preliminary results related on the response of asymmetric

gears under dynamic loading are presesented. The effect

of some design parameters, such as pressure angle or

tooth height on dynamic loads, was shown. Although

asymmetric tooth is emerging as a major concern in gear

researches, in literature, by now there was not any virtual

tool for design of spur gears with asymmetric teeth.

Therefore, in this study, a MATLAB-based virtual tool

called DYNAMIC to analyze dynamic behavior of spur

gears with asymmetric teeth depending on various tooth

parameters. The objective of this paper is to introduce a

developed MATLAB-based virtual tool called DY-

NAMIC and to demonstrate its capabilities.

2. Dynamic Analysis of Gears

2.1. Dynamic Model

During one mesh period the tooth contact load in one

gear pair does not stay constant. This load varies de-

pending on the transition from double tooth contact to

single tooth contact.

To determine the variation of dynamic load as a func-

tion of the contact position (or time), it is necessary to

derive the equations of motion for gear tooth pair in a

mesh. Considering the free body diagrams of the gear

and pinion shown in Figure 1, the equations of motion

can be formulated as:

g

gbgIIIgII IgIIII IIbgD

J

rFFF FrF

&& (1)

p

pbpDb pIIIp IIIp IIIIII

J

rFr F FFF

&&

(2)

where Jp and Jg represent the polar mass moments of in-

ertia of the pinion and gear, respectively. The dynamic

contact loads are FI and FII, while

I and

II are the in-

stantaneous coefficients of friction at the contact points.

p and

g represent the angular displacements of pinion

and gear. The radii of the base circles of the engaged

gear pair are rbp and rbg, while the radii of curvature at

the mating points are

pI,II and

gI,II.

In above equations, if the speed of the pinion tooth is

greater than the speed of the gear tooth, the sign of the

friction force is positive, otherwise it is negative. The

static tooth load is defined as:

p

g

Dbp bg

TT

Frr

(3)

If the angular coordinate is converted into the coordi-

nate along the line of action, the displacements of the

Figure 1. Engaging teeth pairs along the line of action.

Copyright © 2012 SciRes. WJM

F. KARPAT ET AL.

Copyright © 2012 SciRes. WJM

241

undeformed tooth profiles, along the line action, can be

written as: rpg

x

yy

&&&

(7)

g

gg

yr

(4) rpg

x

yy

&&&&&& (8)

Including viscous damping, the equations of motion

reduce to:

p

pp

yr

(5)

The relative displacement, velocity, and acceleration

can then be cast as:

22

2

rrrs

x

xx

&&& x (9)

rpg

x

yy (6) The loaded static transmission errors can be obtained:

g

pDI IpIggIpIIIIpIIggIIp

s

IpIggIp IIpIIggIIp

MMFKSMSM KSMSM

xKSMSMK SMSM

The effective gear masses are:

2

p

pbp

M

Jr (11)

2

g

gbg

M

Jr (12)

The equivalent stiffness of meshing tooth pairs can be

written as:

p

IgI

I

p

IgI

kk

Kkk

(13)

p

II gII

II

p

II gII

kk

Kkk

(14)

The friction experienced by the pinion and the gear

can be expressed as:

1

I

pI

pI bp

Sr

(15)

1

I

gI

gI bd

Sr

(16)

1

I

IpII

pII bp

Sr

(17)

1

I

IgII

gII bd

Sr

(18)

The signs in the above expressions are positive (+) for

the approach and negative (−) for the recess. The dy-

namic contact loads, which include tooth profile error,

can then be written as:

I

Ir I

FKx

(19)

I

IIIrII

FKx

(20)

where

I and

II are the tooth profile errors. In this paper,

the effects of profile errors on the dynamic response of

gears are not considered. Thus, the tooth profile errors

are assumed to be zero. The developed computer pro-

gram has a capability of using any approach for the de-

termination of errors.

The reduced equation of motion in Equation (1) and (2)

is solved numerically using a method previously detailed

in Reference [19-20]. This method employs a linearized

iterative procedure that involves dividing the mesh pe-

riod into many equal intervals. In this study, a MATLAB

program is developed. The flowchart of this computa-

tional procedure used for calculating the dynamic re-

sponses of spur gears, is shown in Figure 2. The time

interval, between initial contact point and the highest

point of single contact is considered as a mesh period. In

the numerical solution, each mesh period is divided into

200 points for good accuracy. Within a small interval,

various parameters of equations of motion are taken as

constants and an analytical solution obtained. The calcu-

lated values of the relative displacement and the relative

velocity after one mesh period are compared with the

initial values xr and vr. Unless the differences between

procedure is repeated by taking the previous calculated

Figure 2. The structure of developed computer program.

F. KARPAT ET AL.

242

them are smaller than a preset tolerance, the iteration xr

and vr at the end point of single pair teeth contact as new

initial conditions. Then the dynamic loads are calculated

by using the calculated relative displacement values.

After the gear dynamic load has been calculated, the

dynamic load factor can be determined by dividing the

maximum dynamic load along contact line to the static

load. The dynamic factor indicates the instantaneous in-

crease of gear tooth load over the static load. It is one of

the most important parameters used for understanding the

dynamic responses of gear drives.

In literature, different methods and empirical equations

are used to calculate the tooth deflections of spur gears.

