Chapter 8 Reva
Chapter 8 Reva
Chapter 8 Reva
8.1 INTRODUCTION
The reliability of a gear along with the other gearbox components is an extremely
important consideration in the design of any power-transmission system, ensuring that
the required loads can be handled over the intended life of the system. Some general
design constraints and requirements need to be given special attention because of their
potential impact on the long-term reliability of the total system. One is the operating
power spectrum and determining the potential requirements for growth. Another is that
changing requirements may cause a configuration change where a misalignment could
cause vibration that could set up stresses and lead to fatigue failure. Another example
is the lubrication system if included as part of the gearbox design, assuring that the
lubrication capacity and filtration system is adequate considering the operating
environment of the total system. The lubricant flow should be designed so that the
particles within the filtration system are removed prior to reentry into the gearbox area.
The involute gear profile is the most commonly used system for gearing. In an
involute gear, the profiles of the teeth are involutes of a circle (contour of gear teeth
curves inward). A very common gear design is the spur gear. Spur gears are
cylindrical in form and operate on parallel axes with the teeth straight and parallel to the
axes. A typical spur gear arrangement is shown in Figure 8.1. In general, the reliability
of drive train spur gears is extremely high due to present design standards. There are,
however, some considerations that should be addressed because in evaluating the
reliability of a gear for specific purposes.
The initial design of the spur gear mesh is normally one of standard proportions and
equal tooth thickness for both pinion and gear. This initial design is then altered to
achieve an optimum configuration to achieve recess action and a balanced bending
stress in pinion and gear. A recess-action gear mesh, shown in Figure 8.2, has a long
addendum pinion and short addendum gear. A recess-action mesh is quieter and
smoother running than standard mesh and has a much lower tendency to score due to
better lubrication within the mesh.
Although the advantage of having balanced bending stresses on a pinion and gear
is primarily lower weight, it does have an indirect effect on reliability. As stated earlier,
whenever there is an inefficient use of weight, reliability is compromised somewhat. For
example, even a fraction of a pound saved in the optimization of a spur gear mesh
could be applied to a bearing where the life could perhaps be doubled. While
Gears and Splines Revision A
8-2
overemphasis of weight reduction can be detrimental to reliability, the carrying of excess
weight can have a far-reaching effect; therefore, a balanced gear system must be the
goal for efficient and reliable systems. Fortunately, it is usually a simple task to achieve
recess-action and balanced bending stress in most spur gear designs. This is
accomplished by experimentally shifting the length of contact up the line of action
toward the driver gear, while increasing the circular tooth thickness of the pinion and
decreasing that of the gear.
To ensure smooth operation of the gear mesh under load, it is generally the practice
to modify the involute profile, usually with tip relief, to correct for the deflection of the
gear tooth under load. The various parameters affecting gear wear are shown in Figure
8.3. Too little tip relief will result in the gear teeth going into mesh early and going out of
mesh late. This condition results in higher dynamic loads with the accompanying
stress, vibration, noise and possible non-involute contact that can lead to hard-lines,
scuffing or scoring of gear teeth. Too much tip relief lowers the contact ratio of the gear
set and again can result in less than optimum performance with respect to stress,
vibration, and noise.
Crowning is generally applied to spur gears to ensure full contact across the face of
the gear without end loading. With insufficient crowning, end loading will occur and
result in higher than predicted vibrational stresses.
Gears and Splines Revision A
8-4
Helical gears, shown in Figure 8.4, are usually quieter and have a greater load-
carrying capacity per inch of face than spur gears. The major disadvantage is that a
thrust load is introduced along the gear shaft, thereby requiring larger and stronger
bearings. Analysis of helical gears is very similar to that used for spur gears. The
stress analysis is performed using an equivalent spur tooth. AGMA standard
procedures have been developed for strength analysis of spur and helical gears.
Spline gears are used to transfer torque between shafts and flanges, gears and
shafts, and shafts and shafts. A typical spline arrangement is shown in Figure 8.4. A
splined shaft usually has equally spaced teeth around the circumference, which are
most often parallel to the shaft’s axis of rotation. These teeth can be straight sided, an
involute form or included angle form (serrations). The teeth on a straight sided spline
have an equal tooth thickness at any point measured radially out from the axis of
rotation. Conversely, the internal parallel spline keys are integral to the shaft and
equally spaced around the circumference. The involute spline has equally spaced teeth
but they have an involute form like a gear tooth. The teeth do not have the same
proportions as a gear tooth. They are shorter in height to provide greater strength.
