Band gaps and dynamics of locally resonant meta-plate with stiffness adjustable low frequency resonator

Abstract

This paper presents the band gaps and dynamics of locally resonant meta-plate with stiffness micro-adjustable high static and low dynamic resonator, in which a Euler-buckled beam provides negative stiffness. The resonator is periodically installed on a thin plate through a threaded linkage. By changing the axial displacement of the buckled beam, the total stiffness of the resonator is adjusted. Through the static analysis, the critical axial compression required for the buckled beam to provide negative stiffness is discussed. With the help of classical thin plate theory, the dynamic equation of the system is obtained. The band gaps of the locally resonant meta-plate are calculated by employing the plane wave expansion method. Considering the simply supported boundary condition, dynamic responses of the finite size locally resonant plate are shown with the aid of numerical simulation. The effects of the axial compression of the buckled beam, the mass of resonator, the damping of oscillators and plates, and spring stiffness of the system on the band gap are analyzed in detail.

Keywords

Locally resonant meta-plate band gaps adjustable stiffness buckled straight beam dynamic response

1. Introduction

As a universal physical phenomenon, vibration widely exists in aerospace, precision instruments, transportation, civil engineering, mechanical engineering, and other fields. A severe vibration of the structure will not only reduce the performance, efficiency, and accuracy of the equipment but also endanger one’s physical and mental health, particularly in the low frequency vibration accompanied by the longer wavelength and the slower energy attenuation in the transmission process.

Vibration isolation is quite a significant and meaningful issue in many engineering fields. In a variety of the vibration isolators, the vibration isolators with the good low-frequency vibration isolation properties have been becoming the focus of the researchers’ attention and have aroused their interests (Carrella et al., 2012; Liu et al., 2020; Zhou et al., 2015). The main ways to achieve a low-frequency vibration isolator are parallel connection of linear vertical spring and diagonal springs, cam spring mechanism, electromagnet, Euler-buckling beam, etc. (Benjamin et al., 2014; Carrella et al., 2009; Hao and Cao, 2015; Kovacic et al., 2008; Lan et al., 2014; Xu et al., 2014).

Among them, Liu et al. (2020) designed an isolation system which consisted of a vertical spring and two lateral springs with three damping generation mechanisms caused by the revolution joints. Zhang et al. (2021) investigated the dynamics of a quasi-zero stiffness structure, in which a vertical spring and a damper formed a positive stiffness system, while two horizontal springs connected by a mass block were used as a negative stiffness system. Gatti (2020, 2022) presented an isolation device including a nonlinear negative stiffness system.Liu et al. (2013) proposed a QZS system through coupling the vertical spring and four piezoelectric buckled beams to suppress the vibration and harvest the energy. Fulcher et al. (2014) took advantage of the negative stiffness property and bi-stability to construct a quasi-zero stiffness vibration and shock isolation system. The dynamic behavior of this system was researched experimentally. Zhou and Liu (2010) developed a semi-active vibration isolator by magnetic spring. Carrella et al. (2009) studied the force transmissibility of a quasi-zero stiffness device including pre-stressed and nonlinear oblique springs. In References. Ding et al. (2019); Hao et al. (2017); Huang et al. (2014b); Sun et al. (2014); Yang et al. (2018); Zhang et al. (2020a), various quasi-zero stiffness isolators were designed and applied in different situations.

In several vibration isolation researches, Euler-buckled beams were applied as a negative stiffness system. Liu et al. (2013) presented a vibration isolator that was composed of Euler-buckled beams negative stiffness corrector and a spring. Huang et al. (2014a) studied the vibration isolation of the single degree of freedom nonlinear isolator including an Euler-buckling beam. Virgin and Davis (2003) presented an axially loaded post-buckled struts vibration isolation system.

Recently, locally resonant metamaterials have generated great interest in vibration isolation (Hou and Assouar, 2008; Lu et al., 2020; Ma et al., 2014; Psarobas et al., 2000; Wang et al., 2021), and have been an emerging domain. In terms of acoustic metamaterials, the Bragg band gaps are formed due to the multiple scattering of the unit cells that are periodically distributed in the propagation medium, as well as the interaction between them (Sharma and Sun, 2016). The wavelength of the Bragg band gaps is the same order as the dimension of the unit cells (Peng and Pai, 2014). Conversely, locally resonant band gaps are generated under the interaction of elastic wave and unit cells (Lu et al., 2020; Sharma and Sun, 2016). Using the locally resonant band gap can achieve the low frequency isolation, and the performance of it can be improved easily, especially for those involving the locally resonant resonators with low dynamic stiffness.

There have been several researches on the locally resonant isolation devices. Among them, Peng and Pai (2014) designed an acoustic meta-plate with two isotropic face sheets and a periodically distributed mass-spring-damper resonator core. Zhang et al. (2020b) designed a small-scale tunable elastic local resonance meta-plate with two negative stiffness and lightweight. Muhammad et al. (2021) presented a pillared-plate structure on the basis of trampoline metamaterial. The influences of the various parameters on the band gaps were studied. Al Ba’ba’a et al. (2017) studied the mechanism of the locally resonant band gap by using transfer functions. Wang et al. (2020) realized a meta-beam by connecting the low-frequency semi-active resonators onto a beam periodically.

