Theoretical Analysis of Capacitive Effect of Roller Bearings on Repeated Starts and Stops of a Machine Operating Under the

Har Prashad 1 Influence of Shaft Voltages B.H.E.L., Corporate R&D Divn.,

Hyberabad, India ^ theoretical analysis has been carried out to study the capacitive effects of roller bearings on repeated starts and stops of a machine operating under the influence of shaft voltages. The analysis gives the time required for the charge accumulation and increase of charge with time on the bearing surfaces based on bearing capaci­tance, resistance of film thickness and the shaft voltage. Also, it investigates the effect of gradual leakage of the accumulated charges with time as the shaft voltage falls as soon as the power supply to the machine is switched off. This paper gives an approach to determine ratio of the number of shaft revolutions required for charge accumulation and gradual discharge of the accumulated charges on the bearing surfaces depending on bearing to shaft voltage. Also, number of repeated starts and stops before initiation of craters on roller track of races are established to restrict the deterioration and damage of bearings.

1 Introduction A number of surveys have indicated that about 30 percent

of all motor failures are due to bearing damage on account of bearing current. Various papers have dealt with the origin of shaft voltage phenomenon in electrical machines and flow of current through rolling-element bearings and lubricated con­tacts [1 to 4]. Investigations and theoretical evaluation have been carried out for the corrugated pattern on the surfaces of a bearing [5]. Also, impedance, capacitance and charge ac­cumulation on the surfaces of a roller bearing operating under the influence of electrical current have been theoretically eval­uated [6]. The damage of a bearing by electrical wear and its effect on deterioration of lubricating greases have been studied [7-9]. Also, theoretical analysis on the effects of instantaneous charge leakage on roller tracks of roller bearings under the influence of electric current has been reported [10].

In this paper, a study is reported on the capacitive effect of roller bearings on repeated starts and stops of a machine op­erating under the influence of shaft voltages to determine the increase in charge accumulation with time and the gradual leakage of the accumulated charges on bearing surfaces as the shaft voltage falls as soon as the power supply to a machine is switched off. Under these conditions the variation of shaft revolutions to accumulate charges and discharge of the ac-

Contributed by the Tribology Division for publication in the JOURNAL OF TRIBOLOGY. Manuscript received by the Tribology Division January 31, 1991; revised manuscript received March 1992. Associate Technical Editor: B. J. Hamrock.

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cumulated charges on the roller tracks of races at various levels of bearing to shaft voltage is analysed. Also, variation of safe limits of starts and stops with the ratio of bearing to shaft voltage is studied.

2 Theoretical Analysis 2.1 Theoretical Background. Magnetic flux develops in

the electrical machines due to dis-symmetry of the magnetic circuits, which closes in the circumference over the yoke and induces the voltage on the shaft. Shaft voltage and flux can occur in the electrical rotating machines due to various reasons which usually results in a localized currents at each bearing rather than a potential difference between shaft ends [5, 11, 12]. Furthermore, under the influence of potential drop across a bearing, the varying film thickness between races and the rollers form capacitor of varying capacitance depending on the permittivity of the lubricant, and offers an impedance to current flow [6].

2.2 Time Required to Accumulate Charges on Bearing Sur­faces After Start of a Machine. At the instant when the ma­chine is started, the potential difference (V) across the inner race and rollers as well as rollers and outer race is zero. But this gradually increases and approaches the shaft voltage (E). While the shaft voltage increases, the charge on the bearing (using high resistivity lubricant 1011 ohm-cm) comprising of charges on inner and outer races (Q, and Q0) build up. Till

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then V is changing, transient current is delivered from the shaft voltage.

To maintain continuity throughout the whole circuit, there is a rate of change of flux (dQb/dT) within the dielectric (lu­bricant), which is given as [11, 13].

I^dQb/dT (1)

And, the stored charges on the bearing surfaces is determined as [6, 13].

Qb = y*Cb (2)

and if the current is varying then [11]

I=Cb*dV/dT (3)

Also, it can be expressed as

I=(E-V)/Rb (4)

So from Eqs. (1) to (4), it is evident that

V = E-Rb*Cb*dV/dT (5)

On intergrating Eq. (5) and applying the initial conditions as T = 0, V = 0, the solution of Eq. (5) is obtained as

\ = E*(l-e-T/cb-Rb) (6)

and so

Q„ = E*Cb(l-e-T/cb-Rb) (7)

The time taken (Tcb) to develop charge Qb on bearing surfaces having capacitance Cb and resistance Rb is expressed as (using Eq. (6)).

