INFLUENCE OF A NON-STANDARD GEOMETRY ... - Dunarea de Jos

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12 THE ANNALS OF UNIVERSITY “DUNĂREA DE JOS “ OF GALAŢI FASCICLE VIII, 2004, ISSN 1221-4590 TRIBOLOGY 2. CURVED FACE WIDTH SPUR GEARS WITH MODIFIED GEOMETRY The curved face width gear geometry is produced using both the traditional conjugate surface generation theory [3] and solid modelling techniques [4]. The simulation of the non-standard gear generation, using solid modelling, was based on the following assumptions: - the virtual gear blank is a cylindrical primitive that is rotated about its axis and is provided with translation motion, tangential to the base circle of the gear, in order to get the rolling motion required for the involute tooth form generation; - the virtual tools, for the gear tooth concave and convex flank generations, are conical solid primitives that substitute the rotational motion of the imaginary rack-cutter flank, with zero pressure angle, about its inclined axis. The modified gear tooth geometry is defined by the tool axis inclination β, and by the radius of the “generating circle”. Fig. 1. Tri-dimensional representation of curved face width spur gear teeth Figure 1 shows the representation of three generated teeth as solid model. The tooth geometry, numerically identified in several sections along the gear face width, is used for further investigations. 3. THE THEORETICAL SLIDING VELOCITY The theory of gearing gives the following relation for the sliding velocity in gear pairs, with the standard involute tooth flank geometry (fig. 2): vs = ( ω1 + ω2 ) ⋅( PM ) (1) where ω 1 and ω 2 are the angular velocities for pinion and gear respectively and (PM) is the distance between the pitch point P and the contact point M. If equation (1) is re-expressed as a function of θ 1 , the pinion roll angle - the independent variable for the computer simulation, it becomes: R1 ⋅ sin θ1 vs = ( ω1 + ω2 ) ⋅ (2) cos( α w + θ1 ) where R 1 is the pinion pitch circle radius and α w - the pressure angle. pinion Rb2 ω 2 gear P M v s ω 1 θ 1f θ 1 θ 1a R b1 Fig. 2. Sliding velocity for a pair of teeth with the standard involute for tooth flank Due to the modified geometry of the nonstandard gears, the sliding velocity variation is modified along the gear face width. To analyse its variation it is assumed that the curved face width gear is made up by a stack of elementary standard spur gear sections. The calculation is developed in the particular case of a gear train with modulus 2 mm, 30 teeth and 24 mm facewidth. Here two identical gears are meshed together giving a gear ratio equal to one. The gear geometry is defined by a tool axis inclination β = 5° and a radius R g = 24 mm. Each elementary gear section, with 2 mm facewidth, positioned by H i - the distance from the centre of the face width, has a particular geometry (table 1) whose parameters are identified using the gear solid model. Also, θ 1 varies, in every section H i , from θ 1a to θ 1f , given by: 2 2 A sin α wi − Ra − Rbi θ1a = α wi − arctg Rbi Rbi θ1 f = arccos − α wi Ra (3) where A is the center distance and R a – the radius of the addendum circle, constant for all the elementary spur gear sections. Section H i [mm] R f Table 1. Gear section geometry R b α w [mm] [°] [mm] 0 27.5 28.19 20 2 27.51 28.20 19.95 4 27.53 28.22 19.84 6 27.56 28.27 19.55 8 27.64 28.34 19.15 10 27.68 28.46 18.44 12 27.77 28.65 17.25 Figures 3 – 6 show the variation of the sliding velocity for several sections along the gear face width. Section 0 (fig. 3) has the standard involute for the gear tooth flank geometry leading to maximum
THE ANNALS OF UNIVERSITY “DUNĂREA DE JOS “ OF GALAŢI FASCICLE VIII, 2004, ISSN 1221-4590 TRIBOLOGY 13 sliding velocity, as known, at the first and last points of contact. It may be seen that in other sections (fig. 4-6) the maximum velocity is gradually increased, reflecting the increase of the length of the line of action due to the increase in gear base circle radius. Section 3, corresponding to the end of the gear face width, with a base circle radius with 1.6% higher than the standard value of the considered geometry, has an increase of 10% for the maximum sliding velocity. The sliding velocity is calculated for a speed of 1000 revs/min. in section 2 (H 2 = 8 mm) Fig. 6. The sliding velocity variation in section 3 (H 3 = 12 mm) Fig. 3. The sliding velocity variation in section 0 (H 0 = 0 mm) Fig. 4. The sliding velocity variation in section 1 (H 1 = 4 mm) Fig. 5. The sliding velocity variation Block [5] was the first who determined the influence of relative sliding velocity on the flash temperature between sliding gear tooth surfaces: 0.5 ϕ f = F( k, ρ,c,w,b,dH ) ⋅v (4) s where F expresses the dependence of flash temperature (ϕ f ) on the thermal conductivity (k), density (ρ), specific heat (c) of the surfaces, Hertzian contact length (w), gear facewidth (b) and the instantaneous energy loss due to friction (dH). The analysis on curve face width gear geometry related to the sliding velocity variation, for the above mentioned dimensional parameters, shows that: - as the gear tooth height gradually decreases; the maximum reduction is about 6%; - the maximum sliding velocity gradually increases; at the end of the gear face width its value is higher with 10% than the standard value recorded for the gear centre section; - the increase in sliding velocity leads to an increase of 4.8% for the gear’s flash temperature, one of the components of the maximum gear surface temperature [8]. 4. MESHING CONDITIONS OF LOADED PLASTIC GEARS Polymer gear teeth, with a relatively low Young’s modulus, will deflect elastically under load, the deflections being larger than those experienced by metal gears. As a result of the applied load, a premature engagement takes place, as well as a delayed disengagement. Hence, the contact point for a plastic gear train is differently positioned compared to the ideal rigid gear pair and is influenced by the amount of tooth deflection. Gear tooth deflections were first calculated by Timoshenko and Bread [11] and the analysis has been refined since the original work by many workers. There are three kinds of deflection to be considered: deflection due to the local surface Hertzian compression and bending and shear deflections.
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INFLUENCE OF A NON-STANDARD GEOMETRY ... - Dunarea de Jos

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