THEORYMECHANICS1902Chapter9 Plane motion of rigid bodyCollege of Mechanical and Vehicle EngineeringWangXiaojun
College of Mechanical and Vehicle Engineering Wang Xiaojun THEORY MECHANICS Chapter9 Plane motion of rigid body
Chapter9 Plane motionof rigidbody> 9.1 Simplification of plane motion of rigid body and itsdecomposition> 9.2 Velocity analysis of particles on a plane graph> 9.3 Acceleration analysis at each particle on a plane graph
➢ 9.1 Simplification of plane motion of rigid body and its decomposition ➢ 9.2 Velocity analysis of particles on a plane graph ➢ 9.3 Acceleration analysis at each particle on a plane graph Chapter9 Plane motion of rigid body
9.1 Simplification of plane motion of rigid body anditsdecomposition1.Definition of plane motion of a rigid bodyO7777777777Observing the motion of the rigid body above, it is found that they have a common feature inthe process of motion, that is, when the rigid body moves, the distance from any particle inthe rigid body to a fixed plane remains constant all the time. The motion of a rigid body withsuch a feature is called plane motion of the rigid body, or plane motion for short
9.1 Simplification of plane motion of rigid body and its decomposition 1. Definition of plane motion of a rigid body O O v O A B O O1 Observing the motion of the rigid body above, it is found that they have a common feature in the process of motion, that is, when the rigid body moves, the distance from any particle in the rigid body to a fixed plane remains constant all the time. The motion of a rigid body with such a feature is called plane motion of the rigid body, or plane motion for short
9.1 Simplification of planemotion of rigid bodyanditsdecomposition2. Simplification ofplane motion of a rigid bodyAAs shown in the figure, when a rigid bodymoves in a plane, the plane figure formed byall particles on the rigid body at the sameSAdistanceas afixed planein space stays inthe元plane where it is movingN元oAfteranalysis,thefollowingconclusionscanbe drawn:The plane motion of a rigid body can be simplified as the motion of a plane figure Sin its ownplane
0 A A1 A2 S As shown in the figure, when a rigid body moves in a plane, the plane figure formed by all particles on the rigid body at the same distance as a fixed plane in space stays in the plane where it is moving. After analysis, the following conclusions can be drawn: The plane motion of a rigid body can be simplified as the motion of a plane figure S in its own plane. 9.1 Simplification of plane motion of rigid body and its decomposition 2. Simplification of plane motion of a rigid body
9.1 Simplification of plane motion of rigid body anditsdecomposition3. Equations of motion for plane motion of a rigid body1MS0SMXo. = fi(t)O'(xo.)yo. = f2(t)0(p= fs(t)2"xo'x0The above equation is called the plane motion equationof a rigid body. Plane motionincludestwo basic forms of motion: translation and fixed axis rotation,namely: planemotionisthecompositemotionoftranslationandrotation
3. Equations of motion for plane motion of a rigid body O x y = = = ( ) ( ) ( ) 3 2 1 f t y f t x f t O O The above equation is called the plane motion equation of a rigid body. Plane motion includes two basic forms of motion: translation and fixed axis rotation, namely: plane motion is the composite motion of translation and rotation. S ( ) O , O O x y M S ( ) O , O O x y M 9.1 Simplification of plane motion of rigid body and its decomposition
9.1 Simplification of plane motion of rigid body and itsdecomposition4.Plane motion is decomposed intotranslation and rotationyMMSMxChoose any particle O' on the plane figure S as the base particle, and then the plane motion(absolute motion) can be decomposed into the translation (implicated motion) with the baseparticle and the rotation (relative motion) with the relative base particle.The velocity and acceleration along with the base particle are related to the choice of thepositionofthebaseparticle
4. Plane motion is decomposed into translation and rotation O x y Choose any particle on the plane figure S as the base particle, and then the plane motion (absolute motion) can be decomposed into the translation (implicated motion) with the base particle and the rotation (relative motion) with the relative base particle. O O M S The velocity and acceleration along with the base particle are related to the choice of the position of the base particle. 9.1 Simplification of plane motion of rigid body and its decomposition S O M O M
