Shock-Induced Vibration Suppression of Composite Truss Core Sandwich Plates via Distributed Nonlinear Energy Sinks

： In order to solve the problem of limited installing space and strict additional quality, the effects of distributed nonlinear energy sinks (NES) on a composite truss core sandwich plate are investigated in this paper. Choose five NESs here and inset them in the different places of the sandwich plate to suppress the vibration of the plate, which is excited by a half-wave shock. The coupled dynamic equations of the system are derived by the principle of conservation of energy. Then, numerical simulation are applied to discuss the vibration control performance of the five NESs with different parameters. The distribution of the five NESs are analyzed and the optimal position distributions are obtained. Based on the optimal location, the transient responses of the system are studied. Moreover, five NESs are compared with a single one in different dimensions. Finally, it is found that the selection of parameters have a great impact on the effectiveness of the five NESs. The new distribution way is introduced to improve the suppression effects of the five NESs in the sandwich plates.


Introduction
Sandwich structures are usually composed of two stiff face skins and thicker core material, such as Kagome, tetrahedral and pyramidal cores. They have a broad application prospect in the engineering field because of their excellent properties, such as high specific strength, high specific stiffness, energy absorption, thermal conductivity and so on [1][2][3][4].
In recent years, many scholars have done a lot on sandwich structures to grasp their characteristics more. Zhang et al. [5] investigated the global and chaotic dynamics of sandwich plates with truss core. Wang et al. [6] studied acoustic transmission of laminated composite sandwich structures with pyramidal truss cores.
Zangana et al. [7] analyzed the dynamic characteristics of composite corrugated core sandwich structures subjected to low-velocity shocks. Li et al. [8] theoretically and experimentally investigated the natural frequencies and the vibration modes of pyramidal truss core sandwich plates with local damage. Xiong et al. [9] analyzed the structural performance of composite sandwich panels under direct shear and three-point bending loads. Wang et al. [10] investigated the mechanical behaviors of composite sandwich plates with 2-D lattice truss cores by out-of-plane compression, shear and three-point bending tests. Huang et al. [11] analyzed the dynamic responses and failures of composite lattice core sandwich beams under impulsive loading. In addition, considering the special core layer of sandwich structures, some scholars investigated its internal space availability, such as, Yin et al. [12] present damping performance and energy absorption capacity of silicone rubber-filled sandwich structures. Zhang et al. [13] studied dynamic responses of pyramidal lattice core sandwich panels, improving the energy absorption and low velocity shock resistances by filling the polyurethane foams. Chen et al. [14] put forward the aerogel-filled sandwich panels to provide both mechanical supports and thermal insulation.
Meanwhile, the vibration control are another important research problem for the sandwich structures, specifically in condition of strong reliability and high vibration suppression requisition. Li et al. [15] analyzed the vibration suppression effects of the active control method on the vibration responses of lattice sandwich beams by the piezoelectric actuator/sensor pairs. Song et al. [16] analyzed the flutter suppression of the lattice sandwich beams by means of active vibration control. Chai et al. [17] investigated the nonlinear responses and vibration control of sandwich plates with different cores.
As one of the popular way to suppress the harmful vibration, nonlinear energy sink (NES) have the characteristics of small additional mass, wider vibration suppression frequency band, and targeted energy transfer [18][19][20][21][22]. Moslemi et al. [23] analyzed the effects of NES on dynamic responses for axially moving beams. Liu et al. [24] investigated the vibration suppression efficiency of NES with geometrically nonlinear damping. Taghipouret al. [25] researched the steady-state dynamic responses of a primary structure with cubic nonlinear stiffness connected with the NES under harmonic excitations. Fang et al. [26] analyzed the vibration suppression of bistable NES on transient responses of a Bernoulli-Euler beam and targeted energy transfer of the system. Zang et al. [27] discussed the influence of lever-type NES on dynamic responses of structures subjected to harmonic excitation. Chen et al. [28] obtained the better vibration reduction by comparing parallel NESs with a single one.
They also present the parallel NES could eliminate the high branch responses of the system due to nonlinear terms. Tian et al. [29] attached NES to a hypersonic 3-D wing to reduce aeroelastic responses of a wing. Li et al. [30] proposed a symmetric single-sided vibro-impact NES to suppress the vibration of cantilever beams. Zhang et al. [31] mitigated the vibration of composite laminated plates using an NES, which subjected to high speed wind loadings. Yao et al. [32] discussed the effects of a grounded NES on the lateral vibration of rotor systems. Zhang et al. [33,34] utilized a NES to suppress the shock-induced vibration of an axially moving beam.
From the above references, the reported researches are mainly on the dynamical characteristic for single NES. Only a few researches have paid attention on the distributed NES, which is a good way to expand the application ranges of NES absorbers. This paper present the vibration control performance of multiple NES absorbers on composite truss core sandwich plates with shock loading. Firstly, the dynamical equations of motion for a composite truss core sandwich plate are built with five inside distributed NESs. These NESs are embedded in the core of the sandwich plate, where one NES is in the center of the composite plate and the other four NESs are symmetrically placed on four sides of the plate. Then, the influence of position distribution of the NESs on the energy dissipation capacity are discussed.
Based on the criteria of the optimal energy dissipation position, it is analyzed with the transient responses for the feasibility of five NESs and the difference of vibration reductions on the first three modes of the plate. In addition, the vibration control performance of single NES and the distributed five NESs are compared to show the suppression advantages of the five NESs with the same structure. Finally, the effects of different parameters for the five NESs on energy dissipation are investigated to improve the performance of the NESs.

