Kinematics of a Rigid Body in Plane Motion

Kinematics of a Rigid Body in Plane Motion - 5 Kinematics...


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5 Kinematics of a rigid body in plane motion 5.1 Introduction A rigid body is defined as being a system of particles in which the distance between any two particles is !ked in magnitude. The number of co-ordinates required to determine the position and orientation of a body in plane motion is three: the system is said to have three degrees of freedom. Figure 5.3 Figure 5.4 time tl and t2 then the average angular speed is Figure 5.1 is shown in Fig. 5*1* The position Of Some representative point such as A and the angle which a line AB makes with the x-axis are three possible co-ordinates. 5.2 Types of motion As (t2-t1)+0, (02-4)+0, and the angular The simplest type of motion is that of rectilinear translation, in which 8 remains constant and A moves in a straight line (Fig. 5.2). It follows that all particles move in lines parallel to the path of A; thus the velocity and acceleration of all points are identical. One way Of describing the position Of the body angle (e2 - el ). If this change takes place between 62 - 81 t2 - tl -- - waverage speed is defined as A0 de w = limAt+o- = - At dt (5.1) The angular velocity vector is defined as having a magnitude equal to the angular speed and a direction perpendicular to the plane of rotation, the positive sense being given by the usual right-hand screw rule. In the present case, o = wk (5.2) It should be noted that infinitessimal rotations This is still true if A is describing a curved path, and angular velocity are vector quantities, whereas finite angular displacement is not. A very important point to note is that the angular speed is not affected by the translation, therefore we do not have to specify any point in the plane about which rotation is supposed to be taking place. Figure 5.2 since if 8 = constant all paths are identical in shape but displaced from each other. This motion is called curvilinear translation (Fig. 5.3). If the angle 0 changes during translation, then this motion is described as general plane motion (Fig. 5.4). In Fig. 5.4 the body has rotated by an
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5.4 Velocity diagrams 55 5.3 Relative motion between two points on a rigid body The definition of the vector product of two vectors has been already introduced in Chapter 4 in connection with the moment of a vector; the same definition is useful in expressing the relative velocity between two points on a rigid body due to rotation. to the graphical solution of plane mechanisms are described in the following sections. 5.4 Velocity diagrams One very simple yet common mechanism is the four-bar chain, shown in Fig. 5.6. It is seen that if the motion of AB is given then the motion of the rest of the mechanism may be determined. Figure 5.6 This problem can be solved analytically, but the solution is surprisingly lengthy and is best left to a computer to solve if a large number of positions of the niechanism are being examined. However, a simple solution may be found by using vector diagrams; this also has an advantage of giving considerable insight into the behaviour of mechanisms.
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