Literatura Lumpen

B.2.1 Relationship to AGMA standard gear rating calculation

The calculation procedure presented in this annex follows the basic principles used in the AGMA gear rating procedure. See [3] and [4] in the bibliography. This AGMA procedure is supplemented here by new features applicable to typical P/M gear design and operating conditions. The influence of each of these conditions is explained below as each new calcula- tion feature is introduced.

Another difference from the AGMA standard form factor calculation has been introduced for reason of calculation convenience. The AGMA procedure, as part of obtaining a non--dimensional form factor, requires that all related gear geometry data first be scaled to unit module equivalents. This is a carry--over from the days that graphical methods were used for such calculations and this increase in scale helped in graphical accuracy. The procedure

used below uses actual gear geometry data and is followed by a simple conversion to the traditional non--dimensional form.

B.2.2 Calculation stages

The calculation process consists of the following stages:

-- critical load: selection of location along the con- tacting tooth surface and selection of direction; -- critical section: fillet definition and section loca-

tion;

-- bending moments and stress: calculation for unit critical load;

-- form factor ratio: calculation of force--stress ratio for critical load with adjustment for tangential load referenced in the form factor definition;

-- non--dimensional factor: conversion to non--di- mensional using module.

B.2.3 Critical load,Wc

The critical load is the load (on the tooth flank) which will produce the maximum tensile stress at the root fillet. The factors which determine this critical load are the direction of the load relative to the tooth outline and the location of the load along the tooth outline . The relationship between the magnitude of this load,Wc, and the transmitted torque is described

in B.2.6.2 and B.3.6. B.2.3.1 Load direction

The load direction is determined first by the geome- try of the active portion of the tooth flank, here understood to be an involute curve associated with the base circle of the gear. Under certain operating conditions common in P/M gear applications, the sliding action between the mating gear teeth will influence the load direction.

B.2.3.1.1 Load force normal to tooth flank The direction of the load transmitted between gear teeth is normal to the involute surface of the tooth flank. If a circle (with its center at the gear axis) is drawn through the load point, the angle between this normal and a tangent to this circle is the involute pressure angle,φWc, at that point. As a normal to the

involute, this direction is also tangent to the gear base circle. See figure B.3.

This load force direction is assumed in AGMA rating calculations. It is also used in the first set of calculations below. See figure B.4.

B.2.3.1.2 Friction force tangent to tooth flank The relative motion between mating gear teeth is a combination of rolling and sliding. The rolling action is generally assumed to offer negligible resistance to the relative motion and is ignored as a direct contributor to the load transmitted between the teeth. The sliding action requires further consideration because it may introduce a significant friction force.

The AGMA gear rating calculation [3] is properly used only for those operating conditions in which some approximation of ideal lubricating conditions exist.1) These operating conditions include an adequate supply of clean lubricant at the gear teeth, an adequate pitch line velocity, and tooth surface geometry accurate enough to permit a well distrib- uted contact area. When these conditions are met, a film of lubricant is forced into the tooth contact area at a pressure which nearly or completely separates the contacting surfaces. The resulting friction force is then small enough to be ignored in the gear bending strength calculations. In many gear applications, and especially in P/M gears with low material density and without sealing of the pores, a full complement of

these lubrication conditions is not present and a significant friction force accompanies the normal tooth force, see figure B.4(b). The role of the friction forces is described in B.3. Load,Wc Base circle Base circle radius,rB φWc Tangency _d Wc

Figure B.3 -- Load normal to involute tooth flank and tangent to base circle

atdOE

(see B.2.3.2.1) at HPSTL (see B.2.3.2.3)

a) No friction, normal force only, alternate locations

b) With friction, normal and tangent forces, shown with sliding inwards (typical of driven gear during approach action) Radial line Friction force Resultant force Normal force δφc φc

φcis load point pressure angle

δφcis load deviation angle

Figure B.4 -- Gear tooth forces B.2.3.2 Critical location

With the gear tooth treated as a loaded cantilever beam, the location of the load producing the maximum fillet bending stress will tend to be as far as

possible from the fillet. This would locate the critical load at the outside diameter of the gear. However, some common mesh geometry conditions help move the location somewhat further down on the

_______________________

1)_{The lubrication port ion of the application clause states, “The ratings determined by these formulas are only valid when}

the gear teeth are operated with a lubricant of proper viscosity for the load, gear tooth surface finish, temperature, and pitch line velocity.”

tooth with a corresponding reduction in the resulting root fillet stresses. See figure B.5.

