Fig. 2: Dynamics of the conventional chiral PnC.
From: Origin and tuning of bandgap in chiral phononic crystals
a, b Theoretical and numerical transmissions of the conventional chiral PnCs. The gray line is the numerical results, and the others are the theoretical results. Therein, “no \({{{{\boldsymbol{k}}}}}_{{{{\boldsymbol{s}}}}}\)” denotes the results of neglecting the stretch mode; the red and blue lines in (b) are the results of considering the stretch mode, where the word “stiffness” in braces denotes that the result is obtained based on Supplementary Eqs. (17)–(28), and the word “inertial” denotes that the theoretical result is obtained based on Supplementary Eqs. (29)–(39). c Theoretical and numerical dispersion spectra (see Supplementary Note 2 for the governing equation of dispersion spectrum). Therein, the gray star-shaped lines are the numerical dispersion curves and the others refers to the theoretical dispersion curves. The gray shaded area refers to the bandgap range. d The relative amplitudes of the longitudinal displacement. Therein, \({{{{\boldsymbol{u}}}}}_{{{{\boldsymbol{b}}}}}\) and \({{{{\boldsymbol{u}}}}}_{{{{\boldsymbol{s}}}}}\) refer to the relative longitudinal displacement induced by the bending and stretch modes, respectively. The relative amplitude is calculated by dividing the absolute amplitude by the input amplitude. The subscript “b” refers to the bending mode and “s” refers to be the stretch mode. e The relative amplitudes of the rotational displacement. Therein, \({{{{\boldsymbol{R}}}}}_{{{{\boldsymbol{b}}}}}\) and \({{{{\boldsymbol{R}}}}}_{{{{\boldsymbol{s}}}}}\) refer to the relative rotational displacement induced by the bending and stretch modes, respectively. The relative amplitude is calculated by dividing the absolute amplitude by the input amplitude. f Displacement contours of the boundaries of the bandgap, where \({{{{\boldsymbol{p}}}}}_{{{{\boldsymbol{l}}}}{{{\boldsymbol{1}}}}}\), \({{{{\boldsymbol{p}}}}}_{{{{\boldsymbol{l}}}}{{{\boldsymbol{2}}}}}\), \({{{{\boldsymbol{p}}}}}_{{{{\boldsymbol{u}}}}{{{\boldsymbol{1}}}}}\) and \({{{{\boldsymbol{p}}}}}_{{{{\boldsymbol{u}}}}{{{\boldsymbol{2}}}}}\) correspond to the frequency points marked in (b). Therein, \({{{{\boldsymbol{p}}}}}_{{{{\boldsymbol{l}}}}{{{\boldsymbol{1}}}}}\) and \({{{{\boldsymbol{p}}}}}_{{{{\boldsymbol{l}}}}{{{\boldsymbol{2}}}}}\) denote the lower-boundary displacement contours of the bandgap; \({{{{\boldsymbol{p}}}}}_{{{{\boldsymbol{u}}}}{{{\boldsymbol{1}}}}}\) and \({{{{\boldsymbol{p}}}}}_{{{{\boldsymbol{u}}}}{{{\boldsymbol{2}}}}}\) denote the upper-boundary displacement contours of the bandgap.
