《理论力学》课程教学资源（PPT课件，英文）Chapter 06 Kinematics of Points

Canto 2KinematicsCollege of Mechanical andVehicle Engineering王晓君

Canto 2 Kinematics College of Mechanical and Vehicle Engineering 王晓君

IntroductionKinematics is the science of studying the geometric properties of motionof objects.Thatis to study the mechanical motion of objects from the pointof view of geometry. Kinematics includes: equations of motion, trajectory,velocity,andacceleration.The significance of learningkinematics:Firstof all,it lays thenecessaryfoundationfor learning dynamics.Second,kinematicsitselfhasitsownindependentapplications.Becausethedescriptionof themotionofthe objectisrelative.Theobject where the observeris called the reference body, and the coordinatesystemconsolidatedon the referencebodyis called the referenceframe.Itonlymakes senseto analyze the motion of an object with a definiteframe ofreference.The concept oftime should be clear: instantaneous and timeinterval.The mechanical models studiedin kinematics are points and rigid bodies

Introduction Kinematics is the science of studying the geometric properties of motion of objects. That is to study the mechanical motion of objects from the point of view of geometry. Kinematics includes：equations of motion, trajectory, velocity, and acceleration. The significance of learning kinematics: First of all, it lays the necessary foundation for learning dynamics . Second, kinematics itself has its own independent applications. Because the description of the motion of the object is relative . The object where the observer is called the reference body , and the coordinate system consolidated on the reference body is called the reference frame . It only makes sense to analyze the motion of an object with a definite frame of reference. The concept of time should be clear: instantaneous and time interval. The mechanical models studied in kinematics are points and rigid bodies

THEORYMECHANICS1902Electronic teaching planKinematics of PointsChapter 6

THEORY MECHANICS Chapter 6 Kinematics of Points Electronic teaching plan

This chapter will introduce three methods of the motion ofthe research point, namely, the radius vector method, therectangular coordinate method and the natural methodWhen a point moves, the curve formed by the continuouschange of its position in space over time is called the motiontrajectory of a point. The motion of a point can be divided intolinear motion and curvilinear motion according to its trajectoryshape. When the trajectory is circular, it is called circularmotion.The mathematicalequation of the position of a point withrespect to time is called the equation of motion of the point.This chapter deals with the equations of motion, trajectories,velocities and accelerations of points, and the relationshipbetween them

This chapter will introduce three methods of the motion of the research point, namely, the radius vector method, the rectangular coordinate method and the natural method. When a point moves, the curve formed by the continuous change of its position in space over time is called the motion trajectory of a point . The motion of a point can be divided into linear motion and curvilinear motion according to its trajectory shape . When the trajectory is circular, it is called circular motion. The mathematical equation of the position of a point with respect to time is called the equation of motion of the point. This chapter deals with the equations of motion, trajectories, velocities and accelerations of points, and the relationship between them

6.1 Vector radius method of point motion一、Equations of motionAs shown in the figure, the moving point MMmoves along its trajectory, and at the instant t, thepoint M is at the illustrated position.If a vector r =OMis made from thereference point O to the moving point MRefererbodythen is called the vector radius.So the motion equation in the form of movingpoint vector radius is.r=r(t)Obviously, the sagittal curve of the vector radius thetrajectory of the point movement.It is simple and intuitive to describe the movementof points by vector diameter method

一、Equations of motion As shown in the figure, the moving point M moves along its trajectory, and at the instant t, the point M is at the illustrated position. Reference body O M r  If a vector is made from the reference point O to the moving point M, then is called the vector radius. r = OM  r  So the motion equation in the form of moving point vector radius is r r(t)   = Obviously, the sagittal curve of the vector radius the trajectory of the point movement. It is simple and intuitive to describe the movement of points by vector diameter method. 6.1 Vector radius method of point motion

