Nonlinear elastic waves in a chain type of metastructure: theoretical analysis and parametric optimization

Abstract

Nonlinear elastic wave propagations in metamaterials and metastructures exhibit much richer dynamic behaviors in comparison with the linear cases. This paper studies the generation of superharmonic wave in a chain type of multi-cell aperiodic structure with nonlinear stiffness. An analytical approach, integrated with the first-order perturbation method and the harmonic balance method, is developed to describe nonlinear wave response with spectrum of higher-order harmonics. The cubic nonlinearity brings the amplitude-dependent bandgap and the nonreciprocal wave phenomenon with the combination of aperiodicity in arrangement, which are presented via both analytical approach and numerical computation. Further, the study introduces a new strategy of parametric optimization on the distribution of multiple local resonances, based on the genetic algorithm method. The optimized metastructures exhibit noteworthy wave properties, including the broadband nonlinear wave suppression, the tunability of wave attenuation on transmissibility level, and the unidirectional wave transmission, by designing different optimization functions in the developed optimization scheme. This work provides an idea to elucidate nonlinear wave responses caused by nonlinear stiffness and to explore wave properties via parametric optimization.

Appendices

Appendix A: Detailed derivation of variables in Sect. 2.2

The intermediate variables in Eq. (12) are given as

$$ L_{11} = \frac{{\overline{v}_{1,s}^{(0)} }}{2}e^{{i\beta_{s} }} e^{i\omega t} + \varepsilon \left( {\frac{{\overline{v}_{1,s}^{(1)} }}{2}e^{{i\varphi_{s} }} e^{i\omega t} + \frac{{9\overline{v}_{3,s}^{(1)} }}{2}e^{{i\psi_{s} }} e^{i \cdot 3\omega t} } \right), $$

(A.1)

$$ L_{12} = \left( {\frac{{\overline{v}_{1,s}^{(0)} }}{2}e^{{i\beta_{s} }} - \frac{{\overline{u}_{1,s}^{(0)} }}{2}e^{{i\alpha_{s} }} } \right)e^{i\omega t} + \varepsilon \left[ {\left( {\frac{{\overline{v}_{1,s}^{(1)} }}{2}e^{{i\varphi_{s} }} - \frac{{\overline{u}_{1,s}^{(1)} }}{2}e^{{i\phi_{s} }} } \right)e^{i\omega t} + \left( {\frac{{\overline{v}_{3,s}^{(1)} }}{2}e^{{i\psi_{s} }} - \frac{{\overline{u}_{3,s}^{(1)} }}{2}e^{{i\theta_{s} }} } \right)e^{i \cdot 3\omega t} } \right], $$

(A.2)

$$ L_{13} = \left( {\frac{{\overline{v}_{1,s}^{(0)} }}{2}e^{{i\beta_{s} }} - \frac{{\overline{u}_{1,s}^{(0)} }}{2}e^{{i\alpha_{s} }} } \right)e^{i\omega t} + \varepsilon \left[ {\left( {\frac{{\overline{v}_{1,s}^{(1)} }}{2}e^{{i\varphi_{s} }} - \frac{{\overline{u}_{1,s}^{(1)} }}{2}e^{{i\phi_{s} }} } \right)e^{i\omega t} + 3\left( {\frac{{\overline{v}_{3,s}^{(1)} }}{2}e^{{i\psi_{s} }} - \frac{{\overline{u}_{3,s}^{(1)} }}{2}e^{{i\theta_{s} }} } \right)e^{i \cdot 3\omega t} } \right]. $$

(A.3)

The intermediate variables in Eq. (13) are given as

$$ L_{21} = \frac{{\overline{u}_{1,N + 1}^{(0)} }}{2}e^{{i\alpha_{N + 1} }} e^{i\omega t} + \varepsilon \left( {\frac{{\overline{u}_{1,N + 1}^{(1)} }}{2}e^{{i\phi_{N + 1} }} e^{i\omega t} + \frac{{9\overline{u}_{3,N + 1}^{(1)} }}{2}e^{{i\theta_{N + 1} }} e^{i \cdot 3\omega t} } \right), $$

(A.4)

$$ \begin{aligned} L_{22} = & \left( {\frac{{\overline{u}_{1,N + 1}^{(0)} }}{2}e^{{i\alpha_{N + 1} }} - \frac{{\overline{u}_{1,N}^{(0)} }}{2}e^{{i\alpha_{N} }} } \right)e^{i\omega t} \\ & + \varepsilon \left[ {\left( {\frac{{\overline{u}_{1,N + 1}^{(1)} }}{2}e^{{i\phi_{N + 1} }} - \frac{{\overline{u}_{1,N}^{(1)} }}{2}e^{{i\phi_{N} }} } \right)e^{i\omega t} + \left( {\frac{{\overline{u}_{3,N + 1}^{(1)} }}{2}e^{{i\theta_{N + 1} }} - \frac{{\overline{u}_{3,N}^{(1)} }}{2}e^{{i\theta_{N} }} } \right)e^{i \cdot 3\omega t} } \right], \\ \end{aligned} $$