Bending load wfc (xfc,yfc) at critical fillet Constant stress parabola Critical section Translated load hfc αWc Critical load,Wc φWc d_Wc X Radial line Compressive load

Figure B.5 -- Data for stress calculation The location is identified by the diameter,dWc, at the

critical load point. The factors which determine this diameter are discussed below.

B.2.3.2.1 Outer load location limit

The first geometry condition which shifts the critical load location from the outside diameter is the tip round which is present on nearly all P/M gears. See figures A.1 and B.4(a). With this tip round, the outer load location limit moves to the point on the tooth at which the involute flank ends and the tip round begins, corresponding to the effective outside diam- eter, dOE. The calculation of this diameter is

described in A.3.1.

B.2.3.2.2 Tooth load sharing

With most spur gear designs, there are two mesh conditions at which two adjacent pairs of teeth are nominally in simultaneous contact. (A pair consists of the mating teeth from each of the two meshing gears.) One such condition corresponds to one pair of teeth just starting to contact with the preceding pair still engaged. The second corresponds to the same pair of teeth nearing the end of contact while the following pair is already engaged.

NOTE: In some gear designs, identified by contact ra- tios of one or less, these conditions of partial overlap- ping of contact between adjacent pairs of teeth is

missing. In other designs, identified by contact ratios of more than two, there are contact intervals at which three pairs of teeth are engaged, with the remaining in- terval having two pairs of teeth engaged. The following remarks do not apply to these conditions, for which the appropriate analyses are beyond the scope of this document.

When the typical overlapping contact is present, there is the potential for the transmitted load to be shared between the two adjacent meshing pairs. If the sharing were equal, the critical load location would not be at the outermost end of the tooth involute since, at this location, the load itself has dropped to one--half. The actual nature of such sharing depends on the accuracy of the involute profiles and the relative stiffness of each pair of teeth at that point in their engagement cycle. A detailed analysis of such conditions is generally too complex for common gear design procedures. As a simplifi- cation, sharing is assumed when both the driver and driven gear’s tooth--to--tooth composite variation meets Q8 or better requirements, or in the case where lesser accuracy prevents load sharing until initial wearing takes place.

If such load sharing is not likely (see B.2.3.2.3), then the critical load location for each gear is at its effective outside diameter, as noted in B.2.3.2.1, and the diameter at the critical load location for each gear is:

d_Wc= d_OE (B.1)

B.2.3.2.3 Highest point of single tooth loading In the typical meshing cycle of a pair of teeth, the stages of the meshing can be identified by a series of points and their corresponding diameters on the two gears, here labelled as the “pinion”,P, for the driving gear and the “gear”,G, for the driven gear, see figure A.8:

-- point 1, the start of the mesh cycle, with the pre- ceding pair still in mesh; diametersd1Pandd1G;

-- point 2, the start of the single pair mesh, with the preceding pair out of mesh; diametersd2Pand

d2G;

-- point 3, the end of the single pair mesh, with the following pair just starting to mesh; diametersd3P

andd3G;

-- point 4, the end of the mesh cycle; diametersd4P

andd4G.

For the calculation for these diameters, see annex A. The selection of points (i.e., 1, 2, 3 or 4) used in calculating the highest point on each tooth at which

full load is transmitted by only a single pair is dependent upon the accuracy of the tooth--to--tooth composite variation (see B.2.3.2.2). If both the driver and the driven gear’s tooth--to--tooth composite variation meets Q8 or better requirements, then the points used are Point 3 and Point 2 respectively. Otherwise, Point 4 and Point 1 are respectively used. Therefore if both driver and driven gears meet or exceed Q8 tooth--to--tooth composite variation re- quirements,

d_{WcP= d3P} (B.2)

and

d_WcG= d_2G (B.3)

If either driver or driven gears do not meet or exceed Q8 tooth--to--tooth composite variation require- ments,

d_WcP= d_4P (B.4)

and

d_WcG= d_1G (B.5)

B.2.3.3 Translation to tooth centerline

To begin the process of calculating the bending moment (see figures B.5 and B.6), the critical load is translated to the tooth centerline. The centerline serves as the neutral axis of the cantilever beam

represented by the tooth. The location of the translated force, expressed as its distance, xWcC,

from the gear center along the centerline, may be calculated by:

Step 1. Once the diameter of the critical load point