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 β from excessive bending stress
ii. Surface pitting / wear β from excessive contact stress
Flank pitting β
surface contact
Root cracking β
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, πΉπ, 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. Lewisβs 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 Lewisβ equation:
9.2 BASIC ANALYSIS OF GEAR-TOOTH BENDING STRESS (LEWIS EQUATION)
β’ The section modulus I/c is Ft2/6, and therefore the bending stress is
π =π
Ξ€πΌ π=
6πΉπ‘β
ππ‘2
β’ The maximum stress in a gear tooth occurs at point a as shown in figure 14-1b.
β’ Using the similarity of triangles, we can write:
π₯
Ξ€π‘ 2=
Ξ€π‘ 2
ββ π₯ =
π‘2
4βl
x
t/2
l x
t/2
t/2
(a)
(b)
9.2 BASIC ANALYSIS OF GEAR-TOOTH BENDING STRESS (LEWIS EQUATION)
β’ From equation (a): we can rewrite it as the following:
π =6πΉπ‘β
ππ‘2=
πΉπ‘π
1
Ξ€π‘2 6β=
πΉπ‘π
1
Ξ€π‘2 4 β
1
Ξ€4 6
β’ Substitute (b) into (c) and multiply the numerator and denominator by the circular pitch p, we find:
π =πΉπ‘π
π( Ξ€2 3)π₯π
β’ Letting y = 2x/3p, we have
π =πΉπ‘πΉπ¦π
β’ The factor y is called the Lewis form factor.
(c)
9.2 BASIC ANALYSIS OF GEAR-TOOTH BENDING STRESS (LEWIS EQUATION)
β’ Most engineers prefer to employ the diametral pitch in determining the stresses. This is done by substituting π = π/π and π¦ = π/π in previous equation. This gives
π =πΉπ‘π
ππ- US Customary
π =πΉπ‘πππ
- SI unit
π =2π₯π
3
where,
β’ Above equation considers only the bending of the tooth. And the effect of the radial load πΉπ is neglected.
9.3 LEWIS FORM FACTOR
Y = 0.334
9.4 GEAR-TOOTH FATIGUE
BENDING ANALYSIS
π =πΉπ‘π
ππ½πΎπ£πΎππΎπ
where,
i. π = Diametral Pitch
ii. πΉπ‘ = Tangential Force
iii. π = Face width
iv. π½ = Spur Gear Geometry Factor [Refer Figure 15.23]
v. πΎπ£ = Velocity or Dynamic Factor [Refer Figure 15.24]
vi. πΎπ = Overload Factor [Refer Table 15.1]
vii. πΎπ = 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
π½ππππππ = 0.235 (N=18) π½ππππ = 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
π·: πΎπ£ =1200 + π
1200
πΈ: πΎπ£ =600 + π
600
π΅: πΎπ£ =78 + π
78
π΄: πΎπ£ =78 + π
78
πΆ: πΎπ£ =50 + π
50
9.4 GEAR-TOOTH FATIGUE
BENDING ANALYSIS
9.4 GEAR-TOOTH FATIGUE
BENDING ANALYSIS
where,
i. πΆπΏ = Load Factor [πΆπΏ= 1.0 for bending]
ii. πΆπΊ = Size or Gradient Factor [πΆπΊ = 1.0 for P > 5 or πΆπΊ = 0.85 for P β€ 5]
iii. πΆπ = Surface Condition Factor
iv. ππ = Reliability Factor [Use Table 15.3]
v. ππ‘ = Temperature Factor [For steel gear, ππ‘ = 1.0 < 160Β°F or ππ‘= 620/(460 + T) for T > 160Β°F]
vi. πππ = Mean Stress Factor [πππ = 1.0 for idler gear and πππ = 1.4 for one-way bending]
vii. ππβ² = Standard R.R Moore endurance limit
ππ = ππβ² πΆπΏπΆπΊπΆπππππ‘πππ
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:
π =πππ
Since factors πΎπ, πΎπ, and ππ have been taken into account separately, the βsafety
factorβ 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. πΆπΏ = 1 (bending load)
ii. πΆπΊ = 1 (since P > 5) *[πΆπΊ = 0.85 for P β€ 5]
iii. πΆπ = 0.68 (Pinion) and = 0.70 (Gear) *machined surface
iv. ππ = 0.814
v. ππ‘ = 1 (Temperature should be < 160 0F)
vi. πππ = 1.4 (One-way bending)
vii. ππβ² = 290/4 = 72.5 ksi (Gear) ππ = 57.8 ksi (Gear)
ππβ² = 330/4 = 82.5 ksi (Pinion) ππ = 63.9 ksi (Pinion)
ππ = ππβ² πΆπΏπΆπΊπΆπππππ‘πππ
modification factors (Empirical Data)
9.4 GEAR-TOOTH FATIGUE
BENDING ANALYSIS
STRENGTH:
9.4 GEAR-TOOTH FATIGUE
BENDING ANALYSIS
π =πΉπ‘π
ππ½πΎπ£πΎππΎπ
The bending fatigue stress is estimated as follow
STRESS:
π = 10 π = 1.25 π½ππππππ = 0.235 (N=18) π½ππππ = 0.28 (N=36)
π =πππππ
12
=π
1810
1720
12
= 811 πππ
= 1.68
πΎπ = 1.25
πΎπ = 1.6
ππ = 114πΉπ‘ ππ = 96πΉπ‘
Therefore,
and
9.4 GEAR-TOOTH FATIGUE
BENDING ANALYSIS
πΎπ£ =1200 + 811
1200
π = 1
63,900 = 114πΉπ‘ , πΉπ‘ = 561 (pinion)
57,800 = 96πΉπ‘ , πΉπ‘ = 602 (gear)
The transmitted power, αΆπ
Hence, the pinion is the weaker member.
9.4 GEAR-TOOTH FATIGUE
BENDING ANALYSIS
αΆπ =πΉπ‘π
33000=
561 811
33000= 13.8 βπ
9.5 GEAR-TOOTH SURFACE
FATIGUE ANALYSIS
πΆπ = 0.5641
π1 β π£π
2
πΈπ+
1 β π£πΊ2
πΈπΊ
πΌ =π πππ πππ π
2
π
π + 1π =
ππ
ππ
ππ» = πΆππΉπ‘
ππππΌπΎπ£πΎππΎπ
ππ» = ππππΆπΏππΆπ
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
ππ» = ππππΆπΏππΆπ
STRENGTH:
πππ = 122 ksi
πΆπΏπ = 0.8 ππππ = 1720 60 40 50 5 = 1.03 Γ 109 ππ¦ππππ
πΆπ = 1 [ 99 % Reliability ]
ππ» = 122 0.8 1 = 97.6 ksi
ππ» = πΆππΉπ‘
ππππΌπΎπ£πΎππΎπ
STRESS:
πΎπ£ = 1.68
πΎπ = 1.25
πΎπ = 1.6
π = 1.25 ππ
ππ = 1.8 ππ
πΌ = 0.107
πΆπ = 2300 ππ π
[πππ = 0.4 π΅βπ β 10 = 0.4 330 β 10 = 122 ππ π]
9.5 GEAR-TOOTH SURFACE
FATIGUE ANALYSIS
ππ» = 2300πΉπ‘
1.25 1.8 0.1071.68 1.25 1.6 = 8592 πΉπ‘
STRESS:
8592 πΉπ‘ = 97600 psi πΉπ‘ = 129 lb
The transmitted power, αΆπ
αΆπ =πΉπ‘π
33000=
129 811
33000= 3.2 βπ
SF: π = π