THEORY MECHANICS2Chapter14 D'alembert'sprincipleCollege of Mechanical and VehicleEngineering王晓君
College of Mechanical and Vehicle Engineering 王晓君 Chapter 14 D'alembert's principle THEORY MECHANICS
Chapter14D'alembert'sprinciple> 14. 1 D' Alembert's principle for particles 14.2 D'Alembert's principle for particle systems> 14.3 Simplification of a rigid body inertial force system
➢ 14. 1 D' Alembert's principle for particles ➢ 14. 2 D'Alembert's principle for particle systems ➢ 14. 3 Simplification of a rigid body inertial force system Chapter 14 D'alembert's principle
Chapter14D'alembert'sprincipleGeneraltheorems of dynamicsDynamics of asystemofnonfreeparticlesD'alembert(Also known asS principlestatic methodCharacteristics: To study the problem of dynamics in thesame way that statics studies the problem of equilibrium
Dynamics of a system of nonfree particles General theorems of dynamics D'alembert' s principle Characteristics: To study the problem of dynamics in the same way that statics studies the problem of equilibrium. (Also known as static method) Chapter 14 D'alembert's principle
14.1D'alembert'sprincipleI,The inertia forceLet the particle M with mass m move along the trajectoryshown in the diagram. The main force acting on the particle Mat a certain instant is , the constraint reaction is N, and itsacceleration is a.According to the basic equations of dynamics, _ F gwe havema=F+NVF+N+(-ma)=0rewrite this asaFg =-maDedineFg It's called the inertial force of the particle.The inertial force: the magnitude of avirtual force acting on aparticle is equal to the product of the mass of the particle and themagnitude of its acceleration, in the opposite direction ofitsacceleration
Let the particle M with mass move along the trajectory shown in the diagram. The main force acting on the particle M at a certain instant is , the constraint reaction is , and its acceleration is . m F N a g F M F N a According to the basic equations of dynamics, we have ma F N = + rewrite this as F + N + (−ma) = 0 Dedine F ma g = − The inertial force: the magnitude of a virtual force acting on a particle is equal to the product of the mass of the particle and the magnitude of its acceleration, in the opposite direction of its acceleration. 14.1 D'alembert's principle Ⅰ. The inertia force g F ——It's called the inertial force of the particle
14.1D'alembert'sprincipleII.D'Alembert's principle ofa particleThe inertialforces are introduced into Newton's secondlaw:F+F+Fg =0-D'Alembert'sprincipleofaparticleThat is: at any instant of a particle's motion, the counterforceof theprincipal dynamic constraint acting on the particle and theforce ofinertia assumed on the particle constitute the formalequilibrium force system, which is D'Alembert's principle of theparticle
Ⅱ. D'Alembert's principle of a particle The inertial forces are introduced into Newton's second law： + + = 0 g F FN F ——D'Alembert's principle of a particle That is: at any instant of a particle's motion, the counterforce of the principal dynamic constraint acting on the particle and the force of inertia assumed on the particle constitute the formal equilibrium force system, which is D'Alembert's principle of the particle. 14.1 D'alembert's principle
14.1 D'Alembert'sprincipleofaparticleExample 1:The pendulum length of the simple pendulum is L,the mass of the pendulum is M. Calculate the differentialequation of the pendulum motion and the tension of the rope
Example 1: The pendulum length of the simple pendulum is L, the mass of the pendulum is M. Calculate the differential equation of the pendulum motion and the tension of the rope. 14.1 D'Alembert's principle of a particle
14.1 D'Alembert'sprincipleofaparticleAnswer:1. Force analysis and motion analysis: Theacceleration of the particleKaaa, =l0, a, =l2202. According to the acceleration analysis, addthe inertial force, and the magnitude is:F°=mlé, Fg =mle?3. By D'Alembert theorem to set up theFgnequilibrium equation:PZF,=0 F°+mgsin=0F8 +mgcos0-F =0ZF, =00+gDifferentialequations of motion for asin 0=01simplependulumFr = Pcos0+mli2Tension of the rope
