COMPUTATIONAL VIBRATION ANALYSIS OF ELASTIC GEARS
IN VEHICLES USING DIFFERENT REDUCTION METHODS AND
NODAL CONTACT CALCULATION
Dennis Schurr, Philip Holzwarth, Pascal Ziegler and Peter Eberhard
Institute of Engineering and Computational Mechanics, University of Stuttgart, Pfaffenwaldring 9, 70569
Stuttgart, Germany
email: dennis.schurr@itm.uni-stuttgart.de
One important source of noise in vehicles is based on the dynamic interaction of gears and the
resulting propagation of vibrations to neighboring components. In vehicles, gears can be part of
devices such as the drive train of an engine. Dynamic phenomena such as gear hammering due to
backfire or other unwanted system responses lead to high contact forces and vibrations. During
gear hammering, the tooth in contact lifts from its counterpart and bounces back and forth within
its backlash. In gear boxes, gears are connected to the housing via shafts and bearings enabling
dynamic effects such as vibrations due to elastic gear body deformations which spread throughout the system. In this work, a mathematically reduced elastic multibody model is introduced
allowing the elastic transient simulation of gear systems. Although the gears are simulated in
reduced space, the introduced model uses a nodal contact calculation allowing the detection of
contact forces with high dynamics in rolling and impact contact situations. The description of the
kinematics uses a floating frame of reference approach allowing large, nonlinearly described rigid
body rotations and small linearly described deformations, as they both naturally occur in gears.
The calculation of the deformation patterns and their associated frequencies are of special interest
in this work since model order reduction is applied. Different kinds of reduction techniques such
as the Craig-Bampton method or the Krylov subspace method are applied. Their influence on
the resulting deformation patterns and frequencies is shown. The resulting dynamics is compared
with corresponding finite element models.
1.
Introduction
Simulation of mechanical systems is an integral part in the product development process nowadays. In the automotive industry, shorter development periods and a larger number of vehicle variants
demand simulations during the design process [1], e.g. in Noise, Vibration and Harshness analysis
(NVH). Accuracy of the model and computational effort of the simulation are of special interest but
contradictory. Usually, the more complex a model, the more time for computation is needed. FiniteElement Analysis (FEA) delivers results of high quality, but computational costs are high. Model
Order Reduction (MOR) methods such as modal truncation, the Craig-Bampton method [2] or alternative methods such as Krylov subspaces [3] enable the reduction of the system size and the usage of
elastic bodies in Multibody System codes. In an Elastic Multibody System (EMBS), elasto-dynamic
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effects of a detailed model of the powertrain of a car can be taken into account using reduced finiteelement bodies and thus increase the quality of the result. Dynamic effects in the power train are
of high interest when it comes to NVH analyses. Obviously, the prompt closure of a clutch leads to
vibrations such as torsional modes in the power train and may lead to tooth hammering or other unwanted effects in the gearings. In most automatic gearings, the gear box consists of several planetary
gear systems placed in serial order. During operation, the deformations of the planetary gear lead to
noise due to staggering of the sun gear or due to a different contact order of gear teeth interacting with
each other. Thereby, the deformation of the carrier gear, which bears the planets, the deformation of
the ring gear, which is either coupled to the housing or rotating, as well as the deformation of the
shafts, which hold the carrier, sun, or ring gear, are of special interest.
In this work, an EMBS of a planetary gear system is set up using one carrier, one sun, four planets,
and one ring gear. The focus is placed on the reduction of the different parts, which is an important
but critical step in the modeling process, as well as on the calculation of tooth contact. The contact
routine has to handle dozens of possible contacts at the same time. The contact force depends not only
on the contact calculation, which is in nodal space, but also on the possible deformation pattern sets
by the used model order reduction technique. For example, in [4], it is shown that modal truncation
with a feasible number of eigenmodes does not enable the monitoring of small, local deformation
patterns on a tooth flank which is in contact. Local approaches with a much higher contact stiffness
are needed if local deformations or stresses are of interest. Thus, the selection of the number and kind
of mode shapes paired with the right contact stiffness is essential in gear contact simulations.
