Study of Contact Strength, Tooth Wear and Metal-Polymer Life of Worm Gears

This paper presents results of a study investigating worm gears consisting of polymer worm wheels and steel invo lute and Archimedes worms. The author uses his own calculation method to predict polymer wheel wear, gear life and maximum contact pressure in mesh. The effect of tooth correction and wear on gear life and contact pressure is considered. Cases of double and triple tooth engagement are analysed. The worm wheel is made of non-reinforced polyamide PA6. Quantitative and qualitative relationships are established between the maximum initial contact pressure along tooth profile and the tooth correction coefficient. Tooth wear causes a considerable decrease in contact pressure, with the highest decrease observed at the exit of engagement. The maximum contact pressure is generated at the exit of engagement. The same trend is observed for tooth wear. The minimum gear life is observed at the exit of engagement. It increases linearly with increasing the coefficient of tooth correction. The gear life significantly increases (by approx. 56%) in triple tooth engagement compared to double tooth engagement.


INTRODUCTION
Worm gears with metal wheels and worms have many different applications in machine building and other industrial areas. These toothed gears have threaded axes where meshing occurs by sliding friction. The most popular are involute and Archimedes worm gears. For their longlasting and reliable operation, it is necessary to ensure boundary lubrication. When lubricant degrades and required lubrication conditions are not met, there occurs galling and the gear is no longer capable of operation. For this reason, worm gears cannot be used under non-lubricating conditions, as this would cause dry friction. To overcome this problem, such gears can be made as a combination of metal (worm) and polymer (worm wheel). Worm wheels can be made of PA polyamides and their composites as well as POM polyacetals. Metal-polymer (MP) worm gears are used in various areas of human activity. They can be operated under low loads and have relatively short operating periods.
The development of methods for estimating load capacity, tooth wear and service life of metal-on-metal and metal-polymer worm gears is of vital importance. Load capacity (contact strength) of metal worm gears is determined in compliance with different standards (ISO / TR 14521: 2010, DIN 3396, BS 721, AGMA 6034, etc.). There exist only very few studies investigating the problem of contact pressure [12,20,21]. In [13,20] the effect of load on pressures under elastohydrodynamic lubrication (EHDL) was investigated. Studies [12,13,23] present methods for calculating contact pressures. The above-mentioned methods can be used for investigating MP worm gears albeit with some limitations. However, the literature provides no information about such investigations.
The application of computational and numerical methods for estimating wear and tribological Study of Contact Strength, Tooth Wear and Metal-Polymer Life of Worm Gears Myron Czerniec 1 , Antoni Świć 2* durability of metal worm gears is reported in very few studies [19,22,24]. Studies [6,18,27] propose methods for investigating gear tooth wear according to Archard's law of abrasive wear with EHDL. The method for estimating wear and life of worm gear proposed in [19] is based on empirical formulas. The numerical method of tooth wear estimation considering lubricant fi lm thickness proposed in [2,6,13] is also based on the Archard wear law. However, it should be emphasized that this wear law does not refl ect wear conditions. The above-mentioned methods do not take into account the eff ects of wheel tooth correction and their wear on the load capacity and service life of the gear. These eff ects have only been investigated by the author in his previous studies [2,4,6]. In [2,4] in gears with two-pair meshing, and in [6] -with two-and three-pair.
The literature review shows that there are no studies investigating the tribological behaviour of MP worm gears, and thus methods for estimating their wear and life at friction with lubrication and under dry friction conditions. The existing studies [9,11,15,26] investigate other aspects of these gears. A study [7,16] proposes a method for calculating load on involute worm gears with localized tooth contact. Elastic strains of gear teeth and bending stresses under diff erent loading conditions were determined by geometric modelling. A method for estimating quasi-static loading of MP worm gears (steel worm -polyamide worm wheel) is presented in [8, 11 15]. In [16] are presented results obtained for seven diff erent types of PA polyamide: PA6, PA6+15GF, PA6+30GF, PA66, PA66+30GF, PA66+60GF, which is widely used as a material for worm gears in cars. Their temporary tensile strength Young's modulus, abrasive wear resistance under dry friction conditions, and impact strength were determined. In [17] von Mises stresses in a worm wheel made of РА66+GF (25 wt.% and 50 wt.% fi berglass) were investigated. Experiments were conducted to determine the life of a prototype worm gear made of these polymer composites. A new method for determining the service life of MP worm gears was presented in [14,15,25].
Experimental results of pin-on-disk tests conducted under fry friction conditions for polyamides and their composites in combination with steel [10,14,17] have been reported in the literature. In [17] a combination of РА6 polyamide and AISI 02 steel was investigated. Results of numerous studies (over 20) conducted on reinforced and non-reinforced polymeric materials are reported in [14]. Studies [1,16,18] investigated the tribological behaviour of PA6 polyamide and steel under dry friction conditions and under friction with lubrication. This paper presents results of a study investigating the eff ect of meshing conditions, tooth correction and wear on the service life and loadbearing capacity of an MP worm gear with an involute and Archimedes worm. The worm wheel was made of PA6 polyamide and PA6+30GF.

