tailieunhanh - Ebook Fundamentals of kinematics and dynamics: Part 2

(BQ) As useful as Mathematica is, however, a tool should not interfere with but enhance one’s grasp of the concepts and the development of analytical skills. The author ensures this with his emphasis on the understanding and application of basic theoretical principles, unified approach to the analysis of planar mechanisms, and introduction to vibrations and rotor dynamics. | Ch5Frame Page 135 Friday, June 2, 2000 8:43 PM 5 Gears INTRODUCTION The kinematic function of gears is to transfer rotational motion from one shaft to another. Since these shafts may be parallel, perpendicular, or at any other angle with respect to each other, gears designed for any of these cases take different forms and have different names: spur, helical, bevel, worm, etc. The fundamental requirement in most applications is that the coefficient of motion transformation (called the gear ratio) remains constant. What is needed to meet this requirement follows from Kennedy’s theorem. The important point is that this requirement imposes a constraint on the suitable geometry of gear teeth profiles. Herein only one such profile is considered, called the involute profile. KENNEDY’S THEOREM The transformation of motion from one shaft to another involves three bodies: a frame (the position of each shaft is fixed in the frame) and two gears. Consider a general case when two disks with arbitrary profiles (Figure ) represent gears 2 and 3. Also assume that disk 2 rotates with the constant angular velocity ω2. The motion is transferred through the direct contact at point P (note that P2 and P3 are the same point P, but the first is associated with disk 2 whereas the second is associated with disk 3). The question is whether or not the angular velocity ω3 of disk 3 will also be constant, and, if not, what is needed to make it constant. The answer is given by Kennedy’s theorem. Kennedy’s theorem identifies the fundamental property of three rigid bodies in motion. The three instantaneous centers shared by three rigid bodies in relative motion to one another all lie on the same straight line. First, recall that the instantaneous center of velocity is defined as the instantaneous location of a pair of coincident points of two different rigid bodies for which the absolute velocities of two points are equal. If one considers body 2 and the frame (represented by point O2) in .