Dynamic Analysis of Rolling Bearing System Using Lagrangian Model vs. FEM Code

7/28/2019 Dynamic Analysis of Rolling Bearing System Using Lagrangian Model vs. FEM Code

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12thIFToMM World Congress, Besanon (France), June18-21, 2007 CK-xxx

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Dynamic analysis of rolling bearing system using Lagrangian model Vs. FEM code

H. Rubio* J. C. Garca Prada C. Castejn E. LaniadoUniversity Carlos III University Carlos III University Carlos III University Carlos III

Madrid, Spain Madrid, Spain Madrid, Spain Madrid, Spain

Abstract The rolling bearings dynamical behaviour

analysis is a critical condition to determine the machine

vibration response. The rolling bearing, with outer ring fixed, is

a multibody mechanical system with rolling elements that

transmit motion and load from the inner raceway to the outer

raceway.In rolling bearing analytical formulation, the contact between

rolling element and raceways is considered as nonlinear springs

and their stiffness are obtained by using Hertzian elastic contact

deformation theory. The contact model among the rolling

element and the raceways will be detailed in the paper due to the

great important in the vibration pattern analysis .

In the work presented, the simulation of kinematics, dynamic and

structural behaviours of the rolling bearings and their vibration

response without faults will be presented, a analytical model

using Lagrange formulation and a simulation model using

Algor

code (events simulation).

To evaluate the suitability and compatibility among analytical

and simulation models, the results from simulation will be

applied to inner ring motion equations of analytical model to

obtain a valuable error signal.

Keywords: Rolling bearing, analytical model, simulation,

multibody and rotor dynamics

I. Introduction

The rolling bearing dynamic behaviour analysis is avery important issue to know the system vibrationresponse. This response is non-linear, mainly when therolling element is a rolling element1 [4, 5]. On condition

operating, rolling bearings generate some vibrations andnoise related to the movement transmission characteristicof these mechanical elements where elastic propertieshave a great influence.

In this work, in a first step, an analytical model isproposed by using Lagrange formulation [9, 10] besides a3D rolling bearing simulation model is developed. Also,

the vibratory signal, obtained from rolling bearingoperation by means of a simulation program (Algor

code), using the "event simulation" technique [1, 12], is

*E-mail: [email protected]

E-mail: [email protected]

E-mail: [email protected] Roller and ball

implemented as input data (rolling element position, inner

race position, contact displacement) solution in theanalytical model.

The main goal of this work will consist ofcharacterizing the error signal in the inner ring motionequations.

II. Analytical model

Figure 1 shows the multibody mechanical system tosimulate: the configuration of a motor connected to arolling bearing, where the outer race is fixed.

Fig. 1. Multibody mechanical system

In rolling bearings, the rolling element-racewaycontact is non-linear, it makes that the faults in the

raceways or the rolling elements origin complexvibrations. For this, in the analytical model presentedLagrange formulation is applied to calculate the innerraceway and rolling elements positions (independent

generalized coordinates) [10].The motor-rolling bearing system motion equationswill be calculated using:

d T T V F

dt p p p

+ =

(1)

where p is generalized coordinate vector, F is

generalized loads (forces), T is total kinetic energyand V is the total potential energy.

The coordinates used in the analytical model arereferenced to the fixed outer race. Figure 2 shows theequivalent geometric model used: rolling element centrepositions and the inner centre position related to the outer

centre position.


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Also, figure 2 describes the non-linear contact rollingelement-raceways. The kinetic and potential energieswere obtained using Koenig kinetic theorem and Hertz

contact theory.

The total kinetic energy of the motor-rolling bearingsystem is the sum of the rolling elements, inner race andthe motor:

1

bN

in ball motor

j

T T T T =

= + + (2)

The total potential energy of the motor-rollingbearing system is the sum of the rolling elements, innerrace, the motor potential energy and the rollingelements-raceways elastic contact energy:

1

bN

in ball motor contact

j

V V V V V =

= + + + (3)

Fig. 2. Geometric model of rolling bearing.

