# Exact Free Vibration Analysis for Plate Built-Up Structures under Comprehensive Combinations of Boundary Conditions

Authors: Xiang Liu; Chen Xie; Han-cheng Dan
Journal: Shock and Vibration
 (2020)

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## Abstract

In this research, an exact dynamic stiffness model for spatial plate built-up structures under comprehensive combinations of different boundary conditions is newly proposed. Dynamic stiffness formulations for plate elements with 16 different types of supported opposite edges and arbitrarily supported boundary conditions along other edges are developed, which makes the dynamic stiffness method (DSM) more applicable to engineering problems compared to existing works. The Wittrick–Williams algorithm of the DSM is applied with the explicit expressions of theJ0 count for plate elements under all above support conditions. In return, there is no need to refine the element in the DSM, and thus, it becomes immensely efficient. Moreover, the present theory is applied for exact free vibration analysis within the whole frequency range of three built-up structures which are commonly encountered in engineering. The results show that the DSM gives exact results with as much as 100-fold computational efficiency advantage over the commercial finite element method. Besides, benchmark results are also provided.


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## Body

## 1. Introduction

Plate built-up structures normally serve as the main structures in a wide range of areas, such as rail transit, aerospace, automotive, and civil engineering. Excessive vibration and noise not only reduce the comfort of people but also may cause fatigue to structures, whereas free vibration properties are one of the most important and fundamental concerns in structural design. Therefore, it is virtually important to find efficient and accurate methods for the free vibration analysis of plate built-up structures. There are many well-developed analysis methods such as the finite element method (FEM) and boundary element method (BEM) for low-frequency vibration analysis and statistical energy analysis (SEA) for the high-frequency range. However, some built-up structures are usually characterized by a complex vibration form in which both long- and short-wavelength deformations occur simultaneously. As a result, both the FEM/BEM and SEA become inapplicable to this midfrequency problem, and alternative methods should be resorted to the study [1].One of the powerful alternatives is the dynamic stiffness method (DSM), whose shape functions are the exact general solutions of the governing differential equations (GDEs). Therefore, the exact natural frequencies and modal shapes of the structures within the whole frequency range can be obtained. Moreover, another benefit of the DSM is that the structure does not need to be meshed unless the geometry and material are discontinuous, which indicates that an infinite number of natural modes can be solved by using extremely few degrees of freedom (DOFs). Thus, it is obvious that the DSM is very efficient compared to the FEM. It is worth mentioning that compared to other analytical methods, the DSM applies an efficient and robust algorithm, the Wittrick–Williams (WW) algorithm [2], which guarantees that no natural frequency is missed. In addition, the DSM elements can be assembled directly just as the FEM to model the complex structure.The DSM was originally proposed by Kolousek [3] in 1941. Subsequently, many authors [4–9] have done numerous works on the DSM of beam structures, which greatly promote the application of the DSM in engineering problems. In addition, there are some studies on the dynamic analysis of plate structures using dynamic stiffness theory. For example, Williams and Wittrick [10] performed vibration and buckling analysis on isotropic and anisotropic plates and compiled computer programs called VIPASA [10], VICON [11], and VICONOPT [12, 13]. The method is later applied by Williams and Banerjee [14] to compute the modal densities of structures. Bercin et al. [15, 16] developed the dynamic stiffness matrices of plate built-up structures and studied the contribution of inplane modes in the mid-high-frequency domain. Boscolo and Banerjee [17] developed the explicit expressions of the dynamic stiffness matrices for out-of-plane free vibration using classical plate theory and first-order shear deformation theory, respectively. Wu et al. [18–20] applied the dynamic stiffness method for power flow analysis of plate built-up structures. However, all above researches are restricted to plates with two opposite sides simply supported in one form, where the opposite edges should be the so-called “SS1” type for inplane vibration (normal stress and tangential displacement are zero) and “S” for out-of-plane vibration (transverse displacement and moment are zero). However, in engineering applications, there are more types of boundary conditions which cannot be modeled by the above boundary conditions. The Ritz method was used to study the free vibration of completely free rectangular shallow shell structures [21]. It is a weak-form-based method because the GDEs and boundary conditions are sometimes satisfied in a variational sense [22]. In contrast, the superposition method adopted by Gorman et al. [23–27] can obtain the solutions that satisfy both GDEs and boundary conditions in a strong manner. This method was applied to study the exact solution of the plates under various boundary conditions and pointed out two types of simply supported boundary conditions for inplane vibration [28]. Xing et al. [29–32] developed the exact modal solutions for inplane vibration of individual plates with opposite edges SS1 and SS2 (tangential stress and normal displacement are zero) supported and transverse vibration with opposite edges simply supported (S) and/or guided (G). However, all existing researches [29–32] focus on single plates and are inapplicable to plate assemblies. To the best knowledge of the authors, there have not been exact dynamic stiffness formulations for plate assemblies with the opposite edges supported by “G” and “SS2” boundary conditions.Furthermore, when applying the DSM to modal analysis, there are many methods for obtaining the eigenvalues from the DS matrix, like the determinantal methods [15, 16, 18–20], but they are inefficient and meanwhile very likely to miss some modal solutions [28]. These shortcomings can be overcome by the Wittrick–Williams (WW) algorithm, which is an extremely efficient and accurate algorithm for the DSM; however, the J0 count in the algorithm is an important and difficult problem [33]. J0 is the number of natural frequencies below the trial frequency when all the nodes of the structure are clamped. A majority of researches [17, 34–36] discretized the structure into a finer dynamic stiffness mesh to ensure that J0 is equal to zero, which greatly reduces the computational efficiency, and the merit of the DSM is not brought into full play.This paper develops the dynamic stiffness formulations for plate elements with opposite edges supported by any combinations of S1, S2 for inplane vibration and S, G for out-of-plane vibration and arbitrarily supported boundary conditions along other edges. That is to say, there are 16 kinds of opposite-edge-supported condition combinations, i.e., four for inplane vibration, S1-S1, S2-S2, S1-S2, and S2-S1, and another four for transverse vibration, S-S, G-G, S-G, and G-S. As a consequence, the number of boundary conditions of a plate element considered by the DSM increases from 100 to 1600, which makes the DSM more applicable to engineering problems. More details about the opposite-edge-supported boundary conditions are given in Section2.1.At the same time, the mode count (J0) of the WW algorithm under all possible opposite-edge-support conditions is formulated analytically in this study. Thus, minimal degrees of freedom are necessary for the DSM to model complex structures, which makes the DSM an efficient analytical method within the whole frequency ranges. This work greatly enhances the superiority of the DSM over the FEM in computational efficiency. The current research can also be used for power flow analysis and puts forward efficient analytical solutions for important parameters (modal density, coupling loss factor [37], etc.) for other methods like the statistical energy analysis (SEA) method.This paper is organized as follows: In Section2.1, the boundary conditions in this research are detailed. Then, the formulations of dynamic stiffness matrices for inplane vibration under different types of opposite-edge-support conditions are developed (Section 2.2). Next, the expressions for three different support conditions of out-of-plane vibration are presented (Section 2.3). Section 2.4 shows the assembly procedure of the plate elements. Then, the J0 formulations are solved in Section 2.5. In Section 3, the natural frequencies of the individual plate for inplane (Section 3.1) and out-of-plane (Section 3.2) vibrations computed by the DSM are presented compared to the FEM solutions. Section 3.3 demonstrates the accuracy and calculational efficiency studies of the DSM on three kinds of plate built-up structures which are widely used in engineering. Finally, some conclusions of this work are drawn in Section 4.


## 2. Theory

This section describes the development of dynamic stiffness (DS) formulations for a plate element under comprehensive combinations of different boundary conditions.Section 2.1 introduces the notations and different opposite-edge-support conditions of the plate element. Section 2.2 and Section 2.3 exhibit the development of elemental DS formulations for inplane vibration and out-of-plane vibration, respectively. The assembly procedure of the elemental matrices is shown in Section 2.4, and the algorithm of the DSM is improved in Section 2.5.


### 2.1. Different Boundary Conditions of a Plate Element

Figure1 shows a plate element with a pair of opposite edges supported. For notational convenience, the boundaries y = 0 and y = L are denoted by supported boundaries (SBs), whereas the boundaries x = 0 and x = b are represented by nodal boundaries (NBs), and the corresponding boundary conditions are abbreviated as SBC and NBC, respectively.Figure 1
Coordinate system and notations for displacements (a) and forces (b).
(a)
(b)Table1 lists the physical meanings of four types of SBCs, namely, S1 and S2 SBCs for inplane vibration (see Figure 2) and S and G for out-of-plane vibration (Figure 3), as well as all possible SBC combinations of a plate. More specifically, an S1 SBC along the edge (either y = 0 or y = L) means inplane vibration is constrained in the x direction but can move freely in the y direction; an S2 SBC means that the edge is fixed in the y direction but can move freely in the x direction.Table 1
Physical meanings of all possible opposite-edge simple supports for a plate.
SBC
y = 0 and y = L
x = 0 and x = b
Combinations
Inplane vibration
S1
u
=
0 and Nyy=0
υ
=
0 and Nxx=0
S1-S1, S2-S2, S1-S2, S2-S1
S2
υ
=
0 and Nyx=0
u
=
0 and Nxy=0
Out-of-plane vibration
S
w
=
0 and M=0
w
=
0 and M=0
S-S, G-G, S-G, G-S
G
ϕ
x
=
0 and T=0
ϕ
y
=
0 and T=0Figure 2
Three types of SBCs for inplane vibration: (a) S1-S1; (b) S1-S2; (c) S2-S2.
(a)
(b)
(c)Figure 3
Three types of SBCs for out-of-plane vibration: (a) S-S; (b) S-G; (c) G-G.
(a)
(b)
(c)It is known that existing dynamic stiffness formulations in the literature could have ten combinations of the NBCs for inplane and transverse vibrations, which are extended in the current research for the NBCs for both inplane and transverse vibrations (see Table2) to 40, respectively, in Table 2. This will no doubt broaden the application scope of the DSM. The letter “F” in Table 2 denotes a free edge and “C” indicates a clamped boundary. The sequence of the SBC combinations is supported boundary1-supported boundary2 and of NBC combinations is nodal boundary1-nodal boundary2.Table 2
All possible BCs of a plate for inplane and out-of-plane vibrations.
NBCs
SBCs
NBC no.
Inplane vibration
F-F, F-C, F-G, F-S,
S-S
10
C-C, C-G, C-S,
S-G
10
G-G, G-S, S-S
G-G
10
G-S
10
Out-of-plane vibration
F-F, F-C, F-S2, F-S1,
S1-S1
10
C-C, C-S2, C-S1,
S1-S2
10
S2-S2, S2-S1, S1-S1
S2-S2
10
S2-S1
10Thus, the possible boundary conditions of a plate element considering both inplane and out-of-plane vibrations are increased from 100 (10 × 10) to 1600 (40 × 40), which greatly improves the engineering applicability of the DSM.


### 2.2. Dynamic Stiffness Formulation for Inplane Vibration of a Plate Element

This section focuses on inplane vibration of plate elements with three different SBCs, as provided in Table2, namely, S1-S1, S2-S2, and S1-S2 (it is easily seen that the DS formulation for a rectangular plate element with S2-S1 SBC should be similar to that with S1-S2 SBC and thus is omitted here for conciseness).By using Hamilton’s principle, the governing differential equation (GDE) in the time domain for the inplane free vibratory motion can be deduced as follows:(1)Eh1−ν2∂2ux,y,t∂x2+1−ν2∂2ux,y,t∂y2+ν+12∂2υx,y,t∂x∂y−ρh∂2ux,y,t∂t2=0,Eh1−ν2∂2υx,y,t∂y2+1−ν2∂2υx,y,t∂x2+ν+12∂2ux,y,t∂x∂y−ρh∂2υx,y,t∂t2=0,which can be transferred into the frequency domain as(2)∂2Ux,y∂x2+1−ν2∂2Ux,y∂y2+ν+12∂2Vx,y∂x∂y+1−ν2Ghρhω2Ux,y=0,∂2Vx,y∂y2+1−ν2∂2Vx,y∂x2+ν+12∂2Ux,y∂x∂y+1−ν2Ghρhω2Vx,y=0,and the natural BCs take the form(3)δu:Nxx=2Gh1−ν∂ux,y,t∂x+ν∂υx,y,t∂y,δυ:Nxy=Gh∂ux,y,t∂y+∂υx,y,t∂x,where E is Young’s modulus, h the thickness, ν the Poisson ratio, ρ the density, ω the angular frequency, and G=E/21+ν the shear modulus.The triangles in Figure2 indicate the boundary constraints of the inplane vibration. Based on Table 1, the general solutions of equation (2) for inplane vibration with three different SBCs should take the following forms:(4)Ux,y=∑m=0∞Umxsinαmy,S1−S1or  S1−S2,∑m=0∞Umxcosαmy,S2−S2,Vx,y=∑m=0∞Vmxcosαmy,S1−S1or  S1−S2,∑m=0∞Vmxsinαmy,S2−S2,with(5)αm=mπL,m=0,1,2,…,S1‐S1or  S2‐S2,2m−1π2L,m=1,2,3…,S1‐S2,where the letter m stands for the half wave number of a plate element in the y direction and L is the length of the plate.Substituting equation (4) into (2) leads to(6)d2Umxdx2−a1αm2−a3ω2Umx∓a2αmdVmxdx=0,a1d2Vmxdx2−αm2−a3ω2Vmx±a2αmdUmxdx=0,with(7)a1=1−ν2,a2=1+ν2,a3=ρ1−ν2E,where “∓” and “±” take the first sign for S1-S1 and S1-S2 and the second sign for S2-S2, and they are the same in the remainder of Section 2.2.By observing equation (6), it is necessary to divide it into two cases to solve the GDE: m≠0 and m=0.(1)
m ≠ 0. Substituting Um=erx, Vm=δerx into equation (6) leads to(8)r2erx−a1αm2−a3ω2erx∓a2αmrδerx=0,a1r2δerx−αm2−a3ω2δerx±a2αmrerx=0.After some fundamental derivations, the exact general solution of equation (6) can be represented as follows:(9)Umx=A1mcoshr1x+A2msinhr1x+A3mcoshr2x+A4msinhr2x,Vmx=∓δ1A2mcoshr1x∓δ1A1msinhr1x∓δ2A4mcoshr2x∓δ2A3msinhr2x,where(10)r1,2=2αm2E+ν−3ν+1ω2ρ±1+ν4ω4ρ22E,(11)δi=a1αm2−a3ω2−ri2a2αmri,i=1,2.By applying equation (9) to (3), the expressions of shear force and bending moment are determined as follows:(12)Nxxmx,y=Nxxmxsinαmy=2Gh1−νA1mr1+ναmδ1sinhr1x+A2mr1+ναmδ1coshr1x+A3mr2+ναmδ2sinhr2x+A4mr2+ναmδ2coshr2xsinαmy,Nxymx,y=Nxymxcosαmy=±GhA1mαm−δ1r1coshr1x+A2mαm−δ1r1sinhr1x+A3mαm−δ2r2coshr2x+A4mαm−δ2r2sinhr2xcosαmy.When the SBC is S2-S2,sinαmy in equation (12) becomes cosαmy, while cosαmy becomes sinαmy.The force and displacement NBCs of the plate element (Figure4) are given as follows:(13)alongx=0,U=U1,V=V1,Nxx=−Nxx1,Nxy=−Nxy1,alongx=b,U=U2,V=V2,Nxx=Nxx2,Nxy=Nxy2.Figure 4
Force and displacement NBCs of a plate element.The relationship between the displacement NBCs and unknown coefficients is determined by applying the BCs for displacements to equation (9) as follows:(14)d=U1V1U2V2=10100∓δ10∓δ2Ch1Sh1Ch2Sh2∓δ1Sh1∓δ1Ch1∓δ2Sh2∓δ2Ch2A1A2A3A4,where Chi=coshrib and Shi=sinhrib.Similarly, inserting force NBCs into equation (12) gives(15)f=Nxx1Nxy1Nxx2Nxy2=Gh0H1∓Ρ100H2∓P20−H1Sh1−H1Ch1±P1Ch1±P1Sh1−H2Sh2−H2Ch2±P2Ch2±P2Sh2A1A2A3A4,where Hi=2ri+αmδiν/ν−1 and Pi=αm−δiriBy eliminating the coefficient vector of equations (14) and (15), the relationship between the force vector f and the displacement vector d can be written as f=Kiωd; thus, the dynamic stiffness matrix for inplane vibration Kiω is derived as follows:(16)Kiω=GhΔik11ik12ik22ik13ik14i−k14ik24iSymk11i−k12ik22i,where Kiω is a symmetric matrix, which is composed of 6 elements k11i, k12i, k13i, k14i, k22i, and k24i. The expressions of these 6 elements are as follows:(17)Δi=2δ1δ21−Ch1Ch2+Sh1Sh2δ12+δ22,k11i=2r2δ1−r1δ2Ch1Sh2δ2−Ch2Sh1δ1ν−1,k12i=∓21−Ch1Ch2r2δ1+r1+2αmνδ1δ2+Sh1Sh2δ1r1+αmνδ1+δ2r2+αmνδ2ν−1,k13i=2r1δ2−r2δ1Sh2δ2−Sh1δ1ν−1,k14i=∓2Ch1−Ch2r1δ2−r2δ1ν−1,k22i=r1δ1−r2δ2Ch1Sh2δ1−Ch2Sh1δ2,k24i=r2δ2−r1δ1Sh2δ1−Sh1δ2.(2)
m = 0. From equation (4), it is easily seen that the solution procedure in the case of m = 0 is different from the case of m ≠ 0 when the SBs of the plate are constrained by S1-S1 and S2-S2. For the sake of brevity, this paper only provides the detailed derivation when the SBC is S2-S2. A similar procedure can be performed when the SBC is S1-S1, but here we only include the final formulation.It is obvious that the plate element can move in the direction ofUx,y with Ux,y=Umxcosαmy and the displacement of Vx,y is always equal to zero by applying m=0 to equation (4). As a consequence, equation (6) can be simplified as(18)d2Umxdx2+a3ω2Umx=0,where a3 is given by equation (7). The general solution of equation (18) can be written in the form(19)Umx=A1′cosrx+A2′sinrx,with r=ωa3.The function for force of inplane vibration is defined by substituting equation (19) into (3) as follows:(20)Nxxmx,y=Nxxmx=Ehr1−v2−A1′sinrx+A2′cosrx.The BCs in this case are given by(21)atx=0,U=U1,Nxx=−Nxx1,atx=b,U=U2,Nxx=Nxx2.Applying the displacement NBCs of equation (21) to (19) leads to(22)U1U2=10CSA1′A2′,where C=cosrb and S=sinrb.Inserting the force NBCs of equation (21) to (20) gives(23)Nxx1Nxx2=Ehr1−v20−1−SCA1′A2′.The dynamic stiffness matrix for inplane vibration in this situation is resolved by eliminating the constantsA1′ and A2′ of equations (22) and (23):(24)Kiω=Ghrk11′ik12′ik12′ik11′i,where the two elements of Kiω are exhibited as follows:(25)k11′i=21−νcotrb,k12′i=−21−νcscrb.Similarly, when the SBC is S1-S1 andm = 0, Ux,y and force of Nxxx,y are always equal to zero. In this case, the dynamic stiffness matrix takes a similar form of equation (24), whereas the entries are slightly different [34].(26)k11′i=cotrb,k12′i=−cscrb,with r=ωa3/a1.


