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

Key Laboratory of Traffic Safety on Track (Central South University), Ministry of Education, Central South University, Hunan, Changsha 410075, China Joint International Research Laboratory of Key Technology for Rail Traffic Safety, Hunan, Changsha 410075, China School of Traffic & Transportation Engineering, Central South University, Hunan, Changsha 410075, China State Key Laboratory of High Performance ComplexManufacturing, Central South University, Hunan, Changsha 410075, China School of Civil Engineering, Central South University, Hunan, Changsha 410075, China


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. erefore, it is virtually important to find efficient and accurate methods for the free vibration analysis of plate built-up structures.
ere 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 shortwavelength 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). erefore, 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). us, 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.
e DSM was originally proposed by Kolousek [3] in 1941. Subsequently, many authors [4][5][6][7][8][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]. e 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 midhigh-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][19][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. e 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][24][25][26][27] can obtain the solutions that satisfy both GDEs and boundary conditions in a strong manner. is 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][30][31][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][30][31][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][19][20], but they are inefficient and meanwhile very likely to miss some modal solutions [28]. ese shortcomings can be overcome by the Wittrick-Williams (WW) algorithm, which is an extremely efficient and accurate algorithm for the DSM; however, the J 0 count in the algorithm is an important and difficult problem [33]. J 0 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][35][36] discretized the structure into a finer dynamic stiffness mesh to ensure that J 0 is equal to zero, which greatly reduces the computational efficiency, and the merit of the DSM is not brought into full play.
is 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 outof-plane vibration and arbitrarily supported boundary conditions along other edges. at 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 Section 2.1.
At the same time, the mode count (J 0 ) of the WW algorithm under all possible opposite-edge-support conditions is formulated analytically in this study. us, 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. is work greatly enhances the superiority of the DSM over the FEM in computational efficiency. e 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.
is paper is organized as follows: In Section 2.1, the boundary conditions in this research are detailed. en, 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. en, the J 0 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.

Theory
is 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. e assembly procedure of the elemental matrices is 2 Shock and Vibration shown in Section 2.4, and the algorithm of the DSM is improved in Section 2.5. Figure 1 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. Table 1 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.

Different Boundary Conditions of a Plate Element.
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 Table 2) to 40, respectively, in Table 2. is will no doubt broaden the application scope of the DSM. e letter "F" in Table 2 denotes a free edge and "C" indicates a clamped boundary. e sequence of the SBC combinations is supported boundary1-supported bound-ary2 and of NBC combinations is nodal boundary1-nodal boundary2.
us, 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.

Dynamic Stiffness Formulation for Inplane Vibration of a Plate Element.
is section focuses on inplane vibration of plate elements with three different SBCs, as provided in Table 2, 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).

Shock and Vibration
, 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 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 U m � e rx , V m � δe rx into equation (6) leads to After some fundamental derivations, the exact general solution of equation (6) can be represented as follows:
e force and displacement NBCs of the plate element ( Figure 4) are given as follows: e relationship between the displacement NBCs and unknown coefficients is determined by applying the BCs for displacements to equation (9) as follows: where C hi � cosh(r i b) and S hi � sinh(r i b). Similarly, inserting force NBCs into equation (12) gives where H i � ((2(r i + α m δ i ]))/(] − 1)) and P i � α m − δ i r i By eliminating the coefficient vector of equations (14) and (15) 6 Shock and Vibration ] is a symmetric matrix, which is composed of 6 elements k i 11 , k i 12 , k i 13 , k i 14 , k i 22 , and k i 24 . e expressions of these 6 elements are as follows: (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 of U(x, y) with U(x, y) � U m (x)cos(α m y) and the displacement of V(x, y) is always equal to zero by applying m � 0 to equation (4). As a consequence, equation (6) can be simplified as where a 3 is given by equation (7). e general solution of equation (18) can be written in the form e function for force of inplane vibration is defined by substituting equation (19) into (3) as follows: e BCs in this case are given by Applying the displacement NBCs of equation (21) to (19) leads to where C � cos(rb) and S � sin(rb). Figure 4: Force and displacement NBCs of a plate element.

