VIBRATIONS GENERATED BY IDEALISED BALL BEARING

Rolling element bearings will generate vibrations during

operation even if they are geometrically and elastically perfect. This is an inherent feature of the

bearing type and is due to the use of a finite number of rolling elements to carry the external

load. A study of a realistic bearing arrangement shows that the number of rolling elements under

load varies with the cage position. This gives rise to a periodical variation of the total stiffness of

the bearing assembly and consequently generates vibrations. The number of rolling bodies under

load also depends on the vertical and horizontal, position of the inner ring relative to the outer

ring, which is assumed to be stationary. The bearing assembly thus constitutes a statically

indeterminate system with time varying and non-linear stiffness coefficients where vertical and

horizontal displacements are strongly coupled. The most convenient way of tackling such a

problem is to assume displacements and subsequently calculate the arising reaction forces.

Force-Deformation relationship will be shown in next sub-section.

In a bearing operating at normal speed, the displacements

generated by the bearing will cause the occurrence of inertia forces proportional to the mass of

the external load. For higher speeds dynamic forces of the same order of magnitude as the dead

weight of the load occur. These dynamic effects have been analysed by including the previously

described stiffness function in the equations of motion of a rotor supported on roller bearings.

Varying Compliance Frequency

The cage ensures the constant angular separation (β ) between rolling elements; hence there is

no interaction between rolling elements. Therefore,

β=2 πNb (2.1)

There is no slipping of balls as they roll on the surface of races. Since there is perfect rolling of

the balls on the surface of races and the two points of ball touching the races have different linear

velocities, the center of the ball has a resultant translational velocity. The translational velocity of

the center is

V cage=V inner+V outer

2 (2.2)

Where

V outer=ωouter×RV innerr=ωinner×r (2.3)

Here the outer race is assumed to be stationary, V outer=0 . Therefore

V cage=V inner

2=

ωinner×r

2 (2.4)

Now the angular velocity of the cage (ωcage ) about the center of the inner race is

ωcage=

V cage

( R+r )/2=

ωinner×r

( R+r ) (2.5)

Since inner race is rigidly fixed to the rotor, henceωinner=ωrotor . Therefore

ωcage=ωrotor ( r

r+ R ) (2.6)

The varying compliance frequency is given as

ωvc=ωcage×Nb (2.7)

Where Nb is the number of balls therefore from equation (2.6) & (2.7)

ωvc=ωrotor×BN (2.8)

Where

BN=( rr+R )×N b

(2.9)

The number BN depends on the dimensions of the bearing, for SKF6002, BN=3.6.from

equation (2.8) we can say that varying compliance frequency is dependent on the bearing

geometry and rotor speed.

Force-Deformation Relationship for Ball Bearings

Force-deformation relationship is given by Hertzian theory. Consider the roller bearing in

Fig.2.2 .The angular position of the cage is defined by ψ j and the gaps between the rollers by

β=2 πNb .

Fig. 2.1 Hertxian contact theory

The bearing has internal radial clearance of γ so that when the inner and outer rings are

concentric, there is no contact between the rollers and the outer ring. Now assume that the center

of the inner ring is moved from O to O' shown in Fig. 2.1. This will cause the two circles to

interfere with each other over a part of the circumference. From fig. 2.2, it becomes clear that

this interference will cause an elastic (for small displacement its deformation of the rollers and

rings. Dowson has shown that for rings mounted firmly against solid steel shaft and bearing

house, the only significant deformations are the local deformations at the contact points between

rollers and rings. He also showed that the local stiffness is near linear and can be well

approximated. These deformations will give rise to the reaction force of the bearing. The zone

over which the inner race and outer race circles interfere with ball will therefore be called the

elastic deformation zone.

