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A semi-analytical method for dynamic analysis of a rectangular plate with general boundary conditions based on FSDT

  • Di Jia , Cong Gao EMAIL logo , Yuliang Yang , Fuzhen Pang , Haichao Li and Yuan Du
Published/Copyright: August 3, 2022
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REVIEWS ON ADVANCED MATERIALS SCIENCE
From the journal REVIEWS ON ADVANCED MATERIALS SCIENCE Volume 61 Issue 1

Abstract

In this paper, the unified Jacobi–Ritz method (JRM) is utilized to analyze the dynamic response of rectangular plates with general boundary conditions. First, the structural energy functional is established in the framework of the first-order shear deformation theory, and the rectangular plate is divided into several equal parts according to the domain decomposition method. Then, the artificial springs are introduced to ensure the continuity of segments and diversified boundary conditions. The Jacobi orthogonal polynomials are expanded to represent the displacement field in one direction. Finally, the free and forced vibration characteristics of the rectangular plate can be obtained by utilizing the Rayleigh–Ritz method, where the Newmark-β integration method is adopted to realize the time-domain solutions for transient vibration response. The results for different structural scale parameters and various boundary conditions are presented, and the validity and accuracy of the presented method are verified by comparing the results from published literature and FEM. The results of the study can provide technical support for vibration control of the plate structure.

Keywords: Jacobi polynomials; steady and transient vibration; rectangular plate; general boundary condition

1 Introduction

The rectangular plate structure has been widely used in practical engineering, which is usually subjected to various steady and transient loads. Therefore, an accurate analysis of the vibration characteristics of the rectangular plate can guide well its design in actual engineering applications.

In recent years, numerous studies have been conducted on the free and steady vibration of various structures, including the dynamic stiffness method [1], the separation variables method [2,3,4], the Ritz method [5,6,7,8,9], the generalized differential quadrature method [10], and the finite-element method (FEM) [11,12]. Yang et al. [13] carried out the exact solutions for the rectangular ceramic plate based on the 3D linear equations. Considering the influence of the initial stresses, Akbarov et al. [14] applied the 3D FEM to obtain the forced vibration response of the rectangular composite plate based on the 3D linearized theory. Wu et al. [15] studied the dynamic response of the rectangular plate under external load by applying the principle of mixed variables. Fan [16] performed the forced vibration of the rectangular plate by means of the damped complex modes analysis method. Based on the Hamilton variation principle, the research studies of Xiao and his colleagues [17,18,19] demonstrated the free and forced vibration characteristics of a thin rectangular plate on the nonlinear elastic foundation by using the Galerkin method. In the framework of Mindlin’s first-order theory, Zhang et al. [20] illustrated the forced vibration characteristics of the rectangular piezoelectric plate, and the trigonometric series solutions were obtained under two types of boundary conditions. Nasirshoaibi and Mohammadi [21] applied the modal expansion method for vibration analysis of an elastically connected rectangular plate. According to the FSDT, a great deal of research studies on composite plates and shells were conducted by Shi et al. [9,22,23], who developed the improved Fourier series method, and the influences of boundary conditions and material parameters on the vibration characteristics of the structure were also discussed. Agarana and Ehigbochie [24] put to use the finite-difference method to analyze the dynamic response of the elastic orthotropic mindlin plate, and the results were compared with those obtained from the published literature. Yahnioglu and Yesi [25] extended the FEM to study the vibration characteristics of a thick composite plate based on the 3D linearized theory of elastic waves. More research studies on the steady vibration response of rectangular plates can be found in the literature [26,27,28,29,30,31].

With regard to the transient vibration, Ou and Mak [32] developed the time-domain FEM and BEM to investigate the transient vibration and sound radiation of the stiffened plate; the stiffeners can be located at any position inside the plate. Fan [33] obtained the transient vibration solutions of the rectangular plate under viscoelastic restraints by using the modal strain energy (method, and the vibration displacement function was approximately selected as a double infinite series. Geng et al. [34,35] dealt with the transient vibration response of an impacted plate by using the interpolated time-domain equivalent source method, and a steel ball impact experiment was carried out on the clamped rectangular steel plate to verify the effectiveness of the method. Considering the nonlinear effect, He and Kashiwagi [36] carried out the transient vibration response of the vertical plate under the impact of a pulse-type wave by using the full-nonlinear boundary-element method. Based on von Karman’s large deflection plate theory, Sheikh and Mukhopadhyay [37] investigated the transient vibration characteristics of stiffened plate structures by using the spline finite strip method, and the spline functions and finite-element shape functions represented the two direction displacement interpolation functions. Hajheidari and Mirdamadi [38] established the transient vibration analysis model of the laminated plate in the framework of the classical lamination plate theory, and the numerical solutions were obtained by using the spectral FEM. Skocilas et al. [39] carried out the transient vibration response of the thin plate with viscoelastic isotropic and anisotropic materials, in which the selected model was defined by applying the constitutive equations related to stress and deformation. Pang et al. [40] conducted a study on the safety of the platform deck structure caused by the transient impact load during the rocket launch process based on the FEM.

From the above analysis, the published literature have ample analysis ways for the steady vibration response of the rectangular plate; however, the research studies on the transient vibration response of rectangular plate structures are relatively lacking, particularly for damped transient attenuation vibration characteristics. Based on this background, this study aims at conducting the dynamic responses of rectangular plate structures subject to general boundary restraints based on the FSDT. The multi-segment principle and unified Jacobi orthogonal polynomials are introduced to ensure the convergence and effectiveness of the algorithm. Meanwhile, the dynamic response characteristics of the rectangular plate are parameterized research.