To study the rolling element bearing structuralvibration characteristics, the rolling element-racewaycontact can be considered as a spring mass system, in

which the outer race is fixed in a rigid support and theinner race is fixed rigidly with the motor shaft. Elastic

deformation between raceways and rolling elementsproduces a non-linear phenomenon between force anddeformation, which is obtained by Hertzian theory. Therolling element bearing is considered as non-linear contact

spring as shown in figure 2.The application of the Hertzian classical theory of

elasticity to the contact point problem between theraceway and ball develops a contact area with ellipticalshape (as figure 3 shows). Some parameters aboutcurvature at contact surfaces are needed in order to obtain

the contact force. The curvature sum parameter isobtained from Harris [8] as:

1 2 1 2A A B B

= + + +

(4)

And the curvature difference parameter is expressedas:

( )( ) ( )1 2 1 2A A B B

F

+ =

(5)

The parameters rA1, rA2, rB1, rB2, A1, A2, B1 and

B2 can be dependent on the inner and outer raceways asshows figure 3. If the inner raceway is considered, then:

1 2 1 2

1 2 1 2

; ; ;2 2 2

2 2 2 1; ; ;

B B IA A B B I

A A B B

B B I I

D D Dr r r r r

D D D r

= = = =

= = = =

(6)

Fig. 3. Geometry considered in the ball-raceway point contact.

If outer raceway is considered, they are given as:

1 2 1 2

1 2 1 2

; ; ;2 2 2

2 2 2 1; ; ;

OB BA A B B O

A A B B

B B O O

DD Dr r r r r

D D D r

= = = =

= = = =

(7)

DB is the ball diameter, DI is the inner racewaydiameter,DO is the outer raceway diameter, rI is the innergroove radius and rO is the outer groove radius. Negative

value denotes a concave surface.The relative approach between steel raceway and steelball, in rolling bearings, is given by [8]:

( )1/ 34 * 2 / 32,79 10 ( )x Q mm = (8)

in which * is a function ofF(). Hence, the contact force

(Q) is

( ) ( )3/ 2 1/ 25 * 3/ 22,15 10 ( )Q x N

= (9)Also, it means a non-linear relation load-

deformation:

3/ 2

( )Q K N= (10)


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Where K is the load-deflection factor for the ballcontact with the inner raceway and for the ball contactwith the outer raceway is:

( ) ( )3/ 2 1/ 2

5 *3/ 22.15 10race race race

NK

mm

= (11)race=(in,out)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

F()

*

Fig. 4. The influence of curvature difference parameter on factor *.

In equation (11), the parameters*

race is obtained from

the figure 4, based on the curvature difference parameter

( )race

F .

In case of the steel roller-steel raceway contact, basedon Harris [8], the process is simplified, so the load-deflection factor is:

4 8/910/97.86 10L

NK L

mm = (12)

And the contact force (Q) is:

10/9 ( )LQ K N= (13)

As previously was considered, to obtain motionequations, the expressions of kinetic and potential energy

for all mechanical components are calculated as:

( )

( )

( )

, ,

, ,

, ,

in in in in

ball j j j

motor in in in

T f x y

T f

T f x y

=

=

=

( )

( )( )

( )

,

,

in in

ball j j

motor in

contact in out

V f y

V f

V f y

V f

=

=

=

=

(14)

And the relative approaches raceways-rolling

elements, in and out(based on figure 2), is expressed as:

( )in jf = (15)

( )out jf = (16)

It allows to calculate Vcontactas:

( ) ( ), , , ,contact in out in in j jV f f x y = = (17)

in which the deformation of spring at inner raceway j(figure 2) is obtained as:

( ), , ,j in in j jf x y = (18)or most clearly as:

2 2 2

2 ( cos sin )j in in j in j in j jx y x y = + + + (19)

The generalized coordinates j (j=1, 2,, Nb,

whereNb is rolling elements number),xin andyin areable to define the mechanical state of the system. We

consider in constant and the arc between centres of two

consecutive rolling elements is also constant. From figure2, the expression (20) is obtained where the angular

position of centres rolling element is localized:

( )

( )

, , ,

21,2,..., 1

j in in j in

j k j bb

f x y

k k NN

+

=

= = (20)

For the generalized coordinate j, where j=1,

2,, Nb, the equations of motion are:

( )

( )[ ] [ ]

[ ] ( )[ ]

2

2

2

sin 2 1

1

2

10

2

1,2,...,

j j j j j j j j

j inin in inLZ LZ

j j

outout out out LZ LZ

j

b

m m g m

kk

kk

j N

+ + + +

+

+ + =

=

(21)

For the generalized coordinatexin the equation is

( ) ( )[ ]1

0bN

j

in R in in in LZj in

m m x k x

=

+ =

(22)

For the generalized coordinate yin the equation is

( ) ( )

