Dispersion Analysis of Plane Wave Propagation in Lattice-Based Mechanical Metamaterial for Vibration Suppression

Abstract

Phononic crystals based on lattice structures provide important wave dispersion characteristics as band structures, showing excellent compatibility with additive manufacturing. Although the lattice structures have shown the potential for vibration suppression, a design guideline to control the frequency range of the bandgap has not been well established. This paper studies the dispersion characteristics of plane wave propagation in lattice-based mechanical metamaterials to realize effective vibration suppression for potential aerospace applications. Triangular and hexagonal periodic lattice structures are mainly studied in this paper. The influence of different geometric parameters on the bandgap characteristics is investigated. A finite element approach with Floquet–Bloch’s principles is implemented to effectively evaluate the dispersion characteristics of waves in lattice structures, which is validated numerically and experimentally with a 3D-printed lattice plate. Based on numerical studies with the developed analysis framework, the influences of the geometric parameters of lattice plate structures on dispersion characteristics can mainly be categorized into three patterns: change in specific branches related to in-plane or out-of-plane vibrations, upward/downward shift in frequency range, and drastic change in dispersion characteristics. The results obtained from the study provide insight into the design of band structures to realize vibration suppression at specific frequencies for engineering applications.

1. Introduction

Mechanical metamaterials are a class of artificial materials with the potential to facilitate structural research and development with unique properties, which may offer preferred structural characteristics by artificially designed microscopic internal mechanisms [1,2,3]. Although those artificial materials have been conceptually studied for decades [4,5,6], their complicated geometries hindered the engineering applications of those conceptual materials due to the difficulty in manufacturing. However, the recent developments in additive manufacturing (AM) technology provide the potential to effectively fabricate such structures at low cost. Mechanical metamaterials with unique properties, such as zero/negative Poisson’s ratio or ultralight weight [7,8], and capabilities of vibration suppression/isolation [9,10,11,12,13,14,15,16], are hoped to realize multifunctional structures, which may extend structural performance. Through design optimization, it is also possible to control the stiffness and mass characteristics of metamaterials in a wide range [17,18]. Finite element analysis (FEA) is commonly used to numerically evaluate the static and dynamic mechanical properties or bandgap characteristics of metamaterials for such design optimizations [19,20,21,22,23].

When the applications of mechanical metamaterials are considered, the concept of mechanical metamaterials is commonly designed or integrated in the form of periodic cellular structures. Phononic crystals are one of the classes of materials, which provide important wave dispersion characteristics due to the periodic Bragg scattering as band structures with bandgaps. Wave control can also be realized based on Helmholtz resonators [24]. Although the characteristics are often exploited to facilitate acoustic wave control [5], they can also be accommodated for vibration suppression [25]. Designing bandgap characteristics with high strength/stiffness of the structure is essential for industrial applications to ensure structural integrity while realizing vibration control capability with metamaterials. Among the concepts of metamaterials, the lattice structures generally offer better strength/stiffness in comparison to other types of metamaterials [26,27,28], while providing the capability of vibration suppression [29,30,31]. Periodic lattice structures also show excellent compatibility with AM. However, the properties of structures fabricated by the AM technique are strongly influenced by the printing process variables [32,33,34] although the AM technique enables to realize complicated geometries or multifunctionality based on lattice structures [35]. Consequently, the influences of the process variables on properties of AM-based structures have to be precisely taken into account for an accurate construction of lattice-based structures by the AM technique [21,36,37].

Periodic structures with unique properties have also shown potential for performance improvements and the realization of specific functionalities in aerospace applications. For example, design concepts with periodic structures in aerospace applications have been shown to realize morphing functionality [38,39,40]. The application of periodic structures, that realize a negative Poisson’s ratio, to morphing systems has shown the potential to achieve performance improvements [41]. With the capability of mechanical metamaterials for vibration suppression, it is also expected to enhance the dynamic stability of structures. The improvements in the stability of aerospace structures can enhance safety and extend flight envelope [42,43,44]. Therefore, various aerospace applications of lattice-based metamaterials have been studied in the literature [14]. For instance, 3D-printed Kagome lattices were integrated into a wing structure to realize a unique combination of high stiffness and competitive vibration properties [45]. It was shown that such a lattice structure could significantly reduce the vibration amplitude of the wing. In addition, two-phase composite metamaterials, consisting of a non-traditional feature design as a star-shaped fiber inserted inside a matrix, were integrated into a turbine blade [46]. It was reported that star-shaped composite material provided an improvement in the energy dissipation and tangent loss properties compared with a classical turbine structure. Similarly, lattice-based metamaterials with bandgap characteristics could further improve the dynamic stability of aerospace structures by realizing effective vibration suppression.

