Construction of a Model of Variable Speed Process for Herringbone Gears Including Friction...

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CONSTRUCTION OF A MODEL FOR VARIABLE

SPEED PROCESS FOR HERRINGBONE GEARS

INCLUDING FRICTION CALCULATED BY

VARIABLE FRICTION COEFFICIENT


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ABSTRACT

A dynamic model that includes friction and tooth profile error excitation for herringbone

gears is proposed for the dynamic analysis of variable speed processes. In this model, the

position of the contact line and the relative sliding velocities are determined by the angulardisplacement of the gear pair. The translational and angular displacements are chosen as the

generalized coordinates to construct the dynamic model. The friction is calculated using a

variable friction coefficient. The tooth profile error excitation is assumed to depend on the

position along the contact line and to vary with the angular displacement of the driving gear.

Thus , the proposed model can be used in the dynamic analysis of the variable speed process

of a herringbone gear transmission system. An example acceleration process is numerically

simulated using the model proposed in this paper. The dynamic responses are compared with

those from the model utilizing a constant friction coefficient and without friction in cases

where the profile error excitations are included and ignored.


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Chapter 1. INTRODUCTION

Herringbone gars offer advantages such as smooth transmission, greater torque transmission

and low axial force; hence, they are widely used in gas turbines, ships and engineering

machinery such as coal mining machinery, many of which work under variable speed

processes. Vibration and noise have significant effects on the reliability and service life of a

gear transmission system. Thus, it is essential to construct a precise dynamic model for a

herringbone gear transmission system used for a variable speed process to obtain a more

accurate dynamic response and improve the reliability and service life. Friction is believed to

be one of the sources of gear vibration and noise, especially under low speed and high torque

conditions, as a result of direct reversal of the friction between the contact teeth. Directly

coupling a tribology model into a dynamic model is an accurate method of calculating the

friction excitation. However, this is time consuming. To overcome this difficulty. Some

friction coefficient formulae are developed to calculate the dynamic friction between thecontact teeth. Drozdov and Gavrikov, ODonoghue and Cameroon, Misharin, and Benedict

and Kelley, all proposed their own friction coefficient formulae; ISO also proposed a

formulae. Xu et al proposed a friction coefficient formulae based on a non-newtonian elasto-

hydrodynamic lubrication (EHL) model that considered the lubricants absolute viscosity at

the oil inlet temperature, effective radius, slide to roll ratio, surface roughness, and maximum

Hertzian pressure and validated it using an experiment.

Some dynamic models that included friction for spur and helical gears have been proposed in

previous studies. Vaisha and Singh reviewed various modeling strategies that have beenhistorically adopted and then illustrated issues for spur gears by assuming equal load shaing

among the contact teeth. Velex and Cahouet proposed a model that included friction for spur

and helical gears based on a finite element procedure and pointed out the potentially

significant contribution of friction to translational vibrations, especially at low to medium

speeds. He et al proposed a more accurate model for spur gears that included the realistic

mesh stiffness and friction, which showed that the dynamic response would even be

amplified in some cases because of interactions between the mesh and friction forces for the

tip relief gears. Furthermore, He et al developed a contact dynamic model for helical gears

that captures the friction reversal due to the relative speed reversal. The assumption was

made explicitly or implicitly that the position of the contact lines and the relative sliding

velocity were determined by the mean angular motion of the gear pair. Therefore, these

models could only be used in the vibration analysis of the transmission sub system operating

at a stable angular velocity, and not for variable speed process. In most of the composed

models the friction is calculated using the coulomb model with a constant friction coefficient.

However, in practical operation, the magnitude of the friction coefficient varies during the

gear meshing process, and the friction direction reverses if the relative speed reverses.

In this paper a dynamic model that includes the friction calculated by Xus variable fric tion

coefficient formulae for herringbone gears is proposed. This Model can be used on thedynamic analysis of the variable speed process. In this model, the position of the contact lines


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and the relative sliding velocity are determined by the angular displacement of the gear pair

rather than the mean angular motion. Moreover, the translational and angular displacements

are chosen as generalized coordinates to construct the dynamic models, which make the

dynamic analysis of a variable speed process possible. The tooth profile error excitation is

usually expressed by a fourier series by a fundamental frequency equal to the tooth meshfrequency, or is not included in dynamic models. However, in this model to analyze the

effects of the tooth profile error excitation on the dynamic behavior of herringbone gears

during a variable speed process, the profile error excitation is assumed to depend on the

position along the contact line and to vary with the angular displacement of the driving gear.

In other words the error excitation is a binary function with variables for both the angular

displacement of the driving gear and the location on the contact line. Finally the acceleration

process of an example involving a pair of herring =bone gears is numerically calculated using

the model proposed in this paper. The dynamic responses are compared with those from a

model utilizing a constant friction coefficient and without friction where the profile error

excitation are included and ignored.


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Chapter 2. LITERATURE SURVEY

Friction is believed to be one of the sources of gear vibration and noise, especially under

high-torque and low-speed conditions, as a result of the direction reversal of the frictionbetween the contact teeth. Directly coupling a tribology model into a dynamic model is an

accurate method for calculating the friction excitation. However, this is time-consuming. To

overcome this difficulty, some friction coefficient formulae are developed to calculate the

dynamic friction between the contact teeth. Drozdov and Gavrikov, ODonoghue and

Cameron, Misharin, and Benedict and Kelley, all proposed their own friction coefficient

formulae; ISO also proposes a formula. Xu et al. proposed a friction coefficient formula

based on a non Newtonian elasto hydrodynamic lubrication (EHL) model that considered the

lubricantsabsolute viscosity at the oil inlet temperature, effective radius of the contact point,

entraining velocity, slide-to-roll ratio, surface roughness, and maximum Hertzian pressure

and validated it using an experiment.

