Negative Stiffness Composite Plate Design for Vibration Suppression

Abstract

This study explores the design and analysis of a negative stiffness plate aimed at enhancing vibration suppression. The concept of the negative stiffness plate is illustrated through the implementation of periodic resonators embedded within a host plate, each incorporating a single vibration absorber. These absorbers generate shear forces and efficiently capture vibration energy when excited resonantly. To validate the effectiveness of these negative stiffness composite plates, numerical experiments were conducted. The findings reveal that the damping of the absorber can significantly influence the frequency stopband created by the absorber’s resonance. Notably, damping has a minor but positive effect on broadening the stopband width and reducing the overall vibration response of the host plate. Furthermore, this study highlights the potential application of negative stiffness plates in mitigating elastic waves induced by earthquakes.

1. Introduction

Negative stiffness structures exhibit the remarkable ability to create a localized vibration “sink”, effectively channeling vibrations into a specific absorber. This mechanism is analogous to the behavior of acoustic/elastic metamaterials (EMMs), which induce bandgap phenomena. Specifically, a bandgap functions as a frequency barrier within which wave propagation is inhibited, thereby facilitating the reduction of vibrations in the structure [1,2,3,4,5].

Over the past two decades, elastic metamaterials (EMMs) have garnered considerable attention due to their distinctive effective material properties [1]. These artificially engineered materials are capable of manipulating and directing the propagation of elastic waves through a localized resonance mechanism that interacts with acoustic/elastic waves. When the frequency of the propagating wave aligns with the localized resonance of the microstructure, the electromagnetic field can exhibit unique dynamic properties, properties like negative effective stiffness and negative effective mass density have been investigated for diverse purposes, encompassing seismic waveguide designs, acoustic absorption mechanisms, and methods for vibration mitigation [6,7,8]. Negative stiffness originates from structural instability or nonlinear geometric deformation, manifested as the negative slope region of the force displacement curve, with the core function of low-frequency energy dissipation. Negative effective mass is achieved through local resonance units or quantum control to reverse the equivalent inertial force, manifested as acceleration direction opposite to external force direction. Typically, electromagnetic fields are constructed by incorporating periodic mechanical subunits into natural materials to induce localized resonance responses and modify wave characteristics such as velocity, direction, and wavelength. However, the geometric shapes of electromagnetic metamaterial (EMM) subunits are often limited, as these models are only applicable to wavelengths exceeding their overall dimensions. Current research primarily focuses on areas such as acoustic resonators [9], subwavelength guiding [10], ultrasonic focusing [11], elastic wave absorption [12], structural vibration reduction [13], and seismic wave manipulation [14].

Negative stiffness structures have emerged as promising alternatives to elastic metamaterials (EMMs) for mitigating vibrations caused by earthquake-induced elastic waves. In recent years, various designs and applications of these structures have been proposed. Nagarajaiah et al. have designed a negative stiffness device (NSD) [15,16,17,18] that incorporates pre-compressed springs and friction pendulum sliding components. Subsequent studies [19,20] have evaluated the effectiveness of these NSDs. Cheng and Shi [21,22,23] have introduced seismic isolation foundations that exhibit distinct band gaps for vertical and horizontal directions, potentially addressing the vertical motion of earthquakes. Choi et al. [24] have proposed a locally resonant metamaterial (LRM) to attenuate noise and vibration of floors in residential buildings. Liu et al. [25] proposed a locally resonant metamaterial sandwich panel design based on impedance matching, which optimizes the frequency response characteristics by encapsulating the core layer resonator. Recently, a range of negative stiffness base isolators [26], dampers [27], and underground structures [28] have been designed and tested for vibration reduction.

Drawing from classical continuum theory, our research team conducted a comprehensive study of negative-stiffness beams and columns [1,29]. Peng and Pai [30,31] first introduced the fundamental mechanism of negative-stiffness beams for structural vibration suppression in 2014. Subsequently, Zhong, Pai, and Zong [3] outlined the design principles of a negative stiffness I-beam using a single stop band in 2018. Recently, ref. [4] significantly broadened the stop band by incorporating multi-stop band negative stiffness composite columns, which can effectively absorb and dampen destructive waves caused by earthquakes.

