Investigation of Factors Influencing the Structural Vibration in Ball Bearings

Hertzian equation for elastic contact is utilized along with lumped parameter approach to obtain the equations that govern the structural vibration of ball bearings. The lumped parameter formulation is obtained by treating various elements with mass lumped at their centers of gravity and the contact as nonlinear springs with nonlinear spring rates.


INTRODUCTION
Rolling element bearings are being used extensively in numerous mechanical systems.They find their use in relatively simple applications in electric motors and compressors to demanding applications in turbines and jet engines.In rolling element bearings, contact deformation is the result of loads that develop between the rolling elements and the inner or outer rings of the bearing.The motion of the balls/rollers relative to the line of action of an external load causes fluctuation in the apparent rigidity of the bearing.This phenomenon is commonly referred to as varying compliance and is known to be the primary cause of vibration in bearings.There are many other factors apart from varying compliance that contribute to vibration in bearings: varying roller diameter, misalignment, fit, surface roughness, flexibility of rotor shaft, etc. Sunnersjo (1978) studied varying compliance of roller bearings and found that it arises because of variation of the number of rolling elements carrying Corresponding author.Tel.: (618) 453-7002.Fax: (618)453-7658.E-mail: farhang@siu.edu.322 K. MEHRA et al.   an external load.Bal'mont et al. (1987) attributed two factors to structural vibrations in bearings: (a) bending vibration of the outer race due to contact force of the balls and (b) variable rigidity of the rotating ball bearing under the action of the radial load component.In their analysis, radial thrust ball bearings were considered.The fluctuation of rigidity was found to occur when the rolling elements moved relative to the line of action of the radial load.The rigidity was determined to vary periodically at frequencies which are multiples of the ball rotational frequency.Tallian and Gustafsson (1965) studied a model of rolling element bearings that described the asso- ciated frequency spectrum and amplitude relation- ships.They attributed two factors to the vibration of the outer ring: (1) cyclic variations in the compli- ance of the bearing and (2) geometric imperfections of the bearing.On examining the geometric imper- fections, the authors found that the balls were the most important influencing elements followed by the inner and the outer rings.Rahnejat and Gohar (1985) considered a rolling element bearing together with the oil film, as nonlinear springs and dampers are rotating about a spindle.When a periodic or step load was applied, squeeze films were developed in addition to the films created by normal rolling motion.The varia- tion of pressure (load) with film thickness and squeeze speed was studied.The authors found that rotating unbalance and surface features introduce further significant frequencies which influence the response.Under the weight of the shaft, the fre- quency observed was a limit cycle.The frequency and amplitude were affected by the number of balls, applied load and radial internal clearance.
In the existing literature on vibration in rolling element bearings, contacts between the rolling ele- ments and the inner and outer races have been considered in order to establish a combined (resultant) bearing stiffness.As a result, the interactions between the individual rolling elements and the rings have not been taken into consideration.The present work addresses such interactions by devel- oping a model to account for the structural vibra- tions in rolling element bearings.Each element is treated separately taking into account its inter- action with the other elements.The equation for Hertzian contact deformation has been used to develop the nonlinear contact stiffness.An ap- proach based on the equivalence between journal and rolling bearings have been used to establish equivalent damping coefficients due to the presence of lubricants.

