Nonlinear Tunability of Elastic Waves in One-Dimensional Mass-Spring Lattices Attached with Local Resonators

Abstract

Vibration propagates in the form of elastic waves. The tuning of elastic waves is of great significance for vibration and noise reduction. The elastic metamaterials (EMs), which can effectively prohibit elastic wave propagation in the band gap frequency range, have been widely studied. However, once the structures of the EMs are determined, the band gap is also determined. In this paper, a discrete nonlinear elastic metamaterial is proposed. The harmonic balance method is used to derive the nonlinear dispersion relation combined with Bloch’s theorem. The low frequency band gap near the linear natural frequency of local resonators is obtained. The theoretical results show that the nonlinearity will change the starting and ending frequencies of the band gap. In addition, amplitude can also influence the band gap. This means that the amplitude can be changed to achieve the tunability of elastic waves in nonlinear elastic metamaterials. Finally, the theoretical results are verified by numerical simulation, and the results are in good agreement with each other.

1. Introduction

In the aerospace field, from unmanned aerial vehicles (UAVs) to manned aircrafts, vibration not only affects the confidentiality of instruments and the comfort of passengers, but also causes hidden dangers for flight safety. Vibration and noise also widely exist in various fields such as mechanical engineering, vehicle engineering and so on. Reducing harmful vibration has always been an urgent problem to be solved in engineering technology [1,2,3]. Phononic crystals (PCs) and elastic metamaterials provide a new way to solve this problem. PCs and EMs are artificially designed periodic structures [4,5,6,7,8,9,10,11,12,13,14]. They possess band gaps (BGs) in which elastic waves are prohibited, while they can propagate in other frequency regions without exhaustion. This makes it possible to achieve wave guiding [15], acoustic filter [16], acoustic cloaking [17], vibration control [18,19], noise reduction [20], acoustic focusing [21,22], non-destructive testing [23], etc. Compared with PCs, EMs can produce band gaps caused by local resonance at lower frequencies.

However, the BGs of EMs are usually fixed once the structures have been designed and fabricated. On the contrary, it is desirable to tune the BGs dynamically or adaptively in most practical applications after they have been fabricated. In recent years, some methods have been proposed to overcome this limitation, such as initial stresses [24,25], piezoelectric materials [26,27], temperature variation [28], external mechanical loads [29] and the space–time modulation elastic metamaterials [30].

Scholars mainly adjust the band gap structure of linear EMs by adjusting their geometric structure and composition. However, in many engineering applications, structures undergo large deformations, such that linear analysis would no longer apply. Nonlinear elastic metamaterials (NEMs) have attracted significant attention recently. Some interesting wave propagation phenomena have been shown in NEMs, such as amplitude-dependent band gaps [31,32,33], bifurcations [34] and chaotic bands [35,36]. These make it possible to tune the BGs by nonlinear tunability. Vakakis and King used standing wave solutions to study the amplitude-dependent attenuation zone of a one-dimensional periodic chain with cubic nonlinearity [37]. Cang He [13] used the string to shift the Bragg and local resonance band gaps. The tension produced by the string is cubic nonlinear. Silva et al. [38] showed that the tunable band gap depended on the amplitude through the metamaterial with the quadratic nonlinear resonators.

In this paper, we proposed a nonlinear elastic metamaterial made of linear mass-spring chains with nonlinear oscillators attached to the masses. Many methods have been developed to study the nonlinear band gap, including multiple timescale approaches [39], perturbation methods [12] and incremental harmonic balance methods [40]. Compared with other methods, the harmonic balance method is suitable for a strong nonlinear system and its solving process is more concise. Hence, we apply the harmonic balance method to calculate the dispersion relation of the proposed structure. A concise amplitude-dependent dispersion relation is obtained. The results show that the mass ratio, linear stiffness ratio, nonlinearity and amplitude can change the position and width of the band gap. This means that when the structure design is completed, the band gap can be changed by changing the amplitude, so that the elastic wave can be tuned.

This paper is organized as follows. In Section 2, we derive the dispersion expressions for one-dimensional linear and nonlinear elastic metamaterials. In Section 3, we discuss the tuning of different parameters on the amplitude-dependent band gap. Numerical simulations are presented in Section 4. The conclusions are drawn in Section 5.