In this paper, a novel design method for negative stiffness plates without the need for external resonators is proposed. The design incorporates mass–spring–damper subsystems periodically within an isotropic plate, with each single-frequency resonator consisting of a central mass flanked by four beams. Dispersion analysis of an infinite negative stiffness plate is conducted using numerical models to derive theoretical stop bands. Subsequently, a finite negative stiffness plate model is established, and its stop bands are determined through frequency response and transient analysis. Finally, numerical experiments validate the effectiveness of the negative stiffness plate design.

2. Negative Effective Stiffness

Negative effective stiffness exhibits parallels with the negative permittivity and permeability observed in electromagnetic materials (EMs). Figure 1 depicts a two-degree-of-freedom (2-DOF) mass-on-spring system, featuring a massless thin plate attached to the ground via two springs of constant stiffness. Beneath the plate, the mass is localized and supported by an auxiliary spring characterized by a particular stiffness $k_1/2$. Harmonic excitation with frequency $\omega$ is applied to the thin plate. The displacement of the thin plate and the mass m₂ are denoted as $u_1$ and $u_2$, respectively. The equations of motion governing the behavior of this spring-mass system are as Equation (1).

$$\left[\begin{array}{cc}0& 0\\ 0& m_2\end{array}\right]\left\{\left.\begin{array}{l}{\ddot{u}}_1\\ {\ddot{u}}_2\end{array}\right\}\right.+\left[\begin{array}{cc}{k}_1+k_2& -k_2\\ -k_2& k_2\end{array}\right]\left\{\left.\begin{array}{l}{u}_1\\ u_2\end{array}\right\}\right.=\left\{\left.\begin{array}{l}F\\ 0\end{array}\right\}\right.$$

(1)

the frequency response functions $H_{i1}(\omega )$ are derived to be Equation (2).

$$H_{11}={\displaystyle \frac{k_2-m_2{\omega }^2}{(k_1+k_2)(k_2-m_2{\omega }^2)-k_2^2}}={\displaystyle \frac{1}{k_{eff}}}$$

(2)

Using Newton’s second law, the effective stiffness of the system can be derived as Equation (3).

$$k_{eff}={\displaystyle \frac{F}{u_1}}=k_1+{\displaystyle \frac{k_2}{1-{({\omega }_2/\omega )}^2}}$$

(3)

we note that the scenario arises when the excitation frequency $\omega$ matches the natural frequency ${\omega }_2$ associated with mass m, the effective stiffness $k_{eff}$ becomes infinitely large, resulting in a displacement $u_1$ that approaches zero. Additionally, within a specific frequency range (denoted as (${\omega }_2\sqrt{k_1/(k_1+k_2)}$, ${\omega }_2$)), the effective stiffness $k_{eff}$ is negative. The mechanism behind this negative effective stiffness involves creating shear forces that absorb vibration energy under resonant excitation.

In the academic field of structural dynamics and vibration control, we have identified a crucial phenomenon: when the external excitation frequency precisely matches the inherent resonance frequency of the auxiliary mass-spring subsystem, it triggers a potent resonance effect. The core advantage of this effect lies in its ability to generate a significant resistance that not only effectively counters the external excitation but also facilitates deep dissipation of vibration energy within the base structure. This specific frequency range, encompassing the resonance frequency and its adjacent spectrum, is termed the “stop band”. Within the meticulously designed stop band, notable changes occur in the propagation characteristics of elastic waves. Specifically, due to the resonance state of the auxiliary mass-spring subsystem, elastic waves experience intense interference effects as they attempt to traverse the base structure. This interference complicates and obstructs the wave’s propagation path, leading to a substantial decrease in propagation efficiency. More importantly, it further promotes energy dissipation within the structure. By meticulously adjusting various parameters of the auxiliary mass-spring subsystem, such as mass distribution, spring stiffness, and damping characteristics, we can achieve precise control over the position and width of the stop band. This control capability enables us to effectively suppress specific vibration frequencies, thereby significantly enhancing the seismic and fatigue resistance of structures. Furthermore, this strategy introduces new theoretical insights and technological approaches to the field of vibration control, exerting a profound impact on the development and application prospects of related disciplines.

