Design of Mechanical Element 1: Gear-Tooth Strength...Gear Horsepower Capacity for Tooth-Bending...
Transcript of Design of Mechanical Element 1: Gear-Tooth Strength...Gear Horsepower Capacity for Tooth-Bending...
Chapter 9: Design of
Mechanical Element 1: Gear-Tooth Strength
DR. AMIR PUTRA BIN MD SAAD
C24-322
[email protected] | [email protected]
mech.utm.my/amirputra
GEAR
9.1 INTRODUCTION
BACK TO NATURE
9.1 INTRODUCTION
Having dealt with gear geometry and force analysis, we now turn to the question
of how much power or torque a given pair of gears will transmit without tooth
failure.
9.1 INTRODUCTION
The two primary failure modes for gears are
i. Tooth breakage โ from excessive bending stress
ii. Surface pitting / wear โ from excessive contact stress
Flank pitting โ
surface contact
Root cracking โ
bending stress
9.2 MODE OF TOOTH FAILURE
9.2 BASIC ANALYSIS OF GEAR-TOOTH BENDING STRESS (LEWIS EQUATION)
An equation for estimating the bending stress in gear teeth in which the toothform entered into the formulation was presented by Wilfred Lewis to PhiladelphiaEngineers Club in 1892.
9.2 BASIC ANALYSIS OF GEAR-TOOTH BENDING STRESS (LEWIS EQUATION)
1. The load is applied to the tip of a single tooth.
2. The radial component of the load, ๐น๐, is negligible.
3. The load is distributed uniformly across the full face width.
4. Stress concentration in the tooth fillet is negligible. Stress concentration factors were unknown in Mr. Lewisโs time but are now known to be important. This will be taken into account later.
5. Force due to tooth sliding friction are negligible.
Assumptions made in deriving Lewisโ equation:
9.2 BASIC ANALYSIS OF GEAR-TOOTH BENDING STRESS (LEWIS EQUATION)
โข The section modulus I/c is Ft2/6, and therefore the bending stress is
๐ =๐
ฮค๐ผ ๐=
6๐น๐กโ
๐๐ก2
โข The maximum stress in a gear tooth occurs at point a as shown in figure 14-1b.
โข Using the similarity of triangles, we can write:
๐ฅ
ฮค๐ก 2=
ฮค๐ก 2
โโ ๐ฅ =
๐ก2
4โl
x
t/2
l x
t/2
t/2
(a)
(b)
9.2 BASIC ANALYSIS OF GEAR-TOOTH BENDING STRESS (LEWIS EQUATION)
โข From equation (a): we can rewrite it as the following:
๐ =6๐น๐กโ
๐๐ก2=
๐น๐ก๐
1
ฮค๐ก2 6โ=
๐น๐ก๐
1
ฮค๐ก2 4 โ
1
ฮค4 6
โข Substitute (b) into (c) and multiply the numerator and denominator by the circular pitch p, we find:
๐ =๐น๐ก๐
๐( ฮค2 3)๐ฅ๐
โข Letting y = 2x/3p, we have
๐ =๐น๐ก๐น๐ฆ๐
โข The factor y is called the Lewis form factor.
(c)
9.2 BASIC ANALYSIS OF GEAR-TOOTH BENDING STRESS (LEWIS EQUATION)
โข Most engineers prefer to employ the diametral pitch in determining the stresses. This is done by substituting ๐ = ๐/๐ and ๐ฆ = ๐/๐ in previous equation. This gives
๐ =๐น๐ก๐
๐๐- US Customary
๐ =๐น๐ก๐๐๐
- SI unit
๐ =2๐ฅ๐
3
where,
โข Above equation considers only the bending of the tooth. And the effect of the radial load ๐น๐ is neglected.
9.3 LEWIS FORM FACTOR
Y = 0.334
9.4 GEAR-TOOTH FATIGUE
BENDING ANALYSIS
๐ =๐น๐ก๐
๐๐ฝ๐พ๐ฃ๐พ๐๐พ๐
where,
i. ๐ = Diametral Pitch
ii. ๐น๐ก = Tangential Force
iii. ๐ = Face width
iv. ๐ฝ = Spur Gear Geometry Factor [Refer Figure 15.23]
v. ๐พ๐ฃ = Velocity or Dynamic Factor [Refer Figure 15.24]
vi. ๐พ๐ = Overload Factor [Refer Table 15.1]
vii. ๐พ๐ = Mounting Factor [Refer Table 15.2]
Geometry factor J for standard spur gears (based on tooth fillet radius of0.35/P).(From AGMA Information Sheet 225.01; also see AGMA 908-B89.)