Vaishya and Singh reviewed various modeling strategies that have been historically adopted

and then illustrated issues for spur gears by assuming equal load sharing among the contact

teeth. Velex and Cahouet proposed a model that included friction for spur and helical gears

based on a finite element procedure and pointed out the potentially significant contribution of

friction to translational vibrations, especially at low to medium speeds. He et al. proposed a

more accurate model for spur gears that included realistic mesh stiffness and friction, which

showed that the dynamic response could even be amplified in some cases because of

interactions between the mesh and friction forces for the tip relief gears. Furthermore, He etal. developed a contact dynamic model for helical gears that captures the friction reversal due

to the relative speed reversal. The assumption was made explicitly or implicitly that the

position of the contact lines and the relative sliding velocity were determined by the mean

angular motion of the gear pair. Therefore, these models could only be used in the vibration

analysis of a transmission system operating at a stable angular velocity, and not for a variable

speed process.

In this paper, a dynamic model that includes the friction calculatedby Xus variable friction

coefficient formula for herringbone gears is proposed. This model can be used in the dynamicanalysis of a variable speed process. In this model, the position of the contact lines and the

relative sliding velocity are determined by the angular displacement of the gear pair, rather

than the mean angular motion. Moreover, the translational and angular displacements are

chosen as generalized coordinates to construct the dynamic model, which makes the dynamic

analysis of a variable speed process possible. The tooth profile error excitation is usually

expressed by a Fourier series with the fundamental frequency equal to the tooth mesh

frequency, or is not included in dynamic models. However, in this model, to analyze the

effects of the tooth profile error excitation on the dynamic behavior of herringbone gears

during a variable speed process, the profile error excitation is assumed to depend on the

position along the contact line and to vary with the angular displacement of the driving gear.\


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Chapter 3. THEORY

1. GEARSA gear or cogwheel is a rotatingmachine part having cut teeth, or cogs, which mesh with

another toothed part in order to transmit torque, in most cases with teeth on the one gear

being of identical shape, and often also with that shape on the other gear. Two or more gears

working in tandem are called a transmission and can produce a mechanical advantage

through a gear ratio and thus may be considered a simple machine. Geared devices can

change the speed, torque, and direction of apower source.The most common situation is for

a gear to mesh with another gear; however, a gear can also mesh with a non-rotating toothed

part, called a rack, thereby producingtranslation instead of rotation.

The gears in a transmission are analogous to the wheels in a crossed belt pulley system. Anadvantage of gears is that the teeth of a gear prevent slippage.When two gears mesh, and one

gear is bigger than the other (even though the size of the teeth must match), a mechanical

advantage is produced, with therotational speeds and the torques of the two gears differing in

an inverse relationship. In transmissions which offer multiple gear ratios, such as bicycles,

motorcycles, and cars, the term gear, as in first gear, refers to a gear ratio rather than an actual

physical gear. The term is used to describe similar devices even when the gear ratio is

continuous rather thandiscrete,or when the device does not actually contain any gears, as in

acontinuously variable transmission.

2. COMPARISION WITH OTHER DRIVE MECHANISMSThe definite velocity ratio which results from having teeth gives gears an advantage over

other drives (such astraction drives andV-belts)in precision machines such as watches that

depend upon an exact velocity ratio. In cases where driver and follower are proximal, gears

also have an advantage over other drives in the reduced number of parts required; the

downside is that gears are more expensive to manufacture and their lubrication requirements

may impose a higher operating cost.

3. CLASSIFICATION OF GEARSGears can be broadly classified into internal and external gears. An external gear is one with

the teeth formed on the outer surface of a cylinder or cone. Conversely, an internal gear is

one with the teeth formed on the inner surface of a cylinder or cone. For bevel gears, an

internal gear is one with thepitch angle exceeding 90 degrees. Internal gears do not cause

output shaft direction reversal.

Further Classification of gears will result in them being classified as

follows:
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Spur Gear

Spur gears or straight-cut gears are the simplest type of gear. They consist of a cylinder or

disk with the teeth projecting radially, and although they are not straight-sided in form (they

are usually of special form to achieve constant drive ratio, mainlyinvolute), the edge of eachtooth is straight and aligned parallel to the axis of rotation. These gears can be meshed

together correctly only if they are fitted to parallel shafts.

Figure no.1 Spur Gear

Helical Gear

Helical or "dry fixed" gears offer a refinement over spur gears. The leading edges of the teeth

are not parallel to the axis of rotation, but are set at an angle. Since the gear is curved, this

angling causes the tooth shape to be a segment of a helix.Helical gears can be meshed in

parallel or crossed orientations. The former refers to when the shafts are parallel to each

other; this is the most common orientation. In the latter, the shafts are non-parallel, and in this

configuration the gears are sometimes known as "skew gears".

The angled teeth engage more gradually than do spur gear teeth, causing them to run more

smoothly and quietly. With parallel helical gears, each pair of teeth first make contact at a

single point at one side of the gear wheel; a moving curve of contact then grows gradually

across the tooth face to a maximum then recedes until the teeth break contact at a single point

on the opposite side. In skew gears, teeth suddenly meet at a line contact across their entire

width causing stress and noise. Skew gears make a characteristic whine at high speeds.

Whereas spur gears are used for low speed applications and those situations where noise
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control is not a problem, the use of helical gears is indicated when the application involves

high speeds, large power transmission, or wherenoise abatement is important. The speed is

considered to be high when the pitch line velocity exceeds 25 m/s.

Figure no. 2 Helical Gear

Bevel Gear

A bevel gear is shaped like aright circular cone with most of its tip cut off. When two bevelgears mesh, their imaginary vertices must occupy the same point. Their shaft axes also

intersect at this point, forming an arbitrary non-straight angle between the shafts. The angle

between the shafts can be anything except zero or 180 degrees. Bevel gears with equal

numbers of teeth and shaft axes at 90 degrees are called mitre gears.

Bevel gears are primarily used to transfer power between intersecting shafts. The teeth of

these gears are formed on a conical surface. Standard bevel gears have teeth which are cut

straight and are all parallel to the line pointing the apex of the cone on which the teeth are

based. Spiral bevel gears are also available which have teeth that form arcs. Hypocycloidbevel gears are a special type of spiral gear that will allow nonintersecting, non-parallel shafts

to mesh. Straight tool bevel gears are generally considered the best choice for systems with

speeds lower than 1000 feet per minute: they commonly become noisy above this point.