Shock and Vibration
Inserting the force NBCs of equation (21) to (20) gives e dynamic stiffness matrix for inplane vibration in this situation is resolved by eliminating the constants A 1 ′ and A 2 ′ of equations (22) and (23): where the two elements of [K i (ω)] are exhibited as follows: Similarly, when the SBC is S1-S1 and m � 0, U(x, y) and force of N xx (x, 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].

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-ofplane 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). e 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.
e GDE in the time domain of out-of-plane vibration is given by which can be transferred into the frequency domain as e natural BCs take the form where D � (Eh 3 /(12(1 − ] 2 ))) is the bending stiffness of the plate. Figure 3 shows the boundary constraints of the transverse vibration. e general solutions of equation (28) with three different SBCs can be written as Substituting equation (30) into (28) gives (except for the case of G-G with m � 0) In the case of the SBC being G-G, when m � 0, equation (32) becomes 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 where where , "±" takes the first sign for i � 1 and the second sign for i � 2, S hi and C hi are the same as in Section 2.2,

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. en, 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.

Coordinate Transformation.
Taking the structure in Figure 5 as an example, assuming that the coordinate system of plate 1 (P 1 of Figure 5) coincides with the global coordinate system, the displacement and force vectors of plate 2 (P 2 of Figure 5) need to be converted to the global coordinate system for the assembly purpose. Supposing U l , V l , W l , ϕ l 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 U g , V g , W g , ϕ g is given by e transformation matrix T of the displacement vector of a plate element can be written as It should be emphasized that, in the case of m � 0, the displacement of V l is equal to 0 when the SBC is S2-S2; then, the displacement vector (U g , W g , ϕ g ) takes the form In this case, T becomes Similarly, the transformation matrix T for the case of S1-S1 and m � 0 is e transformation matrix of the forces can also be defined using the same method, and it is exactly the transposed matrix of T.
us, the spatial transformation function of plate elements is given as follows: e procedure is similar to the finite element method. Take Figure 5 as an example, in which plate1 and plate2 share the same nodal boundary L 4 . Equations (42) and (43) are the dynamic stiffness matrices of plate1 and plate2, respectively. erefore, the assembly of plate1 and plate2 becomes a 12 × 12 matrix by summing the entries at the common nodal boundary L 4 , namely, Shock and Vibration 9 Figure 5: A two-plate built-up structure.

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 L 2 and L 4 are free with L 6 fully clamped, the last 4 values in the displacement vector of equation (44) are zero. en, the global dynamic stiffness matrix of the structure in Figure 5 can be written as By the similar way, various complex plate built-up structures can be assembled in this research.

Wittrick-William Algorithm and J 0 Counts.
Once the dynamic stiffness matrix is developed, the Wittrick-Williams algorithm can be applied to compute the natural frequencies ω of structures. e 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 * : where [K(ω * )] is the elemental dynamic stiffness matrix when ω � ω * , s [K(ω * )] { } is the number of negative diagonal elements after upper triangular transformation by using Gauss elimination of [K(ω * )], and J 0 is the number of natural frequencies between ω � 0 and ω � ω * when the nodal boundaries (NBs) of the plate element are fully clamped.
ere is no doubt that J 0 plays an important role in the Wittrick-Williams algorithm. However, calculating J 0 is generally a difficult problem, and the traditional way is to refine the mesh to make sure J 0 � 0 [17,[34][35][36]. Obviously, it will introduce unnecessary computational cost significantly.
In this study, the J 0 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 J sm 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 where J 0m is J 0 when the half wave number in the y direction is m and [K s m (ω * )] is the dynamic stiffness matrix [K(ω * )] for a plate element with all NBs simply supported (or guided) when m is a certain value. e analytical expression for J sm 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.

