Calculation procedure with friction forces When friction forces are considered, the same basic

calculation procedure is followed with some changes:

-- the critical load direction, relative to the involute tooth flank, will change;

-- the critical load location may change;

-- the critical load magnitude, and therefore the “tooth force adjustment” will change.

To establish the appropriate critical load information, it may be necessary to start with two trial conditions and then perform double sets of calculations.

B.3.1 Coefficient of friction

In all the equations dealing with friction effects, the simplifying assumption is made that the coefficient of friction is constant over the tooth meshing cycle. Values will depend on the material combination (including relative hardness), degree of lubrication, tooth flank surface texture, tooth sliding velocity, tooth contact pressure, and other factors. Values for coefficient of friction may be determined in standard tests, but only apply to the conditions specified in those tests. These may not closely match the operating conditions of the gear mesh. With typical gear materials and with some continuing form of lubrication, the coefficient of friction may fall as low as 0.10, rarely lower. Under less favorable conditions, it may be as high as 0.30, or even higher.

B.3.2 Critical load direction

Without friction, the critical load direction is always normal to the involute flank and its direction relative to the tooth centerline is determined by the load location diameter, as discussed in B.2.3 and its sub--clauses. With friction, the load deviates from this normal direction. The amount of deviation is represented by the load deviation angle, δ_φc de-

scribed in figure B.4(b), which is determined by the coefficient of friction,m, as follows:

δ_φc=arctan m (B.23)

The direction of the deviation is opposite to the direction of relative sliding. The sliding direction generally changes during the meshing cycle, with the change--over taking place when the teeth are contacting at the pitch point, or tangent point of the

operating pitch circles, see figure A.8. On that figure, with the pinion driving, approach action takes place when the contact goes from point 1 to the pitch point. During approach action, the sliding direction on each gear tooth is toward its root and the sign in equation B.23 is plus (+), see figure B.4(b). Recess action takes place when the contact goes from the pitch point to point 4. During recess action, the sliding direction on each gear tooth is toward its tip and the sign in equation B.23 is minus (--).

In some gear designs, the outside diameters are chosen so that the pitch point lies outside the range of contact. In these designs there is no reversal of the sliding direction and the action is all approach or all recess, generally the latter.

When the sliding action in either of the contacting teeth is toward the root, the tooth force direction is such that the bending component is reduced and the compressive component is increased. These act to reduce the tensile stress at the tooth fillet. When the sliding action is toward the tooth tip, the force components undergo reverse changes and the fillet tensile stress is increased. Based on this effect alone, the critical load direction would come from the recess portion of the meshing cycle and the sign in equation B.23 would be negative. However, this choice is not necessarily correct for all meshing conditions, as explained in B.3.3.

B.3.3 Critical load location

Without the friction effect, the critical load location is taken as close to the tooth tip as appropriate for the conditions defined in B.2.3.2. This location produces the maximum bending moment at the base of the cantilevered tooth and the maximum tensile stress at the fillet.

During approach action, friction will reduce the bending stress level compared to the frictionless condition. The opposite is true during recess action where the inclusion of frictional effects will increase the bending stress as compared to the frictionless evaluation.

As a result, when doing the analysis with friction, the selection of the critical load location and the corre- sponding sign (+ or --) ofδφcto use in equation B.23

Table B.1 -- Selection of critical load location points for Y--factor calculation under friction conditions Gear to be evaluated Tooth--to-- tooth accuracy level Critical load location to be evaluated (see A.8.1) Test to de- termine type of ac- tion at crit- ical load location Type of ac- tion at crit- ical load location Effect of action on bending stress Sign ofδφc in equa- tion B.23 2nd location to be evaluated1) Sign ofδφc in equa- tion B.23 for 2nd location evaluation Both driver and driven meet or ex- Point 3 ε3P≤ εA Approach Reduces + NA meet or ex ceed Q8 require- ments Point 3 ε3P>εA Recess Increases -- NA

Driver Either driv- er or driven do not

meet or ex- Point 4

ε4P≤ εA Approach Reduces + NA meet or ex- ceed Q8 require- ments Point 4 ε4P>εA Recess Increases -- NA Both driver and driven meet or ex- Point 2

ε2G>εA Approach Reduces + Pitch point --

meet or ex ceed Q8 require- ments Point 2 ε2G≤ εA Recess Increases -- NA

Driven Either driv- er or driven do not

meet or ex- Point 1

ε1G>εA Approach Reduces + Pitch point --

meet or ex- ceed Q8 require- ments Point 1 ε1G≤ εA Recess Increases -- NA NOTE:

1) _{If the evaluation point on the driver happens to be in approach zone, then the secondary calculation is not required for}

the driven gear since the recess condition will not be present near the pitch point.

