1

ABSTRACT

Vibration induced by gears includes important data about gearbox condition. We can use

dynamic modeling of gear vibration for increasing our information about vibration generating

mechanisms in gearboxes and dynamic behavior of gearbox in the presence of some kind of gear

defects. In this report a six degree-of-freedom nonlinear dynamic model including different gear

errors and defects is developed for investigation of effects of tooth localized defect and profile

modifications on overall gear dynamics. Interactions between tooth modifications and profile

error are studied and the role of profile modification in dynamic response when a localized

defect is incurred by a tooth is shown. It is indicated that although profile modifications and

profile errors are micro-geometrical, they have considerable effects on vibrations of gear pair.

Especially for the case of root relieved teeth that is shown to be more effective in reduction of

vibration in the presence of tooth localized defect. Finally, the simulation results are compared

with results from literature and the model is verified.

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TABLE OF CONTENTS

1. INTRODUCTION ......................................................................................................................................... 4

1.1 DEFINITIONS ............................................................................................................................................ 4

1.2 TYPES OF GEARS ...................................................................................................................................... 5

1.3 ADVANTAGES .......................................................................................................................................... 6

1.4 DISADVANTAGES ..................................................................................................................................... 7

1.5 APPLICATIONS ......................................................................................................................................... 7

1.6 MATERIALS OF SPUR GEAR ..................................................................................................................... 8

2. LINEAR Vs. NONLINEAR ............................................................................................................................. 8

2.1 NONLINEAR DYNAMICS ........................................................................................................................ 10

2.2 DYNAMIC MODELING OF GEARS ........................................................................................................... 11

2.3 HISTORY OF DYNAMIC MODELLING ...................................................................................................... 11

2.4 TOOLS FOR NONLINEAR DYNAMICS ..................................................................................................... 13

2.4.1 FRACTAL ............................................................................................................................................. 13

2.4.2 BIFUCRATION DIAGRAM .................................................................................................................... 14

2.4.3 POINCARE´ MAPS ............................................................................................................................... 15

2.4.4 LYAPUNOV EXPONENT ....................................................................................................................... 16

2.4.5 POWER SPECTRUM ............................................................................................................................ 17

2.4.6 DYNAMIC TRAJECTORIES ................................................................................................................... 18

3. MATHEMATICAL MODELING ................................................................................................................... 18

3.1 SIX-DEGREE-OF-FREEDOM NONLINEAR MODEL ................................................................................... 19

3.2 INCORPORATION OF ECCENTRICITY ERROR .......................................................................................... 21

3.3 INCORPORATION OF PROFILE ERROR ................................................................................................... 21

3.4 CALCULATION OF MESH STIFFNESS ...................................................................................................... 21

3.5 TOOTH PROFILE MODIFICATION ........................................................................................................... 22

3.6 LOCALIZED DEFECT ................................................................................................................................ 23

4. RESULTS ................................................................................................................................................... 24

5. CONCLUSIONS ......................................................................................................................................... 27

6. FUTURE SCOPE ........................................................................................................................................ 28

REFERENCES ................................................................................................................................................ 29

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LIST OF FIGURES

Figure 1 Nomenclature of Spur Gear ............................................................................................................ 5

Figure 2. Spur Gear and Plastic Spur Gears used in a film winding component. .......................................... 7

Figure 3 Spur Gears are used in automatic packing machine ....................................................................... 8

Figure 4 Fractal in bifurcation diagram. .................................................................................................... 14

Figure 5 Bifurcation diagram of logistic map . ( xn+1 = r xn (1- xn))........................................................... 15

Figure 6In Poincaré section S, the Poincaré map P projects point x onto point P(x). ................................. 16

Figure 7 Poincare’ map of journal bearing rotor ......................................................................................... 16

Figure 8 Lyapunov exponent for rotor bearing trajectories. ........................................................................ 17

Figure 9 Power spectrum gear bearing system ............................................................................................ 18

Figure 10 Six -degree-of-freedom dynamic model. .................................................................................... 19

Figure 11 A couple of spur gear teeth pair in contact ................................................................................. 22

Figure 12 The effect of profile modification coefficient on tooth shape ..................................................... 22

Figure 13 Decaying half-sine pulse train .................................................................................................... 23

