#### Abstract

The multilayer plate has a great potential for automotive and aerospace applications. However, the complexity in structure and calculation of the response impede the practical applications of multilayer plates. To solve this problem, this work proposes a new plate finite element and a simplified finite element (FE) model for multilayer plates. The proposed new plate finite element consists of the shear and extension strains in all layers. The multilayer structure with the proposed new plate finite element is regarded as a reference to calculate the reference value of the transverse response. The simplified FE model of multilayer plates is proposed based on the equivalent bending stiffness by curve fitting of the reference value of the transverse response. Numerical study shows that this approach can be used to set up the simplified FE model of multilayer plates.

#### 1. Introduction

In recent decades, the multilayer plates have been demonstrated to be promising engineering structures for automobiles and aerospace vehicles [1–4]. A typical multilayer plate has three layers, including the constraint layer, damping layer, and base layer. The multilayer plate structure is mainly used for vibration suppression with the consumption of the strain energy since the damping layer would deform with the relative motion of the constraint layer and base layer [5–8]. Therefore, the strain in all layers should be well considered when modeling the multilayer plates.

In the literatures, extensive investigations have been conducted on the modeling of the multilayer plates [4, 6, 9–11]. However, an assumption is usually made that only shear train exists in the core layer and extension strain exists in the constraint layer and base layer [12]. This assumption is not valid for certain structures and conditions. Moreover, majority of researchers have studied the multilayer plate structure by presenting the analytical solutions using the modal superposition method. However, the analytical solution is quite complex and not accurate [13–15]. On the other hand, the finite element (FE) modeling is a good numerical method which has been extensively and efficiently applied to investigate the vibrational behavior of structures including the viscoelastic material [13, 14, 16–18]. However, when modeling this plate structure using the FE method, the plate is meshed into many finite elements and the degrees of freedom should be well defined with full consideration of the shear, compression, and extensional damping. For the complex modeling problem, the beam finite element presented by Zapfe and Lesieutre with 11 degrees of freedom (DOFs) can be referred with a very good FE model [19]. Moreover, the plate structure is an extension of beam structure. Considering the efficiency of FE method and accuracy of the structure of plates, the work tries to develop a new plate finite element consisting of the shear and extension strains in all layers as an extension of Zapfe’s beam element.

Even though the plate structure with new plate finite element method can develop a good FE model, there are still plenty of DOFs involved, which makes the FE modeling of multilayer plate structures complicated, especially for the plate structure with a quite thin damping layer. To ease the computational burden and consumption cost, a simplified method should be developed. In [20], a simplified FE model for beam structures with three layers is presented by using single-layer equivalent FE method. The equivalent material properties are calculated and then a regular beam is constructed. Considering the simplicity and accuracy, this work also attempts to develop a simplified FE model for multilayer plates by curve fitting the equivalent bending stiffness.

In this research, a new plate finite element for multilayer plate is developed, and a simplified FE model of multilayer plates is proposed based on the equivalent bending stiffness by curve fitting the response values calculated from the multilayer plate structure with new plate finite element. The rest of this paper is organized as follows. Section 2 presents the proposed new plate finite element as an extension of Zapfe’s beam element with validation. Subsequently, the simplified FE model for multilayer plates is described in Section 3. Finally, conclusions are given in Section 4.

#### 2. New Plate Finite Element for Multilayer Plate

This section presents a new plate finite element for multilayer plate for FE modeling. First, the new plate finite element is proposed with the analysis of degree of freedom (DOF), and then a validation is conducted by comparison with the published data.

##### 2.1. Proposed New Plate Finite Element for Multilayer Plate

The element formulated in the work of Zapfe and Lesieutre was used as the reference in the curve fitting of a transfer function of the multilayer beam [19]. This element can be extended to constrained layer damped plates. The basic formulations in the work of Zapfe and Lesieutre can be followed to develop the new plate elements with more degrees of freedom [19]. Figure 1 gives the multilayer plate studied in this work. It consists of three layers. Layer 1, layer 2, and layer 3 are the base layer, damping layer, and constraining layer, respectively. Figure 2 shows the DOFs of the new plate element. Each of the four nodes in every corner has eight longitudinal degrees of freedom and one transverse degree of freedom. Note that there are five additional midnodes in the element in order to avoid shear locking. These five midnodes have only one transverse degree of freedom.