It can be seen that the traditional local resonator can’t overcome the contradiction between low dynamic stiffness and high static stiffness, it is difficult to further reduce the position of the local resonant band gap further. Consequently, metamaterial beams and plates with the low dynamic stiffness resonators have come into focus. Zhou et al. (2017a, 2017b) proposed a new type of high static and low dynamic stiffness resonator mechanical structure by the vertical and inclined springs. Wu et al. (2019) proposed a meta-beam with low-frequency multimode resonators with quasi-zero stiffness and the theoretical models. Wang (2019) studied the band gaps of meta-plates with high static and low dynamic stiffness cell elements.

The contribution of this work is to propose a locally resonant meta-plate with stiffness micro-adjustable high static and low dynamic resonator, in which the negative stiffness is provided by a buckled beam. Simultaneously, the band gap and displacement transmissibility are studied. The rest of the paper is organized as follows: in Section 2, a stiffness micro-adjustable low frequency resonator is shown, and the dynamic equation and dispersion equation of locally resonant meta-plate are derived by using classical thin plate theory and plane wave expansion method, respectively. In Section 3, the correctness of the numerical calculation method is verified. Then the effects of the various parameters on the band gap and the dynamic performance are analyzed. Section 4 presents the conclusions.

2. Theoretical formulation

2.1. Low frequency resonator with adjustable stiffness

A stiffness micro-adjustable low frequency resonator is depicted in Figure 1. Connected to the lightweight frame is an Euler-buckled-straight-beam with a linear spring and a rigid mass $m$ in the middle. Two stoppers can move along a dovetail groove rail using a threaded rod with screw threads, but the rotation direction is reversed at both ends. The two bearing seats serve not only to support the threaded rod but also to restrict its movement. The buckled straight beam is connected to a linear spring labeled as $k_{1}$ , while the dissipative effects are represented by a linear damper in the vertical direction. The buckled beam provides negative stiffness, which is achieved by adjusting the pre-compression of the buckled beam. The vertical stiffness of the spring is constant; therefore, the resonator’s stiffness can be adjusted to any value within a specific stiffness interval. Assumption that the mass can only move in the vertical direction, we consider a buckled straight beam with a cross-sectional area $A$ , an undeformed beam length $l_{0}$ , and a compressed length $l_{c}$ when in a buckling state under axial pressure $P$ . Here, $h$ and $b$ present the cross sectional thickness and width of the beam, respectively.

The strain energy caused by the axial compressive force $P$ is quantified as

U_{c} = \frac{1}{2} \frac{A E_{c}}{l_{0}} {u_{x}}^{2},

(1)

where

E_{c}

represents Young’s modulus, and

u_{x}

is the total axial displacement, which each end of the beam is compressed by

u_{x} / 2

. The origin is located at the left end of the free beam.

To ensure that only the first-order mode occurs in the clamped buckled beam when it is subjected to a force in the vertical direction, the range of axial load values should be as follows (Du, 2017)

P_{c r 1} = \frac{4 π^{2} E_{c} I}{l_{0}^{2}} < P < P_{c r 2} = \frac{{(2.86 π)}^{2} E_{c} I}{l_{0}^{2}} .

(2)

Additionally, the axial displacement of the beam, $u_{x}$ , should satisfy the following relationship

u_{x, c r 1} = \frac{4 π^{2} I}{A l_{0}} < u_{x} < u_{x, c r 2} = \frac{{(2.86 π)}^{2} I}{A l_{0}} .

(3)

Consider a vertical external force acting on the buckled straight beam shown in Figure 2. The shape of the beam in the first buckling mode can be described as follows

Z (x) = \frac{u_{z}}{2} (1 - \cos (\frac{2 π (x - \frac{u_{x}}{2})}{l_{c}})),

(4)

The bending deformation energy of the buckled straight beam due to the vertical external force is given by

U_{b} = \frac{E_{c} I}{2} \int_{\frac{u_{x}}{2}}^{\frac{u_{x}}{2} + l_{c}} {(\frac{d^{2} Z (x)}{d x^{2}})}^{2} d x = \frac{π^{4} E_{c} I u_{z}^{2}}{l_{c}^{3}},

(5)

where

I

denotes the moment of inertia, and it is defined as

I = b h^{3} / 12

. Thus, the vertical external force

F_{t}

can be determined using the following formula

F_{t} = \frac{\partial U}{\partial u_{z}} = \frac{\partial}{\partial u_{z}} (Δ U_{c} + U_{b}) = 2 π^{4} E_{c} I (\frac{u_{z}}{l_{c}^{3}} - \frac{A u_{z} u_{x}}{4 l_{0} l_{c} I π^{2}} + \frac{A u_{z}^{3}}{16 l_{0} l_{c}^{3} I})

