Analysis of Influence of Excitation Source Direction on Sound Transmission Loss Simulation Based on Alloy Steel Phononic Crystal

Abstract

As a type of locally resonant phononic crystal, alloy steel phononic crystals have achieved notable advancements in vibration and noise reduction, particularly in the realm of low-frequency noise. Their exceptional band gap characteristics enable the efficient reduction of vibration and noise at low frequencies. However, the conventional transmission loss (TL) simulation of finite structures remains the benchmark for plate structure TL experiments. In this context, the TL in the XY-direction of phononic crystal plate structures has been thoroughly investigated and analyzed. Given the complexity of sound wave incident directions in practical applications, the conventional TL simulation of finite structures often diverges from reality. Taking tungsten steel phononic crystals as an example, this paper introduces a novel finite element method (FEM) simulation approach for analyzing the TL of alloy steel phononic crystal plates. By setting the Z-direction as the excitation source, the tungsten steel phononic crystal plate exhibits distinct responses compared to excitation in the XY-direction. By combining energy band diagrams and modes, the impact of various excitation source directions on the TL simulations is analyzed. It is observed that the tungsten steel phononic crystal plate exhibits a more pronounced energy response under longitudinal excitation. The TL map excited in the Z-direction lacks the flat region present in the XY-direction TL map. Notably, the maximum TL in the Z-direction is 131.5 dB, which is significantly lower than the maximum TL of 298 dB in the XY-direction, with a more regular peak distribution. This indicates that the TL of alloy steel phononic crystals in the XY-direction is closely related to the acoustic wave propagation characteristics within the plate, whereas the TL in the Z-direction aligns more closely with practical sound insulation and noise reduction engineering applications. Therefore, future research on alloy steel phononic crystal plates should not be confined to the TL in the XY-direction. Further investigation and analysis of the TL in the Z-direction are necessary. This will provide a novel theoretical foundation and methodological guidance for future research on alloy steel phononic crystals, enhancing the completeness and systematicness of studies on alloy steel phononic crystal plates. Simultaneously, it will advance the engineering application of alloy steel phononic crystal plates.

1. Introduction

Electromagnetic or elastic waves in periodic composites and structures can be inhibited from propagating within one or multiple frequency ranges, known as band gaps. In 1987, S. John [1] and E. Yablonovitch [2] pioneered the concept of utilizing an artificial microstructure with media of differing refractive indices arranged in a periodic fashion. This arrangement creates a band gap for electromagnetic waves, leading to the inception of photonic crystals. Similarly, M. M. Sigalas and E. N. Economou [3] theoretically verified in 1992 that scatterers in a periodic lattice arrangement exhibit elastic band gaps, paralleling the properties of photonic crystals. Subsequently, in 1993, M. S. Kushwaha et al. introduced the notion of phononic crystals, defining them as periodic materials possessing an elastic band gap [4]. In 2000, Liu [5] innovatively embedded aluminum spheres into epoxy resin, marking the debut of local resonance phononic crystals, which are composed of both hetero-crystals and Bragg scattering phononic crystals.

The diverse properties of the material comprising a phononic crystal have a profound impact on its band gap, and variations in these properties can result in disparities in the band gap ranges observed among different phononic crystals [6]. When sound waves propagate through a medium and encounter a phononic crystal, certain frequencies of these waves are unable to propagate further due to the presence of the band gap within the crystal [7]. Currently, the mechanisms responsible for generating band gaps in phononic crystals are broadly classified into two types: Bragg scattering and local resonance. The Bragg scattering mechanism is influenced by the periodic distribution of scatterers within the crystal, causing the sound waves to undergo continuous deflection, ultimately leading to the consumption of acoustic energy. Conversely, local resonance phononic crystals are influenced by the differences in density and modulus between individual scatterers within the crystal and the surrounding matrix. These material property differences result in localized acoustic energy dissipation within the phononic crystal [8,9,10].

With advancements in steel materials, a wide array of alloy steels has found extensive application across numerous fields, attributed to their superior physical and chemical properties. Alloy-steel-based phononic crystals exhibit considerable promise in areas such as negative refraction and super-resolution imaging, with their research and application expanding progressively. Notably, tungsten steels, owing to their exceptional hardness, high strength, robust heat resistance, and chemical stability, are employed in diverse applications. The investigation of tungsten steel phononic crystals encompasses various facets, including the propagation of sound waves, the design and fabrication of phononic crystals, and the impact of phononic crystals on the structural and material properties of the system.