### 2.3. Dynamic Stiffness Formulation for Out-of-Plane Vibration of a Plate Element

In this section, the dynamic stiffness (DS) formulations are developed for the out-of-plane vibration of a plate element with three different SBCs, i.e., S-S, S-G, and G-G (it is easily seen that the DS formulation of the S-G SBC is the same as that of the G-S SBC for transverse free vibration of a rectangular plate element).The DS matrix for out-of-plane vibration can be developed in a similar way as inplane vibration. Since the matrix for plate elements under the S-S SBC has been developed previously, e.g., by Boscolo and Banerjee [17], the expressions under the S-G and G-G SBCs are obtained in this study.The GDE in the time domain of out-of-plane vibration is given by(27)D∂4wx,y,t∂x4+2∂4wx,y,t∂x2∂y2+∂4wx,y,t∂y4+ρh∂2wx,y,t∂t2=0,which can be transferred into the frequency domain as(28)∂4Wx,y∂x4+2∂4Wx,y∂x2∂y2+∂4Wx,y∂y4−ρhω2DWx,y=0.The natural BCs take the form(29)δw:Vx=−D∂3wx,y,t∂x3+2−ν∂3wx,y,t∂x∂y2,δϕy:Mxx=−D∂2wx,y,t∂x2+ν∂2wx,y,t∂y2,where D=Eh3/121−ν2 is the bending stiffness of the plate.Figure3 shows the boundary constraints of the transverse vibration. The general solutions of equation (28) with three different SBCs can be written as(30)Wx,y=∑m=1∞Wmnxsinαomy,S‐S or S‐G,∑m=0∞Wmnxcosαomy,G‐G,with(31)αom=mπL,m=1,2,3…,S‐S,mπL,m=0,1,2,…,G‐G,2m−1π2L,m=1,2,3…,S‐G.Substituting equation (30) into (28) gives (except for the case of G-G with m=0)(32)d4Wmxdx4−2αom2d2Wmxdx2+αom4−ρhω2DWmx=0.In the case of the SBC being G-G, whenm=0, equation (32) becomes(33)d4Wmxdx4−ρhω2DWmx=0.It is found that the dynamic stiffness formulation for a plate element under the above two SBCs is very similar to the case under the S-S SBC. For the sake of completeness, the dynamic stiffness matrix is reported here as(34)Koω=DΔok11ok12ok22ok13ok14o−k14ok24oSymk11o−k12ok22o,where Koω is a symmetric matrix, which is composed of 6 elements k11o, k12o, k13o, k14o, k22o, and k24o; the expressions of these 6 elements are as follows:(35)Δo=r12Sh1Sh2+r22Sh1Sh2+r1r2Ch1−Ch22−Sh12−Sh22,k11o=r2R1−r1R2Ch2Sh1R1−Ch1Sh2R2,k12o=r1R2Ch1Ch2+Sh12−Ch12−R1Sh1Sh2+r2R1Ch1Ch2+Sh22−Ch22−R2Sh1Sh2,k13o=L2−L1r2Ch2Sh1−r1Ch1Sh2,k14o=r2R1−r1R2r2Sh2−r1Sh1,k22o=Ch2−Ch1r2R1−r1R2,k24o=L1−L2r2Sh1−r1Sh2,where ri=αom2±ωρh/D i=1,2, “±” takes the first sign for i=1 and the second sign for i=2, Shi and Chi are the same as in Section 2.2, Ri=ri3−αom2ri2−ν, and Li=ri2−αom2ν.


### 2.4. Coordinate Transformation, Assembly Procedure, and Nodal Boundary Condition Applications

As the dynamic stiffness formulations of inplane and out-of-plane vibrations are derived in the previous sections, the plate elements can be assembled to model plate built-up structures. In this section, a transformation matrix is developed to transform the elemental dynamic stiffness matrix from the local coordinate system to the global coordinate system. Then, all elemental dynamic stiffness matrices of plate elements are assembled directly to obtain a global dynamic stiffness matrix of the complex plate built-up structure, and any boundary conditions can be applied on the nodal boundaries. Finally, modal analysis is performed by using the Wittrick–Williams algorithm.


#### 2.4.1. Coordinate Transformation

Taking the structure in Figure5 as an example, assuming that the coordinate system of plate 1 (P1 of Figure 5) coincides with the global coordinate system, the displacement and force vectors of plate 2 (P2 of Figure 5) need to be converted to the global coordinate system for the assembly purpose.Figure 5
A two-plate built-up structure.
(a)
(b)SupposingUl,Vl,Wl,ϕl is the displacement vector of a nodal boundary in the local coordinate system of plate 2, when m ≠ 0, the displacement vector in the global coordinate system Ug,Vg,Wg,ϕg is given by(36)UgVgWgϕg=cosθ0sinθ00100−sinθ0cosθ00001UlVlWlϕl.The transformation matrixT of the displacement vector of a plate element can be written as(37)T=cosθ0sinθ0000001000000−sinθ0cosθ00000000100000000cosθ0sinθ0000001000000−sinθ0cosθ000000001.It should be emphasized that, in the case ofm = 0, the displacement of Vl is equal to 0 when the SBC is S2-S2; then, the displacement vector Ug,Wg,ϕg takes the form(38)UgWgϕg=cosθsinθ0−sinθcosθ0001UlWlϕl.In this case,T becomes(39)T=cosθsinθ0000−sinθcosθ0000001000000cosθsinθ0000−sinθcosθ0000001.Similarly, the transformation matrixT for the case of S1-S1 and m = 0 is(40)T=1000000cosθ00000010000001000000cosθ0000001.The transformation matrix of the forces can also be defined using the same method, and it is exactly the transposed matrix ofT. Thus, the spatial transformation function of plate elements is given as follows:(41)Keg=TTKelT.


#### 2.4.2. Assembly Procedure

Once all degrees of freedom (DOFs) of plate elements have been transferred into the global coordinate system, the plate elements can be assembled. It should be noted that there are two types of boundaries, namely, nodal boundaries (NBs) and supported boundaries (SBs), and only NBs can be used for the nodal assembly. The procedure is similar to the finite element method. Take Figure5 as an example, in which plate1 and plate2 share the same nodal boundary L4. Equations (42) and (43) are the dynamic stiffness matrices of plate1 and plate2, respectively. Therefore, the assembly of plate1 and plate2 becomes a 12 × 12 matrix by summing the entries at the common nodal boundary L4, namely,(42)Nxx21Nxy21T21M21Nxx41Nxy41T41M41=ΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔU21V21W21ϕ21U41V41W41ϕ41,(43)Nxx42Nxy42T42M42Nxx62Nxy62T62M62=■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■U42V42W42ϕ42U62V62W62ϕ62,(44)NxNxy2T2M2Nxx4Nxy4T4M4Nxx6Nxy6T6M6=ΔΔΔΔΔΔΔΔ0000ΔΔΔΔΔΔΔΔ0000ΔΔΔΔΔΔΔΔ0000ΔΔΔΔΔΔΔΔ0000ΔΔΔΔΔ+■Δ+■Δ+■Δ+■■■■■ΔΔΔΔΔ+■Δ+■Δ+■Δ+■■■■■ΔΔΔΔΔ+■Δ+■Δ+■Δ+■■■■■ΔΔΔΔΔ+■Δ+■Δ+■Δ+■■■■■0000■■■■■■■■0000■■■■■■■■0000■■■■■■■■0000■■■■■■■■U2V2W2ϕ2U4V4W4ϕ4U6V6W6ϕ6.


#### 2.4.3. Application of Nodal Boundary Conditions

In this research, SBs denote a pair of boundaries in which any combinations of simply supported and/or guided supports can be applied, and any arbitrary classical BCs can be applied to the NBs by deleting the certain columns and rows for corresponding fixed DOFs in equation (44). For example, when L2 and L4 are free with L6 fully clamped, the last 4 values in the displacement vector of equation (44) are zero. Then, the global dynamic stiffness matrix of the structure in Figure 5 can be written as(45)KFFCg=ΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔ+■Δ+■Δ+■Δ+■ΔΔΔΔΔ+■Δ+■Δ+■Δ+■ΔΔΔΔΔ+■Δ+■Δ+■Δ+■ΔΔΔΔΔ+■Δ+■Δ+■Δ+■.By the similar way, various complex plate built-up structures can be assembled in this research.


### 2.5. Wittrick–William Algorithm andJ0 Counts

Once the dynamic stiffness matrix is developed, the Wittrick–Williams algorithm can be applied to compute the natural frequenciesω of structures. The following equation is the key equation of the Wittrick–Williams algorithm, which is used to calculate the mode count J when ω is lower than the trial frequency ω∗:(46)J=J0+sKω∗,where Kω∗ is the elemental dynamic stiffness matrix when ω=ω∗, sKω∗ is the number of negative diagonal elements after upper triangular transformation by using Gauss elimination of Kω∗, and J0 is the number of natural frequencies between ω=0 and ω=ω∗ when the nodal boundaries (NBs) of the plate element are fully clamped.There is no doubt thatJ0 plays an important role in the Wittrick–Williams algorithm. However, calculating J0 is generally a difficult problem, and the traditional way is to refine the mesh to make sure J0=0 [17, 34–36]. Obviously, it will introduce unnecessary computational cost significantly.In this study, theJ0 problem of the plate element is resolved by applying an indirect method; it improves the computational efficiency of the dynamic stiffness method. According to the Wittrick–Williams algorithm, the mode count Jsm of the plate element with all NBs simply supported (or guided) when the half wave number in the y direction is m can be given by equation (46), which can be recast as(47)J0m=Jsm−sKmsω∗,where J0m is J0 when the half wave number in the y direction is m and Kmsω∗ is the dynamic stiffness matrix Kω∗ for a plate element with all NBs simply supported (or guided) when m is a certain value. The analytical expression for Jsm is given as follows: the first step is to establish the relationship between m, n, and ω, where n is the half wave number in the x direction.


#### 2.5.1.J0 Formulations for Inplane Vibration

Consider the BC of the plate element is S1S1S1S1, then the general solution of equation (2) for inplane vibration should take the following form:(48)Ux,y=∑m=1∞∑n=1∞Umnsinαmycosnπbx,Vx,y=∑m=0∞∑n=0∞Vmncosαmysinnπbx.Substituting equation (48) into (2), the relationship between m, n, and ω under the S1-S1 or S2-S2 SBC is determined by(49)1−ν22β4−1−ν3−ν4m2π2L2+n2π2b2β2+1−ν2m2π2L2+n2π2b22=0,where β=ωρ/G and b is the width of the plate.Solving equation (49) gives(50)Γ≥n2+ηm2,(51)Γ≥2×n2+ηm21−ν,with(52)η=bL,Γ=bβπ2.SupposeJsmi is Jsm for inplane vibration of a fully simply supported plate. Therefore, Jsmi consists of two parts which can be solved from equations (50) and (51) separately.Equation (50) can be expressed in the form(53)n2+ηm2−Γ≤0.So, natural modes exist for all integersm,n≥1, as shown in Figure 6. Essentially, Jsm1i is the mode count of an S2S2S2S2 or S1S1S2S1 plate when m is fixed which is obtained by solving n of equation (53):(54)Jsm1i=floorΓ−ηm2,where floorx indicates the largest integer not greater than x.Figure 6
J
s
m
1
i of S2S2S2S2 (a) and S1S1S2S1 (b).
(a)
(b)For an S1S1S1S1 plate, natural modes exist for allm,n≥0 except for m=n=0 (Figure 7). Thus, Jsm1i of S1S1S1S1 takes the form(55)Jsm1i=ceilΓ−ηm2,where ceilx is the least integer not less than x. It should be emphasized that m in equation (54) becomes 2m+1/2 for m=0,1,2…, when the BC is S1S1S2S1.Figure 7
J
s
m
1
i of S1S1S1S1.Similar to equation (50), equation (51) can be written as(56)21−νn2+ηm2−Γ≤0.When the BC is S2S2S2S2,m,n≥0 except m=n=0 (Figure 8). By solving equation (56), Jsm2i of S2S2S2S2 is given by(57)Jsm2i=ceil1−νΓ2−ηm2.Figure 8
J
s
m
2
i of S2S2S2S2.In Figure9, when the BC is S1S1S1S1 or S1S1S2S1, for m,n≥1, Jsm2i becomes(58)Jsm2i=floor1−νΓ2−ηm2.Figure 9
J
s
m
2
i of S1S1S1S1 (a) and S1S1S2S1 (b).
(a)
(b)In particular, when the BC is S1S1S2S1,m is replaced by 2m+1/2,m=0,1,2….AddingJsm1i and Jsm2i together gives(59)Jsmi=Jsm1i+Jsm2i.


#### 2.5.2.J0 Formulations for Transverse Free Vibration

Similar toSection 2.5.1, J0 of out-of-plane vibration is derived by an indirect method as well.Consider the BC of the plate element is SSSS, then the general solution of equation (28) for out-of-plane vibration should take the following form:(60)Wx,y=∑m=1∞∑n=1∞Wmnsinαomysinnπbx.Substituting equation (60) into (28), the relationship between m, n, and ω under the S-S SBC and S-S NBC is determined by(61)ωmn2=Dρhπ4L4Lb4n4+2Lb2M2n2+M4,which is also valid for G-G and S-G SBCs with the S-S NBC with(62)M=m,m=1,2,3…,S‐S,m,m=0,1,2,…,G‐G,2m−12,m=1,2,3,…,S‐G.SupposeJsmo is Jsm for out-of-plane vibration of the fully simply supported (or guided) plate and the half wave number in the y direction is m. By solving equation (61), one can solve Jsmo as(63)Jsmo=floorbπωρhD−αom2+iSG,with(64)iSG=0,S−S  or  S−G,1,G−G.


## 2.1. Different Boundary Conditions of a Plate Element

Figure1 shows a plate element with a pair of opposite edges supported. For notational convenience, the boundaries y = 0 and y = L are denoted by supported boundaries (SBs), whereas the boundaries x = 0 and x = b are represented by nodal boundaries (NBs), and the corresponding boundary conditions are abbreviated as SBC and NBC, respectively.Figure 1
Coordinate system and notations for displacements (a) and forces (b).
(a)
(b)Table1 lists the physical meanings of four types of SBCs, namely, S1 and S2 SBCs for inplane vibration (see Figure 2) and S and G for out-of-plane vibration (Figure 3), as well as all possible SBC combinations of a plate. More specifically, an S1 SBC along the edge (either y = 0 or y = L) means inplane vibration is constrained in the x direction but can move freely in the y direction; an S2 SBC means that the edge is fixed in the y direction but can move freely in the x direction.Table 1
Physical meanings of all possible opposite-edge simple supports for a plate.
SBC
y = 0 and y = L
x = 0 and x = b
Combinations
Inplane vibration
S1
u
=
0 and Nyy=0
υ
=
0 and Nxx=0
S1-S1, S2-S2, S1-S2, S2-S1
S2
υ
=
0 and Nyx=0
u
=
0 and Nxy=0
Out-of-plane vibration
S
w
=
0 and M=0
w
=
0 and M=0
S-S, G-G, S-G, G-S
G
ϕ
x
=
0 and T=0
ϕ
y
=
0 and T=0Figure 2
Three types of SBCs for inplane vibration: (a) S1-S1; (b) S1-S2; (c) S2-S2.
(a)
(b)
(c)Figure 3
Three types of SBCs for out-of-plane vibration: (a) S-S; (b) S-G; (c) G-G.
(a)
(b)
(c)It is known that existing dynamic stiffness formulations in the literature could have ten combinations of the NBCs for inplane and transverse vibrations, which are extended in the current research for the NBCs for both inplane and transverse vibrations (see Table2) to 40, respectively, in Table 2. This will no doubt broaden the application scope of the DSM. The letter “F” in Table 2 denotes a free edge and “C” indicates a clamped boundary. The sequence of the SBC combinations is supported boundary1-supported boundary2 and of NBC combinations is nodal boundary1-nodal boundary2.Table 2
All possible BCs of a plate for inplane and out-of-plane vibrations.
NBCs
SBCs
NBC no.
Inplane vibration
F-F, F-C, F-G, F-S,
S-S
10
C-C, C-G, C-S,
S-G
10
G-G, G-S, S-S
G-G
10
G-S
10
Out-of-plane vibration
F-F, F-C, F-S2, F-S1,
S1-S1
10
C-C, C-S2, C-S1,
S1-S2
10
S2-S2, S2-S1, S1-S1
S2-S2
10
S2-S1
10Thus, the possible boundary conditions of a plate element considering both inplane and out-of-plane vibrations are increased from 100 (10 × 10) to 1600 (40 × 40), which greatly improves the engineering applicability of the DSM.


