9.2 Velocityanalysisofparticles on a planegraph2.theorem ofprojection velocitiesIf you project the velocity vector Vg = , +VBa onto AB , you getV[, JBA =[, 1AVBUBThatis,atanyinstant,thevelocityofanytwoxparticlesontheplanegraph is equal totheAAprojection on the line between these twoparticles.That'sthevelocityprojectiontheoremxIt meansthatthe distance between anytwo particles ona rigid body remains constant
2. theorem of projection velocities If you project the velocity vector onto , you get B A BA v v v = + AB B BA A BA v v = That is, at any instant, the velocity of any two particles on the plane graph is equal to the projection on the line between these two particles. That's the velocity projection theorem. It means that the distance between any two particles on a rigid body remains constant. 9.2 Velocity analysis of particles on a plane graph A v B O x y A BA v x y A v B v
9.2Velocityanalysisof particles on a planegraph3.method of instant center1, raise of problemThevelocityofaparticleonaplanefigure iscalculatedbythebaseparticlemethodIf the velocity ofthe base particle is zero, the solutionofthe problem will besimplified2, instantaneous centerAs shown in the figure, there must be a particle and only a particleon the half line perpendicular to it, whose relative velocity andACVimplicated velocity are equal in magnitude but opposite in direction,sothatthe absolute velocity is equalto zero.0VAAThe position of the particleC* satisfiesC*A=-the following relation0That is, if the angular velocity of an instantaneous plane graph is not zero, there will always bea unique particle with zero velocity on the instantaneous graph. This particle is called theinstantaneous velocity center of the figure, or instantaneous center for short
3. method of instant center A A v A v v C A C As shown in the figure, there must be a particle and only a particle on the half line perpendicular to it, whose relative velocity and implicated velocity are equal in magnitude but opposite in direction, so that the absolute velocity is equal to zero. C The position of the particle satisfies the following relation C A v C A = That is, if the angular velocity of an instantaneous plane graph is not zero, there will always be a unique particle with zero velocity on the instantaneous graph. This particle is called the instantaneous velocity center of the figure, or instantaneous center for short. 9.2 Velocity analysis of particles on a plane graph 1、raise of problem The velocity of a particle on a plane figure is calculated by the base particle method. If the velocity of the base particle is zero, the solution of the problem will be simplified. 2、instantaneous center
9.2 Velocityanalysis of particles on a planegraph3, Instantaneous center method to find the velocityIf the instantaneouscenterC*istaken asthebase particle, then the velocity at any particleMoftheplanefigureisMC*VM=VMC=MC*·OM1Millustrate:IThe plane motion ofan instantaneous rigid body can be considered as thefixed axisrotationaroundtheinstantaneouscenter.2、The planefigure has different instantaneous centerof velocity at different instantaneousmoments.The instantaneous center can be in or out of a plane figure, and its position is notfixed.3、 The plane motion of a rigid body can be considered as an instantaneous fixed axisrotationaroundaseriesofinstantaneouscenters
If the instantaneous center is taken as the base particle, then the velocity at any particle M of the plane figure is C = = v v MC M MC 1、The plane motion of an instantaneous rigid body can be considered as the fixed axis rotation around the instantaneous center. 2、The plane figure has different instantaneous center of velocity at different instantaneous moments. The instantaneous center can be in or out of a plane figure, and its position is not fixed. 3、The plane motion of a rigid body can be considered as an instantaneous fixed axis rotation around a series of instantaneous centers. 9.2 Velocity analysis of particles on a plane graph C M M v MC v illustrate： 3、Instantaneous center method to find the velocity
9.2 Velocityanalysisofparticleson a planegraph4、Methods to determine the instantaneous center:Example:AfAL0BCB1B0BBOVB0AY*CAa020A0VBB立AAdb7//
9.2 Velocity analysis of particles on a plane graph 4、Methods to determine the instantaneous center: A A v B B v C a C A B A v B v c A B A v B v b C d Example：