When applied in the field of vibration and noise reduction [11], localized resonance phononic crystals (LRPCs) are often favored due to their ability to suppress large-wavelength, low-frequency sound waves through small cell structures [12,13,14,15,16,17]. The alloy steel phononic crystal plate discussed in this paper falls into the category of local resonance phononic crystals. In terms of noise reduction, when a mechanical wave passes through an LRPC plate, it transforms from a longitudinal wave into an elastic wave, which is a coupling of longitudinal and transverse waves [18,19]. At specific frequencies, this elastic wave induces various vibrations within the scatterers. The energy from these vibrations is localized within the phononic crystal plate by a soft coating layer, effectively controlling low-frequency vibrations and noise. Current research methods for LRPCs predominantly focus on energy band structures. Xiaoling Zhou [20] and colleagues discovered that the band gaps of LRPCs can be expanded to multiple frequency ranges by periodically embedding multilayered coaxial inclusions. They simulated the frequency responses of multilayered periodic structures with varying cell counts. Experimental verification and engineering applications of multilayered LRPCs will be explored in future work. In the study of phononic crystals for vibration and noise reduction, TL can be analyzed based on the band gap. Suobin Li et al. [21] described the design of a broad, locally resonant band gap in a phononic crystal using numerical simulations. They employed the finite element method to calculate dispersion relations, power-transmission spectra, and displacement fields for eigenmodes, providing new avenues for broadening the locally resonant band gaps of phononic crystals at low frequencies. P. R. Saffari et al. [22] investigated sound transmission loss (STL) through air-filled rectangular double-walled sandwich smart magneto-electro-elastic (MEE) plates with a porous functionally graded material (PFGM) core layer, considering an external mean airflow and uniform/non-uniform temperature distributions. Their parameter analysis evaluated the effects of initial electric and magnetic potentials, porosity distributions, incidence angles, acoustic cavity depths, and temperature profile changes on STL. F. Luckluma and M. J. Vellekoop [23] conducted a comprehensive study on 3D phononic crystals arranged in a simple cubic lattice, outlining design strategies for 3D phononic crystals with desired transmission characteristics for various applications.

Given that the phononic crystal consists of a scatterer embedded within a matrix, upon the transmission of a sound wave to the phononic crystal plate, it undergoes a transformation into an elastic wave, which is a composite of both longitudinal and transverse waves. This composite elastic wave subsequently prompts the scatterer to produce vibrations in diverse directions within the matrix, giving rise to various vibration modes. Consequently, in traditional TL studies of phononic crystal plates, a solitary XY-direction excitation source suffices to stimulate the entire spectrum of vibration modes within the plate. Nonetheless, the quintessence of TL simulation transcends mere theoretical and directional guidance for noise reduction endeavors; it aims to mirror real-world scenarios. In the real world, sound waves propagate in intricate patterns that a single-direction excitation source may fail to capture adequately. To comprehensively elucidate the acoustic wave excitation in various directions within alloy steel phononic crystal panels in the context of TL, an enhancement to the conventional approach is proposed. This enhancement involves augmenting the XY-direction excitation with an additional Z-direction excitation source. Utilizing tungsten steel phononic crystals as a case study, and by integrating energy band diagrams with modal analyses, the impact of different excitation source directions on TL simulations is meticulously examined.

2. Simulated Model Energy Band and Simulation Calculation Method for XY-Direction Transmission Loss

2.1. Band Structure

In the realm of analyzing phononic crystals, the dispersion relationship, or more specifically, the band diagram, stands as the paramount mathematical tool. By consulting the energy band diagram, one can accurately and efficiently ascertain the frequency range of the band gap. Notably, the band gap in a phononic crystal signifies a frequency zone devoid of eigenmodes for any arbitrary wave vector K; in simpler terms, it is the region in the band diagram where no viable solutions exist.

Given the periodic nature of phononic crystals, it is feasible to first identify the smallest unit, known as the cell, and then traverse the wave vector K within the irreducible Brillouin zone of this cell (refer to Figure 1 for the cell and the irreducible Brillouin region). By incorporating periodic boundary conditions along with the unit cell boundaries into the calculations, the energy band structure of the phononic crystal can be derived. The relevant equations employed in this process are outlined below.