## 2.2. Dynamic Stiffness Formulation for Inplane Vibration of a Plate Element

This section focuses on inplane vibration of plate elements with three different SBCs, as provided in Table2, namely, S1-S1, S2-S2, and S1-S2 (it is easily seen that the DS formulation for a rectangular plate element with S2-S1 SBC should be similar to that with S1-S2 SBC and thus is omitted here for conciseness).By using Hamilton’s principle, the governing differential equation (GDE) in the time domain for the inplane free vibratory motion can be deduced as follows:(1)Eh1−ν2∂2ux,y,t∂x2+1−ν2∂2ux,y,t∂y2+ν+12∂2υx,y,t∂x∂y−ρh∂2ux,y,t∂t2=0,Eh1−ν2∂2υx,y,t∂y2+1−ν2∂2υx,y,t∂x2+ν+12∂2ux,y,t∂x∂y−ρh∂2υx,y,t∂t2=0,which can be transferred into the frequency domain as(2)∂2Ux,y∂x2+1−ν2∂2Ux,y∂y2+ν+12∂2Vx,y∂x∂y+1−ν2Ghρhω2Ux,y=0,∂2Vx,y∂y2+1−ν2∂2Vx,y∂x2+ν+12∂2Ux,y∂x∂y+1−ν2Ghρhω2Vx,y=0,and the natural BCs take the form(3)δu:Nxx=2Gh1−ν∂ux,y,t∂x+ν∂υx,y,t∂y,δυ:Nxy=Gh∂ux,y,t∂y+∂υx,y,t∂x,where E is Young’s modulus, h the thickness, ν the Poisson ratio, ρ the density, ω the angular frequency, and G=E/21+ν the shear modulus.The triangles in Figure2 indicate the boundary constraints of the inplane vibration. Based on Table 1, the general solutions of equation (2) for inplane vibration with three different SBCs should take the following forms:(4)Ux,y=∑m=0∞Umxsinαmy,S1−S1or  S1−S2,∑m=0∞Umxcosαmy,S2−S2,Vx,y=∑m=0∞Vmxcosαmy,S1−S1or  S1−S2,∑m=0∞Vmxsinαmy,S2−S2,with(5)αm=mπL,m=0,1,2,…,S1‐S1or  S2‐S2,2m−1π2L,m=1,2,3…,S1‐S2,where the letter m stands for the half wave number of a plate element in the y direction and L is the length of the plate.Substituting equation (4) into (2) leads to(6)d2Umxdx2−a1αm2−a3ω2Umx∓a2αmdVmxdx=0,a1d2Vmxdx2−αm2−a3ω2Vmx±a2αmdUmxdx=0,with(7)a1=1−ν2,a2=1+ν2,a3=ρ1−ν2E,where “∓” and “±” take the first sign for S1-S1 and S1-S2 and the second sign for S2-S2, and they are the same in the remainder of Section 2.2.By observing equation (6), it is necessary to divide it into two cases to solve the GDE: m≠0 and m=0.(1)
m ≠ 0. Substituting Um=erx, Vm=δerx into equation (6) leads to(8)r2erx−a1αm2−a3ω2erx∓a2αmrδerx=0,a1r2δerx−αm2−a3ω2δerx±a2αmrerx=0.After some fundamental derivations, the exact general solution of equation (6) can be represented as follows:(9)Umx=A1mcoshr1x+A2msinhr1x+A3mcoshr2x+A4msinhr2x,Vmx=∓δ1A2mcoshr1x∓δ1A1msinhr1x∓δ2A4mcoshr2x∓δ2A3msinhr2x,where(10)r1,2=2αm2E+ν−3ν+1ω2ρ±1+ν4ω4ρ22E,(11)δi=a1αm2−a3ω2−ri2a2αmri,i=1,2.By applying equation (9) to (3), the expressions of shear force and bending moment are determined as follows:(12)Nxxmx,y=Nxxmxsinαmy=2Gh1−νA1mr1+ναmδ1sinhr1x+A2mr1+ναmδ1coshr1x+A3mr2+ναmδ2sinhr2x+A4mr2+ναmδ2coshr2xsinαmy,Nxymx,y=Nxymxcosαmy=±GhA1mαm−δ1r1coshr1x+A2mαm−δ1r1sinhr1x+A3mαm−δ2r2coshr2x+A4mαm−δ2r2sinhr2xcosαmy.When the SBC is S2-S2,sinαmy in equation (12) becomes cosαmy, while cosαmy becomes sinαmy.The force and displacement NBCs of the plate element (Figure4) are given as follows:(13)alongx=0,U=U1,V=V1,Nxx=−Nxx1,Nxy=−Nxy1,alongx=b,U=U2,V=V2,Nxx=Nxx2,Nxy=Nxy2.Figure 4
Force and displacement NBCs of a plate element.The relationship between the displacement NBCs and unknown coefficients is determined by applying the BCs for displacements to equation (9) as follows:(14)d=U1V1U2V2=10100∓δ10∓δ2Ch1Sh1Ch2Sh2∓δ1Sh1∓δ1Ch1∓δ2Sh2∓δ2Ch2A1A2A3A4,where Chi=coshrib and Shi=sinhrib.Similarly, inserting force NBCs into equation (12) gives(15)f=Nxx1Nxy1Nxx2Nxy2=Gh0H1∓Ρ100H2∓P20−H1Sh1−H1Ch1±P1Ch1±P1Sh1−H2Sh2−H2Ch2±P2Ch2±P2Sh2A1A2A3A4,where Hi=2ri+αmδiν/ν−1 and Pi=αm−δiriBy eliminating the coefficient vector of equations (14) and (15), the relationship between the force vector f and the displacement vector d can be written as f=Kiωd; thus, the dynamic stiffness matrix for inplane vibration Kiω is derived as follows:(16)Kiω=GhΔik11ik12ik22ik13ik14i−k14ik24iSymk11i−k12ik22i,where Kiω is a symmetric matrix, which is composed of 6 elements k11i, k12i, k13i, k14i, k22i, and k24i. The expressions of these 6 elements are as follows:(17)Δi=2δ1δ21−Ch1Ch2+Sh1Sh2δ12+δ22,k11i=2r2δ1−r1δ2Ch1Sh2δ2−Ch2Sh1δ1ν−1,k12i=∓21−Ch1Ch2r2δ1+r1+2αmνδ1δ2+Sh1Sh2δ1r1+αmνδ1+δ2r2+αmνδ2ν−1,k13i=2r1δ2−r2δ1Sh2δ2−Sh1δ1ν−1,k14i=∓2Ch1−Ch2r1δ2−r2δ1ν−1,k22i=r1δ1−r2δ2Ch1Sh2δ1−Ch2Sh1δ2,k24i=r2δ2−r1δ1Sh2δ1−Sh1δ2.(2)
m = 0. From equation (4), it is easily seen that the solution procedure in the case of m = 0 is different from the case of m ≠ 0 when the SBs of the plate are constrained by S1-S1 and S2-S2. For the sake of brevity, this paper only provides the detailed derivation when the SBC is S2-S2. A similar procedure can be performed when the SBC is S1-S1, but here we only include the final formulation.It is obvious that the plate element can move in the direction ofUx,y with Ux,y=Umxcosαmy and the displacement of Vx,y is always equal to zero by applying m=0 to equation (4). As a consequence, equation (6) can be simplified as(18)d2Umxdx2+a3ω2Umx=0,where a3 is given by equation (7). The general solution of equation (18) can be written in the form(19)Umx=A1′cosrx+A2′sinrx,with r=ωa3.The function for force of inplane vibration is defined by substituting equation (19) into (3) as follows:(20)Nxxmx,y=Nxxmx=Ehr1−v2−A1′sinrx+A2′cosrx.The BCs in this case are given by(21)atx=0,U=U1,Nxx=−Nxx1,atx=b,U=U2,Nxx=Nxx2.Applying the displacement NBCs of equation (21) to (19) leads to(22)U1U2=10CSA1′A2′,where C=cosrb and S=sinrb.Inserting the force NBCs of equation (21) to (20) gives(23)Nxx1Nxx2=Ehr1−v20−1−SCA1′A2′.The dynamic stiffness matrix for inplane vibration in this situation is resolved by eliminating the constantsA1′ and A2′ of equations (22) and (23):(24)Kiω=Ghrk11′ik12′ik12′ik11′i,where the two elements of Kiω are exhibited as follows:(25)k11′i=21−νcotrb,k12′i=−21−νcscrb.Similarly, when the SBC is S1-S1 andm = 0, Ux,y and force of Nxxx,y are always equal to zero. In this case, the dynamic stiffness matrix takes a similar form of equation (24), whereas the entries are slightly different [34].(26)k11′i=cotrb,k12′i=−cscrb,with r=ωa3/a1.


## 2.3. Dynamic Stiffness Formulation for Out-of-Plane Vibration of a Plate Element

In this section, the dynamic stiffness (DS) formulations are developed for the out-of-plane vibration of a plate element with three different SBCs, i.e., S-S, S-G, and G-G (it is easily seen that the DS formulation of the S-G SBC is the same as that of the G-S SBC for transverse free vibration of a rectangular plate element).The DS matrix for out-of-plane vibration can be developed in a similar way as inplane vibration. Since the matrix for plate elements under the S-S SBC has been developed previously, e.g., by Boscolo and Banerjee [17], the expressions under the S-G and G-G SBCs are obtained in this study.The GDE in the time domain of out-of-plane vibration is given by(27)D∂4wx,y,t∂x4+2∂4wx,y,t∂x2∂y2+∂4wx,y,t∂y4+ρh∂2wx,y,t∂t2=0,which can be transferred into the frequency domain as(28)∂4Wx,y∂x4+2∂4Wx,y∂x2∂y2+∂4Wx,y∂y4−ρhω2DWx,y=0.The natural BCs take the form(29)δw:Vx=−D∂3wx,y,t∂x3+2−ν∂3wx,y,t∂x∂y2,δϕy:Mxx=−D∂2wx,y,t∂x2+ν∂2wx,y,t∂y2,where D=Eh3/121−ν2 is the bending stiffness of the plate.Figure3 shows the boundary constraints of the transverse vibration. The general solutions of equation (28) with three different SBCs can be written as(30)Wx,y=∑m=1∞Wmnxsinαomy,S‐S or S‐G,∑m=0∞Wmnxcosαomy,G‐G,with(31)αom=mπL,m=1,2,3…,S‐S,mπL,m=0,1,2,…,G‐G,2m−1π2L,m=1,2,3…,S‐G.Substituting equation (30) into (28) gives (except for the case of G-G with m=0)(32)d4Wmxdx4−2αom2d2Wmxdx2+αom4−ρhω2DWmx=0.In the case of the SBC being G-G, whenm=0, equation (32) becomes(33)d4Wmxdx4−ρhω2DWmx=0.It is found that the dynamic stiffness formulation for a plate element under the above two SBCs is very similar to the case under the S-S SBC. For the sake of completeness, the dynamic stiffness matrix is reported here as(34)Koω=DΔok11ok12ok22ok13ok14o−k14ok24oSymk11o−k12ok22o,where Koω is a symmetric matrix, which is composed of 6 elements k11o, k12o, k13o, k14o, k22o, and k24o; the expressions of these 6 elements are as follows:(35)Δo=r12Sh1Sh2+r22Sh1Sh2+r1r2Ch1−Ch22−Sh12−Sh22,k11o=r2R1−r1R2Ch2Sh1R1−Ch1Sh2R2,k12o=r1R2Ch1Ch2+Sh12−Ch12−R1Sh1Sh2+r2R1Ch1Ch2+Sh22−Ch22−R2Sh1Sh2,k13o=L2−L1r2Ch2Sh1−r1Ch1Sh2,k14o=r2R1−r1R2r2Sh2−r1Sh1,k22o=Ch2−Ch1r2R1−r1R2,k24o=L1−L2r2Sh1−r1Sh2,where ri=αom2±ωρh/D i=1,2, “±” takes the first sign for i=1 and the second sign for i=2, Shi and Chi are the same as in Section 2.2, Ri=ri3−αom2ri2−ν, and Li=ri2−αom2ν.


## 2.4. Coordinate Transformation, Assembly Procedure, and Nodal Boundary Condition Applications

As the dynamic stiffness formulations of inplane and out-of-plane vibrations are derived in the previous sections, the plate elements can be assembled to model plate built-up structures. In this section, a transformation matrix is developed to transform the elemental dynamic stiffness matrix from the local coordinate system to the global coordinate system. Then, all elemental dynamic stiffness matrices of plate elements are assembled directly to obtain a global dynamic stiffness matrix of the complex plate built-up structure, and any boundary conditions can be applied on the nodal boundaries. Finally, modal analysis is performed by using the Wittrick–Williams algorithm.


### 2.4.1. Coordinate Transformation

Taking the structure in Figure5 as an example, assuming that the coordinate system of plate 1 (P1 of Figure 5) coincides with the global coordinate system, the displacement and force vectors of plate 2 (P2 of Figure 5) need to be converted to the global coordinate system for the assembly purpose.Figure 5
A two-plate built-up structure.
(a)
(b)SupposingUl,Vl,Wl,ϕl is the displacement vector of a nodal boundary in the local coordinate system of plate 2, when m ≠ 0, the displacement vector in the global coordinate system Ug,Vg,Wg,ϕg is given by(36)UgVgWgϕg=cosθ0sinθ00100−sinθ0cosθ00001UlVlWlϕl.The transformation matrixT of the displacement vector of a plate element can be written as(37)T=cosθ0sinθ0000001000000−sinθ0cosθ00000000100000000cosθ0sinθ0000001000000−sinθ0cosθ000000001.It should be emphasized that, in the case ofm = 0, the displacement of Vl is equal to 0 when the SBC is S2-S2; then, the displacement vector Ug,Wg,ϕg takes the form(38)UgWgϕg=cosθsinθ0−sinθcosθ0001UlWlϕl.In this case,T becomes(39)T=cosθsinθ0000−sinθcosθ0000001000000cosθsinθ0000−sinθcosθ0000001.Similarly, the transformation matrixT for the case of S1-S1 and m = 0 is(40)T=1000000cosθ00000010000001000000cosθ0000001.The transformation matrix of the forces can also be defined using the same method, and it is exactly the transposed matrix ofT. Thus, the spatial transformation function of plate elements is given as follows:(41)Keg=TTKelT.


### 2.4.2. Assembly Procedure

Once all degrees of freedom (DOFs) of plate elements have been transferred into the global coordinate system, the plate elements can be assembled. It should be noted that there are two types of boundaries, namely, nodal boundaries (NBs) and supported boundaries (SBs), and only NBs can be used for the nodal assembly. The procedure is similar to the finite element method. Take Figure5 as an example, in which plate1 and plate2 share the same nodal boundary L4. Equations (42) and (43) are the dynamic stiffness matrices of plate1 and plate2, respectively. Therefore, the assembly of plate1 and plate2 becomes a 12 × 12 matrix by summing the entries at the common nodal boundary L4, namely,(42)Nxx21Nxy21T21M21Nxx41Nxy41T41M41=ΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔU21V21W21ϕ21U41V41W41ϕ41,(43)Nxx42Nxy42T42M42Nxx62Nxy62T62M62=■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■U42V42W42ϕ42U62V62W62ϕ62,(44)NxNxy2T2M2Nxx4Nxy4T4M4Nxx6Nxy6T6M6=ΔΔΔΔΔΔΔΔ0000ΔΔΔΔΔΔΔΔ0000ΔΔΔΔΔΔΔΔ0000ΔΔΔΔΔΔΔΔ0000ΔΔΔΔΔ+■Δ+■Δ+■Δ+■■■■■ΔΔΔΔΔ+■Δ+■Δ+■Δ+■■■■■ΔΔΔΔΔ+■Δ+■Δ+■Δ+■■■■■ΔΔΔΔΔ+■Δ+■Δ+■Δ+■■■■■0000■■■■■■■■0000■■■■■■■■0000■■■■■■■■0000■■■■■■■■U2V2W2ϕ2U4V4W4ϕ4U6V6W6ϕ6.


### 2.4.3. Application of Nodal Boundary Conditions

In this research, SBs denote a pair of boundaries in which any combinations of simply supported and/or guided supports can be applied, and any arbitrary classical BCs can be applied to the NBs by deleting the certain columns and rows for corresponding fixed DOFs in equation (44). For example, when L2 and L4 are free with L6 fully clamped, the last 4 values in the displacement vector of equation (44) are zero. Then, the global dynamic stiffness matrix of the structure in Figure 5 can be written as(45)KFFCg=ΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔ+■Δ+■Δ+■Δ+■ΔΔΔΔΔ+■Δ+■Δ+■Δ+■ΔΔΔΔΔ+■Δ+■Δ+■Δ+■ΔΔΔΔΔ+■Δ+■Δ+■Δ+■.By the similar way, various complex plate built-up structures can be assembled in this research.