3. Design of the Negative Stiffness Plate

In the field of vibration control engineering, researchers have continuously explored new technologies and methodologies to address vibration issues arising from complex environments such as traffic loads and seismic activities. Based on the design concept of the two-degrees-of-freedom spring-mass system vibration absorber, this paper innovatively proposes a negative stiffness plate unit, providing a fresh perspective for solving the aforementioned problems. The structural design of the negative stiffness plate unit is illustrated in Figure 2a. Its core lies in the ingenious integration of a mass-spring subsystem. This design enables each unit to be customized according to actual requirements, exhibiting precise local resonance frequencies. When external excitations act on the unit, the mass-spring subsystem responds rapidly and generates a significant amount of out-of-phase inertial forces. These forces cancel each other out at the microscopic scale, effectively reducing the internal stresses within the base plate and achieving the purpose of dissipating vibrations. In the actual production process of negative stiffness plates, rubber can be used as a damping material.

To more clearly demonstrate the mechanical characteristics of the device, we have simplified its representation, as shown in Figure 2b. This simplified model not only retains the key design elements but also makes analysis and calculation more intuitive and efficient. Based on this, we further explored the possibility of assembling multiple replicas of the unit at the microscopic scale. By precisely controlling the layout and arrangement of each unit, we successfully constructed a plate with significant negative stiffness characteristics, as shown in Figure 2c. In addition to innovations in structural design, we also integrated suitable damping materials within the dampers. This measure aims to further enhance the vibration suppression capabilities of the negative stiffness plate. When vibrations occur, the damping materials can absorb and dissipate a significant amount of vibration energy, thereby reducing the amplitude and propagation range of the vibrations. Experimental results indicate that by integrating suitable damping materials, the negative stiffness plate unit exhibits excellent performance in addressing vibrations caused by seismic activities.

As shown in Figure 2b,

$$\left\{\left.\begin{array}{l}{\sigma }_{11}\\ {\sigma }_{22}\\ {\sigma }_{33}\end{array}\right\}\right.={\displaystyle \frac{E}{1-{\nu }^2}}\left[\begin{array}{lll}1& \nu & 0\\ \nu & 1& 0\\ 0& 0& (1-\nu )/2\end{array}\right]\left\{\left.\begin{array}{l}{\epsilon }_{11}\\ {\epsilon }_{22}\\ {\epsilon }_{33}\end{array}\right\}\right.,\left\{\left.\begin{array}{l}{\epsilon }_{11}\\ {\epsilon }_{22}\\ {\epsilon }_{33}\end{array}\right\}\right.=z\left\{\left.\begin{array}{l}{w}_{1xx}\\ w_{1yy}\\ 2w_{1xy}\end{array}\right\}\right.$$

(4)

In Equation (4), ${\sigma }_{11}$ and ${\sigma }_{22}$ represent the normal stresses along the x and y directions, respectively, and E and v are the material’s Young’s modulus and Poisson’s ratio. Subsequently, as shown in Equation (5), we denominate the momentum responses of the plate as M₁, M₂, and M₆, and define its stress responses as N₁, N₂, N₆.

$$\begin{array}{l}{M}_1={\displaystyle {\int }_{-h/2}^{h/2}{\sigma }_{11}zdz}={\displaystyle {\int }_{-h/2}^{h/2}(\frac{E}{1-{\nu }^2}{w}_{1xx}{z}^2-\frac{E\nu }{1-{\nu }^2}{w}_{1yy}{z}^2)dz}=-D(w_{1xx}+\nu w_{1yy})\\ M_2={\displaystyle {\int }_{-h/2}^{h/2}{\sigma }_{22}zdz}={\displaystyle {\int }_{-h/2}^{h/2}(\frac{E\nu }{1-{\nu }^2}{w}_{1xx}{z}^2-\frac{E}{1-{\nu }^2}{w}_{1yy}{z}^2)dz}=-D(\nu w_{1xx}+w_{1yy})\\ M_6={\displaystyle {\int }_{-h/2}^{h/2}{\sigma }_{12}zdz}={\displaystyle {\int }_{-h/2}^{h/2}(-\frac{E\nu }{1-{\nu }^2}{w}_{1xy}{z}^2)dz}=-D(1-\nu )w_{1xy}\\ N_1={\displaystyle {\int }_{-h/2}^{h/2}{\sigma }_{11}dz}={\displaystyle {\int }_{-h/2}^{h/2}(\frac{E\nu }{1-{\nu }^2}{w}_{1xx}z-\frac{E\nu }{1-{\nu }^2}{w}_{1yy}z)dz}=0\\ N_2={\displaystyle {\int }_{-h/2}^{h/2}{\sigma }_{22}dz}={\displaystyle {\int }_{-h/2}^{h/2}(\frac{E\nu }{1-{\nu }^2}{w}_{1xx}z-\frac{E}{1-{\nu }^2}{w}_{1yy}z)dz}=0\\ N_6={\displaystyle {\int }_{-h/2}^{h/2}{\sigma }_{12}dz}={\displaystyle {\int }_{-h/2}^{h/2}(-\frac{E\nu }{1-{\nu }^2}{w}_{1xy}z)dz}=0\\ D={\displaystyle \frac{E{h}^3}{12(1-{\nu }^2)}}\end{array}$$

