A. BASIC CONCEPTS 1.0 VELOCITIES IN GEAR TRAINS 1.1 Introduction Gear trains and speed reducers are mechanical components often used for obtaining a desired angular velocity of an output shaft, while the input shaft rotates at a different angular velocity. The angular-velocity ratio between input and output members must usually remain constant. The value of this ratio can be adjusted in some arrangements (usually with a friction or hydraulic drive and/or clutch arrangement), while in others the ratio is not adjustable. In order, therefore, to design or select a gear train or speed reducer, one of the first tasks Is the determination of the angular velocities. This will now be considered. 1.2 Single Gear Mesh Figures 1, 2 show simple spur-gear meshes with the gears meshing externally (Figure 1) or internally (Figure 2) on fixed centers. Let 1= angular velocity of gear #1, positive counterclockwise; positive counterclockwise; 2= angular velocity of gear #2, N1 = number of teeth on gear #1; N2 = number of teeth on gear #2. In the external mesh (Figure 1) the gears rotate in opposite directions while in the internal mesh (Figure 2) the gears rotate in the same direction. This means that if the angular velocity, 1, is positive, angular velocity 2 will be negative in Figure 1 and positive in Figure 2.
Figure 1 Externally Meshing Spur Gears T58
Figure 2 Internally Meshing Spur Gears The angular velocity ratio, Z21, is defined as the ratio of the angular velocity of gear #2 to that of gear #1, with appropriate attention to sign. Thus: In Figure 1, Z21 = 2 = - N1 (1) 1 N2 and in Figure 2, Z21 = 2 = N1 (2) 1 N2 The opposite signs on the right-hand sides of eqs. (1, 2) follows from the fact that in the external mesh the gears rotate In opposite directions, while in the internal mesh they rotate in the same direction, as has been stated before. In bevel gears the situation is similar, but the angular velocity vectors lie on intersecting, rather than parallel axes. Figure 3 shows an externally meshing bevel gear pair. The notation Is the same as in Figures 1, 2, except for the angular velocity vectors, 1 and 2 These are defined as follows. Let 0 be the point of intersection of the two shaft axes. Then the positive directions of the angular velocity vectors, 1, 2, are directed outward from point 0. The magnitudes of the angular velocity vectors, 1, 2, are denoted by 1, 2 respectively. The direction of rotation of a bevel gear is then obtained from the right-hand rule, as illustrated in Figure 4: if the angular velocity vector, 1, is as shown, the direction of rotation of the associated bevel gear corresponds to that of a right-handed screw advancing in the direction of vector i . The angular velocity ratio, Z21, in Figure 3 is again given by: Z21 = 2 in magnitude. (3) 1 T59
Figure 3 Externally Meshing Bevel Gears
Figure 4 Right-hand Rule
For externally meshing bevel gears, one angular velocity vector is directed away from 0, while the other angular velocity vector is directed towards 0. The case of spur gears may be regarded as a special case in which point 0 recedes to infinity. The angular velocities in helical and worm gears can be analyzed by similar reasoning. For details the reader is referred to paragraphs 5.9.4 and 8.5, respectively, in the Section on GEARS. 1.3 Simple Spur-Gear Trains In a simple gear train, such as shown in Figure 5, only one gear is mounted on each shaft. Suppose the ith gear (i = 1, 2, 3,......., n) has N teeth and rotates with angular velocity, i, measured positive counterclockwise. Then the angular velocity ratio, Zj1, of the jth gear relative to the first gear is given by: Zji = wj = Ni w1 Nj T60 (4), where
the + sign applies when j is odd; and the - sign applies when j is even. This follows directly from the fact that the directions of rotation of adjacent shafts are opposite for externally meshing gears. For internal gear meshes eq. (4) needs to be modified so as to account for the fact that shafts connected by an internal-gear mesh rotate in the same direction. 1.4 Compound Spur Gear Trains Such a gear train is shown in Figure 6. In a compound gear train at least one shaft carries two or more gears. In the typical train shown in Figure 6, all gears are keyed to their respective shafts, so that the angular velocities of all gears are equal to that of the shaft on which they are mounted. The angular velocities of adjacent shafts are governed by the gear ratio of the associated mesh. Let i denote the angular velocity of the jth shaft, going from left to right, measured positive counterclockwise. Then
Hence the angular velocity ratio, Z~, of shaft 5 to shaft 1, is given by:
Hence, for this train, (5) This is also equal to the ratio of the angular velocity of gear 8 to that of gear 1.
