The weak form of the SDOF and MDOF equations of motion are obtained. The original initial conditions problem is transformed into a boundary value problem. The boundary value problem is then solved and transformed back to the initial conditions one. Subsequently, a general method for obtaining numerical methods using an arbitrary number of linearly independent approximating functions is outlined. This is part one of a series of three papers, in the second of which a numerical method is obtained, using Bernstein polynomials of arbitrarily high order. The numerical evidence for the convergence of the method will be presented in the third part paper.

翻译：本文推导了单自由度与多自由度运动方程的弱形式。原始初值问题被转化为边值问题，求解该边值问题后再转换回初值问题。随后，概述了一种利用任意数量线性无关近似函数来构造数值方法的通用框架。这是系列三篇论文的第一部分，第二部分将利用任意高阶的Bernstein多项式导出一类数值方法，而该方法的收敛性数值验证将在第三部分中给出。