THE GOVERNING EQUATIONS
Figure l(a) illustrates the model of the rolling element bearing.The contact stiffness between the ith ball and inner and outer races are represented, respectively, by springs K1; and K2-.The cage ensures constant angular separation between the adjacent rolling elements (Datta, 1991; Datta and  Farhang, 1997a,b).Therefore, the azimuth angle of rolling element (i 2, 3,..., N) is related to that of the first as follows: 0i--0+(i-1)/3, i--1,2,...,N, (1) where, the constant angular separation is given by /3 2rc/U.
The equations of motion for the bearing system are obtained using the Lagrange's equation for a set of independent generalized coordinates d(OT where qk and Qk are, respectively, the k-th generalized coordinate and the corresponding generalized force.The total kinetic energy (T), and the potential energy (V) of the bearing system are the sum of those of the rolling elements, the inner and outer rings, and the cage.Pc represents the dissipation energy due to lubrication.
The total kinetic energy (T) of the bearing may be expressed as T= Tr.e. q-Ti.r.-? To.r.-+-Tc, (3) where Tr.e. represent the kinetic energy sum of the rolling elements.Ti.r., To.. and Tc denote the kinetic energies of the inner and outer rings and the cage, respectively.Neglecting slip, the rolling contact equation for the ith rolling element and the inner ring (Fig. l(b)) can be written as Hence, i 0i _r__ (a 0i), Pr where qSi denotes the rotation of the ith rolling element about its center of mass.Similarly, consid- ering the rolling contact equation between the ith rolling element and the outer ring, Timi [f12 + -Jr-x a -3r-22a/i COS 0 2JCaPiOi sin Oi + j; + 23)a/i sin Oi r (a 0i) (6) + 2f;aPiOi cos Oil + -Ii Oi P --r The kinetic energy expressions for the inner and the outer rings are Ti.r.-1/2ma(:,2a-+-)),2a) + 1/2Ia,2a, The kinetic energy of the cage is calculated by assuming that its center remains coincident with that of the inner ring.Therefore, Tc 1/2mc(22a + )2a) + 1/2IcOn. (9) The potential energy formulation is performed taking datum as the horizontal plane through the global origin.The total potential energy of the bearing is V-rr.e.-ri.r.-3r-ro.r.@ rs --Vc, (0) where V,..o., Ui.r.go.r.Us and Vc are the potential energies due to elevation of the rolling elements, inner and outer rings, springs, and the cage, res- pectively.For the rolling elements, the potential energy due to elevation is given by N Vr.e.Z mig(pi sin Oi + Ya) i=1 N mgNya -+-(miPig sin Oi).The deformation of the springs kli and k2i are (Pi--L10) and (R-xi-L20), respectively.The expression for the potential energy due to the contact deformation is, therefore, (14)   where kli and kzi are the nonlinear stiffnesses due to Hertzian contact between the ball and the inner and outer races, respectively.The potential energy due to the cage elevation is Vc mcgya (15) Energy dissipation in the bearing may be expressed as 1( .c:i2 ) The equations of motion for the bearing are obtained using Eqs.( 2)-( 16).Employing Lagrange's equation, and assuming that the motion of the inner ring is known, the governing equations of motion for the generalized coordinates pi (i 1,2,..., N) are obtained mii + mig sin Oi + mipiO 2 + kii(pi L10) Oxi kzi(R xi L20) + mi2a cos Oi + mifa sin 0 10kli 10kzi )2 The equations of motion for the generalized coordinates X b and Yb are where Fx and Fy are the generalized forces on the outer ring of the bearing, corresponding to the generalized coordinates Xb and Yb, respectively.
When the motion of the inner ring is known, Eqs. ( 17)-( 19) can be solved to obtain the radial displacement of each ball, and the x and y displace ments of the outer ring center.This is a system of N+ 2 second-order, nonlinear, ordinary differen- tial equations with time-dependent coefficients.The reader may refer to Appendix A for details on equivalent spring constants, kli and k2., x; and its derivatives.
The equations of motion may be normalized with respect to the physical and geometrical properties of the rolling elements, i.e. average mass and radius, m m2 mav,

Pr Pr2
Drav, where may and Pav represent the average values.After some manipulations, the normalized equa- tions of motion in terms of the normalized parameters are + g* sin Oi piOi + kli(P Ll0 Ox k2i R* x L20) Op + x a cos Oi + Ya sin Oi 10k*l 10k i=1

RESULTS
Equations ( 20)-( 22) are utilized to predict the motion ofeach ball and the outer ring.In the ensuing sections, vibration response of a ball (hereafter referred to as ball 1), initially located at the zero azimuth angle is presented.In addition, outer ring motion is presented as displacements of its center of mass along the global coordinate axes, X and Y.