A challenge in the engineering application of lattice-based metamaterials for the enhancement of dynamic performance would be to establish an efficient design to realize vibration suppression in a specific frequency range within a limited design space. For this purpose, a deep understanding of the vibration characteristics of lattice-based metamaterials with various design parameters would be the key to achieving an effective integration of lattice structures into aerospace components. Although previous studies have demonstrated the potential and capabilities of such metamaterials [14], the correlation between the vibration characteristics of lattice structures and their design parameters has not been fully explored. Therefore, this paper studies the dispersion characteristics of plane wave propagation in lattice-based metamaterials with various design parameters to realize effective vibration suppression. Triangular and hexagonal periodic lattice structures with various geometric parameters are mainly studied in this paper. This study aims to investigate the influence of different geometric parameters on the bandgap characteristics, hoping to contribute to establishing a guideline to control the frequency range of the bandgap and realize effective designs of lattice structures in aerospace applications. In Section 2, a developed analysis framework for effective dispersion analysis of lattice structures is described. To effectively evaluate the dispersion characteristics of waves in lattice-based mechanical metamaterials, a finite element approach with Floquet–Bloch’s principles is implemented. In Section 3, the developed dispersion analysis framework is verified with numerical transient simulations and validated experimentally. In Section 4, dispersion characteristics of wave propagation in lattice plate structures with different geometric parameters are investigated to evaluate the influence of individual design variables on the dispersion characteristics. Conclusions and a design guideline of lattice plate structures for vibration suppression are discussed by considering the aforementioned influences of design variables in Section 5.

2. Theoretical Formulation

A theoretical formulation to evaluate wave propagations in lattice structures is presented in this section. A finite element approach based on Euler beam elements with Floquet–Bloch’s principles [47,48,49] is implemented for an effective evaluation of the dispersion characteristics of plane wave propagation in lattice structures. The wave dispersion properties of a representative unit cell of the periodically distributed lattice structure are calculated with the analysis framework.

2.1. Periodic Lattice and Floquet–Bloch’s Principles

A heterogeneous plate structure consisting of periodically distributed lattice unit cells was considered in this study. In the current approach, we focused on a representative unit cell of the periodic lattice structures to investigate plane wave propagation with Bloch’s theorem [47]. For effective evaluation, wave propagation is commonly evaluated along the edges of the irreducible region of the first Brillouin zone based on a reciprocal lattice in the wave vector space (k-space) [47,48]. Plane wave propagation in structures was numerically simulated with the finite element method [50] by using Euler beam elements. In this study, we focused on triangular and hexagonal lattice structures. Although the following formulation is mainly described based on a triangular lattice, a similar formulation can be performed for a hexagonal lattice.

Figure 1 describes the lattices in real space and k-space. An entire direct lattice can be expressed as a tessellation of a representative unit cell along a set of basis vectors e_r = (e_r,1, e_r,2)^T. The reference position of the lattice points, that correspond to nodes of the unit cell, is expressed as x_ref = (x_ref, y_ref)^T. An arbitrary position x = (x, y)^T of a lattice point is then given with a combination of the integer pair (n₁, n₂) by

$$x=x_{ref}+n_1{e}_{r,1}+n_2{e}_{r,2}$$

(1)

The displacement of a lattice point in a reference unit cell in the case of a plane wave can be expressed as [48]

$$u\left(x_{ref}\right)=A{e}^{i\left(\omega t-k\cdot x_{ref}\right)}$$

(2)

where A, ω, and k are the amplitude, frequency, and wave vector of the plane wave. Based on Bloch’s theorem, the displacement at a position x can be expressed based on the displacement at the reference point x_ref as

$$u\left(x\right)=u\left(x_{ref}\right)e^{ik\cdot \left(x-x_{ref}\right)}=u\left(x_{ref}\right)e^{i\left(k_1{n}_1-k_2{n}_2\right)}$$

(3)

where k₁ and k₂ are the components of the wave vector k along the e_r,1 and e_r,2 directions:

$${\left[\begin{array}{c}{k}_1\\ k_2\end{array}\right]}^T=k^T\cdot \left[\begin{array}{cc}{e}_{r,1}& e_{r,2}\end{array}\right]$$

(4)

A set of basis vectors e_w = (e_w,1, e_w,2)^T of the reciprocal lattice in k-space is given by

$$\begin{array}{l}{e}_{w,1}=2\pi {\displaystyle \frac{e_{r,2}\times e_{r,3}}{e_{r,1}\cdot e_{r,2}\times e_{r,3}}}\\ e_{w,2}=2\pi {\displaystyle \frac{e_{r,3}\times e_{r,1}}{e_{r,1}\cdot e_{r,2}\times e_{r,3}}}\end{array}$$

(5)

where e_r,3 is a vector parallel to the set of basis vectors e_r = (e_r,1, e_r,2)^T. The basis vectors of the direct and reciprocal lattices have a relationship as

$$e_{w,i}\cdot e_{r,j}=2\pi {\delta }_{ij}$$

(6)

where δ_ij is the Kronecker delta function. The subscript of i and j ranges from 1 to 2.