Spiral bevel gears can be manufactured as Gleason types (circular arc with non-constant tooth

depth), Oerlikon and Curvex types (circular arc with constant tooth depth), Klingelnberg

Cyclo-Palloid (Epicycloide with constant tooth depth) or Klingelnberg Palloid. Spiral bevel

gears have the same advantages and disadvantages relative to their straight-cut cousins as

helical gears do to spur gears. Straight bevel gears are generally used only at speeds below

5 m/s (1000 ft/min), or, for small gears, 1000 r.p.m.
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Figure no. 3 Bevel Gear

Hypoid Gear

Hypoid gears resemble spiral bevel gears except the shaft axes do not intersect. The pitch

surfaces appear conical but, to compensate for the offset shaft, are in fact hyperboloids of

revolution. Hypoid gears are almost always designed to operate with shafts at 90 degrees.

Depending on which side the shaft is offset to, relative to the angling of the teeth, contact

between hypoid gear teeth may be even smoother and more gradual than with spiral bevel

gear teeth, but also have a sliding action along the meshing teeth as it rotates and therefore

usually require some of the most viscous types of gear oil to avoid it being extruded from the

mating tooth faces, the oil is normally designated HP (for hypoid) followed by a number

denoting the viscosity. Also, thepinion can be designed with fewer teeth than a spiral bevelpinion, with the result that gear ratios of 60:1 and higher are feasible using a single set of

hypoid gears. This style of gear is most commonly found driving mechanical differentials;

which are normally straight cut bevel gears; in motor vehicle axles.

Figure no. 4 Hypoid gear
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Worm Gear

Worm gears resemble screws.A worm gear is usually meshed with a spur gear or a helical

gear,which is called the gear, wheel, or worm wheel. Worm-and-gear sets are a simple and

compact way to achieve a high torque, low speed gear ratio. For example, helical gears arenormally limited to gear ratios of less than 10:1 while worm-and-gear sets vary from 10:1 to

500:1. A disadvantage is the potential for considerable sliding action, leading to low

efficiency.

Worm gears can be considered a species of helical gear, but its helix angle is usually

somewhat large (close to 90 degrees) and its body is usually fairly long in the axial direction;

and it is these attributes which give it screw like qualities. The distinction between a worm

and a helical gear is made when at least one tooth persists for a full rotation around the helix.

If this occurs, it is a 'worm'; if not, it is a 'helical gear'. A worm may have as few as one tooth.

If that tooth persists for several turns around the helix, the worm will appear, superficially, to

have more than one tooth, but what one in fact sees is the same tooth reappearing at intervals

along the length of the worm. The usual screw nomenclature applies: a one-toothed worm is

called single thread or single start; a worm with more than one tooth is called multiple thread

or multiple start. The helix angle of a worm is not usually specified. Instead, the lead angle,

which is equal to 90 degrees minus the helix angle, is given.

In a worm-and-gear set, the worm can always drive the gear. However, if the gear attempts to

drive the worm, it may or may not succeed. Particularly if the lead angle is small, the gear's

teeth may simply lock against the worm's teeth, because the force component circumferentialto the worm is not sufficient to overcome friction. Worm-and-gear sets that do lock are called

self locking, which can be used to advantage, as for instance when it is desired to set the

position of a mechanism by turning the worm and then have the mechanism hold that

position. An example is themachine head found on some types ofstringed instruments.

If the gear in a worm-and-gear set is an ordinary helical gear only a single point of contact

will be achieved. If medium to high power transmission is desired, the tooth shape of the gear

is modified to achieve more intimate contact by making both gears partially envelop each

other. This is done by making both concave and joining them at asaddle point;this is called acone-driveor "Double enveloping". Worm gears can be right or left-handed, following the

long-established practice for screw threads.
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Figure no. 5 Worm gear

Rack and Pinion Gear

A rack is a toothed bar or rod that can be thought of as a sector gear with an infinitely large

radius of curvature. Torque can be converted to linear force by meshing a rack with a pinion:

the pinion turns; the rack moves in a straight line. Such a mechanism is used in automobiles

to convert the rotation of the steering wheel into the left-to-right motion of the tie rod(s).

Racks also feature in the theory of gear geometry, where, for instance, the tooth shape of an

interchangeable set of gears may be specified for the rack (infinite radius), and the tooth

shapes for gears of particular actual radii are then derived from that. The rack and pinion gear

type is employed in arack railway.

Figure no. 6 Rack and Pinion Gear

Sun and Planet Gear

In epicyclical gearing one or more of the gear axes moves. Sun and planet gearing was a

method of converting reciprocating motion into rotary motion in steam engines. It was

famously used byJames Watt on his early steam engines in order to get around the patent on

the crank but also had the advantage of increasing the flywheel speed so that a lighter

flywheel could be used. In the illustration, the sun is yellow, the planet red, the reciprocating

arm is blue, theflywheel is green and thedriveshaft is grey.
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Figure no. 7 Sun and Planet gear

Herringbone Gear

A herringbone gear, a specific type of double helical gear, is a special type ofgear which is a

side to side (not face to face) combination of two helical gears of opposite hands. From the

top the helical grooves of this gear looks like letter V. Unlike helical gears they do not

produce an additional axial load.

Figure no. 8 Herringbone gear
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Chapter 4

HERRINGBONE GEAR OR DOUBLE HELICAL GEAR

A herringbone gear, a specific type of double helical gear, is a special type ofgear which is aside to side (not face to face) combination of two helical gears of opposite hands. From the

top the helical grooves of this gear looks like letter V. Unlike helical gears they do not

produce an additional axial load. Like helical gears, they have the advantage of transferring

power smoothly because more than two teeth will be in mesh at any moment in time. Their

advantage over the helical gears is that the side-thrust of one half is balanced by that of the

other half. This means that herringbone gears can be used in torque gearboxes without

requiring a substantial thrust bearing.Because of this herringbone gears were an important

step in the introduction of thesteam turbine to marine propulsion.