9.1 Simplification of planemotionof rigid bodyanditsdecomposition4.Plane motion is decomposed intotranslation and rotationyM2MMAO文OLxAgA0Aβ = △β' but の = limlim△tAt>0△tAt-→0similarly = 'So の=の'That is, at any instant, the angular velocity and angular acceleration of the graph around anyparticle in its plane are the same. Angular velocity and angular acceleration have nothing to dowiththe choiceof base particle position
O M S 4. Plane motion is decomposed into translation and rotation O x y S O M M O = but t t = → 0 lim t t = → 0 lim So = similarly = That is, at any instant, the angular velocity and angular acceleration of the graph around any particle in its plane are the same. Angular velocity and angular acceleration have nothing to do with the choice of base particle position. x y x y 9.1 Simplification of plane motion of rigid body and its decomposition
9.1 Simplification of plane motion of rigid body and itsdecompositionTosum up1,The plane motion of a rigid body can be divided into translation with cardinalparticle and rotation with cardinal particle2, The motion of the figure along with the base particle is related to the choice ofthe position of the base particle. The rotation part has nothing to do with thechoiceofthebaseparticle3、The angular velocity and angular accelerationof a graph around any particle inits plane are the same.The absolute angular velocity and angular accelerationofthe graph
To sum up 1、The plane motion of a rigid body can be divided into translation with cardinal particle and rotation with cardinal particle. 2、The motion of the figure along with the base particle is related to the choice of the position of the base particle. The rotation part has nothing to do with the choice of the base particle. 3、The angular velocity and angular acceleration of a graph around any particle in its plane are the same. The absolute angular velocity and angular acceleration of the graph. 9.1 Simplification of plane motion of rigid body and its decomposition
9.2 Velocity analysis of particles on a plane graph1.method of baseyparticleBAs shown in the figure, know the velocityofa particle A in the graph is Va , theangular velocity of the graph is , andxfind the velocity at any particle B of theAgraph.0xidea:The consolidatedTranslation with thetranslational coordinatebaseparticledecompose1,Analysisofrigid bodysystem is established atthe base particlemotion inplaneThe rotation aboutthe cardinal particle2,Thevelocity synthesistheorem ofparticle
idea： 9.2 Velocity analysis of particles on a plane graph 1、Analysis of rigid body motion in plane Translation with the decompose base particle 2、The velocity synthesis theorem of particle The rotation about the cardinal particle 1. method of base particle A v B O x y A x y As shown in the figure, know the velocity of a particle in the graph is , the angular velocity of the graph is , and find the velocity at any particle B of the graph. A A v The consolidated translational coordinate system is established at the base particle
9.2Velocityanalysis of particles on aplanegraph1.method of baseparticle公PAnd choice B is the moving particle, Ax'y' isVthe dynamic system (with translational motion)公7B1BAVelocity synthesis theorem of particle A=v+Vx'VAA1VNBAVBA = BA·OVABxVB=VA+VBAThat is,the velocityofanyparticlein theplanefigure is equal tothevector sum ofthevelocityofthebaseparticleand the rotational velocity of theparticlerelativetothebase particle. This is the basis particle method for the velocity synthesis of planemotion
1. method of base particle Velocity synthesis theorem of particle A: a e r v v v = + B A BA v v v = + vBA = BA 9.2 Velocity analysis of particles on a plane graph That is, the velocity of any particle in the plane figure is equal to the vector sum of the velocity of the base particle and the rotational velocity of the particle relative to the base particle. This is the basis particle method for the velocity synthesis of plane motion. And choice B is the moving particle, is the dynamic system (with translational motion) Ax y A v B O x y A x y A v B v BA v B v A v BA v