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13.2 Two-Stage Gear Train    A two-stage gear train uses two single-stages in a series. Figure 13-2 represents the basic form of an external gear two-stage gear train. Let the first gear in the first stage be the driver. Then the speed ratio of the two-stage train is:     Speed Ratio = z1 z3 n2 n4            (13-3)                          z2 z4  n1  n3 In this arrangement, n2 = n3     In the two-stage gear train, Figure 13-2. gear 1 rotates in the same direction as gear 4. If gears 2 and 3 have the same number of teeth, then the train simplifies as in Figure 13-3. In this arrangement, gear 2 is known as an idler, which has no effect on the gear ratio. The speed ratio is then:    Speed Ratio = Z1 Z2 = Z1                 (13-4)                          Z2  Z3   Z3 13.3 Planetary Gear System    The basic form of a planetary gear system is shown in Figure 13-4. It consists of a sun gear A, planet gears B, internal gear C and carrier D. The input and output axes of a planetary gear system are on a same line. Usually, it uses two or more planet gears to balance the load evenly. It is compact in space, but complex in structure. Planetary gear systems need a high-quality manufacturing process. The load division between planet gears, the interference of the internal gear, the balance and vibration of

the rotating carrier, and the hazard of jamming, etc. are inherent problems to be solved.    Figure 13-4 is a so called 2K-H type planetary gear system The sun gear, internal gear. and the carrier have a common axis. 13.3.1 Relationship Among the Gears in a Planetary Gear System    In order to determine the relationship among the numbers of teeth of the sun gear A, za, the planet gears B, zb and the internal gear C, zc and the number of planet gears, N, in the system, the parameters must satisfy the following three conditions: Condition No. 1:   Zc= Za+2 Zb     (13-5)    This is the condition necessary for the center distances of the gears to match. Since the equation is true only for the standard gear system, it is possible to vary the numbers of teeth by using profile shifted gear designs.    To use profile shifted gears, it is necessary to match the center distance between the sun A and planet B gears, ax1, and the center distance between the planet B and internal C gears, ax2      ax1 = ax2         (13-6) Condition No. 2:    (za + zc)=integer (13-7)          N    This is the condition necessary For placing planet gears evenly spaced around the sun gear. If an uneven placement of planet gears is desired, then Equation (13-8) must be satisfied.    (za+zc)q=Integer  (13-8)        180 where:    q = half the angle between adjacent planet gears

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