2. Dispersion Relation of One-Dimensional Elastic Metamaterials

2.1. Linear Case

The model of the linear EMs is shown in Figure 1. The unit cells are connected by a spring with a stiffness of $K_1$. In the unit cell, an oscillator with a mass of $m$ is attached to a mass of $M$ by a spring with a stiffness of $K_2$. The distance between the two unit cells a is the so-called lattice constant.

For the n-th unit cell, the motion equations of the mass and the attached oscillator can be written as:

$$M\frac{d^2{u}_n}{d{t}^2}+K_1(2u_n-u_{n-1}-u_{n+1})-K_2(v_n-u_n)=0$$

(1)

$$m\frac{d^2{v}_n}{d{t}^2}+K_2(v_n-u_n)=0$$

(2)

where $u_n$ is the displacement of the n-th mass on the main chain and $v_n$ is the displacement of its additional local resonator, as shown in Figure 1. By introducing the coefficients ${\omega }_0=\sqrt{\frac{K_1}{M}},\alpha =\frac{m}{M}$ and $\beta =\frac{K_2}{K_1}$, Equations (1) and (2) can be written as:

$$\frac{d^2{u}_n}{d{t}^2}+{\omega }_0{}^2(u_n-u_{n-1}-u_{n+1})-\beta {\omega }_0{}^2(v_n-u_n)=0$$

(3)

$$\frac{d^2{v}_n}{d{t}^2}+\frac{\beta }{\alpha }{\omega }_0{}^2(v_n-u_n)=0$$

(4)

Based on Bloch’s theorem, the following equations can be obtained:

$$u_n=C{e}^{i(\omega t-nka)}$$

(5)

$$v_n=D{e}^{i(\omega t-nka)}$$

(6)

where C and D are the amplitudes of $u_n$ and $v_n$, respectively, and $k$ is the wave vector. By combining the equations above, the dispersion relation of the linear EMs can be obtained [41]:

$$cos(ka)=1-\frac{(\frac{\beta }{\alpha }+\beta ){\omega }^2-\frac{{\omega }^4}{{\omega }_0{}^2}}{2(\frac{\beta }{\alpha }{\omega }_0{}^2-{\omega }^2)}$$

(7)

Normalized frequency $\overline{\omega }=\omega /{\omega }_1$ is introduced, where ${\omega }_1=\sqrt{K_2/m}$. Then, Equation (7) can be simplified as:

$$cos(ka)=1-(\frac{\beta }{2\alpha }{\overline{\omega }}^2+\frac{\beta {\overline{\omega }}^2}{2(1-{\overline{\omega }}^2)})$$

(8)

For a given value of $\overline{\omega }$, if the real part of $ka$ exists, the elastic wave can propagate in the structure. The range of $\overline{\omega }$ is the so-called passband and the rest is the bang gap. The imaginary part of $ka$ represents the attenuation characteristic of the band gap. Figure 2 shows the band gap of the linear elastic metamaterial at a mass ratio of $\alpha =0.5$ and stiffness ratio of $\beta =2$. It can be seen that the largest attenuation occurs at the natural frequency of the attached oscillator.

2.2. Nonlinear Case

In this section, we investigate the dispersion relation of the nonlinear EMs as shown in Figure 3. The unit cells are connected by a spring with a stiffness of $K_1$. In the unit cell, an oscillator with a mass of $m$ is attached to a mass of $M$ by a nonlinear spring. The elastic force of the nonlinear spring is expressed as:

$$F=K_{2}x+\gamma K_2{x}^3=K_{2}f(x)$$

(9)

where $x$ is the deformation of the nonlinear spring, $\gamma$ is a parameter controlling the degree of nonlinearity and $f(x)=x+\gamma x^3$. The motion equations of the n-th unit cell and the attached oscillator can be written as:

$$M\frac{d^2{u}_n}{d{t}^2}+K_1(2u_n-u_{n-1}-u_{n+1})-K_{2}f((v_n-u_n))=0$$

(10)

$$m\frac{d^2{v}_n}{d{t}^2}+K_{2}f((v_n-u_n))=0$$

(11)

By introducing $w_n=v_n-u_n$ and the parameters above Equation (3), the motion equations can be rewritten as:

$$\frac{d^2{u}_n}{d{t}^2}+{\omega }_0{}^2(u_n-u_{n-1}-u_{n+1})-\beta {\omega }_0{}^{2}f(w_n)=0$$

(12)

$$\frac{d^2{w}_n}{d{t}^2}+\frac{\beta }{\alpha }{\omega }_0{}^{2}f(w_n)+\frac{d^2{u}_n}{d{t}^2}=0$$