Figure 14 Pitting on a tooth profile ............................................................................................................. 23

Figure 15 Power spectrum of pinion bearing acceleration .......................................................................... 25

Figure 16 Vibration amplitude of pinion bearing for different modifications ............................................ 25

Figure 17 Vibration amplitude of pinion bearing for different modifications in change-over region......... 26

Figure 18 Acceleration near the change-over region with variation of defect width in face direction f .... 27

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1. INTRODUCTION

Spur Gears are the most common means of transmitting power in the modern mechanical

engineering world. They vary from tiny size used in the watches to the large gears used in marine

speed reducers; bridge lifting mechanism and railroad turn table drivers. They form vital

elements of main and ancillary mechanism in many machines such as automobiles, tractors,

metal cutting machine tools, rolling mills, hoisting and transmitting machinery and marine

engines etc.[1]

1.1 DEFINITIONS

1) MODULE:

Module of a gear is defined as ratio of diameter to number of teeth. m= d/N

2) FACE WIDTH

The width along the contact surface between the gears is called the face width.

3) TOOTH THICKNESS

The thickness of the tooth along the pitch circle is called the tooth thickness.

4) ADDENDUM

The radial distance between the pitch circle and the top land of the gear is called the addendum.

5) DEDENDUM

The radial distance between the pitch circle and the bottomland of the gear is called the

dedendum.

6) PRESSURE ANGLE

The angle between the line joining the centers of the two gears and the common tangent to the

base circles.[1]

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Figure 1 Nomenclature of Spur Gear (B.Harish Reddy, 2011)

1.2 TYPES OF GEARS

• Worm

• Non-circular

• Rack and pinion

• Epicyclic

• Sun and planet

• Harmonic gear

• Cage gear

• Magnetic gear

• Spur

• Helical

• Skew gears

• Double helical

• Bevel

• Hypoid

• Crown

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1.3 ADVANTAGES

Gear is one kind of mechanical parts. It can be widely used in industries. A gear is a rotating

machine part having cut teeth, or cogs, which mesh with another toothed part in order to

transmit torque.

Spur gear is the simplest type of gear which consists of a cylinder or disk. Its form is not

straight-sided, thus, the edge of each tooth is straight and aligned parallel to the axis of

rotation. Only gears fit to parallel axles can they rotate together correctly.

As the most common type, spur gears are often used because they are the simplest to design

and manufacture. Besides, they are the most efficient. When compared to helical gears, they

are more efficient. The efficiency of a gear is the power output of its shaft divided by the

input power of its shaft multiplied by 100. Because helical gears have sliding contact

between their teeth, they produce axial thrust, which in turn produces more heat. This causes

a loss of power, which means efficiency is lost.

In addition to these, they also have many other advantages. Spur gears have a much simpler

construction than helical gears because their teeth are straight rather than angular. Therefore,

it is much easier to design and produce them. And they will not fail or break easily. And this

makes them cheaper to purchase and to maintain which then leads to less cost.

SIMPLICITY

Because their teeth are straight rather than angular, spur gears have a much simpler construction

than helical gears. As such, they are easier to produce, and they tend not to break or fail as easily.

This also makes them easier to find.

EFFICIENCY (95% - 98%)

Spur gears are more efficient than helical gears. The efficiency of a gear is the power output of

its shaft divided by the input power of its shaft multiplied by 100. Because helical gears have

sliding contact between their teeth, they produce axial thrust, which in turn produces more heat.

This causes a loss of power, which means efficiency is lost.[1]

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COST

Because spur gears are simpler, they are easier to design and manufacture, and they are less

likely to break. This makes them cheaper to purchase and to maintain.

1.4 DISADVANTAGES

Although they are common and efficient, spur gears have disadvantages as well. Firstly, they

are very noisy when used at some speeds because the entire face engages at once. Therefore,

they're also known as slow-speed gears. Secondly, they can only be used to transfer power

between parallel shafts. They cannot transfer power between non-parallel shafts. Thirdly,

when compared with other types of gears, they are not as strong as them. They cannot handle

as much of a load because the teeth are small and situated parallel to the gear axis, rather than

being large and situated diagonally as the teeth on a helical gear are.