Fig. 2.2 line diagram of radially loaded bearing (courtesy:NPTEL)

If x & y is the displacement of inner race center then ball-race contact deformation in radial

direction at contact is given as,

δ=x cosθi+ y sin θi−γ (2.10)

Where,γ is radial play and

θi=2 πNb

( i−1)+wcage×t i=1, . .. . N b (2.11)

We see that θi is a function of time and imparts the parametric effect to the system

The ball-race contact deformation of the ball generates a restoring force with non-linear

characteristics because of Hertzian contact.

Fθi=Cb δn , n=3 /2

(2.12)

The values of Cb and n are arrived by performing the elastic analysis of the Hertzian contact

between the inner and outer race and the ball.

From equations (2.10) and (2.12)

Fθi=Cb( xcosθ i+ y sin θi−γ )1 .5 U [ δ ]

(2.13)

Obviously a roller can only support compression forces, so for negative values of δ , Fθi should

equal zero. Therefore, multiply the right member with U [ δ ] where U is the Heaviside unit step

function having the value one for positive arguments and the value zero for negative arguments.

The total restoring force is the sum of the restoring force from each of the rolling elements. Thus,

the total restoring force components in X and Y directions are

Fx=Cb∑i=1

Nb

( x cosθi+ y sin θ i−γ )1 . 5 cosθi U [ δ ]

Fy=Cb∑i=1

Nb

(x cosθi+ y sin θ i−γ )1 .5 sin θi U [ δ ] (2.14)

If the rotor is turned very slowly, no inertia forces occur, which makes Fx = 0 and F y=mg .

Inserting these values in equation (2.14) and calculating δ x and δ y as functions of θi , will give

the static VC shaft locus.

Rotor supported by ball bearing

Having derived an expression for the force - deformation relationship for the bearing assembly, it

is now possible to proceed to considering rotor - bearing system. Even at very low speeds

however, the inertia forces have a significant effect. To solve the then occurring dynamic VC

vibration problem, the equations of motion for the mass of the rotor for horizontal and vertical

movements are set up. The damping in this system is represented by an equivalent viscous

damping C. The value of the damping depends on the linearized bearing stiffness as shown in

equation().The system governing equations accounting for inertia, restoring and damping force

and constant vertical force acting on the inner race are

m x..

=−c x.

−F x+W +Fucos (ωt )

m y. .

=−c y.

−F y+Fu sin (ωt ) (2.15)

By rearranging the terms equation of motion is,

m x..

+c x.

+Cb∑i=1

Nb

( x cosθi+ y sin θi−γ )1 .5 cosθi U [ δ ]=W +Fu cos (ωt )

m y. .

+c y.

+Cb∑i=1

N b

( xcosθ i+ y sin θi−γ )1.5sin θi U [ δ ]=Fu sin(ωt ) (2.16)

Here m is the mass of the rotor supported by bearing and mass of inner race. The imbalance

force Fu for this case is zero. System consists of two coupled non-linear ordinary second

order differential equations having a parametric effect in them. The stiffness because of its

step change behavior, the parametric effect with 1·5 non-linearity and the summation term

is non-analytic in nature.

Modified Dynamic model of system

AS Nb =9, so during operation number of balls in loading zone will vary from 4 to 5. So to

remove the ambiguity in problem we assume that on average 4.5 balls will be in loading zone. So

the contribution of each ball for the damping will be given as

c ' =200/4.5

Acording to modified damping the equations of motion are,

m x..

+∑i=1

N b

c ' x.

U [δ ]+Cb∑i=1

Nb

( xcosθ i+ y sin θi−γ )1 .5 cosθi U [ δ ] =W +Fu cos(ωt )

m y. .

+∑i=1

Nb

c ' y.

U [ δ ]+Cb∑i=1

N b

( x cosθi + y sinθ i−γ )1. 5sin θi U [ δ ]=Fusin (ωt ) (2.17)

As we have proposed that damping will be present if U [ δ ] is equal to one. This new damping

formulation has not been reported in any literature. This formulation is based on the fact that, in

no contact zone of roller-race assembly gap is present. So this gap is not contributing to the

damping of the bearing and hence these regions are neglected in modified modeling.