These methods are often based on the classical theory of

elasticity and numerical approaches. However, all of

them are derived for symmetric tooth. Therefore, in this

study, a 2-D finite element model is developed to calcu-

late the deflections of both the asymmetric and the sym-

metric gear teeth (Figure 3). A computer program, which

saves time and provides a means to carry out a paramet-

ric study with the gear parameters, was developed using

MATLAB. This program generates batch files for input

into ANSYS. When this file is executed in ANSYS, the

general procedure of FEA (i.e. 2D modeling, meshing,

loading, solution, and post processing) is automatically

performed. At the end, an output file, that contains nodal

deflection for loaded nodes, is created. This process is

Figure 3. 2-D finite element model.

α

L2

α

L1

α

L3

α

L4

F

F

F

F

F

α

L4

Figure 4. Load application.

repeated for each gear. It should be noted that in this

analysis the loads are applied at five locations on the gear

files, the approximate curves for the single tooth stiffness,

along the contact line, are obtained with respect to the

radius of the gears.

To facilitate the calculation of the Hertzian component

of the deflection at the point loading, the size of the grid

near the point of loading is chosen as recommended by

[19,21] using the following equation:

e0.2 1.2

h

c

be

for 0.9 3

e

c

(21)

where c and e are the length and width of the element,

respectively. And bh is the Hertzian contact width:

2

2.15 pdp d

h

F

bE

(22)

tooth (Figure 4). 8-noded parabolic isoparametric ele-

ments are used for meshing of the 2D model. By using

the nodal deflection values that are read from the output

where F is applied load per unit length and E is Young’s

modulus of gear material.

2.2. Virtual Tool: DYNAMIC

Physics-based modeling and simulation is important in

all engineering problems. The current mature stage of

computer software and hardware makes it possible for

complex mechanical problems, such as gear design, to be

solved numerically. In-house prepared codes to handle

individual research projects, graduate, and/or PhD stud-

ies; commercial packages for engineers in industry are

widely used to solve almost every engineering problem.

Tailored with graphical user interfaces (GUIs) and easy-

to-use design steps, anyone-even a beginner can design a

gear pair and obtain results, e.g. Dynamic Load, Trans-

mitted Torque, Static Transmission Error as a function of

time, and Static Transmission Error Harmonics etc., just

by pressing a command button. Lecturers have been in-

creasingly using these packages to increase their teaching

performance and student understanding.

Based on and triggered by these thoughts, a virtual

tool DYNAMIC is prepared that can be used for educa-

tional and research purposes. The DYNAMIC is a gen-

eral purpose tool for gear analysis (Figure 5).

There are six blocks and a figure block on the front

panel of the tool. Three blocks on the right side of the

front panel, belong to the parameters which will be de-

fined by the users. Pinion and Gear blocks are reserved

for the tooth parameters and Mechanism block is for the

parameters related to the mechanical variables. The two

blocks above the figure are Simulation and Figure Selec-

tion panels. Once the user inputs the needed parameters,

he/she clicks the CALCULATE pushbutton to obtain the

Copyright © 2012 SciRes. WJM

F. KARPAT ET AL.

Copyright © 2012 SciRes. WJM

243

3. Conclusion

solution for the specified parameters. In the Figure Se-

lection block, from the pop-up menu, user can select

which solution to be plotted: Dynamic Load, Transmitted

Torque, Static Transmission Error or Static Transmission

Error Harmonics. Then the required figure can be plotted

with the PLOT button. Once the solutions are calculated,

it is not needed to run the program again and again for

each figure option. CLEAR is to clean the figure axes

before each plot.

In this paper, a MATLAB-based virtual tool, DYNAMIC,

is introduced to analyze dynamic behavior of spur gears

with asymmetric tooth design. The DYNAMIC can be

used to compare conventional spur gears with symmetric

teeth and spur gears with asymmetric teeth. The results

for dynamic load, dynamic factor, transmitted torque,

static transmission error and static transmission error

harmonics can be obtained for various tooth parameters

to show the powerful aspects of asymmetric teeth. Influ-

ence of various parameters (e.g. the pressure angle on

drive side or coast side, addendum, and teeth number) on

the static transmission errors, the amplitudes of harmon-

ics of the static transmission errors and the dynamic

loads can be investigated. By using this program, gear

designers can design a gear pair and obtain results, e.g.

dynamic load, transmitted torque, static transmission

error, and frequency spectra of static transmission error

etc., just by pressing a command button.

The variation of dynamic load, static transmission er-

rors, static transmission error harmonics and, transmitted

torque with respect to time can be seen in Figures 6(a)-

(c). The solutions for different variables can be plotted in

one figure, for comparison. Figure 6(a) shows a sample

result for the variation of the static transmission error

during a mesh period. In Figure 6(b), the frequency

spectra of the static transmission errors obtained by using

the fast Fourier transform (FFT) are depicted. DY-

NAMIC program can also provide transmitted torque

results respect to time for sample gear pairs.

Figure 5. The front panel of the DYNAMIC tool.

F. KARPAT ET AL.

244

(a)

(b)

(c)

Figure 6. The variation of static transmission errors (a),

static transmission error harmonics (b) and, transmitted

torque (c) with respect to time.

4. Acknowledgements

The authors gratefully acknowledge the support of Ulu-

dag University under grant BAP-YDP(M) 2010/10 and

Texas Tech University under grant DE-FG36-06-GO

86092.

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