Wave propagation of the periodic lattice structure is commonly calculated by confining the wave vectors in the Brillouin zone by taking advantage of a periodicity in the reciprocal lattice [48,50,51,52]. The analysis with the first Brillouin zone (i.e., a route OABO) is sufficient in the case of the equilateral triangle. However, the Brillouin zone (a blue region in Figure 1) with a route OABCO is used in this study to allow an analysis for a more general geometry. Dispersion characteristics in any direction can be evaluated by varying the wave vector k along the edge of the Brillouin zone (red lines in Figure 1).

2.2. Plane Wave Propagation

Plane wave propagation in periodic lattice plates can be effectively investigated by focusing on a representative unit cell. The finite element method based on the Euler beam elements was used to numerically evaluate the wave propagation in the representative unit cell. The following formulation is implemented in MATLAB R2020b.

The equations of motion of the representative unit cell for a lattice structure can be expressed as

$$M_{unit}{U}_{unit}^{″}+K_{unit}{U}_{unit}=F_{unit}$$

(7)

where M_unit and K_unit are the global mass and stiffness matrices of the representative unit cell, respectively, while U_unit and F_unit are the nodal displacement and force vectors, respectively. The global mass and stiffness matrices are assemblages of elemental mass and stiffness matrices for the Euler beam elements. The size of the elemental matrices becomes 6 × 6 for the analysis of the in-plane vibration and 12 × 12 for the analysis of the in-plane and out-of-plane vibrations. The propagation of planar harmonic waves at an angular frequency ω within the entire lattice structure is evaluated with Bloch’s theorem by following the approach in Ref. [48] as

$$\left[K_{unit}-{\omega }^2{M}_{unit}\right]u_{unit}=f_{unit}$$

(8)

The following relations with respect to the displacements and forces are given by Bloch’s theorem as

$$\begin{array}{l}{u}_{unit,rt}=e^{i{k}_1}{u}_{unit,lb}\\ u_{unit,lt}=e^{i{k}_2}{u}_{unit,rb}\\ u_{unit,r}=e^{i\left(k_1+k_2\right)}{u}_{unit,l}\\ f_{unit,rt}=e^{i{k}_1}{f}_{unit,lb}\\ f_{unit,lt}=e^{i{k}_2}{f}_{unit,rb}\\ f_{unit,r}=e^{i\left(k_1+k_2\right)}{f}_{unit,l}\end{array}$$

(9)

where the subscripts l, r, t, and b indicate the displacements and forces corresponding to the left, right, top, and bottom nodes from the center of a unit cell (see Figure 2). The subscript i is used for the displacements and forces on the internal nodes of a unit cell, as shown in Figure 2.

With the relations in Equation (9), the transformation into the Bloch-reduced coordinates becomes

$$T_{unit}=\left[\begin{array}{cccc}I& 0& 0& 0\\ e^{i{k}_1}I& 0& 0& 0\\ 0& I& 0& 0\\ 0& e^{i{k}_2}I& 0& 0\\ 0& 0& I& 0\\ 0& 0& e^{i\left(k_1+k_2\right)}I& 0\\ 0& 0& 0& I\end{array}\right]$$

(10)

The components of the governing equations in Equation (7) in the Bloch-reduced coordinates can be obtained with the transformation matrix as

$$\begin{array}{l}{u}_{unit}=T_{unit}{\tilde{u}}_{unit}\\ {\tilde{M}}_{unit}=T_{unit}^H{M}_{unit}{T}_{unit}\\ {\tilde{K}}_{unit}=T_{unit}^H{K}_{unit}{T}_{unit}\\ {\tilde{f}}_{unit}=T^H{f}_{unit}\\ {\tilde{u}}_{unit}=\left[\begin{array}{c}{u}_{unit,lb}\\ u_{unit,rb}\\ u_{unit,l}\\ u_{unit,i}\end{array}\right]\end{array}$$

(11)

where ũ_unit and T^H are the reduced displacements based on Bloch’s theorem and the conjugate transpose of the transformation matrix T. For a free plane wave, the forces f become zero. Therefore, by substituting Equation (11), Equation (8) is reduced to the eigenvalue problem given by

$$\left[{\tilde{K}}_{unit}-{\omega }^2{\tilde{M}}_{unit}\right]{\tilde{u}}_{unit}=0$$

(12)

For a plane wave without any attenuation in the plane of propagation, multiple dispersion surfaces can be obtained by solving Equation (12). A gap, a so-called bandgap, along the ω axis is obtained if the neighboring two surfaces do not intersect each other. This gap indicates that no wave motion occurs in the frequency range, which provides performance of vibration suppression [53].

3. Verification and Validation

Dispersion characteristics of a periodic lattice plate were investigated based on the present dispersion analysis. Transient responses to in-plane and out-of-plane vibrations were also evaluated to verify the formulation and implementation of the present approach of the dispersion analysis. The presented formulation was then validated by comparing solutions of the dispersion analysis with experimental results.

3.1. Dispersion Analysis of Lattice Structures

A dispersion analysis was performed to verify and validate the implementation of the dispersion analysis for lattice structures. For this purpose, a heterogeneous plate with periodic triangular lattices was considered. Figure 3 shows a finite element model of the unit cell used in the following simulation. The cross-section of the beams consisting of the triangular lattices is rectangular. The geometrical parameters of the triangular lattices in the unit cell are given in

Table 1. Geometry of triangular lattices.