## 2.4.1. Coordinate Transformation

Taking the structure in Figure5 as an example, assuming that the coordinate system of plate 1 (P1 of Figure 5) coincides with the global coordinate system, the displacement and force vectors of plate 2 (P2 of Figure 5) need to be converted to the global coordinate system for the assembly purpose.Figure 5
A two-plate built-up structure.
(a)
(b)SupposingUl,Vl,Wl,ϕl is the displacement vector of a nodal boundary in the local coordinate system of plate 2, when m ≠ 0, the displacement vector in the global coordinate system Ug,Vg,Wg,ϕg is given by(36)UgVgWgϕg=cosθ0sinθ00100−sinθ0cosθ00001UlVlWlϕl.The transformation matrixT of the displacement vector of a plate element can be written as(37)T=cosθ0sinθ0000001000000−sinθ0cosθ00000000100000000cosθ0sinθ0000001000000−sinθ0cosθ000000001.It should be emphasized that, in the case ofm = 0, the displacement of Vl is equal to 0 when the SBC is S2-S2; then, the displacement vector Ug,Wg,ϕg takes the form(38)UgWgϕg=cosθsinθ0−sinθcosθ0001UlWlϕl.In this case,T becomes(39)T=cosθsinθ0000−sinθcosθ0000001000000cosθsinθ0000−sinθcosθ0000001.Similarly, the transformation matrixT for the case of S1-S1 and m = 0 is(40)T=1000000cosθ00000010000001000000cosθ0000001.The transformation matrix of the forces can also be defined using the same method, and it is exactly the transposed matrix ofT. Thus, the spatial transformation function of plate elements is given as follows:(41)Keg=TTKelT.


## 2.4.2. Assembly Procedure

Once all degrees of freedom (DOFs) of plate elements have been transferred into the global coordinate system, the plate elements can be assembled. It should be noted that there are two types of boundaries, namely, nodal boundaries (NBs) and supported boundaries (SBs), and only NBs can be used for the nodal assembly. The procedure is similar to the finite element method. Take Figure5 as an example, in which plate1 and plate2 share the same nodal boundary L4. Equations (42) and (43) are the dynamic stiffness matrices of plate1 and plate2, respectively. Therefore, the assembly of plate1 and plate2 becomes a 12 × 12 matrix by summing the entries at the common nodal boundary L4, namely,(42)Nxx21Nxy21T21M21Nxx41Nxy41T41M41=ΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔU21V21W21ϕ21U41V41W41ϕ41,(43)Nxx42Nxy42T42M42Nxx62Nxy62T62M62=■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■U42V42W42ϕ42U62V62W62ϕ62,(44)NxNxy2T2M2Nxx4Nxy4T4M4Nxx6Nxy6T6M6=ΔΔΔΔΔΔΔΔ0000ΔΔΔΔΔΔΔΔ0000ΔΔΔΔΔΔΔΔ0000ΔΔΔΔΔΔΔΔ0000ΔΔΔΔΔ+■Δ+■Δ+■Δ+■■■■■ΔΔΔΔΔ+■Δ+■Δ+■Δ+■■■■■ΔΔΔΔΔ+■Δ+■Δ+■Δ+■■■■■ΔΔΔΔΔ+■Δ+■Δ+■Δ+■■■■■0000■■■■■■■■0000■■■■■■■■0000■■■■■■■■0000■■■■■■■■U2V2W2ϕ2U4V4W4ϕ4U6V6W6ϕ6.


## 2.4.3. Application of Nodal Boundary Conditions

In this research, SBs denote a pair of boundaries in which any combinations of simply supported and/or guided supports can be applied, and any arbitrary classical BCs can be applied to the NBs by deleting the certain columns and rows for corresponding fixed DOFs in equation (44). For example, when L2 and L4 are free with L6 fully clamped, the last 4 values in the displacement vector of equation (44) are zero. Then, the global dynamic stiffness matrix of the structure in Figure 5 can be written as(45)KFFCg=ΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔ+■Δ+■Δ+■Δ+■ΔΔΔΔΔ+■Δ+■Δ+■Δ+■ΔΔΔΔΔ+■Δ+■Δ+■Δ+■ΔΔΔΔΔ+■Δ+■Δ+■Δ+■.By the similar way, various complex plate built-up structures can be assembled in this research.


## 2.5. Wittrick–William Algorithm andJ0 Counts

Once the dynamic stiffness matrix is developed, the Wittrick–Williams algorithm can be applied to compute the natural frequenciesω of structures. The following equation is the key equation of the Wittrick–Williams algorithm, which is used to calculate the mode count J when ω is lower than the trial frequency ω∗:(46)J=J0+sKω∗,where Kω∗ is the elemental dynamic stiffness matrix when ω=ω∗, sKω∗ is the number of negative diagonal elements after upper triangular transformation by using Gauss elimination of Kω∗, and J0 is the number of natural frequencies between ω=0 and ω=ω∗ when the nodal boundaries (NBs) of the plate element are fully clamped.There is no doubt thatJ0 plays an important role in the Wittrick–Williams algorithm. However, calculating J0 is generally a difficult problem, and the traditional way is to refine the mesh to make sure J0=0 [17, 34–36]. Obviously, it will introduce unnecessary computational cost significantly.In this study, theJ0 problem of the plate element is resolved by applying an indirect method; it improves the computational efficiency of the dynamic stiffness method. According to the Wittrick–Williams algorithm, the mode count Jsm of the plate element with all NBs simply supported (or guided) when the half wave number in the y direction is m can be given by equation (46), which can be recast as(47)J0m=Jsm−sKmsω∗,where J0m is J0 when the half wave number in the y direction is m and Kmsω∗ is the dynamic stiffness matrix Kω∗ for a plate element with all NBs simply supported (or guided) when m is a certain value. The analytical expression for Jsm is given as follows: the first step is to establish the relationship between m, n, and ω, where n is the half wave number in the x direction.


### 2.5.1.J0 Formulations for Inplane Vibration

Consider the BC of the plate element is S1S1S1S1, then the general solution of equation (2) for inplane vibration should take the following form:(48)Ux,y=∑m=1∞∑n=1∞Umnsinαmycosnπbx,Vx,y=∑m=0∞∑n=0∞Vmncosαmysinnπbx.Substituting equation (48) into (2), the relationship between m, n, and ω under the S1-S1 or S2-S2 SBC is determined by(49)1−ν22β4−1−ν3−ν4m2π2L2+n2π2b2β2+1−ν2m2π2L2+n2π2b22=0,where β=ωρ/G and b is the width of the plate.Solving equation (49) gives(50)Γ≥n2+ηm2,(51)Γ≥2×n2+ηm21−ν,with(52)η=bL,Γ=bβπ2.SupposeJsmi is Jsm for inplane vibration of a fully simply supported plate. Therefore, Jsmi consists of two parts which can be solved from equations (50) and (51) separately.Equation (50) can be expressed in the form(53)n2+ηm2−Γ≤0.So, natural modes exist for all integersm,n≥1, as shown in Figure 6. Essentially, Jsm1i is the mode count of an S2S2S2S2 or S1S1S2S1 plate when m is fixed which is obtained by solving n of equation (53):(54)Jsm1i=floorΓ−ηm2,where floorx indicates the largest integer not greater than x.Figure 6
J
s
m
1
i of S2S2S2S2 (a) and S1S1S2S1 (b).
(a)
(b)For an S1S1S1S1 plate, natural modes exist for allm,n≥0 except for m=n=0 (Figure 7). Thus, Jsm1i of S1S1S1S1 takes the form(55)Jsm1i=ceilΓ−ηm2,where ceilx is the least integer not less than x. It should be emphasized that m in equation (54) becomes 2m+1/2 for m=0,1,2…, when the BC is S1S1S2S1.Figure 7
J
s
m
1
i of S1S1S1S1.Similar to equation (50), equation (51) can be written as(56)21−νn2+ηm2−Γ≤0.When the BC is S2S2S2S2,m,n≥0 except m=n=0 (Figure 8). By solving equation (56), Jsm2i of S2S2S2S2 is given by(57)Jsm2i=ceil1−νΓ2−ηm2.Figure 8
J
s
m
2
i of S2S2S2S2.In Figure9, when the BC is S1S1S1S1 or S1S1S2S1, for m,n≥1, Jsm2i becomes(58)Jsm2i=floor1−νΓ2−ηm2.Figure 9
J
s
m
2
i of S1S1S1S1 (a) and S1S1S2S1 (b).
(a)
(b)In particular, when the BC is S1S1S2S1,m is replaced by 2m+1/2,m=0,1,2….AddingJsm1i and Jsm2i together gives(59)Jsmi=Jsm1i+Jsm2i.


### 2.5.2.J0 Formulations for Transverse Free Vibration

Similar toSection 2.5.1, J0 of out-of-plane vibration is derived by an indirect method as well.Consider the BC of the plate element is SSSS, then the general solution of equation (28) for out-of-plane vibration should take the following form:(60)Wx,y=∑m=1∞∑n=1∞Wmnsinαomysinnπbx.Substituting equation (60) into (28), the relationship between m, n, and ω under the S-S SBC and S-S NBC is determined by(61)ωmn2=Dρhπ4L4Lb4n4+2Lb2M2n2+M4,which is also valid for G-G and S-G SBCs with the S-S NBC with(62)M=m,m=1,2,3…,S‐S,m,m=0,1,2,…,G‐G,2m−12,m=1,2,3,…,S‐G.SupposeJsmo is Jsm for out-of-plane vibration of the fully simply supported (or guided) plate and the half wave number in the y direction is m. By solving equation (61), one can solve Jsmo as(63)Jsmo=floorbπωρhD−αom2+iSG,with(64)iSG=0,S−S  or  S−G,1,G−G.


## 2.5.1.J0 Formulations for Inplane Vibration

Consider the BC of the plate element is S1S1S1S1, then the general solution of equation (2) for inplane vibration should take the following form:(48)Ux,y=∑m=1∞∑n=1∞Umnsinαmycosnπbx,Vx,y=∑m=0∞∑n=0∞Vmncosαmysinnπbx.Substituting equation (48) into (2), the relationship between m, n, and ω under the S1-S1 or S2-S2 SBC is determined by(49)1−ν22β4−1−ν3−ν4m2π2L2+n2π2b2β2+1−ν2m2π2L2+n2π2b22=0,where β=ωρ/G and b is the width of the plate.Solving equation (49) gives(50)Γ≥n2+ηm2,(51)Γ≥2×n2+ηm21−ν,with(52)η=bL,Γ=bβπ2.SupposeJsmi is Jsm for inplane vibration of a fully simply supported plate. Therefore, Jsmi consists of two parts which can be solved from equations (50) and (51) separately.Equation (50) can be expressed in the form(53)n2+ηm2−Γ≤0.So, natural modes exist for all integersm,n≥1, as shown in Figure 6. Essentially, Jsm1i is the mode count of an S2S2S2S2 or S1S1S2S1 plate when m is fixed which is obtained by solving n of equation (53):(54)Jsm1i=floorΓ−ηm2,where floorx indicates the largest integer not greater than x.Figure 6
J
s
m
1
i of S2S2S2S2 (a) and S1S1S2S1 (b).
(a)
(b)For an S1S1S1S1 plate, natural modes exist for allm,n≥0 except for m=n=0 (Figure 7). Thus, Jsm1i of S1S1S1S1 takes the form(55)Jsm1i=ceilΓ−ηm2,where ceilx is the least integer not less than x. It should be emphasized that m in equation (54) becomes 2m+1/2 for m=0,1,2…, when the BC is S1S1S2S1.Figure 7
J
s
m
1
i of S1S1S1S1.Similar to equation (50), equation (51) can be written as(56)21−νn2+ηm2−Γ≤0.When the BC is S2S2S2S2,m,n≥0 except m=n=0 (Figure 8). By solving equation (56), Jsm2i of S2S2S2S2 is given by(57)Jsm2i=ceil1−νΓ2−ηm2.Figure 8
J
s
m
2
i of S2S2S2S2.In Figure9, when the BC is S1S1S1S1 or S1S1S2S1, for m,n≥1, Jsm2i becomes(58)Jsm2i=floor1−νΓ2−ηm2.Figure 9
J
s
m
2
i of S1S1S1S1 (a) and S1S1S2S1 (b).
(a)
(b)In particular, when the BC is S1S1S2S1,m is replaced by 2m+1/2,m=0,1,2….AddingJsm1i and Jsm2i together gives(59)Jsmi=Jsm1i+Jsm2i.


## 2.5.2.J0 Formulations for Transverse Free Vibration

Similar toSection 2.5.1, J0 of out-of-plane vibration is derived by an indirect method as well.Consider the BC of the plate element is SSSS, then the general solution of equation (28) for out-of-plane vibration should take the following form:(60)Wx,y=∑m=1∞∑n=1∞Wmnsinαomysinnπbx.Substituting equation (60) into (28), the relationship between m, n, and ω under the S-S SBC and S-S NBC is determined by(61)ωmn2=Dρhπ4L4Lb4n4+2Lb2M2n2+M4,which is also valid for G-G and S-G SBCs with the S-S NBC with(62)M=m,m=1,2,3…,S‐S,m,m=0,1,2,…,G‐G,2m−12,m=1,2,3,…,S‐G.SupposeJsmo is Jsm for out-of-plane vibration of the fully simply supported (or guided) plate and the half wave number in the y direction is m. By solving equation (61), one can solve Jsmo as(63)Jsmo=floorbπωρhD−αom2+iSG,with(64)iSG=0,S−S  or  S−G,1,G−G.


## 3. Results

An efficient program for exact modal analysis of individual plates and complex plate built-up structures was compiled in MATLAB based on the dynamic stiffness method described inSection 2. Section 3.1 demonstrates the application of the DSM (Section 2.3) to the out-of-plane free vibration of an individual plate with a comprehensive combination of boundary conditions. Section 3.2 applies the DSM (Section 2.2) to the inplane modal analysis of a single plate under different boundary conditions. Section 3.3 computes the natural modes of three plate built-up structures by the DSM, including a two-plate built-up structure (Section 3.3.1), an I-shaped plate built-up structure (Section 3.3.2), and a section of an extruded aluminum panel (Section 3.3.3), and compares the DSM with the FEM. All DSM and FEM computations are performed on the same computer with an 8 GB Intel Core i5-7200U processor.