(6)

where

Δ U c = 1 / 2 A E_{c} / l_{0} {(u_{x} - {u_{z}}^{2} π^{2} / 4 l_{c})}^{2}

stands for the strain energy of the buckled beam caused by the axial compressive force after bending deformation. Figure 3 displays curves of

F_{t}

against axial compressive displacement of a beam with Young’s modulus

E_{c} = 9.5 \times 10^{9} Pa

, width

b = 0.01 m

, thickness

h = 0.003 m

, and free length

l_{0} = 0.06 m

, respectively. The observation can be expressed as follows when

u_{x} \geq 0.6 mm

, the beam can exhibit negative stiffness. It's worth noting, as indicated in equation (3), that to ensure the first order buckling of the beam, the value of

u_{x}

changes from 0.6 mm to 1.0 mm. The values of the Euler-buckling beam stiffness and the net stiffness of the resonator are presented in Table 1 for various values of

u_{x}

, with the parallel spring having a constant stiffness of

k_{1} = 3 \times 10^{5} N / m

. Specifically, when there is no axial compression (

u_{x} = 0

), the beam is equivalent to a structure with positive stiffness.

Table 1.

Stiffness of the Euler buckling beam and net stiffness of the resonator $(10^{5} N / m)$ .

$u_{x} (m m)$	0	0.6	0.7	0.8	0.9	1.0
Stiffness of the Euler-buckling beam	1.927	−0.381	−0.77	−1.160	−1.553	−1.945
Net stiffness	4.927	2.619	2.230	1.840	1.447	1.055

Furthermore, the stiffness of the Euler-buckling beam with two fixed ends can be obtained by

k_{b} = 2 π^{4} E_{c} I (\frac{1}{l_{c}^{3}} - \frac{A u_{x}}{4 l_{0} l_{c} I π^{2}} + \frac{3 A u_{z}^{2}}{16 l_{0} l_{c}^{3} I}) .

(7)

When a vertical spring is connected in parallel to the buckled beam, as shown in Figure 1, the force–displacements relationship can be rewritten as

F = 2 π^{4} E_{c} I (\frac{u_{z}}{l_{c}^{3}} - \frac{A u_{z} u_{x}}{4 l_{0} l_{c} I π^{2}} + \frac{A u_{z}^{3}}{16 l_{0} l_{c}^{3} I}) + k_{1} u_{z} .

(8)

In summary, the stiffness of the resonator can be altered by adjusting the axial compression of the beam, allowing us to design the necessary micro-adjustable low frequency resonator.

2.2. Locally resonant meta-plate

Figure 4(a) shows a locally resonant meta-plate, which is constructed by periodically attaching low frequency resonators with micro-adjustable stiffness using bolted joints. Because of the translational periodicity and symmetry of the metamaterial structure, the structure has irreducible Brillouin zones. The first irreducible Brillouin zone (Li and Li, 1997; Wang et al., 2018a) of the structure is shown in Figure 4(b). The coordinates of the three corners are $(0,0)$ , $(2 π / a_{1}, 0)$ , and $(2 π / a_{1}, 2 π / a_{2})$ , respectively, where $a_{1}$ and $a_{2}$ are the lattice constants of the periodic structures in $x$ and $y$ directions, respectively. The vector coordinates of an arbitrary resonator are as follows

R = m_{1} a_{1} + m_{2} a_{2},

(9)

where

m_{1}

and

m_{2}

are integers, and

a_{1} = {a_{1}, 0}

and

a_{2} = {0, a_{2}}

are base vectors.

2.3. Dispersion equation of locally resonant meta-plate

Considering influence of bending waves only and using thin plate theory, the governing equation for free vibrations of the present locally resonant meta-plate can be simplified to (Chen, 2020; Qian and Wang, 2022; Wang et al., 2018b; Xiao et al.,2012a, 2012b)

\frac{E_{b} h_{b}^{3}}{12 (1 - ν^{2})} (\frac{\partial^{4}}{x^{4}} + 2 \frac{\partial^{4}}{\partial x^{2} \partial y^{2}} + \frac{\partial^{4}}{y^{4}}) w (r, t) + ρ h_{b} \frac{\partial^{2}}{\partial t^{2}} w (r, t) = \sum_{R} [f_{R} (R, t) δ (r - R)],

(10a)

m \ddot{z} (R, t) + f_{R} (R, t) = 0,

(10b)

f_{R} (R, t) = - k_{t} [w (R, t) - z (R, t)] - c [\dot{w} (R, t) - \dot{z} (R, t)],

(10c)

where

E_{b}

h_{b}

υ

, and

w

are Young’s modulus, thickness, Poisson’s ratio, and transverse displacement of the plate, respectively.

z (R)

and

c

denote the displacement of the mass and coefficient of viscosity of the unit cell, respectively.

f_{R}

is the vertical force applied to the plate by the spring at the location

R

k_{t} = k_{1} + k_{b},

(11)

where the three variables have been explained earlier.