J 0 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: Substituting equation (48) into (2), the relationship between m, n, and ω under the S1-S1 or S2-S2 SBC is determined by where β � ω ����� (ρ/G) and b is the width of the plate. Solving equation (49) gives Suppose J i sm is J sm for inplane vibration of a fully simply supported plate. erefore, J i sm consists of two parts which can be solved from equations (50) and (51) separately. Equation (50) can be expressed in the form So, natural modes exist for all integers m, n ≥ 1, as shown in Figure 6. Essentially, J i sm1 is the mode count of an S2S2S2S2 or S1S1S2S1 plate when m is fixed which is obtained by solving n of equation (53): where floor(x) indicates the largest integer not greater than x.
Adding J i sm1 and J i sm2 together gives

J 0 Formulations for Transverse Free Vibration.
Similar to Section 2.5.1, J 0 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: Substituting equation (60) into (28), the relationship between m, n, and ω under the S-S SBC and S-S NBC is determined by which is also valid for G-G and S-G SBCs with the S-S NBC with Suppose J o sm is J sm 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 J o sm as with

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 in Section 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.  Table 3  It can be seen from Table 3 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. en, the first 8 natural frequencies of the plate under 27 classical BCs in Table 2 are shown in Table 4.

Modal Analysis of Inplane Vibration for an Individual
Plate. e first 8 natural frequencies of inplane vibration for an individual plate (Table 5) 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. e structure is the same as in Section 3.1, while only the inplane motion is considered in this section. e 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. e relative errors between  Figure 9: J i sm2 of S1S1S1S1 (a) and S1S1S2S1 (b).  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.

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).  Table 7. e SBCs are listed in order of L 1 (L 5 )-L 3 (L 7 ).

A Two-Plate
Firstly, the influences of angles on the natural frequencies of the two-plate built-up structure are tabulated in Table 8.
e BCs of this plate built-up structure in Table 8 are listed in an anticlockwise sense of 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%). e 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. erefore, 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. en, Table 9 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°. e BCs of the plate built-up structure in Table 9 are listed in the order of L 1 − L 5 − L 3 − L 7 , 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. is is due to the same reason as in Table 8: low-frequency modes are mainly dominated by out-of-plane vibration. us, the SBCs related to out-of-plane vibration have dominant influence on the low natural frequencies.
All the results of the DSM and FEM match well with each other, as shown in Tables 8 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.  Figure 10, and the material and 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.
IV S1-S1 S1-S2 S2-S2 S2-S1 CPT S-S S-G G-G G-S Table 8: e 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). thickness of all the plates are the same as in the previous example. ere are 10 SBs on the side of y � 0 (front) and y � L (back). e BCs of the SBs on the same side should be the same. ree representative SBCs of (S1 G)-(S1 G), (S1 S)-(S2 G), and (S2 S)-(S2 S) are considered for modal analysis. Table 10 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. e Table 9: e first 8 natural frequencies (ω, rad·s − 1 ) of the two-plate built-up structure with two plates connecting at an angle of 60°under different SBC combinations. 1  2  3  4  5  6  7  8 (S1 S)-(S1 S)-(S1 S)-(S1 S) (S1 S)-(S1 S)-(S1 G)-(S1 G) (S1 G)-(S1 G)-(S1 G)-(S1 G)    Figure 11 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  Table 11 and   modal shapes in Figure 12. e 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. 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 (10-100 th 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.

Conclusions
is 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 the J 0 count for all SBCs discussed above have been developed for the Wittrick-Williams algorithm. With the J 0 problem resolved, there is no need to split a large dynamic stiffness element into smaller ones unnecessarily as the majority of existing works did.
erefore, 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. e 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.
is 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.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest. Table 12: Representative natural frequencies covering the whole frequency range of the first 100 natural frequencies (ω, 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