6.1 Vector radius method of point motionM二、 Velocity17AArXr(t)Asshowninthethefigure,M→*displacement of the moving point M inr(t+ △t)Bthe time interval △t isMM' = △r = r(t+△t)-r(t)△rIt'scalledtheaverage1*velocity of the pointt△rdrThe velocity of the= lim * = lim1moving point at t instant:dtAt→0 △t△t-→>0That is, the velocity of a point is equal to the first derivative ofits vector diameter to time. The direction is along the tangentdirection of the trajectory

二、Velocity t r v   =    It’s called the average velocity of the point The velocity of the moving point at t instant: r dt dr t r v v t t       = =   = =  →   →0 0 lim lim That is, the velocity of a point is equal to the first derivative of its vector diameter to time. The direction is along the tangent direction of the trajectory. MM r r(t t) r(t)     =  = +  − A O B M M  r(t)  r(t + t)  r    v  v  As shown in the figure, the displacement of the moving point M in the time interval t is 6.1 Vector radius method of point motion

6.1 Vector radius method of point motion三、 AccelerationAs shown in the figure, the change amount of the velocity vectorof the moving point M in the time interval △t is=MAvAiva*Average acceleration△tM'avaAcceleration of moving3point at t instant:Avdia = lim a* = lim=rdt△t->0△t-0 △tThat is, the acceleration of a point is equal to the firstderivative of its velocity to time and the second derivative of itsvector radius to time

三、Acceleration M M  v  v  v    v    a  a  As shown in the figure, the change amount of the velocity vector of the moving point M in the time interval is t v v v     = − t v a   =    Average acceleration Acceleration of moving point at t instant: v r dt dv t v a a t t         = = =   = =  →   →0 0 lim lim That is, the acceleration of a point is equal to the first derivative of its velocity to time and the second derivative of its vector radius to time. 6.1 Vector radius method of point motion

6.2 Cartesian coordinate method for themovement of points一、Equation of motionAs shown in the figure, establish a rectangularMcoordinate system onthereference body.Thenx=f(t) y=f(t) z= f(t)tK01This is the eguation of motion of a point in1yrectangular coordinates.XThe equation can be obtained by eliminating time t fromthe equation ofmotion:F(x,y,z)= 0It's called the trajectoryequation of the moving point

一、Equation of motion O x y z i  j  k  r  M x y z As shown in the figure, establish a rectangular coordinate system on the reference body. Then ( ) 1 x = f t ( ) 2 y = f t ( ) 3 z = f t This is the equation of motion of a point in rectangular coordinates. The equation can be obtained by eliminating time t from the equation of motion: F x y z ( , , ) 0 = It’s called the trajectory equation of the moving point. 6.2 Cartesian coordinate method for the movement of points

6.2 Cartesian coordinate method for themovement of points二、VelocityIt can be seen from the figure that theMvector radius of the moving point is:k= xi +yj+zk0idrdxdydzkvydtdtdtdtThen:.i+y.kVyi +y.dxdzxV.VxdtdtdtThat's the velocity of the point in terms of rectangular coordinates.That istheprojectionofthevelocityofa pointontherectangularcoordinateaxisisequal to the first derivative of the corresponding coordinate of a point withrespecttotime

二、Velocity O x y z i  j  k  r  M x y z It can be seen from the figure that the vector radius of the moving point is: r xi yj zk     = + + k dt dz j dt dy i dt dx dt dr v      = = + + v v i v j v k x y z     = + + x dt dx vx = =  y dt dy vy = =  z dt dz vz = =  That's the velocity of the point in terms of rectangular coordinates . That is, the projection of the velocity of a point on the rectangular coordinate axis is equal to the first derivative of the corresponding coordinate of a point with respect to time. Then: 6.2 Cartesian coordinate method for the movement of points

6.2 Cartesian coordinate method for themovement of pointsIf the projection of velocity is known, the magnitude ofvelocity isV= /x? + j? + z2xThe cosine of its direction is cos(v,i)= Vcos(,j) = v.Ncos(v,k)= =v

If the projection of velocity is known, the magnitude of velocity is 2 2 2 v = x + y + z The cosine of its direction is          = = = v z v k v y v j v x v i          cos( , ) cos( , ) cos( , ) 6.2 Cartesian coordinate method for the movement of points