2.
Modelling approach for planetary gears
The kinematics of the EMBS is described by the floating frame of reference approach, which
allows large, nonlinear described rigid body motions as well as small linear described deformations.
Both naturally occur in gear systems. The equations of motion for an elastic body i read as
 i
v̇
i  i
(1)
M · ω̇ = −hie − hiω + hiP + hid ,
q̈i
where M is the mass matrix, he are the internal forces, hω are the inertia forces, hP are the external
surface loads, and hd are the external point forces. The coordinate vector contains translational and
rotational accelerations v̇ and ω̇ as well as the accelerations described by elastic deformations q̈. Gear
bodies easily can have N i > 1000000 elastic degrees of freedom. For orthogonal projections, the task
i
of model order reduction is, to find a projection matrix V ∈ RN ×n which projects q ≈ V · q̄ such
that n = dim(q̄) dim(q) = N i . This leads to the reduced equations of motion for one single
elastic body

  i
v̇
mI
mc̄˜T (q̄)
CTt · V
i
T



˜
ω̇
mc̄(q̄)
J̄(q̄)
Cr · V
(2)
·
= −h̄ie − h̄iω + h̄iP + h̄id
i
T
T
T
¨
V · Ct (q̄) V · Cr (q̄) V · Me · V
q̄
with J̄ being the inertia tensor, c̄ being the center of gravity, Me as the elastic mass matrix, and Ct /Cr
as coupling terms for translations and rotations, respectively. In this work, the treated planetary gear
system, see Fig. 1, is time integrated in reduced space.
Different reduction matrices V are calculated for the carrier gear. The carrier gear is placed on
a shaft and bears four planet gears, see Fig. 1. The four coupling positions on the carrier side are
realized using constraints, e.g. rigid body elements. The defined master nodes of the bearings are at
the center of the four bolts and are connected to the slave nodes, which are given by the bolts surface
nodes, via constraints. The connection from the carrier to each planet is realized using linear bushing
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Figure 1: Planetary gear system with four planets, ring, sun and carrier (left) and part view of planetary gear system without ring (right)
elements. In gear contact simulations, the duration of a tooth contact is about 0.1 ms. Therefore, the
frequency range of interest of the carrier is up to 10 kHz. The transfer function of the carrier’s FEmodel is shown in Fig. 2. This function is approximated using different kinds of reduction methods:
modal truncation, Rubin-McNeal reduction and the Krylov subspace method. For all three methods,
the reduction size is n = 120. This size is given by the Krylov subspaces: four input nodes at the four
carrier shafts with each having six directions and with five matching moments within the interested
frequency range result in n = 4 · 6 · 5 = 120. For the Rubin-McNeal reduction, the four input nodes
result in 24 constraint modes and 96 eigenmodes. The frequency response function of all three
methods is calculated and compared against the original transfer function of the full FE-model. The
relative error of each method is shown in Fig. 2. This error measure is calculated using the Frobenius
norm kHkF and is given as
(3)
rel
F (f ) =
kH − H̄kF
kHE kF
=
, f ∈ [fmin , fmax ].
kHkF
kHkF
For modal truncation, the conformity to the original transfer function, especially for low frequencies,
is poor. Using the Rubin-McNeal method, the relative error becomes much better for the complete
frequency range, especially for low frequencies. The difference between Rubin-McNeal and a classical Craig-Bampton technique is given in the calculation of the eigenmodes. Using the Rubin-McNeal
reduction technique, the input nodes for the calculation of eigenmodes are unconstrained. For the
alternative reduction methods based on Krylov subspaces, the relative error is the best in the complete
considered frequency range.