### 3.1. Modal Analysis of Out-of-Plane Vibration for an Individual Plate

Table3 exhibits the first 8 natural frequencies of the out-of-plane vibration for a square plate (E = 72 GPa, ν=0.3, ρ=2800kg⋅m−3, h = 0.002 m, and L = b = 1 m) calculated by the DSM under three different BCs, namely, S-F-S-F, S-F-G-F, and G-F-G-F. The BCs are listed in order of bottom, right, up, and left in an anticlockwise sense. The results are compared with FEM results with different element sizes (0.1 × 0.1, 0.05 × 0.05, and 0.01 × 0.01) by using SHELL181 elements in ANSYS.Table 3
Natural frequencies (ω,rad⋅s−1) of a square plate under out-of-plane vibration by using the DSM and FEM. The FEM results are calculated in ANSYS by using 3 different element sizes (0.1 × 0.1, 0.05 × 0.05, and 0.01 × 0.01) of SHELL181 elements.
Mode
DSM
FEM (relative error (%))
Size = 0.1
Size = 0.05
Size = 0.01
S-F-S-F
1
29.5593
29.9318 (1.26)
29.6516 (0.31)
29.5624 (0.01)
2
49.5187
49.8420 (0.65)
49.5970 (0.16)
49.5084 (−0.02)
3
112.713
113.996 (1.14)
113.035 (0.28)
112.695 (−0.02)
4
119.525
125.708 (5.17)
121.020 (1.25)
119.575 (0.04)
5
143.442
149.075 (3.93)
144.809 (0.95)
143.470 (0.02)
6
217.106
223.204 (2.81)
218.611 (0.69)
217.103 (0.00)
7
231.050
241.532 (4.54)
233.678 (1.14)
231.108 (0.03)
8
270.037
303.032 (12.22)
277.723 (2.85)
270.315 (0.10)
S-F-G-F
1
7.29859
7.32117 (0.31)
7.30420 (0.08)
7.29855 (0.00)
2
21.1168
21.1404 (0.11)
21.1216 (0.02)
21.1121 (−0.02)
3
66.9707
68.8888 (2.86)
67.4437 (0.71)
66.9850 (0.02)
4
80.9386
81.9516 (1.25)
81.1913 (0.31)
80.9400 (0.00)
5
89.6413
91.3387 (1.89)
90.0569 (0.46)
89.6422 (0.00)
6
158.503
161.070 (1.62)
159.147 (0.41)
158.493 (−0.01)
7
187.214
202.670 (8.26)
190.896 (1.97)
187.346 (0.07)
8
200.157
210.901 (5.37)
202.828 (1.33)
200.245 (0.04)
G-F-G-F
1
29.5593
29.9318 (1.26)
29.6516 (0.31)
29.5624 (0.01)
2
49.5187
49.8433 (0.66)
49.5988 (0.16)
49.5172 (0.00)
3
68.6650
69.7057 (1.52)
68.9265 (0.38)
68.6752 (0.01)
4
112.713
114.002 (1.14)
113.041 (0.29)
112.714 (0.00)
5
119.525
125.708 (5.17)
121.020 (1.25)
119.582 (0.05)
6
143.442
149.081 (3.93)
144.809 (0.95)
143.483 (0.03)
7
189.278
200.258 (5.80)
192.002 (1.44)
189.382 (0.05)
8
217.106
223.204 (2.81)
218.617 (0.70)
217.134 (0.01)It can be seen from Table3 that as the element size becomes smaller, the FEM results converge to the DSM results. When the size is 0.01 × 0.01, the differences of the first 8 natural frequencies between the DSM and the FEM are within ±0.1%. It is worth noting that, in this case, the FEM uses 10,000 elements for a total of 61,206 DOFs for calculation, while only 1 element with 4 DOFs is used for the DSM. It takes 2 seconds for the FEM (mesh size 0.01) to compute the first 10 modes of the individual plate under the S-F-S-F BC, while the DSM only costs 0.14 seconds. It is obvious that the DSM gives exact results with much higher computational efficiency than the FEM. Then, the first 8 natural frequencies of the plate under 27 classical BCs in Table 2 are shown in Table 4.Table 4
The first 8 natural frequencies (ω,rad⋅s−1) of out-of-plane vibration for an individual square plate under 27 classical BCs. All parameters are the same as in Table 3.
BCs
1
2
3
4
5
6
7
8
S-F-S-F
29.5593
49.5187
112.713
119.525
143.442
217.106
231.050
270.037
S-F-S-C
38.9383
101.479
127.986
193.397
222.193
278.092
316.610
343.417
S-F-S-G
29.8811
54.2765
120.271
130.081
147.213
228.724
264.820
271.193
S-F-S-S
35.8606
85.1860
126.435
181.276
189.854
277.118
289.977
334.278
S-C-S-C
88.8520
168.010
212.769
290.288
313.708
396.202
430.297
475.017
S-C-S-G
42.0025
118.754
130.701
203.476
256.231
281.446
340.694
350.978
S-C-S-S
72.5721
158.592
179.989
264.352
307.734
347.504
410.613
432.264
S-G-S-G
30.2904
60.5809
121.162
151.452
151.452
242.324
272.614
302.905
S-G-S-S
37.8631
98.4439
128.734
189.315
219.606
280.187
310.477
340.768
S-S-S-S
60.5809
151.452
151.452
242.324
302.905
302.905
393.776
393.776
S-F-G-F
7.29859
21.1168
66.9707
80.9386
89.6413
158.503
187.214
200.157
S-F-G-C
17.5055
75.7883
76.5541
140.425
195.438
197.653
260.940
262.096
S-F-G-G
7.38983
28.1783
67.5092
93.6371
102.587
172.450
188.166
215.513
S-F-G-S
12.3797
57.7624
73.6886
126.366
162.739
194.231
232.694
250.455
S-C-G-C
73.0917
119.967
194.991
232.763
244.068
352.266
377.279
410.467
S-C-G-G
22.2130
78.4270
99.0506
153.308
198.425
235.793
269.398
291.000
S-C-G-S
53.1923
107.574
159.892
214.567
225.389
326.790
329.678
382.508
S-G-G-G
7.57261
37.8631
68.1535
98.4439
128.734
189.315
189.315
219.606
S-G-G-S
15.1452
75.7261
75.7261
136.307
196.888
196.888
257.469
257.469
G-F-G-F
29.5593
49.5187
68.6650
112.713
119.525
143.442
189.278
217.106
G-F-G-C
10.7909
38.9383
67.6253
101.479
127.986
189.353
193.397
222.193
G-F-G-G
17.1663
29.8811
54.2765
92.7651
120.271
130.081
147.213
228.724
G-F-G-S
35.8606
47.3195
85.1860
126.435
153.345
181.276
189.854
277.118
G-C-G-C
68.6650
88.8520
168.010
189.278
212.769
290.288
313.708
371.060
G-C-G-G
17.1663
42.0025
92.7651
118.754
130.701
203.476
229.072
256.231
G-C-G-S
47.3195
72.5721
153.345
158.592
179.989
264.352
307.734
319.943
G-G-G-G
30.2904
30.2904
60.5809
121.162
121.162
151.452
151.452
242.324


### 3.2. Modal Analysis of Inplane Vibration for an Individual Plate

The first 8 natural frequencies of inplane vibration for an individual plate (Table5) are calculated by using the DSM and FEM under three representative BCs, namely, S1-F-S1-F, S1-F-S2-F, and S2-F-S2-F. The structure is the same as in Section 3.1, while only the inplane motion is considered in this section. The results in Table 5 show that the FEM results converge to the DSM results as the FEM mesh becomes refined from size = 0.1 to size = 0.01, and in the case of size = 0.01, the FEM spends 3 seconds, while the DSM only costs 0.08 seconds to compute the same modes of the individual plate under the S1-F-S1-F BC. It is clear that the DSM is a more efficient and accurate method of inplane vibration analysis than the FEM. The relative errors between the DSM and the FEM are within 0.06% when the element size of FEM is 0.01 × 0.01. Compared with out-of-plane vibration, the DSM and FEM agree better in inplane vibration analysis. It is because the DSM and FEM are based on the same governing differential equation in inplane vibration analysis. Table 6 provides the first 8 natural frequencies of the plate computed by the DSM under 27 classical BCs in Table 2. It is clear that the BCs have a great influence on the natural frequencies of inplane vibration for the plate.Table 5
Natural frequencies (ω,rad⋅s−1) of the plate for inplane vibration analysis using the DSM and FEM. All parameters are the same as in Table 3.
Mode
DSM
FEM (relative error (%))
Size = 0.1
Size = 0.05
Size = 0.01
S1-F-S1-F
1
7482.01
7560.56 (1.05)
7501.49 (0.26)
7482.65 (0.01)
2
9879.85
9961.99 (0.83)
9900.42 (0.21)
9880.94 (0.01)
3
13972.2
14091.3 (0.85)
14001.5 (0.21)
13973.2 (0.01)
4
17268.5
17944.1 (3.91)
17432.7 (0.95)
17275.0 (0.04)
5
17881.5
18078.6 (1.10)
17930.3 (0.27)
17883.2 (0.01)
6
18598.5
18814.4 (1.16)
18652.3 (0.29)
18600.7 (0.01)
7
19759.7
20351.2 (2.99)
19924.0 (0.83)
19766.3 (0.03)
8
19841.5
20436.7 (3.00)
19964.8 (0.62)
19846.7 (0.03)
S1-F-S2-F
1
2732.44
2735.32 (0.11)
2733.19 (0.03)
2732.49 (0.00)
2
7869.56
7886.65 (0.22)
7874.09 (0.06)
7869.69 (0.00)
3
12405.5
12689.5 (2.29)
12475.3 (0.56)
12408.0 (0.02)
4
12991.5
13081.6 (0.69)
13013.7 (0.17)
12992.4 (0.01)
5
15875.7
16080.6 (1.29)
15926.6 (0.32)
15877.6 (0.01)
6
16674.7
16890.5 (1.29)
16727.7 (0.32)
16676.8 (0.01)
7
22053.0
22909.1 (3.88)
22368.8 (1.43)
22065.3 (0.06)
8
22306.7
23377.8 (4.80)
22452.3 (0.65)
22312.2 (0.02)
S2-F-S2-F
1
7482.01
7560.56 (1.05)
7501.49 (0.26)
7482.64 (0.01)
2
13972.2
14091.3 (0.85)
14001.5 (0.21)
13973.2 (0.01)
3
16700.0
16838.9 (0.83)
16734.6 (0.21)
16701.3 (0.01)
4
17268.5
17944.1 (3.91)
17432.1 (0.95)
17275.0 (0.04)
5
17881.5
18078.6 (1.10)
17930.3 (0.27)
17883.2 (0.01)
6
18598.5
18814.4 (1.16)
18652.3 (0.29)
18600.7 (0.01)
7
19841.5
20351.2 (2.57)
19964.8 (0.62)
19846.7 (0.03)
8
26764.4
27484.5 (2.69)
26941.0 (0.66)
26771.4 (0.03)Table 6
The first 8 natural frequencies (ω,rad⋅s−1) of inplane vibration for an individual plate under 27 classical BCs.
BCs
1
2
3
4
5
6
7
8
S1-F-S1-F
7482.01
9879.85
13972.2
17268.5
17881.5
18598.5
19759.7
19841.5
S1-F-S1-C
4939.93
10512.0
14819.8
16274.3
18281.3
22126.5
23713.7
24699.6
S1-F-S1-S2
9879.85
9920.77
16263.1
18197.1
18207.1
19759.7
22636.4
24499.0
S1-F-S1-S1
4939.93
8634.27
13972.2
14819.8
18024.8
18334.7
21462.3
22887.6
S1-C-S1-C
9879.85
17410.3
18576.7
19759.7
23445.5
26351.6
29639.6
29880.9
S1-C-S1-S2
4939.93
14819.8
14940.4
18341.5
22669.5
23002.0
24699.6
27269.8
S1-C-S1-S1
9879.85
11722.7
18190.3
19759.7
20547.6
22544.6
25649.3
28966.7
S1-S2-S1-S2
9879.85
13972.2
16700.0
19759.7
22092.0
22092.0
23617.4
27944.4
S1-S2-S1-S1
4939.93
11046.0
14819.8
17811.2
18671.2
20367.8
24699.6
24699.6
S1-S1-S1-S1
9879.85
9879.85
13972.2
19759.7
19759.7
22092.0
22092.0
23617.4
S1-F-S2-F
2732.44
7869.56
12405.5
12991.5
15875.7
16674.7
22053.0
22306.7
S1-F-S2-C
8496.99
9353.77
14086.9
17210.7
21926.0
22691.3
23326.0
27186.5
S1-F-S2-S2
6986.11
8940.77
13382.2
13856.7
20205.3
20958.3
22661.5
25006.1
S1-F-S2-S1
3741.00
9299.24
13384.8
14757.2
17459.9
19068.9
22600.7
24085.0
S1-C-S2-C
12683.7
16429.9
19933.7
21967.8
24894.1
27510.3
29457.6
31443.9
S1-C-S2-S2
9288.33
14960.9
18673.3
18979.4
24888.1
25239.7
26998.2
27195.8
S1-C-S2-S1
8705.15
13175.8
15941.6
20249.4
22076.1
25302.5
26285.1
27578.2
S1-S2-S2-S2
8350.00
11046.0
17811.2
18671.2
20367.8
24699.6
25050.0
26602.3
S1-S2-S2-S1
6986.11
11808.7
15621.4
15621.4
20958.3
25188.8
25188.8
26405.0
S2-F-S2-F
7482.01
13972.2
16700.0
17268.5
17881.5
18598.5
19841.5
26764.4
S2-F-S2-C
8350.00
10512.0
16274.3
18281.3
22126.5
23713.7
25050.0
25565.0
S2-F-S2-S2
8350.00
9920.77
16263.1
18197.1
18207.1
22636.4
24499.0
25050.0
S2-F-S2-S1
8634.3
13972.2
16700.0
18024.8
18334.7
21462.3
22887.6
27145.9
S2-C-S2-C
16700.0
17410.3
18576.7
23445.5
26351.6
29880.9
29921.8
31883.1
S2-C-S2-S2
14940.4
16700.0
18341.5
22669.5
23002.0
27269.8
29047.9
31526.6
S2-C-S2-S1
8350.00
11722.7
18190.3
20547.6
22544.6
25050.0
25649.3
28966.7
S2-S2-S2-S2
13972.2
16700.0
16700.0
22092.0
22092.0
23617.4
27944.4
31242.8


### 3.3. Modal Analysis of Plate Built-Up Structures

In this section, free vibration analysis by using the DS formulations developed in this paper on three plate built-up structures in engineering applications is performed, namely, a two-plate built-up structure (Section 3.3.1), an I-shaped plate built-up structure (Section 3.3.2), and a section of an extruded aluminum panel (Section 3.3.3).


#### 3.3.1. A Two-Plate Built-Up Structure

Consider a two-plate built-up structure with two plates connecting at an angle ofθ (Figure 5), then the geometry and material properties of the 2 plates are the same (E = 72 GPa, ν=0.3, ρ=2700kg⋅m−3, L=3m, b=2m, and h=0.02m). The SBs are L1, L5, L3, and L7, where the BCs of L1 and L5 should be the same and those of L3 and L7 are the same. Therefore, there are 16 (4 × 4) kinds of SBCs of the plate built-up structure, as shown in Table 7. The SBCs are listed in order of L1L5-L3L7.Table 7
Different SBCs of a two-plate built-up structure.
IV
S1-S1
S1-S2
S2-S2
S2-S1
CPT
S-S
S-G
G-G
G-SFirstly, the influences of angles on the natural frequencies of the two-plate built-up structure are tabulated in Table8. The BCs of this plate built-up structure in Table 8 are listed in an anticlockwise sense of L1−L2−L3−L3−L4−L5−L6−L7. Table 8 lists the first 8 natural frequencies of the two-plate built-up structure with two plates connecting at angles of 30, 60, 90, 120, 150, and 180° under (S1 G)-F-(S1 G)-F-(S1 G)-F-(S1 G) BCs. It can be found from Table 8 that the connection angles have little effect on the natural frequencies except for 180°. In order to explain this phenomenon, this study computed the first 8 natural frequencies of the structure at different angles under the 10 different SBCs in Table 7 with free nodal boundaries. Here are the conclusions: the natural frequencies of the structure at the angle of 180° are significantly different from those of other angles (the differences are much greater than 2%). The rates of change are less than 2% when the structure is at different angles (except 180°) under the SBCs of (S1 S)-(S1 S), (S1 S)-(S1 G), and (S1 G)-(S1 G), and part of the rates are more than 2% under the SBCs of (S1 S)-(S2 S), (S1 G)-(S2 S), (S1 S)-(S2 G), (S1 G)-(S2 G), (S2 S)-(S2 S), (S2 S)-(S2 G), and (S2 G)-(S2 G). It might be due to the reason that the BCs along the opposite sides of the two-plate built-up structure are simply supported (or guided), the two plates are weakly coupled, and the low-frequency modes are mainly dominated by out-of-plane vibration. Therefore, the angles (except 180°) have little effect on the first 30 natural frequencies of the structure in this section. When these two plates are connected at an angle of 180°, the transverse and inplane vibrations are fully coupled.Table 8
The first 8 natural frequencies (ω,rad⋅s−1) of the two-plate built-up structure with two plates connecting at different angles under (S1 G)-F-(S1 G)-F-(S1 G)-F-(S1 G) BCs (RE is short for relative error).
θ (°)
1
2
3
4
5
6
7
8
30
DSM
27.472
47.614
58.418
120.47
151.31
157.32
166.19
172.17
FEM
27.470
47.540
58.359
120.22
151.14
157.20
165.77
172.10
RE (%)
−0.01
−0.15
−0.10
−0.21
−0.11
−0.08
−0.25
−0.04
60
DSM
27.472
47.614
58.418
120.47
151.31
157.32
166.19
172.17
FEM
27.469
47.547
58.359
120.40
151.17
157.20
165.89
172.07
RE (%)
−0.01
−0.14
−0.10
−0.05
−0.09
−0.08
−0.18
−0.05
90
DSM
27.472
47.614
58.418
120.47
151.31
157.32
166.19
172.17
FEM
27.469
47.549
58.358
120.44
151.18
157.19
165.91
172.01
RE (%)
−0.01
−0.14
−0.10
−0.03
−0.09
−0.08
−0.17
−0.09
120
DSM
27.473
47.614
58.418
120.47
151.31
157.32
166.19
172.17
FEM
27.466
47.551
58.354
120.45
151.19
157.17
165.93
171.83
RE (%)
−0.02
−0.13
−0.11
−0.02
−0.08
−0.09
−0.16
−0.20
150
DSM
27.474
47.614
58.418
120.47
151.31
157.32
166.19
172.17
FEM
27.453
47.559
58.332
120.45
151.20
157.09
165.95
170.83
RE (%)
−0.08
−0.11
−0.15
−0.01
−0.07
−0.15
−0.15
−0.78
180
DSM
33.616
43.703
47.614
90.260
120.47
135.66
151.31
166.19
FEM
33.612
43.701
47.579
90.151
120.45
135.62
151.22
166.00
RE (%)
−0.01
−0.01
−0.07
−0.12
−0.01
−0.03
−0.06
−0.11Then, Table9 shows the influence of SBCs on the first 8 natural frequencies of the two-plate built-up structure with two plates connecting at an angle of 60°. The BCs of the plate built-up structure in Table 9 are listed in the order of L1−L5−L3−L7, and the NBCs are free. It can be found that the natural frequencies of the structure have little differences when the SBCs related to out-of-plane vibration are the same although the SBCs for inplane vibration are different. This is due to the same reason as in Table 8: low-frequency modes are mainly dominated by out-of-plane vibration. Thus, the SBCs related to out-of-plane vibration have dominant influence on the low natural frequencies.Table 9
The first 8 natural frequenciesω,rad⋅s−1 of the two-plate built-up structure with two plates connecting at an angle of 60° under different SBC combinations.
SBCs
1
2
3
4
5
6
7
8
(S1 S)-(S1 S)-(S1 S)-(S1 S)
DSM
47.614
58.418
151.31
157.32
166.19
211.89
282.95
320.55
FEM
47.472
58.276
151.08
157.09
165.62
211.39
281.79
319.49
RE (%)
−0.30
−0.24
−0.15
−0.14
−0.34
−0.24
−0.41
−0.33
(S1 S)-(S1 S)-(S1 G)-(S1 G)
DSM
18.769
34.930
91.297
99.132
132.64
182.41
216.87
228.12
FEM
18.709
34.883
91.156
98.992
132.46
182.23
216.17
227.88
RE (%)
−0.32
−0.13
−0.15
−0.14
−0.14
−0.10
−0.32
−0.11
(S1 G)-(S1 G)-(S1 G)-(S1 G)
DSM
27.472
47.614
58.418
120.47
151.31
157.32
166.19
172.17
FEM
27.469
47.547
58.359
120.40
151.17
157.20
165.89
172.07
RE (%)
−0.01
−0.14
−0.10
−0.05
−0.09
−0.08
−0.18
−0.05
(S1 S)-(S1 S)-(S2 S)-(S2 S)
DSM
47.614
58.418
151.31
157.32
166.19
211.89
282.95
320.55
FEM
47.468
58.270
151.08
157.09
165.51
211.26
281.74
319.45
RE (%)
−0.31
−0.25
−0.15
−0.15
−0.41
−0.30
−0.43
−0.34
(S1 S)-(S1 S)-(S2 G)-(S2 G)
DSM
18.769
34.930
91.297
99.132
132.64
182.41
216.87
228.12
FEM
18.706
34.875
91.156
98.992
132.14
181.90
216.17
227.88
RE (%)
−0.34
−0.16
−0.15
−0.14
−0.37
−0.28
−0.33
−0.11
(S1 G)-(S1 G)-(S2 G)-(S2 G)
DSM
27.473
47.614
58.418
120.47
151.31
157.32
166.19
172.17
FEM
27.469
47.544
58.357
120.30
151.17
157.20
165.82
172.01
RE (%)
−0.01
−0.15
−0.11
−0.14
−0.09
−0.08
−0.22
−0.09
(S2 S)-(S2 S)-(S2 S)-(S2 S)
DSM
47.614
58.418
151.31
157.32
166.19
211.89
282.95
320.55
FEM
47.460
58.257
151.07
157.09
165.29
210.97
281.73
319.46
RE (%)
−0.32
−0.28
−0.16
−0.15
−0.54
−0.44
−0.43
−0.34All the results of the DSM and FEM match well with each other, as shown in Tables8 and 9 (differences within 1%). In this section, only 2 elements with 12 DOFs are adopted for the DSM, whereas the FEM uses as much as 30,000 elements (the element size is 0.02×0.02) and 182,106 DOFs to compute the results in Tables 8 and 9. It is obvious that the dynamic stiffness method has incomparable advantages in modal analysis of plate built-up structures.