The 2D Dirac delta function is defined as $δ (r - R) = δ (x - R_{x}) δ (y - R_{y})$ . For the harmonic oscillation of the system, equation (10) becomes

D \nabla^{4} w (r) - ω^{2} ρ h_{b} w (r) = \sum_{R} f_{R} (R) δ (r - R),

(12a)

- ω^{2} m z (R) = f_{R} (R),

(12b)

f_{R} (R, t) = - k_{t} [w (R, t) - z (R, t)] - c [\dot{w} (R, t) - \dot{z} (R, t)],

(12c)

where

D = \frac{E_{b} h_{b}^{3}}{12 (1 - ν^{2})}

ω

is the frequency of planar bending wave and

\nabla^{4} = {(\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}})}^{2}

For the locally resonant meta-plate, the transverse displacement can be described by

w (r) = \sum_{G} W_{G} e^{- i (k + G) r},

(13)

where

k = (k_{x}, k_{y})

is the wave vector,

G

is the 2D reciprocal-lattice vector that can be expanded as

G = n_{1} b_{1} + n_{2} b_{2}

, where both

n_{1}

and

n_{2}

are integers. For a square lattice we have

a_{i} b_{j} = {\begin{cases} 2 π, i = j \\ 0, i \neq j \end{cases} .

(14)

It can also be observed that $w (0) = \sum_{G} W_{G}$ . In addition, since the local resonators are periodically distributed on the plate, the following hold

w (R) = w (0) e^{- i kR},

(15a)

z (R) = z (0) e^{- i kR} .

(15b)

The Dirac delta function has the following properties

\sum_{R} e^{- i kR} δ (r - R) = e^{- i kr} \sum_{R} δ (r - R),

(16a)

\sum_{R} δ (r - R) = \frac{1}{s} \sum_{G} e^{- i Gr},

(16b)

where

s

is area of the unit cell.

Substituting equations (13), (15), and (16) into equation (12) yields

D S {[{(k + G)}_{x}^{2} + {(k + G)}_{y}^{2}]}^{2} W_{G} - ω^{2} S ρ h_{b} W_{G} = - (k_{t} + i c ω) [\sum_{G} W_{G} - z (0)],

(17a)

- ω^{2} m z (0) = (k_{t} + i c ω) [\sum_{G} W_{G} - z (0)],

(17b)

where subscripts

x

and

y

are the two directions of wave propagation, respectively. By truncating the infinite series between (−N, N), the total number of plane waves to be considered is

{(2 N + 1)}^{2}

. For any wave vector, once

ω

is specified, the real energy band diagram can be plotted.

2.4. Dynamic system of locally resonant meta-plate

The dynamic system of a locally resonant meta-plate with $J$ resonators exited by a point harmonic load is given by

{\begin{cases} D \nabla^{4} w (r, t) - ρ h \frac{\partial^{2}}{{\partial t}^{2}} w (r, t) = f_{0} (t) δ (r - r_{0}) + \sum_{R} [f_{R} (R, t) δ (r - R)] \\ m \ddot{z} (R, t) + f_{R} (R, t) = 0 \end{cases},

(18)

where

f_{0} (t) δ (r - r_{0})

indicates that the harmonic excitation applied at location

r_{0}

in meta-plate. For solving the dynamic response, the transverse displacement of the host plate be written as the following finite sum of mode shapes

w (r, t) = \sum_{p = 1}^{P} \sum_{q = 1}^{Q} w_{p q} (t) w_{p q} (r) = \sum_{p = 1}^{P} \sum_{q = 1}^{Q} w_{p q} (t) ϕ_{p q} (x, y),

(19)

where

P

and

Q

are the numbers of the terms in the series, and

ϕ_{p q} (x, y)

is a mode shape. For a simply supported meta-plate with

J

resonators, the mode shapes are assumed to be

ϕ_{p q} (x, y) = \sin (\frac{p π}{a} x) \sin (\frac{q π}{b} y),

(20)

Substituting equations (19) and (20) into equation (18), utilizing the orthogonality of modal function gives

[M] [\begin{array}{l} \ddot{W} (t) \\ \ddot{Z} (t) \end{array}] + [C] [\begin{array}{l} \dot{W} (t) \\ \dot{Z} (t) \end{array}] + [K] [\begin{array}{l} W (t) \\ Z (t) \end{array}], = [\begin{array}{l} [P_{f}] \\ {[0]}_{J} \end{array}] f_{0} (t)

(21)

where

[W (t)] = {[W_{11} (t), W_{12} (t), W_{13} (t), . . ., W_{P Q} (t)]}^{T},

[Z (t)] = {[Z_{1} (t), Z_{2} (t), . . ., Z_{J} (t)]}^{T},

[P_{f}] = \int_{s} [ϕ_{p q} (x, y)] δ (x - x_{0}) δ (y - y_{0}) d x d y,

(22)

and

[M]