For the verification of the reduced models in time domain, a simple intermediate model with
carrier and one planet is set up using the same initial and boundary conditions for both the full FEmodel and the reduced Elastic Multibody model. For the FE reference calculation, the commercial
code Abaqus [5] is used. The result of the kinematics is shown in Fig. 3. Here, the Krylov subspace
method matches the FE reference result very well. Furthermore, the result obtained with the RubinMcNeal method delivers about the same quality as the Krylov subspace based result. Also, the result
obtained using modal truncation, which does not use constraint modes, does not match the FE result as
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Figure 2: Frequency response of carrier and error of different reduced carriers
good as the three aforementioned, especially for larger times. Although the Krylov subspace method
and Rubin-McNeals method deliver about the same quality in time domain, the carrier reduced with
Krylov subspaces is further used because of its better approximation in frequency domain, see Fig 2.
For the reduction of the planets and of the sun gear, the reduced model must be able to handle multiple contact forces at the same time. For example, the planets are in contact with the ring and the sun
gear for most of the time simultaneously. In [6], it was shown that modal truncation combined with
a penalty-contact delivers good results, even for simultaneous multiple teeth contact. Furthermore, it
was shown that the use of additional static modes can reduce the total number of modes used when
only one gear flank is in contact, but the sum of static modes of all flanks together with eigenmodes
does not further reduce system size, [11].
A strategy to circumvent this issue is given by Static-Modes-Switching (SMS), see [7], but SMS
can lead to numerical instabilities due to the switchung of modes.
Another strategy is provided by parametrized model order reduction (PMOR) [8, 9]. This technique interpolates the system matrices between system inputs, e.g. nodes, but some additional preprocessing steps are mandatory. A third technique uses snapshots of equivalent FE simulation results
as ansatz functions for the gears [10]. Those ansatz functions paired with eigenmodes allow small reduction sizes of gears, but slight changes to the design, for example changing the mounting position,
require a reevaluation of all preprocessing steps. In this work, solely eigenmodes are used for the
reduction of the planets and of the sun due to very good results obtained in [6]. For the planets and
for the sun gear, the first 400 eigenmodes, which cover the frequency range up to 180 kHz, were used.
The reduction size of nM od = 400 was chosen in accordance with the contact calculation between
planets and ring gear, which will be discussed later. A further intermediate model is set up for the
verification of the reduced gears and of the contact calculation. Therefore, the first model containing
carrier and one planet is extended with the sun gear and contact is defined between sun and planet.
The equivalent FE model has the same initial and boundary conditions. The comparison with the FE
reference result is shown in Fig. 4. The results are in good accordance with the FE results.
The ring gear is an internal geared wheel, which is in contact with all four planets. Due to its
geometry, the ring gear is, together with the carrier, the most flexible part. In order to account for
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Figure 3: Comparison of different reduced carriers with Abaqus reference result. Reduction size of
each carrier is n=120
both ring body deformations as well as local tooth bendings, global approaches such as eigenmodes
and local approaches such as constraint modes are used. There are 54 input nodes, which are located
in the center of each of the 54 teeth and are used for the constraint mode calculation. The 162
constraint modes are augmented with 238 eigenmodes using the Rubin-McNeal method. This set of
ansatz functions was found to deliver good results for the kinematics and for the contact force when
the ring is in contact with a planet. Furthermore, the frequency response error of the Rubin-McNeal
reduced ring gear is the best compared to a Craig-Bampton or modally reduced ring gear, each of size
400. The frequency response error of all 3 methods is shown in Fig. 5. Reducing the ring gear with
Krylov subspaces results in nKry = 11 · 54 · 3 = 1782, because 11 moments of 54 input nodes for all
3 directions need to be matched within the interested frequency range. Therefore, the much smaller
Rubin-McNeal reduced ring gear of size nR−M cN = 400 is used for the comparison.
3.