Equation of motion
Consider a composite truss core sandwich plate subjected to a shock load F as Fig. 1(a). The sandwich plate is composed of three layers, where the upper and lower face sheets are made of carbon fibers and the middle core is arranged by pyramidal truss core. The five NESs are named as N1, N2, N3, N4, and N5, respectively, which are embedded in five different truss core units. The pyramidal truss core unit with the NES is shown as Fig. 1(b). The mechanical model of the NES gives in Fig. 1(c). The Cartesian coordinate system is built on the central surface of the plate, and u , v and w represent the displacements of any point of the plate in the x, y and z directions, respectively. The shock load F acts on the position 1 2 ( , ) P P . Moreover, the symbolic representation of structural parameters are shown in Table 1.  Based on Allen' s theory [35], the following assumptions are given as (1) The thickness of the truss core sandwich plate remains constant during deformation; (2) Bending deformation only exists in the thin face sheets and shear one happens in the thick truss core of the plate; (3) The deflections of the whole plate are continuous.
Then, the equivalent density is expressed as The shear modulus of the truss core are written as The thin face sheets are made of carbon fiber composite, which are laid in five layers as 0/90/0/90/0. The stress-strain relations for the composite material can be expressed as follows where the equivalent modulus ij Q can be expressed as 4 Here, Qij is the elastic constant and  is the stacking angle of face sheet.
Based on the mechanical condition of the plate, first-order shear deformation theory [36] is applied here to express the displacement fields of the structure as  and y  are the rotations of the transverse normal about the y-and x-axes, respectively.
The relationship of strain and displacement can be expressed as Applying the Hamilton's principle, the motion equations of the sandwich plate coupled with five NESs can be derived as  (7) can be ignored for the small values.
In order to obtain the dimensionless dynamic equation, the following transformation are introduced, For convenience, the "-" on the dimensionless symbols are ignored.
Substituting equation (8) into equation (7), the dimensionless dynamic equations of the system can be written as: Based on the simply supported boundary conditions of the composite truss core sandwich plate, Since the harmful vibration of low-frequency and large amplitude are the main cause for structural damage, here, we focus on the response absorption of the first three natural frequencies for the plate.Using the Galerkin method, the dimensionless dynamic equations of the motion for the coupled system can be obtained as follows.
 and 3  are the first three natural frequencies of the sandwich plate.  The physical and geometrical parameters of the sandwich plate used in this paper are shown in Table 2. In addition, the thickness of each lamina is 0.12mm.
The shock load Γ is in the form of a half-sine pulse as

