The Influence of Design Parameters on Straight Bevel Gear Dynamics

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Mechanical Engineering Research; Vol. 5, No. 2; 2015

ISSN 1927-0607 E-ISSN 1927-0615
Published by Canadian Center of Science and Education

## The Influence of Design Parameters on Straight Bevel Gear Dynamics

Chiu-Fan Hsieh 1 & You-Qing Zhu 1

1 Department of Mechanical and Computer-Aided Engineering, National Formosa University, Taiwan

Correspondence: Chiu-Fan Hsieh, Department of Mechanical and Computer-Aided Engineering, National
Formosa University, 64 Wunhua Road, Huwei, Yunlin, Taiwan. E-mail: cfhsieh@nfu.edu.tw

Received: June 8, 2015  Accepted: June 27, 2015  Online Published: November 13, 2015

doi:10.5539/mer.v5n2p114     URL: http://dx.doi.org/10.5539/mer.v5n2p114

Abstract

This study analyzes the influence of design parameters on the dynamics of straight bevel gears by constructing a
model that allows variation in the shaft angle, pressure angle, and backlash. According to the statistical analysis,
the order of influence of these parameters on weight is shaft angle > pressure angle > backlash. When the shaft
angle is 90°, the statistical results show the drive is stable and the stress fluctuation level is low. The pressure
angle, on the other hand, can affect the gear’s dynamic property by influencing the driving component force on
the gear and the component force on the shaft. The results for the shaft and pressure angles are used to determine
the appropriate backlash. Overall, the analysis not only provides designers with an important reference but
explains the dominance in the market of gear designs with a 90° shaft angle and a 20° pressure angle.

Keywords: bevel gears, shaft angle, pressure angle, backlash, dynamics, stress

1. Introduction

Bevel gears, primarily of spiral, hypoid, and straight types, are widely used on the power transmission of
concurrent axes and alternating axes. In general, the shaft angle of a bevel gear is designed to be 90° with a
maximum reduction ratio of 10, leading to high efficiency and high precision. In particular, the design must meet
the spatial surface meshing condition, and the gear blank and tooth form designs must be able to vary with
different cutting methods (Litvin et al., 1987). The calculation of bevel gears is thus more complicated than for
other gear types (Fong & Tsay, 1991, 1992) and requires the development of special design software. The most
common type of bevel gear is the straight bevel gear, which is applied to power transmissions between
concurrent axes. This gear is characterized by a straight tooth whose depth decreases gradually from large to tip
end. The theoretical tooth form is involute (Al-Daccak et al., 1994), but the tooth form designs vary with
different manufacturing methods. Compared to other bevel gears, however (e.g. the spiral bevel gear), the
configuration is simple and the gear easily made, forged, and shaped. As a result, the bevel differential
mechanism is used extensively in the market.

As regards research on the straight bevel gear, most studies have focused on the theoretical mathematical model
of gear (Tsai & Chin, 1987; Chang & Tsay, 2000; Figliolini & Angeles, 2004), teeth contact analysis (Jiao &
Cao, 2011; Kolivand et al., 2015), finite element modeling and analysis (Nalluveettil & Muthuveerappan, 1993),
teeth deflection analysis (Gagnon et al., 1997), manufacturing techniques and cutting methods (Özel et al., 2005;
Shih & Fong, 2008; Shih, 2012; Fuentes et al., 2011; Tsuji et al., 2013; Kawasaki et al., 2015), measurement
technique (Kawasaki & Shinma, 2009), lubrication regime (Ghahnavieh et al., 2014) and dynamic behavior of
two-stage straight bevel gear (Yassine et al., 2014). To the best of our knowledge, however, no work exists on
the influence of dynamic contact stress and gear design parameters on straight bevel gear dynamics. Hence, this
study assesses the influence of a straight bevel gear’s main design parameters (shaft angle, pressure angle, and
backlash) on its dynamics by constructing a dynamics analysis model that weights the influence of the various
parameters. The results provide a useful reference for straight bevel gear design.