[C]

, and

[K]

are given in (Yang et al., 2022) as

[M] = (\begin{array}{l} \frac{ρ h a b}{4} I_{P \times Q} & 0 \\ 0 & m I_{J \times J} \end{array}),

[C] = (\begin{array}{l} [C_{p}] + [C_{r}] & - [C_{pr}] \\ - [C_{r p}] & [C_{r r}] \end{array}), [K] = (\begin{array}{l} [K_{p}] + [K_{r}] & - [K_{pr}] \\ - [K_{r p}] & [K_{r r}] \end{array})

(23)

Furthermore

[C_{p}] = \frac{ρ h a b}{4} 2 ξ_{p q} [β_{p q}] I_{P \times Q}, [C_{r r}] = 2 ξ_{d} \sqrt{m k_{t}} I_{J \times J}, [C_{r}] = 2 ξ_{d} \sqrt{m k_{t}} \sum_{n = 1}^{J} [ϕ (x_{n}, y_{n})] {[ϕ (x_{n}, y_{n})]}^{T},

[C_{pr}] = {[C_{r p}]}^{T} = 2 ξ_{d} \sqrt{m k_{t}} [[ϕ (x_{1}, y_{1})], . . ., [ϕ (x_{J}, y_{J})]],

[K_{p}] = \frac{ρ h a b}{4} [β_{p q}^{2}] I_{P \times Q}, [K_{r r}] = k_{t} I_{J \times J},

[K_{r}] = k_{t} \sum_{n = 1}^{J} [ϕ (x_{n}, y_{n})] {[ϕ (x_{n}, y_{n})]}^{T},

[K_{pr}] = {[K_{r p}]}^{T} = k_{t} [[ϕ (x_{1}, y_{1})], . . ., [ϕ (x_{Q}, y_{Q})]]

(24)

where

β_{p q} = (\frac{p^{2} π^{2}}{a_{2}} + \frac{q^{2} π^{2}}{b_{2}}) \sqrt{\frac{D}{ρ h}}

, and

ζ_{p q}

and

ζ_{d}

are the damping ratios of the plate and a resonator, respectively. Consider harmonica excitation, equation (21) can be rewritten as follows

[[K] - ω^{2} [M] + i ω [C]] [\begin{array}{l} [W] \\ [Z] \end{array}] = [\begin{array}{l} f_{0} [P_{f}] \\ {[0]}_{Q} \end{array}] .

(25)

From above equation, the steady state dynamic response of the system can be obtained.

3. Results and discussion

3.1. Verification of numerical calculation

To verify the accuracy of the theoretical band gap calculation, consider the locally resonant plate proposed by Wang in Reference. Wang et al. (2019) as an example of the band gap calculation. The parameters of the unit cell are as follows: the lattice constant 0.1 m, thickness 0.002 m, Young’s Modulus $70 \times 10^{9} Pa$ , density $2700 kg / m^{3}$ and Poisson’s ratio 0.3, respectively. A resonator with the mass of 0.027 kg and the spring stiffness of 9.593 × 10⁴ N/m is used. The obtained band gaps are shown in Figure 5. It can be seen that the calculated band gaps are in good agreement with those in the literature. This comparison appears the validity of present computational process, and ensures the accuracy and reliability of the results.

3.2. Tunable band-gap of the meta-plate

In this subsection, the influences of axial displacement

u_{x}

on the tunable band gap of the meta-plate are calculated and studied by numerical simulation of equation (17). The values of

m_{1}

and

m_{2}

have a range of [−5, 5], thus there will be

{(2 L + 1)}^{2} = 121

plane waves. Unless otherwise stated, the values for the parameters used in this calculation are reported in Table 2.

Table 2.

Structural parameters of the locally resonant meta-plate.

Parameters	Meaning	Value
$E_{b}$	Young’s modulus of thin plate	70 × 10⁹ Pa
$h_{b}$	Thickness of the thin plate	1 × 10⁻3 m
$ρ$	Density of rectangular	2700 kg/m³
$υ$	Poisson’s ratio of thin plate	0.3 m
$a$	Square lattice constant	0.1 m
$m$	Mass of the resonator	0.1 kg
$k_{1}$	Vertical spring stiffness	3 × 10⁵ N/m
$ζ_{d}$	Damping ratio of the resonator	0
$ζ_{p q}$	Damping ratios of thin plate	0
$f_{R}$	Point excitation amplitude	1 N