Tcb=-Cb*R„*\oge(l-a) (8)

Similarly time taken to develop charges (Qir and Qor) on a roller and roller tracks of inner as well as outer races is given as

TCjr=-Cir*Rir*\oge(l-a) (9)

7 ,< O T =-C„*/? a r *log e ( l -« ) (10)

Equivalent capacitance and resistance of inner race (C, and R,) as well as outer race (C0 and R0) with rollers ' 'K" in the loaded zone are K Cjn K Con and Rjr/K, Ror/K, respectively. Since rollers within inner and outer races are in parallel, whereas both in inner and outer races are in series to each other for the electric path. So, Tci and Tco are equal to that of Tcir and Tco„ respectively (Eqs. (9) and (10)).

2.3 Current Passing Through Bearing (4), shaft voltage is given as

E = I*Rb + Qb/Cb

Using Eq. (1) and applying initial conditions as T = E/Rb, the solution of Eq. (11) is

I=(E/Rb)*e-T/Rb,Cb

and potential drop across a bearing is given as

. V = I*Rb = E*e-T/Rb'cb

Similarly, potential difference between roller track of inner race as well as outer race and a roller is given as

-T/RivCir

From Eqs. (2) and

(11) 0, /

(12)

(13)

and

V = / * e "

V = I*e~T/Ror'Cor

(14)

(15)

The above Eqs. (14) and (15) are also valid for potential dif­ference between roller tracks of inner as well as outer races and rollers since

and

Ri — Rjr/K, R0 = Ror/K

Cj — K* Cjr, C0 — K* Col

(16)

(17)

2.4 Time Required to Discharge the Accumulated Charges From Bearing Surfaces During Stop. As soon as the power supply to a machine is switched-off, magnetic flux disappears and the shaft voltage becomes zero instantaneously. The rate of discharge of bearing capacitor (formed during bearing op­eration) is determined by differentiating Eq. (13), and given as

-dV/dT=(E/Cb*Rb)>e-T/cb'Rb (18)

At the instant, when machine is stopped (T = 0), the rate of change of bearing potential drop is determined as

dV/dT=-E/Cb*Rb (19)

In time constant (Cb * Rb), the potential drop of bearing capacitor falls and is determined as (by Eq. (13))

V = E*e~ = 0.368E (20)

The time required (Tdb) to drop potential drop V to the value of "a" times the shaft voltage (E) (V = a* E) is given as (by Eq. (13)),

Nomenclature

c c

*-\sn —

c c

c c -

d D E Fs

I K

N- N

ratio of potential difference across bearing to shaft voltage (a = V/E < 1) equivalent bearing capacitance number of cycles before initiation of craters on roller track of inner race and outer race, respectively number of cycles before the machine come to stand still condition after the power supply is put off capacitance between inner race and a roller, and outer race and a roller, respectively equivalent capacitance between inner race and rollers, and outer race and rollers, respectively diameter of roller pitch diameter shaft voltage shaft rotational frequency current passing through bearing number of rollers in the loaded zone number of contacts between inner race and a roller, and outer race and a roller, to

accumulate charges (Q,>), and (Qor), respec­tively number of shaft rotations to accumulate charges (Q,r), and (Qor), respectively

Nj, N0 = number of contacts between roller track of inner race and rollers, and outer race and rollers, to accumulate charges (Qj), and (Q0), respectively

Nm, Non = number of shaft rotations to accumulate charges (Qj), and (Q0), respectively

Nidi Nod = number of contacts between inner race and a roller, and outer race and a roller to dis­charge of the accumulated charges (Q,r), and (Qor), respectively

Nicd, Nocd = number of contacts between roller track of inner race and rollers, and outer race and rollers to discharge of the accumulated charges (Qj), and (Q0), respectively

Nid„, Nodn = number of shaft rotations to discharge of the accumulated charges (Q,>), and (Qor), respectively

Njcdn, Nocdn = number of shaft revolutions to discharge the

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Tdb=-Rb*Cb*\ogea (21)

Similarly time required to discharge the accumulated charges from roller track of inner as well as outer races and a roller/ rollers is given

Tdir = Tdi = -Rir* Cir * lOgetf (22)

Tdor = Td0 = - Ror * Cor * logea (23)

2.5 Duration of Contact Between Roller Track of Races and a Roller. Assuming that the inner race is rotating and the outer race is stationery, the duration of contact between the roller track of the inner race and a roller crossing the width of contact Wir is determined as (10)

tir =2D* Wir/-w*Fs*(3D + d)*(D-d) (24)

Similarly,

tor = 2D*Wor/Tc*Fs* (D2-d2) (25)

Also, the number of contacts between inner race and rollers, and outer race and rollers, in each shaft rotation is determined as j8,>, and j80,., respectively and given as (10).

