2022年周衍柏《理论力学》第五章教案-分析力学 .pdf
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1、第五章分析力学本章要求(1)掌握分析力学中的一些基本概念;(2)掌握虚功原理;(3)掌握拉格朗日方程;(4)掌握哈密顿正则方程。第一节约束和广义坐标一、约束的概念和分类 加于力学体系的限制条件叫约束。按不同的标准有不同的分类:按约束是否与时间有关分类:稳定约束、不稳定约束;按质点能否脱离约束分类:可解约束、不可解约束;按约束限制范围分类:几何约束(完整约束)、运动约束(不完整约束)。本章只讨论几何约束(完整约束),这种约束下的体系叫完整体系。二、广义坐标 1、自由度描述一个力学体系所需要的独立坐标的个数叫体系的自由度。设体系有n 个粒子,一个粒子需要3 个坐标(如 x、y、z)描述,而体系受有
2、K个约束条件,则体系的自由度为(3n-K)2、广义坐标描述力学体系的独立坐标叫广义坐标。例如:作圆周运动的质点只须角度用描述,广义坐标为,自由度为1,球面上运动的质点,由极角和描述,自由度为2。第二节虚功原理本节重点要求:掌握虚位移、虚功、理想约束等概念;掌握虚功原理。一、实位移与虚位移质点由于运动实际上所发生的位移叫实位移;在某一时刻,在约束允许的情况下,质点可能发生的位移叫虚位移。如果约束为固定约束,则实位移是虚位移中一的个;若约束不固定,实位移与虚位移无共同之处。例如图5.2.1 中的质点在曲面上运动,而曲面也在移动,显然实位移与虚位移不一致。二、理想约束设质点系受主动力和约束力的作用,
3、它们在任意虚位移中作的功叫虚功。若约束反力在任意虚位移中对质点系所作虚功之和为零,则这种约束叫理想约束。光滑面、光滑线、刚性杆、不可伸长的绳等都是理想约束。三、虚功原理 1、文字叙述和数学表示:受理想约束的力学体系,平衡的充要条件是:作用于力学体系的诸主动力在任意虚位移中作的元功之和为零。即(1)适用条件:惯性系、理想不可解约束。2、推论设系统的广义坐标为q1,qa,qS,虚位移可写为用广义坐标变分表示的形式:定义:称为相应于广义坐标qa的广义力,则虚功原理表述为:理想约束的力学体系平衡的充要条件为质点系受的广义力为零,即:文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S
4、9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H
5、3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:
6、CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K
7、7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10
8、 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A
9、1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K
10、8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3(2)3、用虚功原理求解平衡问题的方法步骤一般步骤为:(1)确定自由度,选取坐标系,分析力(包括主动力、约束力);(2)选取广义坐标并将各质点坐标表示成广义坐标qa的函数:;(3)求主动力的虚功并令其为零:,由此求出平衡条件。例 见书 P
11、276 例 1第三节拉格朗日方程本节重点要求:(1)掌握拉格朗日方程的两种形式,方程的特点和适用条件等;(2)掌握用拉格朗日方程求解具体问题的步骤;(3)了解循环积分等概念。一、基本形式的拉格朗日方程 1、方程的推导由牛顿第二定律并应用理想约束的条件,可以得到达朗伯拉格朗日方程:(1)将坐标的变分改成用广义坐标q1,qS的变分表示,即:经数学运算,令(称为体系的动能),(称为相应于 qa的广义力),则(1)式变文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K
12、2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D
13、10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS1
14、0A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B
15、3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY
16、5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X
17、2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编
18、码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3为:(2)这就是基本形式的拉格朗日方程,应注意:(2)实际是一组方程。2、方程的适用条件:理想约束。二、保守系的拉格朗日方程设作用于体系的力全为保守力,则广义力可由(V为势能)求得:在普遍形式的拉氏方程(2)中,由于 V不包含广义速度,可令:(动能与势能的差)为拉格朗日函数,则(2)式变为:(3)应指出(3)的适用条件为保守系,理想约束,且(3)应用很
19、普遍。三、应用拉格朗日方程求解问题的步骤,例一般步骤:画草图,确定自由度 s 和广义坐标 qa;分析主动力,若为保守系,则求出势能 V;若为非保守力,则计算广义力 Qa;求动能 T=T();对保守系,求出 L=T-V,进而代入方程(3),写出运动方程;对非保守系,将 T和广义力 Q 代入方程(2),写出运动方程。解方程,求出q(t)。例 1 P265 4.10题圆环在光滑圆圈上运动,而圆圈绕垂直圆面的轴作匀角速运动,求圆环运动规律。文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2
20、H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码
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