Cartesian coordinate systems are parallel. In Fig. 2, the x'- and y'-axes define the plane of the plate, and the z'-axis denotes the out-of-plane coordinate. The origins O' and O are assumed to be located at one of plate edges and one of container corners, respectively. The location of the plate is determined by the coordinate of point O' (x=x₀, y=y₀, z=z₀) in the coordinate system (O, x, y, z). Let w(x', y', t) denote the displacement of the mid-plane of the plate in the z'-direction from the static equilibrium (w=0). Moreover, the plate is considered to be traveling in the y'-direction at a constant axial speed V. In addition, the plate is assumed to be subject to a pretension per unit width N_y0 in the y'-direction.

2.2 Plate-liquid interaction

In the present paper, we neglect nonlinear effects for the dynamic pressure at the liquid-structure interface. Nonlinear terms in the boundary conditions at this interface are also omitted. It is shown that the linear model is valid according to the study given by Amabili^[38]. Additionally, the liquid in this study is simplified to be inviscid and incompressible. The plate pre-stress owing to the fluid gravity is also neglected. Using the assumptions above, we may utilize the potential flow theory to describe the fluid-structure interaction.

We may utilize the velocity potential function to describe the liquid velocity in the whole liquid field, which satisfies Laplace's equation in the Cartesian coordinate,

(1)

where is related to the liquid velocity by , and .

Boundary conditions of the liquid in the x-direction and at the bottom of the liquid domain represent rigid-wall conditions, as shown in Fig. 2. These boundary conditions correspond to the zero-frequency conditions which satisfy

(2)

(3)

Assume that the elastic plate vibrates at relatively high frequencies so that the effect of surface waves can be neglected^[39]. The following condition may be applied to the velocity potential at the free liquid surface:

(4)

Additionally, when the distance to the vibrating plate is very far in the z-direction, there will be no pressure disturbance for the liquid. Thus,

(5)

The velocity component of the liquid normal to the structure must coincide with that of the structure due to the fact that there exists permanent contact between the plate surface and the peripheral liquid layer. The compatibility condition of the plate surface can be expressed by

(6a)

(6b)

where w is the transverse displacement of the moving plate, and

It should be noticed that one cannot use trigonometric functions that are usually used for non-moving plates, to describe the dynamic displacement w of moving plates. The reason is that in the compatibility condition (6) , there is a first-order derivative of w with respect to t, which will result in the permanent existence of time t. Therefore, trigonometric functions do not work in the Rayleigh-Ritz method.

To solve this problem, we introduce an exponential function to depict the transverse displacement of the moving plate. It is expressed in the form of

(7)

Thus, the total differential in Eq. (6) can be written as

(8)

where V is the longitudinally moving speed of the plate.

Utilizing the method of separation of variables and considering Eq. (6) , the following velocity potential function is assumed:

(9)

in which ω denotes the circular natural frequency of the immersed moving plate.

Firstly, by substituting Eq. (8) into Eq. (1) , one can get

(10)

where u_x and u_y are non-negative real numbers.

For the simplification in expression, the following non-dimensional coordinates and parameters are adopted:

(11)

According to the boundary conditions expressed by Eqs. (2) -(5) , we express the velocity potential in the following form:

(12)

It can be certified that Eq. (12) exactly satisfies Eqs.(2) -(5) . Here, A_mj is an unknown constant.

Substituting Eqs. (8) and (12) into Eq. (6) gives

(13)

(14)

Multiplying both sides of Eqs. (13) and (14) by cos (mπξ) cos (α_jπη) and integrating in the domain 0≤ξ≤1 and 0≤η≤1 yield

(15)

(16)

Applying the orthogonality of normal modes yields the following equations:

For m=0,

(17)

(18)

and for m≠0,

(19)

(20)

in which ξ₁ =x₁ /d=ξ₀ +β, η₁=y₁ /h=min {1, η₀ +γ}, ξ₀=x₀ /d, and η₀ =y₀ /h.

From Eqs. (17) -(20) , we can derive the exact expression of A_mj as follows:

(21)

where

(22)

(23)

(24)

(25)

Thus, the liquid velocity-potential function can be written as

(26)

where the formula I_mj is the plate-liquid coupling coefficient, depicting coupling relationships between the moving plate and liquid.