2 , F ml F g ml n g = = FT P g Fn g Fτ an a 2. According to the acceleration analysis, add the inertial force, and the magnitude is: 3. By D'Alembert theorem to set up the equilibrium equation: 1. Force analysis and motion analysis; The acceleration of the particle 2 , a l a l = n = F = 0 F + mgsin = 0 g = 0 + cos − T = 0 g Fn Fn mg F Answer: + sin = 0 l g 2 cos F P ml T = + ——Differential equations of motion for a simple pendulum ——Tension of the rope 14.1 D'Alembert's principle of a particle
14.1 D'Alembert'sprincipleofaparticleExample 2: The cylinder of the ball mill rotates aroundthe horizontal axis O at the uniform angular velocity 0, FgFand the steel ball and materials to be crushed are installedinside. The steel ball is carried to a certain height by thecylinder wall to escape from the cylinder wall, and thenOmgfalls freely along the parabolic trajectory to smash thematerials. Set the radius of the inner wall of the cylinderas r in the figure, and try to find the escape Angle α ofthe steel ball.Answer: A steel ball that has not been detached from the cylinder walis taken as the research object, and the force is shown in the figure.Before the ball is detached from the cylinder wall, it moves ina circle, and its acceleration isa, =0a, =ro?The magnitude of the inertial force F isFg =mro
M F g F mg N r O Example 2: The cylinder of the ball mill rotates around the horizontal axis O at the uniform angular velocity , and the steel ball and materials to be crushed are installed inside. The steel ball is carried to a certain height by the cylinder wall to escape from the cylinder wall, and then falls freely along the parabolic trajectory to smash the materials. Set the radius of the inner wall of the cylinder as in the figure, and try to find the escape Angle of the steel ball. r Answer: A steel ball that has not been detached from the cylinder wall is taken as the research object, and the force is shown in the figure. Before the ball is detached from the cylinder wall, it moves in a circle, and its acceleration is a = 0 2 an = r The magnitude of the inertial force is g F 2 F mr g = 14.1 D'Alembert's principle of a particle
14.1 D'Alembert'sprincipleofaparticleSuppose to add the inertial forces, by D'Alembert'sprinciple, we have:ZF,=0N+mgcos0-Fg=0LroSo:N=mcosO)gThis is the normal reaction force suffered by the steel ballat any position . Obviously, when the steel ball is separatedfrom the cylinder wall, N = O, and thus, its escape angle αcan be obtained as:α=arccog
So： ( cos ) 2 = − g r N mg This is the normal reaction force suffered by the steel ball at any position . Obviously, when the steel ball is separated from the cylinder wall, , and thus, its escape angle can be obtained as: N = 0 arccos( ) 2 g r = Suppose to add the inertial forces, by D'Alembert's principle, we have: Fn = 0 + cos − = 0 g N mg F 14.1 D'Alembert's principle of a particle
14.2 D'Alembert's principle forparticle systemsMass m,,AcceleratedvelocityaNparticlesParticlelforma systemExternal force F, internalforceF,,F=-ma,of particles:Whatkind ofD'Alembert'sFe + Fi + F8 =0force systemprinciple ofaparticle:is formed?Whatkind ofZF ++ZF8 =0force systemNparticlesis formed?constitutetheparticle system:Zmo(Fe)+Em.(F)+Zmo(F°)=0ZFe +ZF8 =0Zmo(Fe)+Emo(F°)= 0
N particles form a system of particles: Particle i 0 e i g F F F i i i + + = 14.2 D'Alembert's principle for particle systems + + = 0 g i i i e Fi F F External force , internal force , e Fi i Fi Mass mi , Accelerated velocity i a i i g Fi m a = − D'Alembert's principle of a particle： N particles constitute the particle system: What kind of force system is formed？ ( ) + ( ) + ( ) = 0 g O i i O i e mO Fi m F m F ( ) ( ) 0 e g + = m F m F O i O i0 e g + = F F i i What kind of force system is formed？