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Dynamic Simulation of Meshing Force in Broken Tooth Involute Gear

Meshing Process Based on ADAMS

CHEN cai, SHI quan, WANG Guangyan, GE Hongyu, HE Zewen

Mechanical Engineering College, Shijiazhuang 050003, ChinaE-mail: [email protected]

Abstract: In order to obtain the changing pattern of meshing force for broken tooth gear meshing and evaluate quantitatively the battle damage degree of involute gear transmission system, a computation method based on dynamic simulation was proposed.The 3D model of broken gear is established through Pro/E and the dynamic model of involute gear transmission system is

established by using the dynamic simulation software ADAMS based on the Hertz contact theory. The model of the gear withdifferent number and position of broken tooth is established when the coincidence degree is known. The simulation results show

the meshing force fluctuation frequency and fluctuation amplitude of the gear transmission system show significant differenceswhen the broken tooth number is different or the tooth breaking in different position, and the broken tooth leads the meshingforce appears the lag phenomenon.

Key Words: Gear meshing, Tooth Broken, Simulation

1 Introduction

Involute gear is a kind of transmission device widely appliedin mechanical equipment. Its fatigue life, stiffness and

strength are all closely related to gear meshing force. In the

battlefield environment, gear grinding and fracture damage

will occur because of shell fragments penetration. In this

case, the gear meshing and equipment performance will be

influenced if the damaged gear is not replaced timely. In the past simulation research of gear meshing force, scholars

only consider the gear in the normal state and didn’t make a

research of the changing regularity of meshing force if thegear is damaged [1~3]. Therefore, it is necessary to analyze

the meshing force of damaged gears to get the changing

regularity of meshing force. It will lay the foundation ofquantitative assessment of equipment performance after

gears be damaged. There are many difficulties in actual test

of gear meshing force, because the gear meshing force has

big fluctuation and the gear is always in the rotation motion.

The paper uses the ADAMS virtual simulation technology,focuses on the study of the changing regularity of meshing

force when the gear tooth is completely fractured. The

virtual prototype model of damaged gear transmission

system is established and the changing regularity of meshing

force is obtained through simulation research.

2 Calculation Method of Meshing Force

In the gear transmission process, the meshing force is

produced by the gear tooth contact mutually. Assume the

gear is a rigid body and ignore the elastic wave and

kinematic pair clearance of the gear. Gear meshing force isdefined as:

( , 0, 0, , ) , 0

0, 0

e

s Kx F x d C x x

F x

1

Where K is meshing force, e is nonlinear index, F s is the Step

function, C is damping coefficient, and d is the maximum

penetration depth.

When x0, two gears can’t contact, the meshing force is 0;

when x<0, two gears mutual contact and meshing force F is

related to stiffness, contact deformation, the nonlinear

exponent, damping coefficient and penetration depth.

Formula (1) shows that the meshing force F is composed of

two parts:Elastic component Kxe, similar to a non-elastic

spring.Damping component ( , , , , ) s F x o o d C x . In order to

avoid the damping component mutation makes the function becomes discontinuous, the Step function F s with transition

curve is used.

0 0

2

0 0 1 1 0 0 1

1 1

,

( , , , , ) (3 2 ) ,

,

s

h x x

F x x h x h h a x x x

h x x

2

In the formula,1 0

a h h 0

1 0

x x

x x

; x is independent

variable; x0h0 x1and h1 are all real; x0 is the initial value of

the independent variable; h0 is the initial value of the

dependent variable; x1 is the final value of the independent

variable; h1 is the final value of the dependent variable.Analysis shows that the stiffness coefficient, nonlinearindex, damping coefficient and maximum penetration depth

should be determined before the gear meshing force

simulation. The key of the meshing force simulation is to

determine the stiffness coefficient. In the Hertz contact

theory [4], it can be found that the distance of corresponding

point of two mutual contacted rotating body can becalculated by using the following formula:

21 3

*2

9( )16

P x

RE 3

Where P is the load applied on the object, R is compositive

curvature radius, and E * is integrated elasticity modulus.

Proceedings of the 34th Chinese Control Conference

July 28-30, 2015, Hangzhou, China

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1 2

1 2

R R R

R R

4

R1 and R2 represent the equivalent radius of two gear contact

point respectively. Because the deformation of gear tooth is

very small, so the paper took the pitch radius for R1 and R2

values.

* 1 22 2

1 2 2 1(1 ) (1 )

E E E E v E v 5

Where E * is comprehensive elastic variable; v1 and v2 are

respectively the Poisson’s ratio of two gear ’s material. E 1

and E 2 are respectively the elastic modulus of two gear ’s

material.