**(a)**

**(b)**

As described in Figure 1, the DOF vector of each node can be given in the following equations. For nodes *m*, *k*, *p*, and *q*, there are four DOFs in the u and v directions, respectively, and one DOF in the w direction. For nodes 1, 2, 3, 4, and 5, only one DOF exists in the w direction. To fully characterize the shear strain and extension strain and clearly represent the transverse displacement at each location of the plate, the DOF vectors are represented as

By combining all of the DOFs of each node, the DOF vector of the proposed new finite element for the multilayer plate can be expressed in the following equations:

It can be found that each finite element for the multilayer plate has 41 DOFs. During the FE modeling, the multilayer is meshed into many elements with and nodes in the longitudinal directions. The overall DOF in the multilayer plate structure can be expressed as

For each meshed element, a shape function matrix for each layer can be defined aswhere the shape function in matrix **N** can be written as , , , and . The variables and are the longitudinal coordinates; and are the length and width of each meshed element, respectively. By calculating the differences of matrices , , and , a new shape function can be obtained:

Meanwhile, a modulus matrix with Young’s modulus E and shear modulus G can be formulated as

Then, the corresponding stiffness matrix and mass matrix of each meshed element in each layer can be set aswhere is the density of the plate. Since the multilayer plate is meshed in many plate finite elements, the stiffness matrix and mass matrix can be obtained by integrating the stiffness matrix and mass matrix of each plate finite element in the plate structure. By exciting a given force vector, the response can be obtained according to the Lagrange formula [21]:where stands for the excited force vector; represents the inherent frequency of the multilayer plate structure; and is the response matrix of the multilayer plate with meshed plate finite elements. The response matrix can be obtained by combining the matrix in each plate finite element.

##### 2.2. Validation of Proposed New Plate Finite Element

After the plate element is formulated, it should be validated to illustrate its effectiveness. In the existing literature, Kung and Singh provided some results calculated by the analytical model developed for the beam element in their work [22]. Naturally, these data can be used here to validate the plate element. Table 1 gives the configuration of a plate to be validated which is simply supported along all four edges.

The comparison of the frequency and loss factor between the published data [22] and the data calculated by the new plate element is presented in Tables 2 and 3, respectively. From Tables 2 and 3, it can be seen that the new plate element provides close results to the published data in the work of Kung and Singh by predicting the natural frequencies and loss factors. The difference of frequency between the data in Kung and Singh and new plate finite element at each mode is less than 9%, while the difference of loss factor is less than 16%. The biggest differences of frequency and loss factor are 8.8% and 15.9%, respectively. The biggest difference appears at the mode (1, 1) with the lowest frequency and highest loss factor. Moreover, both the differences of frequency and loss factor decrease along with the increase of the mode. Thus, the proposed new finite element matches well with the published data, especially for high modes. As a result, the multilayer with the proposed new finite element can be used as the reference to regress the equivalent plate bending stiffness as described.

#### 3. Simplified FE Model for Multilayer Plates

This section presents the simplified FE model for multilayer plates. First, the equivalent bending stiffness is derived by using curve fitting method. Then, a simplified FE model is developed and simulated by applying the equivalent bending stiffness to a regular single-layer plate element.

##### 3.1. Identification of Equivalent Bending Stiffness

Since there is no closed-form solution for the multilayer plate structure, an example is given here based on the curve fitting of the response, instead of the transfer function, in order to derive the equivalent plate bending stiffness. At first, the transverse vibration response can be calculated analytically by using a modal superposition method [23]. Then, this response can be regressed according to the reference value calculated by the proposed new plate element in Section 2. For a clamped multilayer plate considered here, Table 4 gives the parameters. A unit amplitude harmonic force is applied on the center of the multilayer plate, as shown in Figure 3. Figure 4 shows the response at the center of the multilayer plate under different frequencies. This response is used as the reference to regress the response calculated by the analytical modal superposition method. In this work, an undamped multilayer plate is used in the numerical example of regression of the equivalent bending stiffness of the plate structure.

