Influence of the cylindrical gear pairs parameters to the transverse contact ratio - Dobratić, Predrag S.; Ristivojević, Mileta R. (Vojnotehnički glasnik / Military Technical Courier, 2014)
In the field of transmission and power transformation from the power engine to the working machine, gear pairs, are mostly used in mechanical engineering due to their compactness of the structure, high reliability and capacity. One way of improving the performance characteristics of gear pairs, and thus the gear transmitters, is to increase the number of simultaneously meshed pairs of teeth, or increasing the transverse contact ratio. To this end, this paper analyzes in detail the partial and simultaneous influence of the number of teeth and tooth profile shapes, moving through shifting coefficient and pressure angle, to the number of simultaneously meshed pairs of teeth. The obtained results allow us to define the optimum parameters of cylindrical gear pairs, in terms of the size of the transverse contact ratio. Introduction The kinematic indicator of the existence of the transmitting continuity of a rotary movement is the total contact ratio. In order to achieve the transmitting continuity of a rotary movement, the total contact ratio must be higher than one. Based on the total contact ratio, the information on the number of simultaneously meshed pairs of the teeth which rotate during the contact period is obtained. For example, if the  – one and two pairs of teeth alternately take turn, and if the  – two and three pairs of teeth alternately take turn, etc. It should be noted that the  is a necessary but not sufficient condition for achieving multiple meshes. The next condition to meet is the compatibility between the accuracy of teeth production, the teeth rigidity and the intensity of the total load of the teeth pair. The influence of the addendum coefficient on the transverse contact ratio is shown in the paper (Li, 2008). It is shown that changes in the addendum coefficient in conjunction with the corresponding values of the shifting coefficient and gear ratio can achieve the transverse contact ratio value higher than four. The paper (Imrek, 2009) showed that the appropriate structural solution of the form of the teeth of spur gears can achieve a double mesh of the teeth at all points of the lenght of the line of action, when the value of the transverse contact ratio is less than two. The mathematical model of the transverse contact ratio This section presents a mathematical model (equations from 1 to 4) and the algorithm (Fig. 1) to determine the transverse contact ratio. The effect of the pressure angle and the number of teeth on the transverse contact ratio This section analyzes the effect of the pressure angle and the number of the teeth meshed on the transverse contact ratio using equation 4. Figs. 2, 3 and 4 show the effect of the pressure angle and the number of the teeth meshed on the size of the transverse contact ratio when the shifting coefficient and , and when the number of the teeth on the pinion is in the interval from . The analysis shows that the influence of the number of the teeth on the transverse contact ratio is much more pronounced at smaller values of the pressure angle. It can be concluded that the transverse contact ratio increases with the number of teeth and with a reduction in the pressure angle. The gradient of enhancement of the transverse contact ratio is more expressed at lower values of the gear ratio and at lower pressure angles. The effect of the pressure angle, the number of teeth and the shifting coefficient on the transverse contact ratio This section analyzes the effect of the pressure angle, the number of teeth meshed and the shifting coefficient on the transverse contact ratio using equation 3. Shifting coefficients This section shows the influence of the pressure angle and the number of the teeth meshed on the size of the transverse contact ratio when the shifting coefficients are and when the number of the teeth on the pinion is in the interval from (Figs. 5, 6 and 7). The influence of the number of the teeth on the transverse contact ratio is significantly higher when the shifting coefficients are higher than zero, but at the same time the value of the transverse contact ratio is lower. Shifting coefficients and This section shows the influence of the pressure angle and the number of the teeth meshed on the size of the transverse contact ratio when the shifting coefficients are and , and when the number of the teeth on the pinion is in the interval from (Figs. 8, 9 and 10). When the shifting coefficients of the pinion are positive and the shifting coefficients of the gear are negative, the influence of the number of the teeth on the transverse contact ratio is lower than in the case when both shifting coefficients are positive. At the same time, the interval number of the teeth of the pinion is increased, when the triple mesh is achieved. Also, the value of the transverse contact ratio is increased. Shifting coefficients and This section shows the influence of the pressure angle and the number of the teeth meshed on the size of the transverse contact ratio when the shifting coefficients are  and , and when the number of the teeth on the pinion is in the interval from (Figs. 11, 12 and 13). When the shifting coefficients of the pinion are negative and the shifting coefficients of the gear are positive, the influence of the number of the teeth on the transverse contact ratio is much lower compared to the case when both coefficients are positive or when the shifting coefficient of the pinion is positive, a the gear's one is negative. At the same time, the interval number of the teeth of the pinion is increased, when the triple mesh is achieved. In this case, the highest values of the transverse contact ratio were obtained.  The interval of the pressure angle was increased when the triple mesh of the teeth is achieved. The change of the transverse contact ratio by changing the addendum coefficient The paper (Li, 2008) analyzed the impact of the addendum coefficient () on the transverse contact ratio of the cylindrical gear pair. The analysis was conducted for two gear pairs, when the gear ratio  and , and when the shifting coefficients . What was shown in both variants is that the transverse contact ratio increases with the growth of the addendum coefficient. To perceive the simultaneous influence of the addendum coefficient and the shape of teeth profiles on the transverse contact ratio, the shifting coefficient was changed in the paper, and the results are shown in Tables 1 and 2. Conclusion Higher values of the number of teeth meshed gears and higher values of the gear ratio correspond to higher values of the transverse contact ratio. In contrast, the lower values of the pressure angle correspond to higher values of the transverse contact ratio. Thereby, the gradient of the influence of the number of teeth and the pressure angle on the  transverse contact ratio depends on the value of the shifting coefficient. This gradient has the highest value when the shifting coefficients of the pinion and the gear are negative. The negative values of the shifting coefficient of the pinion and the positive values of the shifting coefficient of the gear cause the smallest gradient influence of the number of teeth and the pressure angle on the transverse contact ratio. At the same time, the combination of shifting coefficients causes the highest value of the  transverse contact ratio. The analysis has shown that by varying the number of teeth of the gear pair, the pressure angle and the shifting coefficient, the value of the transverse contact ratio higher than three cannot be accomplished. This means that during the contact period only a double and triple mesh of teeth can be achieved. For achieving the quadruple mesh of teeth, it is necessary to change the addendum coefficient.
Keywords: spur gears; mesh of the teeth; contact ratio