Precision herringbone gears are more difficult to manufacture than equivalentspur or helical

gears and consequently are more expensive. They are used in heavy machinery. Where the

oppositely angled teeth meet in the middle of a herringbone gear, the alignment may be such

that tooth tip meets tooth tip, or the alignment may be staggered, so that tooth tip meets tooth

trough. The latter alignment is the unique defining characteristic of aWuest type herringbone

gear,named after its inventor.

A disadvantage of the herringbone gear is that it cannot be cut by simple gear hobbing

machines, as the cutter would run into the other half of the gear. Solutions to this have

included assembling small gears by stacking two helical gears together, cutting the gears with

a central groove to provide clearance, and (particularly in the early days) by casting the gears

to an accurate pattern and without further machining. With the older method of fabrication,

herringbone gears had a central channel separating the two oppositely-angled courses of

teeth. This was necessary to permit the shaving tool to run out of the groove. The

development of the Sykes gear shaper made it possible to have continuous teeth, with no

central gap. The W.E. Sykes and Farrel Gear Machine companies dissolved in 1983-84. It has

been standard industry practice to obtain an older machine and rebuild it if necessary to

create this unique type of gear. There is at least one machine, a chevron (herringbone) gear

cutter made by Farrel, that is fully functional and operated by Precision Boring Company ofClinton, Michigan. (www.precisionboring.com/press-parts-repair.php) Recently, the Bourn

and Koch company has developed a CNC controlled derivation of the W.E. Sykes design

called the HDS1600-300. This machine, like the Sykes Gear Shaper, has the ability to

generate the true apex without the need for a clearance groove cut around the gear. This

allows the gears to be used in positive displacement pumping applications, as well as power

transmission.
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Chapter 5

CONSTRUCTION OF A DYNAMIC MODEL

1. Analysis of mesh process of a Helical GearA herringbone gear can be considered to be a combination of two helical gears with opposite

hands. Thus, for the clarity of illustration, the meshing process for a pair of helical gears is

investigated to obtain formulae for the effective radius and tangential velocity of the contact

point. The meshing process of the helical gears is illustrated in Fig. 9, where 1and 2are

the angular velocities of the right-hand driving gear and left-hand driven gear, respectively.

N1N1N2N2 is theoretical plane of action, and B1B1 B2B2 is the transverse contact area.

K1K2is the contact line, and bis the base helix angle. Figure 9(b) illustrates the motion of

the contact line when ab, whereas Fig. 9(c) illustrates the motion of the contact line whena


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The instantaneous contact of the helical gears could be approximately simulated using two

truncated cones with opposite orientations, as shown in Fig. 10. Here, t1and t2are the radii

of curvature of the equivalent circular truncated cones of the driving and driven gears,

respectively. N1N1 and N2N2 , which are the same as in Fig. 9, are the spin axes of the

equivalent circular truncated cones, and L is the position coordinate of point K on the contactline.

Figure no. 10 Contact between circular truncated cones with oppositeOrientations

The transverse radii of curvature of K for the driving and driven gears are derived as Eqs. (1)

and (2), respectively.

..1 .. 2The normal radii of curvature of K for the driving and driven gears are derived as Eqs. (3)

and (4), respectively.

.3

.4


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Utilizing Eqs. (3) and (4), the effective radius of curvature of K can be expressed as below

. 5

. 6 . 7Where

() . 8

is the angular displacement of the driving gear. When

is equal to

, which denotes

the initial angular displacement of the driving gear, the first meshing tooth pair begins to

mesh. The meshing tooth pairs are shown in Fig. 11 and denoted by i, i = 1, 2ceil(),where the ceil function rounds to the nearest (higher) integer value; mod(x, y)=x-yfloor(x/y), where the oor function rounds the value of x/y down to the nearest (lower)

integer value.is the total contact ratio, which can be calculated using = + , rb1andtn1are the base radius and tooth number of the driving gear, respectively, whereas rb2is the

base radius of the driven gear. , pb, b, and bdenote the transverse pitch pressure angle,transverse base pitch, face width, and base helix angle, respectively.

Figure no. 11 Snapshot of pair of helical gears in plane perpendicularto axis of rotation and containing point K


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Figure 11 shows a snapshot of a pair of helical gears in a plane that is perpendicular to the

axis of rotation and contains point K.For the sake of clarity, only the velocity of the driving

gear at point K is illustrated in Fig. 11.

The transverse pressure angles of K for the driving and driven gears are given by Eqs. (8) and(9), respectively.

8 ... 9The tangential velocities of K for the driving and driven gears are given by Eqs. (10) and

(11), respectively.

. 10 .. 11Utilizing Eqs. (10) and (11), the entrainment velocity and slide-to roll ratio of K can be

defined as Eqs. (12) and (13), respectively.

... 12

13The above geometrical and motion analyses are conducted for a case in which the driving

gear is right-handed. To expand the application range of the above analysis results to a left-

handed driving gear, the positive rotating orientation of the driving gear and the starting point

of L are defined in Fig. 12.

2. Model FormulationThe lumped-parameter dynamic model for herringbone gears, as shown in Fig. 13, will be

considered in this paper. In this paper, gears are represented by rigid wheels that are

supported by compliant bearings and connected to each other through elastic elements. The

effects of damping and gravity are ignored, along with the tilting motion. The coordinate

system is chosen as follows: the x-axis is in the off-line-of-action direction, the y-axis is

parallel to the line of action, and the z-axis is parallel to the gear rotational axis. If the

stiffness of the gear substrate is large, the relative displacement between sides can be ignored,

and it is not necessary to construct the dynamic model by separating the two sides of theherringbone gear.