(5)

In this context, D represents the bending stiffness of the material. Assuming that the sums of forces and moments along the x, y, and z directions must always be zero, the following relationships can be derived as Equation (6).

$$\left\{\begin{array}{l}\sum F_x=0\\ \sum F_y=0\\ \sum F_z=0\\ \sum M_x=0\\ \sum M_y=0\\ \sum M_z=0\end{array}\right.\Rightarrow \left\{\begin{array}{l}{N}_{1x}dxdy+N_{6y}dxdy=0\\ N_{2y}dxdy+N_{6x}dxdy=0\\ Q_{1x}dxdy+Q_{2y}dxdy+Fdxdy=\rho dxdy\\ -M_{6x}dxdy-M_{2y}dxdy+Q_{2}dxdy=0\\ M_{6y}dxdy+M_{1x}dxdy-Q_{1}dxdy=0\\ 0=0\end{array}\right.\Rightarrow \left\{\begin{array}{l}{N}_{1x}+N_{6y}=0\\ N_{2y}+N_{6x}=0\\ Q_{1x}+Q_{2y}+F=\rho dxdy\\ -M_{6x}-M_{2y}+Q_2=0\\ M_{6y}+M_{1x}-Q_1=0\\ 0=0\end{array}\right.$$

(6)

where

$$\begin{array}{l}{Q}_1=M_{6y}+M_{1x}=-D(w_{1xxx}+v{w}_{1yyx}+(1-v)w_{1xyy})=-D(w_{1xxx}+w_{1yyx})\\ Q_2=M_{6x}+M_{2y}=-D(w_{1yyy}+v{w}_{1xxy}+(1-v)w_{1xyx})=-D(w_{1yyy}+w_{1xxy})\end{array}$$

(7)

These signify the intensities of the lateral shear forces acting within the yz and xz planes, respectively.

Additionally, the kinetic energy $T$, elastic potential energy $\Pi$, and non-conserved work $W_{nc}$ arising from external forces can be determined as outlined below Equations (8)–(10).

$$\delta T={\displaystyle {\int }_{-a}^a{\displaystyle {\int }_{-b}^b(\rho h{\ddot{w}}_1\delta w_1)dydx}}$$

(8)

$$\begin{array}{cc}\hfill \delta \prod & ={\displaystyle {\int }_{-a}^a{\displaystyle {\int }_{-b}^b{\displaystyle {\int }_{-h/2}^{h/2}({\sigma }_{11}\delta {\epsilon }_{11}+{\sigma }_{22}\delta {\epsilon }_{22}+{\sigma }_{12}\delta {\epsilon }_{12})}dzdydx}}\hfill \\ & ={\displaystyle {\int }_{-a}^a{\displaystyle {\int }_{-b}^b(-M_{1xx}\delta w-M_{2yy}\delta w-2M_{6xy}\delta w)dydx}}\\ & +{\displaystyle {\int }_{-b}^b\{{\left[-M_1\delta w_x+(M_{1x}+M_{6y})\delta w-M_6\delta w_y\right]}_{x=-a}^{x=0^{-}}+{\left[-M_1\delta w_x+(M_{1x}+M_{6y})\delta w-M_6\delta w_y\right]}_{x=0^{+}}^{x=a}\}dy}\\ & +{\displaystyle {\int }_{-a}^a\{{\left[-M_2\delta w_x+(M_{2y}+M_{6x})\delta w-M_6\delta w_x\right]}_{x=-b}^{x=0^{-}}+{\left[-M_2\delta w_x+(M_{2y}+M_{6x})\delta w-M_6\delta w_x\right]}_{x=0^{+}}^{x=b}\}dx}\end{array}$$

