Vibration and Damping Analysis of a Multilayered Composite Plate with a Viscoelastic

Based on the theory of Donnell and Kirchhoff hypothesis and by using the complex constant model of viscoelastic materials, the vibration equations of five-layered constrained damping plate are established.The transfer matrix method (TMM) is improved and used to solve equations.The improved TMM ismore effective to solve complex structural vibration.The influence of layer numbers, thickness of each layer, and arrangement of materials on vibration behavior are discussed. It is proved that multilayered plates can more effectively reduce natural frequency and obtain higher structural loss factor. The loss factor increases with the number of whole layers. Symmetrical structure can obtain higher structural loss factor than one-direction structure. Uniform arrangement of viscoelastic materials and constrained materials can obtain higher structural loss factor than nonuniform arrangement. There is different optimum frequency with different material thickness, and the optimum frequency is not dependent from layer numbers.


Introduction
Viscoelastic material, with its excellent damping characteristics, can greatly dissipate vibration energy, and better damping effect can be obtained when a constrained layer is attached to them, forming a passive constrained layer damping (PCLD).The PCLD is widely used in aviation, aerospace, submarines, vehicles, and so forth, due to its simple structure, low cost, high reliability, and effectiveness in a wide temperature and frequency range.However, traditional PCLD is a three-layer constrained damping structure (i.e., host layer, viscoelastic material layer, and constrained layer), which can obviously lead to relatively large attached mass, and it is not suitable for places where light weight is required, such as in aerospace engine drum and blade.Thus optimized design on constrained damping is necessary to achieve optimal damping property with the least attached mass.
Constrained damping materials have been applied to the aerospace structures since 1960s.The pioneer work in this field could be traced back to Kerwin [1] and Ross et al. [2]; their works were extended by Mead and Markus [3], who presented the fourth-and sixth-order equation of motion governing the axial and bending vibrations of PCLD beams.Rikards et al. [4] studied the vibration and damping of laminated composite beams by using a simple Timoshenko beam finite element.Cupiał and Nizioł [5] discussed the threelayered composite plate with a viscoelastic core layer and two laminated face layers by the first-order shear deformation theory.Markuš [6] used the unconstrained passive damping treatments to damp the vibrations of thin cylindrical shell.Pan [7] studied the axisymmetrical vibration of a finite length cylindrical shell with a viscoelastic core.Alam and Asnani [8,9] considered the vibration and damping analysis of a general multilayered cylindrical shell having an arbitrary number of orthotropic material layers and viscoelastic layers.
With the development of material science and engineering, increasing thinner damping materials can be fabricated, and multilayer constrained damping structures began to be applied to engineering.Moreira et al. [10], Matter et al. [11], and Amichi and Atalla [12] studied finite element modeling and computational method of damping structures.Xie and Shepard Jr. [13] investigated the contribution of parameters to damping.Chazot et al. [14] and Abdoun et al. [15] examined the responses of multilayer constrained damping structures.Yang et al. [16] considered strain energy and damping characteristics of five-layer structure while Li et al. [17] conducted experimentations on its damping property.Researches of multilayer constrained damping structures mainly focused on finite element method and experimental study.The dynamic behavior and internal mechanism of multilayer PCLD treatments in vibration suppression of structures are not entirely clear due to the complexity in composite structure and difficulty in describing the damping characteristics in viscoelastic material accurately.
TMM is proposed in 1920s and widely used to solve dynamic problems in structural vibration with various boundary conditions.Li et al. [18], by means of TMM, carried out theoretical analysis for the natural vibration characteristics of three-layer constrained damping plate with various boundary conditions.Xiang et al. [19,20] extended TMM and make it convenient and effective to analyze the circular sandwich cylindrical shells partially or fully covered with PCLD treatments under almost all boundary conditions.Based on the basic assumptions of constrained damping structures, this paper developed the general equation of multilayer damping plate and improved TMM by introducing an association matrix to solve the equations.Also, the influence of the number of structure layers, thickness of materials, and various layouts on the vibration characteristics are discussed.