According to the above, we can conclude that spur gears have many advantages as well as

some disadvantages. Although sometimes, its disadvantages may affect them a lot, their

advantages still outweigh their disadvantages. That is to say, spur gears are still popular

among many industries. And they can have good performances to meet people's

requirements[1]

1.5 APPLICATIONS

Figure 2. Spur Gear and Plastic Spur Gears used in a film winding component.[7]

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Figure 3 Spur Gears are used in automatic packing machine[7]

1.6 MATERIALS OF SPUR GEAR

While coming to manufacturing materials for Spur gears, a wide variety is available. These

includes [1]

2. LINEAR Vs. NONLINEAR

Linear relationships are often the first approximation used to describe any relationship, even

though there is no unique way to define what a linear relationship is in terms of the underlying

nature of the quantities. For example, a linear relationship between the height and weight of a

person is different than a linear relationship between the volume and weight of a person. The

second relationship makes more sense, but both are linear relationships, and they are, of course,

• Cast iron

• Phenolic

• Bakelite

• Plastics

• Steel

• Nylon

• Aluminium

• Bronze

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incompatible with each other. Medications, especially for children, are often prescribed in

proportion to weight. This is an example of a linear relationship.

Nonlinear relationships, in general, are any relationship which is not linear. What is important in

considering nonlinear relationships is that a wider range of possible dependencies is allowed.

When there is very little information to determine what the relationship is, assuming a linear

relationship is simplest and thus, by Occam's razor, is a reasonable starting point. However,

additional information generally reveals the need to use a nonlinear relationship.

A linear dynamical system is one in which the rule governing the time-evolution of the system

involves a linear combination of all the variables.

EXAMPLE

A nonlinear dynamical system is simply not linear

EXAMPLE: 𝜃

+ sin 𝜃 = 0 (nonlinear pendulum)

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2.1 NONLINEAR DYNAMICS

Nonlinear dynamics, study of systems governed by equations in which a small change in one

variable can induce a large systematic change; the discipline is more popularly known as chaos

(see chaos theory). Unlike a linear system, in which a small change in one variable produces a

small and easily quantifiable systematic change, a nonlinear system exhibits a sensitive

dependence on initial conditions: smaller virtually immeasurable differences in initial conditions

can lead to wildly differing outcomes. This sensitive dependence is sometimes referred to as the

"butterfly effect," the assertion that the beatings of a butterfly’s wings in Brazil can eventually

cause a tornado in Texas. Historically, in fact, one of the first nonlinear systems to be studied

was the weather, which in the 1960s Edward Lorenz sought to model by a relatively simple set of

equations. He discovered that the outcome of his model showed an acute dependence on initial

conditions. Later work revealed that underlying such chaotic behavior are complex but often

aesthetically pleasing geometric forms called strange attractors. Strange attractors exist in an

imaginary space called phase space, in which the ordinary dimensions of real space are

supplemented by additional dimensions for the momentum of the system under investigation. A

strange attractor is a fractal, an object that exhibits self-similarity on all scales. A coastline, for

instance, looks much the same up close or far away. Nonlinear dynamics has shown that even

systems governed by simple equations can exhibit complex behavior. The evolution of nonlinear

dynamics was made possible by the application of high-speed computers, particularly in the area

of computer graphics, to innovative mathematical theories developed during the first half of the

20th cent. Three branches of study are recognized: classical systems in which friction and other

dissipative forces are paramount, such as turbulent flow in a liquid or gas; classical systems in

which dissipative forces can be neglected, such as charged particles in a particle accelerator; and

quantum systems, such as molecules in a strong electromagnetic field. The tools of nonlinear

dynamics have been used in attempts to better understand irregularity in such diverse areas as

dripping faucets, population growth, the beating heart, and the economy.[13]

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2.2 DYNAMIC MODELING OF GEARS

Dynamic modeling of gears is a useful tool for studying vibration response of gear systems

under various gear parameters and working conditions. The earlier of gear dynamic models were

presented in 1990’s. A thorough and comprehensive review of gear mathematical models used in

gear dynamics before 1986 has been presented by Ozguven and Houser. In this review gear

dynamic models without defects have been discussed [2]. Many of single and multi-degree-of-

freedom gear dynamic models have been developed, most of which were single or two degree of-

freedom models that only included torsional vibration. Kahraman and Singh examined the

nonlinear dynamics of a spur gear pair with backlash in frequency domain, with the nonlinearity

originating mainly from backlash.