From the formula (3), the relationship between the normalcontact force P and deformation x can be deduced as

follows:3

1 2 * 24

3 P R E x 6

So the stiffness coefficient K is depends on the contacted

body’s shape and material.1 2 *4

3 K R E 7

3 The Establishment of the Virtual Prototype

Model of Gear Transmission System

3.1 Determine the Maximum Threshold of the Number

of Broken Tooth

The gear tooth will be fractured when the fragment penetrates it. The gear will completely loses transmission

capacity when the fracture tooth reaches a certain number.

In order to research the changing regularity of meshing force

of the fault gears, the maximum number of broken toothshould be determined. The paper sets the broken tooth

number by calculating the gear ’s coincidence degree which

will ensure that the gear transmission system is not

completely losing its function when it has broken tooth.

Coincidence degree is an important evaluation index of

gear ’s drive continuous and load transfer uniformity. The

coincidence degree is defined as follows:

1l

p 8

Where l is actual meshing length; p is base pitch of the gear.

The value of coincidence degree is greater than or equal to 1.

When the coincidence degree is higher, the more tooth aremeshing at the same time and the time that a couple of tooth

meshing is longer. So the load on the each tooth is smallerand the bearing capacity of the gear is greatly improved.

The following formula can be used to calculate the

coincidence degree:

1 1 2 2

1(tan tan ) (tan tan )

2 a a z z

9

11

1

arccos ba

a

r

r 10

22

2

arccos ba

a

r

r 11

Where is pressure angle of gear pitch circle,a

is

pressure angle of addendum circle,ar is the radius of

addendum circle, andbr is the radius of base circle.

In this paper, the determination principles of broken teeth

number are as follows:

1 2, 22 3, 3

...

1 , ( 2,3, 4,...)

nn

q n q n q q

12

When 1 2 , it means at least one pair of teeth

meshed in the process that gears turn a base circle pitch and

part of the time also has two pairs of teeth meshed. So the

maximum number of broken teeth is two. When 2 3 ,

it means at least two pairs of teeth meshed in the process that

gears turn a base circle pitch and part of the time also has

three pairs of teeth meshed. So the maximum number of broken teeth is three. According to this principle to

determine the number of broken tooth, it can guarantee the

gear transmission system doesn’t completely lost the ability

of transmission.

3.2 Establish Solid Modeling

Using the powerful function of parametric modeling of

Pro/E, the solid model of gear transmission system is

constructed (Fig.1). The specific parameters are shown in

table 1. Then the information of solid model is injected inthe dynamics simulation software ADAMS. To determine

the relative location of parts and eliminate the freedom ofrigid body, it is required to use constraints and motions to

connect them. In this paper, the material of the gears is steel.

The revolute is added in the geometric of two gears to ensure

that the gear can relatively rotate around the axis. The entity

to entity contact force is added between the two gears.

Fig. 1: The model of gear transmission system in the Pro/E

Table 1: Gear Parameters

GearModulus

(mm)The Number

of ToothPressure

Angle(degree)Tooth

Width(mm)

Small

Gear2 24 20 20

BigGear

2 30 20 20

Through formula (9) ~ formula (12), the coincidence degree

in this paper is 1.628, so the number of broken tooth is set as

two. Considering the diversity of tooth fracture position,

three different solid models are constructed as shown in Fig.2 to Fig. 4.

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Fig. 2: Gear transmission system with one broken tooth

Fig. 3: Gear transmission system with two nonadjacent brokenteeth

Fig. 4: Gear transmission system with two adjacent broken teeth

4 Validation and Analysis of the Model

4.1 Validation of the Model

In this paper, the gear model takes Poisson’s radio as

v1=v2=0.29, elastic modulus as E 1=E 2=2.07105 N/mm

2 and

get K =5.5105 N/mm

2 by formula (7). According to the

experience, take collision force index e=2.2, damping

coefficient 100 /C N s mm and set input torque as static

load T =1.5×105 Nmm. In order to make the simulation

environment close to the real environment, the Step function

is used to define the load which will let the force increasing

smoothly in 0.2 second. It is shown in figure 5, namely step

time000.2150000

Fig. 5: Torque load

Set the input speed of capstan as 60r/min, the simulation

time as t =0.5s and step as 720. The simulation result of

driven gear ’s angular velocity is shown in figure 6.

Fig. 6: Time domain graph of driven gear’s angular velocitychange

It can be seen from the figure 6 that the angular velocity of

driven gear exist fluctuation. In the initial operation stage,

the speed of capstan increases from 0 to 60r/min in a

moment, but the load in the driven gear is small at the same

time. So the angular speed exist fluctuation. The average

value of driven gear ’s speed is 288 / s which is same as

theoretical value. So the model that has been constructed has

higher accuracy.