Vibration Equation of Multilayer Constrained Damping Plate
For a thin plate whose length is  and width is , when  layers of damping materials and constrained materials are alternately attached to its one side or both sides, it becomes 2 + 1 layers damping structure, as Figure 1 shows.
Here the thickness, elastic modulus, and Poisson's ratio of each layer are denoted as ℎ  ,   ,   ( = 1, 2, . . ., 2 + 1), respectively.For the damping layers, it is assumed that (1) only shear effect is considered; (2) there is no slippage between layers; (3) displacements between layers are continuous and the value of vertical deflection of each layer is the same.The deformation (2th layer) in  and  directions is shown in Figures 2 and 3, where   and V  denote each layer's displacements in  and  directions, and  is the displacement in  direction.Because each layer is too thick, in the equations we take the middle placements as the whole layer's, as shown in Figures 2 and 3 The strain column vector of the viscoelastic layer  2 = { 2  2 }  is expressed as where ℎ  2 = (ℎ 2−1 + ℎ 2+1 )/2 + ℎ 2 .The internal stress of the host layer, damping layer, and each constrained layer in the middle can be shown as in Figures 4-6, respectively.The internal stress of the outermost layer is the same as the host layer.The 1st layer is the host layer, and the th ( = 3, 5, . . ., 2+ 1) layer is the constrained layer.The internal stress can be expressed as ( The vibration equations of the constrained layer are as follows: where =1   , and  = ∑ 2+1 =1   .There are 2 + 3 equations in the equations above. Substituting (1) and ( 2) into (3), we have where there are 2 + 3 equations and 2 + 3 variables in (4).

Solutions of the Equations.
The TMM is widely used in the structural vibration for its good boundary conditions adaptability.The improved TMM is more convenient to obtain the first-order derivative of the state vector.Therefore it is adopted to solve the equations in this paper.
When the boundary condition of  direction is simply support condition at two ends, the solution of (4) can be written as where  is the vibration wave number in  direction,  * is the complex frequency,  * = 2(1 + ), and  is natural circular frequency of vibration. is the structural loss factor.  () and   () are the displacement functions of the th layer in  and  directions, and () is the displacement function in  direction.Substituting ( 5) into (4), we have Boundary conditions are as follows: In [19,20], state vectors are selected based on boundary conditions, which is convenient when the state vector has relatively less elements.However, when the state vector has more elements, the first-order differential of the state vector is hard to obtain.In this paper, the state vector and the boundary conditions are defined separately, and an association matrix is introduced to correlate them.
State vector is defined as Boundary condition vector is Then we have where  is the association matrix.
where  and  are 4 + 8-order square matrices and can be derived from ( 6) and ( 7) and detail format of  and  can be seen in Appendix.From ( 12) and ( 13), the transfer matrix  can be obtained:
The damping layer and the constrained layer are evenly divided into two layers, respectively, and they are attached to the host layer alternately in order to form the fivelayer constrained damping plate.The overall amount of constrained damping material attached to the five-layer plate is the same as that of three-layer plate which is the numerical example.The natural frequency and loss factor of both threelayer plate and five-layer plate are calculated, respectively.
The natural frequency and loss factor are calculated by both the proposed method and commercial finite element method (FEM) software ANSYS, and the results are shown in Table 1.
As can be seen from Table 1, (1) the results from the proposed method are very close to that of the ANSYS software, demonstrating the correctness of the proposed method; (2) compared to the three-layer plate, on condition that the overall thickness and attached mass are the same, the five-layer plate can achieve lower natural frequency and higher loss factor and hence better damping effect.

Influence of Material Parameters on the Vibration
In order to examine the damping performance of multilayer constrained plate, influence of various material layers and different kinds of layouts on the vibration characteristics is calculated, and vibration characteristics of multilayer constrained damping plate under various boundary conditions are also worked out, which accordingly demonstrated the universality of the results.