Apart from extracorporeal forces, when an elastic wave propagates freely within a periodic lattice medium, the corresponding characteristic equation is expressed as

$$-\rho \left(r\right){\omega }^2\mathrm{v}\left(\mathrm{r}\right)=\left[\lambda \left(r\right)+\mu \left(r\right)\right]\nabla \nabla \cdot \mathrm{v}\left(r\right)+\mu \left(r\right){\nabla }^2\mathrm{v}\left(r\right),$$

(1)

where λ and μ are the Lame parameters of the material.

The relationship between the Young’s modulus of elasticity E and Poisson’s ratio ν is

$$\lambda ={\displaystyle \frac{E\nu }{\left(1-2\right)\left(1+\nu \right)}}, \mu ={\displaystyle \frac{E}{2\left(1+\nu \right)}}$$

(2)

where μ is numerically equal to the shear modulus of the material and ∇ is a vector differential operator.

Due to the periodicity inherent in photonic crystals, in accordance with Bloch’s theorem, the calculations can be efficiently conducted within a representative unit cell. This unit cell is meshed adaptively, taking into account the variations in the structures, and is subsequently divided into finite elements interconnected by nodes. Following the division of the network using the FEM, the discrete formulation of the eigenvalue equations within the unit cell can be expressed as

$$\left(\mathrm{K}-{\omega }^2\mathrm{M}\right)\mathrm{U}=0$$

(3)

where

$$\mathrm{K}=\int {\mathrm{N}}^{\mathrm{T}}\mathrm{C}\left(\mathrm{r}\right)\mathrm{N}\mathrm{d}{V}_e$$

(4)

are the stiffness of the unit cell, and

$$\mathrm{M}=\int {\mathsf{\rho }\left(\mathrm{r}\right)\mathrm{N}}^{\mathrm{T}}\mathrm{N}\mathrm{d}{V}_e$$

(5)

are the mass matrices of the unit cell, and

$$\mathrm{U}={\left[{\mathrm{U}}_1{\mathrm{U}}_2\cdots {\mathrm{U}}_n\right]}^{\mathrm{T}}$$

(6)

is the displacement matrix of the unit cell, with

$${\mathrm{U}}_i={\left[u_i{v}_i\cdots w_i\right]}^{\mathrm{T}}(i=\mathrm{1,2},\cdots ,n)$$

(7)

being the displacement at the nodes. $V_e$ is the whole domain of the unit cell, $\mathrm{N}$ is the matrix of the shape function, $\mathrm{C}\left(\mathrm{r}\right)$ is the elastic tensor, and $\mathsf{\rho }\left(\mathrm{r}\right)$ is the mass density tensor.

According to the propagation principle of the medium wave of the periodic structure, that is, the Bloch theorem, the wave field is applied as follows:

$$\Psi \left(\mathrm{r}\right)=e^{i\left(k\bullet \mathrm{r}\right)}{\Psi }_k\left(\mathrm{r}\right)$$

(8)

where $\Psi$ = $u_x, u_y, u_z, p$, k is the wave vector, and $k=\left(k_x,k_y,k_z\right),\left|k\right|={\displaystyle \frac{2\mathsf{\pi }}{\lambda }}={\displaystyle \frac{2\mathsf{\pi }}{\mathrm{c}\mathrm{T}}}={\displaystyle \frac{\omega }{\mathrm{c}}}$. In two dimensions, $k_z=0$, ${\Psi }_k\left(\mathrm{r}\right)$ is a periodic field function.

From the above equation, the cellular outside should be met:

$$\mathrm{U}\left(\mathrm{r}+{\mathrm{R}}_n\right)=\mathrm{U}\left(\mathrm{r}\right)e^{i\left(k\bullet {\mathrm{R}}_n\right)},$$

(9)

where ${\mathrm{R}}_{\mathrm{n}}$ is a positive vector, simultaneous with (1) and (6), and the characteristic frequency $\omega$ can be solved given a wave vector $k$.

By substituting the characteristic frequency $\omega$ into Equation (1), the corresponding U(r) for that frequency can be derived. COMSOL Multiphysics is employed to directly solve the eigenvalue equation under complex boundary conditions. Subsequently, the wave vector is swept across the edges of the irreducible Brillouin zone, allowing us to obtain the eigenvalue problem that describes the relationship between the wave frequency and the wave vector. This relationship, known as the dispersion relation, is also referred to as the band structure.

