| 摘要: | 板殼振動與克拉尼圖在近百年內已被科學家們大量研究與應用，然而在過去的研究
中模擬平板振動特性大多採用 FEM 方法，而對於克拉尼圖的探討大多採用物理實驗，
本研究第一次提出採用雙向耦合離散元素法(DEM)與有限元素法(FEM)，模擬顆粒體在
彈性矩形板上的動態行為，探討顆粒體在不同無因次加速度(Γ)下的顆粒聚集情況，並與
對應實驗比較。研究中採用粒子面積佔有率、粒子平移速度、粒子旋轉速度、粒子擾動
速度及粒子溫度，進一步分析顆粒體在矩形板上運動時的內部物理行為。本研究也有考
慮各種不同參數對顆粒體聚集圖樣造成的影響，包括矩形板有無受重力效應影響、顆粒
楊氏係數及顆粒恢復係數。
本研究結果摘要如下：
(1)不考慮矩形板的重力效應，當無因次加速度(Γ)小於 1 時，顆粒體往腹點聚集，形成
反克拉尼圖，當無因次加速度(Γ)大於等於 1 時，顆粒體往節線聚集，形成克拉尼圖，
且在無因次加速度(Γ)甚大於 1 時，形成克拉尼圖所需時間大幅減少，Γ值越大，圖樣
形成時間越短。
(2)考慮矩形版的重力效應，矩形板受重力影響，易於產生預變形，使顆粒體滾向變形較
大位置，較難形成反克拉尼圖，但當無因次加速度(Γ)值超過某個筏值時，仍會形成克
拉尼圖。
(3)隨著無因次加速度(Γ)的增加，顆粒體平移速度增加速率較快，形成克拉尼圖的時間
較短。
(4)形成反克拉尼圖時，顆粒體會朝波腹滾動，形成克拉尼圖時，顆粒體會朝節線滾動。
(5)形成反克拉尼圖時，顆粒體間碰撞現象較微弱，形成克拉尼圖時，顆粒碰撞現象較劇
烈，且碰撞多集中於腹點，無因次加速度(Γ)越大，碰撞趨勢越劇烈。
關鍵字：克拉尼圖，反克拉尼圖，雙向耦合離散元素法與有限元素法，無因次加速
度，顆粒體內部物理性質
;Vibration of plate structures and the phenomenon of clustering and inversion of Chladni
patterns have been extensively studied by scientists in the past century. However, prior
investigations predominantly employed the Finite Element Method (FEM) to simulate plate
vibration characteristics and relied on physical experiments for chladni patterns. This study
pioneers the application of a bidirectional coupled Discrete Element Method (DEM) and Finite
Element Method (FEM) to simulate the dynamic behavior of particles on an elastic rectangular
plate. The proposed coupled model was validated against corresponding experimental
observations. The aggregation behavior of particles was explored under various dimensionless
accelerations (Γ). Particle area fraction, particle translational velocity, particle rotational
velocity, particle perturbation velocity, and granular temperature are employed to further
analyze the internal physical behavior of particles on the rectangular plate. Various parameters
are considered in this study to understand their impact on the patterns of particle aggregation,
including the influence of gravity on the rectangular plate, particle Young′s modulus, and
particle restitution coefficient. The main findings are summarized below
(1) Disregarding the effect of gravity on the rectangular plate, when the dimensionless
acceleration (Γ) is less than 1, particles aggregate towards the nodal lines, forming an
inverse Chladni patterns. When Γ is greater than or equal to 1, particles aggregate towards
the anti-nodal line, forming a Chladni patterns. Moreover, as Γ significantly exceeds 1, the
time required to form a Chladni patterns substantially decreases. Larger values of Γ reduce
formation time for Chladni patterns.
(2) Considering the gravitational effect on the rectangular plate, its susceptibility to gravity
leads to pre-deformation, causing particles to roll towards areas with higher deformation,
and making it difficult to form an inverse Chladni patterns. However, when the
dimensionless acceleration (Γ) exceeds a certain threshold, a Chladni pattern still emerges.
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(3) As the dimensionless acceleration (Γ) increases, the rate of increase in particle translational
velocity is faster, resulting in a shorter formation time for the Chladni patterns.
(4) When forming the inverse Chladni patterns, particles roll towards the anti-nodal regions,
while when forming the Chladni patterns, particles roll towards the nodal regions.
(5) When forming the inverse Chladni patterns patterns, there is a weaker occurrence of particle
collisions, while during the formation of the Chladni pattern, particle collisions are more
intense, concentrated largely around the nodal points. Moreover, with larger values of the
dimensionless acceleration (Γ), the tendency for collisions becomes more pronounced.
Keywords: Chladni patterns, inverse Chladni patterns, bidirectional coupled DEM and FEM,
dimensionless acceleration, internal physical properties of particles |