(A.5)

$$ \begin{aligned} L_{23} = & \left( {\frac{{\overline{u}_{1,N + 1}^{(0)} }}{2}e^{{i\alpha_{N + 1} }} - \frac{{\overline{u}_{1,N}^{(0)} }}{2}e^{{i\alpha_{N} }} } \right)e^{i\omega t} \\ & + \varepsilon \left[ {\left( {\frac{{\overline{u}_{1,N + 1}^{(1)} }}{2}e^{{i\phi_{N + 1} }} - \frac{{\overline{u}_{1,N}^{(1)} }}{2}e^{{i\phi_{N} }} } \right)e^{i\omega t} + 3\left( {\frac{{\overline{u}_{3,N + 1}^{(1)} }}{2}e^{{i\theta_{N + 1} }} - \frac{{\overline{u}_{3,N}^{(1)} }}{2}e^{{i\theta_{N} }} } \right)e^{i \cdot 3\omega t} } \right]. \\ \end{aligned} $$

(A.6)

The intermediate variables in Eq. (14) are given as

$$ L_{31} = \frac{{\overline{u}_{1,s}^{(0)} }}{2}e^{{i\alpha_{s} }} e^{i\omega t} + \varepsilon \left( {\frac{{\overline{u}_{1,s}^{(1)} }}{2}e^{{i\phi_{s} }} e^{i\omega t} + \frac{{9\overline{u}_{3,s}^{(1)} }}{2}e^{{i\theta_{s} }} e^{i \cdot 3\omega t} } \right), $$

(A.7)

$$ L_{32} = \left( {\frac{{\overline{u}_{1,s}^{(0)} }}{2}e^{{i\alpha_{s} }} - \frac{{\overline{v}_{1,s}^{(0)} }}{2}e^{{i\beta_{s} }} } \right)e^{i\omega t} + \varepsilon \left[ {\left( {\frac{{\overline{u}_{1,s}^{(1)} }}{2}e^{{i\phi_{s} }} - \frac{{\overline{v}_{1,s}^{(1)} }}{2}e^{{i\varphi_{s} }} } \right)e^{i\omega t} + \left( {\frac{{\overline{u}_{3,s}^{(1)} }}{2}e^{{i\theta_{s} }} - \frac{{\overline{v}_{3,s}^{(1)} }}{2}e^{{i\psi_{s} }} } \right)e^{i \cdot 3\omega t} } \right], $$

(A.8)

$$ L_{33} = \left( {\frac{{\overline{u}_{1,s}^{(0)} }}{2}e^{{i\alpha_{s} }} - \frac{{\overline{v}_{1,s}^{(0)} }}{2}e^{{i\beta_{s} }} } \right)e^{i\omega t} + \varepsilon \left[ {\left( {\frac{{\overline{u}_{1,s}^{(1)} }}{2}e^{{i\phi_{s} }} - \frac{{\overline{v}_{1,s}^{(1)} }}{2}e^{{i\varphi_{s} }} } \right)e^{i\omega t} + 3\left( {\frac{{\overline{u}_{3,s}^{(1)} }}{2}e^{{i\theta_{s} }} - \frac{{\overline{v}_{3,s}^{(1)} }}{2}e^{{i\psi_{s} }} } \right)e^{i \cdot 3\omega t} } \right], $$

(A.9)

$$ \begin{aligned} L_{34} = & \left( {\overline{u}_{1,s}^{(0)} e^{{i\alpha_{s} }} - \frac{{\overline{u}_{1,s - 1}^{(0)} }}{2}e^{{i\alpha_{s - 1} }} - \frac{{\overline{u}_{1,s + 1}^{(0)} }}{2}e^{{i\alpha_{s + 1} }} } \right)e^{i\omega t} \\ & + \varepsilon \left[ {\left( {\overline{u}_{1,s}^{(1)} e^{{i\phi_{s} }} - \frac{{\overline{u}_{1,s - 1}^{(1)} }}{2}e^{{i\phi_{s - 1} }} - \frac{{\overline{u}_{1,s + 1}^{(1)} }}{2}e^{{i\phi_{s + 1} }} } \right)e^{i\omega t} + \left( {\overline{u}_{3,s}^{(1)} e^{{i\theta_{s} }} - \frac{{\overline{u}_{3,s - 1}^{(1)} }}{2}e^{{i\theta_{s - 1} }} - \frac{{\overline{u}_{3,s + 1}^{(1)} }}{2}e^{{i\theta_{s + 1} }} } \right)e^{i \cdot 3\omega t} } \right], \\ \end{aligned} $$

