are presented at the end of the
paper.

INTRODUCTION

Ancient engineers successfully used Direct Gear Design. They knew the desired gear ratio and center
distance, and available power source (e. g. water current, wind, horse power). They used them to
define the gear parameters: diameters, number and shape of the teeth for each gear. Then they
manufactured gears and carved their teeth using available materials, technology, and tools.


Fig.1 Ancient Direct Gear Design

It is important to note that the gear and tooth geometry were defined (or designed) first. The
manufacturing process and tools then formed or cut this geometry in wood, stone, or metal. In other
words, gear parameters were primary and the manufacturing process and tool parameters were
secondary. This is the essence of Direct Gear Design.

Modern gear generating process was developed during the technological revolution of the 19
th
century,.
This process uses a gear rack profile as a cutting edge of the hob that is in mesh with the gear blank.



Fig.2 Gear hobbing

Gear hobbing is a reasonably accurate and highly productive manufacturing process. With some
exceptions, gears that are cut by the same tool can mesh together. Hobbing machines require
complicated and expensive tools. Common parameters of the cutting tool (generating rack), such as
the profile (pressure) angle, diametral pitch, tooth addendum and dedendum (Fig.3), were standardized
and have become the foundation for gear design. This has made gear design indirect, depending on
the pre-selected (typically standard) set of cutting tool parameters.


Fig.3 Generating rack parameters

This traditional gear design approach has its benefits including:
Interchangeability of the gears
Low tool inventory
Simple gear design process

Once the tool is chosen, the only way to affect the gear tooth profile is to change the position of the tool
relative to the gear blank. This will change the tooth thickness, root diameter, outer diameter, and
strength of the tooth. This tool positioning is called addendum modification or X-shift. It is used to
balance the gear strength and reduce sliding.

Table 1 presents a typical drawing specification, describing a traditionally designed spur gear. Most of
its parameters belong to the tool or generating process parameters. Very few parameters actually
belong to the gear.


Table 1
Tool
or
Generating
Process
Parameter
Gear
Parameter
NUMBER OF TEETH

X
STANDARD NORMAL PITCH
X

PRESSURE ANGLE
X

STANDARD PITCH
DIAMETER
X
BASE DIAMETER

X
ADDENDUM MODIFICATION
(X-SHIFT)
X
FORM DIAMETER
X

ROOT DIAMETER

X
OUTSIDE DIAMETER

X
TOOTH THICKNESS ON
STANDARD PITCH
DIAMETER
X
ADDENDUM X

WHOLE DEPTH
X



Traditional gear design based on standard tool parameters provides universality which is acceptable
for many gear applications. However, it does not provide the best possible performance for any
particular gear application because it is constrained by predefined tooling parameters.
Traditional tool based gear design is not the only available approach to designing gears. There is
another approach - Direct Gear Design.
The theoretical foundation of modern Direct Gear Design was developed by Dr. E.B. Vulgakov in
his Theory of Generalized Parameters [1]. Practical engineering implementation of this theory
was called Direct Gear Design [2]. Direct Gear Design is an application driven gear
development process with primary emphasis on performance maximization and cost
efficiency without concern for any predefined tooling parameters.
The Direct Gear Design method typically includes:
Gear Mesh Synthesis
FEA Modeling, Load Sharing, and Stress Calculation
Efficiency Maximization
Bending Stress Balance
Fillet Profile Optimization

1. GEAR MESH SYNTHESIS
1.1. Gear Tooth
Direct Gear Design defines the gear tooth without using the generating rack parameters like diametral
pitch, module, or pressure angle. The gear tooth (Fig.4) is defined by two involutes of the base circle d
b

and the circular distance (base tooth thickness) S
b
between them. The outer diameter d
a
limits tooth
height to avoid having a pointed tooth tip and provides a desired tooth tip thickness S
a
. The non-involute
portion of the tooth profile, the fillet, does not transmit torque, but is a critical element of the tooth
profile. The fillet is the area with the maximum bending stress, which limits the strength and durability of
the gear.