Property	Value
Thickness, ttri, mm	3.0
Width, wtri, mm	1.0
Length, Ltri, mm	20
Angle, θtri, deg	60

. The length L_tri of each diagonal beam is 20 mm. The width and thickness of the lattice beams are 1.0 mm and 3.0 mm. The angles between the neighboring lattice beams are 60°. The material properties used for the simulation are given in

Table 2. Material properties.

Property	Value
Young’s modulus E, GPa	2.944
Poisson’s ratio ν	0.2984
Density ρ, kg/m3	1166.0

. Young’s modulus, Poisson’s ratio, and density were set to be 2.944 GPa, 0.2984, and 1166.0 kg/m³, respectively. The finite element model was constructed with 30 Euler beam elements based on a convergence study.

Figure 4 shows dispersion curves for the triangular lattices obtained by the dispersion analyses for the in-plane and out-of-plane vibrations. The blue lines indicate the branches corresponding to the in-plane vibrations, while the red lines correspond to the out-of-plane vibrations. The shaded regions indicate the bandgaps, in which no branches exist for each vibration. The bandgap for the in-plane vibration was expected to be in the range of 4.084–5.032 kHz between the third and fourth branches. For the out-of-plane vibration, the frequency range of the bandgap was from 4.109 to 5.671 kHz.

3.2. Verification of Dispersion Analysis with Transient Simulations

The present approach for the dispersion analysis was verified by comparing the solutions of the dispersion analysis with the ones of transient structural simulations. The transient vibration responses of the triangular lattices were evaluated by MSC.Marc 2017.1.0 [54]. A lattice plate with the same unit cell used in the dispersion analysis was considered. Figure 5 shows a lattice plate. To ensure a sufficient number of periodicities, triangular lattices were distributed with 12 cells in the x_val direction and 14 cells in the y_val direction. The same material properties in Table 2 were used. The plate was modeled with two-node beam elements (element type 98). Each edge of the triangles was discretized into 10 elements.

The wave propagation of the in-plane vibration in the y_val direction was first investigated. As shown in Figure 5, a sine wave was excited at input nodes along the top edge. The in-plane sinusoidal wave Y in the y_val direction was applied as

$$Y=A\sin \left(2\pi ft\right)$$

(13)

where A, f, and t are the amplitude, frequency, and time, respectively. The generalized-α method was used for the transient simulations. The time step was divided into 40 segments for one vibration period. The amplitude A was 0.10 mm. Transient responses of the lattice plate with different frequencies ranging from 1 to 9 kHz were evaluated with an increment of 0.1 kHz. Each simulation was performed until 100 cycles of vibration were input.

Figure 6 shows in-plane deformations with the vibration inputs with frequencies of 3.0, 4.5, and 6.0 kHz at t = (100 − 1/4)T, which was a sufficient time for the waves to propagate. The counter indicates displacements in the y_val direction. Note that the frequency of 4.5 kHz is within the bandgap of the triangular lattices for in-plane vibrations. In-plane wave propagations were observed with the vibration inputs at 3.0 and 6.0 kHz. On the other hand, the vibration of the plate model was suppressed at 4.5 kHz due to the existence of the bandgap. Accelerations in the y_val direction at the input node and the output node (a node on the first cell from the bottom edge, as shown in Figure 5) up to 0.01 s are shown in Figure 7. While peaks of acceleration magnitudes on the output node were larger than those of inputs at 3.0 and 6.0 kHz, a decrease in amplitude was observed at 4.5 kHz.

The wave propagation transmittance was also evaluated to investigate the effectiveness of the bandgap for vibration suppressions. Transmittance Tr is defined as

$$Tr=20{\log }_{10}{\displaystyle \frac{a_{out}}{a_{in}}}$$

(14)

where a_in and a_out are average amplitudes of the accelerations at input and output nodes. Figure 8 shows the transmittance with the inputs of different frequencies. According to the solution of the dispersion analysis, the bandgap of the triangular lattices existed in the range of 4.084–5.032 kHz (the gray region in Figure 8) for in-plane vibrations. The low transmittance around 4–5 kHz in the transient solutions agreed with the bandgap frequency range obtained by the dispersion analysis. There were other drops in the transmittance outside of the predicted bandgap (e.g., around 2.7, 6.7, and 7.7 kHz); however, this was due to the location of the output measurement. The output measurement node was closer to the nodes in the stationary wave at those frequencies. This will be further discussed in the following section.