2. Dynamics Analysis Model for a Straight Bevel Gear

_2.1 Straight Bevel Gear Modeling_

First, using Solidworks, a straight bevel gear CAD model is built in which the tooth form is created based on a
mathematical model. Here, the modulus is 2, the number of teeth is 15, the pressure angle is 20°, the shaft angle
is 90°, and the backlash is 0.2. The building of this model is described below and linked to the illustrations in
Figure 1:

www.ccsenet.org/mer Mechanical Engineering Research Vol. 5, No. 2; 2015

Step1: A blank is created with a shaft angle of 90° (a);

Step2: The involute profile is drawn on the gear plane (b);

Step3: A straight bevel gear tooth form is cut along the conical point of involute profile (c);

Step4: The required teeth are cut in a ring array to complete a straight bevel gear (d); and then

Step5: Another straight bevel gear (45 teeth) is built following the above steps, and the two straight bevel gears
are combined (e).

(a) Blank is created (b) Involute profile is drawn

(e) The straight bevel gear is assembled

Figure 1. Construction of the straight bevel gear

_2.2 Dynamics Analysis Model_

To construct the dynamic analysis model, it employs a systematic dynamics calculation model for an
experimentally validated cycloidal speed reducer (Hsieh, 2014), whose applications include testing traditional,
improved, and novel designs and identifying their shortcomings and advantages (Hsieh, 2014). These modeling
and calculation techniques (Hsieh, 2014) allow constructing a dynamic analysis model of the straight bevel gear.
In this paper, the gear modulus is fixed at 2, the driving gear is set at 15 teeth, and the driven gear is at 45 teeth.
The simulated conditions include total run duration of 1.5 sec and the driving gear rotation speed set at a
constant 120 rpm (720 deg/s). The contact pattern is a 3D collision mode, and gravity acceleration is 9.81
m/sec^2. The material is alloy steel whose mechanical properties are given in Table 1.

In the stress analysis, stress can be calculated by the finite element method using SolidWorks (Hsieh, 2014).
During bevel gear motion, the variations in gear contact stress can be measured over time. First, the triangular
mesh is calculated. If the values obtained when calculating gear engagement are to be accurate, the grid density

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in the place without gear contact can be low, but the grid density in the place with gear contact must be high.
Grid computation is shown in Figure 2(a) and stress calculation in Figure 2(b).

The material used in all models and the friction factor are the same. Assuming the loading at the driven gear is
fixed as 5 N-m, the effect on dynamics was carried out by giving different driving torque, which includes 15
N-m, 30 N-m, 45 N-m, 60 N-m, 75 N-m and 90 N-m, respectively. Thus, different parameter design values are
assigned to the straight bevel gear model that result in different motion and stress variations. The tooth profile
design parameters are pressure angle, shaft angle, and backlash, whose parameter value settings are listed in
Table 2. To assess the influence of parameter design on the straight bevel gear’s dynamics, these parameter
values are cross related, producing 27 groups as simulation subjects in each driving torque case.

Table 1. Mechanical properties of the material

Material

Mechanical properties N/mm^2

Elastic modulus Shear modulus Tensile strength Yield strength

Alloy steel 210,000 79,000 723.1025 620.422

Table 2. Gear design parameter values in each driving torque case

Numerical

Parameter

Pressure angle Backlash Shaft angle

1 14.5° 0.1 45°

2 20° 0.3 60°

3 25° 0.5 90°

(a) Grid chart (b) Schematic diagram of stress

Figure 2. Stress analysis model

_2.3 Dynamics Analysis_

In the dynamics analysis, for example, the driving torque is 15 N-m, the fixed pressure angle is 20°, the backlash
is 0.1 mm, and the variable shaft angle is 45°, 60°, and 90° respectively. The following results are obtained based
on the data from one revolution of the driving gear. The time interval of calculation is 0.0013889 s. Figures 3
and 4 show the analytic results of the angular velocity and angular acceleration of the driven gear. The velocity
fluctuation is regular in all three shaft angle designs, but the 60° shaft angle has the maximum fluctuation. The
angular acceleration analysis further reveals that the 60° shaft angle has the maximum peak values. Although
one explanation could be larger vibrations, when the shaft angle is 90°, the minimum angular velocity fluctuation
and acceleration reach peak values, so the motion is relatively stable and the vibration slight.