2.2. Transmission Loss

The energy band structure corresponds to the infinite periodicity of a phononic crystal. However, since the practical implementation of the phononic crystal’s structure in engineering is limited, the energy band structure serves merely as an important reference in engineering applications related to vibration and noise reduction. It is necessary to integrate the transmission loss characteristics to comprehensively determine the vibration and noise reduction performance of the phononic crystal. The simulation and calculation method for the transmission loss of a conventional XY-direction excitation source is shown in Figure 2.

As shown in Figure 2, the periodic plates are comprised of 5 × 5 periodic cells. In order to prevent reflections, two perfectly matched layers (PMLs), each with a length equivalent to twice the lattice constant (2a), are implemented in the X-direction. Meanwhile, periodic boundaries are employed in the Y-direction. A deformation node is set up to distort the plot based on a vector quantity, allowing for the visual observation of pressure transfer.

3. Z-Direction Transmission Loss Simulation Model and Z-Direction Excitation Source Setting

A two-dimensional, three-component, three-parameter local resonance phononic crystal, featuring inclusions composed of soft silicone rubber and hard tungsten embedded in a resin substrate, was investigated using the finite element method-perfectly matched layer (FEM-PML). The unit cell diagram is presented in Figure 3. The unit cell dimensions are as follows: a = 0.1 m. The substrate is made of epoxy resin, while tungsten, with a radius of 0.046 m, serves as the scatterer. A silicone gel layer, with a thickness of 0.002 m, is applied between the scatterer and the substrate. Material properties, including density and Lame constants, influence the band gap frequency and overall vibration characteristics of phononic crystals. Variations in density directly impact the band gap frequency of the phononic crystal and the attenuation effect of vibrations within the band gap frequency range. Typically, an increase in material density shifts the first band gap towards lower frequencies, effectively modulating the band gap frequency range of phononic crystals. In finite-layered multi-periodic structures, vibrations are significantly attenuated within the band gap frequency range, contributing to vibration damping. Changes in density may influence this attenuation effect, subsequently altering the overall vibration characteristics of the phononic crystal. Lame constants play a crucial role in determining the existence and size of the band gap. Large differences in Lame constants favor the formation of a complete band gap. An increase in the elastic modulus generally shifts the first gap towards higher frequencies. Additionally, variations in the elastic constant also affect the width of the band gap, modifying the phononic crystal’s ability to control elastic waves within a specific frequency range. The material density and Lame constants of the models are listed in

Table 1. Material parameters.

Material	$\mathit{\rho }$ (kg/m3)	$\mathit{\lambda }$ (Pa)	$\mathit{\mu }\left(\mathbf{P}\mathbf{a}\right)$
Resin	1180	4.52 × 109	1.59 × 109
Rubber	1300	6.051 × 105	4 × 104
Tungsten	19,350	3.06 × 1011	1.311 × 1011

.

The simulation using a conventional XY-direction excitation source cannot accurately reflect the primary incident direction of sound waves in real engineering applications. When attempting to simulate a standing wave tube, accurately replicating its periodic structure is not feasible. To address this issue, this paper innovatively proposes exciting a 10 × 10 unit cell structure phononic crystal plate in the Z-direction. By configuring a reasonable PML and boundary conditions, a more precise simulation of the TL in real engineering application environments can be achieved. The model is illustrated in Figure 4.

A pressure with a value of P0 is applied to the Z-direction excitation source, and this pressure is integrated to obtain P1. The integration of the pressure (denoted as Pm) on the receiving boundary is calculated using the following formula:

$$\mathrm{P}\mathrm{m}=-\left(SL11+SL22+SL33\right)/3$$

(10)

Among them, SL11, SL22, and SL33 are the 11th component, the 22nd component, and the 33rd component of the stress tensor in the local coordinate system, respectively. The parameters of the PMLs (perfectly matched layers) are identified automatically by the software (COMSOL Multiphysics 6.2). The rigid constraint boundary condition is given by u = 0. Therefore, the transmission can be calculated by 20 × log(P1/Pm).

4. Numerical Results and Discussion

4.1. Band Gap Numerical Results and Energy Band Data Analysis

The simulated energy band diagram is shown in Figure 5.

The phononic crystal plate exhibits a directional band gap in the XM-direction within the frequency ranges of 34.05–48.77 Hz, 49.13–56.09 Hz, 56.16–94.45 Hz, and 208.93–953.38 Hz, as indicated by the yellow region in Figure 5. There is a complete band gap spanning from 98.57 Hz to 208.93 Hz, as illustrated by the purple area in the figure. Specific values can be found in

Table 2. Phononic crystal plate band gap.

