The 14th IFToMM World Congress, Taipei, Taiwan, October 25-30,2015 DOI Number: 10.6567/IFToMM.14TH.WC.OS6.029

Simulation of Gear Systems with Dynamic Analysis

Y. Temis1 E. Kozharinov

2 D. Kalinin

3

CIAM, BMSTU CIAM CIAM

Moscow, Russia Moscow, Russia Moscow, Russia

Abstract: In the present paper, two different

methodologies for simulate the dynamic behavior of gears

are described and compared. The nonlinear torsion

vibrations of geared systems are studied at resonance

frequency with the contact loss of the meshing teeth. In the

proposed model a time-varying mesh stiffness of tooth

pairs and a viscous damping proportional to the meshing

stiffness are considered. Gear errors of each meshing tooth

pair are also included. Dynamic motion of the spur gear

pair over wide range of excitation frequency using the

finite element method was simulated. Results of bending

stress distribution during overall operation speeds were

obtained by FEA. It may serve as a tool for aiding the gear

fault diagnosis.

Keywords: gear mesh, dynamic model, FEM, friction damper

I. Introduction.

Dynamic loads on the teeth, resulting from the operation of

gears are one of the important factors determining the

reliability and durability of the transmission. Therefore, the

analysis of dynamic loads in gear mesh and the level of

vibration in aircraft transmissions and drives is relevant. A

qualitative assessment of the dynamic state of heavily gear

necessary to develop mathematical models of meshing

gears helps evaluate dynamic processes in gear mesh,

transmission errors, stiffness of supports and structural

elements, and other factors that are sources of vibration

excitation in the transmissions.

Theoretical description of dynamic processes in meshing

of gears is usually based on two principal methods. In

shock method for estimating the dynamic loads, the

proposed A.I. Petrusevich and M.D. Genkin [1] and further

developed M.S. Polotskiy, the process of meshing is

considered as two independent phenomena - the medial

edge and strikes, which is typical for transmission at

relatively low speeds. Vibrating method for calculating

dynamic loads in mesing developed in papers of N.A.

Kovalev [2], suggests as the main source of excitation - a

periodic variation in the meshing stiffness intermating and

related parametric torsional vibration. Based on this

method were explained resonance phenomena in gears

found the zone of instability and the effects of transmission

parameters shown to decrease with the approach of

parametric oscillations overlap factor to an integer and

supercritical range. In papers Kahraman [4] and Parker [3]

used a model with lumped parameters in which the gearing

seems as hard drives connected to the elastic- damping

coupling. With modern methods of designing aircraft

gears, vibration excitation sources such as impact

processes, caused by premature meshing tooth top as a

result of strain and deflection of the basic step are

sidelined. This is due to an increase in manufacturing

precision of aviation gear (now modern machines are

moving to accuracy class 3), as well as the development of

methods for selecting tooth profile modification,

compensating the deviation of the deformed shape of a

tooth from the theoretical. As a result, the main source of

excitation of vibration in geared system is a time-varying

mesh stiffness characterized primarily contact ratio ε.

II. Model formulation.

In the present section the analytical model of motion

of a spur gear system is described.

The most difficult problem in gear dynamic modeling is to

evaluate the level of dynamic loads in the transmission

elements. Most precisely such a solution can be obtained

by a numerical method for solving dynamic elasticity

problems, such as finite element method in a dynamic

setting. However, using the finite element method is

effective only for the simplest model of a gear pair.

Evaluation of the dynamic properties of the entire

powertrain is usually carried out by means of analytical

models of dynamic systems with lumped parameters [12].

The properties of each subsystem of such a model for a

more accurate result is obtained by dynamic simulation in

the FEM.

A dynamic model of gear train in generally represents a

system of the inertial masses connected to each other

through elastic and damping elements (fig. 1). Stiffness

characteristic for elastic coupling usually evaluated

through nonlinear finite element model or based on results

of experimental data.

Fig. 1. The lumped-parameter dynamic

model of a spur gear.

To assess the qualitative behavior of a nonlinear system of

gears meshing considered two mass dynamic model of a

pair of gears (lumped-parameter dynamic model), in which

the gear pair is represented by two rigid discs with mass m1

and m2 and inertia moment J1 and J2 respectively [3]. Rigid

disks are connected by a spring-damper set along the line

of contact. The system is balanced attached in opposite

directions to the wheels of the external torques load M1

and M2 respectively (fig. 1).