Effect of Preload
The vibration responses of ball and the outer ring in the presence of a preload of 10 and N are shown in Fig. 2. Comparison of the figures suggests that the frequency of oscillation increases in the presence of preload.This is due to the increased stiffness of the bearing.In the absence of preload, the response amplitude of vibra-tion of ball is almost three times larger than that in the presence of preload.Also, the outer ring center shows amplitudes that are almost an order of magnitude larger than those in the presence of preload.
When the preload is reduced from 10 to N, no significant change is observed in the motions of ball and the outer ring center (Fig. 2).The fre- quency remains almost the same and the amplitude in the case of N preload increases very slightly, as expected.

Effect of Interference Check
The springs representing the Hertzian contact are allowed to act only in compression, and to ensure this, the interference checks are performed.In the absence of preload, responses of ball are observed for the cases in which interference checks are performed (Fig. 2) and are neglected (Fig. 3).The frequency and amplitudes of oscillation are quite different in the two cases.In contrast, a comparison of the same cases in the presence of preload, Fig. 2 with Fig. 3, illustrates strong agreement between the corresponding responses.The preload seems to en- sure the ball-race contact, thereby making the inter- ference checks unnecessary.This effect is of special significance since in practice, bearings are expected to have some amount of preload after assembly.

Effect of Ball Rotational Speed
Figure 4 shows the state of contact when the bearing is run at an extremely high speed of rotation.It is assumed that there is no preload, and the ball rotational speed is 74,990RPM.
Figure 4(a) illustrates the state of contact between the eight balls and the inner race, whereas Fig. 4(b) shows the corresponding contact conditions between the eight balls and the outer race.In the aforementioned figures, each ball is assigned two states indicated by odd and even numbers.Odd numbers on the ordinate indicate a no-contact condition, while even numbers represent a state of contact.Since there are 8 balls in bearing 6204, there are 16 numbers in all.For example, numbers and 2 are associated with ball 1, numbers 3 and 4 with ball 2, and so on.A plotted line corresponding to a state signifies the existence of that state.
Figure 4(a) shows that at almost all time instants, the dark band appears for an odd number on the y axis, indicating that there is no contact between the balls and the inner race.The dark band appears momentarily against an even number, signifying that all the balls have contact with the inner race for  FIGURE 3 Normalized radial displacement of ball and the x and y displacement of outer ring center interference checked: Left column is ball motion, right column is outer ring motion, top row corresponds to no preload, and bottom row preload of 10 N (-x-displacement, y-displacement).VIBRATION IN BALL BEARINGS 327 a very brief duration of time.On the other hand, Fig. 4(b) shows the dark band for all even numbers on the ordinate, indicating contact between the balls and the outer race at all times.This phenomenon is primarily due to the presence of high centrifugal forces on the balls, causing the balls to experience a radially outward thrust, thus main- taining contact with the outer race.

Effect of Fluid Film Lubrication
Damping corresponding to a film thickness equal to 10-3mm, and a dynamic oil viscosity of 3.4 10-2kg/ms is introduced in the mathematical and are model.The damping coefficients c li Czi estimated using Eqs.(A.16) and (A.17) and refer- ring to the relevant constants in Table I and Eschmann et al. (1985).
Figures 5 depict the vibration response of ball 1.It is seen that the transient vibrations decay, and a vibration response of smaller frequency and magni- tude persists.Although the ball oscillation is dampened considerably, it is periodic in nature and repeats after almost one cycle of rotation.The motion of the outer ring center exhibits decay in vibrations.The x and the y displacements show a steady-state behavior after nearly 50 of rotation of the outer ring.The negative value of the average amplitude of oscillations (in the y direction) of the outer ring is attributed to the effect of the gravitational term appearing in the governing equation of motion, Eq. ( 22).When the damping is increased by an order of magnitude, there is hardly any oscillation in the motion of ball 1, as shown in Fig. 6.Similarly, there is a rapid decay in the oscillations of the outer ring center, as depicted in Fig. 6.Thus, introduction of fluid film lubrication helps in dampening the oscillations to a large extent.

