Abnormal Stop Band Behavior Induced by Rotational Resonance in Flexural Metamaterial

This paper investigates abnormal stop band behavior of resonance-based flexural elastic metamaterials under the rotational resonance motion. Due to the unique physics of flexural waves, we found that the stop band generated by the rotational resonance motion exhibits peculiar behavior which are quite different from general belief – it is shown that the negativity due to the rotational resonance does not provide any stop bands and the stop band generation due to the rotational resonance is governed by totally different band gap condition. To explain the peculiar behavior, a discrete Timoshenko beam model with both effective mass and rotational inertia as independent variables is introduced, and the wave behaviors of resonance-based flexural elastic metamaterial are precisely and fully described. The unique band gap condition, including the peculiar behavior, is derived with numerical validations. We expect our new model can provide a strong background for various flexural elastic metamaterials which can be effectively applied in various vibration devices.

A. Analytic investigation for the discrete flexural metamaterials without inner resonators. Figure S1. Equivalent mass-spring system for general flexural wave, two typical wave dispersion curves of flexural wave.
According to the classical Timoshenko theory [S1], the flexural wave can be viewed as a coupled dynamic behavior of the vertical and rotational motions. Thus, the mass-spring system used for longitudinal or shear wave cannot be used to describe the flexural wave. Here, we introduce the mass-spring system in Fig. S1 which can well-describe the general flexural wave [S2]. As can be seen in Fig. S1, each mass has two degrees of freedom, the vertical displacement n u and the rotational displacement n  . Accordingly, two kinds of springs are used to connect each mass, the spring  that transfers the vertical force, and the rotational spring  that transfers the rotational moment. Note that the x-directional displacement is ignored since the x-directional motion is related to the longitudinal elastic wave, which is not covered by this paper.
First, based on the mass-spring system shown in Fig. S1, the wave dispersion equation for -3-flexural metamaterials without inner resonators can be analytically derived. As can be seen in Fig. S1, each mass has two degrees of freedom, the vertical displacement n u and the rotational displacement n  . Accordingly, two kinds of springs are used to connect each mass, the spring  that transfers the vertical force, and the rotational spring  that transfers the rotational moment.
Here, a special attention should be paid for the rotational motion of each mass, n  . Due to the rotational motion n  , the vertical displacement of the n th mass is not same as n u . Instead, the vertical displacement of the n th mass can be written as The left side of the n th mass: 2 The right side of the n th mass: 2 Accordingly, the total vertical force acting on the n th mass is Also, considering that the forces acting on the left and right side of the n th mass also generate moment, the total moment acting on the n th mass is which is the same equations with equations (1a, b) and (7) used in our main manuscript.
In fact, the above procedures are exactly same as the classical Timoshenko theory [S1]. To clarify this point, the spring and mass coefficients  ,  , m , and I in Equations (S6, 7) are replaced by the equivalent coefficients in a continuum medium. For a bending beam, the spring coefficients are known as [S2]: I is the shear modulus, Young's modulus, the cross-sectional area -5-and the cross-sectional momentum of inertia, respectively. Also,  is the shear correction factor. In the same manner, m , and I can be written as; Note that for a continuum medium, the periodicity a is extremely small and it can be assumed which is exactly same as the dispersion equation of the classical Timoshenko beam theory [S1].
Therefore, the analytic procedure carried out here is only valid for low frequency ranges. At high frequencies, the flexural wave should be considered as the anti-symmetric guided Lamb wave mode, which cannot be described with the proposed discrete system.
The left side of the inner n th mass: The right side of the inner n th mass: Based on Equation (S11), the total forces and moments acting on each mass can be written as For the outer n th mass: For the inner n th mass: (S13d) Again, by Floquet-Bloch condition, Re-arranging Equations (S16a, b) by assembling the mass and inertia terms yields 2 22 1 2 22 which is the same equations with Equations (5a, b) used in our main manuscript. First, let us start from the well-known negative mass case shown in Fig. S3 (b). Assume that the incident wave forces the unit cell to move along the upward direction. To form the band gap, the internal resonator's motion should compensate this upward motion so that the unit cell would not move. Thus, the internal resonator should move downward -the out-of-phase motion is required. Considering that the negative mass is due to the large out-of-phase motion of the internal resonator, this explains why the negative mass generates band gap. Now, assume that the incident wave forces the unit cell to rotate along the clockwise direction. This indicates that the upper part of the unit cell moves along the positive horizontal direction, while the lower part, moves along the negative horizontal direction. To compensate this motion, the internal resonator should exhibit elongation at the upper side and compression at the lower side, as shown in Fig. S3 (c). Therefore, the internal resonator should also rotate along the clockwise direction, i.e., the in-phase motion is required to form the band gap. This explains why the negative rotational inertia does not generate the band gap; the negative rotational inertia can be achieved when the out-of-phase motion takes place. , the periodic structure exhibit the rotational standing wave shown in Fig. S4 (a), forming the Bragg gap. Thus, if 2 I is larger than 4 , stop band is formed due to the Bragg scattering. However, the periodic structure can also have another rotational standing wave shown in Fig. S4