A Simplified Finite Element Approach for Modeling of Multilayer Plates

.e 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..e proposed new plate finite element consists of the shear and extension strains in all layers. .e multilayer structure with the proposed new plate finite element is regarded as a reference to calculate the reference value of the transverse response..e simplified FEmodel 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.


Introduction
In recent decades, the multilayer plates have been demonstrated to be promising engineering structures for automobiles and aerospace vehicles [1][2][3][4].A typical multilayer plate has three layers, including the constraint layer, damping layer, and base layer.e 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][6][7][8].erefore, 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][10][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]. is 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][14][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][17][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 simpli ed method should be developed.In [20], a simpli ed FE model for beam structures with three layers is presented by using single-layer equivalent FE method.e equivalent material properties are calculated and then a regular beam is constructed.Considering the simplicity and accuracy, this work also attempts to develop a simpli ed FE model for multilayer plates by curve tting the equivalent bending sti ness.
In this research, a new plate nite element for multilayer plate is developed, and a simpli ed FE model of multilayer plates is proposed based on the equivalent bending sti ness by curve tting the response values calculated from the multilayer plate structure with new plate nite element.e rest of this paper is organized as follows.Section 2 presents the proposed new plate nite element as an extension of Zapfe's beam element with validation.Subsequently, the simpli ed FE model for multilayer plates is described in Section 3. Finally, conclusions are given in Section 4.

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

Proposed New Plate Finite Element for Multilayer Plate.
e element formulated in the work of Zapfe and Lesieutre was used as the reference in the curve tting of a transfer function of the multilayer beam [19]. is element can be extended to constrained layer damped plates.e 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 ve additional midnodes in the element in order to avoid shear locking.
ese ve midnodes have only one transverse degree of freedom.
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 U j (j m, k, p, k, 1, 2, 3, 4, 5) are represented as ( By combining all of the DOFs of each node, the DOF vector of the proposed new nite element for the multilayer plate can be expressed in the following equations: • v q3 , v q4 , w q . (2) It can be found that each nite element for the multilayer plate has 41 DOFs.During the FE modeling, the multilayer is meshed into many elements with n x and n y nodes in the longitudinal directions.e overall DOF in the multilayer plate structure can be expressed as For each meshed element, a shape function matrix N(N ∈ R 3×12 ) for each layer can be de ned as where the shape function in matrix N can be written as n en, the corresponding sti ness matrix KK and mass matrix MM of each meshed element in each layer can be set as where ρ is the density of the plate.Since the multilayer plate is meshed in many plate nite elements, the sti ness matrix K and mass matrix M can be obtained by integrating the sti ness matrix KK and mass matrix MM of each plate nite element in the plate structure.By exciting a given force vector, the response can be obtained according to the Lagrange formula [21]: where F stands for the excited force vector; ω represents the inherent frequency of the multilayer plate structure; and X is the response matrix of the multilayer plate with meshed plate nite elements.e response matrix X can be obtained by combining the matrix U in each plate nite element.

Validation of Proposed New Plate Finite Element.
After the plate element is formulated, it should be validated to illustrate its e ectiveness.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 con guration of a plate to be validated which is simply supported along all four edges.e 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.
e di erence of frequency between the data in Kung and Singh and new plate nite element at each mode is less than 9%, while the di erence of loss factor is less than 16%.
e biggest di erences of frequency and loss factor are 8.8% and 15.9%, respectively.e biggest di erence appears at the mode (1, 1) with the lowest frequency and highest loss factor.Moreover, both the di erences of frequency and loss factor decrease along with the increase of the mode.us, the proposed new nite element matches well with the published data, especially for high modes.As a result, the multilayer with the proposed new nite element can be used as the reference to regress the equivalent plate bending sti ness as described.

Simplified FE Model for Multilayer Plates
is section presents the simpli ed FE model for multilayer plates.First, the equivalent bending sti ness is derived by using curve tting method.en, a simpli ed FE model is developed and simulated by applying the equivalent bending sti ness to a regular single-layer plate element.