Note that on the driving gear, only one position needs to be evaluated close to the tooth tip; either Points 3 or 4 depending on the tooth--to--tooth accuracy level of the gear (see B.2.3.2.3). For the driven gear, either Points 1 or 2 close to the tooth tip need to be evaluated depending upon the tooth--to--tooth accu- racy level of the gear. In addition, the driven gear may need a secondary position evaluated, which is just inside of the operating pitch circle at the start of recess action where higher friction level dominates. Both positions (where necessary) need to be used in separateY--factor calculations. The position which results in the lowestY--factor value will be used in all subsequent calculations.

B.3.4 Calculation of Y--factor for friction conditions

Once the critical load location and direction are selected, as discussed above, the calculation proce-

dure is the same up to step 2 of B.2.3.3. Here, equation B.6, for the direction angle of the critical tooth force, is changed to the following:

γ_{Wc= φWc+ δφc− αWc} (B.24) where

δ_φc is load deviation angle (see B.3.2,

equation B.23), degrees.

The remaining calculations are the same except for the tooth force adjustment ratio described in B.2.6.2 for non--friction conditions and described in B.3.5 for friction conditions.

B.3.5 Tooth force adjustment for friction conditions

In B.2.6.2, a tooth force adjustment is necessary to allow for the difference between the tooth force value used in the stress calculations and the force value in whichY--factors are generally expressed, the former

acting normal to the tooth surface and the latter tangent to the operating pitch circle. Under friction conditions, the tooth force used for stress calcula- tions is no longer normal to the tooth surface and this difference is reflected in a modified adjustment ratio. The adjustment ratio, as defined in equation B.20, is modified to account for friction as follows:

=



cos φA cosφ_Wc



cos



φWc+ δφc



(B.25) m_cA=WA Wc= d_Wc d_A



cos



φWc+ δφc



where

dA is operating pitch diameter (see A.7.2),

mm;

φA is operating pressure angle (see A.7.1),

degrees.

B.4 Symbols

See table B.2.

Table B.2 -- Symbols, terms and definitions

Symbol Definition Units

Where first found

AWs Form factor force--stress ratio -- -- B.2.6

dWc Diameter at critical load location mm B.2.3.2.2

dWcP,dWcG Highest point at which full load is transmitted by single pair mm B.2.3.2.3

fBS Bending stress factor -- -- B.2.4.2

fBSX Maximum bending stress factor -- -- B.2.4.3

hf Height of translated load force above fillet section mm B.2.4.2

m Module mm B.2.7

mcA Tooth force adjustment ratio -- -- B.2.6.2

sBS Bending tensile stress at fillet section N/mm2 B.2.4.2

sCS Compressive stress N/mm2 B.2.5.2

st Combined tensile stress N/mm2 B.2.5.3

WA Tangential force acting at operating pitch circle N B.2.6.2

Wc Critical load N B.2.3

Wcx Force component parallel to tooth axis N B.2.3.4.2

Wcy Force component normal to tooth axis N B.2.3.4.1

wf Width of fillet section mm B.2.4.2

xWcC Distance from gear center of translated critical tooth force mm B.2.3.3

xWc,yWc Coordinates of critical load point mm B.2.3.3

Y Non--dimensionalY--factor -- -- B.2.7

αWc Half tooth thickness angle degrees B.2.3.3

δ_φc Load deviation angle degrees B.3.2

γWc Direction angle of critical tooth force degrees B.2.3.3

φWc Pressure angle at critical load point degrees B.2.3.3

φWc Involute pressure angle degrees B.2.3.1.1

Annex C (informative)

Calculation of the stress correction factor,Kf

[This annex is provided for informational purposes only and should not be construed as a part of AGMA930--A05,

Calculated Bending Load Capacity of Power Metallurgy (P/M) External Spur Gears.]

C.1 Introduction

The stress correction factor,Kf, which considers the

effect of stress concentration at the sharply changing cross--section of the tooth at its root fillet, is another component for determining the bending strength geometry factor,J. The geometry element in stress concentration is the curvature (radius) at the fillet in relation to the overall tooth size and shape and the location of the load. The calculation presented here for this element is the Dolan and Broghamer method as used in other AGMA gear rating calculations.

In calculating the influence of stress concentration on gear tooth bending strength, elements other than geometry must also be considered. These non--ge- ometry elements are the type of loading (repeated or occasional) and the material condition (degree of ductility or brittleness). Together, they can influence the long term sensitivity of the gear to the initial level of stress concentration.

These elements are also considered in the AGMA gear rating calculations. Their treatment here follows a different format.

In document AGMA 930-A05 (Page 40-43)