Transient analysis with out-of-plane vibrations was also performed for verification purposes. Out-of-plane sinusoidal displacements were applied at the input nodes. The amplitude, time step, and simulation time were the same as the case of the in-plane vibrations. Note that out-of-plane vibration waves also propagated in the y_val direction. Snapshots of the out-of-plane deformations by the vibration inputs with frequencies of 3.0, 4.5, and 7.0 kHz at t = (100 − 1/4)T are shown in Figure 9. The counter indicates displacements in the z_val direction. Out-of-plane wave propagations were observed with the vibration inputs at 3.0 and 7.0 kHz. On the other hand, the vibration of the plate model was attenuated rapidly at 4.5 kHz after two or three unit cells from the input nodes. Accelerations in the z_val direction at the input and output nodes up to 0.006 s are shown in Figure 10. The amplitude of the out-of-plane vibration at 4.5 kHz was significantly reduced on the output node. On the other hand, the amplitudes of vibration at 3.0 and 7.0 kHz were comparable to those of the input vibrations.

Similarly to the in-plane cases, the transmittance of out-of-plane vibrations was evaluated with Equation (14). Here, a_in and a_out were average amplitudes of the accelerations in the z_val direction. The transmittance of each frequency is shown in Figure 11. The gray region indicates the bandgap, which was predicted by the dispersion analysis, existing from 4.109 to 5.671 kHz. The transmittance within the bandgap was drastically decreased in the transient solutions. The fluctuations of the transmittance at other frequencies were also related to the positions of steady-state wave nodes. Therefore, the solutions of the transient and dispersion analyses agreed well.

3.3. Experimental Validation

A vibration experiment was performed to validate the dispersion analysis with the present approach and demonstrate the feasibility of lattice plates for vibration suppression. For the vibration tests, the lattice plate model in the previous section was fabricated by using a 3D printer (Raise 3D Pro3, CA, USA) with PLA filaments. The plate model was layered in the z_val direction. The layer height was set to 0.20 mm. The model was built with an infill sequence of 0, 60, and 120°. Nozzle and heated bed temperatures were 205 °C and 55 °C, respectively. Material properties of the plate model were the values in Table 2, which were obtained by a series of tensile tests in accordance with Japan Industrial Standards K7161 [55] with specimens fabricated with the printing process parameters. A base fixture of the lattice plate was integrated at the edge of the input side of the fabricated model, as shown in Figure 12. Figure 13 describes the experimental setup for in-plane vibrations. In-plane sinusoidal waves were excited by Mini SmartShaker K2007E01 (The Modal Shop, Inc., OH, USA) to the input edge with a mount, as shown in Figure 13. In-plane accelerations were obtained with two uniaxial accelerometers 2250A-10 (Endevco Inc., NC, USA) on the input and output nodes (the same locations in Figure 5). The residual noise of the accelerometers was 0.002 g rms. The acceleration data were transferred to a Fast Fourier Transform (FFT) analyzer CF-9400 (ONO SOKKI Co., Ltd., Kanagawa, Japan) through a signal conditioner Model 133 (Endevco Inc., NC, USA). The frequency range of 1.0–9.0 kHz with a 0.1 kHz increment was evaluated with the frequency resolution Δf = 50 Hz and sampling frequency f_s = 25.6 kHz.

Figure 14 shows the transmittance over the frequency range obtained from the in-plane vibration experiment. The gray region indicates the bandgap obtained by the dispersion analysis. The reduction of the transmittance around 4.5 kHz, which was within the bandgap frequency, could be observed although there was a shift in the low transmittance range toward the higher frequency. The transmittance values at other frequencies such as 4.0 and 6.0 kHz were larger than zero, which meant that the input vibrations were not attenuated at the output measurement point. The discrepancy between the low transmittance range and the predicted bandgap was due to the imperfection in the 3D-printed model, which will be further discussed in the following section. As observed in the transient simulations, other drops in the transmittance were observed around 2.7, 6.7, and 7.7 kHz, which were related to the positions of steady-state wave nodes. Figure 15 shows the vibration spectra of input and output measurements. By comparing Figure 15a,b, the peaks of the spectra in the input and output data were observed at 2.7 kHz outside of the bandgap, indicating that the cause of the decline in transmittance around 2.7 kHz was not due to the bandgap but the node of steady-state vibration (i.e., the input and output nodes vibrated at the same phase). The same responses could also be observed in the spectra in the input and output measurements at 6.7 and 7.7 kHz. On the contrary, the vibration spectrum of the output measurement disappeared around the bandgap frequency (i.e., 5.0 kHz), as shown in Figure 15d. In general, all vibration modes would be suppressed only if an input frequency was within the bandgap; otherwise, a certain vibration mode would exist in a band structure. Therefore, it was concluded that the experimental results agreed with the solutions of the dispersion analysis.

Out-of-plane vibration tests with the same lattice plate were also performed for validation purposes. The fixing direction of the mount and the installation of acceleration measurements were modified to excite and measure out-of-plane vibrations, as shown in Figure 16. The direction of the input vibration was in the z_val direction. The input and output accelerations were measured on the mount and near the bottom tip of the plate model, respectively.