Identi cation of Equivalent Bending Sti ness.
Since there is no closed-form solution for the multilayer plate structure, an example is given here based on the curve tting of the  Shock and Vibration 3 response, instead of the transfer function, in order to derive the equivalent plate bending sti ness.At rst, the transverse vibration response can be calculated analytically by using a modal superposition method [23].en, 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 di erent frequencies.is 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 sti ness of the plate structure.e approach for obtaining equivalent properties is described in the following.e analytically calculated response is regressed based on the reference value at the rst two resonances.Subsequently, with equation ( 8), the equivalent bending sti ness can be calculated with corresponding sti ness matrix.In this example, the resonances are selected at the frequencies of around 75 Hz and 285 Hz. en, a line/ curve can be obtained by tting the two obtained equivalent bending sti ness at the two resonant frequencies.Figure 5 demonstrates how a smooth curve ts to the two resonant frequencies in order to determine bending sti ness.It should be noted that this example just shows the basic idea of extracting equivalent properties.In reality, the curve tting should be conducted for more resonant frequencies.

Development and Simulation of Simpli ed FE Model.
After the equivalent multilayer plate bending sti ness is derived, it is applied to the regular single-layer plate structure to set up the simpli ed 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 nite element for single-layer plate can be expressed in the following equations: • w s2 , w s3 , w s4 , u ps1 , u ps2 , v ps1 , v ps2 , w ps , w s5 , u qs1 , u qs2 , • v qs1 , v qs2 , w qs . ( Based on the de ned DOF vectors, it can be seen that each nite element for the single-layer plate has 25 DOFs.For the meshed multilayer plate structure, the overall DOF is   ( After de ning the DOF vector of the proposed new nite element for the single-layer plate, the corresponding sti ness matrix and mass matrix can be set based on the boundary conditions and equivalent multilayer bending sti ness.With the given sti ness 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).

Simulation of Simpli ed FE Model.
Figure 7 shows the comparison between the reference and the response calculated with a simpli ed FE model.It can be seen that they match well in most frequency ranges except in the frequency range of antiresonance.e frequencies of resonances calculated by a simpli ed FE model are in good agreement with the reference.e di erence of frequency at the antiresonance is less than 10%.Even though the frequency of antiresonance is a little di erent, the response values obtained by two methods are almost the same.e goal of this example is to show the possible solution of setting up the simpli ed FE model for multilayer plates.As a result, this approach can be used to e ectively set up the simpli ed FE model of multilayer plates.Besides, the approach can be improved by curve tting over more resonances or deriving approximate solutions of the plate transverse displacements in order to derive the transfer function.e latter can be

Shock and Vibration
better because the transfer function contains more structural characteristics than the simple response, so it will be considered as a future work.

Conclusions
is 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.en, 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.
e 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.

Figure 2 :
Figure 2: Analysis of DOF for the proposed new plate nite element for multilayer plate.(a) Proposed new plate nite element for multilayer plate.(b) Analysis of DOF for each node for multilayer plate.

Figure 4 :
Figure 4: Response of the center of the multilayer plate calculated with the new plate element.

Figure 6 :Figure 5 :Figure 7 :
Figure 6: Analysis of DOF for the proposed new plate nite element for the single-layer plate.(a) Proposed new plate nite element for the single-layer plate.(b) Analysis of DOF for each node for the single-layer plate.
1 y/l y − xy/l x l y , n 2 xy/l x l y , n 3 1 − x/l x − y/l y + xy/l x l y , and n 4 x/l x − xy/l x l y .e variables x and y are the longitudinal coordinates; l x and l y are the length and width of each meshed element, respectively.By calculating the differences of matrices P 1 , P 2 , and P 3 , a new shape function B(B ∈ R 5×12 ) can be obtained:

Table 1 :
Plate parameters for validation.

Table 4 :
Plate parameters for curve tting.

Table 2 :
[22]arison of frequency between the data in Kung andSingh[22]and new plate nite element.

Table 3 :
[22]arison of loss factor between the data in Kung and Singh[22]and new plate nite element.