振動系統數值模擬時間步長設定之最佳化

Abstract

振動現象普遍存在於各類工程系統中，從最基本的彈簧–質量模型，到車輛懸吊系統、工具機結構、精密設備、運輸設施、大型風力發電機，以及先進的AI機器人結構等，皆呈現由慣性、剛性與阻尼交互作用所形成的多樣化動態特性。近年的研究將振動分析應用於影像訊號處理（image signal processing）、疲勞損傷推估(fatigue damage estimation, FDE)、智慧感測（intelligent sensing）與結構健康監測（structural health monitoring, SHM）等領域，使時間域模擬成為理解系統行為、預測結構反應與進行控制設計的重要基礎工具。 在上述振動研究與數值模擬中，時間步長或取樣頻率的設定，直接影響模擬結果的解析度、導數計算品質以及整體運算成本。然而，若取樣訊號僅滿足奈奎斯特取樣定理所規範的「不失真」最低要求，雖可避免頻域混疊，卻無法完整還原振動的物理行為，時間域波形之曲率特性難以重現。由於缺乏可量化的判定準則，現行時間步長多半依賴經驗法則設定，使得模擬結果雖趨於保守與安全，卻往往伴隨不必要的計算成本，使得模擬結果偏向安全但低效。 基於此，本研究旨在建立一套可量化的時間步長設定準則。從離散時間節點連線之幾何平滑度出發，利用二階差分均方根與相對平滑度指標，定義「最小且足夠」的取樣節點數，透過單自由度與雙自由度振動系統驗證其可行性。研究結果顯示，提出的方法能有效反映離散時間節點的連線品質，並在精確度與計算效率之間提供明確取捨依據，可作為各類振動模擬與時間域分析中之客觀時間離散選擇機制。

Vibration phenomena are ubiquitous in engineering systems, ranging from the most fundamental spring–mass models to vehicle suspension systems, machine tool structures, precision equipment, transportation infrastructure, large-scale wind turbines, and advanced AI-driven robotic structures. These systems exhibit diverse dynamic characteristics arising from the coupled interactions of inertia, stiffness, and damping.In recent years, vibration analysis has been increasingly applied to fields such as image signal processing, fatigue damage estimation (FDE), intelligent sensing, and structural health monitoring (SHM). As a result, time-domain simulation has become a fundamental tool for understanding system behavior, predicting structural responses, and performing control design. In vibration research and numerical simulations, the selection of time step or sampling frequency directly affects the resolution of simulation results, the quality of derivative evaluations, and the overall computational cost. However, when the sampling process merely satisfies the minimum “distortion-free” requirement prescribed by the Nyquist sampling theorem, frequency-domain aliasing can be avoided, but the physical behavior of the vibration cannot be fully reconstructed. In particular, the curvature characteristics of time-domain waveforms cannot be faithfully captured. Due to the lack of quantitative evaluation criteria, time-step selection is often based on empirical rules, leading to simulations that are conservative and numerically stable but accompanied by unnecessary computational expense, resulting in low overall efficiency. Motivated by these limitations, this study aims to establish a quantitative criterion for time-step selection. Starting from the geometric smoothness of vibration time histories, the proposed approach employs the root mean square of second-order differences and a relative smoothness index to define the “minimum yet sufficient” number of sampling points. The feasibility of the method is validated through single-degree-of-freedom and two-degree-of-freedom vibration systems. The results demonstrate that the proposed criterion effectively reflects the connection quality of discretized time points and provides a clear trade-off between accuracy and computational efficiency, serving as an objective guideline for time discretization in vibration simulations and time-domain analyses.