Fig.4 Tooth parameters

1.2. Gear Mesh
Two involute gears can mesh together (Fig.5) if they have the same base circle pitch.
Other parameters of a gear mesh are:
Center distance a
w

Operating pitch diameters d
w1
and d
w2
(diameters with pure rolling action and zero sliding)
Tooth thicknesses on the operating pitch diameters S
w1
and S
w2

Operating pressure angle w
(involute profile angle on the operating pitch diameters)
Contact ratio




Fig.5 Mesh parameters

There is a principal difference in the pressure angle definitions in traditional and Direct Gear Design. In
traditional gear design, the pressure angle is the tooling rack profile angle. In Direct Gear Design, the
pressure angle is the mesh parameter. It does not belong to one gear. If the mesh condition (the center
distance, for example) changes, the pressure angle changes as well.

2. FEA MODELING, LOAD SHARING, AND STRESS CALCULATION

The Direct Gear Design approach results in a wide variety of tooth profiles, depending on the particular
gear drive performance priorities. For this reason, the Lewis equation and experimentally defined stress
concentration factors, traditionally used for bending stress calculation of rack-generated gears, do not
provide reliable results for direct designed gears. FEA is chosen as the Direct Gear Design stress
analysis tool for bending stress and deflection calculations. For calculating contact stress and
deflection, the Hertz equation is used. The load sharing operation defines the force distribution between
the simultaneously meshed pairs of teeth, and calculates bending and contact stress in every phase of
the gear mesh. FEA and the Hertz equation are used here in combination to

define bending and
contact deflection. Fig. 6 presents the typical load sharing, bending and contact stress charts for
conventional and high contact ratio gears.


Fig. 6. Load sharing. a. conventional gears; b. high contact ratio gears, solid line load distribution along
the tooth; dash line contact stress along the tooth; dashdot line maximum bending stress at the tooth
fillet, I one pair of tooth mesh area; II two pairs of tooth mesh area; III three pairs of tooth mesh area.

Fig. 7 presents an FEA tooth model and a bending stress isograms chart.

a

b

Fig. 7. a. The FEA mesh; b. the bending stress isograms
.

3. EFFICIENCY MAXIMIZATION

In gear transmissions, almost all inefficiency or mechanical losses are transferred into heat, reducing
gear performance, reliability, and life. This is especially critical for plastic gears. Plastics do not conduct
heat as well as metal. Heat accumulates on the gear tooth surface, leading to premature failure.
The well-known [3] gear efficiency equation for spur gears is:

E
100
1
f
cos ( ) 2
H
1
( )
2
H
2
( )
2
+
H
H
+
1
2






% :=


Where:
H
1
and H
2
are the maximum specific sliding velocities of the pinion and the gear;
f is the friction coefficient;
is the operating pressure angle.

Direct Gear Design maximizes gear efficiency by equalizing the maximum specific sliding velocities for
both gears. Unlike in traditional gear design, it can be done without compromising gear strength or
stress balance.

4. BENDING STRESS BALANCE

Mating gears should be equally strong. If the initially calculated bending stresses for the pinion and the
gear are significantly different, the bending stresses should be balanced [4].




Fig.8 Balance of the maximum bending stresses

Direct Gear Design defines the optimum tooth thickness ratio S
p1
/S
p2
(Fig.8), using FEA and an iterative
method, providing a bending stress difference of less than 1%. If the gears are made out of different
materials, the bending safety factors should be balanced.


5. FILLET PROFILE OPTIMIZATION
Traditional gear design is based on predefined cutting tool parameters; the fillet is defined by the trace
of the cutting tools edge. The cutting tool typically provides a fillet profile with an increased radial
clearance in order to avoid root interference, resulting in high teeth with large radial clearance and small
fillet radii in the area of maximum bending stress.


Fig.9 Fillet profile optimization;
1. involute profiles; 2. form diameter; 3. initial fillet profile; 4. optimized fillet profile.

Direct Gear Design optimizes the fillet profile for any pair of gears in order to minimize the bending
stress concentration [4]. The initial fillet profile is a trace of the mating gear tooth tip. The optimization
process is based on FEA and a random search method (Fig.9). The Direct Gear Design software
program sets up the center of the fillet and connects it with the FEA nodes on the fillet. Then it moves
all the nodes along the beams and calculat