Some New Gears Aspects

Corresponding Author: Florian Ion Tiberiu Petrescu ARoTMM-IFToMM, Bucharest Polytechnic University, Bucharest, (CE), Romania Email: scipub02@gmail.com Abstract: In today's mechanical transmissions most widely used, the gears, are spread across all industries. For this reason, their importance has become overwhelming, which is why we want to recall in this paper some important aspects regarding toothed wheels. It is the geometry, cinematic, the forces and the yields of these mechanisms, which will be presented in the work in the form of newly synthesized relations on modern bases. Another important aspect in toothed wheels is their synthesis by modern methods that avoid tooth interference during operation. To avoid the interference between teeth, we must know the minimum number of teeth of the driving wheel, in function of the pressure angle (normal on the pitch circle, alpha0), in function of the tooth inclination angle (beta) and in function of the transmission ratio (i). In optimal and high-efficiency gearing, gears require a modern design with increased coverage. These achievements can only be achieved today in the context of lowering the value of the alpha engagement angle. Through all the aspects presented, which relate to the dynamics of gears, the work can be considered among those of the optimal dynamic synthesis of the gears.


Introduction
Gears have spread today in all areas. They have the advantage of working with very high efficiency. In addition, tools can transmit large tasks. Regardless of their size, tools need to be synthesized carefully according to specific conditions. This paper tries to present the main conditions that must be met for the correct synthesis of a tool.
The beginning of the use of pinion gears should be sought precisely in ancient Egypt at least a thousand years before Christ, where for the first time wheel drive units were used for irrigation and worm gear worm gears for cotton processing.
Then, 230 years BC, in Alexandria, Egypt, the toothed wheel was used again.
These tools have been built and used since ancient times to handle heavy anchors and catapults used on battlefields. These were then introduced into the wind and water mills (as a reduction or multiplication in wind or water pumps) (Fig. 1).
The Antikythera Mechanism is a name given to a complex astronomical device, a 32×16×10 cm device discovered in 1900 in a sunken ship near the coast of Antikythera, an island between Crete and the Greek continent, for which several types of evidence undoubtedly point to around 80 BC. for the date of the shipwreck. The device was made of bronze gears mounted in a wooden box, but due to the fact that it was crushed in the wreck, various parts of the faces were lost and the remainder was then covered with a hard limestone deposit in time at the same time as the corroded metal to a thin core covered with strong metal salts that retains much of the previous bronze shape during the 2000 years of the dive (See Antikythera 1 in Fig. 2).
The modern adventure of the toothed wheel began with the toothed wheel created by Leonardo da Vinci in the fifteenth century. He is also the founder of a new cinema and dynamics, stating, among other things, the principle of overlapping independent movements (Fig. 3).
Benz has created an original toothed and transmission chain engine (patented after 1882, Fig. 4), but the first patent of a toothed gear belongs to British British Starley & Hillman in 1870 (12 years before the Germans) being designed and built to be used for bicycle transmissions and later for motored tricycles.
In Cleveland (USA), begin after 1912 to produce industrial specialized wheels and gears (cylindrical, worm, conical, with straight teeth, inclined or curved; Fig. 5).       The paper presents how to accurately determine the mechanical performance of a gearbox. Based on these relationships, an optimal synthesis of the performance of a classic, mechanical, manual gearshift can be achieved regardless of its operating status (Frăţilă et al., 2011;

Materials and Methods; Gearings Synthesis
In a cylindrical gearing, forces, speeds, powers and efficiency can be determined using relationships 2.1-2.6 and can be seen in Fig. 7:

Gearing today.
Where: F m = The motive force (the driving force) F τ = The transmitted force (the useful force) F ψ = The slide force (the lost force) v 1 = The velocity of element 1, or the speed of wheel 1 (the driving wheel) v 2 = The velocity of element 2, or the speed of wheel 2 (the driven wheel) v 12 = The relative speed of the wheel 1 in relation with the wheel 2 (this is a sliding speed) The consumed power (in this case the driving power): The useful power (the transmitted power from the profile 1 to the profile 2) will be written: The lost power will be written: The momentary efficiency of couple will be calculated directly with the next relation: The momentary losing coefficient, will be written: It can easily see that the sum of the momentary efficiency and the momentary losing coefficient is 1. Now, one can determine the geometrical elements of gear. These elements will be used in determining the couple efficiency, η.
The main geometric elements belonging to the external cylindrical gear (for straight teeth, beta = 0) can still be determined.
The radius of the basic circle of the wheel 1 (of the driving wheel), (2.7): The radius of the outside circle of wheel 1 (2.8):  And now one determines the same parameters for the wheel 2, the radius of basic circle (2.10) and the radius of the outside circle (2.11) for the wheel 2:  Now we can determine, for the external gear, the minimum (2.13) and the maximum (2.9) pressure angle for the right teeth. For the external gear with bended teeth (β ≠ 0) it uses the relations (2.14, 2.15 and 2.16): The mechanical efficiency of the cylindrical gear shall be determined by integrating the instantaneous efficiency across all gear sections of the gear unit starting from the minimum pressure angle and going up to the maximum pressure angle as defined in expression (2.21): External gears are the most common, not because they are the best, but because they are easier to design and build (Fig. 8a). Internal gears can be much more efficient and more reliable if and only if properly designed ( Fig. 8b). At inner engagement the teeth in contact make better contact not only on a point or line as on the outside but on a curve or surface, contact being larger, more natural, stronger, more complete and without wear, noises, shocks, like in external gears. However, due to the fact that the internal gears are provided with additional conditions for avoiding the teeth interference in contact, the correct design is much more difficult and from a technological point of view it sometimes does not work correctly, leading to slight random interferences during the operation of the builtin gear, which in the course of time lead to premature wear, noises, or even blocking in operation, although their operation should have been much superior theoretically and for this reason the design difficulties most often give up the superiority internal gears preferring the choice of the outer ones.
To an external gearing, contact between profiles shall only be made to a single point, while at the internal gearing the contact between profiles is by winding each other (Fig. 9). Line of action Line of action n2 C

External gearing
Internal gearing

Results; Gears Synthesis by Avoid the Interferences
In order to avoid interference phenomenon, point A must lie between C and K 1 (the addendum circle of the wheel 2, C a2 need to cut the line of action between points C and K 1 and under no circumstances does not exceed the point K 1 ). Similarly, C a1 addendum circle must cut the action line between points C and K 2 , resulting in point E, which in no circumstances, does not exceed the point K 2 .
The conditions to avoid the phenomenon of interference can be written with the relations (3.1).
The basic conditions of interference, are the same (CA<K1C; CE<K2C), but the originality of this new presented method consist in the mode in which it was solved the classical relationship (see the system 3.1) (Fig. 10).
The system (3.3) represents a simple, unitary and general relationship capable of generating functional solutions for gears, giving the minimum number of teeth of wheel 1 (motor wheel) to avoid interference. In the appendix Table 1-15 an alpha0 value (35°) will be chosen and the beta angles (from 0° to 40°) and the transmission ratio i (from 1 to 80) are incrementally incremented in order to thus getting the minimum number of teeth correctly.
Then, the alpha value (from 35° to 5°) will be decreased successively.
At the internal gearbox, the interference avoidance condition is the same as for the external gear (relationship 3.3): CA K C and CE K C When we have inclined teeth, one takes z min →z min /cosβ and α 0 →α 0t and the relationship (3.2) takes the form (3.3). The minimum number of teeth of the driving wheel 1, is a function on some parameters: The pressure angle (normal on the pitch circle, α 0 ), the tooth inclination angle (β) and the transmission ratio (i = |i 12 | = |-z 2 /z 1 | = z 2 /z 1 ), (see the relationship 3. In addition, the inner gear can also write the additional condition of the wheel with internal teeth (systems 3.4 and 3.5). If the mechanism is designed and built without checking these two additional conditions for the existence of an internal gear, it will not work properly. As it has already shown, the inner gear is much superior in operation to the outside, but only when rigorous design and construction, its manufacturing technology being much more difficult than that of the classic outer gear: It should also be mentioned that additional relations (3.6) have also been used.

