本論文主要研究古典力學石墨烯結構的晶格在非厄米特系統的狀況下，其邊緣態所發生的變化。為了實現非厄米特系統，我們將阻尼考慮進連接晶格的彈簧裡，以此來得到遞迴矩陣的關係式。
首先我們用推導出的遞迴矩陣，模擬出了在無阻尼之下，幾種不同晶格形狀的頻譜圖以及邊緣態。接著分別討論加入阻尼後的系統，以及具有PT對稱 (PT symmertry) 的阻尼系統，發現在給予特定大小範圍的阻尼後，特徵頻率的實部與虛部兩者之間的數量級相同。在此範圍下調整阻尼的大小，觀察到邊緣態被虛部頻譜擾動產生而變化，虛部頻譜輕則影響邊緣態的衰減速度；重則使邊緣態不再衰減而成為簡併態。

This thesis primarily investigates the behavior of edge states in the lattice structure of classical mechanical graphene under non-Hermitian systems. To realize a non-Hermitian system, damping is incorporated into the springs connecting the lattice, allowing us to derive a recursive matrix relation.
First, using the derived recursive matrix, we simulate the spectral diagrams and edge states for various lattice geometries without damping. We then discuss both the system with damping and the system with PT symmetry (parity-time symmetry) damping. It is observed that when the damping is set within a specific range, the real and imaginary parts of the eigenfrequencies become comparable in magnitude. By tuning the damping strength within this range, we observe that the edge states are modified due to perturbations from the imaginary part of the spectrum: in milder cases, the imaginary part affects the decay rate of the edge states; in more severe cases, the edge states no longer decay and instead become degenerate states.

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