#### 3.3.2. I-Shaped Plate Built-Up Structure

Now we consider an I-shaped plate built-up structure (E = 72 GPa, ν=0.3, ρ=2700kg⋅m−3, L=3m, b=1m, Δz=2m, and h=0.01m), as shown in Figure 10, and the material and thickness of all the plates are the same as in the previous example. There are 10 SBs on the side of y = 0 (front) and y = L (back). The BCs of the SBs on the same side should be the same. Three representative SBCs of (S1 G)-(S1 G), (S1 S)-(S2 G), and (S2 S)-(S2 S) are considered for modal analysis.Figure 10
I-shaped plate built-up structure.Table10 shows the comparisons of the results computed by the DSM and FEM. Only 5 dynamic stiffness elements with 24 DOFs are used for the I-shaped plate built-up structure, whereas the FEM uses 120,000 elements (the element size is 0.01×0.01) with a total of 724,206 DOFs. The DSM gives exact results, while the computational cost is extremely inexpensive.Table 10
The first 8 natural frequenciesω,rad⋅s−1 of the I-shaped plate built-up structure under three representative SBCs. E = 72 GPa, ν=0.3, ρ=2700kg⋅m−3, L=3m, b=1m, Δz=0.2m, and h=0.01m.
SBCs
1
2
3
4
5
6
7
8
(S1 G)-(S1 G)
DSM
32.016
59.626
79.283
106.44
116.73
127.10
144.20
172.84
FEM
32.011
59.556
79.275
106.30
116.62
127.01
144.09
172.50
RE (%)
−0.02
−0.12
−0.01
−0.13
−0.09
−0.08
−0.08
−0.19
(S1 S)-(S2 G)
DSM
40.350
85.128
86.563
131.30
135.77
155.44
166.31
217.16
FEM
40.198
84.999
86.507
129.89
135.47
155.31
166.02
216.63
RE (%)
−0.38
−0.15
−0.06
−1.07
−0.22
−0.08
−0.17
−0.24
(S2 S)-(S2 S)
DSM
59.626
106.44
116.73
144.20
172.84
197.58
202.07
234.47
FEM
59.495
106.24
116.55
143.99
172.35
197.08
201.89
232.69
RE (%)
−0.22
−0.19
−0.15
−0.15
−0.29
−0.25
−0.09
−0.76


#### 3.3.3. A Section of an Extruded Aluminum Panel

Figure11 shows a section of an extruded aluminum panel which has wide engineering applications such as the airplane fuselage and high-speed train body structures. In this section, the modal analysis of the structure in Figure 1 (E = 72 GPa, ν=0.3, ρ=2700kg⋅m−3, L=0.2m, be=0.1m, b=0.5m, Δz=0.1m, and h=0.002m) is carried out by using the DSM and FEM under three different SBCs, and the material and thickness of all the plates are the same. Some representative natural frequencies are shown in Table 11 and modal shapes in Figure 12. The DSM uses 21 elements with 52 DOFs to obtain the results, while 16,800 elements (the element size is 0.005×0.005) for a total of 101,352 DOFs are required for the FEM. It can be concluded that the results computed by the DSM and FEM agree well. Obviously, the DSM is applicable to the complex plate built-up structures, and therefore, the DSM is suitable for optimization design and parameter analysis due to its high efficiency and analytical nature.Figure 11
A section of an extruded aluminum panel.Table 11
The first 8 natural frequenciesω,rad⋅s−1 of the extruded aluminum panel under three representative SBC combinations. E = 72 GPa, ν=0.3, ρ=2700kg⋅m−3, L=0.2m, be=0.1m, b=0.5m, Δz=0.1m, and h=0.002m.
SBCs
1
2
3
4
5
6
7
8
(S1 G)-(S1 G)
DSM
2761.7
2764.6
3462.0
3552.8
3558.0
3558.8
3651.5
3864.5
FEM
2763.1
2763.2
3470.0
3554.2
3555.0
3560.3
3657.9
3868.7
RE (%)
0.05
−0.05
0.23
0.04
−0.09
0.04
0.18
0.11
(S1 S)-(S2 G)
DSM
2963.3
2966.4
3626.0
3717.2
3806.0
4010.9
4122.3
4475.2
FEM
2957.4
2957.7
3630.6
3717.7
3807.9
3997.2
4105.9
4456.2
RE (%)
−0.20
−0.29
0.13
0.01
0.05
−0.34
−0.40
−0.42
(S2 S)-(S2 S)
DSM
3558.0
3558.8
4131.0
4222.7
4285.0
4465.2
4591.9
4920.4
FEM
3546.2
3546.9
4131.1
4220.1
4283.4
4448.8
4576.9
4906.5
RE (%)
−0.33
−0.33
0.00
−0.06
−0.04
−0.37
−0.33
−0.28Figure 12
The first 3 modal shapes of the extruded aluminum panel computed by the DSM (a) and FEM (b).
(a)
(b)Another advantage of the DSM for vibration analysis lies in its high efficiency and accuracy over the whole frequency range. Some representative natural frequencies covering the midfrequency and high-frequency ranges (10–100th modes) of the I-shaped plate built-up structure and extruded aluminum panel are listed in Table 12, where the SBC is (S2 S)-(S2 S). It is found that the computational efficiency of the DSM is more than 100 times that of the FEM. For example, the FEM takes 662 seconds to compute the first 100 modes of the I-shaped plate built-up structure, while the DSM only takes 3.28 seconds.Table 12
Representative natural frequencies covering the whole frequency range of the first 100 natural frequenciesω,rad⋅s−1 of the I-shaped plate built-up structure and extruded aluminum panel in Figures 10 and 11 under the SBC combination of (S2 S)-(S2 S).
10
30
50
70
90
100
I-shaped plate built-up structure
DSM
250.24
575.30
1008.3
1302.5
1563.8
1771.4
FEM
249.61
574.35
1005.2
1295.1
1591.0
1767.8
RE (%)
−0.25
−0.16
−0.31
−0.57
1.74
−0.21
Extruded aluminum panel
DSM
5403.4
7288.5
10458
14332
16015
16882
FEM
5369.2
7262.1
10447
14323
16052
16926
RE (%)
−0.63
−0.36
−0.10
−0.07
0.23
0.26


## 3.1. Modal Analysis of Out-of-Plane Vibration for an Individual Plate

Table3 exhibits the first 8 natural frequencies of the out-of-plane vibration for a square plate (E = 72 GPa, ν=0.3, ρ=2800kg⋅m−3, h = 0.002 m, and L = b = 1 m) calculated by the DSM under three different BCs, namely, S-F-S-F, S-F-G-F, and G-F-G-F. The BCs are listed in order of bottom, right, up, and left in an anticlockwise sense. The results are compared with FEM results with different element sizes (0.1 × 0.1, 0.05 × 0.05, and 0.01 × 0.01) by using SHELL181 elements in ANSYS.Table 3
Natural frequencies (ω,rad⋅s−1) of a square plate under out-of-plane vibration by using the DSM and FEM. The FEM results are calculated in ANSYS by using 3 different element sizes (0.1 × 0.1, 0.05 × 0.05, and 0.01 × 0.01) of SHELL181 elements.
Mode
DSM
FEM (relative error (%))
Size = 0.1
Size = 0.05
Size = 0.01
S-F-S-F
1
29.5593
29.9318 (1.26)
29.6516 (0.31)
29.5624 (0.01)
2
49.5187
49.8420 (0.65)
49.5970 (0.16)
49.5084 (−0.02)
3
112.713
113.996 (1.14)
113.035 (0.28)
112.695 (−0.02)
4
119.525
125.708 (5.17)
121.020 (1.25)
119.575 (0.04)
5
143.442
149.075 (3.93)
144.809 (0.95)
143.470 (0.02)
6
217.106
223.204 (2.81)
218.611 (0.69)
217.103 (0.00)
7
231.050
241.532 (4.54)
233.678 (1.14)
231.108 (0.03)
8
270.037
303.032 (12.22)
277.723 (2.85)
270.315 (0.10)
S-F-G-F
1
7.29859
7.32117 (0.31)
7.30420 (0.08)
7.29855 (0.00)
2
21.1168
21.1404 (0.11)
21.1216 (0.02)
21.1121 (−0.02)
3
66.9707
68.8888 (2.86)
67.4437 (0.71)
66.9850 (0.02)
4
80.9386
81.9516 (1.25)
81.1913 (0.31)
80.9400 (0.00)
5
89.6413
91.3387 (1.89)
90.0569 (0.46)
89.6422 (0.00)
6
158.503
161.070 (1.62)
159.147 (0.41)
158.493 (−0.01)
7
187.214
202.670 (8.26)
190.896 (1.97)
187.346 (0.07)
8
200.157
210.901 (5.37)
202.828 (1.33)
200.245 (0.04)
G-F-G-F
1
29.5593
29.9318 (1.26)
29.6516 (0.31)
29.5624 (0.01)
2
49.5187
49.8433 (0.66)
49.5988 (0.16)
49.5172 (0.00)
3
68.6650
69.7057 (1.52)
68.9265 (0.38)
68.6752 (0.01)
4
112.713
114.002 (1.14)
113.041 (0.29)
112.714 (0.00)
5
119.525
125.708 (5.17)
121.020 (1.25)
119.582 (0.05)
6
143.442
149.081 (3.93)
144.809 (0.95)
143.483 (0.03)
7
189.278
200.258 (5.80)
192.002 (1.44)
189.382 (0.05)
8
217.106
223.204 (2.81)
218.617 (0.70)
217.134 (0.01)It can be seen from Table3 that as the element size becomes smaller, the FEM results converge to the DSM results. When the size is 0.01 × 0.01, the differences of the first 8 natural frequencies between the DSM and the FEM are within ±0.1%. It is worth noting that, in this case, the FEM uses 10,000 elements for a total of 61,206 DOFs for calculation, while only 1 element with 4 DOFs is used for the DSM. It takes 2 seconds for the FEM (mesh size 0.01) to compute the first 10 modes of the individual plate under the S-F-S-F BC, while the DSM only costs 0.14 seconds. It is obvious that the DSM gives exact results with much higher computational efficiency than the FEM. Then, the first 8 natural frequencies of the plate under 27 classical BCs in Table 2 are shown in Table 4.Table 4
The first 8 natural frequencies (ω,rad⋅s−1) of out-of-plane vibration for an individual square plate under 27 classical BCs. All parameters are the same as in Table 3.
BCs
1
2
3
4
5
6
7
8
S-F-S-F
29.5593
49.5187
112.713
119.525
143.442
217.106
231.050
270.037
S-F-S-C
38.9383
101.479
127.986
193.397
222.193
278.092
316.610
343.417
S-F-S-G
29.8811
54.2765
120.271
130.081
147.213
228.724
264.820
271.193
S-F-S-S
35.8606
85.1860
126.435
181.276
189.854
277.118
289.977
334.278
S-C-S-C
88.8520
168.010
212.769
290.288
313.708
396.202
430.297
475.017
S-C-S-G
42.0025
118.754
130.701
203.476
256.231
281.446
340.694
350.978
S-C-S-S
72.5721
158.592
179.989
264.352
307.734
347.504
410.613
432.264
S-G-S-G
30.2904
60.5809
121.162
151.452
151.452
242.324
272.614
302.905
S-G-S-S
37.8631
98.4439
128.734
189.315
219.606
280.187
310.477
340.768
S-S-S-S
60.5809
151.452
151.452
242.324
302.905
302.905
393.776
393.776
S-F-G-F
7.29859
21.1168
66.9707
80.9386
89.6413
158.503
187.214
200.157
S-F-G-C
17.5055
75.7883
76.5541
140.425
195.438
197.653
260.940
262.096
S-F-G-G
7.38983
28.1783
67.5092
93.6371
102.587
172.450
188.166
215.513
S-F-G-S
12.3797
57.7624
73.6886
126.366
162.739
194.231
232.694
250.455
S-C-G-C
73.0917
119.967
194.991
232.763
244.068
352.266
377.279
410.467
S-C-G-G
22.2130
78.4270
99.0506
153.308
198.425
235.793
269.398
291.000
S-C-G-S
53.1923
107.574
159.892
214.567
225.389
326.790
329.678
382.508
S-G-G-G
7.57261
37.8631
68.1535
98.4439
128.734
189.315
189.315
219.606
S-G-G-S
15.1452
75.7261
75.7261
136.307
196.888
196.888
257.469
257.469
G-F-G-F
29.5593
49.5187
68.6650
112.713
119.525
143.442
189.278
217.106
G-F-G-C
10.7909
38.9383
67.6253
101.479
127.986
189.353
193.397
222.193
G-F-G-G
17.1663
29.8811
54.2765
92.7651
120.271
130.081
147.213
228.724
G-F-G-S
35.8606
47.3195
85.1860
126.435
153.345
181.276
189.854
277.118
G-C-G-C
68.6650
88.8520
168.010
189.278
212.769
290.288
313.708
371.060
G-C-G-G
17.1663
42.0025
92.7651
118.754
130.701
203.476
229.072
256.231
G-C-G-S
47.3195
72.5721
153.345
158.592
179.989
264.352
307.734
319.943
G-G-G-G
30.2904
30.2904
60.5809
121.162
121.162
151.452
151.452
242.324