Discussion; Determining the Gearing Performance Depending on the Degree of Coverage
In this section, there is briefly presented a completely original method of determining the efficiency of parallel gear gears. Based on the computational relationships presented, the dynamic synthesis of the gears can be made so as to result in mechanisms with high efficiency in operation.
The originality of the method consists in determining the yield (which does not take into account the friction coefficient in the coupling, this being considered only an additional effect and not the main cause that produces the effective mechanical efficiency, the mechanical efficiency of a machine depends on the authors' mainly by the transmission angle of the main coupler of the mechanism).
Calculate the yield of a gear with a fixed spindle gear, considering that at a certain moment there are several pairs of drive drums, not just one.
It starts from the idea of having four pairs of drums in engagement (simultaneous). The first pair of teeth (which go on the right-to-left engagement line as it engages) are the engagement point i, defined by the radius of the r i1 and the angle (pressure) of the position a i1 ; the forces at this point are the force of the F mi motors, perpendicular to the point and position of the vector and the force transmitted from the wheel 1 to the second wheel by the point i, F ti , parallel to the engagement line and pointing from the wheel 1 to the wheel 2, the transmission force being basically the projection of the drive force on the engagement axis (line); the defined speeds are similar to the forces (for the original cinematic, precision); the same parameters will also be defined for the other three points of engagement, j, k, l (following the drawing in Fig. 11).
Write the relationships between speeds (4.1) first: From relations (4.1) one obtains the equality of tangential speeds (4.2) and we express the motor speeds (4.3): The forces simultaneously transmitted at the four points must be equal to each other (4.4): Engine forces shall be deducted (4.5): ; ; ; cos cos cos cos Instant yield is written as (4.6): The help lines used shall be marked with (4.7) and (4.8): ( ) ( )  Relationships (4.8) were retained where the plus sign is for the gears to which the driving wheel 1 has external gear (external or internal gearing) and the minus sign is used when the driving wheel 1 has internal gearing, i.e., when the driving wheel 1 is a ring (only at the inner engagement). The instantaneous yield in the expression (4.6) uses relations (4.8) and takes the form (4.9): One starts in the relation (4.9) with the expression of the yield (4.6) for 4 pairs of engaging teeth, but immediately (even within the relation) we make a generalization by replacing the number 4 (four pairs of engaging teeth) with the variable E, which represents the full side of the coverage +1 and after the expressions written in the form of sums narrow down, the variable E is replaced with the degree of coverage, thus reaching the final shape. The average yield is more interesting than the instantaneous one and is calculated (precisely by integrating the instantaneous one from the minimum pressure to the maximum angle) simply by the approximation which determines the average yield by replacing in the expression of the instantaneous yield of the variable pressure angle (α 1 ) with its average value given by the normal pressure angle (standardized, α 0 ), (4.10), where ε 12 represents the degree of coverage and is calculated with the expression (4.11) for the external engagement and the relation (4.12) for the inner engagement:  There are wheels of the helical gears, which are used very often (relations 4.13, 4.14, 4.15). For helical gears, the calculations show a decrease in the efficiency of the with the tilt angle on the rise of the teeth (β). For given angle which does not exceed 25°, the efficiency of fishing gear is good enough. However, when the tilt angle is greater than 25°, speeds will suffer a significant decrease in the yield:

Conclusion
There are wheels of the helical gears, which are used very often. For helical gears, the calculations show a decrease in the efficiency of the with the tilt angle on the rise of the teeth (β). For given angle which does not exceed 25°, the efficiency of fishing gear is good enough. However, when the tilt angle is greater than 25°, speeds will suffer a significant decrease in the yield.
The highest yield that can be achieved with two gears is that of the inner gear, with the inner driving gear (the wheel becomes the driver and the smaller wheel with external gear will be driven); Conversely, when we form an inner gear with the small (outer gear) driving wheel, the resulting yield is the smallest possible; When gearing is external, the output is higher for the high steering wheel; The more the normal engagement angle, α 0 , decreases, the degree of coverage increases and with it the engagement efficiency; when the normal angle of engagement drops to 5 degrees, the coverage reaches 6.5-7.3 and the output reaches theoretical values of 99-99.5%, meaning that the gear will actually work at 100%. Yield increases also with the number of teeth of the driving wheel.
In order to avoid interference phenomenon, point A must lie between C and K 1 (the addendum circle of the wheel 2, C a2 need to cut the line of action between points C and K 1 and under no circumstances does not exceed the point K 1 ). Similarly, C a1 addendum circle must cut the action line between points C and K 2 , resulting in point E, which in no circumstances, does not exceed the point K 2 .
The conditions to avoid the phenomenon of interference can be written with the relations (3.1).
The basic conditions of interference, are the same (CA<K1C; CE<K2C), but the originality of this new presented method consist in the mode in which it was solved the classical relationship (see the system 3.1).
The system (3.3) represents a simple, unitary and general relationship capable of generating functional solutions for gears, giving the minimum number of teeth of wheel 1 (motor wheel) to avoid interference. In the appendix Table 1-15 of Figure 12 an alpha0 value (35°) will be chosen and the beta angles (from 0° to 40°) and the transmission ratio i (from 1 to 80) are incrementally incremented in order to thus getting the minimum number of teeth correctly.
Then, the alpha value (from 35° to 5°) will be decreased successively.
Appendix, Figure 12 with 15 Tables