2 Mathematical formulation

2.1 Rectangular plate

Figure 1 shows the analytical model of rectangular plate, where a, b, and h represent the length, width, and thickness of the structure, respectively. The cartesian coordinate system (x, y, z) is adopted to describe the model, in which x, y, and z, respectively, represent the length, width, and thickness directions of the structure, u, v, and w, respectively, denote the displacement fields in the x-, y-, and z-directions of the middle surface. The rectangular plate boundary conditions can be obtained by introducing three groups of linear springs (k u , k v , and k w ) and two groups of rotational springs (k x and k y ) at ends x = 0, x = a, y = 0, and y = b. Various boundary conditions can be simulated by setting different spring stiffness values. The rectangular plate is divided into H segments based on the DDM, where the ith segment is connected to the i + 1th segment by artificial springs. Normally, the stiffness of the connecting spring is set to infinity, indicating a direct strong coupling effect. Figure 2 exhibits the excitation point and the test point of the rectangular plate, and the load acts at the center of the structure in the z-direction. There are three vibration test points and one excitation point. The excitation point and test point are signified by first alphabet EP and TP, respectively, for subsequent analysis.

Figure 1 Calculation model of rectangular plate structure.
Figure 1

Calculation model of rectangular plate structure.

Figure 2 The excitation point and the examination point of the rectangular plate.
Figure 2

The excitation point and the examination point of the rectangular plate.

2.2 Expressions of rectangular plate’s energy

The structural energy functional is established based on FSDT [41,42,43], and the assumed displacement field of the ith segment is expressed as

(1a) u i ( x , y , z , t ) = u 0 i ( x , y , t ) + z ϕ x i ( x , y , t ) ,

(1b) v i ( x , y , z , t ) = v 0 i ( x , y , t ) + z ϕ y i ( x , y , t ) ,

(1c) w i ( x , y , z , t ) = w 0 i ( x , y , t ) .

The strains of the rectangular plate are taken as follows:

(2a) ε x x i = ε x x 0 , i + z χ x x i , ε y y i = ε y y 0 , i + z χ y y i ,

(2b) γ x y i = γ x y 0 , i + z χ x y i , γ y z i = γ y z 0 , i , γ x z i = γ x z 0 , i ,

where ε x x i , ε y y i , γ x y i , γ y z i , and γ x z i , respectively, represent normal, shear, and transverse shear strains, and then,

(3a) ε x x 0 , i = ∂ u 0 i ∂ x , ε y y 0 , i = ∂ v 0 i ∂ y ,

(3b) γ x y 0 , i = ∂ u 0 i ∂ y + ∂ v 0 i ∂ x , γ y z 0 , i = ∂ w 0 i ∂ y + ϕ y i , γ x z 0 , i = ∂ w 0 i ∂ x + ϕ x i ,

(3c) χ x x i = ∂ ϕ x i ∂ x , χ y y i = ∂ ϕ y i ∂ y , χ x y i = ∂ ϕ x i ∂ y + ∂ ϕ y i ∂ x .

The stress of the rectangular plate is shown as

(4) σ x x i σ y y i τ x y i τ y z i τ x z i = Q 11 Q 12 0 0 0 Q 12 Q 22 0 0 0 0 0 Q 66 0 0 0 0 0 Q 66 0 0 0 0 0 Q 66 ε x x i ε y y i γ x y i γ y z i γ x z i ,

where σ x i , σ y i , τ x y i , τ x z i , and τ y z i , respectively, represent normal and shear stresses. Q ij (i,j = 1, 2, 6) can be expressed as

(5) Q 11 = Q 22 = E 1 − μ 2 , Q 12 = μ E 1 − μ 2 , Q 66 = E 2 [ 1 + μ ] ,

where E and μ represent elastic modulus and Poisson’s ratio, respectively, and the forces and moments can be expressed as,

(6) N x i N y i N x y i = ∫ − h / 2 h / 2 σ x x σ y y σ x y d z , M x i M y i M x y i = ∫ − h / 2 h / 2 σ x x σ y y σ x y z d z , Q x z i Q y z i = ∫ − h / 2 h / 2 κ ¯ σ x z σ y z d z .

The following equation can be obtained by substituting equation (4) into equation (6):

(7a) N x x i N y y i N x y i M x x i M y y i M x y i = A 11 A 12 0 B 11 B 12 0 A 12 A 22 0 B 12 B 22 0 0 0 A 66 0 0 B 66 B 11 B 12 0 D 11 D 12 0 B 12 B 22 0 D 12 D 22 0 0 0 B 66 0 0 D 66 ε x x 0 , i ε y y 0 , i γ x y 0 , i χ x x i χ y y i χ x y i ,

(7b) Q x i Q y i = κ ¯ A 55 A 45 A 45 A 44 γ x z 0 , i γ y z 0 , i ,

where N x , N y, N xy , M x , M y , M xy , Q xz , and Q yz , respectively, represent the force, moment resultants, and transverse shear force of the structure, κ ¯ is the shear correction factor, and the value of the study is set as 5/6. A ij , B ij , and D ij (i, j = 1, 2, 6) are the extensional, extensional-bending coupling, and bending stiffness of the rectangular plate, which are shown as

(8) ( A i j , B i j , D i j ) = ∫ − h / 2 h / 2 Q i j ( 1 , z , z 2 ) d z .