The transmittance of the out-of-plane vibration at different frequencies is shown in Figure 17. The gray region indicates the out-of-plane bandgap of 4.109–5.671 kHz, which was predicted by the dispersion analysis. The experimental result indicated a lower transmittance within the predicted bandgap frequency range, while the transmittance outside of the bandgap showed higher values. The minimum value of transmittance was observed in the experiment at 5.1 kHz, which was close to the center of the bandgap. The other low transmittance values were observed around 7.0 and 8.6 kHz. These drops outside of the bandgap were also caused by the influence of vibration nodes, similar to the in-plane vibration tests. As shown in Figure 18, the low transmittance outside of the bandgap (e.g., 7.0 kHz) was obtained because the input and output nodes vibrated in the same phase. The peak waveform of the output spectrum disappeared only within the bandgap, as seen in Figure 18b.

3.4. Comparison of Transient Simulations and Experimental Results

To further investigate the discrepancy between the experimental result and the solution of dispersion analysis for the in-plane vibration, the transmittances of the transient solution and experimental result were compared, as shown in Figure 19. Although slight differences in the frequency range of low transmittance existed, the trends were qualitatively consistent. One reason for the difference in the frequency ranges of low transmittances was the differences in the geometry and structural characteristics of the 3D-printed plate model, as mentioned in the preceding section. It is known that the geometrical precision and material properties of a 3D-printed structure depend on the size and geometry of the model as well as the printing process parameters [56]. The width of the fabricated plate model was an average of 1.091 mm although the width of triangular lattices in the simulation model was 1.0 mm. Moreover, Young’s modulus of the fabricated plate model could have differed from the one of the simulation model due to the print path. For example, the fabricated plate model consisted of shell layers (i.e., contours along the outline surfaces) and infill regions. Although the fabricated model was built with the infill sequence of 0, 60, and 120°, the shell layers were always parallel to the sides of the triangle shape (see Figure 20). The percentage of shell layers for each lattice beam, compared to the infill region, was high for the lattice plate with the thin width of the triangular lattices in the current plate model. This configuration provided a higher tensile stiffness of triangular lattices than the stiffness obtained from the aforementioned tensile tests with standard tensile specimens. The higher in-plane stiffness caused the shift in the transmittance curve to the higher frequencies. Figure 21 compares the transmittances obtained from the transient analysis and the experiment for the out-of-plane vibration. For the out-of-plane vibration case, the frequency ranges of low transmittance obtained from the numerical solution and the experimental result agreed well. A small difference in the thickness of the fabricated and simulated models contributed to the agreement. The error in the average thickness of the fabricated lattice beams to the simulation model was 0.024 mm. In addition, the influence of the print path would have a higher impact on the material properties in the printing plane (i.e., in-plane properties for the plate). Consequently, the numerical and experimental results for the out-of-plane vibration agreed well.

Therefore, it was confirmed that the present method for dispersion analysis with the beam elements was properly implemented to obtain sufficiently accurate solutions for the following numerical studies to investigate the influences of geometric parameters for lattice structures on bandgap characteristics. However, further investigations, especially for the in-plane vibration, will be performed in our future works to obtain better agreements between the numerical analysis and experimental results based on fabricated lattice structures by AM. Also, further experimental studies with fabricated plates with other lattice geometries, such as hexagonal lattices, would be performed as a further validation of the present approach in future works.

In addition, the present approach could enable extremely efficient analysis of plane wave dispersion properties of lattice structures. For example, the simulations by the present approach were a hundred times faster than the trial-calculated simulations with the shell elements by COMSOL Multiphysics version 6.1 [57], which is a commercial software offering the capability of dispersion analysis.

4. Numerical Studies

The influences of geometric parameters on the generation of bandgaps with lattice structures were explored by performing wave dispersion analysis with the present implementation. In this study, unit cells with triangular lattices and hexagonal lattices with different geometric parameters were investigated.

4.1. Influences of Geometric Parameters on Vibration Suppression with Lattice Structures

Figure 22 describes the unit cells of triangular and hexagonal lattices with rectangular beams and the corresponding geometric parameters for the study. The basic Brillouin zones of each unit cell were defined, as shown in Figure 22. For the unit cell of triangular lattices, the beam length L was defined as one of the diagonal lines in the Brillouin zone. Another parameter was the angle θ between the neighboring diagonal lines. Note that the short side of the diagonal line could be obtained based on the definition of the Brillouin zone (i.e., the periodic condition). Similarly, the three geometric parameters of the hexagonal lattice were defined, as shown in Figure 22b. The beam length was defined differently. An additional height parameter H was defined for hexagonal lattices to specify the geometry of the unit cell.

Table 3. Baseline of the geometric parameters.

Property	Triangular Lattice	Hexagonal Lattice
Beam thickness t, mm	3.0	2.0
Beam width w, mm	1.0	1.2
Beam length L, mm	20	20
Beam height H, mm	-	15
Beam angle θ, deg	60	30

lists the baseline values for the geometric parameters for the following parametric study. Although the thickness and width of the lattice beams could also be varied, those parameters were fixed to the baseline values in this section. The variable parameters were altered individually within the ranges given in

Table 4. Variations in the geometric parameters.