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Figure no.12 Definition of positive rotating orientation of driving gearand starting point of L: (a) right hand and (b) left hand

Figure no. 13 Lumped-parameter dynamic model of herringbone

gears, including friction


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Therefore, four degrees of freedom can depict the motion of the gears, including the

translation motion along the x, y, and z axes and the rotating motion about the z-axis, as

shown in Fig. 13. Fy, Fz, and f denote the y-direction and z-direction components of the

meshing force and the tooth friction, respectively. Figure 13 shows how these act on gear 1.

kxj, kyj, and kzjare the x-direction, y-direction, and z-direction supporting stiffness values forgear j(j=1, 2), where gear 1 is the driving gear, and gear 2 is the driven gear. xj,yj, zj, and j

denote the x-direction, y-direction, and z-direction translational displacements and the

angular displacement of gear j(j=1,2), respectively. With [x1, y1, z1, 1, x2, y2, z2, 2] chosen

as generalized coordinates, the equations of motion are given as Eq. (14), where T1and T2are

the driving and driven moments, respectively, whereas M1 and M2 are the moments of

friction acting on gear 1 and gear 2, respectively. The positive directions of the generalized

coordinates are depicted as shown in Fig. 13.

Figure no. 14 Illustration of motion of contact line of Herringbone

Gears


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{

}

.. 14

3. Calculation of meshing forces, frictions and moment of frictions

The motion of the contact line of the herringbone gears is illustrated in Fig. 14, wheresubscripts l and r represent the left-hand and right-hand half teeth of the herringbone gears,

respectively. B1B1B2B2and B1B1B2B2are the transverse contact areas. K1K2and K1

K3 are the lines of contact. K1K2 , K1K2, K1 K3, and K1 K3 represent different

locations of the contact lines in the plane of action.

3.1Calculation of meshing forcesyiaand zia(a=l, r) denote the deformations of the ith (i=1,2ceil()) meshing tooth pair in

the y-direction and z-direction, respectively. eyiaand ezia(a=l, r) denote the y-direction and z-

direction components of the profile error excitation of the ith meshing tooth, respectively. yia

and zia(a=l, r) can be derived as below equation

{

}. 15

The normal deformation ia(a=l,r) of the ith (i=1,2ceil()) tooth pair is defined as

..16Here, kua(a=l, r) denote the mesh stiffness per unit length along the contact line of the left-

hand and right-hand halves of the meshing tooth pair, respectively. The y-direction

component of the meshing force of the ith meshing tooth pair Fyi is derived as follows:

\


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If

.. 17

The y-direction component of the meshing force between two gears Fyis given as

19The z-direction component of the meshing force of the ith meshing tooth pair Fziis derived as

follows:

If

{

}

.. 20

where s, a1, b1, a2, b2, a3, and b3are the same as those in Eq. (17).


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If

21


The z-direction component of the meshing force between two gears Fzis given as

22If the mesh stiffness per unit length is expressed as a binary function with variables for both

the angular displacement of the driving gear and the position on the contact line, the

variability of the mesh stiffness per unit length can be included in the model. If different

mesh stiffness values per unit length are set for the two sides of the herringbone gears, the

mesh stiffness difference between the sides will also be included.

3.2Calculation of tooth frictionIn this paper, Xusformula based on a thermal, non Newtonian EHL model, as shown in Eq.

(23), is utilized to calculate the friction. If the RMS value is large, the friction coefficient

obtained by the formula may be considerably larger or unrealistic. Therefore, before the

formula is utilized, the friction coefficient value obtained by the formula should be checked.

||. 23

where

() || || ..24Here, is the absolute viscosity at the oil inlet temperature in centipoises; S is the RMScomposite surface roughness in micrometers; R is the effective radius of the curvature in

meters; Veis the entrainment velocity in meters per second; SR is the slide-to roll ratio; Phis

the Hertzian pressure in gigapascals calculated using Eq. (25); and bk= -8.92, 1.03,1.04, -0.35, 2.81, -0.10, 0.75, -0.39, and 0.62 for k=1, 2,,9, respectively.


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25where E is the effective modulus of elasticity, and W is the load per unit length along the

contact line.Equation (23) calculates the absolute value of the friction coefficient. To capture the direction

reversal of the friction, the friction coefficient should be defined using Eq. (26). A typical

friction coefficient curve for Xus formula is shown in Fig. 15.

. 26Here if x>0, sgn(x)=1; if x=0, sgn(x)=0; and if x


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If

{

}


The friction between two gears f is given as

293.3Calculation of moment of friction

The values of the moment arm of the friction acting on gear j (j=1,2) at point Ka(a=l, r) are

denoted by tkaj, which can be computed using Eqs. (1) and (2). Mji denotes the moment of

friction between the ith meshing tooth pair acting on gear j. Mjican be derived as follows:

If

{

}

. 30


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If

31

The moment of friction acting on gear j is given as

.. 32


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Chapter 6.

SIMULATING PROFILE ERROR EXCITATION WITH PSEUDO RANDOM

NUMBERS

The normal deviation from a perfect teeth profile can introduce excitation, namely, profile

error excitation, into the dynamic model, which can be projected in the directions of the

action line and rotating axis of the herringbone gears. The profile error excitation is usually

expressed using a Fourier series with the fundamental frequency equal to the tooth mesh

frequency, or is not included in dynamic models and is assumed to be invariable along the

contact line. The profile error excitations are usually variable along the contact line because

of the randomness of manufacture error or longitudinal modification. Ajmi and Velex

considered the profile error excitations to depend on the position along the face width and to

vary with time. Because the model proposed here is designed for use with a variable speed

process, during which the mesh frequency is variable, the profile error excitations are

assumed to vary with both the angular displacement of the driving gear (denoted by 1) andthe position along the contact line (denoted by L). In other words, the error excitation is a

binary function with variables of 1and L. A realistic binary error excitation function can bemeasured using an advanced instrument. However, such an instrument is not normally

available.

Therefore, in this paper, the profile error excitation is assumed to be normally distributed and

simulated with pseudo-random numbers to include the effect of profile error excitation on the

dynamic response of herringbone gears. If different seeds are used for the random number

generator, different error excitations can be obtained for the left-hand and right-hand half

parts. A profile error excitation matrix is generated in the MATLAB software, as shown in

Fig. 19, and then the profile error excitation value of (1, L) is obtained by interpolation. Theprofile error excitation function varies with 1at a cycle of 2lcm (tn1,tn2)/ tn1, where l cm isthe function of the least common multiple of tn1and tn2, which are the tooth numbers of the

driving and driven gears, respectively. Therefore, only a cycle of the profile error matrix is

generated in practical application.


