含van der Pol型自激项的单摆系统是典型的自激机械系统,本文研究了该系统的张弛振荡特性.首先通过引入新的时间尺度和变量,把原系统表示成标准的快慢系统.然后基于几何奇异摄动理论,求得系统的慢变流形及其结构,从而证明了张弛振荡解的存在性,并进一步求得了张弛振荡解及其周期的近似表达式.理论结果表明,发生张弛振荡时,单摆快速通过其平衡位置,而在远离平衡位置的一段区域上停留较长时间,且存在两个分界点把快速运动和慢速运动分隔开.数值算例证明了理论分析的正确性.
《高等数学》是工科院校本科各专业的重要基础课之一。本文针对《高等数学》课程思政教学中的“两张皮、硬融入、难考核”等问题,运用“融”思维提出了三维融合的解决方案——思政元素与教学内容融合、思政育人与教学策略融合、思政考核与课程考核融合。通过两角度挖思政元素、双线融通教学设计实现思政元素与教学内容的融合;借助混合式对分课堂、情境探索性数学实验实现思政育人与教学策略的融合;通过“反思之慧、感受之深、设问之疑”作业及“数学之美、数学之用”作业实现思政考核与课程考核的融合。Advanced mathematics is one of the important foundational courses for undergraduate majors in engineering colleges. This article proposes a three-dimensional integration solution for the problems of “two skins, hard integration, and difficult assessment” in the ideological and political teaching of the course “Advanced Mathematics”. Three-dimensional integration is the integration of ideological and political elements with teaching contents, the integration of ideological and political education with teaching strategies, and the integration of ideological and political assessment with course assessment. The integration of ideological and political elements and teaching contents is achieved by exploring ideological and political elements from two perspectives and integrating the teaching design of dual lines;The integration of ideological and political education with teaching strategies is realized by the blended PAD Class and situational exploratory mathematics experiments;The integration of ideological and political assessment and curriculum assessment is achieved by the “wisdom of reflection, depth of feeling, doubt of questioning” homework and the “Beauty and Application of Mathematics” homework.