The strain energy is expressed as

(9) U V i = 1 2 ∫ 0 a ∫ 0 b ( N x x i ε x x 0 , i + N y y i ε y y 0 , i + N x y i γ x y 0 , i + M x x i χ x x i + M y y i χ y y i + M x y i χ x y i + Q x i γ x z 0 , i + Q y i γ y z 0 , i ) d x d y .

The following equation can be obtained by substituting equations (3) and (7) into equation (9)

(10a) U S i = 1 2 ∫ 0 a ∫ 0 b A 11 ∂ u 0 i ∂ x 2 + A 22 ∂ v 0 i ∂ y 2 + A 66 ∂ u 0 i ∂ y + ∂ v 0 i ∂ x 2 + 2 A 12 ∂ u 0 i ∂ x ∂ v 0 i ∂ y + κ ¯ A 44 ∂ w 0 i ∂ y + ϕ y i 2 + κ ¯ A 55 ∂ w 0 i ∂ x + ϕ x i 2 + 2 κ ¯ A 45 ∂ w 0 i ∂ x + ϕ x i ∂ w 0 i ∂ y + ϕ y i d x d y ,

(10b) U B S i = 1 2 ∫ 0 a ∫ 0 b B 11 ∂ u 0 i ∂ x ∂ ϕ x i ∂ x + B 12 ∂ u 0 i ∂ x ∂ ϕ y i ∂ y + ∂ v 0 i ∂ y ∂ ϕ x i ∂ x + B 22 ∂ v 0 i ∂ y ∂ ϕ y i ∂ y + B 66 ∂ v 0 i ∂ x + ∂ u 0 i ∂ y ∂ ϕ x i ∂ y + ∂ ϕ y i ∂ x d x d y ,

(10c) U B i = 1 2 ∫ 0 a ∫ 0 b D 11 ∂ ϕ x i ∂ x 2 + 2 D 12 ∂ ϕ x i ∂ x ∂ ϕ y i ∂ y + D 22 ∂ ϕ y i ∂ y 2 + D 66 ∂ ϕ x i ∂ y + ∂ ϕ y i ∂ x d x d y .

The boundary potential energy U b for a rectangular plate is shown as

(11) U b = 1 2 ∫ 0 b [ k u , x 0 u 0 2 + k v , x 0 v 0 2 + k w , x 0 w 0 2 + k x , x 0 ϕ x 2 + k y , x 0 ϕ y 2 ] x = 0 + [ k u , x L u 0 2 + k v , x L v 0 2 + k w , x L w 0 2 + k x , x L ϕ x 2 + k y , x L ϕ y 2 ] x = a d y + 1 2 ∫ 0 a [ k u , y 0 u 0 2 + k v , y 0 v 0 2 + k w , y 0 w 0 2 + k x , y 0 ϕ x 2 + k y , y 0 ϕ y 2 ] y = 0 + [ k u , y L u 0 2 + k v , y L v 0 2 + k w , y L w 0 2 + k x , y L ϕ x 2 + k y , y L ϕ y 2 ] y = b d x .

The potential energy of the connective spring is given by

(12) U s i = 1 2 ∫ 0 b k u ( u 0 i − u 0 i + 1 ) 2 + k v ( v 0 i − v 0 i + 1 ) 2 + k w ( w 0 i − w 0 i + 1 ) 2 + k x ( ϕ x i − ϕ x i + 1 ) 2 + k y ( ϕ y i − ϕ y i + 1 ) 2 i , i + 1 d y .

The total potential energy is expressed as

(13) U B S = U b + ∑ i = 1 H − 1 U s i .

The kinetic energy of the ith segment is shown as below

(14) T i = 1 2 ∫ 0 a ∫ 0 b I 1 ∂ u 0 i ∂ t 2 + ∂ v 0 i ∂ t 2 + ∂ w 0 i ∂ t 2 + I 3 ∂ ϕ x i ∂ t 2 + ∂ ϕ y i ∂ t 2 + 2 I 2 ∂ u 0 i ∂ t ∂ ϕ x i ∂ t + ∂ v 0 i ∂ t ∂ ϕ y i ∂ t d x d y ,

where

(15) ( I 1 , I 2 , I 3 ) = ∫ − h / 2 h / 2 ρ ( 1 , z , z 2 ) d z .

The work done by the external concentrated load on the ith rectangular plate segment is

(16) W i = ∫ f w , i w i d x ,

where f w , i is the external concentrated load inflicted in the z-direction.

2.3 The displacement function and the characteristic equation of the system

Based on the multi-segment partitioning principle, the Jacobi orthogonal polynomials [44,45,46] are expanded to represent the displacement field in one direction. The recurrence formulas of Jacobi polynomials are derived as

(17a) P 0 ( α , β ) ( ϕ ) = 1 ,

(17b) P 1 ( α , β ) ( ϕ ) = α + β + 2 2 ϕ − α − β 2 ,

(17c) P i ( α , β ) ( ϕ ) = ( α + β + 2 i − 1 ) { α 2 − β 2 + ϕ ( α + β + 2 i ) ( α + β + 2 i − 2 ) } 2 i ( α + β + i ) ( α + β + 2 i − 2 ) × P i − 1 ( α , β ) ( ϕ ) − ( α + i − 1 ) ( β + i − 1 ) ( α + β + 2 i ) i ( α + β + i ) ( α + β + 2 i − 2 ) P i − 2 ( α , β ) ( ϕ ) ,

where ϕ ∈ [ − 1 , 1 ] , α , β > − 1 , and i = 2 , 3 , …

Each of the displacement function components of the plate is written in the form of Jacobi polynomials and trigonometric series as follows:

(18a) u 0 = ∑ m = 0 M ∑ n = 0 N A m P m ( α , β ) ( x ) cos n π y b e i ω t ,

(18b) v 0 = ∑ m = 0 M ∑ n = 0 N B m P m ( α , β ) ( x ) sin n π y b e i ω t ,

(18c) w 0 = ∑ m = 0 M ∑ n = 0 N C m P m ( α , β ) ( x ) sin n π y b e i ω t ,

(18d) ϕ x = ∑ m = 0 M ∑ n = 0 N D m P m ( α , β ) ( x ) sin n π y b e i ω t ,

(18e) ϕ y = ∑ m = 0 M ∑ n = 0 N E m P m ( α , β ) ( x ) cos n π y b e i ω t ,

where A m , B m , C m , D m , and E m indicates the expansion coefficients.

The Lagrangian energy function of a rectangular plate is given by

(19) ℒ = ∑ 1 N − 1 ( W i + T i − U V i ) − U B S .

Minimizing the energy function with respect to unknown expansion coefficients

(20) ∂ ℒ ∂ ϑ = 0 , ϑ = A m , B m , C m , D m , E m .

Therefore, the dynamic response equation of rectangular plate is derived as

(21) ( K − ω 2 M ) Q = F ,

where K, M, and Q respectively signify the stiffness, mass, and coefficients matrix. By solving equation (21), the free vibration results for the rectangular plate can be obtained.

2.4 Vibration response solution

The unknown coefficients matrix of the rectangular plate under external excitation is derived as

(22) Q = ( K − ω 2 M ) − 1 F ,

where F denote the external excitation matrix, and the steady forced vibration response of rectangular plate can be obtained by substituting the above results into equation (18).

The Rayleigh damping is used in this study

(23) C = a M + b K ,

where C denotes the damping matrix, a and b are denoted as follows:

(24) a = 2 w i w j w i + w j ξ , b = 2 ξ w i + w j ,

where w i and w j denote the ith and jth frequency of the structure and ξ is the damping ratio. The Rayleigh damping is set as a = 2, b = 0.00003 in this study.

Assuming that the acceleration is constant within the time range of [t, t + Δt], two parameters β and γ are introduced at the same time:

(25a) Δ x ̇ i = [ ( 1 − β ) x ̈ i + β x ̈ i + 1 ] Δ t ,

(25b) Δ x i = x ̇ i Δ t + 1 2 − γ x ̈ i + γ x ̈ i + 1 Δ t 2 .

In this study, β = 1/2, γ = 1/4, and the above formula can be solved to obtain

(26a) x ̈ i + 1 = 1 γ Δ t 2 Δ x i − 1 γ Δ t x ̇ i − 1 2 γ − 1 x ̈ i ,

(26b) x ̇ i + 1 = β γ Δ t Δ x i − 1 − β γ x ̇ i + 1 − β 2 γ x ̈ i Δ t .

The incremental balance equation is shown as

(27) M x ̈ i + 1 + C x ̇ i + 1 + K x i + 1 = F i + 1 .

Substituting equation (26) into equation (27) can be obtained

(28) [ K ¯ ] x i + 1 = F ¯ i + 1 .

where

(29) [ K ¯ ] = K + 1 γ Δ t 2 M + β γ Δ t C,

(30) F ¯ i + 1 = F i + 1 + M 1 γ Δ t 2 x i + 1 γ Δ t x ̇ i + 1 2 γ − 1 x ̈ i + C β γ Δ t x i + β γ − 1 x ̇ i + β 2 γ − 1 x ̈ i Δ t .

The response of the structure at any time can be obtained by repeating the iteration until the end of the time.

3 Validation and discussion

The free, clamped, simply support, and elasticity support boundary conditions are signified by F, C, S, and E, respectively. Unless otherwise indicated, the geometrical dimensions and material properties in this study are as follows: a = 1 m, b = 1 m, h = 0.05 m; E = 210 GPa, ρ = 7,800 kg·m−3, μ = 0.3; M = 8, α = 1, β = 1, H = 6. The nondimensional frequency is defined as Ω = ω b 2 / π 2 ρ h / D .

3.1 Convergence study

Combined with the specific conditions of simply supported square plates on four sides, the convergence of the method is discussed. It is worth noting that the following conclusions are also valid for rectangular plates with other boundary conditions. As mentioned earlier, the artificial springs and DDM are utilized in the proposed method; therefore, the convergence of the algorithm depends on the spring stiffness, Jacobi parameters, and the number of segments.

The frequency parameters of rectangular plates with different spring stiffnesses are shown in Figure 3. As the spring stiffness changes in the range of 10−1–1015, the boundary conditions and continuity conditions change from free state to clamped state. Obviously, the stiffness value can be selected as zero for the free boundary condition, and the stiffness values can be selected in the range of 1012–1015 for the clamped boundary condition. Whether its a boundary spring or the connective spring, the dimensionless frequency parameter obviously increases rapidly, with the spring value increasing in the range of 106–1010. The boundary conditions of the plate used in this paper are shown in Table 1 [47,48]. It should be noted that although different coupling and boundary conditions can be easily simulated by adjusting the stiffness of the artificial springs, modeling rigid constraints with high stiffness values may generate boundless results [49,50]; the penalty method with negative stiffness can be used to determine the appropriate stiffness to obtain accurate results.