2.3 Liquid energy

For an ideal, inviscid, and incompressible liquid without the consideration of free surface wave, the liquid kinetic energy T_f may be written as

(27)

in which ρ_f is the liquid mass density, is the liquid domain, and is the velocity vector of liquid. By utilizing Green's theorem, one can transform the volume integration to the surface integration surrounding the liquid domain. With the consideration of boundary conditions expressed by Eqs. (2) -(5) , one obtains

(28)

Applying Eqs. (6) and (26) in Eq. (28) , the expression for the maximum kinetic energy of the liquid is found to be

(29)

2.4 Elastic strain energy and kinetic energy of moving plate

The total kinetic energy T_p of the longitudinally moving plate neglecting rotary inertia is given by

(30)

where the first term represents the kinetic energy due to the longitudinal motion of the plate, and the other term denotes the kinetic energy due to the plate's transverse vibration.

By utilizing Eq. (8) , we derive the maximum kinetic energy of the longitudinally moving plate as

(31)

The elastic strain energy U_T due to the effect of in-plane tensile force per unit width N_y0 on the transverse deformation is written by considering that the area element △x'△y' changes as ^[40]. Thus, the elastic strain energy in the moving plate can be written as

(32)

According to Kirchhoff's hypothesis, the normal stresses in the z'-direction are negligible, and the strains linearly change along the plate thickness direction. Thus, the elastic strain energy of the moving plate is given by

(33)

The total strain energy of the moving plate is then written as

(34)

Considering Eq. (7) in Eq. (34) , we can get the maximum elastic strain energy for the moving plate as follows:

(35)

2.5 Rayleigh-Ritz solution

Generally speaking, numerical methods such as the finite element method and the boundary element method can be chosen to solve coupling vibrations of plate-liquid systems for general cases, whereas analytical methods can only be used for limited special cases. Nevertheless, numerical methods also need a huge amount of computation, and their disadvantage is that they cannot describe the dynamic characteristics of plate-liquid systems qualitatively. Therefore, analytical methods still play an important role in learning the qualitative characteristics of such systems.

The Lagrangian function for the immersed moving plate has the form of

(36)

The mode function of the plate can be assumed as

(37)

in which C_nl are the unknown coefficients to be determined, L and N denote the orders of the series to be truncated, and f_n(x') and gl(y') are admissible functions in the x'- and y'-directions, respectively.

In this study, we choose the beam eigen-functions as admissible functions of the plate, which have the following non-dimensional forms^[41]:

(38)

(39)

where a_i, b_i, c_i, d_i, and k_i(i=n, l) are different constants and can be determined by imposing boundary conditions of beams.

Take f_n(ξ') as an example. The boundary conditions of a beam are

(i) at the clamped end,

(40)

(ii) at the simply supported end,

(41)

(iii) at the free end,

(42)

where I is the area moment of inertia of the beam cross-section.

Boundary conditions for the admissible function g_l(η') can be gained analogously as f_n (ξ'), and they are omitted here.

Introducing Eq. (37) in Eq. (36) and then using the Rayleigh-Ritz method yield

(43)

From Eq. (43) , we may derive the following equation:

(44)

where

(45)

in which

(46)

In Eq. (44) , the elements form a matrix , which is termed as the AVMI matrix^[42-44]. It indicates that the liquid effect on the vibration of the plate is equivalent to the added mass of the moving plate.

To possess non-trivial solutions for C_nl(n=1, 2, …, N;l=1, 2, …, L) in Eq. (44) , the determinant of the coefficient matrix must be zero. Thus, one can obtain the eigen-frequencies. Then, by substituting the results into Eq. (44) and solving the equation numerically, the mode functions of the system are determined.

3 Result analysis

The numerical solution of the problem is obtained by programming in MATHEMATICA^[45]. Firstly, a comparison study is carried out for a moving plate in vacuum. To compare with the results given by Tang and Chen^[28], the geometric and physical parameters of the moving plate are set the same as the reference. Additionally, the variables V and ω_i in this study should be transformed to the same non-dimensional quantities as Ref. [28] by the relationships and . The results calculated by the present method and those given by Tang and Chen^[28] are listed in Table 1. One can find that perfect agreement has been achieved, showing the validity of the present method in this study.