Simulation of a planetary gear system
Using the reduced elastic parts, the planetary gear system shown in Fig. 1 is assembled and integrated. In planetary gear systems, different kinds of operational modes exist depending on specific
characteristics such as the transmission ratio, purpose of application or design parameters. If the ring
gear is stagnant, sun and carrier rotate. For a stagnant sun or carrier, the other two parts rotate. The
movement of the planets is inherent. Here, the ring gear stands still. Its bore diameter is larger compared to the bore diameter of the sun. This allows the sun to be fixed on an axle. Here, the input torque
of Tin = 600 Nm is directly applied to the bore of the carrier gear. The output torque is directly applied to the bore of the sun and is velocity proportional in order to get a constant velocity after a short
bedding-in process. The initial rotational velocity of the carrier is ωcarrier = 85 rad/sec. Time integration is performed using the in-house matlab-based code GTM [13], which uses the explicit one-step
Runge-Kutta integrator ode45. In Fig. 6, results showing the rotational velocities of planet one, ring
and sun gear are shown. After about 1 ms for the bedding-in process, the rotational velocities do not
change anymore. Furthermore, there is an overlapping oscillation in the rotational velocities. This is
the result of the planets interacting with the carrier, sun and ring gear. Multiple tooth contacts exist
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Figure 4: Comparison between FE reference results obtained with Abaqus and the reduced elastic
model using 400 eigenmodes
at all times accelerating and decelerating the planets and the sun. Also, the resulting deformations
play an important role in the resulting gear dynamics. In Fig. 7, the deformation of the ring gear is
shown for three time steps. The deformation is the result of the superposition of all used ring modes
multiplied with the elastic amplitudes at selected time steps. The magnitude of the deformation is
displayed qualitatively. It is notable at t = 0.033 sec, the interaction with the four planets leads to
four buckling regions in the ring with one region being stronger deformed. The same is noticeable for
the snapshot at t = 0.066 sec. At t = 0.1 sec, this distinct pattern is still observable, although some
different modes are overlapping. The buckling of the four regions can be traced back to the first two
modes. Both are excited the strongest during simulation.
4.
Conclusion
A reduced EMBS of a planetary gear system was set up and simulated. Each gear was reduced and
assessed according to the frequency response error, actual behavior in time domain using intermediate
models as well as the reduction size n. For the carrier, the Krylov subspace method and the RubinMcNeal method showed the best behavior in time domain. But as the frequency domain behavior was
the best for the Krylov subspace reduced carrier, this reduced carrier was further used for simulation
of the planetary gear system. For the ring gear, three different reduction methods were compared.
Similar to the carrier gear, the Rubin-McNeal method delivers a much better behavior in frequency
domain compared to modal truncation. Also, the Craig-Bampton reduced ring gear shows a better
behavior in frequency domain as modal truncation, but a worse behavior as Rubin-McNeal reduced
ring. Therefore, the Rubin-McNeal method was used for the reduction of the ring gear. For the
planets and the sun, it was argued that normal modes resulting from an eigenanalysis already deliver
good results for external geared wheels. Here, advanced techniques such as SMS or PMOR are
not neccessary. Finally, the planetary gear system was time integrated. The EMBS shows a nonstationary behavior at all times. This is due to the continuous excitation of all elastic bodies, resulting
in a nonlinearly described rigid body motion with an overlapping linearly described deformation. The
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Figure 5: Frequency response error of reduced ring gears with sizes n = 400, and mode shapes from
Rubin-McNeal reduction: constraint mode 1 (top), eigenmode 4 (middle) and eigenmode 20 (bottom)
Figure 6: Rotational velocities of planet one, carrier and sun
Figure 7: Snapshots of ring gear at times t =0.033 s (left), t =0.066 s (middle) and t =0.1 s (right)
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three snapshots of the deformation of the ring gear show a buckling of the four regions interacting
with the four planets. This is an expected behavior and illustrates the capability of the used method
for vibration analysis in planetary gear systems.
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