Figure 3 Frequency parameters Ω of rectangular plate with different boundary parameters.
Figure 3

Frequency parameters Ω of rectangular plate with different boundary parameters.

Table 1

The spring stiffness values of the general edge conditions

Boundary conditions k u k v k w k x k y
F 0 0 0 0 0
E 108 108 108 108 108
S 1015 1015 1015 0 0
C 1015 1015 1015 1015 1015

Figure 4 exhibits the variation of the dimensionless results for a different number of segments and the highest order of the Jacobi polynomial. As expected, the frequency parameters stable quickly when H ≥ 3 and M ≥ 4. However, the matrix pathological can be obtained if the convergence parameters are too large. Therefore, the number of segments H is chosen as 6, and the polynomial truncation number M is selected as 8 in this study.

Figure 4 Frequency parameters Ω of rectangular plate with different convergence parameters.
Figure 4

Frequency parameters Ω of rectangular plate with different convergence parameters.

As mentioned earlier, the Jacobi polynomial can be transformed into the Chebyshev polynomial or the Legendre polynomial by selecting different values of Jacobi parameters. The relative percentage errors of rectangular plates with different Jacobi parameters are shown in Figure 5, and α = β = 1 is selected as the reference value. Obviously, the maximum relative percentage error is almost negligible in the existing method regardless of the values of α and β, which means that the introduction of the Jacobi polynomial enriches the selection and diversity of displacement functions.

Figure 5 Relative percentage error of rectangular plate with different Jacobi parameters.
Figure 5

Relative percentage error of rectangular plate with different Jacobi parameters.

3.2 Free vibration characteristics of the rectangular plate

In this section, the free vibration solutions of the rectangular plate under general boundary constraints are compared with the existing literature. Table 2 shows the comparison of dimensionless frequency parameters between the proposed method and the published literature [51]. The comparison study indicates that the current method has higher accuracy for free vibration analysis of the rectangular plate. Figure 6 displays the mode shape of a rectangular plate; it is apparent that the proposed method in this paper can well display the mode shapes of the rectangular plate under general boundary conditions.

Table 2

Comparison of the natural frequency of present method and published literature

Boundary conditions SSSS SSSC CSCS
Mode.no Ref. [51] Present Ref. [51] Present Ref. [51] Present
a/b = 0.4 1 7.2500 7.2494 7.4410 7.4447 7.6842 7.6863
2 10.2500 10.2482 10.8840 10.8832 11.6290 11.6299
3 15.2500 15.2472 16.4120 16.4167 17.7090 17.7157
4 22.2490 22.2462 23.9560 23.9567 25.8040 25.8022
5 25.9990 25.9981 26.0930 26.0977 26.2020 26.2064
a/b = 0.6 1 3.7780 3.7775 4.0540 4.0604 4.4270 4.4310
2 6.7780 6.7772 7.5750 7.5700 8.5095 8.5052
3 11.7780 11.7769 12.2540 12.2515 12.4280 12.4219
4 12.1110 12.1106 13.1200 13.1131 14.6050 14.6161
5 15.1110 15.1098 15.6220 15.6236 16.2170 16.2180
Figure 6 The mode shape of rectangular plate.
Figure 6

The mode shape of rectangular plate.

Table 3 and Figure 7 show the dimensionless frequency parameters of a rectangular plate with different shear correction factors under general boundary restraints. It is not hard to find that the shear correction factor has little influence on the dimensionless frequency parameters. At the same time, the boundary constraint has a significant impact on the vibration characteristics of the rectangular plate, and the frequency parameter gradually increases with the enhancement of the boundary conditions, which means that the enhancement of the boundary conditions increases the overall stiffness of the structure.

Table 3

Nondimensional frequency parameters of rectangular plate subject to general boundary conditions

Mode.no CSCS SSSS FSFS ESES CSFS
κ = 3/6 1 2.8321 1.9450 0.9677 1.8392 1.2649
2 5.2943 4.7939 1.5970 3.1285 3.2270
3 6.5499 4.7949 3.5849 4.6285 4.0913
4 8.8248 7.4947 3.8355 5.3910 6.0497
5 9.6420 9.3375 4.5561 6.3000 6.8901
κ = 4/6 1 2.8547 1.9542 0.9692 1.8753 1.2687
2 5.3465 4.8322 1.6029 3.2055 3.2498
3 6.6503 4.8331 3.6058 4.6904 4.1175
4 8.9775 7.5860 3.8574 5.4840 6.1128
5 9.7823 9.4634 4.5889 6.4489 6.9747
κ = 5/6 1 2.8685 1.9601 0.9702 1.9023 1.2710
2 5.3789 4.8561 1.6067 3.2643 3.2641
3 6.7130 4.8569 3.6191 4.7326 4.1336
4 9.0738 7.6436 3.8708 5.5526 6.1523
5 9.8699 9.5420 4.6094 6.5559 7.0275
κ = 1 1 2.8779 1.9642 0.9709 1.9237 1.2727
2 5.4009 4.8725 1.6093 3.3118 3.2739
3 6.7558 4.8733 3.6285 4.7638 4.1446
4 9.1400 7.6834 3.8799 5.6066 6.1795
5 9.9298 9.5958 4.6234 6.6381 7.0636
Figure 7 Frequency parameters Ω of rectangular plate with different shear correction factors.
Figure 7

Frequency parameters Ω of rectangular plate with different shear correction factors.