Next, we study the longitudinally moving plate coupled with a liquid, as shown in Fig. 2. The liquid domain is finite in the x- and y-directions, but is infinite in the z-direction. A convergence study for an entirely simply-supported (SSSS) moving plate submersed in a liquid is performed first. The system characteristics are given as follows: E=210 GPa, υ =0.3, δ=0.01 m, z₀ =0.2 m, λ =1, ξ₀ =0.5(1-β), η₀ =0.5(1-γ), ρ_p =7 850 kg/m³, ρ_f =1 000 kg/m³, N_y0 =10 N, V=20 m/s, and β =γ. Thus, the case is related to a moving square plate, the projection of which on the xOy-plane is in the center of the rigid wall. The first four natural frequencies of the system for different plate-liquid size ratios are shown in Table 2. For the purpose of comparison, exact natural frequencies of the moving plate are also given in this table. It is clear that the natural frequencies of the moving plate in liquid are smaller than those of plate in vacuum due to the added virtual mass effect of the liquid. One can also see that the convergence rate is quite fast. The first four natural frequencies of the immersed moving plate can be obtained with good accuracy by adopting three beam mode shapes. Additionally, the natural frequencies of the moving plate increase with the decrease of the plate-liquid size ratios. What is more, the plate-liquid size ratio has an insignificant effect on the truncated numbers of the beam mode function. Therefore, three beam mode shapes are enough for different plate-liquid size ratios. However, the necessary truncated numbers of m and j in the velocity potential function are related to the plate-liquid size ratio. The number should increase as the plate-liquid size ratio decreases. Another phenomenon can also be observed, i.e., natural frequencies of the moving plate immersed in a liquid tend to be constants as the plate-liquid size ratio decreases gradually, particularly for the higher-order natural frequencies. This phenomenon demonstrates that if a longitudinal moving plate immerses in a liquid with infinite depth or width, it is reasonable to approximate the natural frequencies with those where the liquid domain is finite whereas its depth or width is large.

The real and imaginary parts of the first four natural frequencies of an immersed SSSS plate with respect to the moving velocity are plotted in Figs. 3-Figs. 4, respectively. The parameters are E=210 GPa, υ=0.3, δ=0.01 m, ξ₀ =0.5(1-β ), η₀ =0.5(1-γ), z₀ =0.2 m, a=0.6 m, b=0.6 m, ρ_p=7 850 kg/m³, ρ_f=1 000 kg/m³, d=0.8 m, and h=0.8 m. One can see from Fig. 3 that as the moving speed increases, all the natural frequencies of the plate decrease. If the moving speed does not get to the critical speed, the real parts of corresponding natural frequencies are positive, while the imaginary parts are zero. It is interesting that when the plate speed reaches a critical value, the fundamental natural frequency decreases to zero first. After this critical value, the real part of the fundamental frequency keeps zero all the time. With the continual increase of the speed, the real parts of other natural frequencies become zero one by one. It is also noticed that when the real parts of natural frequencies are positive numbers, their imaginary parts are zero. When the real parts of natural frequencies get to zero at the critical points, their imaginary parts change to be different from zero thereafter. Specially, ω₃ and ω₄ have the same value in a small speed range (353m/s-357m/s), indicating that there may exist internal resonance between these two modes. The nonlinear vibration analysis in this area can be carried out in future research. It is also interesting that after this coupling speed range, the values of ω₃ and ω₄ both experience sudden change, and ω₃ falls to zero quickly, while ω₄ falls to zero slowly, just like they change their trajectories. This phenomenon has not been detected in vibrations of moving plates in vacuum before.

In this study, if the beam mode functions in vacuum are adopted as admissible functions and the AVMI matrix is diagonal, the mode shapes of the plate in liquid have the same forms as those of the plate in vacuo. This will give a great simplification for the natural frequency calculation of immersed moving plates. In this case, natural frequencies of the immersed moving plate can be expressed as , where ω_i are the natural frequencies of moving plates in vacuum, and and M_ii are the ith main diagonal elements of matrices and M, respectively. Here, is termed as the AVMI factor. Adopting the beam mode shapes as stated in Eqs. (38) -(39) , one can calculate the AVMI matrix for an SSSS moving plate immersed in liquid. For β =γ=3/4,

(47)

and for β =γ=3/14,

(48)

It is seen from the above AVMI matrices that they are both diagonal dominant, i.e., . Particularly, for a smaller plate-liquid size ratio (β =γ=3/14), the diagonal dominant phenomenon is more obvious. This implies that neglecting off-diagonal elements of the matrix does not result in big errors. Therefore, if we ignore all the off-diagonal elements, the natural frequencies of the system can be approximately expressed as (termed as AVMI factor solutions^[42,44,46]). The error between the results from the AVMI factor solutions and Rayleigh-Ritz solutions is quite small due to the diagonal dominant AVMI matrix. In addition, the error tends to get smaller and smaller with the decrease of the plate-liquid size ratio.