Computational considerations

The equations of motion, Eq. (2.17) are solved using the modified Runge kutta method to obtain

the radial displacement and velocity of the rolling elements. In order to eliminate the effect of the

free response an artificial damping was introduced into the system. With this damping, transient

vibrations are eliminated and peak steady state amplitudes of vibration can be estimated. The

longer the time it takes to reach steady state vibrations, the longer the CPU time that is needed

and hence the more expensive the computation; we chose a value of c = 200 N s/m. To observe

the nonlinear behavior of the system, parameters of the ball bearing are selected and are shown

in Table 2.

Table 2 Geometrical properties of bearing

Ball diameter (Db) 4.762 mm

Inner race diameter(Di=2r) 18.738 mm

Outer race diameter(Do=2R) 28.262 mm

Pitch diameter(Dp) 23.5 mm

Radial Load(W) 6 N

Mass of the rotor(m) 0.6 kg

Damping factor(c) 200 N s/m

Raidial clearance(γ ) 20 μm

Results & Discussion

The frequency response of the dynamic model is obtained to study the VC vibrations. The

system is simulated at 2120 r.p.m. and BN=3.6 (for SKF 6002), therefore from equation (2.8)

ωvc =126.8 Hz. The frequency spectrum shows the VC frequency and its harmonics as shown in

Fig. 2.3. The frequency spectrum has a band structure as shown in between spikes of VC and its

multiples.

Fig. 2.3 Frequency response at 2120 r.p.m

Speed response plots

Speed response plots are obtained for the combination of the above parameters under study.

These plots are generated by numerical integration to reach steady state when peak-to-peak

values of x and y displacements are obtained. For reaching steady state for the first speed the

initial conditions are taken as the fixed point solutions. For successive speeds, the initial

conditions are taken as the steady state solution obtained for the preceding speed. For a non-

linear system the response plots have regions of multivalued solution which are generally the

high-amplitude regions. Generating the response curve so that the i th steady state speed solution

VC=126.8 VC=126.8

is near the (i - l) th speed solution ensures that the same response curve is plotted. Throughout,

otherwise there is a danger of the solution jumping from one response curve to another.

0 2 4 6 8 10 120

0.5

1

1.5

2

2.5x 10

-5

Speed in krpm

Peak

to p

eak

ampl

itude

in m

12 14 16 18 20 22 240

0.5

1

1.5

2

2.5x 10

-5

Speed in krpm

Peak

to p

eak

ampl

itude

in m

Fig.2.4 Response plot ,c=200 Ns/m, ӿ - vertical response, + - horizontal response

The overall response plot for the system is shown in Fig. 2.4. The peak-to-peak (pp) vertical

response is higher than the peak-to-peak horizontal response. The overall response plot has a

very rough appearance. Three regions can be identified which have high pp response. Three

regions of stable solutions can be easily identified in fig. 2.4, region I(7000-8250) ,region II

(11000-13400) and region III(22900 onwards).

Response plot for the modified dynamic model based on new damping formulation is

shown in Fig. 2.5. The peak-to-peak (pp) vertical response is almost equal to the peak-to-peak

horizontal response. Three regions of high pp response can be easily identified in the dynamic

model too, infact it is more clear in the modified model. Three regions of stable response is also

observed in response plot but the region is shifted compared to response plot of earlier model as

shown in Fig. 2.4. Region I(7150-8200), region II(11800-16500), region III(22800 onwards)

shows the stable behavior as depicted from response plot. Third region of stability is observed in

both the models.

