Archive > 2015 > August 2015 > Effects of Contact Ratios on Mesh Stiffness of Helical Gears for Lower Noise Design

# Effects of Contact Ratios on Mesh Stiffness of Helical Gears for Lower Noise Design

Lan Liu, Yunfei Ding, Liyan Wu and Geng Liu

In this paper, the influences of various gear parameters on the mesh stiffness are systematically investigated by using the finite element method. The comprehensive analysis shows that contact ratios are the key factors affecting the fluctuation value of mesh stiffness.

Introduction

Gear transmission has been widely used in mechanical equipment as one of the most important transmission modes. The vibration and noise of gearsets is directly related to the whole characteristics of the transmission system. In order to improve the performance and reduce the noise of gear transmission system, more attention should be paid to the gear dynamics. The fluctuation of mesh stiffness is one of the most important internal excitations of gear transmission, so it is crucial issue to find the influence factors of mesh stiffness fluctuation (Ref. 1). Many scholars have performed a lot of work to investigate the mesh stiffness for general gears using theoretical and experimental methods (Refs. 2-5). The main result told that when the contact ratio is an integer that the stiffness is approximately constant, which has a low effect on the dynamic characteristics (Ref. 4). Liu (Ref. 6) and Bu (Ref. 7) discussed the influence of design parameters such as helix angle, pressure angle, tooth face width, etc. on the mesh stiffness. Related research focused on the relationship between mesh stiffness and gears basic parameters, which were not involved how to adjust the parameters to reduce the mesh stiffness fluctuations and achieve a lower noise level.

In this paper, the mesh stiffness and its fluctuation value of helical gears with different parameters are calculated respectively by using the finite element method. The gear parameters concerned include pressure angle, helical angle, addendum coefficient and face width and etc.. Since the mesh stiffness fluctuation is closely related to the loads variation on the contact lines, the model for solving the mean length, total length and time-varying length of contact lines is also established. Then the influences of various gear parameters on the mesh stiffness are systematically investigated. The comprehensive analysis of the mesh stiffness shows that contact ratios are the key factors affecting the fluctuation value of mesh stiffness when the gear parameters are changed. By optimizing the basic parameters of helical gears, the fluctuation of the mesh stiffness of helical gears can be reduced.

Method for Calculating Mesh Stiffness

A modified method for determining the time-varying mesh
stiffness and actual load distribution based on linear programming
is used in this research referring to the literature
(Ref. 8). Using *Pro/E*. software, a 3-D model of gear with true
involutes profile was established based on the gear manufacturing
technology firstly. Then the flexibility coefficient
matrix of gear tooth surface was obtained using the substructure
method by *ANSYS* software. Finally the time-varying
meshing stiffness was solved by using linear programming
method. The 3-D geometry and finite element models are
presented in Figure 1. The advantage of this method is that
the whole process is parameterized. In this method, the load distribution along the contact lines and mesh stiffness during
the whole meshing period can be evaluated simultaneously.

Figure 2 shows the time-varying mesh stiffness in a mesh
period calculating by the finite element method mentioned
above. The *x* coordinate t means dimensionless time which is
the mesh time divided by one mesh period. The *y* coordinate
c_{γ} and *L (t)* are the time-varying mesh stiffness and the timevarying
contact-line length respectively. The meshing stiffness
decreases at the instantaneous position where the teeth
enter contact or exit contact.

In order to study the laws of mesh stiffness, its fluctuation
η_{cγ} is defined as follows:

where c_{γm} is the mean value of the time-varying mesh
stiffness in one whole mesh period. The symbol Δc_{γ} is the
difference value between the maximum value c_{γmax} and
minimum value c_{γmin} of the time-varying mesh stiffness,
i.e. — Δc_{γ} = c_{γmax} − c_{γmin}.

Fluctuation of Contact-Line Length

Figure 3 shows the action plane of a pair of helical gears and
the contact lines at different meshing times. Gear teeth begin
to meshing from A position and out of meshing at C position.
The line *AD* represents the actual action line, and line
*CD* means the tooth face width *B*, where ε_{α} and ε_{β}, being the
transverse contact ratio and overlap contact ratio, respectively,
and *p _{bt}* and

*p*being the transverse base pitch and axle base pitch, respectively. β

_{ba}_{b}is the base helix angle.

The formulas of contact-line length are derived based on
Figure 3, which include the time-varying total contact-line
length within the action plane *L(t)*, the mean value of the total
contact-line length L_{m}, the maximum value *L _{max}*, and minimum
value

*L*. Here we define

_{min}*E*and

_{α}*E>*as representing the integer part of ε

_{β}_{α}and ε

_{β}, while we define e

_{α}and e

_{β}as representing the decimal part of ε

_{α}and ε

_{β}respectively.