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Angular velocity of the driven gear

242

241.5

241

240.5

240

239.5

239

2000

1600

1200

800

400

0 60 120 180 240 300 360
Degree

Figure 3. Angular velocity of the driven gear

Angular acceleration of the driven gear

0 60 120 180 240 300 360

Degree

Figure 4. Angular acceleration of the driven gear

0.1-45

0.1-60

0.1-90

0.1-45

0.1-60

0.1-90

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Stress of the driving gear

0.1-45 0.1-60 0.1-90

76000

72000

68000

64000

60000

56000

52000

0 60 120 180 240 300 360

Degree

(a)

Stress of the driving gear

0.1-45 Stress of the driving gear

63990

63985

63980

63975

63970

63965

63960

63955

63950

63945

63940

63935

0.1-60

57320

57300

57280

57260

57240

57220

57200

0 60 120 180 240 300 360

Degree

0 60 120 180 240 300 360

Degree

Stress of the driving gear

0.1-90

72070

72060

72050

72040

72030

72020

72010

72000

0 60 120 180 240 300 360

Degree

(b)

Figure 5. Stress variation of driving gear

According to the stress analysis, the axial component force increases with the shaft angle, while the stress of the
driving gear increases gradually (Figure 5(a)). As Figure 5(b) shows, the three stress curves of the driving gear
fluctuate regularly but slightly. When the shaft angle is 90°, the average stress is maximum about 72000 N/m^2;
when the shaft angle is 60°, the average stress is about 64000 N/m^2; and when the shaft angle is 45°, the
average stress is minimum about 57000 N/m^2.

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0.1-45 0.1-60 0.1-90

45000

40000

35000

30000

25000

20000

15000

10000

5000

Stress of the driven gear

0 60 120 180 240 300 360

Degree

(a)

Stress of the driven gear

0.1-45 Stress of the driven gear

39200

39000

38800

38600

38400

38200

38000

37800

37600

0.1-60

5400

5200

5000

4800

4600

4400

4200

4000

0 60 120 180 240 300 360

Degree

0 60 120 180 240 300 360

Degree

Stress of the driven gear

0.1-90

28800

28700

28600

28500

28400

28300

28200

28100

28000

27900

27800

0 60 120 180 240 300 360

Degree

(b)

Figure 6. Stress variation of driven gear

The stress of the driven gear is as shown in Figure 6. The stress peaks are all regular (Figure 6(b)). In this case,
the maximum average stress is the shaft angle 60° after that is the shaft angle 90°, and the minimum average
stress is the shaft angle 45°. Since the stress ripple coefficient (Hsieh, 2014) can indicate the stress fluctuation
level, the calculation result shows the hierarchy of the stress ripple level is as shaft angle 45°>shaft angle 60°
>shaft angle 90°. This analysis thus suggests a shaft angle of 90° as an ideal design.

The stress analysis can indicate the contact stress condition, thus would affect the life of the components. For
comparing, each driving torque case includes 27 sets of stress data.

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Since the rotation speed setting is on the driving gear, only the stress of the driven gear is analyzed and discussed.
The equation of standard deviation can be expressed as

_Sd_ ( _i_ ) = _N_ 1 _N_ Σσ2 −Σ( σ)2, _i_ = 1~ 27 (1)

where _N_ is the number of stress data for one revolution of the driving gear. σ is the stress data of the driven
gear.

In order to compare the stress variation level, the coefficient of variation can be defined as

_Cv_ ( _i_ ) = ~~σ~~ _Sd_ ×100, _i_ = 1~ 27 (2)

In which, ~~σ~~ means the average stress of the driven gear.

Here, the influences of the backlash, pressure angle and the shaft angle parameters, the equations are defined
respectively as

_C_ ( _j_ )

9
# =  Cv (, jm ), m =0.1, 0.3, 0.5 (3)

_j_ =1

# Bm =  Cv (, jm )

_j_ =1

_v_,

_C_ ( _j_ )

9
# =  Cv (, jn ), n =14.5, 20, 25  (4)

_j_ =1

# Pn =  Cv (, jn )

_j_ =1

_v_,

_C_ ( _j_ )

9
# =  Cv (, jo ), o =45, 60, 90  (5)

_j_ =1

# So =  Cv (, jo )

_j_ =1

_v_,

In this paper, the tendency of a whole influence is calculated by sum of variation coefficient of each driving
torque case as follows:

_SBm_ = _Bm_ (15) + _Bm_ (30) + _Bm_ (45) + _Bm_ (60) + _Bm_ (75) + _Bm_ (90), _m_ =0.1, 0.3, 0.5 (6)

_SPn_ = _Pn_ (15) + _Pn_ (30) + _Pn_ (45) + _Pn_ (60) + _Pn_ (75) + _Pn_ (90), _n_ =14.5, 20, 25 (7)

