BEAM STRENGHT OF SPUR GEAR TOOTH - Weebly



BEAM STRENGTH OF SPUR GEAR TOOTH

The accurate stress analysis of a gear tooth for a particular application is a complex problem because of the following reasons:

1. There is continuous change in the point of application of load on the tooth profile.

2. The magnitude and direction of applied load also change.

3. In addition to static load, the dynamic load due to inaccuracy of the tooth profile, error in machining and mounting, tooth deflection, acceleration and stress concentration also act on the tooth which are all difficult to model mathematically.

Wilfred Lewis, in a paper titled, “The investigation of the strength of gear tooth” published in 1892, derived an equation for determining the approximate stress in a gear tooth by treating it as a cantilever beam of uniform strength. The beam strength calculation is based upon the following assumptions:

1. The tangential component of the force on gear tooth, Ft, is uniformly distributed across the face width; however, in actual force distribution it is found to be non-uniformly distributed. This assumption is valid for small face widths b, i.e.

2. The effect of the radial component Fr, which produces direction compressive stress, is neglected.

9.5m ( b ( 12.5m

3. The maximum stress is assumed to occur when the entire load is at the tip of the tooth. This is however, not true because when more than one pair of teeth are in contact, the load is shared between all of them. Further, as the tooth moves through its path, the magnitude of the force and its moment are changed.

4. The tooth is assumed to be a simple cantilever beam

5. The effect of stress concentration and manufacturing errors are neglected.

Equation (12.6) gives the relation between the tangential force and the bending stress produced in the gear tooth. When the bending stress reaches the permissible magnitude of bending strength of the material, (d, the corresponding force Ft is called the beam strength Fbeam. Thus beam strength is the maximum value of the tangential force than can be resisted by a tooth without bending failure. Therefore, Eq.(12.6) can be rewritten as

Fbeam = bm(dY

In any gear drive, the average tangential force on a gear tooth can be computed form Eql (12.4). However, in many practical applications the actual force may be greater than the average force on account of the poor service conditions and the dynamic load on the tooth due to profile being inaccurate. In order to include the effect of the above two conditions, two factors, namely the service factor Cs and the velocity factor Cv are introduced and the maximum force between the two mating teeth is computed by the following relation:

[pic]

The values of the factors Cs and Cv may be found from Tables 12.3 and 12.4, respectively.

In order to avoid failure of the gar tooth due to bending, the beam strength, Fbeam, should be greater than the maximum force on the gear tooth. Thus,

Fbeam ( Fmax

Table 12.3 values of service factor (Cs)

|Type of load |Type of service |

| |Intermittent or 3 h/day |8-10h/day |Continuous |

|Steady |1.0 |1.0 |1.25 |

|Light shock |1.0 |1.25 |1.5 |

|Medium shock |1.25 |1.5 |1.8 |

|Heavy shock |1.50 |1.8 |2.0 |

Dynamic load on gear

The maximum force acting on a gear tooth during power transmission is called the dynamic load, Fdyn, which is the sum of the tangential force Ft and incremental force Fi. The reasons behind the incremental force Fi are the following.

1. Inaccuracies of the tooth profile.

2. Error in tooth spacing resulting in excessive backlash

3. elastic deformation of a tooth under load, affecting kinematic perfection

4. inertial of the rotating masses

5. Misalignment between bearings supporting gears.

Buckingham had found that the maximum effect of the above factors occurs when the contact is transferred from one pair to the next pair of teeth. According to him, the dynamic load Fdyn is given by

Fdyn = Ft + Fi

The increment force Fi for average conditions may be found from the following equation

[pic]

where

c is the dynamic load factor [pic]

e is the profile error in action between gears (refer Table 12.5 and 12.6)

K1 = constant = 9 for 20o full depth teeth

K3 = 20.67, a constant.

Further it is suggested that for satisfactory design of the gear under dynamic load conditions, the dynamic load should be equal to endurance strength Fen.

Endurance strength is computed form the modified Lewis equation as

Fen = Fdyn X FoS = bm(enY

Where

(en is the endurance limit (=1.75 BHN)

BHN is the Core Brinel hardness number

WEAR STRENGTH

Due to rolling and sliding actions of the gear teeth, the following types of surface destructions (wear) may occur:

Abrasive wear. Scratching of the tooth surface due to the presence of foreign materials in the lubricant is called abrasive wear.

Corrosive wear. Chemical reactions on the surface of a gear cause corrosive wear.

Pitting. Repeated application of the stress cycle, known as pitting, cause fatigue failure.

Scoring. Inadequate lubrication between metal-to metal contact cause scoring.

It is observed from the results of various experiments that abrasion, corrosive, the scoring are caused by improper lubrication, whereas pitting usually occurs because of repeated application of Hertz contact stress on the portion of a gear tooth which has relatively little sliding motion compared to rolling motion. Clearly, the spur gear will have pitting near the pitch lien where motion is almost all of rolling.

In designing a gear against wear, the material should be able to resist the repeated contact stresses. In order to obtain a formula for actual surface stress that exists between tow mating gear teeth, the Hertz equation for contact stress between two cylinders in rolling contact is used (figure 12.8)

In order to apply the Hertz contact stress equation to spur gears, the force F is replaced by the allowable wear load Fwear, radii r1 and r2 are considered as pitch radii of pinion and gear and contact stress is replaced by the allowable surface endurance limit (es.

The limiting wear load equation for spur gear pairs is given as

Fwear = d1bQK

Where

Q is the ratio factor (=2Z2/(Z1+Z2)

K is the load stress factor [pic]

For steel gears the surface endurance limit may be calculated by the following formula

(es = (2.75 BHN – 70) N/mm2

Buckingham has reported that the allowable wear load as obtained by Eq. (12.16) is clearly a normal force that must be greater than the dynamic load for satisfactory design.

................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download