(A.10)

$$ \begin{aligned} L_{35} = & \left( {\overline{u}_{1,s}^{(0)} e^{{i\alpha_{s} }} - \frac{{\overline{u}_{1,s - 1}^{(0)} }}{2}e^{{i\alpha_{s - 1} }} - \frac{{\overline{u}_{1,s + 1}^{(0)} }}{2}e^{{i\alpha_{s + 1} }} } \right)e^{i\omega t} \\ & + \varepsilon \left[ {\left( {\overline{u}_{1,s}^{(1)} e^{{i\phi_{s} }} - \frac{{\overline{u}_{1,s - 1}^{(1)} }}{2}e^{{i\phi_{s - 1} }} - \frac{{\overline{u}_{1,s + 1}^{(1)} }}{2}e^{{i\phi_{s + 1} }} } \right)e^{i\omega t} + 3\left( {\overline{u}_{3,s}^{(1)} e^{{i\theta_{s} }} - \frac{{\overline{u}_{3,s - 1}^{(1)} }}{2}e^{{i\theta_{s - 1} }} - \frac{{\overline{u}_{3,s + 1}^{(1)} }}{2}e^{{i\theta_{s + 1} }} } \right)e^{i \cdot 3\omega t} } \right], \\ \end{aligned} $$

(A.11)

$$ \begin{aligned} L_{36} = & \left( {\frac{{\overline{u}_{1,s}^{(0)} }}{2}e^{{i\alpha_{s} }} - \frac{{\overline{u}_{1,s - 1}^{(0)} }}{2}e^{{i\alpha_{s - 1} }} } \right)e^{i\omega t} \\ & + \varepsilon \left[ {\left( {\frac{{\overline{u}_{1,s}^{(1)} }}{2}e^{{i\phi_{s} }} - \frac{{\overline{u}_{1,s - 1}^{(1)} }}{2}e^{{i\phi_{s - 1} }} } \right)e^{i\omega t} + \left( {\frac{{\overline{u}_{3,s}^{(1)} }}{2}e^{{i\theta_{s} }} - \frac{{\overline{u}_{3,s - 1}^{(1)} }}{2}e^{{i\theta_{s - 1} }} } \right)e^{i \cdot 3\omega t} } \right], \\ \end{aligned} $$

(A.12)

$$ \begin{aligned} L_{37} = & \left( {\frac{{\overline{u}_{1,s}^{(0)} }}{2}e^{{i\alpha_{s} }} - \frac{{\overline{u}_{1,s + 1}^{(0)} }}{2}e^{{i\alpha_{s + 1} }} } \right)e^{i\omega t} \\ & + \varepsilon \left[ {\left( {\frac{{\overline{u}_{1,s}^{(1)} }}{2}e^{{i\phi_{s} }} - \frac{{\overline{u}_{1,s + 1}^{(1)} }}{2}e^{{i\phi_{s + 1} }} } \right)e^{i\omega t} + \left( {\frac{{\overline{u}_{3,s}^{(1)} }}{2}e^{{i\theta_{s} }} - \frac{{\overline{u}_{3,s + 1}^{(1)} }}{2}e^{{i\theta_{s + 1} }} } \right)e^{i \cdot 3\omega t} } \right]. \\ \end{aligned} $$

(A.13)

Appendix B: Fitting curves of third harmonic amplitude with respect to the excitation amplitude $a_{0}$.

Since it is reasonable to recognize the relationship between $3\omega$ amplitude and excitation amplitude $a_{0}$ as a cubic polynomial by expanding Eq. (34) in the analysis, this appendix presents the fitting curves of higher harmonic responses in the numerical computation. The coefficients of fitting cubic polynomial denote as $\left[ {\begin{array}{*{20}c} {B_{0} } & {B_{1} } & {B_{2} } & {B_{3} } \\ \end{array} } \right]$, with the subscript standing for the order of each term in the polynomial. For simplicity, the selection of system parameters remains the same in Table 1 while the vector of inner mass ratios in the aperiodic system is identical to that in Table 2. The excitation frequency is selected as 15 Hz and small parameter is given as $\varepsilon = 0.1$. Figure 

Fig. 14

Fitting curves of third harmonic amplitude with respect to the excitation amplitude $a_{0}$ in the case of: a aperiodicity; b periodicity and c a single cell

14a presents the fitting curve of data with R-squared almost equal to 1, representing the good performance of cubic polynomial fitting. Table

Table 6 Polynomial coefficients of fitting cubic curves for different types of system

6 shows the corresponding polynomial coefficients, predicting the rapid growing of third harmonics with an increase of excitation amplitude $a_{0}$.

In addition, this appendix also carries out the fitting on the data of periodic system and single cell, as shown in Fig. 14b and c. The periodic inner mass is selected based on the equality of gross inner masses in comparison to the aperiodic system, as shown in Table 6. The good results of polynomial fitting validate the similar relationship between $3\omega$ amplitude and excitation amplitude $a_{0}$ for the periodic system as well as the single cell when introducing cubic nonlinearity. However, the differences of the obtained polynomial coefficients in three example cases confirm the influence of unit-cell arrangements on the higher harmonics, especially when increasing input amplitude to a larger extent. In that case, the value of $B_{3}$ dominates the amplitude of third harmonics, but would not be considered in the current work due to strong nonlinearity.