4.2

Analysis of the Simulation Results

Take the simulation time t =1sstep=1440, the angular

speed of capstan w=120r/min.

Figure 7 shows the changing of meshing force of the normalgears. In the moment when the capstan rotates, a great

impact is produced because the speed of capstan increases

from 0 to 120r/min in a moment. From 0 to 0.2s, with the

load gradually increasing, the meshing force also increases

accordingly. After 0.2s, the load is not increases and the

meshing force fluctuates near a stable value. The fluctuation

amplitude is 554.68N~637.94N which presents the obvious

periodic variation and the period is 0.0208s. This result is

corresponding to the set value.

Fig. 7: Change curve of meshing force of gear transmission systemunder normal condition

In order to obtain more accurate simulation data, ignore the

initial stage that the impact has a sharp increase caused by

the sharp increase of angular velocity. The simulation resultis shown in figure 8 to figure 10.

Figure 8 shows the meshing force change of geartransmission system whose broken tooth number is one. It

can be seen from the figure: from 0 to 0.2s, broken tooth part

didn’t enter the meshing state. The meshing force increases

with the increase of driven gear load and it also has obvious

fluctuation with a periodic variation. After 0.2s, when the

broken tooth part didn’t enter the meshing state, the meshing

force didn’t have any changes. Then the broken tooth part

enters the meshing state, the meshing force also has a few

changes. But when the broken tooth just leaves the meshing

state, the meshing force first fluctuation amplitude decreases,

and then produces an obvious fluctuation. The maximumvalue is 8518.70N which is 1.08 times than other time’s

maximum value.

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Fig. 8: Change curve of meshing force of gear transmission systemwith one broken tooth

Figure 9 shows the meshing force change of gear

transmission system that has two nonadjacent broken teeth.

It can be seen from the figure: from 0 to 0.2s, broken tooth

part didn’t enter the meshing state. The meshing force

increases with the increase of driven gear load and it also has

obvious fluctuation with a periodic variation. After 0.2s, the

meshing force changes are similar to the situation which is

shown in figure 8. When the broken tooth just leaves the

meshing state, two obvious fluctuations are produced. The

maximum value is 8519.33N which is 1.07 times than othertime’s maximum value.

Fig. 9: Change curve of meshing force of gear transmission system

with two nonadjacent broken teeth

Figure 10 shows the meshing force change of gear

transmission system that has two adjacent broken teeth. Itcan be seen from the figure: from 0 to 0.2 s, broken teeth part

didn’t enter the meshing state. The meshing force increases

with the increase of driven gear load and it has obvious

fluctuation with a periodic variation. After 0.2s, when the

broken teeth enters the meshing state, the meshing force

didn’t have any change. But when the broken teeth just leavethe meshing state, the meshing force first has a relatively

stable value, then the gears violent impact. The meshing

force rises sharply to 34836.12N which is 4.45 times than

other time’s maximum value.

Fig. 10: Change curve of meshing force of gear transmission

system with two adjacent broken teeth

By comparing the figure 8, figure 9 and figure 7, it can be

seen that the meshing force of gear transmission system

which has broken gears is affected by the number of broken

tooth. If the broken tooth number is one, the meshing force

values have small fluctuations when the broken tooth leaves

the meshing state. If the broken tooth number is two, themeshing force’s frequency increases and fluctuation

amplitude increases sharply. By comparing figure 9 and

figure 10, it can be seen the meshing force is affected by

broken tooth position. If broken teeth are not adjacent, the

fluctuation frequency of meshing force decreases obviously,

but the amplitude doesn’t increases obviously. If brokenteeth are adjacent, the fluctuation frequency decreases

obviously and the amplitude increases obviously at the same

time. Finally, through the comparison of figure 8 to figure

10 and figure 6, it can be seen that the influence of broken

tooth to the meshing force exist lag phenomenon. When the

broken tooth part enters the meshing state, the meshing forcedoesn’t change significantly. While when it leaves the

meshing state, fluctuation frequency and fluctuation

amplitude of the meshing force change immediately.

5 Conclusion

The damaged model of gear transmission system isestablished by Pro/E and ADAMS. And the change

regularity of meshing force has been researched. Through

analyzing the results of simulation, it can be seen there is a

mapping relationship between the meshing force and broken

tooth number or broken tooth position. This result will

provide basic information for quantitative assessment of battlefield damage. But the model established in this paper is

not very accurate and quantitative evaluation of battlefield

damage for gear transmission system is a great and arduouswork, so the next work needs to consider the gear

transmission system’s change regularity under differentdamage nodes or work conditions.

References

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[3]

Q. Z. Li, K. Liu. Calculation and simulation of gear meshingforce based on virtual prototyping technology[J].Heavymachiner,2006,(6):49~51.

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