Pir = (3D + d)*(D-d)/4d*D (26)

and,

t30r = (D2-d2)/4D*d (27)

2.5.1 Number of Shaft Rotation for the Charge Accu­mulation on Bearing Surfaces After Start of a Machine. The number of contacts between roller track of inner race as well as outer race with a roller (Nic, Noc) for the charge accumulation Qir and Qon respectively, after start of a machine can be de­termined as the ratio of time taken (T^and Tcor) to accumulate charges to the time taken for each contact between roller track of inner race (tir) and a roller, and similarly between outer race and a roller (tor). Thus, the number of contacts (Nic, Noc) can be determined as

K = Tcir/tir, and Noc = Tcor/tor (28)

By using Eqs. (9), (10), (24), and (25), the Nic and Noc are expressed as

K = -w*Fs* (3D + d)* (D-d)*Cir

*Rir*\oge(l-a)/2D* Wir (29)

and

Nomenclature (cont.)

accumulated charge (Q,), and (Q0), respec­tively

Ki, Nsso = number of starts and stops before the formation of craters on the roller track of inner race, and outer race, respectively

Qb = electric charge accumulation on bearing Qir, Qor = charge accumulation between roller track of

inner race and a roller, and outer race and a roller, respectively

Qi, Qo — charge accumulated between inner race and rollers, and outer race and rollers, respec­tively

Rb = equivalent bearing resistance Rin Ror = resistance between roller track of inner race

and a roller, and outer race and a roller, respectively

Rj, R0 = resistance between roller tracks of inner race and rollers, and outer race and rollers, respectively

tin tor = duration of each line contact between roller track of inner race and a roller, and outer race and a roller, respectively

T = time

Noc=-ir*Fs*(D2-d2)*Cor*Ror

*\oge(\-a)/2D*Wor (30)

The number of contacts (N, and Na) to accumulate charges Qi and Q0 will be the same as that of A^ and Noc, respectively (Eqs. (16) and (17)). The number of shaft rotations to accu­mulate the charges (Qir and Qor) are determined as the ratio of number of contacts between a roller and inner race, as well as outer race and a roller to the number of respective contacts in each shaft rotation (Eqs. (26) and (27)). Thus Nic„ and Nocn

are expressed as follows using Eqs. (26) to (30).

K„ = -[2ir*Fs*d*Cir*Rir*loge(l-a)]/Wir (31)

Ken = ~ [2ir * Fs * d * Cor * Ror * loge(l - a)]/ Wor (32)

Nicn and Noc„ will be equal to Nin and Non, similar to that of Nic and 7Voc as equal to N, and N0, respectively (Eqs. (16) and (17)).

2.6 Number of Shaft Rotations for the Discharge of the Accumulated Charges After the Stop of a Machine. The number of contacts between roller track of inner as well as outer races with a roller to discharge of the accumulated charges from the bearing surfaces (Nid and Nod) after the power supply to a machine is switched off, are determined similar to Eq. (28) and given as

Kd = Tdir/tin and Nod = Tdor/tor (33)

Similar to the logic discussed above, relations for number of contacts between inner race and a roller and outer race and a roller (Nid, Kd), and Nidn and Nodn are developed using Eqs. (22) to (27), and given as

Kd = - [Rir * Cir * logea/2D * Wir]

*ir*Fs*(3D + d)*(D-d) (34)

Kd = - [Ror * Cor * log,a/2D * Wor]

*-K*Fs*(D2-d2) (35)

Nidn= -2ir*Fs*d*Rir*Cir*logea/Wir (36)

and

Nod„ =-2ir*Fs*d*Ror*Cor* logea/ Wor (37)

The Nkd and Nocd are equal to Nid and Nod, and Nicd„ and Kcdn are equal to that of Nidn and Nodn, respectively (Eqs. (16) and (17)).