If e_{α} + e_{β} ≤ 1, the time-varying length *L(t)* can be expressed
as:

While, e_{α} + e_{β} > 1:

Where: *L _{1} = E_{α} E_{β} l_{1}(t) + E_{β} l_{2}(t) + E_{α} l_{3}(t), l_{1}(t) = p_{ba}/cos β_{b},
l_{2}(t) = p_{ba} e_{α}/cos/β_{b}, l_{3}(t) = p_{ba} e_{β}/cos β_{b}, e_{1} = minv(e_{α}, e_{β})* and

*e*= max

_{2}*(e*.

_{α}, e_{β})The mean value L_{m} of the contact-line length can be given
by:

According to Equations 2 and 3, the maximum value L_{max} of
the contact-line length can be derived as:

When e_{α} + e_{β} ≤ 1, the minimum value *L _{min}* of the contact-line
length can be expressed as:

while e_{α} + e_{β} > 1, the minimum value L_{min} of the contact-line
length can be expressed as:

In order to measure the fluctuation of contact-line length
during one whole meshing period, the changing ratio of the
total length of contact lines η_{L}, defined as the relative difference
between the maximum value L_{max} and the minimum
value L_{min} to the mean value L_{m} of the total length of contact
lines. The formula of ηL is expressed as:

On the basis of Equations 4–7, η_{L} can also be expressed as:

Figure 4 shows the effects of helix angle β on the time-varying
contact lines. Table 1 displays the initial parameters of the
helical gears that are discussed in Figure 5. The helix angle β
is varied from 16° – 35°. It is seen that the variation of curves
has the same trend — but with different amplitudes. Comparing
contact ratios at different helix angles, as seen in Table 2, it is found that when the overlap contact ratio of a helical gear
is close to an integer, such as when β is 20° or is 31°, the amplitude
of *L(t)* is very low, and the changing ratio of the total
length of contact lines η_{L} is approximate to zero.

In order to reveal the rules of the length of contact lines,
considering the general conditions, the surface chart about
the changing ratio of the total length of contact lines η_{L} vs.
different transverse contact ratios and overlap contact ratios
is plotted in Figure 5; this curved surface chart is obtained by
the Equation 9.

From Figure 5 the influences of contact ratios, including
transverse contact ratio, overlap contact ratio and total contact
ratio to the length of contact line are exhibited. The results
show that contact ratios are the key factor affecting the
fluctuation value of contact-line length. The fluctuation value
of η_{L} has an extreme maximum when the total contact ratio
is an integer, while it has a minimum, i.e. — zero — when the
transverse contact ratio or face contact ratio is an integer.

The Influential Factors of Mesh Stiffness

In order to discuss the influential factors of mesh stiffness
and its fluctuation, a series of mean values of the time-varying
mesh stiffness c_{γm} and their fluctuation values η_{cγ}, mean
values of contact-line length L_{m} and their changing ratios η_{L} of
helical gears with different parameters were solved, respectively,
using the method mentioned above.

* Helix angle*. Figure 6 shows the effect of helix angle β on
the mean value of mesh stiffness cγm and contact-line length
L

_{m}. The helix angle β is varied from 14° – 42°. It is seen that the mean values of mesh stiffness and lengths of contact lines decrease in the same trend while helix angle β increases.

Figure 7 shows that the changing ratio of the
total length of contact lines η_{L} and mesh stiffness
η_{cγ} change with the contact ratios when
helix angle β increases. The overlap contact ratio is varied
from 1.41 – 3.91, while the total contact ratio varying from
3.11 – 5.06 when β increased from 14° – 42°. It is seen that
mesh stiffness and lengths of contact lines have the same
trend, while helix angle β or contact ratios increase.

The graph shows that the minimum value of η_{cγ}, as well as
η_{L}, appears when the overlap contact ratio is close to an integer.
However the maximum value of η_{cγ} and η_{L} appears when
the total contact ratio is close to an integer.

* Addendum coefficient*. Figure 8 shows the effect of addendum
coefficient h

_{an}on the mean value of mesh stiffness c

_{γm}and contact-line length L

_{m}; the addendum coefficient h

_{an}is varied from 0.4 – 1.4. It is seen that the mean values of mesh stiffness and lengths of contact lines increase in the same trend when addendum coefficient han increases. The increasing values of Lm and c

_{γm}are 144 mm and 8.37 N/(μm·mm), respectively.