CONCLUSIONS
A lumped parameter model has been introduced in this paper to investigate structural vibrations in ball bearings.Using the model, effects of preload, operational speed, and fluid film damping on the vibration response of the bearing have been studied.The foregoing results provide the following 5 conclusions: E The presence of preload results in oscillations of g relatively higher frequency and at comparatively kli lower amplitudes.Furthermore, the vibration response is found to be highly sensitive to the mere kzi existence of preload and not as much to its amount, as preloads of 100-1 N do not cause appreciable K difference in the response.So long as there exists preload in the bearing, the interference check has L little influence on the vibration response.In con- trast, in the absence of preload, the interference L10 checks have a profound influence on the response.A significant implication of this is reflected in the L20 linearization of the equations of motion: preload facilitates linear model representation of a rolling element bearing, rn When the bearing is run at an extremely high ball m rotational speed, contact between the balls and the mb outer race is maintained, while there is intermittent M contact of the balls with the inner race for very brief Mb periods of time.N Introduction of a lubricant is found to dampen the oscillations considerably.For higher lubricant / viscosity, the oscillations exhibit an almost steady-Pf state behavior after a short time interval.With an increase in lubricant viscosity, the decay is observed qk to be quite rapid.Young's modulus of elasticity acceleration due to gravity stiffness of spring between inner race and the i-th element stiffness of spring 2 between outer race and the ith element coefficient in Hertz equation for defor- mation in point contact length or the major axis of the elliptical contact region undeformed length ofspring (length at equilibrium) ball/inner race contact undeformed length of spring 2 (length at equilibrium) ball/outer race contact Poisson ratio mass of the inner ring mass of the outer ring inner ring center of mass outer ring center of mass number of rolling elements in the bearing oil viscosity dissipation energy term in the Lagrange equation kth generalized coordinate in the Lagrange equation generalized force in the Lagrange equation external radius of inner ring internal radius of outer ring position vector of the inner ring center of mass position vector of the outer ring center of mass position vector of the outer ring center of mass with respect to the inner ring center of mass total kinetic energy of the bearing total potential energy of the bearing vector magnitude defining length of spring 2 x-displacement, velocity, and accelera- tion of the inner ring center of mass Ya, ))a, j)a Xb, -b,-b Yb, b, j)b Ox Pi, bi y-displacement, velocity, and accelera- tion of the inner ring center of mass x-displacement, velocity, and accelera- tion of the outer ring center of mass y-displacement, velocity, and accelera- tion of the outer ring center of mass coefficient in Hertz equation for defor- mation in point contact spin displacement of the ith rolling element angular displacement and velocity of the inner ring angular displacement and velocity of the cage orientation of spring 2, representing contact stiffness between the ball and the outer ring radial position and velocity of the ith rolling element

E EN NE ER RG GY Y M MA AT TE ER RI IA AL LS S Materials Science & Engineering for Energy Systems
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FIGURESchematic drawing of the rolling element bearing.

FIGURE 2
FIGURE 2 Normalized radial displacement of ball and the x and y displacement of outer ring center interference checked: Left column is ball motion, right column is outer ring motion, top row corresponds to no preload, middle row preload of 10 N and bottom row preload of N (m x-displacement, --y-displacement).

FIGURE 5 FIGURE 6
FIGURE 5 Normalized radial displacement of ball and the x and y displacement of the outer ring: Left column is ball motion, right column is outer ring motion (viscosity 0.034 kg/m s) (m x-displacement,--y-displacement).

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TABLE
Geometric and physical parameters of ball bearing 6204