In many of earlier gear dynamic models, tooth defects have not been taken into account. And in

some cases, assembly, eccentricity, and manufacturing errors have been included in dynamic

models. In past few years, researchers have been working on gear dynamic models including

gear defects like surface pitting, spalling, wear, crack, and broken tooth. Kuang and Lin studied

the effects of tooth wear on vibration spectra of a spur gear pair, their findings showed that the

dynamic loading and vibration spectra of spur gear pair may vary significantly due to tooth wear.

Their study revealed that this variation could be a symptom for monitoring tooth wear. A study

on dynamic models including gear defects has been performed by Parey and Tandon [3]. The

review suggested that although much work had been done on gear vibration modeling in the

presence of gear defects, yet a precise analytical procedure for prediction of gear vibration

considering tooth local defect could be developed. Most of gear defects are incurred by gear

teeth for which tooth profile is perturbed from the original involute.[1]

2.3 HISTORY OF DYNAMIC MODELLING

Although many researchers consider mesh stiffness variations and transmission error as the main

contributors to gear systems vibration, tooth localized defects or errors can add considerable

amount of vibration or noise to the gear system. Different procedures are used by researchers to

simulate the effects of tooth localized defects. Parey. Developed a six-degree- of-freedom

dynamic model including local tooth defect, considering the effects of excitations due to gear

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errors and defects and tooth profile modifications. They used impulse phenomenon to simulate

the effect of localized defect. In a different approach, some of authors like Fakh fakh

investigated the effect of tooth defect through the change it makes in mesh stiffness. Lin and

Kuang also performed a simulation to investigate the effects of tooth profile defects on dynamic

contact loads. Chen and Shao proposed an analytical method for calculation of tooth mesh

stiffness with root crack and successfully examined the effects of crack propagation on dynamic

response by calculating some statistical indicators, their work could assist the researchers who

aim to simulate tooth root crack in dynamic modeling without a need to FEA models of meshing

gears.

Often, tooth profile is intentionally modified to reduce the effects of some incipient defects. Tip

or root relief is considered as a method for prevention of undercutting gear teeth. This

modification has another important effect and it is the reduction of transmission error which

leads in reduction of gear system noises. Gear tooth profile has a great influence on the vibration

and noise of gear system. Tavakoli and Houser were of the first who investigated the

optimization of tooth profile to minimize the static transmission error. Lin. Implemented an

analytical approach to study the effect of profile modifications on static transmission error and

dynamic loading of spur gears, one of the important results drawn from their work was that

increasing applied load reduces the sensitivity of spur gears to changes in length of profile

modifications. Cai and Hayashi developed a method to optimize the modification of tooth profile

for a pair of spur gears to make its rotational vibration become zero by using a vibration model.

Optimization of static transmission error characteristics for a range of torques by introducing

appropriate tooth profile modifications and micro geometries to minimize the static transmission

error variations was considered as a method for gear system noise and vibration. In an attempt by

Kuang and Lin, an analytical formulation for dynamics of a spur gear pair is developed in which

a harmonically varying input torque has been used. They showed that an increase in contact ratio

and addendum modification causes the transmitted torque variation and peaks of its

corresponding frequency spectra to decay. Wang adopted a mathematical model to identify the

relationship between the rotational movement of gears and acoustical measured noise signals

from a gear tester, and then they could optimize the profile modification by the use of identified

model. Through the use of dynamic modeling, Barbieri showed that the vibration level of the

profile, optimized using modifications was lower than that of pure involute even if profile errors

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were taken into account. An original model combining the analysis of crack initiation and

propagation in relation to dynamic tooth loads has been developed by Osman and Velex. They

found the role of profile relief to be crucial as a substantial reduction of fatigue risk could be

expected in the part of the teeth close to the engagement area.[1]

To our knowledge based on the published literature, none of authors have considered the

interactions between tooth profile modification and tooth localized defect on vibrations of a gear

system. In this report the effect of different profile modifications and profile error on dynamic

response of gear system in the presence of tooth localized defect, both in time and frequency

domain will be studied. The role of profile modification in reduction of vibration especially in

change-over region will be investigated. At the end, simulation results are presented and through

the comparison with results included in literature, modeling procedure is verified.