CALCULATION METHOD FOR MP WORM GEARS
The proposed modifi ed calculation method for MP worm gears ( Fig.1) is based on the previous method developed for metal worm gears [2,3,5,6] that are operated under boundary friction or EHDL conditions. The proposed calculation method is based on phenomenological method for estimating material wear at sliding friction [1,4,7] due to fatigue wear. The application of this friction mechanism for materials such as polymers is more justifi ed than the use of the adhesive-abrasive wear mechanism according to the Archard linear wear law.
MP worm gears are usually operated under dry friction conditions, because polyamides and polyacetals used for worm wheels have very good lubricating properties. Nevertheless, these gears can also be used with lubrication.
Below is shown a modifi ed calculation method for MP worm gears. The function of linear wear of worm wheel teeth 2 during engagement under initial (constant) contact pressure has the following form: where: t j ' = 2b j / v j is the time of contact of the elements in mesh at j-th point on friction path with a length of 2b j , Q r is the contact area width based on the Hertz theory of contact stresses, v j is the sliding velocity at a point j of engagement at a height of worm threads in the axial section, w = 1, 2, 3 is the number of pairs of teeth carrying the load, f is the sliding friction factor, C,m are the indicators of abrasive wear resistance of materials of the worm teeth and worm wheel, determined via experiments conducted according to the method described in [19], τ S2 = 0.5R m is the shear strength of the polymer R m is the temporary tensile (compression) strength of the polymer is the maximum contact pressure, are respectively Poisson's ratio and Young's modulus for the material of the worm (k = 1) and worm wheel (k = 2), b is the width of the worm wheel, j denotes points on the worm thread profile and worm wheel teeth.

Maximum contact pressures
are determined using the Hertz equation where: N′ is the tooth load, ρ j is the equivalent curvature radius at a point j of engagement.
( ) where: N n is the torque on the wormshaft, n 1 is the number of revolutions of the worm,

( )
/ cos arctg f ′ ρ = α is the apparent friction angle, N is the power transmitted by the gear.
The equivalent curvature radius ρ j at a point j of engagement is calculated from profile curvature radii of the worm 1 j ρ and worm wheel 2 ρ j Hence, • for an involute worm gear ( ) • for an Archimedes worm gear with trapezoid profile it is necessary to consider where: , cos b 1 c r 0 5d = α is the radius of the main cylinder of the worm, Other parameters: • for an involute worm gear: • for an Archimedes worm gear: The total sliding velocity during engagement where: j v′ is the velocity due to rotational motion of the worm (helical component), j v ′ ′ is the velocity of a contact point along the tooth profile (velocity component resulting from rotating about the teeth). where: are the angular velocities of the worm and worm wheel, respectively.
The gear life t * for a given permissible wear h 2* of worm wheel teeth, assuming that The application of gear tooth correction leads to changes in interaxial distance and reference diameter where: a w = r 1 + r 2 is the interaxial distance in a non-corrected gear, x 2 is the tooth correction coefficient. Now using Equation (3) which describes tooth load and the equation describing e pAj it is necessary to deduce a radius of r w1 = 0.5d w1 .
Worm wheel tooth correction does not cause any changes in worm dimensions.
Worm wheel teeth undergo wear with operation, which leads to reduction in maximum contact pressure and, as a result, to increased gear life. This relationship must therefore be taken into consideration. The change in the initial curvature  Since wear of the steel worm can be omitted, the variable curvature radii are:  (14) According to the rotary-cumulative and block-cumulative procedure the total linear wear Gear life for the total number of worm wheel revolutions 2 n * resulting in reaching the permissible tooth wear 2* h is calculated as: 2 2 / 60 t n n * * = (16)