The approach for obtaining equivalent properties is described in the following. The analytically calculated response is regressed based on the reference value at the first two resonances. Subsequently, with equation (8), the equivalent bending stiffness can be calculated with corresponding stiffness matrix. In this example, the resonances are selected at the frequencies of around 75 Hz and 285 Hz. Then, a line/curve can be obtained by fitting the two obtained equivalent bending stiffness at the two resonant frequencies. Figure 5 demonstrates how a smooth curve fits to the two resonant frequencies in order to determine bending stiffness. It should be noted that this example just shows the basic idea of extracting equivalent properties. In reality, the curve fitting should be conducted for more resonant frequencies.

##### 3.2. Development and Simulation of Simplified FE Model

After the equivalent multilayer plate bending stiffness is derived, it is applied to the regular single-layer plate structure to set up the simplified FE model. For the single-layer plate structure, the same nodes are used to present the DOF as the multilayer plate structure, as shown in Figure 6. By combining all of the DOFs of each node, the DOF vector of the proposed new finite element for single-layer plate can be expressed in the following equations:

**(a)**

**(b)**

Based on the defined DOF vectors, it can be seen that each finite element for the single-layer plate has 25 DOFs. For the meshed multilayer plate structure, the overall DOF is

After defining the DOF vector of the proposed new finite element for the single-layer plate, the corresponding stiffness matrix and mass matrix can be set based on the boundary conditions and equivalent multilayer bending stiffness. With the given stiffness matrix and mass matrix, the response of the regular single-layer plate structure can be obtained according to the Lagrange formula presented in equation (4).

##### 3.3. Simulation of Simplified FE Model

Figure 7 shows the comparison between the reference and the response calculated with a simplified FE model. It can be seen that they match well in most frequency ranges except in the frequency range of antiresonance. The frequencies of resonances calculated by a simplified FE model are in good agreement with the reference. The difference of frequency at the antiresonance is less than 10%. Even though the frequency of antiresonance is a little different, the response values obtained by two methods are almost the same. The goal of this example is to show the possible solution of setting up the simplified FE model for multilayer plates. As a result, this approach can be used to effectively set up the simplified FE model of multilayer plates. Besides, the approach can be improved by curve fitting over more resonances or deriving approximate solutions of the plate transverse displacements in order to derive the transfer function. The latter can be better because the transfer function contains more structural characteristics than the simple response, so it will be considered as a future work.

#### 4. Conclusions

This paper proposes a new plate finite element as an extension of Zapfe’s beam element and a simplified FE model of multilayer plates. First, the new plate finite element is developed as a reference. Then, the curve fitting approach is applied to multilayer plates with the reference value to derive the equivalent bending stiffness. Finally, a simplified FE model of multilayer plates is proposed by applying the derived equivalent bending stiffness to a regular single-layer plate structure. The numerical example shows that this approach can be used to effectively set up the simplified FE model of multilayer plates.

In terms of applications, the proposed method can provide guidance in model development for the active vibration control of high-performance light weight smart structures, such as wind turbine, helicopter and aircraft structures, and so on. In addition, the approach can be further improved by curve fitting over more resonances or deriving approximate solutions in the future. Also, the plates with different boundary conditions and loading modes should be studied in future works. In addition, the simplified modeling of the multilayer plate with rotation will be further investigated.

#### Data Availability

The data used to support the findings of this study are included within the article.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

This work was supported by the Science and Technology Development Fund of Macau S.A.R. under grant no. 012/2015/A, the Research Grant of the University of Macau under grant nos. MYRG2016-00212-FST and MYRG2017-00135-FST, and the National Natural Science Foundation of China under grant no. 51705084. This work was also supported by the Natural Science Foundation of Guangdong Province of China (grant no. 2018A030313999) and Fundamental Research Funds for the Central Universities (grant no. 2018MS46).