( )[ ]1

b

in R in in R

Nj

in in LZj in

m m y m m g

k Wy

=

+ + +

=

(23)

The result is a system of (Nb+2) second-order,

non-linear differential equations. Then the algebraicequations (21) is necessary to calculate the centres rolling

element angular position. The LZ (Load Zone) in theseequations indicates if compression in the contact pointexists or not. Then the deformation at the contact pointswill be calculated in (24) and (25) as:

( )0

j in j

j in

If LZ f

If LZ

=

= (24)

( )0

j out j

j out

If LZ f

If LZ

=

= (25)


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The results obtained from the generalized coordinates

(xin , yin) determine the vibrations of the motor-rollingbearing system.

The results obtained from the generalized coordinates

(1, 2, , Nb) determine the load at the contactpoints (rolling element-outer raceway and rolling

element-inner raceway).

III. 3D simulation model

In order to test the analytical formulation presented, a

simulation model of rolling bearing, using Algor

code,has been developed. The software Algor uses the eventsimulation technique that combines finite elementmethod FEM in time domain with dynamic events [1, 12].

The analysis and simulations of its kinematics,

dynamic and structural behaviours, in different serviceconditions, are considered. Finally, the study will beapplied in a rolling bearing model that contains allmechanical parameters: materials, friction factors, loads,contact elements, geometrical restrictions, etc. (see figure5).

Fig. 5. Rolling bearing Algor model in initial position.

The geometric model is designed with thesecharacteristics:

Number of bearing rollers: 13.Roller diameter: 10 mm.Roller length: 10 mm.Inner raceway diameter: 36 mm.Outer raceway diameter: 56 mm.Pitch diameter: 46 mm.Angular velocity of the inner ring: 4500 rpm.

Mass of the roller: 0.006 kg.

Mass of the inner ring and shaft: 0.078 Kg.Load in the centre of the inner ring: 2000 N.

By processing simulation model the temporal

evolution of the parameters which define the analytical

model generalized coordinates is obtained. We mean:(xin ,yin) Position of the inner ring centre, in X

and Y (figure 6).

j ( j=1, 2,, Nb) Radial position of theroller centre.

0 0.01 0.02 0.03 0.04 0.05 0.06

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

Time (s)

X(innerring)(mm)

0 0.01 0.02 0.03 0.04 0.05 0.06-0.22

-0.2

-0.18

-0.16

-0.14

-0.12

-0.1

-0.08

Time (s)

Y(innerring)(mm)

Fig. 6. Inner ring centre position, in X and Y.

Data obtained from simulation model are

processed and included in the analytical modeldeveloped. The distance between centres of twoconsecutive rollers is not considered as constant and the

angular position of the rolling elements j ( j=1, 2,, Nb) are included.

In order to obtain partial derivatives of thegeneralized coordinates, the temporal sequences arederived (see figure 7).

0 0.01 0.02 0.03 0.04 0.05 0.0622.98

23

23.02

23.04

23.06

23.08

Time (s)

(mm

)

0 0.01 0.02 0.03 0.04 0.05 0.06-400

-200

0

200

400

Time (s)

d()(mm

/s)

0 0.01 0.02 0.03 0.04 0.05 0.06

-5

0

5

x 106

Time (s)

dd()(mm

/s2)

Load Zone Load Zone

Fig. 7. Position, velocity and acceleration of the roller centre.


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To obtain the deformation at the contact point of

roller-inner raceway and roller-outer raceway (in and

out), is calculated as:

= + in in roller jr r (26)

= +out out roller jr r (27)

where rin is the inner raceway radius, rout is the outerraceway radius and rrolleris the roller radius.

IV. Results and discussion

Time domain vibration signals of good roller bearings

obtained from Algor code have been considered toanalyze and check the Lagrangian model of inner ring

motion. Figure 6 describes the vibration components X eY of inner ring. The corresponding equations of the inner

ring motion are calculated based on the numericalsimulation results obtained from Algor code. The time

to reach steady state vibrations needs at least 300 hours ofCPU simulation time which means more of two cycles ofspindle.

The model presented analyzes the different weight of

energetic levels of rolling element and inner racewaykinetics and potential terms related with generalized

coordinates. The Lagrangian model considers the Hertzcontact theory without damping to obtain rolling elementdeformation on raceways.

