Bevel gears are often used to transmit power between non-parallel
shafts. They comprise several geometries, the most significant of
which are the spiral bevel gears (SBG). The vibration affects the
stress distribution, the contact pressure, and the fatigue life of a
gearset. Nonlinearity is affected by a number of factors, including
backlash and time-varying mesh stiffness. It is to note that gear
mesh transmission error is one of the most significant sources of
driveline vibration and noise. A variety of approaches is used to
study gear pairs. [1-5].
The total displacement of gears during meshing is given by the
sum of different deformations, such as deflection due to bending,
shear, and surface indentation. Sánchez et al [6] studied the
mesh stiffness of pairs of spur gears taking into account the
Hertzian effect and presented a formula for approximate mesh
stiffness. Sánchez considered bending, shear, compression and
contact deflection to obtain the mesh stiffness. For spur gears,
the equation proposed in Ref.[6] calculated the load sharing ratio
and the load at any point along the contact path. Cheng et al [7],
used the finite element method to calculate time-varying mesh
stiffness considering the influence of Hertzian deflection and
various loads.
Recently, the research on SBGs has focused on tooth contact
analysis to obtain static transmission error (STE), which is
regarded as the primary cause of vibration. Kickbush et al [8]
proposed two finite element models (two-dimensional and threedimensional)
to approximate MS. Tang et al [9] investigated how
two distinct STEs affect an SBG’s dynamic response. The two
STEs evaluated were a pre-designed parabolic function and a sine
function. Other research has investigated the influence of faults on
the dynamic behavior of gear pairs [10]. Peng et al [11] proposed
a new method for estimating load transmission error (LTE) which
considers the effect of bearing supports. Wang et al. [12] used a
finite element analysis technique to investigate the time-varying
mesh stiffness of a gear pair with cracked teeth.
The main objective of the present study is to calculate the mesh
stiffness using LTCA. For this purpose, two powerful software,
based on FEM, are used to calculate the mesh stiffness with the
highest possible accuracy. After obtaining the mesh stiffness, we
are able to analyze the behavior of spiral gear pairs in different
situations, e.g. in the presence of misalignments.
Loaded tooth contact analysis
Mesh stiffness and dynamic investigation are two of the most
prominent and interesting topics that attract great attention
from researchers. Investigation of mesh stiffness is significant
to understand the behavior of the gear system. Obtaining mesh
stiffness is mandatory to simulate the dynamic model of the
geartrain.
To calculate the mesh stiffness of a gear pair, a loaded tooth
contact analysis (LTCA) must be conducted [11, 13- 15]. Finite
element analysis, experimental testing, and simplified analytical
models are all suitable methods for performing LTCA. Comparing
these methods, one could find that experimental methods [5]
require expensive test rigs; and that while the analytical solutions
developed for spur/helical gear pairs, they are not well suited to
gear pairs with complicated geometries, such as spiral gears. In
this context, FEM has been widely used in recent years owing to
its reduced costs and its ability to model gear pairs with different
geometries [1, 16, 17].
Several software have been developed to perform LTCA, two of the
most powerful of which are MASTA and Calyx. This study focuses
primarily on obtaining the mesh stiffness as accurately as possible.
Both Calyx and MASTA are FEM-based, so comparing their results
would be likely to ensure their accuracy. With respect to accuracy,
the most sensitive output of a LTCA is the contact pressure. Fig.
1 shows the pressure distribution on the tooth surface along the
pinion profile. The pressure distribution is parabolic, as predicted
by the theory.
The next step, which is undertaken to check the accuracy of the
results, is to compare them with the FEM data present in literature,
see Bibel et al [18];it should be pointed out that they validated
their results with experimental data.
Fig. 2 shows the maximum principal stress on the pinion root at
the middle surface. The solid line shows the simulation results,
and the solid red circles are the results of Bibel it should be: [18];
we observe a very good match.
As mentioned, earlier, two reliable software with powerful solvers
were used in this study. The static transmission error is expanded
through the Fourier series and the coefficients are shown in Fig.3.
The black bars represent the Fourier coefficient from the Calyx
simulation results, and the red bars show the MASTA simulation
results.
Dynamic investigation
In this section, a model is provided to study the dynamics of
an SBG gear pair. Consider the bevel gear pair in Fig. 4: the
translational degrees of freedom for both the driver and driven
gears are constrained in all directions; the gears can rotate about
their axes, but no further rotation is allowed. In addition, the
following assumptions are made: a pure involute profile with dry
contact and no friction; and thermal effects are ignored.