Property	Triangular Lattice	Hexagonal Lattice
Beam length L, mm	1.0–59 (1.0)	1.0–59 (1.0)
Beam height H, mm	-	0.75–44.25 (0.75)
Beam angle θ, deg	3.0–177 (3.0)	−21–88.5 (1.5)

by fixing the other parameters to the baseline values to evaluate individual influences. The parentheses indicate the increments in sweep simulations for the parametric study. The material properties used for the study are given in Table 2.

A series of parametric sweep simulations for the dispersion analysis were then performed. The geometry of the Brillouin zones was categorized into two cases based on the angle θ_w between the basis vectors, e_w,1 and e_w,2, to achieve parametric sweep simulations for the study, as shown in Figure 23. In the parametric sweep simulations, the angle θ_w was first calculated when the variable parameters were changed. Periodic boundary conditions were then applied based on the wave vector k, which were along the edges of the corresponding Brillouin zone, OABCO, in Figure 23.

Figure 24 shows the dispersion curves of triangular and hexagonal lattices with the baseline parameters. The unit cell of triangular lattices with the parameters exhibited a bandgap in the range of 4.07–5.03 kHz. The bandgap was a complete gap through the whole wave vectors, which would provide suppressions for omnidirectional vibration. On the other hand, a partial gap between the OA direction within the frequency of 4.83–5.90 kHz was observed for the unit cell of hexagonal lattices. Therefore, the unit cell of hexagonal lattices with the parameters could provide a capability of vibration suppression in the y_unit direction since the OA direction in the k-space corresponds to the y_unit direction in real space. The unit cell of triangular lattices with the baseline parameters could provide robust vibration suppression in any direction, while the unit cell of hexagonal lattices with the baseline parameters having the partial bandgap could offer vibration suppression in a specific direction. It is important to note that such a characteristic could be exploited to control wave propagation in a specific direction or used as a waveguide [58].

The changes in the frequency range of bandgap with different variable parameters for the unit cell of triangular lattices are given in Figure 25. A large decrease in the bandgap frequency could be observed as the beam length L increased. Only the scale of the unit cell changed with the change in the beam length. Therefore, the sparse lattice structure with the larger beam length resulted in the reduction in the effective stiffness, which led to the downward shift in the bandgap frequency. On the other hand, the angle θ changed the geometry of the unit cell. Consequently, the eigenvalues (i.e., dispersion curves) corresponding to each wave vector changed drastically. As a result, the bandgap, which was obtained with the baseline parameters, disappeared with a drastic increase/decrease in the angle parameter.

The changes in the bandgap frequency with different variable parameters for the hexagonal lattices are shown in Figure 26. Herein, only the partial bandgaps in the y_unit direction, as shown in Figure 24b, were evaluated. According to Figure 26a, a decrease in the bandgap frequency could again be observed as the beam length L increased. However, unlike the triangular lattice, the geometry of the hexagonal lattice was not simply scaled with the change in the beam length. Therefore, the generation and vanishment of the partial bandgaps occurred. The change in angle θ gave a more complicated influence on the bandgap characteristics. When the angle was close to zero or negative, as shown in Figure 27, unique properties such as negative Poisson’s ratio were exhibited, which resulted in being reluctant to propagate waves in the y_unit direction. Consequently, the large areas of bandgap were obtained in the lower angles. The beam height H exhibited an influence on the bandgap frequency similar to the beam length. As the height increased, the bandgap frequency decreased slightly. However, the influence of the height variation was smaller than the one of the length change as the beam length had a larger impact on the scaling of the unit cell. The ranges of the bandgaps, as shown in Figure 26, were wider than the ones of the triangular lattices since the partial bandgap was evaluated in the case of the hexagonal lattices. Therefore, if it is sufficient to suppress vibrations in a specific direction, a lattice structure could be effectively designed with a partial bandgap corresponding to the direction with wider effective ranges for vibration suppression.

4.2. Design Approach for a Bandgap in a Specific Frequency Range

Based on the results of the parametric study in the preceding section, a design approach to obtain a bandgap in a specific frequency range could be considered. For this purpose, the influences of the individual geometric parameters on the dispersion curves were further investigated.

The influences of the cross-sectional geometry (i.e., beam width w and thickness t) were first evaluated. Figure 28 shows the dispersion curves with three different cross-sectional parameters: baseline, t = 3.6 mm, and w = 1.2 mm. The other geometric parameters were the same as the baseline values. The branches that correspond to the in-plane and out-of-plane vibrations are denoted with the blue and red lines, respectively. When the thickness of the beams with a rectangular cross-section was increased, the frequency-wise increments in the branches that corresponded to the out-of-plane vibrations were observed. In this case, the branches related to the in-plane vibrations were preserved. This effect occurred because the components of stiffness and mass matrices that corresponded to the in-plane and out-of-plane vibrations could be separated. For example, the in-plane stiffness and mass characteristics of the lattices were linearly related to the beam thickness, while the out-of-plane stiffness characteristics changed with a factor of cubic with respect to the beam thickness. Therefore, only the eigenvalues related to the out-of-plane vibrations changed. Similarly, when the beam width was increased, only the branches that corresponded to the in-plane vibrations were altered without changing the branches related to the out-of-plane vibrations. Based on the observations, it was found that the cross-sectional parameters could control individual branches that corresponded to the in-plane or out-of-plane vibrations. This is very useful when one wants to tune the frequency range of an obtained bandgap. Note that the current study has limitations due to the use of the Euler beam elements. Therefore, nonlinearities such as geometric stiffness due to large deformation cannot be considered.

