力学专业英语_7.pdf

Lesson 8 The planar motion of a rigid body The planar motion of a rigid body occurs when the path of each point lies wholly in a plane,with the planar paths for all points in the body being parallel to each other.A simple example is a wheel mounted on a rotating shaft.The plane of the motion of any point in that system is perpendicular to the axis of the shaft.The restriction to planar motion substantially simplifies the kinematical description in comparison to a general three-dimensional motion.One of the attributes of this simplification is that it will be possible to depict the motion by means of a planar diagram.Without loss of generality,we will always let the xy plane coincide with the plane of motion.By definition,a rigid body is a system of particles that are always at a fixed distance from each other.This geometric restriction means that the movement of any point in the body relative to any other is limited.The existence of such kinematical relations is contrasted by the situation where particles in a system are not rigidly connected.All materials deform when forces are applied to them.The concept of a rigid body is merely a model that we create as an approximation.When we use the rigid body model,we are assuming that the movement of points due to deformation is negligible.Let us consider three arbitrarily selected points A,B and C to be csribed onto a body.The triangle ABC depicted in Figure 1 is useful for characterizing the kinematics of a rigid body motion.To describe where the triangle ABC is situated in space,it would be sufficient to know the coordinates locating point A,and the angle between the x axis and side AB.From this information,the position of a rigid body at any instant is defined by the absolute position of a point in the body and the angular orientation of any line in the body.Our knowledge of the information required to locate the position of a rigid body permits us to describe how the position change with time.Figure 1 depicts the position of triangle ABC at an instant subsequent to the original position of the triangle indicated by the dashed lines.The distances x and y describe the movement of point A,and describes the angle of rotation of line AB.By the definition of a rigid body we know that the angles between all sides of the triangle are constant.It follows then that each line undergoes the same rotation.Hence,angular motion is an overall property of the motion of the body.It is the same regardless of which points in the body are being discussed.Now,suppose that the movement of point A is specified.In addition to this movement,points B and C may move relative to point A.Because the angular motion is an overall property,the radial lines from point A to points B and C undergo the same rotation.In other words,each point moves in a circular path relative to point A.Therefore,we may conclude that The motion of a rigid body consists of a superposition of two movements.The first par consists of a movement of all points following the motion of an arbitrarily selected point in the body.The second movement is a rotation about the selected point in which all lines rotate by the same amount.This statement is known as Chasles theorem.Special terms are used to describe the motion of rigid bodies.One simple type of motion occurs when every line in the body retains its original orientation,that is,there is no rotation.This is called a translation.Another simple type of motion occurs when one point in the body is fixed in space.Chasles theorem states that the motion of the body may be considered to consists solely of a rotation about the fixed point,this is called a pure rotation,Chasles theorem may be reworded to state that the motion of a rigid body is the superposition of a translation of an arbitrary point and a pure rotation about that point.Suppose that the motion of point A in the rigid body shown in Figure 2 is known,as is the angular rotation of the body,.Point B is an arbitrarily chosen point whose motion we seek.As shown in Fugure 2 the vectors and denote the positions of points A and B respectively.Note that is independent of the choice for points A and B,because Chasles theorem states that the rotation is the same for all lines in a rigid body.The absolute value of is called the angular speed of the rigid body.The qualitative insight provided by Chasles theorem will enable us to form algebraic relations for determining the velocities and accelerations of points on a rigid body.The relationship between the absolute motions of points A and B is readily formed by referring to the position vectors depicted in figure 2,i.e.In the above equations(2)and(3),the velocity and acceleration of point A are denoted by VA and aA respectively.The symbols VB and aB are similarly defined for point B.The vector is the angular velocity of the body,and a is the angular acceleration.Both vectors are perpendicular to the plane of motion,so they are parallel to the axis for the rotational portion of the motion.As usual,the angular unit for and a must be radians.