Lagrange equations with generalized coordinates φ1 and

φ2, corresponding rotational angular displacement of gears,

can be expressed in the form:

𝐽1φ̈1 + 𝑟b1𝑘𝑧(𝑡)[𝑟b1𝜑1 + 𝑟b2𝜑2 + 𝑒(𝑡)]+ 𝑟b1𝑐𝑧[𝑟b1φ̇1 + 𝑟b2φ̇2 + �̇�(𝑡)] = 𝑀1(𝑡),

(1)

𝐽2φ̈2 + 𝑟b2𝑘𝑧(𝑡)[𝑟b1𝜑1 + 𝑟b2𝜑2 + 𝑒(𝑡)]+ 𝑟b2𝑐𝑧[𝑟b1φ̇1 + 𝑟b2φ̇2 + �̇�(𝑡)] = 𝑀2(𝑡).

where φ̈1, φ̈2, φ̇1, φ̇2 – angular acceleration, angular

velocity of the input pinion and the output wheel

respectively; kz (t) – mesh stiffness; сz – damping

coefficient (based on the results of experiments); 𝑟b1, 𝑟b2 –

radius of base circle of gears; e(t) – displacement

excitation from gear transmission errors. In practice, the

deviation of the district step differs for each tooth, so this

function varies with the frequency of the rotor. The study

applies the law changes or harmonic law with an amplitude

corresponding to the degree of accuracy of manufacture of

the gears or assumed to be constant for each tooth, which is

permissible under the assumption made above the small

effect of this excitation source for gears with a high degree

of accuracy. Mesh stiffness kz (t, M) - is a time-varying and

load-varying function, and depends primarily on the

number of tooth in meshing.

For pure evaluating of teeth geometry influence on

vibration behavior the finite element methods (FEM) were

used to determine the tooth spring stiffness by modeling

the elastic behavior of gear tooth. The results of a series of

static calculations by finite element method was used to

obtain relationship between tooth geometry and operation

forces. On fig. 2 shows a graph of the mesh stiffness for

gears, tested in a pilot plant in CIAM [8].

Fig. 2. FEM modelling gear mesh stiffness curve simulated at speed 10 rpm.

On figure 2 shows the simulation results of gear mesh

stiffness for spur gear pair with the contact ratio of the

overlap ε <2 and HCR gears with the same parameters of

the tooth profile. Mesh stiffness function obtained by

modeling meshing of gear pairs at a speed of 10 rpm using

a finite element dynamic analysis package Ansys

Transient. As can be seen from the figure, gears with high

contact ratio have a smaller range of variation function of

mesh stiffness. The variable component of mesh stiffness

for gear with ε <2 is 15-20%, for gear with HCR is 7-8%.

In finite model of gear take to account not only stiffness of

meshing teeth meshing teeth. In this model also modeling

stiffness of gear body or ring gear, and the complete

structure of the gears. This allows more accurately

determine all three components of the mesh stiffness: teeth

bending stiffness kzT, contact stiffness kzC and torsional

stiffness of the diaphragm and the crown kZB. The three

main component of gear stiffness (body, teeth and contact)

can be considered to act like three springs in a row, which

means that the combined stiffness 𝑘zΣ for each pinion and

gear is calculated as:

1

𝑘zΣ=

1

𝑘zT+

1

𝑘zC+

1

𝑘zB ; (2)

On fig.3 shows the result of solving the contact task for

meshing teeth using FEM, namely, the equivalent stress

distribution pattern, scaling for clarity on a scale of 400:1.

The angular displacement of the wheel points caused by

strain in the contact area of the teeth, tooth bending under

the force engaged and partial deformation of body gear. As

can be seen from figure 3, the ratio of the bending and

strain of the contact depends on the force application point

to a tooth, and the presence of the tip contact (in this case,

the stress state in a contact zone not defined by the Hertz

formula), whereby at two adjacent teeth of have a different

ratio 𝑘zC/𝑘zT, while compliance diaphragm is practically

constant over the angle of rotation of the wheel.

Fig. 3. Gear tooth deformation - FEA simulation of Von-Mises stress. I – Total body and hub strain; II- local

strain caused by tooth bending; III - strain in tooth contact point.

The studies of the deformational behavior of the gear

wheels show that the stiffness of gear body and teeth are

almost load-independent while the contact deformation is

non-linear, as a Hertzian contact occurs between the teeth

of the gears.