Tcb = time required to accumulate charge (Qb) on bearing surfaces

Tan Tcor = time taken to accumulate charges (Q,>), and {Qor), respectively

TCh Tco = time taken to accumulate charges (Qi), and (Qo), respectively

Tdir, Tdor = time required to discharge accumulated charges from roller track of inner race and a roller, and outer race and a roller, respectively

Tdb = time required to discharge of accumulated charges from bearing surfaces

Tdi, Tdo = time required to discharge accumulated charges from roller track of inner race and rollers, and outer race and rollers, respectively

V = potential drop across bearing Wjr, Wor = width of line contact on roller track of inner

race, and outer race, respectively Ay, Por = number of line contacts at one single posi­

tion on roller track of loaded zone of inner race and outer race, respectively, by a single line position on rollers in each shaft rotation

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2.7 Determination of the Ratio of Number of Contacts for Charge Accumulation to Discharge of the Accumulated Charges. The ratio of the number of contacts and shaft rev­olutions for the charge accumulation to the discharge of the accumulated charges from the inner as well as outer races is determined using Eqs. (31) to (37), and given as

1 v ten

Nid„ 'Nn, _ N,„ _

^ icdn

Nol, log e ( l -a )

Nn, logea (38)

2.8 Number of Starts and Stops before the Initiation of Craters on the Bearing Surfaces. The iVSJ, and Nsso can be determined as the ratio of the net time required to initiate craters (Csi/Fs and Cs0/Fs) to the time lapse for charge and discharge of the accumulated charges (Tcir and Tdir) in each start and stop of a machine to the number of cycles (Csp) required before the machine comes to standstill condition after the power supply to the machine is off. Thus Nssi and Nsso can be determined as

and

Nssi - - Cs/Fs * (Tcir + Tdir) * C„

N„

(39)

is* = - Cso/Fs * (Tcor + Tdor) * C„, (40)

The number of cycles Csp, before the machine comes to stand­still condition, depend on the machine inertia, friction in bear­ings etc. Various experimental investigations reveal that, in general Csp vary between 20FS to 9QFS. Using Eqs. (9), (10), (22) and (23), the Nssi and Nsso are determined as

Nssi = ~ C^/Fs * Cir * Rir * logea (\-a)*Csp

and

Nsso = - Cso/Fs * Cor * Ror * logea (l-a)*CSf

And the ratio of A S1- to Nsso is given as

Nssi/NSSo = Csi * C,r * R;r/Cs0 * Cor * Ror

(41)

(42)

(43)

3 Data Deduction The ratio Nicn/N!dn is determined for different values of

bearing to shaft voltage (a = V/E) varying from 0.1 to 0.9, and is shown in Fig. 1. The same variation is valid for Nocn/ Nodm Ni„/Nkdn and Non/Nocdn (Eq. (38)).

Also, the number of starts and stops of a machine before the initiation of craters on the bearing surfaces are determined for a = 0.1 to 0.9 using Eqs. (41) and (42). Figure 2 indicates the variation of number of starts and stops (N,s,- and Nsso) at various levels of bearing to shaft voltage (V/E) of the NU 330 bearing operating at 1200 rpm (Fs = 20 s"1) having C,>, Cor, Rir, and Ror as 57.98 (pf), 69.58 (p<j>), 38.6 X 107 ohm and 32.2 x 107 ohm, respectively [6, 10]. Csp for the machine has been experimentally determined as approximately 20 Fs. And CSj and Cso based on the temperature rise and thermal stress calculations [10, 14] under frequent start and stop regimes are approximately established as 2 x 105 cycles.

4 Results and Discussions

4.1 Time and Number of Shaft Rotations Required to Accumulate Charges on Bearing Surfaces and Discharge of the Accumulated Charges. Time required to accumulate the charges on the roller tracks of inner race and outer race depends on capacitance (C,-, C0) and resistance (7?,- and Ra). Also, this is a function of natural logarithm of the difference of shaft and bearing voltage to the shaft voltage (Eqs. (9) and (10)). Similarly, Tcb is a function of Rb, Cb, and loge (1 - a) (Eq. (8)). Similarly, time required to discharge of the accumulated charges depends on C,, Cot Rit and R0, and is a function of natural logarithm of the ratio of bearing to the shaft voltage [Eq. (22), (23)]. Also, Tdb is a function of Rb, Cb, and loge a (Eq. (21)).

