理论力学拉格朗日方程.ppt

《理论力学拉格朗日方程.ppt》由会员上传分享，免费在线阅读，更多相关内容在行业资料-天天文库。

1、Chapter17:Lagrange'sequationsTheoreticalMechanics1第十七章拉格朗日方程2OnthebasisofD‘Alembert’sPrincipleandoftheTheoremofVirtualDisplacementsinthischapterthegeneralequationofdynamicsandtheLagrange'sequationsofthesecondkind(abbreviatedasLagrange'sequations)isdeduced.本章在达朗伯原理和虚位移原理

2、的基础上，进一步导出动力学普遍方程和拉格朗日第二类方程（简称拉格朗日方程）。Dynamics动力学3动力学普遍方程和拉格朗日方程是研究动力学问题的有力手段，在解决非自由质点系的动力学问题时，显得十分简捷、规范。ThegeneralequationofdynamicsandtheLagrange'sequationsareeffectivemeanstostudydynamicalproblems.Theyprovideverysimple,directandstandardwaystosolvedynamicalproblemsofu

3、nfreesystemsofparticles.Dynamics动力学4Chapter17:Lagrange'sequations第十七章拉格朗日方程§17–1Generalequationofdynamics§17–1动力学普遍方程5§17–3拉格朗日第二类方程的积分§17–3IntegralsoftheLagrange'sequationsofthesecondkind§17–2Lagrange'sequationsofthesecondkind§17–2拉格朗日第二类方程6§17-1Generalequationofdynami

4、cs§17-1动力学普遍方程Supposethattherearenparticlesinasystem,particleibeingdescribedby.Thenwehave设质点系有n个质点，第i个质点Ifthesystemofparticleisundertheactionoftheidealconstraints,thecanbetreatedaspositiveforcesandweget若质点系受有理想约束，将作为主动力处理，则：Dynamics动力学7解析式：Theexplicitformis动力学普遍方程。Thisi

5、sthegeneralequationofdynamics.Dynamics动力学8Undertheactionoftheidealconstraintsthesumofthevirtualworksofthepositiveforcesandoftheinertialforces,actingoneveryparticleofthesystem,alonganarbitraryvirtualdisplacementiszeroatanymoment.在理想约束的条件下，质点系的各质点在任一瞬时受到的主动力与惯性力在任意虚位移上所作的

6、虚功之和为零。[Example1]AtriangularprismBisslippingalongthesmoothinclinedplaneofthetriangularprismA.ThetriangularprismAisplacedonthesmoothhorizontalplane.TheweightsofAandBareMandm,theangleofinclinationis.DeterminetheaccelerationofA.例1三棱柱B沿三棱柱A的光滑斜面滑动，三棱柱A置于光滑水平面上，A和B的质量分别为M和m

7、，斜面倾角为。试求三棱柱A的加速度。Dynamics动力学9解：研究两三棱柱组成的系统。该系统受理想约束，具有两个自由度。SolutionInvestigatethesystemconsistingofbothtriangularprisms.Thissystemisundertheactionofidealconstraints.Ithastwodegreeoffreedom.Dynamics动力学10Thesystemhastwodegreeoffreedom,wecanchooseasindependentvirtualdis

8、placement.Moreover,.SowegetFromthegeneralequationofdynamicswehave由动力学普遍方程：Dynamics动力学11系统为二自由度，取互不相关的为独立虚位移，且，