(9)

$$\delta W={\displaystyle {\int }_{-b}^b{\left[-M_1\delta w_x+Q_1\delta w-M_6\delta w_y\right]}_{x=-a}^{x=a}dy}+{\displaystyle {\int }_{-a}^a{\left[-M_2\delta w_y+Q_2\delta w-M_6\delta w_x\right]}_{x=-b}^{x=b}dx}+k(u-w_0)\delta w_0$$

(10)

Then, based on the expanded Hamiltonian principle, the controlling equations for the infinite negative stiffness plate unit are deduced as Equation (11).

$$\begin{gathered} \begin{array}{l}0={\displaystyle {\int }_0^t(\delta T-\delta \prod +\delta W)}dt\\ ={\displaystyle {\int }_0^t\left\{{\displaystyle {\int }_{-a}^a{\displaystyle {\int }_{-b}^b(-\rho h\ddot{w}+M_{1xx}+M_{2yy}+2M_{6xy}+\left[\tilde{Q}+k(u-w_0)\right]\delta (x,y)\delta wdydx}}\right\}}dt\end{array} \\ \tilde{Q}=(Q_1^{x={\epsilon }_1/2}-Q_1^{x=-{\epsilon }_1/2}){\epsilon }_2+(Q_2^{y={\epsilon }_2/2}-Q_1^{y=-{\epsilon }_2/2}){\epsilon }_1,\hspace{1em}{\epsilon }_1,{\epsilon }_2\approx 0 \end{gathered}$$

(11)

It is important to note that the discontinuity solely arises in the internal transverse shear forces, and this discontinuity is mathematically represented using a two-dimensional Dirac function. As a result, the governing equations Equation (12) for the base plate can be derived based on this representation.

$$-\rho h\ddot{w}+M_{1xx}+M_{2yy}+2M_{6xy}+\left[\tilde{Q}+k_2(w_2-w_0)\right]\delta x,y)=0$$

(12)

Integrating Equation (7) and yields:

$$\begin{array}{l}0={\displaystyle {\int }_{-a}^a{\displaystyle {\int }_{-b}^b\{-\rho h\ddot{w}+M_{1xx}+M_{2yy}+2M_{6xy}+\left[\tilde{Q}+k_2(w_2-w_0)\right]\delta x,y)\}}}dxdy\\ ={\displaystyle {\int }_{-a}^a{\displaystyle {\int }_{-b}^b(-\rho h\ddot{w})}}dxdy+{{\displaystyle {\int }_{-a}^a(M_{2y}+M_{6x})}}_{y=-b}^{y=b}dx+{{\displaystyle {\int }_{-b}^b(M_{1x}+M_{6y})}}_{x=-a}^{x=a}dy+k(u-w_0)\end{array}$$

(13)

It should be noted that the loads $\tilde{Q}$ have been removed in the process of integration. According to Equation (13), the base plate can be considered as a rigid body undergoing accelerated motion, while experiencing transverse shear forces and a focused force acting on its four boundaries.

The control equation for m₂ can be formulated as Equation (14) through the application of Newton’s second law of motion.

$$m_2{\ddot{w}}_2+k_2{w}_2-k_2{w}_0=0$$

(14)

Illustrated in Figure 2b, when an infinite plate comprising repetitive unit cells is impacted by a harmonic wave of a single frequency, the displacements of both the plate $w_1(x,y,t)$ and the vibration dampener $w_2(t)$ can be represented as Equation (15).

$$\begin{array}{l}{w}_1=A{e}^{(\alpha x+\beta y-\omega t)}\\ w_2=B{e}^{-j\omega t}\end{array}$$

(15)

In this context, A and B stand for the magnitudes of vibrational oscillation, $\alpha$ and $\beta$ represent the spatial frequencies along the x and y axes, respectively, while ω indicates the frequency of the wave. By incorporating Equation (15) into Equations (13) and (14), the resultant expressions can be recast in a matrix notation.

