Contributions to the Synthesis of Fixed Axle Gears by Avoiding the Interference Phenomenon

Email: rvvpetrescu@gmail.com Abstract: It can be seen that the minimum number of teeth required to avoid interference for the standard pressure angle, normally on the alpha0 = 20° decomposition circle, is 13, corresponding to a transmission ratio i = 1 and increases with the transmission ratio i reaching to the maximum value of 18 teeth for i>100. For normal transmission ratios, values ranging from 13 to 17 teeth for standard pressure angle. If alpha0 decreases to 4°, zmin varies between 275 and 410 teeth. When alpha0 increases to 35°, zmin varies between 5 and 6 teeth. By lowering the number of teeth of the drive wheel 1, the coverage and gearing decreases as well as increases the pressure angle, increases effort, wear and reduces the life span of the gear. If we increase in turn, the minimum number of input wheel teeth increases the coverage, increases the gear efficiency, reduces the pressure angles and effort in the coupling, increases gear reliability and operates with much lower vibrations and noise, with high yields and for a longer time. The minimum number of teeth required to avoid interference is basically a function of the transmission ratio i = |i12| = z2/z1 and the normal pressure angle of the alpha circle and the angle of inclination of the beta teeth. This is practically also maintained in internal gear gears, where there are still two additional types of interference. It is noted that zmin decreases when i decreasing and when alpha0 and/or beta increases.


Introduction
According to the standards in force (see Standard 915/2-81), the gear is defined as an elementary mechanism consisting of two gears (wheels, sectors, or toothed racks) in absolute/relative rotation/translation, in which one of the elements trains the other through the action of the teeth in successive and continuous contact.
Gears, or gears with gears, are basically upper couplings (generally C4th grade), which have the function of transmitting and/or turning the movement by reducing the speed (with the torque increase) or by increasing the angular velocity (with lowering the load), from the input to the exit, with an almost constant power holding (with very small losses, mechanical and friction due to the large and very high yields of the gears).
The oldest, most used (more widespread), more reliable and performing better, are fixed axle gears that will be presented in this chapter.
There are also movable axles (subject to a separate chapter), or mixed, which, although lighter and more compact, work in return with lower yields than fixed axes and are less rigid and reliable.
From a structural-geometric-cinematic (and constructive) point of view, fixed axle gears are classified into three major categories, depending on the relative position of the axles of the two wheels making up the gear: • A-parallel (cylindrical), • B-competing conical) • C-crosses (spindle-worm, spatial, toroidal).
The cylindrical gears (A) may be external (between two external gears) or inside (between an external gear and one with internal toothing).
They can also be combined, an element with rotating motion (a toothed wheel with external teeth) and the other one (rack).
The geometric elements of a toothed wheel and a gear can be seen in Fig. 1 and 2 (according to international standards). If the rotation axes are parallel, the gear unit is called cylindrical. When the tooth line has the same direction as the axis of rotation, it is said that the gear has straight teeth.  The main parameters of such gear are shown in Fig. 1, in which the teeth of an unpainted wheel are represented in an outer cylindrical gear not rigid with straight teeth.
The starting element of a wheel is the splitting circle (or step -on which the pitch is measured), a circle that defines the position of the other circles of the wheel. The diameter of the splitting circle is one of the first elements to be calculated on a wheel and on a gear (at a gear we will have two wheels, so two division diameters).

Materials and Methods
The number of gear pairs engaged simultaneously (for good gearing) is the degree of coverage. So the gear ratio ("contact ratio" in English) marked with (shows how many pairs of teeth are engaged at the same time; Fig. 3).
In order for the engagement to take place without shocks, without sliding, no noise and no play, the gear is designed so that when a pair of teeth is out of engagement, the next pair is already engaged.
The circle on the base circle shows how long a pair is engaged.
Whenever it encompasses in the actual AE engagement segment, so many pairs of engagement will fit simultaneously into the AE segment on which the actual engagement is made. Practically, the degree of coverage will be the ratio of AE to p b .
It must be overhead to have multiple pairs in simultaneous engagement so that no "dead times", interruptions of engagement, gaming and gambling collisions occur due to gaming, which also produces vibrations and noises.
A higher degree of coverage also brings increased mechanical efficiency.
The engagement segment AE is calculated directly with relation (1)  The degree of coverage ε is determined by dividing AE to the step p b (relation 2):