2.4 TOOLS FOR NONLINEAR DYNAMICS

There are various parameters which are used to represent nonlinear dynamics results which help

the designer to take some steady actions before high vibrations. The tools which are used for

nonlinear dynamics are as follows:

2.4.1 FRACTAL

A fractal is a natural phenomenon or a mathematical set that exhibits a repeating pattern that

displays at every scale. If the replication is exactly the same at every scale, it is called a self-

similar pattern. Fractals can also be nearly the same at different levels. This latter pattern is

illustrated in Figure 4. Fractal also includes the idea of a detailed pattern that repeats itself.

Fractals are different from other geometric figures because of the way in which they scale.

Doubling the edge lengths of a polygon multiplies its area by four, which is two (the ratio of the

new to the old side length) rose to the power of two (the dimension of the space the polygon

resides in). Likewise, if the radius of a sphere is doubled, its volume scales by eight, which is

two (the ratio of the new to the old radius) to the power of three (the dimension that the sphere

resides in). But if a fractal's one-dimensional lengths are all doubled the spatial content of the

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fractal scales by a power of two that is not necessarily an integer. This power is called the fractal

dimension of the fractal, and it usually exceeds the fractal's topological dimension.[2]

• Characteristics:

– A fractal is a geometric object which can be divided into parts, each of which is

similar to the original object.

– They possess infinite detail, and are generally self-similar (independent of scale)

– They can often be constructed by repeating a simple process (a map, for instance)

ad infinitum.

– They have a non-integer dimension.

Figure 4 Fractal in bifurcation diagram.

2.4.2 BIFUCRATION DIAGRAM

A bifurcation diagram summarizes the essential dynamics of a gear-train system and is therefore

a useful means of observing its nonlinear dynamic response. In the present analysis, the

bifurcation diagrams are generated using two different control parameters, namely, the

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dimensionless unbalance coefficient, β, and the dimensionless rotational speed ratio, s,

respectively. In each case, the bifurcation control parameter is varied with a constant step and the

state variables at the end of one integration step are taken as the initial values for the next step.

The corresponding variations of the x(nT) coordinates of the return points in the Poincaré map

are then plotted to form the bifurcation diagram.[2]

Figure 5 Bifurcation diagram of logistic map . ( xn+1 = r xn (1- xn))

2.4.3 POINCARE´ MAPS

A Poincaré section is a hyper-surface in the state-space transverse to the flow of the system of

interest. In non-autonomous systems, points on the Poincaré section represent the return points of

a time series corresponding to a constant interval T, where T is the driving period of the

excitation force. The projection of the Poincaré section on the x(nT) plane is referred to as the

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Poincaré map of the dynamic system. When the system performs quasi-periodic motion, the

return points in the Poincaré map form a closed curve. For chaotic motion, the return points form

a fractal structure comprising many irregularly-distributed points. Finally, for nT-periodic

motion, the return points have the form of n discrete points.[11]

Figure 6 In Poincaré section S, the Poincaré map P projects point x onto point P(x). (Shivakumar

Jolad,2005)

Figure 7 Poincare’ map of journal bearing rotor(C.K. Chen, 2007)

2.4.4 LYAPUNOV EXPONENT

The Lyapunov exponent of a dynamic system characterizes the rate of separation of

infinitesimally close trajectories and provides a useful test for the presence of chaos. In a chaotic

system, the points of nearby trajectories starting initially within a sphere of radius ε0, initial

radius, after time t an approximately ellipsoidal distribution with semi-axes of length εj(t).

• The Lyapunov exponents of a dynamic system are defined by

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j(t)| = еλjt

|ε0|

• λj denotes the rate of divergence of the nearby trajectories.