$$\left[\begin{array}{cc}{\displaystyle \frac{\sin (a\alpha )\sin (b\beta )}{\alpha \beta }}\left[4\rho h{\omega }^2-4D{({\alpha }^2+{\beta }^2)}^2\right]-k_2& k_2\\ k_2& m_2{\omega }^2-k_2\end{array}\right]\left\{\begin{array}{l}A\\ B\end{array}\right\}=0$$

(16)

Equation (16) poses an eigenvalue challenge. By equating the determinant of the matrix within Equation (16) to zero, we can establish a connection between the wave numbers $\alpha$, $\beta$ and the frequency $\omega$ of the wave [3].

$$\{{\displaystyle \frac{\sin (a\alpha )\sin (b\beta )}{\alpha \beta }}\left[4\rho h{\omega }^2-4D{({\alpha }^2+{\beta }^2)}^2\right]-k_2\}(m_2{\omega }^2-k_2)-k_2^2=0$$

(17)

4. Stopband of the Negative Stiffness Plate

In the interdisciplinary field of structural dynamics and materials engineering, the negative effective stiffness property revealed by Equation (17) opens up a novel design approach for engineers, particularly in the development of negative stiffness plates with unique vibration suppression capabilities. Negative stiffness, this unconventional mechanical characteristic, when cleverly incorporated into structural design, can significantly reshape the dynamic behavior of structures. Notably, when the effective stiffness exhibits negative characteristics, the propagation mode of elastic waves within the structure undergoes a fundamental transformation. In this context, the phase velocity of the elastic waves (i.e., the advancement rate of the wave front) may run counter to the direction of the group velocity (representing the average rate of energy transmission), a trait that is rarely observed in conventional material systems.

Furthermore, when the frequency of external excitation approaches the natural resonant frequency of the mass-spring subsystem embedded within the structure (serving as an additional component), the subsystem triggers intense resonant phenomena. This resonant effect not only generates an effective damping force against external excitation, thereby reducing the dynamic stress borne by the structure, but also effectively disperses and dissipates the vibration energy within the base structure. This mechanism results in a significant suppression of elastic wave propagation within a specific frequency range (which we refer to as the “stop band”), preventing it from transmitting smoothly as in other frequency ranges. Therefore, by finely tuning the parameters of the mass-spring subsystem, we can precisely manipulate the location and width of the stop band, thereby achieving detailed control over the vibration characteristics of the structure.

Assuming a positive real-valued frequency, Equation (17) can be resolved to determine equilibrium conditions. Referencing Figure 1, the unit cell dimensions are set at 2a = 0.2 m, 2b = 0.2 m, and h = 10 mm, with a Young’s modulus of E = 69 GPa. Additional parameters such as the Poisson’s ratio $\nu =0.33$, mass density $\rho =2800 \mathrm{kg}/{\mathrm{m}}^3$, attached mass m2 = 0.1 kg, and the natural frequency ${\omega }_2=30 \mathrm{Hz}$ are also considered. By computing, the relationship between wave number $\alpha$, $\beta$ and wave frequency $\omega$ is obtained. As depicted in Figure 3, a stop band emerges between 30 Hz and 32.86 Hz, within which wave propagation is prohibited. This stop band aligns to the right of the local resonance frequency, attributed to the local vibration absorber’s ability to generate phase-shifted inertial forces, effectively absorbing vibrational energy.

5. Frequency Response Analysis of Negative Stiffness Plate

5.1. Frequency Response Analysis

A crucial utilization of negative stiffness plates involves mitigating and dampening the impact of destructive waves that propagate due to events such as earthquakes, winds, noise, or explosions. To showcase the effectiveness of our novel frequency-responsive vibration absorber, we have crafted a negative stiffness plate with precisely tailored dimensions and material characteristics, as depicted in Figure 4. The dimensions of the plate are La = 2 m along the X-axis, Lb = 3 m along the Y-axis, and h = 10 mm in thickness. The material properties include a Young’s modulus of E = 69 GPa, a Poisson’s ratio $\nu =0.33$, and a mass density $\rho =2800 \mathrm{kg}/{\mathrm{m}}^3$. Additionally, the plate has attached masses of m2 = 0.1 kg, a natural frequency ${\omega }_2=30 \mathrm{Hz}$, and a damping ratio $\epsilon =(0.001-0.01)$.