The dynamic equations of motion of this system (Fig. 4) are given
by [16, 19-23]:
Due to assembly and manufacturing errors, or changes in tooth
profile, the backlash between mating teeth varies; this is called
geometric transmission error, or e(t). The linear dynamic
transmission error (DTE) along the line of action is defined as
λ=rb1θ1 - rb2θ2. The two equations (1) are merged, and the following
equation is obtained
Equation (2) presents the equivalent displacement of the gear
mesh.
f(λ-e) is the backlash function of the displacement (Equation
(3)). The backlash function multiplied by the stiffness returns
the restoring force [2]. Whenever λ-e is between -b and +b, the
contact loss occurs [24]. For λ-e>b, the contact occurs in forward
flank, while if λ-e<-b, unwanted contact occurs at the back. In
addition, the torsional mesh stiffness of the gear set is a timevarying
function, which is periodic with the fundamental mesh
frequency ωm=2π/60 N1γs.
The equivalent mesh stiffness can be expanded by using the
The equivalent mesh stiffness can be expanded by using the
Where k0, aj, and bj are the coefficients of the Fourier series. Fig.
5 shows the rotational deflection of the Gear.
In order to normalize the governing equation, new parameters are
introduced as follows:
In order to normalize the governing equation, new parameters are
introduced as follows:
Consequently, Equations (2), (3), and (4) can be rewritten as
follows:
Equation (6) is a second-order nonlinear differential equation with
time-varying parameters.
This governing equation is solved numerically via the “RADAU
scheme”, which is an implicit scheme of the Runge–Kutta
algorithm with step control, capable of handling problems that
require a high numerical stability; see, e.g. [1, 2, 16, 17, 23]*.
Numerical results
The design data of the SBG pair considered is listed in Table 1.
Note that the gear pair considered was previously used by Bibel
et al. [18]. The bifurcation diagrams and root mean square (RMS)
are obtained by varying the excitation frequency, i.e., the rotational
speed of the pinion.
Table 1. Geometric parameters of the SBG pair
Simulations are performed in the range of ω⁄ω_n ϵ (0.1,2.5),
forward and backward.
For each frequency, the final condition of previous set is considered
as the initial condition for the new step. For the backward
simulation, the vibration is periodic (see Fig. 6), except in the
interval ω⁄ωn ϵ (2,1.3) when subharmonic bifurcation occurs.
Fig. 7 shows the amplitude–frequency diagram. The primary
resonance as well as, super-harmonic resonance, and a parametric
resonance (ω⁄ωn =0.5 and 2, respectively) are identified. These
results show that the system experiences steady behavior until it
approaches super-harmonic, primary and parametric resonances.
Conclusion
This study was conducted to calculate mesh stiffness and
transmission error. The results were compared using two
important FEM software programs (MASTA and Calyx), and finally
the accuracy of the results was evaluated considering the data
from a recent study by Bibel et al. As the results showed, the
accuracy of the output results was acceptable.
Furthermore, the dynamic response of the system was also
evaluated in this study. Subharmonic and super harmonic
phenomena are illustrated for some special harmonics. Backside
contact and contact loss was observed during the dynamic
investigation of the SBG.
The dynamic equation presented is derived after some
simplifications, namely: one degree of freedom and neglecting the
effect of friction between two coupled teeth. It would be insightful
for future studies to increase the number of DOFs or to consider
the effect of friction on the dynamic response in order to analyze
the dynamic behavior of the system in greater detail.
The output of this study may be a significant step towards further
investigation of the dynamic behavior of an SBG system.
In other words, the comparison made in this study allows the
effect of different parameters (for instance, misalignment, tip/
root modification, andcrowning) on mesh stiffness, as well as
the dynamic behavior of SBG sets a result of these effects to be
investigated.
Funding: The authors thank the Emilia Romagna region of Italy for
supporting this research through the project “DiaPro4.0 – Sistema
‘cost effective’ multisensore di Diagnostica- Prognostica integrato
in azionamenti meccanici dell’Industria 4.0” – (PG/2018/632156).
Acknowledgments: The authors would like to thank EnginSoft’s
team for their support in this research.
Fig. 1 - Pressure distribution on the tooth surface
Fig. 2 - Maximum principal stress on the root of the tooth relative to the pinion mid plate
(- FEM simulation, • verified FEM results [18]).
Fig. 3 - Fourier coefficients for rotational deflection (• MASTA, • Calyx)
Fig. 4 - Dynamic model of a gear system with rotational degrees of freedom
Fig. 5. Rotational deflection of the Gear
Fig. 6 - Bifurcation diagram based on backward simulation
Fig. 7 – RMS diagram of forward and backward simulations
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