RESULTS AND DISCUSSION
The following were determined: the initial contact pressure The calculations were made using the following load-related, kinematic, geometrical, material and tribological parameters: Dry friction occurs in the tested gears.
The following contact points j on the worm wheel tooth profile were selected: Results ar e given in Figs The contact pressure is the lowest at the entry of engagement while its highest values are located at the exit of engagement. It increases up to 2.067 times. A similar qualitative change can also be observed along tooth profile. The increase value depends on the tooth correction coefficient x 2 . Tooth correction reduces the pressure by approx. 1.55 times. The contact pressure in the MP involute worm gear is up to 1.34 times higher than that in the Archimedes worm gear, particularly at the exit of engagement. The maximum contact pressure in the gear with a PA6 worm wheel is up to 1.03 times lower than that obtained for the gear with a PA6+30GF wheel.
It can be observed that the maximum contact pressure ( ) Increasing the number of teeth in mesh from w = 2 to w = 3 leads to a decrease in the pressure by up to 1.22 times. This demonstrates a close relationship between tooth wear and reduction in initial pressure at the exit of engagement point (it is reduced up to 2.06 times).
Between the entry and exit of engagement, the pressures ( ) max w jh p increase in a linear fashion. The minimum gear life t min (p j = const) and t Bmin (p j = var), considering the eff ect of tooth correction, is shown in Figs. 9 and 10.  Tooth correction leads to a linear increase in minimum gear life (at j = 5). The life of the gear after tooth correction is about 1.2 times longer than that of the gear with non-corrected teeth for p j = const (t min) and about 1.25 times for p j = var (t Bmin ) due to tooth wear. The minimum gear life is observed at the exit of engagement. The life of the Archimedes worm gear is approx. 1.28 times longer than that of the involute worm gear. The life of the gear with a PA6+30GF wheel is approx. 1.5 times longer than that of the gear with a PA6 wheel. It can also be observed that the gear life greatly depends on the tooth engagement type (by approx. 1.55 times).
The linear wear h 2j of gear teeth along their profi le is shown in Figs. 11 and 12.
Lower wear is located at the entry of engagement while the permissible wear 5 0 2 , h * = mm can be observed at the exit of engagement. The wear increases linearly along the tooth profi le.
The linear wear 2(5) h of corrected teeth at the exit of engagement per one hour of operation is shown in Figs. 13 and 14. The results show the fi nal hour of gear operation for the models p j = const and p j = var.
Tooth correction leads to a decrease in the linear wear 2 (5) h (at exit of engagement). The decrease depends on the type of gear and meshing as well as worm wheel polymeric material.

CONCLUSIONS
According to the presented analytical method of testing metal-polymer involute and archimedes worm gears, quantitative and qualitative regularities of contact pressures in meshing, wear of non-metal worm gear teeth, gear service life were established.
It can be observed that the maximum contact pressure increases as the worm wheel teeth enter in mesh, which results from a decrease in equivalent curvature radius. The maximum contact pressure is located at the exit of engagement. The maximum contact pressure is signifi cantly reduced due to the wear of worm wheel teeth. Worm wheel tooth correction leads to reduced contact pressures and increased gear life. The minimum gear life of the worm wheel teeth is observed at the exit of engagement, i.e. at tooth root. It is slightly higher at the addendum. An increase in tooth pairs in engagement from two to three leads to a considerable reduction in contact pressure and, at the same time, to a signifi cant increase in gear life.