9.4 GEAR-TOOTH FATIGUE
BENDING ANALYSIS
๐ฝ๐๐๐๐๐๐ = 0.235 (N=18) ๐ฝ๐๐๐๐ = 0.28 (N=36)
Geometry factor J for standard spur gears (based on tooth fillet radius of0.35/P).(From AGMA Information Sheet 225.01; also see AGMA 908-B89.)
9.4 GEAR-TOOTH FATIGUE
BENDING ANALYSIS
๐ท: ๐พ๐ฃ =1200 + ๐
1200
๐ธ: ๐พ๐ฃ =600 + ๐
600
๐ต: ๐พ๐ฃ =78 + ๐
78
๐ด: ๐พ๐ฃ =78 + ๐
78
๐ถ: ๐พ๐ฃ =50 + ๐
50
9.4 GEAR-TOOTH FATIGUE
BENDING ANALYSIS
9.4 GEAR-TOOTH FATIGUE
BENDING ANALYSIS
where,
i. ๐ถ๐ฟ = Load Factor [๐ถ๐ฟ= 1.0 for bending]
ii. ๐ถ๐บ = Size or Gradient Factor [๐ถ๐บ = 1.0 for P > 5 or ๐ถ๐บ = 0.85 for P โค 5]
iii. ๐ถ๐ = Surface Condition Factor
iv. ๐๐ = Reliability Factor [Use Table 15.3]
v. ๐๐ก = Temperature Factor [For steel gear, ๐๐ก = 1.0 < 160ยฐF or ๐๐ก= 620/(460 + T) for T > 160ยฐF]
vi. ๐๐๐ = Mean Stress Factor [๐๐๐ = 1.0 for idler gear and ๐๐๐ = 1.4 for one-way bending]
vii. ๐๐โฒ = Standard R.R Moore endurance limit
๐๐ = ๐๐โฒ ๐ถ๐ฟ๐ถ๐บ๐ถ๐๐๐๐๐ก๐๐๐
9.4 GEAR-TOOTH FATIGUE
BENDING ANALYSIS
9.4 GEAR-TOOTH FATIGUE
BENDING ANALYSIS
SAFETY FACTOR
The safety factor for bending fatigue can be taken as the ratio of fatigue strength
to fatigue stress:
๐ =๐๐๐
Since factors ๐พ๐, ๐พ๐, and ๐๐ have been taken into account separately, the โsafety
factorโ need not be as large as would otherwise be necessary. Typically, a safety
factor of 1.5 might be selected, together with a reliability factor corresponding to
99.9 percent reliability.
9.4 GEAR-TOOTH FATIGUE
BENDING ANALYSIS
SAMPLE PROBLEM
Gear Horsepower Capacity for Tooth-Bending Fatigue Failure
Figure above shows a specific application of a pair of spur gears, each withface width, b = 1.25 in. Estimate the maximum horsepower that the gearscan transmit continuously with only a 1 percent chance of encounteringtooth-bending fatigue failure.