• A positive value of the maximum Lyapunov exponent λ is generally taken as an

indication of chaotic motion.[10]

Figure 8 Lyapunov exponent for rotor bearing trajectories.( C.K. Chen, 2007)

2.4.5 POWER SPECTRUM

The spectrum components of the motion performed by the gear-bearing system are analyzed

using the Fast Fourier Transformation method to derive the power spectrum of the displacement

of the dimensionless dynamic transmission error. In the analysis, the frequency axis of the power

spectrum plot is normalized using the rotational speed, ω [10]

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Figure 9 Power spectrum gear bearing system(C.K. Chen, 2007)

2.4.6 DYNAMIC TRAJECTORIES

The dynamic trajectories of the gear-bearing system provide a basic indication as to whether the

system behavior is periodic or non-periodic. However, they are unable to identify the onset of

chaotic motion. Accordingly, some other form of analytical method is required. In the present

study, the dynamics of the gear-bearing system are analyzed using Poincaré maps derived from

the Poincaré section of the gear system.[10]

3. MATHEMATICAL MODELING

In many of gear systems, the coupling between torsional vibration modes is controlled by mesh

stiffness. So in most of practical cases a two-degree-of-freedom model which only includes

torsional vibrations may yield accurate results. However, when torsional mode which is

controlled by mesh stiffness, is coupled with other vibration modes, in such cases increasing the

degrees-of-freedom of the model is more important than elaborating it with some secondary

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effects such as tooth friction. The model presented here, is the same developed by Ozguven with

some modifications, which includes the effects of gear teeth profile modification and localized

tooth defect.[10]

3.1 SIX-DEGREE-OF-FREEDOM NONLINEAR MODEL

The model considered here, consists of a pair of spur gears on the two shafts that are connected

to the load and prime mover. Torsional compliances of shafts and transverse compliances of

bearings and consequently torsional and transverse vibrations of the gears along the line of action

have been taken into account. By the use of this model, the response including modulations due

to torsional and transverse vibrations can be calculated.[5]

Figure 10 Six -degree-of-freedom dynamic model.( H.N. Ozguven,1988)

The 6-degree-of-freedom nonlinear model, which is shown in Fig. 1, has four angular rotations

of prime mover, pinion, gear and load) and two translations (of pinion and gear) along the line of

action. The effects that are included in the mathematical model and thus considered in the

dynamic analysis are: Time varying mesh stiffness and damping; torsional compliances of pinion

and gear shafts; material damping in shafts (linear viscous); bearing compliances and damping’s

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(linear viscous); transverse compliances of shafts; inertia of prime mover and load; drive and

load torques; separation of teeth in mesh; backlash and backside collision that may follow tooth

separation. The dynamic analysis is done in the plane of gears and any out-of-plane motion is

neglected. As the excitation is basically along the line of action, transverse vibration of gears in

the direction perpendicular to the line of action is neglected. Another simplifying assumption

made is to neglect off-line of action and the friction between the teeth in contact. The main

advantage of this model is that the vibration transmitted to the bearings through the gears can be

used for further investigations and comparison with experimental test data.[4]

According to Fig. 1 equations of motion (1-6) can be derived.[11]

D �̈�D + ct1 (�̇�D - �̇�1) + kt1 (𝜃D - 𝜃1) = TD (1)

1 �̈�1 + ct1 (�̇�1 - �̇�D) + kt1 (𝜃1 – 𝜃D) = -F0R1 (2)

2 �̈�2 + ct2 (�̇�2 - �̇�L) + kt2 (𝜃2 – 𝜃L) = F0R2 (3)

L �̈�L + ct2 (�̇�L - �̇�2) + kt2 (𝜃L – 𝜃2) = -TL (4)

m1 ̈1 + c1 ̇1 + k1y1 = F0 (5)

m2 ̈2 + c2 ̇2 + k2y2 = -F0 (6)

where F0, is dynamic mesh force and can be calculated by

F0 = km (θ1R1 – θ2R2 + y2 – y1) + cm (�̇�1R1 - �̇�2R1 + ̇2 - ̇1) (7)

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3.2 INCORPORATION OF ECCENTRICITY ERROR

Gears after assembling on shafts, may have some eccentricity. The effect of eccentricity can be

investigated by incorporating it in the error function. The eccentricities can be modeled as[4]

Eec = е1 sin 𝜃1 + е2 sin 𝜃2 (8)

where e1, e2 are the eccentricity errors of gear 1 and gear 2, respectively.