Next, the influences of the beam length on the triangular lattices were further evaluated. Figure 29 shows the dispersion curves of triangular lattices with the baseline length and L = 22 mm. The corresponding geometries of the lattice structures with the triangular lattices are also described in Figure 29. According to Figure 29c,d, it could be confirmed that the lattice structure was scaled up with the increment in the beam length. The scaling of the unit cell led to a downward shift in the branches of dispersion curves. This effect would help to tune an arbitrary bandgap in a specific frequency range. The influences of the angle θ on the triangular lattice were also evaluated. Figure 30 shows the dispersion curves of triangular lattices with the baseline angle and θ = 75°. The corresponding geometries of the lattice structures are also given in Figure 30. It could be seen that the lattice structure was tilted with the increment of the angle from the baseline, which gave an equilateral triangle. The difference in the geometry of the unit cell caused a drastic change in the dispersion curves. This also led to a large difference in the bandgap width. Therefore, this parameter has a huge impact on the generation/vanishment of bandgaps.

The influences of the beam length on the hexagonal lattice were then evaluated. Figure 31 shows the dispersion curves of hexagonal lattices with the baseline length and L = 22 mm. The corresponding geometries of the lattice structures with the hexagonal lattices are also given in Figure 31. The lattice structure with the hexagonal lattices did not simply scale up with the increment in the beam length but varied the geometry of the unit cells. In the current case, the hexagonal shape was elongated in the x_unit direction as the height H is fixed. The shape and order of each branch changed slightly with the change in the geometry of the unit cells, as shown in Figure 31a,b. Therefore, the beam length for the hexagonal lattices can be used to modify dispersion characteristics, while the parameter could also offer the capability to tune the range of bandgap frequency as observed in the triangular lattices. As seen in the previous section, a change in the angle θ would have a similar effect on the dispersion curves of the hexagonal lattices as the angle θ of the triangular lattices.

From the results, the influences of each parameter on dispersion characteristics and the generation/vanishment of bandgaps were confirmed. Mainly, the influences of the geometric parameters can be categorized into three patterns: change in specific branches related to in-plane or out-of-plane vibrations, upward/downward shift in frequency range, and drastic change in dispersion characteristics. These effects can be used to control bandgap for engineering applications.

5. Conclusions

This paper studied the dispersion characteristics of plane wave propagation in mechanical metamaterials to realize effective vibration suppression. Triangular and hexagonal periodic lattice structures with various geometric parameters were mainly studied in this paper. In the part of developing the dispersion analysis framework, formulations for dispersion analysis of periodic lattice structures were described. The developed analysis framework was verified and validated with numerical transient simulations and vibration experiments using the additively manufactured model. Dispersion analysis of lattice plate structures with different geometric parameters was then performed with the presented analysis approach.

The presented dispersion analysis for the designed lattice plate model identified the bandgap, which would not allow any vibration modes to be excited in the structure. In both the transient analysis and experimental results, a significant decrease in transmittance was also observed within the bandgap, which demonstrated the capability of the lattice plate for vibration suppression. Although differences in frequency between the numerical solutions and experimental results were identified, the transmittance curves were qualitatively consistent. These results suggested that the proposed approach could effectively identify dispersion characteristics of lattice structures to investigate the influences of the geometric parameters on the bandgap characteristics.

The correlation between the geometric parameters and the bandgap characteristics of lattice structures consisting of triangular and hexagonal periodic lattices was investigated in the numerical studies. Triangular lattices could generate bandgap in all directions and offer robust vibration suppression. Hexagonal lattices produced bandgap in a specific direction, but they could be obtained in a wider frequency range, which had the advantage of making it easier to achieve a targeted performance of vibration suppression. In addition, a decoupled effect was observed in thickness t and width w that changed only specific dispersion curves for specific vibrations.

This work provides insight into the design of band structures to realize vibration suppression at specific frequencies and would contribute to establishing a guideline to control the frequency range of the bandgap and realize effective designs of lattice structures in engineering applications. In future works, further investigations will be performed to better understand potential causes of differences in vibration characteristics of fabricated lattice structures by the AM technique and the numerical solutions with the present analysis approach. Also, further experimental studies with fabricated plates with other lattice geometries, such as hexagonal lattices, would be performed as a further validation of the present approach in future works. Investigations considering not only the vibration characteristics but also the strength/stiffness of lattice structures will also be performed as a course toward realizing high-performance aerospace structures with lattice-based metamaterials.