Results and Discussion
In order to avoid the interference phenomenon ( Fig. 4), point A must be between C and K 1 (i.e., the wheel head wheel 2, Ca 2 must cut the engagement segment between the points C and K 1 and in no shape not exceed K 1 ). Similarly, the circle Ca1 must cut the right-hand drive between points C and K 2 , determining the point E, which in no way must pass K 2 . These conditions of avoidance of interference are written with relations (3):     The relation that generates it always gives lower values than the relationship that generates it, so that the condition (4) is sufficient to find the minimum number of teeth required to avoid the interference of the gear teeth; in other words, the initial condition is that point A should be between points C and K: Table 1 shows the values obtained with the relationship (4) for different standardized values of the transmission ratio i and for three different values assigned to the pressure angle.

Conclusion
It can be seen that the minimum number of teeth required to avoid interference for the standard pressure angle, normally on the alpha0 = 20° decomposition circle, is 13, corresponding to a transmission ratio i = 1 and increases with the transmission ratio i reaching to the maximum value of 18 teeth for i> 100. For normal transmission ratios, values ranging from 13 to 17 teeth for standard pressure angle. If alpha0 decreases to 4°, z min varies between 275 and 410 teeth.
When alpha0 increases to 35°, z min varies between 5 and 6 teeth.
By lowering the number of teeth of the drive wheel 1, the coverage and gearing decreases as well as increases the pressure angle, increases effort, wear and reduces the life span of the gear.
If we increase in turn, the minimum number of input wheel teeth increases the coverage, increases the gear efficiency, reduces the pressure angles and effort in the coupling, increases gear reliability and operates with much lower vibrations and noise, with high yields and for a longer time.
The minimum number of teeth required to avoid interference is basically a function of the transmission ratio i = | i 12 | = z 2 /z 1 and the normal pressure angle of the alpha circle and the angle of inclination of the beta teeth. This is practically also maintained in internal gear gears, where there are still two additional types of interference.
It is noted that z min decreases when i decreasing and when alpha0 and/or beta increases.

Acknowledgement
This text was acknowledged and appreciated by Dr. Veturia CHIROIU Honorific member of Technical Sciences Academy of Romania (ASTR) PhD supervisor in Mechanical Engineering. Antonescu, P. and F.I.T. Petrescu, 1989. Contributions to cinetoelastodynamic analysis of distribution mechanisms. Bucharest. Antonescu, P., M. Oprean and F.I.T. Petrescu, 1985a. Contributions to the synthesis of oscillating cam mechanism and oscillating flat stick. Proceedings of the 4th International Symposium on Theory and Practice of Mechanisms, (TPM' 85), Bucharest. Antonescu, P., M. Oprean and F.I.T. Petrescu, 1985b. At the projection of the oscillate cams, there are mechanisms and distribution variables. Proceedings of the 5th Conference of Engines, Automobiles, Tractors and Agricultural Machines, (AMA' 58), I-Motors and Cars, Brasov. Antonescu, P., M. Oprean and F.I.T. Petrescu, 1986. Projection of the profile of the rotating camshaft acting on the oscillating plate with disengagement. Proceedings of the 3rd National Computer-aided Design Symposium in the field of Mechanisms and Machine Parts, (MMP' 86), Brasov. Antonescu, P., M. Oprean and F.I.T. Petrescu, 1987. Dynamic analysis of the cam distribution mechanisms. Proceedings of the 7th National Symposium on Industrial Robots and Space Mechanisms, (RSM' 87), Bucharest. Antonescu, P., M. Oprean and F.I.T. Petrescu, 1988. Analytical synthesis of Kurz profile, rotating the flat cam. Mach, Build. Rev. Antonescu, P., F.I.T. Petrescu and O. Antonescu, 1994. Contributions to the synthesis of the rotating cam mechanism and the tip of the balancing tip. Brasov. Antonescu, P., F.I.T. Petrescu and D. Antonescu, 1997. Geometrical synthesis of the rotary cam and balance tappet mechanism. Bucharest, 3: 23-23. Antonescu, P., F.I.T. Petrescu and O. Antonescu, 2000a. Contributions