In figure 7, it presents radial position, velocity andacceleration of roller centre to outer ring centre relatedwith level of contact deformation (inner or outer raceway

roller). The magnitude of deformation at inner and outercontact can be seen. The elastic contact lets obtain thelevel of load at roller, we can see the load zone (LZ)variation vs. time, we detect over 6-7 rolling elements onload zone every time.

Applying the kynematic data from Algor code to

equations 22 and 23, a modulated error signal around 0 N,coordinate X, and anothermodulated error signal around2000 N, level of initial load, coordinate Y. The dampingand natural frequencies generate transient vibrations that

must be considered in more complete models in order tostudy the error signal behaviour (balance equation 22 and

23).In figure 8 and 9, the unbalanced in equations 22 and

23 is presented. We obtain a unbalanced dynamicactuation forces involving inner ring rotationalcoordinates X, Y around 22 Hz. (figures 10 and 11), butthe tendency is correct.

The effect of the bearing cage has also not beenconsidered, experimental evidence suggests that the cagecontacts do not play a large role in the roller bearing, butthe residual perturbations obtained with the simulated datamakes necessary a new and more completed model to

study the influence of geometry and material cage.

0 0.01 0.02 0.03 0.04 0.05 0.06-3000

-2000

-1000

0

1000

2000

3000

Time (s)

Fx(N)

Fig. 8. Error signal of first term eq. 22.

0 0.01 0.02 0.03 0.04 0.05 0.06-3000

-2000

-1000

0

1000

2000

3000

Time (s)

W

(N)

Fig. 9. Error signal of first term eq. 23.

0 50 100 150 200 250 3000

0.5

1

1.5

2

2.5

3x 10

10

Frequency (Hz)

PowerSpectralDensity

Fig. 10. Power spectrum of error signal of first term eq. 22.

0 50 100 150 200 250 3000

0.5

1

1.5

2x 10

7

Frequency (Hz)

PowerSpectralDensity

Fig. 11. Power spectrum of error signal of first term eq. 23.

V. Conclusions

A model of the motion of roller bearing (roller

bearings and inner ring) has been developed usingLagrange equations. Using the Lagrangian model we can

calculate the radial position of the rolling elements and thecentral point of the inner ring, but a system of fifteenequations with nonlinear and partial differentials is neededto solve.

Time domain vibration signals of good roller bearings

obtained from Algor code have been considered to


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analyze and check the Lagrangian model of inner ringmotion.

Experimental data obtained using a simulation model

agrees reasonably well with predictions. The application

of the analytical model to obtain the centre coordinates ofthe spindle using the data signal from the Algor codesatisfied the equations, but a unbalanced error force isdetected at 22 Hz possible related with natural frequenciesand damping phenomena at contact rolling element-inner

raceway. The consideration of damping in the simulationstudy makes necessary a more detailed analysis in futuremodels and studies.

The model is general and available to other rollerbearings types, as long as the corresponding FE model canbe accurately constructed.

The effect of damping has not been studied. When

analyze the orbit of the spindle we observed a erratic

evolution, we must increased the Lagrangian model toanalyze the impact effect over natural or dampingfrequencies.

Using the analytic model is possible study theinfluence of bearing faults at the inner or outer raceway

and obtains the vibration effects at the spindle studyinganalytic model vs. FEM model, and the relative slidingbetween the bearing surfaces.The authors would thank the funds provided by theSpanish Government through the Project MCYTDPI2003-084790-C02-01.

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vibrations. ASME paper 77-WA/DE-5 1977;18.[4] Y. Cao, Y. Altintas. A general method for the modeling of

spindle-bearing systems. Journal of Mechanical Design,126, 1089-1104, 2004.

[5] N. S. Feng, E. J. Hahn, and R. B. Randall. Simulation ofvibration signals from a rolling element bearing defect.DSTO International Conference on Health and Usage

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[6] J. C. Garca-Prada, H. Rubio, et alt. Simulacin dedefectos en elementos rodantes: aplicacin a rodamientos.Anales de Ingeniera Mecnica, XV CNIM, 1478-1583,

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[10] S. P. Harsha. Non-Linear dynamic response of a balancedrotor supported on rolling element bearings. MechanicalSystems and Signal Processing. 19, 551-578, 2005.

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[14] M. Tiwari, K. Gupta and O. Prakash.. Dynamic responseof an unbalanced rotor supported on ball bearings. Journalof Sound and Vibration. 238(5), 757-779, 2000.

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Documents

Dynamic Analysis of Rolling Bearing System Using Lagrangian Model vs. FEM Code