For a completely clamped (CCCC) moving plate immersed in liquid, the AVMI matrices with different plate-liquid size ratios are obtained by using the clamped-clamped beam mode shapes from Eqs. (38) -(39) . For β =γ=3/4,

(49)

and for β =γ=3/14,

(50)

Compared with the corresponding AVMI matrices for the SSSS moving plate, the AVMI matrices for the CCCC plate are less diagonal dominant. This is because that, as well known, we cannot obtain the exact mode shape for the clamped-clamped beam, and we can only calculate the approximate mode shape instead. However, the AVMI matrices for the CCCC moving plate are still diagonal dominant. The first four natural frequencies for the moving plate immersed in liquid are calculated by the AVMI factor method. The errors between the results and those obtained by the Rayleigh-Ritz method are denoted as

which are listed in Table 3.

From Table 3, we may find that the errors are quite small and they are all within 1%. This demonstrates that the AVMI factor method can be used not only in stationary plate-liquid systems^[43-44,46] but also in moving plate-liquid systems.

Fix the following variables: E=210 GPa, υ =0.3, δ=0.01 m, ρ_p =7 850 kg/m³, ρ_f =1 000 kg/m³, z₀ =0.2 m, ξ₀ =0.5(1-β), η₀ =0.5(1-γ), N_y0=10 N/m, V=20 m/s, a=0.6 m, b=0.6 m, and d=0.8 m. Table 4 shows the first four natural frequencies of a partially immersed moving plate with different immersion depths γ. Four different boundary conditions, namely, SSSS, CCCC, SCSF, and CCCF, are taken into account. Here, S, C, and F stand for simply-supported, clamped, and free constraints, respectively. From this table, one can see that with the decrease of γ (for a fixed width b, it means that the liquid depth gets larger), the natural frequencies of the partially immersed moving plate decrease.

Table 5 depicts the first four natural frequencies of a partially immersed moving plate with different γ. The plate considered here is a rectangular plate with λ=1/2. The other parameters are the same as Table 4. It should be noted that the plate position is fixed all the time for this table, and the change of γ can be realized by altering the liquid depth. As γ decreases, the natural frequencies of the partially immersed moving rectangular plate decrease. Under the same conditions, the CCCC moving plate has the highest natural frequencies, and the SCSF plate has the lowest ones, as shown in Table 4. Moreover, we may find that for the CCCC moving plate, ω₂ nearly equals ω₃, which indicates that there may be interesting 1:1 internal resonances between these two modes.

Table 6 investigates the plate position effect on the vibration characteristics of an immersed moving plate. Here, the selected parameters are E=210 GPa, υ =0.3, δ=0.01 m, a=0.5 m, b=0.5 m, z₀ =0.2 m, _,ρ_p =7 850 kg/m³, ρ_f =1 000 kg/m³, N_y0=10 N/m, V=20 m/s, h=0.8 m, and d=0.8 m. It should be noticed that the liquid domain is constant and the plate is totally immersed. Without loss of generality, two common boundary conditions, namely, SSSS and CCCC boundary conditions, are considered here. By analyzing Table 6, one may see that when the plate is closer to the free liquid surface, the natural frequencies are higher. Additionally, when the plate is further to the rigid wall in the x-direction, the natural frequencies are higher. Specially, when the plate is located at the corner of the rigid container, the natural frequencies are the lowest. When the plate is in the center of the liquid domain and near the free liquid surface, the natural frequencies are the highest. These results may give engineers and designers some useful references in the designing of moving plate-liquid coupling systems.