## 3.2. Modal Analysis of Inplane Vibration for an Individual Plate

The first 8 natural frequencies of inplane vibration for an individual plate (Table5) are calculated by using the DSM and FEM under three representative BCs, namely, S1-F-S1-F, S1-F-S2-F, and S2-F-S2-F. The structure is the same as in Section 3.1, while only the inplane motion is considered in this section. The results in Table 5 show that the FEM results converge to the DSM results as the FEM mesh becomes refined from size = 0.1 to size = 0.01, and in the case of size = 0.01, the FEM spends 3 seconds, while the DSM only costs 0.08 seconds to compute the same modes of the individual plate under the S1-F-S1-F BC. It is clear that the DSM is a more efficient and accurate method of inplane vibration analysis than the FEM. The relative errors between the DSM and the FEM are within 0.06% when the element size of FEM is 0.01 × 0.01. Compared with out-of-plane vibration, the DSM and FEM agree better in inplane vibration analysis. It is because the DSM and FEM are based on the same governing differential equation in inplane vibration analysis. Table 6 provides the first 8 natural frequencies of the plate computed by the DSM under 27 classical BCs in Table 2. It is clear that the BCs have a great influence on the natural frequencies of inplane vibration for the plate.Table 5
Natural frequencies (ω,rad⋅s−1) of the plate for inplane vibration analysis using the DSM and FEM. All parameters are the same as in Table 3.
Mode
DSM
FEM (relative error (%))
Size = 0.1
Size = 0.05
Size = 0.01
S1-F-S1-F
1
7482.01
7560.56 (1.05)
7501.49 (0.26)
7482.65 (0.01)
2
9879.85
9961.99 (0.83)
9900.42 (0.21)
9880.94 (0.01)
3
13972.2
14091.3 (0.85)
14001.5 (0.21)
13973.2 (0.01)
4
17268.5
17944.1 (3.91)
17432.7 (0.95)
17275.0 (0.04)
5
17881.5
18078.6 (1.10)
17930.3 (0.27)
17883.2 (0.01)
6
18598.5
18814.4 (1.16)
18652.3 (0.29)
18600.7 (0.01)
7
19759.7
20351.2 (2.99)
19924.0 (0.83)
19766.3 (0.03)
8
19841.5
20436.7 (3.00)
19964.8 (0.62)
19846.7 (0.03)
S1-F-S2-F
1
2732.44
2735.32 (0.11)
2733.19 (0.03)
2732.49 (0.00)
2
7869.56
7886.65 (0.22)
7874.09 (0.06)
7869.69 (0.00)
3
12405.5
12689.5 (2.29)
12475.3 (0.56)
12408.0 (0.02)
4
12991.5
13081.6 (0.69)
13013.7 (0.17)
12992.4 (0.01)
5
15875.7
16080.6 (1.29)
15926.6 (0.32)
15877.6 (0.01)
6
16674.7
16890.5 (1.29)
16727.7 (0.32)
16676.8 (0.01)
7
22053.0
22909.1 (3.88)
22368.8 (1.43)
22065.3 (0.06)
8
22306.7
23377.8 (4.80)
22452.3 (0.65)
22312.2 (0.02)
S2-F-S2-F
1
7482.01
7560.56 (1.05)
7501.49 (0.26)
7482.64 (0.01)
2
13972.2
14091.3 (0.85)
14001.5 (0.21)
13973.2 (0.01)
3
16700.0
16838.9 (0.83)
16734.6 (0.21)
16701.3 (0.01)
4
17268.5
17944.1 (3.91)
17432.1 (0.95)
17275.0 (0.04)
5
17881.5
18078.6 (1.10)
17930.3 (0.27)
17883.2 (0.01)
6
18598.5
18814.4 (1.16)
18652.3 (0.29)
18600.7 (0.01)
7
19841.5
20351.2 (2.57)
19964.8 (0.62)
19846.7 (0.03)
8
26764.4
27484.5 (2.69)
26941.0 (0.66)
26771.4 (0.03)Table 6
The first 8 natural frequencies (ω,rad⋅s−1) of inplane vibration for an individual plate under 27 classical BCs.
BCs
1
2
3
4
5
6
7
8
S1-F-S1-F
7482.01
9879.85
13972.2
17268.5
17881.5
18598.5
19759.7
19841.5
S1-F-S1-C
4939.93
10512.0
14819.8
16274.3
18281.3
22126.5
23713.7
24699.6
S1-F-S1-S2
9879.85
9920.77
16263.1
18197.1
18207.1
19759.7
22636.4
24499.0
S1-F-S1-S1
4939.93
8634.27
13972.2
14819.8
18024.8
18334.7
21462.3
22887.6
S1-C-S1-C
9879.85
17410.3
18576.7
19759.7
23445.5
26351.6
29639.6
29880.9
S1-C-S1-S2
4939.93
14819.8
14940.4
18341.5
22669.5
23002.0
24699.6
27269.8
S1-C-S1-S1
9879.85
11722.7
18190.3
19759.7
20547.6
22544.6
25649.3
28966.7
S1-S2-S1-S2
9879.85
13972.2
16700.0
19759.7
22092.0
22092.0
23617.4
27944.4
S1-S2-S1-S1
4939.93
11046.0
14819.8
17811.2
18671.2
20367.8
24699.6
24699.6
S1-S1-S1-S1
9879.85
9879.85
13972.2
19759.7
19759.7
22092.0
22092.0
23617.4
S1-F-S2-F
2732.44
7869.56
12405.5
12991.5
15875.7
16674.7
22053.0
22306.7
S1-F-S2-C
8496.99
9353.77
14086.9
17210.7
21926.0
22691.3
23326.0
27186.5
S1-F-S2-S2
6986.11
8940.77
13382.2
13856.7
20205.3
20958.3
22661.5
25006.1
S1-F-S2-S1
3741.00
9299.24
13384.8
14757.2
17459.9
19068.9
22600.7
24085.0
S1-C-S2-C
12683.7
16429.9
19933.7
21967.8
24894.1
27510.3
29457.6
31443.9
S1-C-S2-S2
9288.33
14960.9
18673.3
18979.4
24888.1
25239.7
26998.2
27195.8
S1-C-S2-S1
8705.15
13175.8
15941.6
20249.4
22076.1
25302.5
26285.1
27578.2
S1-S2-S2-S2
8350.00
11046.0
17811.2
18671.2
20367.8
24699.6
25050.0
26602.3
S1-S2-S2-S1
6986.11
11808.7
15621.4
15621.4
20958.3
25188.8
25188.8
26405.0
S2-F-S2-F
7482.01
13972.2
16700.0
17268.5
17881.5
18598.5
19841.5
26764.4
S2-F-S2-C
8350.00
10512.0
16274.3
18281.3
22126.5
23713.7
25050.0
25565.0
S2-F-S2-S2
8350.00
9920.77
16263.1
18197.1
18207.1
22636.4
24499.0
25050.0
S2-F-S2-S1
8634.3
13972.2
16700.0
18024.8
18334.7
21462.3
22887.6
27145.9
S2-C-S2-C
16700.0
17410.3
18576.7
23445.5
26351.6
29880.9
29921.8
31883.1
S2-C-S2-S2
14940.4
16700.0
18341.5
22669.5
23002.0
27269.8
29047.9
31526.6
S2-C-S2-S1
8350.00
11722.7
18190.3
20547.6
22544.6
25050.0
25649.3
28966.7
S2-S2-S2-S2
13972.2
16700.0
16700.0
22092.0
22092.0
23617.4
27944.4
31242.8


## 3.3. Modal Analysis of Plate Built-Up Structures

In this section, free vibration analysis by using the DS formulations developed in this paper on three plate built-up structures in engineering applications is performed, namely, a two-plate built-up structure (Section 3.3.1), an I-shaped plate built-up structure (Section 3.3.2), and a section of an extruded aluminum panel (Section 3.3.3).


### 3.3.1. A Two-Plate Built-Up Structure

Consider a two-plate built-up structure with two plates connecting at an angle ofθ (Figure 5), then the geometry and material properties of the 2 plates are the same (E = 72 GPa, ν=0.3, ρ=2700kg⋅m−3, L=3m, b=2m, and h=0.02m). The SBs are L1, L5, L3, and L7, where the BCs of L1 and L5 should be the same and those of L3 and L7 are the same. Therefore, there are 16 (4 × 4) kinds of SBCs of the plate built-up structure, as shown in Table 7. The SBCs are listed in order of L1L5-L3L7.Table 7
Different SBCs of a two-plate built-up structure.
IV
S1-S1
S1-S2
S2-S2
S2-S1
CPT
S-S
S-G
G-G
G-SFirstly, the influences of angles on the natural frequencies of the two-plate built-up structure are tabulated in Table8. The BCs of this plate built-up structure in Table 8 are listed in an anticlockwise sense of L1−L2−L3−L3−L4−L5−L6−L7. Table 8 lists the first 8 natural frequencies of the two-plate built-up structure with two plates connecting at angles of 30, 60, 90, 120, 150, and 180° under (S1 G)-F-(S1 G)-F-(S1 G)-F-(S1 G) BCs. It can be found from Table 8 that the connection angles have little effect on the natural frequencies except for 180°. In order to explain this phenomenon, this study computed the first 8 natural frequencies of the structure at different angles under the 10 different SBCs in Table 7 with free nodal boundaries. Here are the conclusions: the natural frequencies of the structure at the angle of 180° are significantly different from those of other angles (the differences are much greater than 2%). The rates of change are less than 2% when the structure is at different angles (except 180°) under the SBCs of (S1 S)-(S1 S), (S1 S)-(S1 G), and (S1 G)-(S1 G), and part of the rates are more than 2% under the SBCs of (S1 S)-(S2 S), (S1 G)-(S2 S), (S1 S)-(S2 G), (S1 G)-(S2 G), (S2 S)-(S2 S), (S2 S)-(S2 G), and (S2 G)-(S2 G). It might be due to the reason that the BCs along the opposite sides of the two-plate built-up structure are simply supported (or guided), the two plates are weakly coupled, and the low-frequency modes are mainly dominated by out-of-plane vibration. Therefore, the angles (except 180°) have little effect on the first 30 natural frequencies of the structure in this section. When these two plates are connected at an angle of 180°, the transverse and inplane vibrations are fully coupled.Table 8
The first 8 natural frequencies (ω,rad⋅s−1) of the two-plate built-up structure with two plates connecting at different angles under (S1 G)-F-(S1 G)-F-(S1 G)-F-(S1 G) BCs (RE is short for relative error).
θ (°)
1
2
3
4
5
6
7
8
30
DSM
27.472
47.614
58.418
120.47
151.31
157.32
166.19
172.17
FEM
27.470
47.540
58.359
120.22
151.14
157.20
165.77
172.10
RE (%)
−0.01
−0.15
−0.10
−0.21
−0.11
−0.08
−0.25
−0.04
60
DSM
27.472
47.614
58.418
120.47
151.31
157.32
166.19
172.17
FEM
27.469
47.547
58.359
120.40
151.17
157.20
165.89
172.07
RE (%)
−0.01
−0.14
−0.10
−0.05
−0.09
−0.08
−0.18
−0.05
90
DSM
27.472
47.614
58.418
120.47
151.31
157.32
166.19
172.17
FEM
27.469
47.549
58.358
120.44
151.18
157.19
165.91
172.01
RE (%)
−0.01
−0.14
−0.10
−0.03
−0.09
−0.08
−0.17
−0.09
120
DSM
27.473
47.614
58.418
120.47
151.31
157.32
166.19
172.17
FEM
27.466
47.551
58.354
120.45
151.19
157.17
165.93
171.83
RE (%)
−0.02
−0.13
−0.11
−0.02
−0.08
−0.09
−0.16
−0.20
150
DSM
27.474
47.614
58.418
120.47
151.31
157.32
166.19
172.17
FEM
27.453
47.559
58.332
120.45
151.20
157.09
165.95
170.83
RE (%)
−0.08
−0.11
−0.15
−0.01
−0.07
−0.15
−0.15
−0.78
180
DSM
33.616
43.703
47.614
90.260
120.47
135.66
151.31
166.19
FEM
33.612
43.701
47.579
90.151
120.45
135.62
151.22
166.00
RE (%)
−0.01
−0.01
−0.07
−0.12
−0.01
−0.03
−0.06
−0.11Then, Table9 shows the influence of SBCs on the first 8 natural frequencies of the two-plate built-up structure with two plates connecting at an angle of 60°. The BCs of the plate built-up structure in Table 9 are listed in the order of L1−L5−L3−L7, and the NBCs are free. It can be found that the natural frequencies of the structure have little differences when the SBCs related to out-of-plane vibration are the same although the SBCs for inplane vibration are different. This is due to the same reason as in Table 8: low-frequency modes are mainly dominated by out-of-plane vibration. Thus, the SBCs related to out-of-plane vibration have dominant influence on the low natural frequencies.Table 9
The first 8 natural frequenciesω,rad⋅s−1 of the two-plate built-up structure with two plates connecting at an angle of 60° under different SBC combinations.
SBCs
1
2
3
4
5
6
7
8
(S1 S)-(S1 S)-(S1 S)-(S1 S)
DSM
47.614
58.418
151.31
157.32
166.19
211.89
282.95
320.55
FEM
47.472
58.276
151.08
157.09
165.62
211.39
281.79
319.49
RE (%)
−0.30
−0.24
−0.15
−0.14
−0.34
−0.24
−0.41
−0.33
(S1 S)-(S1 S)-(S1 G)-(S1 G)
DSM
18.769
34.930
91.297
99.132
132.64
182.41
216.87
228.12
FEM
18.709
34.883
91.156
98.992
132.46
182.23
216.17
227.88
RE (%)
−0.32
−0.13
−0.15
−0.14
−0.14
−0.10
−0.32
−0.11
(S1 G)-(S1 G)-(S1 G)-(S1 G)
DSM
27.472
47.614
58.418
120.47
151.31
157.32
166.19
172.17
FEM
27.469
47.547
58.359
120.40
151.17
157.20
165.89
172.07
RE (%)
−0.01
−0.14
−0.10
−0.05
−0.09
−0.08
−0.18
−0.05
(S1 S)-(S1 S)-(S2 S)-(S2 S)
DSM
47.614
58.418
151.31
157.32
166.19
211.89
282.95
320.55
FEM
47.468
58.270
151.08
157.09
165.51
211.26
281.74
319.45
RE (%)
−0.31
−0.25
−0.15
−0.15
−0.41
−0.30
−0.43
−0.34
(S1 S)-(S1 S)-(S2 G)-(S2 G)
DSM
18.769
34.930
91.297
99.132
132.64
182.41
216.87
228.12
FEM
18.706
34.875
91.156
98.992
132.14
181.90
216.17
227.88
RE (%)
−0.34
−0.16
−0.15
−0.14
−0.37
−0.28
−0.33
−0.11
(S1 G)-(S1 G)-(S2 G)-(S2 G)
DSM
27.473
47.614
58.418
120.47
151.31
157.32
166.19
172.17
FEM
27.469
47.544
58.357
120.30
151.17
157.20
165.82
172.01
RE (%)
−0.01
−0.15
−0.11
−0.14
−0.09
−0.08
−0.22
−0.09
(S2 S)-(S2 S)-(S2 S)-(S2 S)
DSM
47.614
58.418
151.31
157.32
166.19
211.89
282.95
320.55
FEM
47.460
58.257
151.07
157.09
165.29
210.97
281.73
319.46
RE (%)
−0.32
−0.28
−0.16
−0.15
−0.54
−0.44
−0.43
−0.34All the results of the DSM and FEM match well with each other, as shown in Tables8 and 9 (differences within 1%). In this section, only 2 elements with 12 DOFs are adopted for the DSM, whereas the FEM uses as much as 30,000 elements (the element size is 0.02×0.02) and 182,106 DOFs to compute the results in Tables 8 and 9. It is obvious that the dynamic stiffness method has incomparable advantages in modal analysis of plate built-up structures.


### 3.3.2. I-Shaped Plate Built-Up Structure

Now we consider an I-shaped plate built-up structure (E = 72 GPa, ν=0.3, ρ=2700kg⋅m−3, L=3m, b=1m, Δz=2m, and h=0.01m), as shown in Figure 10, and the material and thickness of all the plates are the same as in the previous example. There are 10 SBs on the side of y = 0 (front) and y = L (back). The BCs of the SBs on the same side should be the same. Three representative SBCs of (S1 G)-(S1 G), (S1 S)-(S2 G), and (S2 S)-(S2 S) are considered for modal analysis.Figure 10
I-shaped plate built-up structure.Table10 shows the comparisons of the results computed by the DSM and FEM. Only 5 dynamic stiffness elements with 24 DOFs are used for the I-shaped plate built-up structure, whereas the FEM uses 120,000 elements (the element size is 0.01×0.01) with a total of 724,206 DOFs. The DSM gives exact results, while the computational cost is extremely inexpensive.Table 10
The first 8 natural frequenciesω,rad⋅s−1 of the I-shaped plate built-up structure under three representative SBCs. E = 72 GPa, ν=0.3, ρ=2700kg⋅m−3, L=3m, b=1m, Δz=0.2m, and h=0.01m.
SBCs
1
2
3
4
5
6
7
8
(S1 G)-(S1 G)
DSM
32.016
59.626
79.283
106.44
116.73
127.10
144.20
172.84
FEM
32.011
59.556
79.275
106.30
116.62
127.01
144.09
172.50
RE (%)
−0.02
−0.12
−0.01
−0.13
−0.09
−0.08
−0.08
−0.19
(S1 S)-(S2 G)
DSM
40.350
85.128
86.563
131.30
135.77
155.44
166.31
217.16
FEM
40.198
84.999
86.507
129.89
135.47
155.31
166.02
216.63
RE (%)
−0.38
−0.15
−0.06
−1.07
−0.22
−0.08
−0.17
−0.24
(S2 S)-(S2 S)
DSM
59.626
106.44
116.73
144.20
172.84
197.58
202.07
234.47
FEM
59.495
106.24
116.55
143.99
172.35
197.08
201.89
232.69
RE (%)
−0.22
−0.19
−0.15
−0.15
−0.29
−0.25
−0.09
−0.76


### 3.3.3. A Section of an Extruded Aluminum Panel

Figure11 shows a section of an extruded aluminum panel which has wide engineering applications such as the airplane fuselage and high-speed train body structures. In this section, the modal analysis of the structure in Figure 1 (E = 72 GPa, ν=0.3, ρ=2700kg⋅m−3, L=0.2m, be=0.1m, b=0.5m, Δz=0.1m, and h=0.002m) is carried out by using the DSM and FEM under three different SBCs, and the material and thickness of all the plates are the same. Some representative natural frequencies are shown in Table 11 and modal shapes in Figure 12. The DSM uses 21 elements with 52 DOFs to obtain the results, while 16,800 elements (the element size is 0.005×0.005) for a total of 101,352 DOFs are required for the FEM. It can be concluded that the results computed by the DSM and FEM agree well. Obviously, the DSM is applicable to the complex plate built-up structures, and therefore, the DSM is suitable for optimization design and parameter analysis due to its high efficiency and analytical nature.Figure 11
A section of an extruded aluminum panel.Table 11
The first 8 natural frequenciesω,rad⋅s−1 of the extruded aluminum panel under three representative SBC combinations. E = 72 GPa, ν=0.3, ρ=2700kg⋅m−3, L=0.2m, be=0.1m, b=0.5m, Δz=0.1m, and h=0.002m.
SBCs
1
2
3
4
5
6
7
8
(S1 G)-(S1 G)
DSM
2761.7
2764.6
3462.0
3552.8
3558.0
3558.8
3651.5
3864.5
FEM
2763.1
2763.2
3470.0
3554.2
3555.0
3560.3
3657.9
3868.7
RE (%)
0.05
−0.05
0.23
0.04
−0.09
0.04
0.18
0.11
(S1 S)-(S2 G)
DSM
2963.3
2966.4
3626.0
3717.2
3806.0
4010.9
4122.3
4475.2
FEM
2957.4
2957.7
3630.6
3717.7
3807.9
3997.2
4105.9
4456.2
RE (%)
−0.20
−0.29
0.13
0.01
0.05
−0.34
−0.40
−0.42
(S2 S)-(S2 S)
DSM
3558.0
3558.8
4131.0
4222.7
4285.0
4465.2
4591.9
4920.4
FEM
3546.2
3546.9
4131.1
4220.1
4283.4
4448.8
4576.9
4906.5
RE (%)
−0.33
−0.33
0.00
−0.06
−0.04
−0.37
−0.33
−0.28Figure 12
The first 3 modal shapes of the extruded aluminum panel computed by the DSM (a) and FEM (b).
(a)
(b)Another advantage of the DSM for vibration analysis lies in its high efficiency and accuracy over the whole frequency range. Some representative natural frequencies covering the midfrequency and high-frequency ranges (10–100th modes) of the I-shaped plate built-up structure and extruded aluminum panel are listed in Table 12, where the SBC is (S2 S)-(S2 S). It is found that the computational efficiency of the DSM is more than 100 times that of the FEM. For example, the FEM takes 662 seconds to compute the first 100 modes of the I-shaped plate built-up structure, while the DSM only takes 3.28 seconds.Table 12
Representative natural frequencies covering the whole frequency range of the first 100 natural frequenciesω,rad⋅s−1 of the I-shaped plate built-up structure and extruded aluminum panel in Figures 10 and 11 under the SBC combination of (S2 S)-(S2 S).
10
30
50
70
90
100
I-shaped plate built-up structure
DSM
250.24
575.30
1008.3
1302.5
1563.8
1771.4
FEM
249.61
574.35
1005.2
1295.1
1591.0
1767.8
RE (%)
−0.25
−0.16
−0.31
−0.57
1.74
−0.21
Extruded aluminum panel
DSM
5403.4
7288.5
10458
14332
16015
16882
FEM
5369.2
7262.1
10447
14323
16052
16926
RE (%)
−0.63
−0.36
−0.10
−0.07
0.23
0.26