3.3 INCORPORATION OF PROFILE ERROR

It can be said that every gear has some deviation from ideal involute profile. It has to be taken

into account as an error. For a given pair of spur gear teeth, the value of profile error is random

and it can be denoted by

e1 (random) = [1- d (1- li)] e (9)

where e is the maximum error value, d is coefficient of error variation, range (0 - 1), li is a

random value, range(0 - 1), whose elements are randomly distributed for z1 pair of teeth; z1 is

the number of pinion teeth.[4]

3.4 CALCULATION OF MESH STIFFNESS

The mesh stiffness between an engaged gear pair consists of two parts: One associated with the

local Hertzian deformation and the other associated with the tooth bending deflection. The unit

width Hertzian stiffness resulting from the tooth surface contact was first approximated by Yang

as

h =

(10)

where E is young modulus and v is Poisson’s ratio. An equation has been proposed for

calculation of bending stiffness of an addendum modified gear tooth [5].

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Figure 11 A couple of spur gear teeth pair in contact (M. Divandari,2012)

3.5 TOOTH PROFILE MODIFICATION

Meshing of the next tooth pair does not occur on involute curve. Because as a result of elastic

deformation, a sudden interference will arise and the tip of tooth will enter into flank of meshing

tooth. So it is necessary to modify tooth profile. Tooth profile modification coefficient which is

called profile shift coefficient or addendum modification coefficient is shown by X , and its value

can be positive or negative. The effect of different addendum modification coefficients on tooth

profile is shown in Fig. 12. It is obvious that the change in tooth shape will affect its bending

stiffness and consequently will generate variations in the mesh stiffness of the mating gears.[5]

Figure 12 The effect of profile modification coefficient on tooth shape(Gitin M Maitra, 1985)

.

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3.6 LOCALIZED DEFECT

Different procedures have been used by authors to simulate the effect of surface pitting. Parey

used impulse phenomenon for simulation of pitting in gear dynamics. Original work was

presented, in which it was claimed that the severity, extent and age of damage can be better

represented by pulses. In a gear pair system, due to the deformation and elasticity of the

contacting components, half-sine pulses are considered to model the impulse forcing signal more

representatively. The response of the practical system to a half-sine pulse can be best presented

by decaying sinusoid as shown in Fig. 13 [5]

Figure 13 Decaying half-sine pulse train

Figure 14 Pitting on a tooth profile(A. Parey, 2006)

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(

) sin √ 0t + sin 2)} (11)

where 2) and k is the height of the pulse; is the damping ratio for linear

deformation; ω0 = frequency of the generated pulse; a pulse width; b is the defect

width in profile direction; and va is the relative velocity at the defect point.

After including the profile error, eccentricities and defect, the dynamic mesh force F0 is modified

as follows:

F = F0 + ka (E1 + Eec) + kb (E2 + Eec) + ca ( ̇1 + ̇ec) + cb ( ̇2 + ̇ec) + kh* x(t)*f (12)

where ka, kb, ca, cb are the mesh stiffness and damping at contact points A, B respectively (See

Fig. 11), E1, E2are profile errors for tooth pairs at points A, B respectively which are calculated.

f is defect width in face direction, kh is the unit width Hertzian stiffness and a dot denotes

differentiation with respect to time. Since equations of the model with inclusion of backlash and

eccentricities, are non-linear, therefore these equations could not be solved analytically. Then the

set of equations are solved using the Runge-Kutta 4th

order method.

4. RESULTS

Time domain signals (acceleration (m/sec2)), representing one complete revolution of

gear/pinion set from simulations. To gain more insight into the signals, the acceleration signal is

shown in a smaller time interval. The parameters of the gear system used in computer

simulations presented in Table 1 are determined based on reference. To demonstrate the effect of

profile modification coefficient ton gear vibration, simulated signals for different values of X are

first compared in frequency domain (Fig. 15).Power spectral densities were calculated using

Welch’s averaged, modified periodogram method. Spectrum comparisons for the gears with

different values of profile shift coefficients are shown in Fig. 16. It can be seen that at

frequencies lower than 1kHzwhere meshing frequency and its multiples are present, and

frequencies higher than 4kHz, vibration level is the highest for the tip relieved teeth and the

lowest for root.

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Figure 15 Power spectrum of pinion bearing acceleration (B. H. Aghdam,2012)

relieved ones. It is interesting, because it follows the same behavior as mesh frequency for the

same tooth modifications. Actually it can be said that an increase in mesh stiffness due to profile

modification causes an increase in the level of vibration, in frequency domain, where an increase

in addendum modification value caused a decrement in vibration level both in time and

frequency domains.[5]

Holding the maximum error value at a fixed level and changing the profile modification value

from minus to positive, the vibration amplitude at mesh frequency and its multiples increases.

