動量平衡運動方程式在衝擊載重與波傳問題上的應用

Application of Momemtum Equations of Motion to Impact and Wave Propagation Problems

利用傳統逐步積分法來分析衝擊載重或波傳問題時，一般都是針對動力平衡運動方程式來進行逐步積分，但由於脈衝的作用時間往往是非常的短，因而進行逐步積分時所採用的積分時間步長就必須遠小於脈衝的作用時間才能有效地抓住外力的變化。如此一來，積分時間步長可能遠小於精確度的要求而大幅度增加電腦計算所需的時間。為了克服此一困難，可以將動力平衡運動方程式對時間積分一次而轉換成動量平衡運動方程式。先前所發展的積分型逐步積分法即是針對此動量平衡運動方程式來進行逐步積分以獲得結構系統的動態歷時反應。在此積分法中，運動方程式之外力已因對時間的一次積分而轉換成動量。由於積分所特別具有的平滑與累加效應，將使動量在逐步積分的過程中比外力更為容易掌握，甚至於使用大於脈衝作用時間的積分時間步長仍可被完全正確的掌握住，並且與脈衝的形狀幾乎完全無關。相反的，傳統逐步積分法則可能需採用小於脈衝作用時間的101為積分時間步長才有可能得到較為可靠的數值解，並且隨脈衝的形狀而有相當大的差異。是以積分型逐步積分法非常適用於衝載重擊或波傳問題的歷時分析中而可大量地節省計算所需的時間。

The use of a step-by-step integration method to obtain the shock responsefrom an impulse is usual to solve the force equation of motion.Since the loading time is very short, the time step used for step-by-step integrationmust be smaller than the loading time, so that the variation ofimpulse can be effectively captured. Thus, this time step may be muchsmaller than the time step needed for accuracy and thus the cost of computingis increased. To overcome this difficulty, the momentum equationsof motion are obtained from the time integration of force equations of motion.An integral form step-by-step integration method, which has beendeveloped previously, is used to solve the momentum equation of motion.In this method, an external force is integrated with time and becomes an external momentum. Thus, it can be easily captured due to the smoothingeffect of time integration. In fact, an external momentum can be reliablycaptured even if the time step is larger than the loading time for any shapeof impulse. Whereas, when using an original form of step-by-step integrationmethod, the time step might be as small as or less than 110 of theloading time to allow reliable solutions to be achieved. The choice of thistime step is very shape-dependent. Thus, the integral form step-by-step integrationmethod is very suitable to measure shock response from animpulse or in wave propagation problems.