Belt Position	Starting and Ending Frequency (Hz)	Band Gap Direction
1–2	34.05–48.77	XM-direction
3–4	49.13–56.09	XM-direction
4–5	56.16–94.45	XM-direction
6–7	98.57–208.93	Full band gap
6–7	208.93–953.38	XM-direction

. Since the various band gaps in the XM-direction are composed of flat bands, they can be combined into a single range of 34.05–953.38 Hz, yielding a total bandwidth of 919.33 Hz, which demonstrates the characteristic of ultra-wide band gaps. The occurrence of numerous straight bands may be attributed to the constraint of wave vector variation in the Brillouin zone of the crystal by phase matching conditions. In particular, in the XM-direction, the wave vector may lie close to the boundary of the Brillouin zone, causing the wave propagation velocity to approach zero. Consequently, in this direction, the wave frequency remains constant regardless of wave vector variations, resulting in the formation of a flat frequency band. As depicted in Figure 6, the band gap mechanism is analyzed through a modal diagram, showing the first six mode shapes of the convex phononic crystal. To clearly visualize the vibration direction of each mode, long red arrows have been added to the surface, and blue cone arrows have been incorporated within the body.

The simulation of the vibration mode diagram at each characteristic frequency shows that the 34.05 Hz scatterer longitudinal vibration mode and the 49.15 Hz scatterer rotational vibration mode correspond to the start and stop frequencies of the first XM-direction band gap, respectively, and the 34.05 Hz vibration mode corresponds to the initial vibration frequency of the main vibration energy localized in the longitudinal movement of the scatterer. The 49.15 Hz mode corresponding to the termination frequency mainly focuses on the rotational motion of the scatterer along the z axis, which is not only the termination frequency of the first XM-direction band gap but also the starting frequency of the second XM-direction band gap. In addition, 56.10 Hz and 56.50 Hz correspond to the rotational vibration mode of the scatterer along the x and y axes, respectively, which is the termination frequency of the second XM-direction band gap and also the starting frequency of the third XM-direction band gap, and its energy locality is the scatterer vibrating along the x, y axis. The values 94.45 and 98.58 Hz correspond to the transverse vibration mode of the scatterer, corresponding to the termination frequency of the band gap of the third XM-direction and also the starting frequency of the complete band gap. The complete band gap termination frequency of the phononic crystal plate is 208.93 Hz, which corresponds to the longitudinal vibration mode of the substrate. From 208.93 Hz to 953.38 Hz is the fourth XM-direction band gap, and its termination frequency is the rotational vibration mode of the substrate along the z axis. Although 591 Hz and 594.37 Hz are not the start and stop frequencies of the band gap, in the energy band diagram, the lower edge of the corresponding frequency after the eighth-order sweeping of the Brillouin zone is considered, which may result in the generation of different excitation source directions of TL in the next step. So, the modality is analyzed and found to correspond to the transverse vibration mode of the substrate.

4.2. TL Data Analysis Under Different Excitation Sources

To illustrate the influence of different excitation source directions on the calculation of the transmission loss of the phononic crystal plate, this paper first calculates the transmission loss of the phononic crystal plate in the conventional XY-direction and obtains Figure 7.

As shown in Figure 7, the conventional XY-direction transmission loss calculation of the phononic crystal plate reveals that the transmission loss drops sharply from −20 to −298 dB near the lower edge of the full band gap at 90 Hz. Combining the modal analysis (Figure 6) and the vibration displacement (Figure 8), the mechanism is determined to be the generation of matrix XY-direction vibration in the vicinity of 90 Hz, leading to the localization of acoustic energy. The directional band gap region gradually expands until the transmission loss at the upper edge of the directional band gap, at 955 Hz, stabilizes near a high position. It is observed that the band structure aligns well with the transmission loss in the XY-direction. However, in real engineering applications, sound wave energy does not solely propagate through the XY-direction; rather, more energy passes through the phononic crystal plate via the Z-direction or at an angle relative to the Z-direction. Therefore, a simulation calculation of the transmission loss is performed for a Z-direction excitation source.