(13)

Here, we adopt the harmonic balance method to solve the nonlinear equations. $w_n(t)$ is assumed to be complex Fourier series:

$$\begin{array}{l}{w}_n(t)={\displaystyle \sum _j{\epsilon }^{(j-1)/2}}{A}_{j,n}{e}^{ij\omega t}+{\epsilon }^{(j-1)/2}{\overline{A}}_{j,n}{e}^{-ij\omega t}\\ j=1,3,5\dots \end{array}$$

(14)

where $\epsilon$ represents a small parameter. By substituting Equation (14) by Equation (13) and integrating twice with respect to time, the time response of mass n can be obtained as follows:

$$\begin{array}{l}{u}_n(t)=A_{1,n}{e}^{i\omega t}(-1+\frac{\beta }{\alpha {\omega }^2}+\frac{\gamma \beta }{\alpha {\omega }^2}(3A_{1,n}{\overline{A}}_{1,n}+\frac{3\epsilon {\overline{A}}_{1,n}^2{A}_{3,n}}{A_{1,n}}))+\\ \epsilon A_{3,n}{e}^{3i\omega t}(-1+\frac{\beta }{9\alpha {\omega }^2}+\frac{\beta \gamma }{9\alpha {\omega }^2}(\frac{A_{1,n}^3}{\epsilon A_{3,n}}+6A_{1,n}{\overline{A}}_{1,n}))+c.c.+O({\epsilon }^2)\end{array}$$

(15)

Based on Bloch’s theorem, the displacement of the mass n can be expressed as:

$$u_n(t)={\displaystyle \sum _j{B}_{n,j}{e}^{-inka}{e}^{ij\omega t}}+c.c.$$

(16)

where $B_{n,j}$ is the coefficient in front of $e^{ij\omega t}$ in Equation (15). We assume the nonlinearities to be small, so the contributions to the solution of no less than third harmonics are neglected. Substituting Equations (14) and (16) by (12), equating the coefficient in front of $e^{ij\omega t}$ to zero, the dispersion relation for the nonlinear EMs is obtained as follows:

$$cos(ka)=1-\frac{{\overline{\omega }}^2\beta }{2\alpha }(1+\frac{\alpha (1+3\gamma A_{1,n}{\overline{A}}_{1,n})}{1+3\gamma A_{1,n}{\overline{A}}_{1,n}-{\overline{\omega }}^2})$$

(17)

It is evident that the dispersion relationship is affected by nonlinear parameters and amplitude. It is the same as the dispersion relation for the linear case in Equation (8), when the nonlinear parameter $\gamma$ is equated to zero.

3. Tuning on the Nonlinear Band Gap

In Equations (8) and (14), let $cos(ka)=1$ or −1, and the starting and ending frequencies of the band gap can be obtained, respectively. For the linear case:

$${\omega }_{l-start}=\sqrt{\frac{(1+\alpha +\frac{4\alpha }{\beta })-\sqrt{{(1+\alpha +\frac{4\alpha }{\beta })}^2-\frac{16\alpha }{\beta }}}{2}}$$

(18)

$${\omega }_{l-end}=\sqrt{1+\alpha }$$

(19)

In the nonlinear case:

$${\omega }_{n-start}=\sqrt{\frac{c-\sqrt{c^2-16\beta (\alpha +3\alpha \gamma A_{1,n}{\overline{A}}_{1,n})}}{2\beta }}$$

(20)

$${\omega }_{n-end}=\sqrt{(1+\alpha )(1+3\gamma A_{1,n}{\overline{A}}_{1,n})}$$

(21)

where $c=4\alpha +\beta +\alpha \beta +3\beta \gamma A_{1,n}{\overline{A}}_{1,n}+3\alpha \beta \gamma A_{1,n}{\overline{A}}_{1,n}$.

Figure 4 shows the effect of mass ratio on the band gap of the linear EMs and Figure 5 shows the nonlinear case. It shows that both the ending and starting frequencies of the stopband increase with the increase in the attached mass, regardless of whether it is linear or nonlinear. Additionally, the width of the band gap also increases. A more detailed comparison is shown in Figure 6. For the nonlinear EMs, the ending frequency of the band gap is about 26% higher than that of the linear EMs, while the starting frequency is slightly increased. This indicates that nonlinearity can significantly enhance the influence of the mass ratio on the band gap.