For the sake of convenience, damping is not considered temporarily when the plane wave expansion method is used to calculate the theoretical band gap. Ignoring the mass of the lightweight frame, the band structures is shown in Figure 6, where areas filled with light blue indicate band gaps, and capital letters $Γ - Χ - M - Γ$ on the horizontal axis denote the direction of wave propagation in the first irreducible Brillouin zone. In Figure 6(a), the band structures being calculated in the case of $u_{x} = 0$ are presented. Figure 6(b) shows the band structures of the system with the first order buckling beam. Keep in mind that if the waves can’t propagate in any direction within this band gap, it is called a complete band gap. In this case, no matter which direction the bending wave propagates, the plate will not vibrate. From this, “BG-E” is the complete band gap, while “BG-D1” is the directional band gap corresponding $Γ - Χ$ . Furthermore, as observed from Figure 6 that with the increase of $u_{x}$ , the complete band gap BG-E is narrowing and moving downward. This is because the stiffness of the resonator is decreasing gradually, which reveals that complete band gap BG-E is a locally resonant band gap. The directional band gap “BG-D1” is also narrowing due to the fact that the frequency corresponding to the point g₃ is a fixed value related to the Bragg band gap, however, the end frequency of “BG-D1” is decreasing. What is more intriguing is that the upper and lower limits of the complete band gap are determined by the values of g₂ and g₁ after the beam is buckled, respectively. Generally, by making small adjustment to $u_{x}$ , different band gaps can be achieved.

In Figure 7, four cases with different $u_{x}$ are utilized to analyze the influence of the mass of the resonator on the band gap of the meta-plate. Figure 7(a) illustrates how the varying mass leads to a varying of the complete band gap of the meta-plate with the value of $u_{x}$ fixed at 0.0 mm. With the increases of mass, an intersection point emerges between $g_{3}$ and $g_{2}$ at m = 0.15, as in point A. The result is that before the point A the complete band gap is determined by the values of $g_{1}$ and $g_{3}$ , after that the band gap will depend on $g_{1}$ and $g_{2}$ . When considering the other three values of $u_{x}$ (0.6, 0.8 and 1.0) which make the buckled beam shows negative stiffness, the solution of $g_{3}$ always has the same value, and is unaffected by the mass, the complete band gaps are determined by the values of $g_{1}$ and $g_{2}$ . This shows that the formation of the directional band gap is independent of the locally resonant mechanism but is related to the Brag dispersion of the periodic structures. It can be seen from Figure 7(b) that with the increase of mass, both $g_{1}$ and $g_{2}$ shift towards lower frequency, and the band width is gradually widens.

3.3. Displacement transmissibility of the meta-plate

The foregoing study fails to reflect the bending wave propagation characteristics of the system. In this subsection, the bending wave transmissibility of the system under harmonic point excitation ( $f_{0} = f_{r} \sin (ω t) δ (r - 0)$ , $ω = 0 - 1000 Hz$ ) at the origin is analyzed. The displacement transmissibility is applied to estimate the attenuation characteristic of the bending wave. Defining the displacement transmissibility as

ξ = 20 \log \frac{| w_{1} |}{| w_{2} |},

(26)

where

w_{1}

and

w_{2}

denote the transverse displacements of the excitation test point (0.1, 0.1) and response test point (0.7, 0.7), respectively, as shown in Figure 8.

Figure 9 compares the displacement transmissibility of the system with and without buckled straight-beam resonators for the different $u_{x}$ values, in which the red solid line represents the displacement transmissibility of the plate without resonators, while the blue solid line denotes the displacement transmissibility of the plate with resonator. As expected, it can be seen that the band gap forms within a specific frequency region for the meta-plate equipped with resonators. These band gaps are in good agreement with those obtained via the plane wave expansion method. Moreover, as $u_{x}$ increases, the band gap shifts towards the lower frequency region and narrows.

Figure 10 shows the influence of the damping ratio of the resonator on the band gap of the meta-plate when the compression $u_{x} = 1.0$ . It can be seen that the amplitude of the displacement transmission curve within the band gap decreases with the increase of $ξ_{d}$ , which indicates that the attenuation ability of the locally resonant plate to bending wave is weakening. Meanwhile, the lower bound of the band gaps goes down, while the upper bound of band gaps increases. This is because that the damping ratio of the resonator causes the natural frequency of the resonator to reduce. On the other hand, damping will consume energy and make the vibration response less violent. Thus, the upper band gap limit increases, and the role of damping in vibration reduction is fully reflected. Figure 11 shows the influence of the damping ratio of plate on the band gap. It is observed that the damping of the plate has little effect on the band gap.

To investigate the influence of the stiffness of the parallel spring on the band gap, the maximum negative stiffness that the buckled beam can provide $1.945 \times 10^{5} N / m$ is selected. The stiffness level of the spring are evaluated as $1.95 \times 10^{5} N / m$ , $2.1 \times 10^{5} N / m$ , $2.25 \times 10^{5} N / m$ and $2.45 \times 10^{5} N / m$ , respectively. Figure 12 presents the band gap and displacement transmissibility. One can see that increasing stiffness of the spring represents increasing band-gap width and its location.

4. Conclusion

In this paper, a stiffness micro-adjustable resonator structure composing an Euler-buckling straight beam, linear spring, mass block and movable stoppers is introduced into the locally resonant meta-plate. The statics of the resonator is studied firstly. Subsequently, the effects of the axial compression of the beam, the stiffness of the parallel spring, the mass of the resonator and the damping of the system on the band gap and dynamic response are studied.