Figures 5-8 describe the first two natural frequencies of a immersed moving plate with different liquid domain widths. Four different boundary conditions, namely, SSSS, CCCC, CCCF, and SCSF, are treated here. The parameters selected here are as follows: E=210 GPa, υ =0.3, δ=0.01 m, z₀ =0.2 m, ξ₀ =0.5(1-β ), η₀ =0.5(1-γ), a=0.6 m, b=0.6 m, ρ_p =7 850 kg/m³, ρ _f =1 000 kg/m³, N_y0 =10 N/m, V=20 m/s, and h=0.6 m. It is clear that when the liquid domain width is within a relatively small value range, the natural frequencies of the system are highly sensitive to the width change. As the liquid domain width increases, the natural frequencies increase rapidly. However, this sensitivity becomes weaker and weaker with the increase of the liquid domain width. When the liquid domain is wide enough, the natural frequencies of the system tend to reach constant values, as seen from Figs. 5-8. Therefore, if a plate is located in an infinitely wide liquid domain, for example, a moving plate in ocean, it is reasonable to use the natural frequencies of a plate in a relatively wide liquid domain to approximate those in an infinitely wide liquid domain.

Figure 9 shows the first four natural frequencies versus the in-plane tensile force for a simply supported moving plate immersed in liquid. The parameters used here are E=210 GPa, υ =0.3, δ=0.01 m, z₀ =0.2 m, ξ₀ =0.5(1-β ), η₀ =0.5(1-γ), a=0.6 m, b=0.6 m, ρ_p =7 850 kg/m³, ρ _f =1 000 kg/m³, V=20 m/s, h=0.8 m, and d=0.8 m. A detailed check of the figure shows that all the natural frequencies increase with the increasing in-plane tensile force. However, for the moving plate-liquid system studied here, the effect of the in-plane tensile force on the natural frequencies is insignificant, and the frequency curves are nearly horizontal.

Figure 10 shows variations of the first two frequency ratios of an immersed SSSS moving plate with different liquid depths. The y-axis is denoted by , where ω_i is the wet natural frequency and ω_i is the natural frequency of the same moving plate in vacuum. The parameters selected here are E=210 GPa, υ =0.3, δ=0.01 m, z₀ =0.2 m, ξ₀ =0.5(1-β ), a=0.6 m, b=0.6 m, ρ _p =7 850 kg/m³, ρ_f =1 000 kg/m³, N_y0=10 N/m, V=20 m/s, and d=0.8 m. In this calculation, the plate position is constant, i.e., at the bottom of the rigid container all the time, and it is totally submersed in liquid. Initially, when the liquid depth increases, the fundamental frequency ratio r₁ =ω₁ /ω₁ decreases. Subsequently, this ratio increases with the increasing liquid depth and tends to be constant when the liquid depth is large enough. As for the second frequency ratio r₂ =ω₂ /ω₂, it increases with the increasing liquid depth and then tends to be constant for a large liquid depth.

In Fig. 11, variations of the first two frequency ratios of an immersed CCCC moving plate with different liquid depths are plotted. The parameters here are the same as Fig. 9. As can be seen from Fig. 11(a), in the beginning, the fundamental frequency ratio decreases with the increase of liquid depth. Then, it increases and tends to be constant as the liquid depth increases continually. The second frequency ratio has analogous tendency as that for the SSSS plate. However, this ratio gets to be constant more quickly with the increase of the liquid depth, at about 1/γ=21, as shown in Fig. 11(b).

4 Conclusions

In this study, the vibration of a longitudinally moving plate submersed in an infinite liquid domain is investigated by the Rayleigh-Ritz method for the first time. The Kirchhoff's plate theory together with the velocity potential theory is adopted to model the moving plate-liquid system. The AVMI method is also adopted. The results are compared with the solutions from the Rayleigh-Ritz method. The conclusions are drawn as follows:

(ⅰ) The dynamic deformation depicted by an exponential function can overcome the disadvantage of trigonometric functions for the immersed longitudinally moving plate. Thus, the exponential function can be used to calculate natural frequencies of moving plate-liquid systems.

(ⅱ) It is shown that the present method has a quick convergence speed. The method not only has very good accuracy but also avoids a huge amount of computation.

(ⅲ) For the immersed moving plate, numerical results demonstrate that the AVMI factor method can get solutions close to those solved by the Rayleigh-Ritz method. Therefore, the AVMI factor method is also adoptable to solve the vibration problems of moving plate-liquid systems.

(ⅳ) The results show that key system parameters, including the traveling speed, the plate location, the plate-liquid ratio, the liquid depth, and the boundary condition, all have significant effects on the vibration characteristics of the immersed moving plate.