結構動力精細時間積分法之研究

A Study of Precise Integration Method in Structural Dynamic Problems

邱佳聖(Jia-Sheng Qiu)

指導教授 ： 莊清鏘

中原大學/工學院/土木工程研究所/碩士(2012年)

https://doi.org/10.6840/cycu201200082

摘要

結構動力模擬時常見的方式是先建對應的動力平衡方程式，再用逐步時間積分法求解，滿足的是離散時間點動力平衡。時間步長的選用和時間積分法有關，除了要滿足自由振動穩定性和精確性的要求外，也要考慮載荷變化。系統的自然振動頻率很高或載荷變化較劇烈時，經常要用很小的時間步長才能獲得滿意結果。本研究改用結構動力動量平衡方程式和精細時間積分法模擬結構動力反應，此方式可以大幅降低載荷劇烈變化時間步長選取的敏感性，並保有精細時間積分法的精確性和穩定性。除了時間步長內載荷衝量線性變化的假設外，本研究也用多項式描述時間步程載荷衝量變化，由數值算例可看出，多項式假設的動量平衡方程，可進一步降低精細時間積分法時間步長選用的限制。
此外本研究也從頻率域角度探討時間積分法穩態響應的精確性。從載荷反應可看出，載荷正弦或餘弦變化時，精細時間積分法在自然振動頻率附近沒有假共振，中央差分法和紐馬克法(平均加速度法)則存在假共振，同時精細時間積分法比中央差分法、紐馬克法有更好的載荷反應精確性，動量平衡精細時間積分法比動力平衡精細時間積分法載荷反應精確性佳，且載荷用多項式描述時隨著項數增加精確性提升。

關鍵字

載荷反應 ； 時間步長 ； 動量平衡運動方程式 ； 精細時間積分法

並列摘要

A very popular approach to conducting structural dynamic response analysis is to first formulate its dynamic equilibrium equations of motion, and then employ a step-by-step time integration scheme to solve the equations such that dynamic equilibrium is satisfied at discretized time instants. The selection of time step size depends on the features of the time integration approach, and should consider its numerical stability, desired accuracy, predominant frequencies of the analyzed structure as well as the major characteristics of the external loadings. In case of high dominant structural frequencies or dramatic loading variation, a sufficiently small time step is usually favored in order to achieve satisfactory numerical accuracy. This study chooses to employ momentum equations of motion working together with a so-called Precise Integration Method (PIM) to solve for dynamic structural response. The proposed method is insensitive to the selection of time step size in case of large loading variation and is capable of keeping superior numerical stability and accuracy characteristics of the original PIM. This study also investigates the influence of linear functions and a higher degree of polynomial functions in describing the variation of loading momentum within an integration time step. Illustrative numerical examples show that the use of polynomial function can further loosen the constraint of time step size selection but keeps the desired level of numerical accuracy at the same time.
In addition, this study also investigates the accuracy of stationary structural response under periodic forced vibration using analytical and calculated transfer functions from frequency domain viewpoint. In this regard, PIM does not produce spurious resonance at structural natural period under forced vibrations when sine and cosine functions are employed as the source of external loadings, but unfortunately such an advantage was not observed in the central difference and Newmark’s (average acceleration) methods. Among the three methods, PIM provides the best prediction accuracy. This study concludes that solving structural dynamic problems using momentum equilibrium description in PIM is more advantageous than using force equilibrium. The parametric study also confirms that a higher polynomial degree of loading description is advantageous in reaching better prediction accuracy should the temporal variation of external loading be highly nonlinear.

並列關鍵字

external loading effects ； time step size ； momentum equation of motion ； Precise integration method

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