When the stiffness of the parallel spring is slightly greater than the negative stiffness provided by the buckling beam, the resonator is close to quasi-zero stiffness, resulting in an ultra-low frequency band range. Increasing the axial compression, the band gap broadband decreases, and the starting cutoff frequency of the band gap is approaching the low frequency. Moreover, as the mass of the resonator increases, the range of the band gap increases with the starting and ending frequencies move to the low frequency. The damping of the plate has no effect on the band gap range, while the damping of the resonator can widen the width of bang dap.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors acknowledge the financial support of National Natural Science Foundation of China through grant nos. 12272056, 11832002, the Tianjin Natural Science Foundation grant number 19JCZDJC32300.

ORCID iDs

Y Xu

YX Hao

References

Al Ba’ba’a

Nouh

Singh

(2017) Formation of local resonance band gaps in finite acoustic metamaterials: a closed-form transfer function model. Journal of Sound and Vibration 410: 429–446.

Fulcher

Shahan

Haberman

, et al. (2014) Analytical and experimental investigation of buckled beams as negative stiffness elements for passive vibration and shock isolation systems. Journal of Vibration and Acoustics 136(3): 1–12.

Carrella

Brennan

Kovacic

, et al. (2009) On the force transmissibility of a vibration isolator with quasi-zero-stiffness. Journal of Sound and Vibration 322(4–5): 707–717.

Carrella

Brennan

Waters

, et al. (2012) Force and displacement transmissibility of a nonlinear isolator with high-static-low-dynamic-stiffness. International Journal of Mechanical Sciences 55: 22–29.

Chen

(2020) Research on Low-Frequency Vibration Reduction Characteristics of Locally Resonant Plate. Master’s Dissertation, East China University of Science and Technology, Shanghai, China.

Ding

Chen

(2019) Nonlinear vibration isolation for fluid-conveying pipes using quasi-zero stiffness characteristics. Mechanical Systems and Signal Processing 121: 675–688.

(2017) Low Frequency Broadband Lightweight Vibration Reduction Design of Local Resonance Beam. Master’s Dissertation, National University of Defense Science and Technology, Changsha, China.

Fulcher

Shahan

Haberman

Gatti

(2020) Statics and dynamics of a nonlinear oscillator with quasi-zero stiffness behaviour for large deflections. Communications in Nonlinear Science and Numerical Simulation 83: 105143.

10.

Gatti

Shaw

Gonçalves

PJP

, et al. (2022) On the detailed design of a quasi-zero stiffness device to assist in the realisation of a translational lanchester damper. Mechanical Systems and Signal Processing 164: 108258.

11.

Hao

Cao

(2015) The isolation characteristics of an archetypal dynamical model with stable-quasi-zero-stiffness. Journal of Sound and Vibration 340: 61–79.

12.

Hao

Cao

Wiercigroch

(2017) Nonlinear dynamics of the quasi-zero-stiffness SD oscillator based upon the local and global bifurcation analyses. Nonlinear Dynamics 87(2): 987–1014.

13.

Hou

Assouar

(2008) Modeling of Lamb wave propagation in plate with two-dimensional phononic crystal layer coated on uniform substrate using plane-wave-expansion method. Physics Letters A 372(12): 2091–2097.

14.

Huang

Liu

Sun

, et al. (2014a) Effect of the system imperfections on the dynamic response of a high-static-low-dynamic stiffness vibration isolator. Nonlinear Dynamics 76(2): 1157–1167.

15.

Huang

Liu

Sun

, et al. (2014b) Vibration isolation characteristics of a nonlinear isolator using euler buckled beam as negative stiffness corrector: a theoretical and experimental study. Journal of Sound and Vibration 333: 1132–1148.

16.

Kovacic

Brennan

Waters

(2008) A study of a nonlinear vibration isolator with a quasi-zero stiffness characteristic. Journal of Sound and Vibration 315(3): 700–711.

17.

Lan

Yang

(2014) Design and experiment of a compact quasi-zero-stiffness isolator capable of a wide range of loads. Journal of Sound and Vibration 333(20): 4843–4858.

18.

. (1997) Brillouin zone and inverted lattice primitive cell. College Physics 16(1): 29–31.

19.

Liu

Huang

Hua

(2013) On the characteristics of a quasi-zero stiffness isolator using euler buckled beam as negative stiffness corrector. Journal of Sound and Vibration 332: 3359–3376.

20.

Liu

Tang

(2020) New insights into the damping characteristics of a typical quasi-zero-stiffness vibration isolator. International Journal of Non-linear Mechanics 124: 103511.

21.

Zhou

Gao

, et al. (2020) Flexural vibration bandgaps of the multiple local resonance elastic metamaterial plates with irregular resonators. Applied Acoustics 159: 107115.

22.