Figure 7 and Figure 8 show the effect of stiffness ratio $\beta$ on the band gap in the linear and nonlinear EMs, respectively. Figure 9 shows the trends of the starting and ending frequencies with respect to the stiffness ratio. The greater the ratio of linear stiffness ratio $\beta$ is, the greater the band gap is. Different from the mass ratio, the change of stiffness ratio almost does not affect the ending frequency of the band gap, but it will make the starting frequency smaller. Compared with the linear EMs, both the starting and ending frequencies of the nonlinear case are increased, but the width of the band gap is not significantly increased.

Figure 10 shows the effect of nonlinear parameter $\gamma A_{1,n}{\overline{A}}_{1,n}$ on the stopband in the EMs. Figure 11 shows the trends of the starting and ending frequencies with the change in the nonlinear parameter. It can be seen that when the nonlinear parameters increase, the ending frequency of the band gap increases greatly, while the starting frequency increases marginally. As a result, the width of the band gap increases significantly. In addition, the position of the maximum value of the attenuation is shifted above the linear natural frequency.

4. Numerical Simulation

The above solution of the vibration band gap of the one-dimensional mass-spring chain is carried out under the infinite period structure. In this section, the numerical calculation method is adopted to calculate the vibration transmission properties of the mass-spring structure with finite period. A diagram of the finite structure is shown in Figure 12. The transmission properties of the finite lattices are defined as follows:

$$tr=20{\log }_{10}\left|\frac{u_{out}}{u_{in}}\right|$$

(22)

where $u_{out}$ is the response of the output point and $u_{in}$ is the amplitude of the input point. Here, we apply sine sweep displacement excitation, with a frequency band from 1 to 20 Hz and an amplitude $U$ to the first unit cell and measure the response at the n-th cell. Specific parameters are shown in

Table 1. Parameters of finite lattices.

K1	K2	M	m
1000 N/m	2000 N/m	1 kg	0.5 kg

.

Figure 13, Figure 14 and Figure 15 show the effects of unit cell number, amplitude of the displacement excitation and nonlinearity on the transmission properties of the mass-spring chain attached with nonlinear local resonators. Figure 13 shows that the increase in the cell number does not change the band gap width, but the vibration attenuation in the band gap range becomes increasingly larger. Figure 14 and Figure 15 show that the increase in excitation amplitude and nonlinearity have little effect on the lower boundary of the band gap, but will increase the upper boundary. This indicates a good agreement with the theoretical analysis in Section 3. However, in the numerical simulation, the influence of amplitude and nonlinearity on the upper boundary of the band gap is not as great as that in the theoretical analysis. This mainly takes into account the fact that the amplitude in Section 3 is not consistent with the amplitude of displacement excitation. Although the simulation method has disadvantages, the trend of the band gap change caused by nonlinearity and amplitude still coincides with the analytical results.

We also choose two typical frequencies, f = 4 Hz and f = 10 Hz, to represent the corresponding responses in the passband and band gap, respectively. Figure 16 and Figure 17 are time domain and frequency domain diagrams of excitation and response at f = 4 Hz , respectively. Figure 18 and Figure 19 are the case at f = 10 Hz. It can be clearly seen that the elastic wave can propagate without attenuation in the passband, while it delays significantly in the band gap.

5. Conclusions

In this paper, we focus on the nonlinear tunability of elastic waves in one-dimensional mass-spring lattices attached with local resonators. The harmonic balance method is applied to derive the nonlinear dispersion relation and a concise expression of wave vector and normalized frequency is obtained. The results show that the band gap can be adjusted by mass ratio, stiffness ratio, nonlinearity and displacement amplitude. The mass ratio can increase both the upper and lower boundary of the band gap. However, it has little effect on the width of the band gap. The linear stiffness ratio has little effect on the upper boundary of the band gap, but the lower boundary decreases significantly with the increase in the stiffness ratio. This means that the width of the band gap increases. By changing the nonlinearity or displacement amplitude, the bang gap can be shifted and the width of the band gap visibly increases. With the increase in nonlinear parameters, the maximum value of attenuation shifts above the linear natural frequency of local resonators. Then, a numerical simulation is carried out to obtain the transmission properties of the finite mass-spring lattices attached with local resonators. The result shows that the analytical predictions are in good agreement with the simulations. Our research discusses the tuning effect of various parameters on elastic waves in nonlinear EMs in detail, which provides a potential application for vibration reduction.