Figure 5 Simple Gear Trains Extended with Idler Gears T61
Figure 6 A Compound Gear Train
1.5 Reverted Gear Trains A reverted gear train Is a compound gear train at least one gear of which rotates freely on the shaft on which it is mounted, so that its speed is different from that of its shaft. In Figure 7, for example, gear #1 is rigidly connected to shaft A and gears #2, 3 are rigidly connected to shaft B. Gear #4, however, is free to rotate on shaft A. In this case,
Hence, the gear ratio, Z41, is given by:
While this formula is analogous to eq. (5), the difference is that angular velocity, w4, is not generally equal to 1, even though both gears are mounted on the same shaft.
Figure 7 Reverted Gear Train 1.6 Simple Planetary Spur Gear Trains A planetary or epicyclic gear train is one in which the axis of at least one of the gears is moving. The simplest such train consists of three members, such as shown in Figure 8. It consists of a stationary or sun gear (#1), a planet gear (#2) rolling externally on the sun gear and an arm (#3). The motion of planet 2 consists of a rotation about its axis (02), while that axis is rotating about the axis, 01, of the sun gear. In this sense the motion is like that of a planet around the sun. The path of a general point on the planet (inside the pitch circle) Is an epicycloidal path and hence, this gear train is also called an epicyclic gear train. T62
Figure 8 Simple Planetary Spur Gear Train For purposes of kinematic analysis we consider the more general case In which all three members are moving, but point O1 remains fixed. Let the angular velocities of the sun gear, planet and arm be denoted by 1, 2, 3, respectively, the positive sense being counterclockwise. We consider the actual motion of the system as composed of two motions: a. Motion with the arm: the gears and arm rotate as one integral unit about the center of the sun gear; b. Motion relative to the arm: the gears rotate about fixed centers while the arm is stationary. We then sum both motions in such proportion that the motion is the real motion of the system. For example, if gear #1 is stationary, its angular velocity when summed must vanish. This procedure can be conveniently carried out in tabular form and hence is known as the tabular method: Motion Rotation with arm Rotation relative to arm (Sum) Hence Gear #1 x Y Gear #2 x z21Y O Arm #3 x
1 = X + Y 1 = X + Y 2 = X+Z21Y 3 = XT63
2= X + Z21Y(i) (ii) (iii)
3 = X
Eliminating X, V from eqs. (i, ii, iii), we have: (7) This is the general relationship between the angular velocities of the gears and arm. It follows that when all members are free to rotate, it is a two-degree-of-freedom system. Now we specialize to the case of Figure 8 by setting w1 equal to zero. Eq. (7) then becomes: (8) From eq. (8) we note that the angular velocity of the planet is the difference of two angular velocities: the angular velocity of the arm and the angular velocity of the planet relative to the arm. For this reason such gear trains are also sometimes called differential gear trains. In the case of an internal spur gear mesh eq. (7) remains valid, the value of Z21 now being positive. The analysis of planetary bevel gear trains can be developed in a similar manner, with careful attention to the vector directions of each of the angular velocities. 1.7 Compound Planetary Spur Gear Trains These are also known as coupled epicyclics. Such trains consist of several Interconnected simple planetary spur gear trains. The interconnections can result in complex systems and practice is required in order to gain an understanding of their characteristics. A number of methods have been proposed for the kinematic analysis of compound planetary gear trains. A method which seems relatively straightforward and general is based on the following considerations. Any compound planetary gear train, as stated above, consists of a set of interconnected simple planetary gear trains. We can write the general equation (7) for each simple gear train and solve these for the various angular velocities. In order to carry Out this procedure we need to identify the two gears and the arm for each simple train, as well as the fixed member, if any. Planets in parallel are discarded inasmuch as these are kinematically redundant. They are used for strength and torque capacity. We illustrate the procedure in the case of the planetary gear train shown in Figure 9. This consists of an input arm, 2; a sun gear, 1; a floating link, 3, mounted on the arm and integral with the coaxial planets 3', 3", and an output gear, 4, coaxial with the axis of the arm. This is a compound planetary gear train consisting of two simple planetary trains. The first consists of arm 2, planet 3' and sun gear 1; the second consists of arm 2, planet 3M and gear 4. We rewrite eq. (7) in the general form: Zjii - j (1 - Zji)k = 0 (9), where i, j, denote the gears and k denotes the arm. The number of teeth on each gear and the angular velocities are defined as follows: Gear 1: N1 teeth; 1 A