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Figure no. 17Time domain responses of angular acceleration of driving gear when profile error

excitations are included (a) shows results for model with variable friction coefficient; (b) shows

results for model with constant friction coefficient of 0.02; (c) shows results for model without

friction

Figure no. 19 Simulated profile error excitation with pseudo-random

Numbers


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Figure no. 20 Time domain responses of translational displacements of driving gear when profile error

excitations are included ((a), (d), and (g) show results for model with variable friction coefficient; (b),

(e), and (h) show results for model with constant friction coefficient of 0.02; c, f, and i show results

for model without friction)


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Figure no. 21 WignerVille distributions of translational displacements of driving gear when profile

error excitations are included: (a), (d), and (g) show results for translational displacements of

drivinggear in x-direction, y-direction, and z-direction for model with variable friction coefficient,

respectively; (b), (e), and (h) show results for translational displacements of driving gear in xdirection,

y-direction, and z-direction for model with constant coefficient of 0.02, respectively; (c), (f), and (i)

show results for translational displacements of driving gear in x-direction, y-direction, and z-direction

for model without friction, respectively


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Figure no. 22 WignerVille distributions of angular accelerations of driving gear when profile error

excitations are included: (a) shows results for model with variable friction coefficient; (b) shows

results for model with constant friction coefficient of 0.02; (c) shows results for model withoutfriction

Figure no. 23 Time domain responses of angular velocity of driving gear when profile error

excitations are included (blue solid line shows results for model with variable friction coefficient; red

dashed line shows results for model with constant friction coefficient of 0.02; black dotted line shows

results for model without friction)


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Chapter 7.

COMPUTATIONAL AND ANALYSIS OF ACCELERATION PROCESS FOR

EXAMPLE

The acceleration process of an example of herringbone gears is computed and analyzed. The

parameters are listed in Table 1. To simplify the calculation, the mesh stiffness per unit

length along the contact line is assumed to be constant by referencing the literature . The

initial angular displacements of the driving and driven gears, h10 and h20, should satisfy the

relationship: 10=(z2/ z1)20. The dynamics responses are compared with those of a model

utilizing a constant friction coefficient and without friction in cases where the profile error

excitations are included and ignored.

Figure 20 illustrates the time domain responses of the translational displacements of thedriving gear in the x-direction (offline- of-action direction), y-direction (line-of-action

direction), and z-direction (rotational axis), which are denoted by x1, y1, and z1, respectively,

when the profile error excitations are included. The x-direction displacement is zero if the

friction is ignored, as shown in Fig. 20(c). The amplitude of x1 for a variable friction

coefficient is smaller than that for a constant friction of 0.02, maybe because the mean value

of the variable friction coefficient is slightly smaller than 0.02, which will be validated by

Fig. 23. Compared with the model without friction, the amplitudes of y 1 for variable and

constant friction coefficients are smaller, maybe because the friction suppresses the vibration

in the line-of-action direction, like damping. The amplitudes of y1 for variable friction

coefficient are larger than that for constant friction, maybe because the variable friction

coefficient enhances the fluctuation of friction and reduces the friction damping effect. The

differences among the amplitudes of z1for the variable and constant friction coefficients and

without friction are not very significant or as regular as those for x 1 and y1 in the time

domain, because the vibration in the z-direction is mainly excited by profile error, as shown

in(Figs. 24(g)24(i)). If the profile error excitations are ignored, the displacement in the z-

direction will disappear. In a word, friction affects the time domain responses of the

translational displacements for the driving gear. Furthermore, the variable friction coefficient

has a different effect than the constant friction coefficient.


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Figure no. 25Time domain responses of translational displacements and angular acceleration of

driving gear when profile error excitations are ignored (a), (d), (g), and (j) show results for model with

variable friction coefficient; (b), (e), (h), and (k) show results for model with constant frictioncoefficient of 0.02; (c), (f), (i), and (l) show results for model without friction

Figure 21 illustrates WignerVille distributions (WVD) of the translational displacements of

the driving gear in the x-direction, y-direction, and z-direction when the profile error

excitations are included. The dynamic responses during the acceleration process are unstable.

From the viewpoint of engineering signal processing, they are unsteady signals. Therefore,

the frequency components are investigated using WVD. A main frequency of 365 Hz

emerges in the WVD of x1. There are two peak values in the WVD of the variable friction

coefficient, whereas there are three for a constant friction coefficient of 0.02. The variable

friction coefficient decreases the energy density globally and changes the emerging time and

continuing duration of the peak values. Three main frequencies of 137 Hz, 255 Hz, and 380

Hz exist in the WVD of y1. At the frequency of 255 Hz, there is nearly no difference between

the variable friction coefficient, constant friction coefficient of 0.02, and the case without

friction. Compared with the model without friction, the amplitudes of y1for the variable and

constant friction coefficients are smaller at 137 Hz. The variable friction coefficient enhances

the energy density all along the time axis compared with the constant friction coefficient at

137 Hz. At the frequency of 380 Hz, compared with the model without friction, the

amplitudes of y1 for the variable and constant friction coefficients are smaller. The variable

friction coefficient improves the magnitude and changes the emerging time and continuingduration of the peak value of energy density compared with the constant friction coefficient at


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380 Hz. Only one main frequency of 130 Hz emerges in the WVD of z1. The energy density

distributions for the variable and constant friction coefficients are obviously different from

that in the case without friction. The energy density increases along the time axis in the WVD

of thevariable friction coefficient and decreases for the constant friction coefficient, whereas

it nearly remains stable in the WVD without friction. The energy density peak values of z 1forthe variable and constant friction coefficients are larger than that for the case without friction,

maybe because friction enhances the vibration in the z-direction, acting as an excitation. The

energy density peak value of z1for the variable friction coefficient is larger than that for the

constant friction coefficient, maybe because the variation in the friction coefficient enhances

the fluctuation of friction excitations. In a word, the energy distributions for the models with

friction are distinguished from that without friction in the WVDs of the translational

displacements, which has a direct effect on the vibration and noise of the transmission

system. Moreover, compared with the constant friction coefficient, the variable friction

coefficient changes the energy density distribution.