0 2 4 6 8 10 120

0.5

1

1.5

2

2.5

3x 10

-5

Speed in krpm

Peak

to p

eak

ampli

tude

in m

12 14 16 18 20 22 240

0.5

1

1.5

2

2.5

3x 10

-5

Peak

to p

eak

ampl

itude

in m

Speed in krpm

Fig. 2.5 Response plot ,c=200 Ns/m, ӿ - vertical response, + - horizontal response

Route to chaos

Ball bearings are a non-negligible source of vibration in many types of rotating machine. Fukata

studied the dynamic behavior of a radially loaded bearing under a varying rotor speed. They

noted that, away from the two critical speeds, the solutions were periodic, showing the ball pass

frequency / 6 and its harmonics. Around the first critical speed, they noticed some subharmonics

of the ball pass frequency whereas, around the second one, the solutions failed to remain periodic

and were described by what they called beat and chaos-like behavior The subharmonic route is

associated with the first critical speed. It generates more and more subharmonics of the ball pass

frequency, and is characteristic of the instability of the bearing. The quasi-periodic route is

associated with the second critical speed. The competition between two basic components results

in an increasing number of combinations and it finally leads to chaos by overlap of resonances.

When defects are introduced in the bearing, a third route to chaos is noticeable and looks like

intermittency. While the vertical stiffness is always greater than zero, this is not true for the

horizontal one. There are some positions for the ball complement that resolve the horizontal

stiffness to zero, and so, periodically, the rotor location is unstable .The two critical speeds of the

bearing assembly are defined by a ball pass frequency.

The horizontal stiffness is subject to large variations as the

cage rotates and therefore, when it is excited by the ball pass frequency, ωvc , cannot take control

of the motion and the bearing is unstable. The well-known cascade of subharmonics is the

characteristic route to the chaos of unstable behaviours; to improve its stability, one system

generates more and more subharmonics of the driving frequency, and this phenomenon leads to

chaos after an infinite number of bifurcations or period doubling. An increasing number of points

in the bifurcation diagram indicates that more bifurcations occur, but these are very close to each

other.

Whatever the route described, the loss of contact is always present when

chaos takes place, and so the occurrence of loss of contact is not a criterion for chaos; it is

necessary but not sufficient.

POINCARE MAPS

Poincare maps are obtained by sampling the four-dimensional flow (x, x', y, y'), once per forcing

period T=1/ f vc . Resulting attractors are then projected onto the phase plane (x, x'). The

response of the dynamic models represented by equations (2.16) and (2.17) is compared by using

the poincare maps. As discussed earlier, in modified model the response has been shifted in rpm

band as seen in pp response Fig. 2.5 and same is proved by using Poincare maps, Fig. 2.6.

(a) (b)

Fig.2.6 (a)-c’(modified damping) , (b)- c = 200 Ns/m

In the earlier model the chaos at 6400 is through a route of pitchfork bifurcations.As speed

increase, stability returns by a torus solution which is clear form Poincare map at 6850 r.p.m.

Similar route to stability is observed in the modified too, but as we can observe the modified

model is somewhat more choatic, which is quite related to actual response.

As speed is increased further system response becomes stable as

seen in response plots (2.4) and (2.5). From 6900 to 8250 r.p.m. there is period one stable

response. From 8300 r.p.m.again pitchfork bifurcation takes place and that leads to chaotic

region from 8300 to 10500 r.p.m..The chaotic region extends up to 10500 r.p.m. As seen if Fig

2.8 after 10500 r.p.m. system goes to period one stable response through torus solution.

As we can see, in modified model system stability is reached at the speeds slight

higher than the earlier model. For the modified model the response is shifted in the sense of

r.p.m. In modified model from 7200 to 8400 r.p.m. there is period one stable response.

(a) (b)

Fig. 2.7 (a) -c’(modified damping) , (b) - c = 200 Ns/m

(a) (b)

a-c’(modified damping) , b- c = 200 Ns/m

(a) (b)

Fig. 2.8 (a)-c’(modified damping) , (b)- c = 200 Ns/m

From 8400 r.p.m. same phenomenon is observed as basic model till 11500 r.p.m. The Poincare

maps of both the model are compared as shown in Fig. 2.8. from the above comparison we can

say the modified model gives the good results and is the correct justification for the ball-bearing

dynamic model.