_SSo_ = _So_ (15) + _So_ (30) + _So_ (45) + _So_ (60) + _So_ (75) + _So_ (90), _o_ =45, 60, 90 (8)

3. Statistical Results and Discussion

_3.1 Influence on Dynamics of the Backlash Parameter_

As shown in Figure 7 (a), when the backlash is small (0.1 mm), the gear collision frequency and gear vibration
will increase and the gear contact travel will extend to increase the stress. Thus, the variation coefficients in most
driving torque cases are the largest. The whole tendency shows that the stress ripple level would relieve with the
increasing of backlash as shown in Figure 8(a). When the backlash is large, in contrast, the collision between
gears can be reduced, although there will be energy loss in the power transmission, reducing transmission
efficiency. The maximum variation coefficient in backlash parameter is the smallest (see Figure 8) among the
three parameters, thus it has the minimum influence comparing to the other parameters. Accordingly, the main
influence on stress and fluctuation occurs at the shaft angle and pressure angle, an appropriate backlash value
should be determined based on their design and working condition.

_3.2 Influence on Dynamics of the Pressure Angle Parameter_

The variation coefficient results show the influence degree of the pressure angle parameter is between the
backlash and shaft angle parameters in Figures 7 and 8. In the driven gear, the stress ripple level would be
increased with the decreasing of the pressure angle especially at 14.5°, which load and fluctuation are the largest.
Apparently, the gear driving component force and stress fluctuation increase as the pressure angle decreases. On
the other hand, the larger the pressure angle (25°), the lower the gear driving component force, and the stress
fluctuation decreases while the load on the shaft increases. Hence, according to the stress statistical result, a
pressure angle of 20° is a better choice.

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Backlash analysis

120

100

120

100

15N m 30N m 45N m 60N m 75N m 90N m

(a) Backlash

Pressure angle analysis

15N m 30N m 45N m 60N m 75N m 90N m

(b) Pressure angle

Shaft angle analysis

15N m 30N m 45N m 60N m 75N m 90N m

Figure 7. Statistical results in each driving torque case

0.1

0.3

0.5

14.5

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Backlash analysis (Sum of 15Nm~90Nm)

240

230

220

210

200

190

180

170

500

450

400

350

300

250

200

150

100

600
550
500
450
400
350
300
250
200
150
100
50
0

0.1 0.3 0.5

(a) Backlash

Pressure angle analysis (Sum of 15Nm~90Nm)

25 20 14.5

(b) Pressure angle

Shaft angle analysis (Sum of 15Nm~90Nm)

45 60 90

Figure 8. Statistical results by sum of variation coefficient of each driving torque case

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_3.3 Influence on Dynamics of the Shaft Angle Parameter_

Since the maximum variation coefficient in shaft angle parameter is the largest as shown in Figures 7 and 8, it
places the maximum influence on the shaft angle design. The variation coefficient of shaft angle at 45° is the
largest in each driving torque case (see Figure 7(c)). When the shaft angle increases to 60°, the stress fluctuation
frequency would obviously reduce. The shaft angle at 90° is the most stable one and thus has the least stress
variation coefficient in most driving torque cases. Furthermore, the shaft angle at 90° can obtain more stable
operation than the case at 60° under a larger driving torque. Besides, Figure 8 (c) represents the whole tendency
in the smaller shaft angle (like shaft angle at 45°), however, its combination with other two parameters may not
be a better condition. The shaft angle at 90° can get the lowest variation coefficient. Hence, unless there are
special power requirements, the shaft angle at 90° is recommended as the design basis.

4. Conclusions

By constructing a dynamics analysis model for straight bevel gears, this analysis generates accurate shaft angle,
pressure angle, and backlash design parameters and then measures the influence of each parameter on the gear’s
dynamics. From the statistical analysis, the influence of the backlash is not obvious because the variation
coefficient is quite similar in different backlash size. However, the size of the backlash still can influence the
gear’s dynamic properties. The analysis also reveals that the pressure angle can be designed according to load.
That is, when the pressure angle is large, the stress and stress fluctuation on the gear can be reduced, but the
axial component force is increased, placing a burden on the bearing. The component having the maximum
influence on dynamics is the shaft angle, and the shaft angle at 90° with other two design parameters would be
recommended as a better design.

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