Table 4 and Figure 8 display the dimensionless frequency parameters of the rectangular plate with various a/b ratios under general boundary restraints. The results indicate that the nondimensional frequency parameters gradually decrease in general with the a/b ratio increase. That is to say, when the width (b) of the rectangular plate is fixed, the overall stiffness of the structure gradually decreases as the length (a) increases.

Table 4

Nondimensional frequency parameters of rectangular plate with different a/b radio

Mode.no CSCS SSSS FSFS ESES CSFS
a/b = 0.5 1 8.9615 4.8231 0.9582 2.8350 2.2526
2 10.8082 7.5613 2.6810 5.9866 4.9644
3 14.4478 12.0863 3.8261 6.0269 9.4592
4 19.9047 15.7744 6.2370 10.0635 9.5061
5 22.1490 18.1974 8.4869 10.7104 12.3241
a/b = 1 1 2.8685 1.9601 0.9702 1.9023 1.2710
2 5.3789 4.8561 1.6067 3.2643 3.2641
3 6.7130 4.8569 3.6191 4.7326 4.1336
4 9.0738 7.6436 3.8708 5.5526 6.1523
5 9.8699 9.5420 4.6094 6.5559 7.0275
a/b = 2 1 1.3756 1.2385 0.9812 1.2771 1.0497
2 2.3614 1.9711 1.1739 1.9323 1.5786
3 3.8323 3.1883 1.7673 2.7605 2.5689
4 4.2299 4.1654 2.7593 3.8094 3.9586
5 5.1036 4.8734 3.8973 4.1584 4.0161
Figure 8 Frequency parameters Ω of rectangular plate with different a/b ratio.
Figure 8

Frequency parameters Ω of rectangular plate with different a/b ratio.

3.3 Steady vibration characteristics of the rectangular plate

In this section, the steady forced vibration analysis of the rectangular plate subjected to external excitation load is carried out. The excitation load is the unit concentrated force along the z-direction, the analysis frequency band is from 2 to 1,000 Hz, and the interval is 2 Hz.

Figure 9 displays the comparison results for the steady vibration response of the present method and FEM. Obviously, the proposed method is in good agreement with the FEM results, and the trends of the two curves are basically the same; there is only a small deviation at the single peak. In other words, the current approach can effectively analyze the steady vibration of the rectangular plate.

Figure 9 Comparison of the steady vibration response of the present method and FEM results.
Figure 9

Comparison of the steady vibration response of the present method and FEM results.

After verifying the effectiveness of the proposed method for the steady forced vibration of the rectangular plate, a parametric study on the steady vibration characteristics of the rectangular plate is presented. First, Figure 10 shows the comparison results for the steady vibration response of the rectangular plate under various boundary conditions. The results indicate that the boundary conditions have a significant influence on the steady vibration response of the structure. In addition, the first peak frequency of the steady vibration gradually increases with the increase of the boundary conditions, while the number of peaks gradually decreases. At the same time, it is not difficult to find that the peak of the steady vibration appears at the natural frequency of the rectangular plate. It can be concluded that the natural frequency of the structure gradually increases with the enhancement of boundary conditions, while the number of natural frequencies of the structure within a certain frequency range decrease.

Figure 10 Comparison of the steady vibration of rectangular plate with different boundary conditions.
Figure 10

Comparison of the steady vibration of rectangular plate with different boundary conditions.

Figures 11–12 show the comparison results for the steady vibration characteristics of the rectangular plate with different h/a and a/b ratios. The results demonstrate that the first peak frequency gradually increases with increasing h/a ratio, which indicates that the increase of thickness of the rectangular plate has an obvious influence on the structural stiffness. In addition, the first peak frequency of the structure gradually decreases as the length (a) increases when the width (b) of the rectangular plate is fixed, while the number of peaks within a certain frequency range increases, which is consistent with the conclusion of free vibration.

Figure 11 Comparison of the steady vibration of rectangular plate with different h/a ratios.
Figure 11

Comparison of the steady vibration of rectangular plate with different h/a ratios.

Figure 12 Comparison of the steady vibration of rectangular plate with different a/b radios.
Figure 12

Comparison of the steady vibration of rectangular plate with different a/b radios.

3.4 Transient vibration characteristics of rectangular plate

For transient vibration response, three different types of time-dependent loads are considered in the paper, which is shown in Figure 13.

  1. Rectangular pulse load

    (31) f ( t ) = q 0 , t 0 ≤ t ≤ t 0 + τ 0

  2. Sine pulse load

    (32) f ( t ) = q 0 ⁎ sin π τ ⁎ t − π τ ⁎ t 0 , t 0 ≤ t ≤ t 0 + τ 2 q 0 ⁎ sin − π τ ⁎ t + π τ ⁎ t 0 + π , t 0 + τ 2 ≤ t ≤ t 0 + τ 0

  3. Triangular pulse load

(33) f ( t ) = q 0 2 τ ⁎ t − 2 τ ⁎ t 0 , t 0 ≤ t ≤ t 0 + τ 2 q 0 − 2 τ ⁎ t + 2 τ ⁎ t 0 + 2 , t 0 + τ 2 ≤ t ≤ t 0 + τ 0

Figure 13 Sketch of applied load for rectangular plate with three different types of time-dependent loads.
Figure 13

Sketch of applied load for rectangular plate with three different types of time-dependent loads.

First, the transient response analysis of the undamped system is carried out. Figure 14 displays comparison results for the undamped transient vibration response of present method and FEM under rectangular pulse load, in which q 0 = 1 N, t 0 = 0.005 s, τ = 0.005 s, Δt = 0.0001 s. Obviously, the proposed method is in good agreement with the FEM calculation results, and the trends of the two curves are basically the same; in other words, the present method can effectively analyze the transient vibration response for the rectangular plate.

