Mathematical and physical aspects in the analysis of pendulum motion in the presence of quadratic and linear damping are presented. New results are obtained from the analysis which are of research and educational interest. Closed-form expressions for the velocity vs large angle of swing are derived in the case of quadratic damping for both forward and backward swings. In the case of linear damping, there is no closed-form expression for the velocity vs large angle of swing, but an implicit relationship between the velocity and small angle of swing is derived and discussed. The derived formulas are useful for the determination of the amplitudes of the pendulum swings corresponding to given initial velocities and for the evaluation of the dissipated energy at any stage of the pendulum motion. Instructive project-based activities are suggested, which are suitable for collaborative learning and undergraduate student research.

介绍了在存在二次和线性阻尼的情况下钟摆运动分析中的数学和物理方面。从分析中获得了具有研究和教育意义的新结果。在向前和向后摆动的二次阻尼的情况下，推导出速度与大摆动角度的闭合形式表达式。在线性阻尼的情况下，没有速度与大摆角的闭式表达式，但推导出并讨论了速度与小摆角之间的隐式关系。推导出的公式可用于确定与给定初始速度相对应的摆摆幅，以及评估摆运动任何阶段的耗散能量。建议开展基于项目的指导性活动，