9.4 GEAR-TOOTH FATIGUE
BENDING ANALYSIS
where,
i. ๐ถ๐ฟ = 1 (bending load)
ii. ๐ถ๐บ = 1 (since P > 5) *[๐ถ๐บ = 0.85 for P โค 5]
iii. ๐ถ๐ = 0.68 (Pinion) and = 0.70 (Gear) *machined surface
iv. ๐๐ = 0.814
v. ๐๐ก = 1 (Temperature should be < 160 0F)
vi. ๐๐๐ = 1.4 (One-way bending)
vii. ๐๐โฒ = 290/4 = 72.5 ksi (Gear) ๐๐ = 57.8 ksi (Gear)
๐๐โฒ = 330/4 = 82.5 ksi (Pinion) ๐๐ = 63.9 ksi (Pinion)
๐๐ = ๐๐โฒ ๐ถ๐ฟ๐ถ๐บ๐ถ๐๐๐๐๐ก๐๐๐
modification factors (Empirical Data)
9.4 GEAR-TOOTH FATIGUE
BENDING ANALYSIS
STRENGTH:
9.4 GEAR-TOOTH FATIGUE
BENDING ANALYSIS
๐ =๐น๐ก๐
๐๐ฝ๐พ๐ฃ๐พ๐๐พ๐
The bending fatigue stress is estimated as follow
STRESS:
๐ = 10 ๐ = 1.25 ๐ฝ๐๐๐๐๐๐ = 0.235 (N=18) ๐ฝ๐๐๐๐ = 0.28 (N=36)
๐ =๐๐๐๐๐
12
=๐
1810
1720
12
= 811 ๐๐๐
= 1.68
๐พ๐ = 1.25
๐พ๐ = 1.6
๐๐ = 114๐น๐ก ๐๐ = 96๐น๐ก
Therefore,
and
9.4 GEAR-TOOTH FATIGUE
BENDING ANALYSIS
๐พ๐ฃ =1200 + 811
1200
๐ = 1
63,900 = 114๐น๐ก , ๐น๐ก = 561 (pinion)
57,800 = 96๐น๐ก , ๐น๐ก = 602 (gear)
The transmitted power, แถ๐
Hence, the pinion is the weaker member.
9.4 GEAR-TOOTH FATIGUE
BENDING ANALYSIS
แถ๐ =๐น๐ก๐
33000=
561 811
33000= 13.8 โ๐
9.5 GEAR-TOOTH SURFACE
FATIGUE ANALYSIS
๐ถ๐ = 0.5641
๐1 โ ๐ฃ๐
2
๐ธ๐+
1 โ ๐ฃ๐บ2
๐ธ๐บ
๐ผ =๐ ๐๐๐ ๐๐๐ ๐
2
๐
๐ + 1๐ =
๐๐
๐๐
๐๐ป = ๐ถ๐๐น๐ก
๐๐๐๐ผ๐พ๐ฃ๐พ๐๐พ๐
๐๐ป = ๐๐๐๐ถ๐ฟ๐๐ถ๐
STRESS:
STRENGTH:
9.5 GEAR-TOOTH SURFACE
FATIGUE ANALYSIS
9.5 GEAR-TOOTH SURFACE
FATIGUE ANALYSIS
9.5 GEAR-TOOTH SURFACE
FATIGUE ANALYSIS
9.5 GEAR-TOOTH SURFACE
FATIGUE ANALYSIS
9.5 GEAR-TOOTH SURFACE
FATIGUE ANALYSIS
9.5 GEAR-TOOTH SURFACE
FATIGUE ANALYSIS
For the gears in above problem, estimate the maximum horsepower thatthe gears can transmit with only a 1 percent chance of a surface fatiguefailure during 5 years of 40 hours/week, 50 weeks/year operation.
Gear Horsepower Capacity for Tooth Surface Fatigue Failure
9.5 GEAR-TOOTH SURFACE
FATIGUE ANALYSIS
๐๐ป = ๐๐๐๐ถ๐ฟ๐๐ถ๐
STRENGTH:
๐๐๐ = 122 ksi
๐ถ๐ฟ๐ = 0.8 ๐๐๐๐ = 1720 60 40 50 5 = 1.03 ร 109 ๐๐ฆ๐๐๐๐
๐ถ๐ = 1 [ 99 % Reliability ]
๐๐ป = 122 0.8 1 = 97.6 ksi
๐๐ป = ๐ถ๐๐น๐ก
๐๐๐๐ผ๐พ๐ฃ๐พ๐๐พ๐
STRESS:
๐พ๐ฃ = 1.68
๐พ๐ = 1.25
๐พ๐ = 1.6
๐ = 1.25 ๐๐
๐๐ = 1.8 ๐๐
๐ผ = 0.107
๐ถ๐ = 2300 ๐๐ ๐
[๐๐๐ = 0.4 ๐ตโ๐ โ 10 = 0.4 330 โ 10 = 122 ๐๐ ๐]
9.5 GEAR-TOOTH SURFACE
FATIGUE ANALYSIS
๐๐ป = 2300๐น๐ก
1.25 1.8 0.1071.68 1.25 1.6 = 8592 ๐น๐ก
STRESS:
8592 ๐น๐ก = 97600 psi ๐น๐ก = 129 lb
The transmitted power, แถ๐
แถ๐ =๐น๐ก๐
33000=
129 811
33000= 3.2 โ๐
SF: ๐ = ๐