Figure 14 Comparison of the undamped transient vibration response of the present method and FEM.
Figure 14

Comparison of the undamped transient vibration response of the present method and FEM.

Figure 15 displays the comparison of the undamped transient vibration with different boundary conditions under a rectangular plate. The amplitude of the undamped vibration response increases with the weakening of boundary conditions. Undoubtedly, the weakening of boundary conditions reduces the stiffness of the structure and makes the structure more prone to vibration. Then, the comparison of the undamped transient vibration with different types of time-dependent loads under different boundary conditions is shown in Figure 16, and the vibration pickup points are chosen as point 2; the action time and amplitude of those three kinds of loads are precisely the same. The influence of the rectangular pulse load on the structure is always more significant than that of sine pulse load and triangular pulse load from the perspective of the amplitude of the displacement response at the initial moment of load application. The displacement response of the sine pulse load and the triangle pulse load is basically the same under SSSS boundary conditions, which is smaller than the displacement response caused by the rectangular pulse load with the passage of time. For CSCS boundary conditions, the response caused by the rectangular pulse load is greater than the response caused by the sine pulse load, while the response caused by the triangular pulse load is the smallest with the passage of time, owing to the amplitude of rectangular pulse load acting on the structure is larger than other loads within a certain period.

Figure 15 Comparison of the undamped transient vibration with different boundary conditions.
Figure 15

Comparison of the undamped transient vibration with different boundary conditions.

Figure 16 Comparison of the undamped transient vibration with different types of time-dependent loads.
Figure 16

Comparison of the undamped transient vibration with different types of time-dependent loads.

For the damped transient vibration response, Figure 17 displays the comparison of the damped transient vibration with different boundary conditions under the rectangular pulse load. The amplitude of the damped vibration response increases with the weakening of the boundary condition. The comparison of the damped transient vibration with different types of time-dependent loads is shown in Figure 18. Regardless of the boundary condition, the maximum vibration displacement at the initial moment is always caused by the rectangular pulse load, followed by the sine pulse load, and the response caused by the triangle pulse load is the smallest, owing to the amplitude of the rectangular pulse load acting on the structure is larger than other loads within a certain period, which is consistent with the conclusion of the undamped vibration response.

Figure 17 Comparison of the damped transient vibration response with different boundary conditions.
Figure 17

Comparison of the damped transient vibration response with different boundary conditions.

Figure 18 Comparison of the damped transient vibration with different types of time-dependent loads.
Figure 18

Comparison of the damped transient vibration with different types of time-dependent loads.

4 Conclusion

In this paper, the unified Jacobi–Ritz method (JRM) is utilized to analyze the dynamic response of the rectangular plate with different boundary conditions. The rectangular plate is divided into several equal sections in the framework of the DDM, and the structural energy functional is established based on the FSDT; five groups of artificial springs are introduced to ensure the continuity of the segments and diversified boundary conditions. The unified Jacobi orthogonal polynomials are expanded to represent the displacement field of the plate in one direction, and the other direction is expressed by the trigonometric series. Polynomials such as Legendre and Chebyshev are special forms of Jacobi polynomial; in other words, the introduction of Jacobi polynomials enriches the selection and diversity of displacement functions.

The free, steady, and transient vibration characteristics of the rectangular plate have been well analyzed by utilizing the Rayleigh–Ritz method, in which the Newmark-β integration method is adopted to realize the time-domain solutions for transient vibration response, especially for the damped transient vibration response, as well as the good convergence and feasibility of the present method have been proved. The effectiveness of the proposed method is verified by comparing it with the existing literature and FEM results.

The calculation results of the vibration characteristics for the rectangular plate under different boundary conditions, structural scale parameters, and load types are given, indicating that the constraint conditions and geometric dimensions have a significant impact on the dynamic response of the rectangular plate. The results of the study can support the vibration control of the plate structure and further enrich the research data of the JRM.

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Acknowledgments

This research was supported by Harbin Engineering University and Naval Research Institute.

  1. Funding information: This study was funded by the National Natural Science Foundation of China (U2006229, 52101351), Ph.D. Student Research and Innovation Fund of the Fundamental Research Funds for the Central Universities (3072020GIP0102), and “Quick Response” Special Project of Harbin Engineering University (3072020CFT0111).

  2. Author Contributions: Di Jia: Writing and Editing. Cong Gao: Methodology and investigation. Yuliang Yang: Editing and Revision. Fuzhen Pang: Conceptualization and Methodology. Haichao Li: Revision. Yuan Du: Validation.

  3. Conflict of interest: The authors declare that there is no conflict of interest regarding the publication of this paper.