Fig. 1 Variation of the ratio of shaft revolutions to accumulate and discharge of accumulated charges (Wjc„/A/W„) at various levels of bearing to shaft voltages (V/E) on roller tracks of inner and outer races of a roller bearing operating under the influence of electrical current

400

.7000001

V/E

Fig. 2 Variation of number of starts and stops of a motor before for­mation of craters on roller track of inner and outer races (Nssi and Nsso) at various levels of bearing to shaft voltages V/E of the NU330 bearing operating under the influence of electrical currents

The number of shaft rotations to accumulate and discharge of the accumulated charges depends on diameter of rolling-element, shaft rotational frequency, width of contact between roller track of inner and outer races and a rolling element, and natural logarithm of the shaft to bearing voltage, besides Cin

Con Rin and Ror (Eqs. (28) to (37)). However, the ratio of shaft revolutions to accumulate and discharge of the accu­mulated charges from inner race and outer race with a roller/ rollers is independent of all the parameters and depends only on the ratio of difference of natural logarithm of shaft and bearing voltage to the voltage across bearing, to the ratio of the natural logarithm of shaft and the bearing voltage (Eq. (38)).

The ratio of shaft rotations to accumulate and discharge of the accumulated charges (NjC„/Nirtn) increases with V/E. As the (V/E) increases from 0.1 to 0.9, the (Nic„/Nid„) increases from 0.05 to 20.91 (Fig. 1). It is evident that for the successive higher value of (a), the ratio of Nicn/Nidn is higher than for the immediate lower value of V/E (Fig. 1). The ratio of shaft rotations to accumulate/discharge of the accumulated charges by a single roller and by all the rollers on a bearing is identical (Eq. (38)). Also, complete charge accumulation take the same number of shaft revolutions equivalent to that of the charge

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accumulation by a single rolling-element and roller track of races (Eqs. (16), (17) and (31), (32)).

5.2 Number of Start and Stop Cycles Before the Initiation of Craters on the Roller Tracks of Races Due to Discharge of the Accumulated Charges. The number of starts and stops before the initiation of craters on the roller track of inner and outer races depend on the number of cycles required to initiate craters, shaft rotational frequency, respective capacitance and resistance of the races and (V/E) (Eqs. (41) and (42)). However, the ratio of Nssi and Nsso is independent to shaft voltage and shaft rotational frequency (Eq, (43)).

The number of starts and stops to initiate craters on the roller tracks decreases as the ratio of bearing to shaft voltage increases. For the NU 330 bearing number of starts and stops to initiate craters on the roller track of inner and outer races decreases from 803.63 to 463.5 as the bearing to shaft voltage (a) increases from 0.5 to 0.9 (Fig. 2).

6 Conclusions 1. Time required to accumulate and discharge of the ac­

cumulated charges on the roller track of races depends on the capacitance and resistance between the rollers and roller track of races, and depends on the natural logarithm of the ratio of bearing to shaft voltage.

2. The number of shaft rotations to accumulate and dis­charge of the accumulated charges on roller track of races depend on shaft rotational frequency, diameter of rolling ele­ment and width of contact between roller track of races and rolling-element besides capacitance and resistance of races, and ratio of bearing to shaft voltage.

3. The number of rotations required to accumulate and discharge of the charges between a roller and roller track of races of a bearing is the same as that of all rollers and roller track of races.

4. As the ratio of bearing to shaft voltage increases, the ratio of shaft rotation to accumulate and discharge of the accumulated charges increases.

5. With increase of bearing to shaft voltage, the number of starts and stops to initiate craters on the roller track of races decreases.

This analysis besides establishing the effect of capacitive

response of the bearings, is useful for transient performance analysis under the effect of shaft voltages.

Acknowledgment The author would like to thank management of Bharat Heavy

Electricals Limited, Corporate R and D, for their permission to publish/present this paper.

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10 Prashad, H., "Theoretical Analysis of the Effects of Instantaneous Charge Leakage of Roller Tracks of Roller Bearings Lubricated with High Resistivity Lubricants under the Influence of Electric Current," ASME JOURNAL OF TRI­BOLOOY, Vol. 112, Jan. 1990, pp. 37-43.

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12 Bradford, M., "Prediction of Bearing Wear due to Shaft Voltage in Elec­trical Machines," ERA Technology, Ltd., 1984.

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14 Prashad, H., "Analysis of the Effects of an Electric Current on Contact Temperature, Contact Stresses and Slip Bands Initiation on Roller Tracks of Roller Bearings," Wear, Vol. 131, 1989, pp. 1-14.

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