Figure 9 shows the changing ratio of the total length of contact
lines η_{L} and mesh stiffness η_{cγ} change with the contact
ratios when addendum coefficient h_{an} increases. The transverse
contact ratio is varied from 0.63 – 2.05, while the overlap
contact ratio remains unchanged, and the total contact ratio
varying from 3.26 – 4.68 when han increases from 0.4 – 1.4. The
graph shows that the minimum value of η_{cγ}, as well as η_{L}, appear
when the transverse contact ratio is close to integer,
which is 1 or 2 here. However, the maximum value of η_{cγ} and
η_{L} appears when the total contact ratio is close to integer 4.

* Tooth face width*. Figure 10 shows the effect of tooth face
width on the mean value of mesh stiffness c

_{γm}and contact-line length L

_{m}. The face width B is varied from 52 mm – 118 mm. It is seen that c

_{γm}and L

_{m}increase in the same trend when the face width

*B*increases. The increasing values of L

_{m}and c

_{γm}are 109 mm and 3.19 N / (μm·mm), respectively.

The results of η_{cγ} and η_{L}, by varying the tooth face width B
from 52mm to 118mm, are plotted in Figure 11, which shows
that ηL and ηcγ change with the contact ratios when face width
*B* increases. The overlap contact ratio is varied from 1.48 – 3.37
while the transverse contact ratio remains the same, and the
total contact ratio varying from 2.99 – 4.87.

The graph shows that the minimum value of η_{cγ} — as well
as η_{L} — appears when the overlap contact ratio is close to an
integer — 2 or 3 in this case. But the maximum value of η_{cγ} and
η_{L} appears when the total contact ratio is close to integer 4.

Pressure angle. Figure 12 shows the effect of gear pressure
angle α_{n} on the mean value of mesh stiffness c_{γm} and contactline
length L_{m}. The pressure angle α_{n} is varied from 16° – 26°. It
is seen that the mean values of mesh stiffness and lengths of
contact lines decrease in the same trend when pressure angle
α_{n} increases. The decreasing values of L_{m2 and cγm are 46.98mm
and 1.41N/ (μm·mm), respectively.}

The results of η_{cγ} and η_{L} by varying the pressure angle α_{n}
from16° to 26° are plotted in Figure 13, which shows that η_{L}
and η_{cγ} change with the contact ratios when pressure angle
α_{n} increases. The transverse contact ratio is varied from
1.73 – 1.29, while the overlap contact ratio is unchanged, and
the total contact ratio varying from 4.36 – 3.92.

Here the graph doesn’t show that the minimum value of
η_{cγ} or η_{L} appears when the overlap contact ratio is an integer
because of the pressure angle range. However, the maximum
value of η_{cγ} and η_{L} appears when the total contact ratio is close
to integer 4. Regardless of calculating errors, the trend of η_{cγ} is
completely the same as η_{L}.

Conclusions

In this paper the mesh stiffness and its fluctuation value of helical gears with different parameters, respectively, are calculated by using the finite element method. The influences of various gear parameters on the mesh stiffness are systematically investigated. The gear parameters concerned here include pressure angle, helical angle, addendum, co-efficient, face width, etc. The comprehensive analysis of the mesh stiffness shows that contact ratios are the key factors affecting the fluctuation value of mesh stiffness when the gear parameters are changed. The fluctuation value of mesh stiffness attains a minimum when the transverse contact ratio or overlap ratio is close to an integer, while it has an extreme maximum when the total contact ratio is approximate to an integer.

Since mesh stiffness fluctuation is closely related to the load variations on the contact lines, the model for solving the mean length, total length and time-varying length of contact lines is also established. By calculating the length of contact lines of various helical gear pairs with different basic parameters, the results show that the total length of contact lines doesn’t change when the transverse contact ratio or overlap ratio is an integer, while it fluctuates more intensively when the total contact ratio is indeed an integer.

In comparing the fluctuation amplitude of the total length of contact lines with the fluctuation amplitude of mesh stiffness, it is found that the fluctuation amplitudes of both contact lines and mesh stiffness have the same trend when gear parameters are changed. So it is proposed that the length and fluctuation value of contact line can be used to approximately measure the trend of mesh stiffness — but the values of mesh stiffness still need special calculation software.

According to the above discussion, it can be predicted that by optimizing the basic parameters of helical gears, the fluctuation of the mesh stiffness of helical gears can be reduced and the gear transmission system with appropriate contact ratios can achieve a lower vibration and noise level.

Acknowledgment. This work is supported by the 111 project (Grant No.B13044) and the Engineering Research Center of Expressway Construction & Maintenance Equipment and Technology (Chang’an University), MOE (2013G1502057).

References

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The article "Effects of Contact Ratios on Mesh Stiffness of Helical Gears for Lower Noise Design" appeared in the August 2015 issue of Power Transmission Engineering.