## 3.3.1. A Two-Plate Built-Up Structure

Consider a two-plate built-up structure with two plates connecting at an angle ofθ (Figure 5), then the geometry and material properties of the 2 plates are the same (E = 72 GPa, ν=0.3, ρ=2700kg⋅m−3, L=3m, b=2m, and h=0.02m). The SBs are L1, L5, L3, and L7, where the BCs of L1 and L5 should be the same and those of L3 and L7 are the same. Therefore, there are 16 (4 × 4) kinds of SBCs of the plate built-up structure, as shown in Table 7. The SBCs are listed in order of L1L5-L3L7.Table 7
Different SBCs of a two-plate built-up structure.
IV
S1-S1
S1-S2
S2-S2
S2-S1
CPT
S-S
S-G
G-G
G-SFirstly, the influences of angles on the natural frequencies of the two-plate built-up structure are tabulated in Table8. The BCs of this plate built-up structure in Table 8 are listed in an anticlockwise sense of L1−L2−L3−L3−L4−L5−L6−L7. Table 8 lists the first 8 natural frequencies of the two-plate built-up structure with two plates connecting at angles of 30, 60, 90, 120, 150, and 180° under (S1 G)-F-(S1 G)-F-(S1 G)-F-(S1 G) BCs. It can be found from Table 8 that the connection angles have little effect on the natural frequencies except for 180°. In order to explain this phenomenon, this study computed the first 8 natural frequencies of the structure at different angles under the 10 different SBCs in Table 7 with free nodal boundaries. Here are the conclusions: the natural frequencies of the structure at the angle of 180° are significantly different from those of other angles (the differences are much greater than 2%). The rates of change are less than 2% when the structure is at different angles (except 180°) under the SBCs of (S1 S)-(S1 S), (S1 S)-(S1 G), and (S1 G)-(S1 G), and part of the rates are more than 2% under the SBCs of (S1 S)-(S2 S), (S1 G)-(S2 S), (S1 S)-(S2 G), (S1 G)-(S2 G), (S2 S)-(S2 S), (S2 S)-(S2 G), and (S2 G)-(S2 G). It might be due to the reason that the BCs along the opposite sides of the two-plate built-up structure are simply supported (or guided), the two plates are weakly coupled, and the low-frequency modes are mainly dominated by out-of-plane vibration. Therefore, the angles (except 180°) have little effect on the first 30 natural frequencies of the structure in this section. When these two plates are connected at an angle of 180°, the transverse and inplane vibrations are fully coupled.Table 8
The first 8 natural frequencies (ω,rad⋅s−1) of the two-plate built-up structure with two plates connecting at different angles under (S1 G)-F-(S1 G)-F-(S1 G)-F-(S1 G) BCs (RE is short for relative error).
θ (°)
1
2
3
4
5
6
7
8
30
DSM
27.472
47.614
58.418
120.47
151.31
157.32
166.19
172.17
FEM
27.470
47.540
58.359
120.22
151.14
157.20
165.77
172.10
RE (%)
−0.01
−0.15
−0.10
−0.21
−0.11
−0.08
−0.25
−0.04
60
DSM
27.472
47.614
58.418
120.47
151.31
157.32
166.19
172.17
FEM
27.469
47.547
58.359
120.40
151.17
157.20
165.89
172.07
RE (%)
−0.01
−0.14
−0.10
−0.05
−0.09
−0.08
−0.18
−0.05
90
DSM
27.472
47.614
58.418
120.47
151.31
157.32
166.19
172.17
FEM
27.469
47.549
58.358
120.44
151.18
157.19
165.91
172.01
RE (%)
−0.01
−0.14
−0.10
−0.03
−0.09
−0.08
−0.17
−0.09
120
DSM
27.473
47.614
58.418
120.47
151.31
157.32
166.19
172.17
FEM
27.466
47.551
58.354
120.45
151.19
157.17
165.93
171.83
RE (%)
−0.02
−0.13
−0.11
−0.02
−0.08
−0.09
−0.16
−0.20
150
DSM
27.474
47.614
58.418
120.47
151.31
157.32
166.19
172.17
FEM
27.453
47.559
58.332
120.45
151.20
157.09
165.95
170.83
RE (%)
−0.08
−0.11
−0.15
−0.01
−0.07
−0.15
−0.15
−0.78
180
DSM
33.616
43.703
47.614
90.260
120.47
135.66
151.31
166.19
FEM
33.612
43.701
47.579
90.151
120.45
135.62
151.22
166.00
RE (%)
−0.01
−0.01
−0.07
−0.12
−0.01
−0.03
−0.06
−0.11Then, Table9 shows the influence of SBCs on the first 8 natural frequencies of the two-plate built-up structure with two plates connecting at an angle of 60°. The BCs of the plate built-up structure in Table 9 are listed in the order of L1−L5−L3−L7, and the NBCs are free. It can be found that the natural frequencies of the structure have little differences when the SBCs related to out-of-plane vibration are the same although the SBCs for inplane vibration are different. This is due to the same reason as in Table 8: low-frequency modes are mainly dominated by out-of-plane vibration. Thus, the SBCs related to out-of-plane vibration have dominant influence on the low natural frequencies.Table 9
The first 8 natural frequenciesω,rad⋅s−1 of the two-plate built-up structure with two plates connecting at an angle of 60° under different SBC combinations.
SBCs
1
2
3
4
5
6
7
8
(S1 S)-(S1 S)-(S1 S)-(S1 S)
DSM
47.614
58.418
151.31
157.32
166.19
211.89
282.95
320.55
FEM
47.472
58.276
151.08
157.09
165.62
211.39
281.79
319.49
RE (%)
−0.30
−0.24
−0.15
−0.14
−0.34
−0.24
−0.41
−0.33
(S1 S)-(S1 S)-(S1 G)-(S1 G)
DSM
18.769
34.930
91.297
99.132
132.64
182.41
216.87
228.12
FEM
18.709
34.883
91.156
98.992
132.46
182.23
216.17
227.88
RE (%)
−0.32
−0.13
−0.15
−0.14
−0.14
−0.10
−0.32
−0.11
(S1 G)-(S1 G)-(S1 G)-(S1 G)
DSM
27.472
47.614
58.418
120.47
151.31
157.32
166.19
172.17
FEM
27.469
47.547
58.359
120.40
151.17
157.20
165.89
172.07
RE (%)
−0.01
−0.14
−0.10
−0.05
−0.09
−0.08
−0.18
−0.05
(S1 S)-(S1 S)-(S2 S)-(S2 S)
DSM
47.614
58.418
151.31
157.32
166.19
211.89
282.95
320.55
FEM
47.468
58.270
151.08
157.09
165.51
211.26
281.74
319.45
RE (%)
−0.31
−0.25
−0.15
−0.15
−0.41
−0.30
−0.43
−0.34
(S1 S)-(S1 S)-(S2 G)-(S2 G)
DSM
18.769
34.930
91.297
99.132
132.64
182.41
216.87
228.12
FEM
18.706
34.875
91.156
98.992
132.14
181.90
216.17
227.88
RE (%)
−0.34
−0.16
−0.15
−0.14
−0.37
−0.28
−0.33
−0.11
(S1 G)-(S1 G)-(S2 G)-(S2 G)
DSM
27.473
47.614
58.418
120.47
151.31
157.32
166.19
172.17
FEM
27.469
47.544
58.357
120.30
151.17
157.20
165.82
172.01
RE (%)
−0.01
−0.15
−0.11
−0.14
−0.09
−0.08
−0.22
−0.09
(S2 S)-(S2 S)-(S2 S)-(S2 S)
DSM
47.614
58.418
151.31
157.32
166.19
211.89
282.95
320.55
FEM
47.460
58.257
151.07
157.09
165.29
210.97
281.73
319.46
RE (%)
−0.32
−0.28
−0.16
−0.15
−0.54
−0.44
−0.43
−0.34All the results of the DSM and FEM match well with each other, as shown in Tables8 and 9 (differences within 1%). In this section, only 2 elements with 12 DOFs are adopted for the DSM, whereas the FEM uses as much as 30,000 elements (the element size is 0.02×0.02) and 182,106 DOFs to compute the results in Tables 8 and 9. It is obvious that the dynamic stiffness method has incomparable advantages in modal analysis of plate built-up structures.


## 3.3.2. I-Shaped Plate Built-Up Structure

Now we consider an I-shaped plate built-up structure (E = 72 GPa, ν=0.3, ρ=2700kg⋅m−3, L=3m, b=1m, Δz=2m, and h=0.01m), as shown in Figure 10, and the material and thickness of all the plates are the same as in the previous example. There are 10 SBs on the side of y = 0 (front) and y = L (back). The BCs of the SBs on the same side should be the same. Three representative SBCs of (S1 G)-(S1 G), (S1 S)-(S2 G), and (S2 S)-(S2 S) are considered for modal analysis.Figure 10
I-shaped plate built-up structure.Table10 shows the comparisons of the results computed by the DSM and FEM. Only 5 dynamic stiffness elements with 24 DOFs are used for the I-shaped plate built-up structure, whereas the FEM uses 120,000 elements (the element size is 0.01×0.01) with a total of 724,206 DOFs. The DSM gives exact results, while the computational cost is extremely inexpensive.Table 10
The first 8 natural frequenciesω,rad⋅s−1 of the I-shaped plate built-up structure under three representative SBCs. E = 72 GPa, ν=0.3, ρ=2700kg⋅m−3, L=3m, b=1m, Δz=0.2m, and h=0.01m.
SBCs
1
2
3
4
5
6
7
8
(S1 G)-(S1 G)
DSM
32.016
59.626
79.283
106.44
116.73
127.10
144.20
172.84
FEM
32.011
59.556
79.275
106.30
116.62
127.01
144.09
172.50
RE (%)
−0.02
−0.12
−0.01
−0.13
−0.09
−0.08
−0.08
−0.19
(S1 S)-(S2 G)
DSM
40.350
85.128
86.563
131.30
135.77
155.44
166.31
217.16
FEM
40.198
84.999
86.507
129.89
135.47
155.31
166.02
216.63
RE (%)
−0.38
−0.15
−0.06
−1.07
−0.22
−0.08
−0.17
−0.24
(S2 S)-(S2 S)
DSM
59.626
106.44
116.73
144.20
172.84
197.58
202.07
234.47
FEM
59.495
106.24
116.55
143.99
172.35
197.08
201.89
232.69
RE (%)
−0.22
−0.19
−0.15
−0.15
−0.29
−0.25
−0.09
−0.76


## 3.3.3. A Section of an Extruded Aluminum Panel

Figure11 shows a section of an extruded aluminum panel which has wide engineering applications such as the airplane fuselage and high-speed train body structures. In this section, the modal analysis of the structure in Figure 1 (E = 72 GPa, ν=0.3, ρ=2700kg⋅m−3, L=0.2m, be=0.1m, b=0.5m, Δz=0.1m, and h=0.002m) is carried out by using the DSM and FEM under three different SBCs, and the material and thickness of all the plates are the same. Some representative natural frequencies are shown in Table 11 and modal shapes in Figure 12. The DSM uses 21 elements with 52 DOFs to obtain the results, while 16,800 elements (the element size is 0.005×0.005) for a total of 101,352 DOFs are required for the FEM. It can be concluded that the results computed by the DSM and FEM agree well. Obviously, the DSM is applicable to the complex plate built-up structures, and therefore, the DSM is suitable for optimization design and parameter analysis due to its high efficiency and analytical nature.Figure 11
A section of an extruded aluminum panel.Table 11
The first 8 natural frequenciesω,rad⋅s−1 of the extruded aluminum panel under three representative SBC combinations. E = 72 GPa, ν=0.3, ρ=2700kg⋅m−3, L=0.2m, be=0.1m, b=0.5m, Δz=0.1m, and h=0.002m.
SBCs
1
2
3
4
5
6
7
8
(S1 G)-(S1 G)
DSM
2761.7
2764.6
3462.0
3552.8
3558.0
3558.8
3651.5
3864.5
FEM
2763.1
2763.2
3470.0
3554.2
3555.0
3560.3
3657.9
3868.7
RE (%)
0.05
−0.05
0.23
0.04
−0.09
0.04
0.18
0.11
(S1 S)-(S2 G)
DSM
2963.3
2966.4
3626.0
3717.2
3806.0
4010.9
4122.3
4475.2
FEM
2957.4
2957.7
3630.6
3717.7
3807.9
3997.2
4105.9
4456.2
RE (%)
−0.20
−0.29
0.13
0.01
0.05
−0.34
−0.40
−0.42
(S2 S)-(S2 S)
DSM
3558.0
3558.8
4131.0
4222.7
4285.0
4465.2
4591.9
4920.4
FEM
3546.2
3546.9
4131.1
4220.1
4283.4
4448.8
4576.9
4906.5
RE (%)
−0.33
−0.33
0.00
−0.06
−0.04
−0.37
−0.33
−0.28Figure 12
The first 3 modal shapes of the extruded aluminum panel computed by the DSM (a) and FEM (b).
(a)
(b)Another advantage of the DSM for vibration analysis lies in its high efficiency and accuracy over the whole frequency range. Some representative natural frequencies covering the midfrequency and high-frequency ranges (10–100th modes) of the I-shaped plate built-up structure and extruded aluminum panel are listed in Table 12, where the SBC is (S2 S)-(S2 S). It is found that the computational efficiency of the DSM is more than 100 times that of the FEM. For example, the FEM takes 662 seconds to compute the first 100 modes of the I-shaped plate built-up structure, while the DSM only takes 3.28 seconds.Table 12
Representative natural frequencies covering the whole frequency range of the first 100 natural frequenciesω,rad⋅s−1 of the I-shaped plate built-up structure and extruded aluminum panel in Figures 10 and 11 under the SBC combination of (S2 S)-(S2 S).
10
30
50
70
90
100
I-shaped plate built-up structure
DSM
250.24
575.30
1008.3
1302.5
1563.8
1771.4
FEM
249.61
574.35
1005.2
1295.1
1591.0
1767.8
RE (%)
−0.25
−0.16
−0.31
−0.57
1.74
−0.21
Extruded aluminum panel
DSM
5403.4
7288.5
10458
14332
16015
16882
FEM
5369.2
7262.1
10447
14323
16052
16926
RE (%)
−0.63
−0.36
−0.10
−0.07
0.23
0.26


## 4. Conclusions

This paper has developed new formulations for both dynamic stiffness modelling and the associated algorithm for complex plate built-up structures for more general cases. In terms of modelling, dynamic stiffness formulations for plate elements with four different types of opposite-edge-support conditions and arbitrarily supported boundary conditions along other edges have been developed for both inplane and out-of-plane vibrations. As a result, the present formulations cover 16 types of opposite-edge-support boundary conditions (SBCs) with inplane and out-of-plane vibrations coupled, which is in a sharp contrast to existing research applicable to only one type of SBC and has greatly expanded the application range of the DSM. In terms of the algorithm, analytical expressions of theJ0 count for all SBCs discussed above have been developed for the Wittrick–Williams algorithm. With the J0 problem resolved, there is no need to split a large dynamic stiffness element into smaller ones unnecessarily as the majority of existing works did. Therefore, very few DOFs are required for modelling complex plate built-up structures, which has made the DSM to be highly efficient for the whole frequency range. An efficient and accurate program has been developed based on the DS model for individual plates and plate built-up structures. The program has been applied to a couple of complex plate built-up structures in engineering applications, and exact natural frequencies and mode shapes within the whole frequency range are computed extremely efficiently.This study focuses on the free vibration analysis of the structures; however, the formulations can also be used directly for forced vibration analysis, or extended to compute accurate key parameters for other methods. For example, it can compute the modal density and coupling loss factor which are key parameters for the statistical energy analysis (SEA) method especially within the midfrequency range. A drawback of the present study lies in that there are some limitations for opposite-edge-support conditions and the elements can only be assembled in one direction. In order to address these limitations, the spectral dynamic stiffness method [33, 38] can be used as a powerful alternative, but it will involve more degrees of freedom for more complex boundary conditions.



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*Source: 5305692-2020-03-20.xml*