Determination of the optimal position of five NESs
The optimal position of vibration suppression for the five NESs are discussed firstly. Generally speaking, it is agreed that the vibration suppression is effective when the responses of the controlled system can be reduced to 5% of the initial amplitude.
We also use this criterion to test the vibration absorption capacity of the five NESs.
Let the parameters of the five NESs be equal, which include stiffness, damping and mass. Moreover, the specific parameters are set to 0. The energy dissipation ratio is introduced to analyze the overall efficiency of the five NESs with different distribution. The related equation is written as w q a ,b +w q a ,b +w q a ,b w d = w q p ,p +w q p ,p +w q p ,p d   Fig. 4(a) is much better than that in Fig. 4(b) and the optimal position can make the the five NESs work efficiently.

Research on transient responses based on the optimal position
Here, two different amplitudes of the load are considered, including f=5 and f=10.
The relative displacements between the NESs and the sandwich plate are shown in Figs. 6 and 7, respectively. With the increase of force, the duration of relative displacements between the sandwich plate and the NESs are longer. Due to different positions, the maximum relative displacement between each substructure and the sandwich plate is different, but they are in a reasonable range, i.e. / 2 / 2 ( 1,2,3, 4,5) (e) the plate and the N5 Fig. 7 The relative responses between the plate and the each substructure of five NESs when f=10:  (14) where T and Tn respectively represent the times for reaching to 5% of the initial amplitudes of the sandwich plate without and with the five NESs.     Table 3, which indicts that the vibration suppression of the five NESs gradually decreases on the first-order mode, while the effects on the second and third-order modes increases first and then decreases.

Comparison of single NES and five NESs
In this part, the performance of the five NESs in the optimal vibration suppression area is compared with that of a single NES. Firstly, the optimal suppression position for the single NES is obtained by successively locating it in the different unit of the sandwich plate, and the energy dissipation ratios of the single NES are obtained with different excitation amplitudes in Fig. 11.

The effect of five NESs parameters
The aim of this paper is to obtain the optimal vibration reduction of the five NESs with the minimum mass, so t  is chosen as 0.04 for the following parameter optimization based on the Fig 12. The amplitude of the shock force is set as 5 f= and the parameters of the five NESs keep the same as Fig. 3. Fig. 13 shows the effects of nonlinear stiffness of the NESs on the vibration absorption, which indicts that with the increase of nonlinear stiffness, the suppression of the five NESs increase firstly and decrease later. The effects of the damping of the five NESs on the vibration absorption are consistent with that of the stiffness, as shown in Fig. 14.
The discrepancy of the center one and the other four NESs are also considered. Let the mass ratio of N1 be 1  , the ratios of the other four NESs (Nn, n=2, 3

Conclusions
In this paper, the vibration absorption of the five NESs on the composite truss core sandwich plate are investigated with shock excitations. The nonlinear dynamical equations of motion for the coupled system are built by the first-order shear deformation theory and Galerkin method, and then the vibration control performance of the five NESs are analyzed by numerical methods.
Firstly, the results find that the position distribution of both the five NESs and the shock excitation have a great influence on the vibration suppression, but the excitation amplitudes have little effects on the optimal area for the five NESs. Then, based on the optimal energy absorption area, feasibility of the scheme and the action of five NESs on the first three modes of the sandwich plate are discussed. The performance of the single and the five NESs in the optimal position are compared and the five NESs present the much better suppression than the single one. It is found that for a larger range of excitation amplitudes, the best suppression quality of the five NESs can decrease 60% compared with the single NES. The smaller the excitation amplitude, the more obvious the quality advantage of the five NESs. Therefore, the reasonable distribution of multi NESs in multi-dimensional space have great values of small total mass and high efficiency in vibration control fields.
In addition, the influence of the parameters on the efficiency of the five NESs are also studied to further improve the vibration suppression. Finally, two different layout ways are selected to discuss the effects of the position N1 on the characters of the five NESs, which show that the varied position of N1 can improve the performance of the five NESs.