Figure 22 illustrates the time domain responses of the angular acceleration of the driving

gear, denoted by 1, when the profile error excitations are included. The amplitudes of 1for

thevariable and constant friction coefficients are smaller than that without friction, maybe

because the friction suppresses the vibration in the peripheral direction like damping. The

amplitude for the variable friction coefficient is larger than that for the constant friction

coefficient, maybe because the variable friction coefficient enhances the fluctuation of

friction and reduces the friction damping effect. Figure 22 illustrates the WVD of the angular

acceleration of the driving gear. Three main frequencies of 137 Hz, 255 Hz, and 380 Hz

emerge in the WVD of 1. The energy densities for the variable and constant frictioncoefficients are slightly smaller than that for the case without friction at 137 Hz. Moreover,

the energy density of the variable friction coefficient is higher and decreases more slowly

than that of the constant friction coefficient. The differences at frequencies of 255 Hz and 380

Hz are not as distinct as those at 137 Hz between the variable friction coefficient, constant

friction coefficient, and the case without friction. In a word, friction affects the dynamic

responses of the angular motion. Furthermore, the effects of the variable and constant friction

coefficients are different.

Figure 23 illustrates the time domain responses of the angular velocity of the driving gear,denoted by x1, when the profile error excitations are included. At the beginning of the

simulation, there is nearly no distinction between the variable friction coefficient, the

constant friction coefficient of 0.02, and the case without friction. However, over time, the

difference becomes more distinct. The mean values of x1 increases almost linearly with

fluctuation, validating the feasibility of using the proposed model in the analysis of a variable

speed process for herringbone gears. The slope of the curve without friction is slightly larger

than that with friction, which reflects the damping effect of friction. The slope of the curve

for the variable friction coefficient is slightly larger than that for the constant friction

coefficient of 0.02, which validates that the mean value of the variable friction coefficient is

only slightly smaller than 0.02.


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Figure 24 illustrates the time domain responses of the translational displacements and angular

acceleration of the driving gear in the case where the profile error excitations are ignored. A

comparison of Figs. 20(a) and 20(b) and Figs. 24(a) and 24(b) shows that the difference in the

amplitudes of x1between the variable and constant friction coefficients is more significant

than that in the case where the profile error excitations are included. It is also observed thatthe amplitudes of y1 and 1with friction are obviously smaller than those without friction,

maybe because the damping effect of friction becomes more obvious when the profile error

excitations are ignored. The difference between the amplitudes of y1 for the variable and

constant friction coefficients is more obvious than that in the case where the profile error

excitations are included; the same trend is also found for 1. If the profile error excitations are

ignored, the axial displacements z1for herringbone gears will disappear, just like spur gears,

as shown in Figs. 24(g)24(i). Figure 25 illustrates the WVD of the translational

displacements and angular acceleration of the driving gear. In Fig. 25, the same trend for

energy density distribution variation is observed as in the time domain responses. It can be

observed from Figs. 2023 and Figs. 24 and 25 that the profile error excitation enhances the

vibration, whether the friction (calculated using a variable or constant friction coefficient) is

considered or not. Moreover, the differences between the variable friction coefficient,

constant friction coefficient, and the case without friction are not very obvious when the

profile error excitations areincluded. Thus, it is concluded that the contribution of the friction

to vibration is lower than that of the profile error excitation. In a word, the friction significantly

affects the dynamic responses, and the differences between the variable and constant friction

coefficients are obvious, especially when the error excitations are ignored.


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Figure. no 25 WignerVille distributions of translational displacements and angular acceleration of

driving gear when profile error excitations are ignored (a), (d), and (g) show results for translational

displacements in x-direction and y-direction and angular acceleration of driving gear for model with

variable friction coefficient, respectively; (b), (e), and (h) show results for translational displacements

in x-direction and y-direction and angular acceleration of driving gear for model with constant

coefficient of 0.02, respectively; (c), (f), and (i) show results for translational displacements in x-

direction and y-direction and angular acceleration of driving gear for model without friction,

respectively


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Figure no. 26 Time domain responses of meshing force (a)(c) show results for model with variable

friction coefficient, with constant friction of 0.02, and without friction, respectively, when profile

error excitations are included; (d)(f) show results for model with variable friction coefficient, withconstant friction of 0.02, and without friction, respectively, when profile error excitations are

Ignored

Figure no. 27 WignerVille distributions of meshing force (a)(c) show results for model with

variable friction coefficient, with constant friction of 0.02, and without friction, respectively, when

profile error excitations are included; (d)(f) show results for model with variable friction coefficient,

with constant friction of 0.02, and without friction, respectively, when profile error excitations are

ignored


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Figure 26 illustrates the time domain responses of the meshing force. The amplitudes of the

meshing forces for variable and constant friction coefficients are smaller than that in the case

without friction, maybe because the friction suppresses the meshing force fluctuation, like

damping. The amplitude for the variable friction coefficient is larger than that for the constant

friction coefficient, probably because the variable friction coefficient enhances the fluctuationof friction and reduces the friction damping effect. Figure 27 illustrates the WVD of the

meshing force. A main frequency of 137 Hz emerges in the WVD. The energy densities for

the variable and constant friction coefficients are smaller than that for the case without

friction at 137 Hz. Moreover, the energy density of the variable friction coefficient is larger

and decreases more slowly than that of the constant friction coefficient. By comparing Figs.

26(a)26(c) and Figs. 26(d) and 26(e) or Figs. 27(a)27(c) and Figs. 27(d) and 27(e), it can

also be observed that the meshing force differences between the variable friction coefficient,

constant friction coefficient, and the case without friction are more obvious when the profile

error excitations are ignored. Moreover, the profile error excitation enhances the meshing

force fluctuation, whether the friction (calculated by a variable or constant friction

coefficient) is considered or not. In a word, the friction suppresses the meshing force

fluctuation, and the differences between the variable friction coefficient, constant friction

coefficient, and the case without friction are obvious, especially when the error excitations

are ignored


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Chapter 8.

