本文首先討論彈簧力學蜂窩晶格 (honeycomb lattice) 以及正方晶格於不同邊界 條件下的邊界態，並解釋其存在原因。而後加入科氏力變量，破壞時間反演對稱性 (Time Reversal symmetry) 產生手徵邊緣態 (chiral edge state)，探討相關的能帶以及 拓樸相變。最後改變球與彈簧的質量，破壞 PT 對稱，藉此打開帶隙產生谷拓樸邊 緣態 (valley edge state)，討論能帶與拓樸邊界態的變化以並藉由計算貝里曲率 (berry curvature) 觀察整個布里渊區中的局部陳數是否符合預期。最後發現引入旋轉以後對邊 緣態有較強的干預作用，除此之外，外部邊界的選擇也會導致內部所產生的谷拓樸邊 緣態產生不同程度的影響。

This thesis first explores the boundary states of spring-mechanical honeycomb and square lattices under various boundary conditions, explaining the reasons for their ex- istence. Next, the inclusion of the Coriolis force term breaks time-reversal symmetry (TRS), leading to the emergence of chiral edge states. The study examines the associated band structures and topological phase transitions.
Finally, by altering the masses of the balls and springs to break PT symmetry, a band gap is introduced, resulting in the formation of valley topological edge states. The paper discusses the changes in band structures and topological edge states and calculates the Berry curvature to observe whether the local Chern number across the entire Brillouin zone aligns with expectations.
This thesis concludes that the introduction of rotation significantly interferes with the edge states. Furthermore, the selection of external boundaries also affects the topological valley edge states generated within the system, leading to varying degrees of impact.

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