As shown in Figure 9, it can be observed that there is a significant difference in the transmission loss between the XY- and Z-directions. Notably, the transmission loss graph excited in the Z-direction lacks a flat area, as seen in the XY-direction transmission loss. Possible reasons for this include the anisotropy of phononic crystals, band structure, boundary effects, differences in wave modes, and local resonance phenomena. Propagation in the Z-direction involves more reflection, scattering, and resonance, leading to frequent peak-and-valley changes in the spectrum. In contrast, propagation in the XY-direction is more regular, exhibiting a relatively flat transmission loss area. This phenomenon underscores the complex control capabilities of phononic crystals over sound wave propagation in different directions. The maximum transmission loss in the Z-direction is 131.5 dB at 685 Hz, which is 166.5 dB lower than the maximum transmission loss of 298 dB in the XY-direction. Additionally, the peak distribution in the Z-direction is relatively regular. Considering that local resonance phononic crystals are primarily applicable in the mid-to-low-frequency range, combined with the energy band range of 34.05–953.38 Hz, the research area is selected as 0–1000 Hz. The peak frequencies in the transmission loss graph for the Z-direction are 15, 40, 240, 345, 405, 565, 685, 740, 910, and 990 Hz, corresponding to transmission losses of 72 dB, 68 dB, 104 dB, 96 dB, 114 dB, 85 dB, 131.5 dB, 102 dB, 43 dB, and 99 dB, respectively.

In order to study the influence of the two excitation source directions on the phononic crystal TL simulation, the vibration displacement of all peaks in the Z-direction excitation is plotted. This is then compared and analyzed with the peak vibrational displacement diagram of the XY-direction excitation. By combining the energy band diagram and modal analysis, the mechanism of vibration reduction and noise reduction is judged.

As shown in Figure 10, the first effective transmission loss peak of the phononic crystal plate was found to be at 15 Hz, and this frequency is not included in the band. This is because the energy band can only reflect the vibration mode of the unit cell under the free boundary. The first-order vibration mode of the whole plate, according to the mass law, must have a frequency lower than that of a mode of the unit cell oscillator. The study of phononic crystal plates is not limited to the width of the frequency domain energy bands but also considers the TL, which will bring more reliable simulation data for future engineering applications. Moreover, the conventional TL from an XY-direction excitation source does not reflect the first-order vibration mode in the Z-direction of the entire plate. It can be observed from Figure 8 that the first peak at 20 Hz in the XY-direction excitation source corresponds to the first order of the overall plate in the XY-direction. Obviously, in practical engineering applications, the acoustic energy propagating through the XY-direction is extremely small. Therefore, to simulate more reliable data, relying solely on the excitation source in the XY-direction is insufficient. By contrast, we need Z-direction excitation, as acoustic energy propagates much more significantly through the Z-direction than the XY-direction. At 25 Hz, there is the second-order vibration mode of the entire plate in the Z-direction. The vibration displacement cloud diagram clearly shows that under Z-direction excitation at 25 Hz, the acoustic energy transmitted by the entire plate is larger than the energy incident on the surface of the plate, with a difference of 14 dB. Additionally, 40 Hz is the first-order resonance mode of the scatterer and matches the mode of 34.05 Hz in the cell band structure under the free boundary, confirming each other’s correctness.

The 90 Hz frequency appears in both the Z-direction excitation and the XY-direction excitation TL and is therefore more representative for comparative analysis at the 90 Hz excitation. First, by examining the energy band diagram under free boundary conditions, it can be determined that 94.45 Hz and 98.575 Hz correspond to vibration modes in the lateral direction, i.e., the XY-direction. Therefore, it can be confirmed that the 90 Hz frequency, which is simultaneously present in both the XY-direction and the Z-direction excitation TL, is generated due to lateral vibration of the scatterer. Secondly, the vibration displacement map also supports this conclusion. In Figure 8, the 90 Hz vibration displacement map in the XY-direction clearly shows that the lateral vibration of the scatterer causes most of the acoustic energy to be localized in the first cycle, with only rare partial energy localization occurring in the second cycle. Consequently, the TL exhibits a significant downshift at 90 Hz, dropping from −20 to −298 dB, a decrease of 278 dB. In Figure 10, the lateral vibration of the scatterer can also be observed through the vibration displacement map. However, at this point, there is no acoustic energy localized in the phononic crystal plate because, for the Z-direction, the upper and lower surfaces of the plate do not exhibit a significant gap due to the lateral vibration of the scatterer. The fundamental reason is that even when a z-incident acoustic wave produces a coupling of longitudinal and transverse waves in the elastic body, most of the acoustic energy propagates along the longitudinal wave in the Z-direction, while only a small portion of the transverse wave propagates through the XY-direction. This decline in the Z-direction by only 1 dB under a 90 Hz excitation can be verified. From the energy band diagram and modal analysis, it can be seen that the substrate generates a longitudinal vibration at 208.93 Hz, which corresponds to the upper edge of the full band gap. In the TL excited in the Z-direction, 205 Hz represents an ineffective transmission loss peak, meaning that in the vicinity of 205 Hz, the phononic crystal plate cannot effectively provide sound insulation. To explore its mechanism, the vibration displacement map at 205 Hz under Z-direction excitation is analyzed. In Figure 10e, it can be observed that since 205 Hz is an ineffective TL peak, there is no vibration displacement information on the upper surface. To more clearly see the transmission characteristics at 205 Hz, the vibration displacement of the lower surface (-Z-direction) is displayed, revealing that the substrate has a significant vibration displacement in the -Z-direction at 205 Hz. Therefore, it can be concluded that the phononic crystal plate is not effectively damped at 205 Hz.