Figure 16 Vibration amplitude of pinion bearing for different modifications (M. Divandari,2012)

In order to investigate the effect of profile error in change-over region, where double tooth pair

contact changes to single tooth contact, profile error value is increased for two types of profile

shift, positive and minus. The results are shown in Fig. 16. It is indicated that increasing the

profile error amplitude, generally increases the oscillations in the system in case of tip relief

(negative value of X) or root relief (positive value of X) and has an effect on discontinuity in this

26

region. But the effect of addendum modification is mainly important at change-over region,

where the smoother meshing occurs in case of tip relief (X = 0.1)and the lowest profile error

(Fig. 16). A closer view of the change-over region is demonstrated in Fig. 17.[5]

Figure 17 Vibration amplitude of pinion bearing for different modifications in change-over region.

(M. Divandari,2012)

For investigation of effects of tooth surface pitting, dimensions of pitting are varied for different

modifications and the results are presented in Figs. 17 and 18.In Fig. 17, the effect of increment

of defect width in face direction f for two kinds of tooth modifications i.e. tip relief (positive X

value) or root relief (negative X value), is studied. The left part of these curves shows the region

where a single pair of teeth is in contact just near the pitted zone and the right part is for double

teeth pairs’ contact. It id indicated that discontinuity value is smaller for the case of root relief

and an increase in defect dimension causes an increase in the discontinuity for both cases. It

means that a sudden impact due to localized defect is well tolerated in case of minus addendum

modification coefficient (root relief). But when either side of the change-over region is

considered, it is seen that the impact due to pitting has more effect on root relieved tooth pair. If

the simple case of single pair of tooth contact at the left side of the region is considered, a clear

insight can be achieved.

With respect to Fig.12(d), root relieved tooth is similar to a cantilever beam that is weakened at

its supported end, so the impact to a pair of these teeth in mesh may have more significant effect

due to lower stiffness, in comparison with tip relieved tooth pair.

27

Figure 18 Acceleration near the change-over region with variation of defect width in face direction f

(R. Barzamini,2012)

In another case, defect width in profile direction b is altered and the results are shown in Fig. 18.

The same effects as of Fig. 17 can be seen here, except that because of an increment in impulse

duration, the impact to the system is a little smoother in comparison with the case of reduced

defect width in profile direction. In this case, again the impact is better absorbed with gear

shaving root relieved teeth.

5. CONCLUSIONS

This report presented a combined gear/bearing dynamic model for a gearbox to study the effect

of tooth profile modification on gear vibration. The dynamic model considered here, has the

ability of modeling different types of gear errors and defects also showed a good ability in

modeling tooth profile modification interaction with profile error and tooth localized defect and

its effect on vibration response of the system. Results obtained from simulations showed that the

change in mesh stiffness due to profile modifications (root/tip relief), although they are micro-

geometrical, causes the change in vibration response of gear pair in a manner similar to the

change in mesh stiffness. The simulated signals showed the same behavior as the measured

signals. Two interesting results were drawn from the simulations:

First is that increasing the profile error, while profile modification coefficient is kept constant,

increases the overall oscillations of the system, and also affects the meshing at change-over

region. But when comparing the effect of profile modification to that of profile error, addendum

28

modification could have an important advantage and it is smoother meshing at change-over

region in lower profile errors. So the important result is that if the gears are root relieved, and the

profile error is decreased, the vibration level of the system, both in time and frequency domain

decreases significantly and meshing occurs more smoothly.

Second is that impulse due to pitting causes smaller discontinuity in the change-over region for

the case of root relieved teeth. It means that the localized defect may affect the gear pair

vibration less in the case of tip relieved teeth.

So in turn it will allow the designer to take actions against chaotic motion by observing the tools

for nonlinear analysis viz. bifurcation diagram, poincare’ map, fractals etc.

6. FUTURE SCOPE

This model can be further extended to consider the effect of the other tooth defects, such as wear

with varying load in presence of tooth modifications and specially profile error which seems to

be an important factor in gear systems simulations.

29

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