Experimental studies of the dynamics of gear show that

resonant modes of dynamic loads in gears increase of 3 ÷ 5

times, and meshing processes may be accompanied by loss

of gear contact. Opening the possibility of teeth simulates

loss of stiffness and damping characteristics of engagement

at the time of the negative value of the difference of mutual

displacement of teeth before sampling backlash 𝑗𝑛 𝑚𝑖𝑛, and

then restored transmission contact nonworking side

surfaces of teeth:

𝐹дин(𝑡, 𝜑1, 𝜑2) =

=

{

𝐤𝐳(𝐭) ∙ (𝛗𝟏 ∙ 𝐫𝐛𝟏+𝛗𝟐 ∙ 𝐫𝐛𝟐) + 𝐜𝐳 ∙ (�̇�𝟏 ∙ 𝐫𝐛𝟏+�̇�𝟐 ∙ 𝐫𝐛𝟐)

𝑤ℎ𝑒𝑛 (𝜑1 ∙ 𝑟𝑏1+𝜑2 ∙ 𝑟𝑏2) > 0

𝟎 𝑤ℎ𝑒𝑛 (𝜑1 ∙ 𝑟𝑏1+𝜑2 ∙ 𝑟𝑏2) ≤ 0

𝑎𝑛𝑑 |(𝜑1 ∙ 𝑟𝑏1+𝜑2 ∙ 𝑟𝑏2)| ≤ 𝑗𝑛 𝑚𝑖𝑛𝐤𝐳(𝐭) ∙ (𝛗𝟏 ∙ 𝐫𝐛𝟏+𝛗𝟐 ∙ 𝐫𝐛𝟐 − 𝐣𝐧 𝐦𝐢𝐧) + 𝐜𝐳 ∙ (�̇�𝟏 ∙ 𝐫𝐛𝟏+�̇�𝟐 ∙ 𝐫𝐛𝟐)

𝑤ℎ𝑒𝑛 (𝜑1 ∙ 𝑟𝑏1+𝜑2 ∙ 𝑟𝑏2) ≤ 0

𝑎𝑛𝑑 |(𝜑1 ∙ 𝑟𝑏1+𝜑2 ∙ 𝑟𝑏2)| ≥ 𝑗𝑛 𝑚𝑖𝑛

(3)

The magnitude of the frictional force is directly related to

the friction coefficient and the normal tooth load.

Therefore, accurate determination of the friction

coefficient is required.

In this study, an empirical formula developed by Xu and

Kahraman [10] was adopted as it was found to accurately

model the instantaneous coefficient of friction along the

path of contact of a pair of gears in mesh.

Ш. Results and discussion.

The software package Matlab Simulink® used to solve the

system of nonlinear differential equations with the

construction of a mathematical model describing the gear

meshing dynamics in the form of block diagrams and

numerical solution of the system based on the one-step

explicit Runge-Kutta method of 4th and 5th order.

Parametric resonance in the meshing frequency has a

typical form for nonlinear systems with smooth increase of

the dynamic component when passing downward turns the

system and step amplitude growth with an increase in

turnover. This phenomena is a result of unstable behavior

of nonlinear system.

Experimentally shown that the resonance phenomena

in gears are largely at frequencies that are multiples of 0.5,

1, 1.5, 2 of main resonance, which is a consequence of the

parametric nature of resonant vibration in the system. The

results of dynamic modeling of gear meshing also secrete

half tooth harmonic and amplitude growth when passing

through its corresponding frequency. Figure 4 shows the

simulation result for the torsional vibration by developed

analytical model with the passage of the system through its

natural frequency and multiples thereof frequency.

Fig. 4. Frequency response curve for dynamic factor.

Fig. 4 shows a unstable range at critical frequency of

system, which characterized by a sharp increasing of

amplitude at growth of gear rotation speed and its further

gradual decrease. Also noted an increase in vibrations

when passing system through the "half" of the tooth mesh

frequency.

The important result of the simulation is to confirm the

existence of discontinuous vibration in the transmission at

critical range, repeatedly discovered during experiments.

On fig. 5 shows the frequency response for gear with

different contact ratio: contact ratio ε=2,08 (blue line) and

contact ratio ε=1,68 (red line). Gears with contact ratio

greater than two have a dynamic loads smaller by 30% the

level of dynamic loads in gears with standard profile, as

well as a narrow band of resonant oscillations zone.

Fig. 5. Frequency response curve of dynamic factor for

different type of gears: contact ratio ε=2,08 (blue line)

and contact ratio ε=1,68 (red line).

Unstable behavior of the dynamical system of the gears at

the critical frequency, confirmed by mathematical

modeling and experiment, create uncertainty in the

determination of the dynamic component of the load in this

area and can lead to unstable operation of gear system.

Obviously, it is necessary considered when designing gear

system with operate parameters at critical and subcritical

range with unstable region and find possibility to operation

at supercritical range.

IV. Finite element model of geared pair.

In order to verification results of presented analytical

model and a detailed analysis of the processes occurring in

meshing teeth on the various modes of transmission, the

numerical dynamic simulation through finite element

package Ansys® Transient Structural has been conducted.

