Abstract The symmetric propagation of topological edge states in two-dimensional (2D) mechanical systems cannot be directly applied to a one-dimensional (1D) system such as a beam. In this work, we adopt symmetry principle of single- and double-sided pillared phononic beams to realize asymmetric transmission of a topological edge state. The design is constituted by assembling a two-sided pillared beam (two symmetry planes) with a one-sided pillared beam (one symmetry plane). With symmetry plane consideration, we find that: 1) in a single-sided pillared beam, the torsional and the shear waves on one hand, and the longitudinal and the flexural waves on the other hand, are mixed together. 2) in a double-sided pillared beam, all the four beam's modes are decoupled from each other so that an appropriate choice of the parameters can produce for each pair (torsional-shear and longitudinal-flexural) a Dirac cone for one of the modes and a bandgap for the other mode in the same frequency range. The Dirac cone is lifted to form a trivial or nontrivial bandgap with band inversion by tuning the interval between pillars, and a topological shear/flexural edge state can be achieved consequently at the interface between two topologically different phases. Asymmetric topological edge state of the torsional/longitudinal incident wave can be achieved by assembling these two beams. It is further demonstrated a high robustness of the topological edge state against the disorder perturbation in pillar's position/frequency and the defects. The asymmetric topological state can be extended to other beam's motions following the symmetry principle and opens opportunities for designing topological solid devices with high performance, such as robust filter.

中文翻译：

---

基于对称原理的弹性梁非对称拓扑态

摘要 二维 (2D) 机械系统中拓扑边缘状态的对称传播不能直接应用于一维 (1D) 系统，例如梁。在这项工作中，我们采用单面和双面柱状声子束的对称原理来实现拓扑边缘态的非对称传输。该设计是由一个两侧立柱梁（两个对称平面）与一个侧立柱梁（一个对称平面）组装而成。考虑对称面，我们发现： 1）在单面柱状梁中，一方面是扭转波和横波，另一方面是纵波和弯曲波。2）在一个双面柱梁中，所有四个梁' s modes are decoupled from each other so that an appropriate choice of the parameters can produce for each pair (torsional-shear and longitudinal-flexural) a Dirac cone for one of the modes and a bandgap for the other mode in the same frequency range. 通过调整柱之间的间隔，狄拉克锥被提升以形成具有带反转的平凡或非平凡带隙，并且因此可以在两个拓扑不同的相之间的界面处实现拓扑剪切/弯曲边缘状态。通过组合这两个光束可以实现扭转/纵向入射波的非对称拓扑边缘状态。进一步证明了拓扑边缘状态对柱子位置/频率和缺陷的无序扰动的高度鲁棒性。