As shown in Figure 5, If the damping of the absorber $\epsilon =0.001$, excitation force $F_0=1\times {\displaystyle \sum _{i=0}^{200}\sin ({\omega }_{i}t+\beta )}N$ at $x=10 \mathrm{mm}$, ${\omega }_i=0.1\times i \mathrm{Hz}$, The variables $\beta$ are uniform random numbers within the range of [−1, 1], it is shown in Figure 5a. Then, the frequency response function (FRF) of excitation force can be predicted by frequency response analysis (see Figure 5b), the displacement of the plate can be calculated based on the FE Model (see Figure 5c–e). Furthermore, if the damping of the absorber $\epsilon =0.01$, through the application of finite element (FE) modeling and frequency response analysis, the frequency response function (FRF) of displacement can be accurately determined, as depicted in Figure 5f. Upon analyzing the FRF, it becomes apparent that augmenting the damping coefficients of the absorbers results in a broadening of the stopband region and a notable decrease in the amplitudes of vibrations.

5.2. Harmonic Excitation

To validate the effectiveness of the negative stiffness plate, it is necessary to evaluate its dynamic responses under various excitation force frequencies, both within and outside the stopbands. For $\epsilon =0.001$, excitation force $F_0=\mathrm{10,000}\sin (\omega t+\beta )\mathrm{kN}$ at $x=10 \mathrm{mm}$, and time step $\Delta t=0.005 s$, Figure 5 present the transient responses of the plate under excitation frequencies of 20 Hz, 30 Hz, and 40 Hz, respectively. Additionally, Figure 6b,c depict the results obtained from harmonic excitations, showing the scenarios where the frequency is beneath and above the local resonance frequency of the vibration absorber. In these cases, the absorbers are unable to effectively stop wave propagation due to the mismatch between the excitation frequency and the resonant frequency of the absorbers. Conversely, Figure 6a demonstrates the results for harmonic excitation $\omega =30 \mathrm{Hz}$ at a frequency within the stopband. In this scenario, the vibration absorbers are able to absorb the structural vibration, effectively mitigating its impact. To further investigate the effect of damping on vibration mitigation, if the damping of the absorber $\epsilon =0.01$, Figure 7 present the transient responses of the plate under the same excitation frequencies (20 Hz, 30 Hz, and 40 Hz) but with increased damping in the absorbers. By comparing Figure 6a with Figure 7a, it becomes evident that increasing the damping of the absorbers can reduce their efficiency in mitigating vibrations.

6. Earthquake Excitation

The El-centro wave is selected as seismic excitations, as shown in Figure 8a. The spectral characteristics of its seismic waves are shown in Figure 8b. Because the seismic load is not a single frequency of harmonic excitation, so how to design the vibration frequency of the substructure of the negative stiffness plate is the key to the structural shock absorption.

According to previous research [3], when the resonance frequency of the substructure is set close to the main frequency of the main structure, it can minimize the structural vibration caused by earthquakes. As shown in

Table 1. Ratio of the relative displacement reduction.

Displacement	D1, (m2 = 0)	D2, (m2 ≠ 0)	(D1 − D2)/D1, Ratio
Dmax	7.85	5.39	31.34%
Davg	0.89	0.72	19.10%

and Figure 9, this article compares the vibration of negative stiffness plates with and without substructures and finds that compared to ordinary plates, the overall average displacement of negative stiffness plates under seismic action is reduced by 31.34%, and the maximum displacement is reduced by 19.10%.

7. Conclusions

The preceding information provides a comprehensive overview of the working mechanism and modeling technique of the negative stiffness plate. Based on this analysis, several key conclusions can be drawn:

The negative stiffness plate operates on the principle of conventional metamaterial plates with attached mass-spring-damping subsystems. These subsystems generate locally resonant vibrations that produce inertial forces that counteract external loads, effectively inhibiting elastic wave propagation in the host plate.

For a single-stopband negative stiffness plate, a stopband emerges near the resonant frequency of the unit cell. Increasing the damping of the resonators within the plate can slightly broaden this stopband and effectively reduce the response amplitude of the host plate, enhancing its vibration-damping capabilities.

When subjected to seismic waves with a broad distribution of external excitation frequencies, the negative stiffness plate demonstrates significant potential for reducing structural vibration. Specifically, when the natural frequency of the substructure aligns with the vibration frequency of the main structure, the plate can significantly mitigate the vibration caused by earthquakes.