GeWe consider a model of two spur gears in the bulk

formulation (Fig. 6), with a torque on the inner cylindrical

surface of the driven wheels and set speed for the revolute

joint of the drive wheel. The system is driven with a

smooth loading and overclocking pinion to eliminate the

influence of dynamic factors in overclocking. Given the

considerable amount of computation to simulate

acceleration gears from rotation speed 10 rpm up to 9000

rpm. The gears was modelled for analysis along contact

lines using quadratic hexahedral and tetrahedral 18-x series

elements with a mesh size of 1 mm. Contact elements were

used for contact sides of teeth. Geometry parameters and

loading conditions are shown in Table 1.

Table 1.

Number of teeth z1=49 z2=51

Normal module, mm 4

Face width, mm 15

Pressure angle 20 deg

Range of load torque 100…2000 Nm

Range of rotation

speed 10…9000 rpm

For graphs of bending stresses at low speeds (Fig. 7) was

used element size of 0.5 mm. Solves the problem of taking

into account the friction engagement and structural

damping in the material. The simulation results are

presented in Figures 7, 8 and 9.

Fig. 6. Finite element model of spur gear pair in Ansys Transient Structural.

Fig. 7. Load distribution of bending stress for each tooth at speed 10 rpm.

Fig. 8. Load distribution of bending stress for single tooth

at speed 40 rpm.

Fig. 7 shows graphs of bending stress distribution in tooth

root along the one mesh phase at rotation speed 10 rpm.

This result is of great interest in the evaluation of both

static and fatigue strength of the teeth, because existing

methods for evaluating stress in gear dynamic allow to

obtain the average value for a series of meshing tooth and

strain measurement results do not give a high precision at

high speeds. The graph clearly shows the zone single- tooth

meshing corresponding peak site and zone of double

contact meshing with the increase of the root stress at the

start of meshing and decreasing at end of meshing as a

result of load transfer between the teeth.

Figure 8 shows the dynamic bending stresses in the tooth at

a speed of 50 rpm. This results is in good agreement with

the experimental data [11].

Fig. 9 shows a static transmission error of gears caused

by various deformations of the teeth in the meshing

action and which is the main excitation source in

transmission. Analysis of relationship between character

of static transmission error and the geometry of tooth

profile allows to select the optimal

modification parameters for gear with low level of

vibration to reduce dynamic stresses in the transmission.

Fig. 9. Transmission error of geared system modeled by

Ansys Transient

Fig. 10 shows the simulation results for the bending

stresses in teeth roots during increase of pinion rotation

speed until ω> ω0 (red line) and then reduction of rotation

speed to 10 rpm. In the this graphs clearly identified point

of resonance frequency, as well as point of stress increasing

at half the speed of resonance.

Fig. 10. Ansys Transient simulation results: bending stress distribution with increasing rotation speed (blue)

and reducing (red).

Fig. 10 shows the difference between resonance frequency

with the maximum values of the bending stresses in the

root when rotation speed is increased (blue) and resonance

frequency when rotation speed decrease (red graph). Also

showed a significant reduction of dynamic stresses in

supercritical range. This result may be used when

designing the operation condition of transmission. Also

modeling confirmed the presence of discontinuous

vibration with loss of contact of the teeth in the critical

range.

Analysis of the influence of stiffness and transmission

gearing overlap factor to dynamic loads, has been

conducted using developed model. Decrease of dynamic

loads in gears in the programs with overlap ratio ε ≥ 2 (3 to

5 times ) with two-pair stiffness ratio and tooth contact

from three-pair 1.1 to 1.3 times was showed . A similar

reduction of vibration and dynamic loads in gears with a

high contact ratio was shown in CIAM experiments.

The simulation results showed a reduction dynamic stress

level in the gear train on critical range with increasing of

external loads.

V. Conclusions

In the above sections, the parametrically excited

vibration of the gear-pair system in mesh was investigated.

A comparison between the analytical lumped parameter

model results and simulation of gear system through finite

element model was also presented.

Results of modeling show confirmed the presence of

discontinuous vibration with loss of contact of the teeth in

the critical range. Time-varying mesh stiffness function

examines as main source of vibration and dynamic

behavior of a gear system and evaluate through nonlinear

finite element model. Dynamic motion of the spur gear pair

over wide range of excitation frequency using the finite

element method was simulated. Simulation of gear

meshing through finite element model was obtained results

of bending stresses distribution in the depressions of teeth

per meshing phase in various modes of transmission, and

also obtained character of static transmission error

distribution. The simulation results showed a reduction

dynamic stress level in the gear train on critical range with

increasing of external loads. It may serve as a tool for

aiding the gear fault diagnosis.

References

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Dynamic loads in spur gears. IMASH, 1956.

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