Involute splines provide a more smooth transition through a radius as opposed to
straight sided splines decreasing the possibility of fatigue cracks. Involute splines are
usually crowned. The serration type of spline has a tooth that is non-involute. The male
teeth are in the form of an included angle, with the female serration having spaces of
the same included angle. Serrations are generally used on smaller diameter shafts
where the included angle form permits more teeth to be used on a smaller
circumference, providing a greater contact area.
The most common problem associated with splines is wear due to fretting;
particularly, with loose splines. It is important that the reliability analysis confirms that
the bearing stress is below the allowable limit. Tight splines should have an adequate
length pilot to react with bending loads. Lubrication is a particular factor in the reliability
of loose splines and, if at all possible, should remain flooded with oil at all times.
Crowning is usually required to prevent excessive wear.
The definition of failure for a gear or spline is not very precise because of the
wearing pattern of the gear. During the initial period of operation, minor imperfections in
the gear will be smoothed out, and the working surfaces will polish up, provided that
proper conditions of installation, lubrication and application are being met. Under
continued normal conditions of operation, the rate of wear will be negligible. A gear has
failed when it can no longer “efficiently” perform the job for which it was designed. Thus
the definition of failure may be determined by the amount of vibration, noise, or results
of a physical inspection.
The more common modes of gear and spline failure are wear, surface fatigue,
plastic flow and breakage. The following paragraphs describe each of these failure
modes. Table 8-1 provides a summary of possible failure modes for gears and splines.
In the shear mode, the gear immediately ceases to transmit power while in the wear
mode it degrades gradually before complete failure
8.3.1 Wear
Wear is the removal of metal, worn away normally in a uniform manner from the
contacting surface of the gear teeth. The first stage of wear is the polishing phase
during wear-in of the gear when asperities of the contacting surfaces are gradually worn
off until very fine, smooth surfaces develop. Moderate wear of the gear occurs during
its design life. Excessive wear occurs when the gear is operating in or near the
boundary lubrication regime where the load is being carried by surface asperities rather
than by the lubricant. Excessive wear is similar to moderate wear but the gear teeth are
experiencing a considerable amount of material being removed from the surfaces.
Contamination in the lubrication system can accelerate this wear. During this phase the
tooth-surface profile is being destroyed so that high dynamic loads are encountered
which in turn accelerates the wear rate until the gear is no longer usable.
Plastic flow is the cold working of the tooth surfaces, caused by heavy loads and the
rolling and sliding action of the gear mesh. The result of these high contact stress
levels is the yielding of the surface and subsurface material and surface deformation.
This same failure mode in a slow speed operation combined with an inadequate
lubricating film can result in a rippled surface. The cold working action of the gear
surface leads to deteriorated gear box operation.
The previous paragraphs have provided an insight into the specific characteristics
and failure modes of the more common gear types. Gears, fortunately, are designed to
a specification and through the standardization of the American Gear Manufacturer's
Association (AGMA), gears of various manufacturers and designs can be compared.
The best approach for the calculation of failure rates for a gear system is to use the
manufacturer's specification for each gear as the base failure rate, and adjust the failure
rate for any difference in the actual usage from that purpose for which the gear was
designed. If the manufacturer’s failure rate is not available, a gear or spline is usually
designed for a life of 100 million revolutions for the particular application, the application
including such factors as operating speed, temperature, lubrication and torque. Either
way, the gear failure rate can be expressed as:
∗ λG,B can usually be obtained from the manufacturer and it will be expressed in
failures/operating hour at a specified speed, load, lubricant, and temperature. Also, a
service factor will usually be provided to adjust the normal usage factor for certain
specific conditions found in typical industries. These factors include such things as
vibration, shock and contaminates. Failure data for similar equipment may also be
available or a base failure rate of one failure/108 revolutions can be used:
[λG,B = (RPM x 60) x 1 / design life (revolutions)]
.
The speed deviation multiplying factor, CGS, can be calculated using the relationship
established in Equation (8-1) noting that the lubrication film thickness varies with speed
to the 0.7 power. Therefore:
0.7
⎛V ⎞
CGS =k +⎜ o ⎟ (8-3)
⎝ Vd ⎠
The gear loading multiplying factor, CGP, has a lubricant and a fatigue impact.