The 240 Hz frequency corresponds to the first-order vibration mode of the substrate, but unlike the vibration displacement at 205 Hz, it becomes an effective TL peak. At 285 Hz, we observe the second-order vibration mode of the matrix. By comparing the vibration displacement in the positive Z-direction with that in the -Z-direction, it can be determined that 285 Hz represents an ineffective TL peak. The vibration displacement maps for the remaining peaks are similar to those analyzed previously and are therefore not described here. The significance of adding these vibration displacement maps is that they provide an intuitive understanding of how the phononic crystal plates exhibit different vibration displacements under various excitation sources.

5. Conclusions

The use of XY-direction excitation source simulation alone fails to capture the complexity of acoustic propagation paths in real-world engineering applications. Consequently, when simulating with Z-direction excitation in a standing wave tube, accurately simulating the periodic structure becomes infeasible. To address this issue, this paper innovatively establishes a Z-direction excitation simulation model for a phononic crystal plate featuring a 10 × 10 cell structure. By implementing reasonable boundary conditions and a PML, the simulation of the TL of the phononic crystal plate in a real engineering application environment becomes more accurate.

The calculation of transmission loss in the Z-direction of the phononic crystal plate differed significantly from that in the XY-direction. Notably, the transmission loss map excited in the Z-direction lacked the flat region observed in the XY-direction transmission loss map. Furthermore, the maximum transmission loss in the Z-direction was 131.5 dB, which is 166.5 dB lower than the maximum transmission loss of 298 dB in the XY-direction. Additionally, the peak distribution in the Z-direction exhibited a more regular pattern.

The Z-direction excitation is plotted against the vibration displacement of all peaks within the study range and is compared with the peak vibration displacement diagram of the XY-direction excitation. This comparison incorporates energy band diagrams and a modal analysis to evaluate its vibration and noise reduction mechanisms. When Z-direction excitation is observed, the first effective transmission loss peak for the phononic crystal plate is at 15 Hz, a frequency not included in the band range. This is due to the fact that the energy band can only reflect the vibration mode of the unit cell under free boundary conditions. According to the mass law, the first-order vibration mode of the entire plate is lower than the first-order mode of the unit cell. In the TL of the conventional XY-direction excitation source, the first peak at 20 Hz corresponds to the first-order vibration mode of the plate’s XY-direction and does not reflect the first-order Z-direction vibration mode of the entire plate, highlighting the necessity of considering Z-direction excitation as a source.

The alloy steel phononic crystal plate exhibits significant application potential in vehicle engineering, machinery manufacturing, and construction. During the investigation of sound wave propagation characteristics within this plate, incorporating a Z-direction excitation source brings the study closer to practical sound insulation and noise reduction engineering applications. This approach can provide valuable guidance for the application of alloy steel phononic crystal plates in the aforementioned fields.

Phononic crystals, as a novel type of functional material, hold vast application potential in the realm of vibration and noise control. Future research endeavors will diversify, encompassing deeper investigations into band gap mechanisms, the optimization of phononic crystal structures, the expansion of the band gap frequency range, and the incorporation of novel materials. Additionally, the integration of Bragg scattering and local resonance mechanisms merits further exploration to devise phononic crystal structures with broader band gaps and enhanced noise reduction capabilities.