Vibration Characteristics of Constrained Damping Plate
with Various Layers.Dividing the damping material in the numerical example evenly into 2 layers, 3 layers, 4 layers, and 5 layers and attaching them to the host layer, then 5-layer, 7layer, 9-layer, and 11-layer constrained damping plates come into being.From Figures 7-12, we can see that (1) with the number of layers increasing, the natural frequency of the multilayer structure becomes slightly lower; (2) the loss factor increases with the number of layers going up, which is more obvious in the high-order vibrations.

Influence of Various Arrangement on the Vibration Characteristics of Constrained Damping Plate.
For 5-layer constrained damping plate, 5 layouts can be obtained depending on attaching constrained layers and damping layers onto one side (single side) or two sides (double sides) of the host layer and on whether the thicknesses of the 2 constrained layers and 2 damping layers are equal or not.They are uniform layout, single side nonuniform layouts (including the thicker outer layer layout and thicker inner layer layout), symmetrically uniform layout, and symmetrically nonuniform layout.The natural frequency and loss factor vary with different arrangement, as shown in Figures 13 and 14.

Influence of the Thickness of Damping Material and Con-
strained Material on the Damping Performance.Constrained damping design can enhance the vibration resistance of structure; meanwhile, it brings attached mass to the structures.Particularly, the increase of the constrained layer can greatly raise the weight of the structures, which is adverse to some light weight structures.Therefore, it is necessary to study the    (1) it is not that the thicker the materials, the larger the loss factor; there exists reasonable material thickness combination for different layers; (2) with reasonable thickness, higher loss factor can be obtained with smaller attached mass.

Conclusion
The vibration and damping characteristics of multilayered constrained damping plate are investigated by using improved TMM.The natural frequencies and loss factor of constrained damping plate with various layers are calculated.The influence of layer numbers, thickness of each layer, and arrangement of materials on vibration behavior are discussed.The following conclusions can be drawn from this paper: (1) Multilayer structure can achieve better damping effect.Although the more the layers are, the better the damping effect is, the loss factor increases only slightly when the number of layers attains a certain value.Thus we do not need to design too many layers.
(2) When choosing the thickness of damping material and constrained material, one should first determine the frequency range he is aiming at and obtain the optimal thickness by calculation with three-layer structure.
(3) When arranging the damping material and constrained material, the thickness of each viscoelastic layer should be equal and the thickness of each constrained layer should be equal too.
(4) The proposed method shows some advantages in terms of accuracy and efficiency over FEM and traditional numerical computation, especially in solving multispan and multilayer structure problems.

Appendix
Taking the 5-layer plate as an example,  = 2 (2 layers of viscoelastic materials),  and  are both 16-order matrices, in which

Figure 4 :
Figure 4: Internal stress of the host layer.

Figure 5 :Figure 6 :
Figure 5: Internal stress of the damping layer.

Figure 7 :
Figure 7: The natural frequency of constrained damping plate with various layers ( = 5).
wave number of x directions

Figure 8 :
Figure 8: The loss factor of constrained damping plate with various layers ( = 5).

Figure 9 :
Figure 9: The natural frequency of constrained damping plate with various layers ( = 10).
wave number of x directions

Figure 10 :
Figure 10: The loss factor of constrained damping plate with various layers ( = 10).

Figure 11 :
Figure 11: The natural frequency of constrained damping plate with various layers ( = 20).
wave number of x directions

Figure 12 :
Figure 12: The loss factor of constrained damping plate with various layers ( = 20).

Single side uniform 5
-layer plate Single side nonuniform 5-layer plate Single side nonuniform 5-layer plate Symmetrically uniform 5-layer plate Symmetrically nonuniform

Figure 13 :
Figure 13: The natural frequency of five-layered constrained damping plate with various arrangement.

Figure 14 :
Figure 14: The loss factor of five-layered constrained damping plate with various arrangement.

2Figure 18 :
Figure 18: Influence of thickness on the loss factor in 11-layer structure.

Table 1 :
Dynamical results of three-and five-layered constrained damping plate.
in this table stands for the vibration order in  direction.