From Equation (8-1), the impact of load or torque can be expressed as:
k k
Change in expected life = 0.13
= 0.13 (lubricant impact) (8-4)
W ⎛ LO ⎞
⎜ ⎟
⎝ LD ⎠
Gears and Splines Revision A
8-9
Where: W = Load Parameter
LO = Operating Load, lbs
LD = Design Load, lbs
k = Constant
and the expression for torque or load on the fatigue rate of the component is:
4.56
⎛L ⎞
Change in expected life = k ⎜ D ⎟ (fatigue impact) (8-5)
⎝ LO ⎠
4.69
⎛ L /L ⎞
CGP =⎜ O D⎟ (8-6)
⎝ k ⎠
The alignment of gears, bearings and shafts can be critical in the operation of a
system. CGA, the misalignment factor, can be expressed as:
2.36
⎛ A ⎞
CGA = ⎜ E ⎟ (8-7)
⎝ 0.006 ⎠
The lubricant factor CGL is a function of the viscosity of the lubricant used in a gear
system. CGL can be expressed as:
0.54
⎛ν ⎞
CGL =⎜ O ⎟ (8-8)
⎝ν L ⎠
460 + TAT
CGT = for TAT > 160O F (8-9)
620
o
Where: TAT = Operating temperature, F
The AGMA has developed service factors for most industrial applications of gears,
bearings, and gearbox designs whereby the expected extent of usage in vibration and
shock environments can be taken into account when a gear system is selected for use.
This service factor can be used as a multiplying factor for determining the inherent
reliability or expected failure rate (CGV) for a specific gearbox or bearing in a particular
environment. Most manufacturers provide service factor data for each of their products.
An example of a service factor for a speed-decreasing drive is shown in Table 8-1.
The failure rate in failures per million revolutions of spline gears can be calculated
by:
10 6
λGS , B = (8-11)
θ
⎛ φ GL ⎞
4.56
−2.36
θ = 7.08 x10 −10
⎜ ⎟ ( AE ) (8-12)
⎝ GD ⎠
Substituting the expression for the spline gear base failure rate into Equation (8-11)
yields:
where CGS, CGL, CGT, and GGV are calculated by Equations (8-3), (8-8), (8-9), and (8-
10) respectively.
2.2
Gear Velocity Multiplying Factor, C GS
2.0
1.8
1.6
1.4
1.2
1.0
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
Vo / Vd
0.7
⎛V ⎞
CGS = 1.0 + ⎜ o ⎟
⎝ Vd ⎠
10
Gear Load Multiplying Factor, C GP
0.1
0.01
0.001
0.0001
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3
L O /LD
4.69
⎛ L /L ⎞
CGP =⎜ O D ⎟
⎝ 0.5 ⎠
10.0
1.0
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2
Misalignment Angle, A E , Degrees
1.4
Lubricant Multiplying Factor, C GL
1.2
1.0
0.8
0.6
0.4
0.2
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
νO /νL
0.54
⎛ν ⎞
CGL =⎜ O ⎟
⎝ν L ⎠
1.8
Temperature Multiplying Factor, C GT
1.6
1.4
1.2
1.0
0.8
-50 0 50 100 200 300 400 500 600 700
Operating Temperature, TAT oF
460 + TAT
CGT = for TAT > 160O F
620
and:
o
Where: TAT = Operating temperature, F
8.6 REFERENCES
10. “Boston Gear Catalogue”, Catalogue 100, INCOM International Inc., Quincy,
Massachusetts
13. Cormier, K.R., “Helicopter Drive System R&M Design Guide”, United Technologies
Corp., Stanford, Connecticut, Report ADAD69835 (April 1979)
19. Hindhede, U., et al, “Machine Design Fundamentals”, John Wiley & Sons, NY,
1983
53 Rumbarger, John H., “A Fatigue Life and Reliability Model for Gears”, American
Gear Manufacturers Association, Report 229.16 (January 1972)
54. AGMA Standard for Surface Durability Formulas for Spiral Bevel Gear Teeth,
American Gear Manufacturers Association, Report 216.01 (January 1964)
55. AGMA Standard Nomenclature of Gear Tooth Failure Modes, American Gear
Manufacturers Association Report 110.04 (August 1980)
Gears and Splines Revision A
8-19
58. Parmley, R.O., Mechanical Components Handbook, McGraw-Hill Book Co., NY
1985
70. “Validation of Gearbox Reliability Models from Test Data”, Eagle Technology, Inc.,
Report No. 87-D-0075 (October 1987)
71. Dennis N. Pratt, “Investigation of Spline Coupling Wear”, Report No. SY-51R-87,
Naval Air Warfare Center, Patuxent River, MD (December 1987)
98. Raymond J. Drago, “Rating the Load Capacity of Involute Splines”, Machine
Design, February 12, 1976
102. Dan Seger, Niagara Gear Corporation, “Inside Splines” , Gear Solutions, January
2005