  4. Data availability statement: The data used to support the findings of this study are included within the article.

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Received: 2021-12-15
Revised: 2022-02-26
Accepted: 2022-03-29
Published Online: 2022-08-03

© 2022 Di Jia et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/rams-2022-0033
Keywords for this article
Jacobi polynomials; steady and transient vibration; rectangular plate; general boundary condition
Creative Commons
BY 4.0

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  16. Multiple wide band gaps in a convex-like holey phononic crystal strip
  17. Response analysis and optimization of the air spring with epistemic uncertainties
  18. Molecular dynamics of C–S–H production in graphene oxide environment
  19. Residual stress relief mechanisms of 2219 Al–Cu alloy by thermal stress relief method
  20. Characteristics and microstructures of the GFRP waste powder/GGBS-based geopolymer paste and concrete
  21. Development and performance evaluation of a novel environmentally friendly adsorbent for waste water-based drilling fluids
  22. Determination of shear stresses in the measurement area of a modified wood sample
  23. Influence of ettringite on the crack self-repairing of cement-based materials in a hydraulic environment
  24. Multiple load recognition and fatigue assessment on longitudinal stop of railway freight car
  25. Synthesis and characterization of nano-SiO2@octadecylbisimidazoline quaternary ammonium salt used as acidizing corrosion inhibitor
  26. Perforated steel for realizing extraordinary ductility under compression: Testing and finite element modeling
  27. The influence of oiled fiber, freeze-thawing cycle, and sulfate attack on strain hardening cement-based composites
  28. Perforated steel block of realizing large ductility under compression: Parametric study and stress–strain modeling
  29. Study on dynamic viscoelastic constitutive model of nonwater reacted polyurethane grouting materials based on DMA
  30. Mechanical behavior and mechanism investigation on the optimized and novel bio-inspired nonpneumatic composite tires
  31. Effect of cooling rate on the microstructure and thermal expansion properties of Al–Mn–Fe alloy
  32. Research on process optimization and rapid prediction method of thermal vibration stress relief for 2219 aluminum alloy rings
  33. Failure prevention of seafloor composite pipelines using enhanced strain-based design
  34. Deterioration of concrete under the coupling action of freeze–thaw cycles and salt solution erosion
  35. Creep rupture behavior of 2.25Cr1Mo0.25V steel and weld for hydrogenation reactors under different stress levels
  36. Statistical damage constitutive model for the two-component foaming polymer grouting material
  37. Nano-structural and nano-constraint behavior of mortar containing silica aggregates
  38. Influence of recycled clay brick aggregate on the mechanical properties of concrete
  39. Effect of LDH on the dissolution and adsorption behaviors of sulfate in Portland cement early hydration process
  40. Comparison of properties of colorless and transparent polyimide films using various diamine monomers
  41. Study in the parameter influence on underwater acoustic radiation characteristics of cylindrical shells
  42. Experimental study on basic mechanical properties of recycled steel fiber reinforced concrete
  43. Dynamic characteristic analysis of acoustic black hole in typical raft structure
  44. A semi-analytical method for dynamic analysis of a rectangular plate with general boundary conditions based on FSDT
  45. Research on modification of mechanical properties of recycled aggregate concrete by replacing sand with graphite tailings
  46. Dynamic response of Voronoi structures with gradient perpendicular to the impact direction
  47. Deposition mechanisms and characteristics of nano-modified multimodal Cr3C2–NiCr coatings sprayed by HVOF
  48. Effect of excitation type on vibration characteristics of typical ship grillage structure
  49. Study on the nanoscale mechanical properties of graphene oxide–enhanced shear resisting cement
  50. Experimental investigation on static compressive toughness of steel fiber rubber concrete
  51. Study on the stress field concentration at the tip of elliptical cracks
  52. Corrosion resistance of 6061-T6 aluminium alloy and its feasibility of near-surface reinforcements in concrete structure
  53. Effect of the synthesis method on the MnCo2O4 towards the photocatalytic production of H2
  54. Experimental study of the shear strength criterion of rock structural plane based on three-dimensional surface description
  55. Evaluation of wear and corrosion properties of FSWed aluminum alloy plates of AA2020-T4 with heat treatment under different aging periods
  56. Thermal–mechanical coupling deformation difference analysis for the flexspline of a harmonic drive
  57. Frost resistance of fiber-reinforced self-compacting recycled concrete
  58. High-temperature treated TiO2 modified with 3-aminopropyltriethoxysilane as photoactive nanomaterials
  59. Effect of nano Al2O3 particles on the mechanical and wear properties of Al/Al2O3 composites manufactured via ARB
  60. Co3O4 nanoparticles embedded in electrospun carbon nanofibers as free-standing nanocomposite electrodes as highly sensitive enzyme-free glucose biosensors
  61. Effect of freeze–thaw cycles on deformation properties of deep foundation pit supported by pile-anchor in Harbin
  62. Temperature-porosity-dependent elastic modulus model for metallic materials
  63. Effect of diffusion on interfacial properties of polyurethane-modified asphalt–aggregate using molecular dynamic simulation
  64. Experimental study on comprehensive improvement of shear strength and erosion resistance of yellow mud in Qiang Village
  65. A novel method for low-cost and rapid preparation of nanoporous phenolic aerogels and its performance regulation mechanism
  66. In situ bow reduction during sublimation growth of cubic silicon carbide
  67. Adhesion behaviour of 3D printed polyamide–carbon fibre composite filament
  68. An experimental investigation and machine learning-based prediction for seismic performance of steel tubular column filled with recycled aggregate concrete
  69. Effects of rare earth metals on microstructure, mechanical properties, and pitting corrosion of 27% Cr hyper duplex stainless steel
  70. Application research of acoustic black hole in floating raft vibration isolation system
  71. Multi-objective parametric optimization on the EDM machining of hybrid SiCp/Grp/aluminum nanocomposites using Non-dominating Sorting Genetic Algorithm (NSGA-II): Fabrication and microstructural characterizations
  72. Estimating of cutting force and surface roughness in turning of GFRP composites with different orientation angles using artificial neural network
  73. Displacement recovery and energy dissipation of crimped NiTi SMA fibers during cyclic pullout tests
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