Hou

Assouar

(2014) Opening a large full phononic bandgap in thin elastic plate with resonant units. Journal of Applied Physics 115(9): 093508.

23.

Muhammad Hussain

Lim

(2021) Composite trampoline metamaterial with enlarged local resonance bandgap. Applied Acoustics 184: 108353.

24.

Peng

Pai

(2014) Acoustic metamaterial plates for elastic wave absorption and structural vibration suppression. International Journal of Mechanical Sciences 89: 350–361.

25.

Psarobas

Stefanou

Modinos

(2000) Phononic crystals with planar defects. Physical Review B 62(9): 5536–5540.

26.

Qian

Wang

(2022) Studies of a new-style resonator to control electro-mechanical coupling bandgap of a locally resonant piezoelectric/elastic phononic crystal double-layer nonlocal nanobeam. Applied Mathematical Modelling 102: 786–796.

27.

Sharma

Sun

(2016) Local resonance and bragg bandgaps in sandwich beams containing periodically inserted resonators. Journal of Sound and Vibration 364: 133–146.

28.

Sun

Jing

, et al. (2014) A quasi-zero-stiffness-based sensor system in vibration measurement. IEEE Transactions on Industrial Electronics 61(10): 5606–5614.

29.

Virgin

Davis

(2003) Vibration isolation using buckled struts. Journal of Sound and Vibration 260: 965–973.

30.

Wang

(2019) Low frequency band gap mechanism and wave characteristics of quasi zero stiffness local resonance superstructure. Doctoral Dissertation, Hunan University, Changsha, China.

31.

Wang

(2018a) Micro-vibration isolation performance of a clamped-clamped buckled beam quasi-zero-stiffness isolator. Journal of Vibration and Shock 37(15): 124–129.

32.

Wang

Gong

Cheng

, et al. (2018b) Analytic inverted lattice space and wave vector space. Journal of Bengbu University 7(5): 72–75.

33.

Wang

Zhou

Cai

, et al. (2019) Mathematical modeling and analysis of a meta-plate for very low-frequency band gap. Applied Mathematical Modelling 73: 581–597.

34.

Wang

Zhou

Ouyang

, et al. (2020) A semi-active metamaterial beam with electromagnetic quasi-zero-stiffness resonators for ultralow-frequency band gap tuning. International Journal of Mechanical Sciences 176: 105548.

35.

Wang

Zhang

, et al. (2021) Bandgap properties in metamaterial sandwich plate with periodically embedded plate-type resonators. Mechanical Systems and Signal Processing 151: 107375.

36.

Huang

Liu

, et al. (2019) Low-frequency multi-mode vibration suppression of a metastructure beam with two-stage high-static-low-dynamic stiffness oscillators. Acta Mechanica 230(12): 4341–4356.

37.

Xiao

Wen

(2012a) Broadband locally resonant beams containing multiple periodic arrays of attached resonators. Physics Letters A 376: 1384–1390.

38.

Xiao

Wen

(2012b) Flexural wave band gaps in locally resonant thin plates with periodically attached spring–mass resonators. Journal of Physics D: Applied Physics 45: 195401.

39.

Zhang

Zhou

, et al. (2014) On the analytical and experimental assessment of the performance of a quasi-zero-stiffness isolator. Journal of Vibration and Control 20(15): 2314–2325.

40.

Yang

Zheng

, et al. (2018) Structural design and isolation characteristic analysis of new quasi-zero-stiffness. Journal of Vibration Engineering & Technologies 8: 47–58.

41.

Yang

Chen

Guo

, et al. (2022) Analytical solution of band gap in an attached locally resonant phononic crystal thin plate. Journal of Vibration and Shock 41(8): 238–243.

42.

Zhang

Xia

, et al. (2020a) A torsion-translational vibration isolator with quasi-zero stiffness. Nonlinear Dynamics 99(2): 1467–1488.

43.

Zhang

Gao

(2020b) New mechanism of tunable broadband in local resonance structures. Applied Acoustics 169: 107482.

44.

Zhang

Cao

Huang

(2021) Bursting oscillations in an isolation system with quasi-zero stiffness. Mechanical Systems and Signal Processing 161: 107916.

45.

Zhou

Liu

(2010) Atunable high-static-low-dynamic stiffness vibration isolator. Journal of Sound and Vibration 329: 1254–1273.

46.

Zhou

Wang

, et al. (2015) Nonlinear dynamic characteristics of a quasi-zero stiffness vibration isolator with cam-roller-spring mechanisms. Journal of Sound and Vibration 346: 53–69.

47.

Zhou

Wang

, et al. (2017a) Local resonator with high-static-low-dynamic stiffness for lowering band gaps of flexural wave in beams. Journal of Applied Physics 121(4): 044902.

48.

Zhou

Wang

, et al. (2017b) Multi-low-frequency flexural wave attenuation in euler–bernoulli beams using local resonators containing negative-stiffness mechanisms. Physics Letters A 381(37): 3141–3148.