ADVANTAGES AND DISADVANTAGES OF A DYNAMIC MODEL

Can be used in the dynamic analysis of variable speed process of a herringbone geartransmission

Analysis can be carried out including various friction coefficients Analysis based on tooth profile error excitation can be carried out The position of contact line and relative sliding velocity are determined by the angular

displacement

The friction is calculated using variable friction coefficient A more accurate dynamic response is obtained Improves the reliability and service life of the transmission system Captures the friction reversal due to the relative speed reversal.

Based on various assumption such as profile error excitation is assumed to depend onthe position along contact line and to vary with the angular displacement of the gear

pair

Takes into account only Xu et al Coefficient based on a non NewtonianElasto Hydrodynamic Lubricant

Expensive to carry out compared to constant friction analysis


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Chapter 9.

ADVANTAGES AND DISADVANTAGES OF HERRINGBONE GEARS

Herringbone gears are a type of double helical gear, meaning that they have both a lefthand and right hand element to cancel thrust or side loading on their bearings. A true

herringbone has a 30 helix angle, 20 transverse pressure angle, and an AGMA stub

whole depth factor. They may have a center groove between the two hands, but this is

not absolutely necessary. Herringbone gears are cut on specialized machines that cut

both hands at the same time, precisely locating the apex of the two helixes.

Herringbone gears, overcome the problem of axial thrust presented by 'single' helicalgears by having teeth that set in a 'V' shape. Each gear in a double helical gear can be

thought of as two standard, but mirror image, helical gears stacked. This cancels out

the thrust since each half of the gear thrusts in the opposite direction. They can bedirectly interchanged with spur gears without any need for different bearings.

Double helical gears give an efficient transfer of torque and smooth motion at veryhigh rotational velocities.

A disadvantage of herringbone gears is the high cost due to special gear shapingequipment and special cutting tools.

Also during assembling a gear box ,aligning is to be given special attention.


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Chapter 10.

CONCLUSION

In this paper, a dynamic model that included the friction and tooth profile error excitation forherringbone gears was proposed. This model could be used in the dynamic analysis of a

variable speed process. From the numerical simulation of the dynamic model, the following

conclusions were obtained:

Friction obviously affects the dynamic characteristics during the variable

process of herringbone gears, such as by acting as a source of vibration in the off-line-of-

action direction and suppressing the meshing force fluctuation and vibration in the line-of-

action and circumference directions like damping. These effects are especially significant

when the profile error excitations are ignored.

Although the contribution of the friction to vibration is lower than that of

the profile error excitation, the dynamic responses are especially affected by friction when the

profile error excitations are ignored.

The dynamic responses obtained using a variable friction coefficient are

different from those obtained using a constant friction coefficient. The variable friction

coefficient enhances the fluctuation of friction and reduces the friction damping effect. The

differences between the variable and constant friction coefficients are especially obvious in

the case where the profile error excitations are ignored. Therefore, the friction should be

included in a variable speed process dynamic model, and the variation in the friction

coefficient should not be ignored, especially when the profile error excitations are ignored.

The profile error excitation is the source of vibration in the axial

direction. If the profile error excitations are ignored, the axial displacements of herringbone

gears will disappear, just like spur gears.

The dynamic model proposed in this paper can also be used for spur gears by setting the helix

angle to zero and for helical gears by setting either half width of the herringbone gears to

zero.


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Chapter 11.

REFERENCES

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8186.

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Borner, J., and Houser, D. R., 1996, Friction and Bending Moments as GearNoiseExcitations, SAE Technical Paper No. 961816, 105(6), pp. 16691676.

He, S., Singh, R., and Pavic, G., 2008, Effect of Sliding Friction on Gear NoiseBased on a Refined Vibro-Acoustic Formulation, Noise Control Eng. J., 56(3),pp.

164175. Velex, P., and Sainsot, P., 2002, An Analytical Study of Tooth Friction Excitations

in Errorless Spur and Helical Gears, Mech. Mach. Theory, 37(7), pp.641658.

He, S., Gunda, R., and Singh, R., 2007, Inclusion of Sliding Friction in Contact Dynamics Model for Helical Gears, ASME J. Mech. Des., 129(1), pp. 4857.

Velex, P., and Cahouet, V., 2000, Experimental andNumerical Investigations on theInfluence of Tooth Friction in Spur and Helical Gear Dynamics, ASME J. Mech.

Des., 122(4), pp. 515522.

Xu, H., Kahraman, A., Anderson, N. E., and Maddock, D. G., 2007, PredictionofMechanical Efficiency of Parallel-Axis Gear Pairs, ASME J. Mech. Des.,129(1), pp.

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ODonoghue, J. P., and Cameron, A., 1966, Friction and Temperature in RollingSliding Contacts, ASLE Trans., 9(2), pp. 186194.

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Benedict, G. H., and Kelley, B. W., 1961, Instantaneous Coefficients of GearToothFriction, ASLE Trans., 4(1), pp. 5970.

ISO TC 60, DTR 13989. Vaishya, M., and Singh R., 2003, Strategies for Modeling Friction in Gear

Dynamics, ASME J. Mech. Des., 125(2), pp. 383393.

Vaishya, M., and Singh, R., 2001, Sliding Friction-Induced Non-Linearity andParametric Effects in Gear Dynamics, J. Sound Vib., 248(4), pp. 671694.

Vaishya, M., and Singh, R., 2001, Analysis of Periodically Varying Gear MeshSystems With Coulomb Friction Using Floquet Theory, J. Sound Vib.,


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He, S., Gunda, R., and Singh, R., 2007, Effect of Sliding Friction on the Dynamics of Spur Gear Pair With Realistic Time-Varying Stiffness, J. Sound Vib., 301(35),

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Ajmi, M., and Velex, P., 2005, A Model for Simulating the Quasi-Static andDynamic Behaviour of Solid Wide-Faced Spur and Helical Gears, Mech. Mach.Theory, 40(2), pp. 173190.

Xu, H., 2005, Development of a Generalized Mechanical Efficiency PredictionMethodology for Gear Pairs, Electronic thesis or Dissertation, Ohio State University,

Columbus, OH.

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