Fig. 1

Schematics of the diatomic periodic lattices with an interface model (color online)"

Fig. 2

The dispersion relation of periodic lattices, where $ f_{\rm Ⅰ} $ to $ f_{\rm Ⅳ} $ are the boundaries of the acoustic and optical bands, and the parametric values are assumed to be $ \alpha_1=1.5 $, $ \alpha_2=0.5 $, $ \mu_1=1.5 $, $ \mu_2=0.5 $, $ \omega_{\rm e}=10 $, and $ \beta=0.1 $ (color online)"

Fig. 3

Band transition in a 1D lattice system: (a) the band structures for different stiffnesses of connection springs $ \alpha_1 $ and masses of the element mass $ \mu_1 $, (b) the band transition at $ \mu_1=\mu_2 =1 $ with the increasing dimensionless stiffness $ \alpha_1 $, (c) the band transition at $ \alpha_1=\alpha_2 =1 $ with the increasing dimensionless stiffness $ \mu_1 $, and (d) the band transition at $ \mu_1 =1.5 $ and $ \mu_2=0.5 $ with the increasing dimensionless stiffness $ \alpha_1 $ (color online)"

Fig. 4

Trend of the transition point frequencies, where the $ x $-axis represents the parametric excitation frequency, the $ y $-axis represents the transition frequency, and the red points indicate the transition point frequencies corresponding to the parametric excitation frequencies $ \omega_{\rm e} $ of 0, 10, 20, and 30, respectively (color online)"

Table 1

Transition point frequencies at different parametric excitation frequencies"

Table 2

The modified Zak phase on the band boundaries for Case 1"

Table 3

The modified Zak phase on the band boundaries for Case 2"

Table 4

The modified Zak phase on the band boundaries for Cases 3 and 4"

Fig. 5

(a) The frequency response function showing the interface mode within the band gap, when $ \alpha_1=1.5 $, $ \alpha_2=0.5 $, $ \mu_1=1 $, $ \mu_2=1 $, $ \beta=0.1 $, and $ \omega_{\rm e}=20 $. (b) The displacement distribution along the entire topological combination lattices, where the $ x $-axis represents the number of elements based on the global metamaterial structure, and the $ y $-axis denotes the amplitudes of element responses (color online)"

Fig. 6

The frequency response function showing the interface mode within the band gap with varying parametric excitations for Case 1, where the solid lines represent the frequency responses for the CL, the dotted lines denote the frequency responses for the SL, the yellow stars denote the frequency and response of the topological transition points, the wax yellow dotted dash lines denote the transition points of the analytical predictions, and different colored lines denote the frequency responses at the excitation frequencies $ \omega_{\rm e} =0, 10, 20, 30 $, respectively (color online)"

Fig. 7

The frequency response function showing the interface mode within the band gap with varying parametric excitations for Case 2, where the wax yellow dotted-dash lines denote the transition points of the analytical prediction, and different colored lines denote the frequency responses at the excitation frequencies $ \omega_{\rm e} =0 $ and 30, respectively (color online)"

Fig. 8

The frequency response function showing the interface mode within the band gap with varying parametric excitations for Case 3, where the wax yellow dotted-dash lines denote the transition points of the analytical prediction, and different colored lines denote the frequency responses at the excitation frequencies $ \omega_{\rm e}=0 $ and 30, respectively (color online)"

Fig. 9

The frequency response function showing the interface mode within the band gap with varying parametric excitations for Case 4, where the wax yellow dotted-dash lines denote the transition points of the analytical prediction, and different colored lines denote the frequency responses at the excitation frequencies $ \omega_{\rm e}=0 $ and 30, respectively (color online)"

Fig. 10

The frequency response function showing the interface mode within the band gap with varying middle masses and parametric excitations for Case 1, where the wax yellow dotted-dash line denotes the transition point of the analytical predictions, different symbols denote the variable middle mass with $ \mu_{\rm c}=0.5 $, 1, and 1.5, respectively, and different colored lines denote the frequency responses at the excitation frequencies $ \omega_{\rm e}=0 $ and 20, respectively (color online)"

Fig. B1

The schematic of internal force and displacement transfer from left to right ends of the unit cell, where $ F_{\rm L} $ and $ U_{\rm L} $ are the force and displacement on the left, respectively, $ F_{\rm I} $ and $ U_{\rm I} $ are the force and displacement on the middle, respectively, and $ F_{\rm R} $ and $ U_{\rm R} $ are the force and displacement on the right, respectively (color online)"