chapter 3 stability analysis of hydrodynamic journal...

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CHAPTER 3

STABILITY ANALYSIS OF HYDRODYNAMIC JOURNAL BEARING AND ITS

EXPERIMENTAL VERIFICATION

Journal bearings have been widely used in high-speed rotating machinery. The dynamic

coefficients of oil-film force affect the machine unbalance response and machine

stability. With some ambiguous understandings on the oil-film bearing theory, such as the

boundary conditions, cavitation and whirl phenomena, it is difficult to calculate the

dynamic coefficients accurately. Therefore both of the experimental and theoretical

investigations on the dynamic coefficients of journal bearing are indispensable.

If the shaft is considered to be rigid mass there in connections with the fluid film spring

there will be natural frequency of vibration. There is also disturbing force coming from

residual unbalance in the system. Therefore the resonant vibration will be at shaft rotation

speed called synchronous whirl and has been observed as a precession or orbiting of the

center of shaft about the center of the bearing [3].

The influence of fluid film bearing on the dynamic behavior of the journal is studied

theoretically. The stiffness of the shaft itself combined with the stiffness of bearing that

support the journal determines several forms of natural frequencies of vibration called

critical speed or threshold of whirl instability or stability of journal bearings.

The stiffness of the bearing film is non linear, but for small displacements of shaft about

the equilibrium film thickness, the film stiffness may be taken as a constant. Synchronous

speed (Stability speed) is determined theoretically for different operating conditions and

verified experimentally on Journal bearing test rig.

3.1 Pressure distribution (Theoretical Analysis) [18]

Bearing under consideration.(Journal Bearing Test Rig)

Diameter of journal( Dj ) : 39.96 mm

Length of bearing ( L ) : 40.00 mm

Clearance ( C ) : 0.185 mm

C / Rj : 0.005

Speed ( N ) : 800 rpm

Load ( W ) : 150 N

Lubricant : SAE 20W40, µ = 0.0981 Pa-sec.

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3.1.1Calculations for theoretical pressure [1]

1) Bearing Pressure

(11)

=

2) Sommerfeld Number

(

)

(

) (12)

= (108) 2

(0.0981 * 800) / (60* 0.09 * 106)

= 0.16951

From Raimondi and Boyd chart ( Refer Appendix A and B) ε = 0.5413

3) Pressure Distribution :

6 2 2 2 2 1 2 (13)

6 * [(2* Π * 800 / 60) (108 )] 2

( 0.52 * sin 10 )*( 2 + 0.52 sin10) /

[(2 +0.52 2) (1 + 0.52 * cos10)

2]

= 25.156kPa

A MATLAB program to find and plot the pressure distribution of Journal Bearing

A Matlab program is done to find pressure distribution using Reynolds equation.

Refer Appendix C for program.

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3.2 Pressure distribution (Tables and plots)

3.2.1 Pressure distribution at speed of 800rpm & load of 150N

Eccentricity ratio=0.5413

Table 3.1 Theoretical Pressure distribution at speed of 800 rpm and load of 150 N

Angle

(degree)

Pressure distribution,

(kPa)

0 0

10 25.434

20 51.231

30 77.761

40 105.401

50 134.532

60 165.521

70 198.679

80 234.164

90 271.803

100 310.770

110 349.048

120 382.622

130 404.521

140 404.223

150 368.747

160 287.193

170 158.865

180 0.0872

Fig. 3.1 Plot of pressure distribution at speed of 800rpm & load of 150N

Fig. 3.1 shows that maximum pressure of 404.521 kPa is developed at 130 degrees for a

load of 150N and speed of 800 rpm.

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Fig. 3.2 shows that maximum pressure of 639.681kPa is developed at 150 degrees for

load of 300N and speed of 800 rpm , as load is increased from 150N to 300N the

maximum pressure is increased.

Angle

(degree)


(kPa)

0 0

10 26.488

20 53.547

30 81.780

40 111.865

50 144.600

60 180.943

70 222.075

80 269.428

90 324.685

100 389.605

110 465.456

120 551.507

130 641.566

140 717.361

150 739.681

160 647.614

170 390.921

180 0.2219

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3.2.3Pressure distribution at speed of 900rpm & load of 150N






Angle

(degree)


(kPa)

0 0

10 28.191

20 56.745

30 86.025

40 116.392

50 148.197

60 181.747

70 217.251

80 254.707

90 293.701

100 333.070

110 370.378

120 401.202

130 418.391

140 411.871

150 370.104

160 284.433

170 155.878

180 0.0853

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3.2.4Pressure distribution at speed of 900rpm & load of 300N




Fig. 3.4 shows that maximum pressure of 736.494 kPa is developed at 150 degrees for

load of 300N and speed of 900 rpm , as load is increased from 150N to 300N the

maximum pressure is increased and angle of maximum pressure is shifted from 130 to

150 degrees.

Angle

(degree)


(kPa)

0 0

10 29.731

20 60.070

30 91.658

40 125.207

50 161.537

60 201.615

70 246.593

80 297.807

90 356.712

100 424.609

110 501.916

120 586.486

130 670.150

140 732.900

150 736.674

160 628.618

170 371.931

180 0.2095

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Angle

(degree)


(kPa)

0 0

10 30.774

20 61.898

30 93.718

40 126.566

50 160.744

60 196.485

70 233.879

80 272.750

90 312.439

100 351.470

110 387.059

120 414.513

130 426.721

140 414.284

150 367.199

160 278.853

170 151.565

180 0.0827

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Fig. 3.6 shows that maximum pressure of 740.635 kPa is developed at 140 degrees for

load of 300N and speed of 1000 rpm, as load is increased from 150N to 300N the

maximum pressure is increased and the angle of maximum pressure is shifted from 130 to

140 degrees.

Angle

(degree)


(kPa)

0 0

10 32.909

20 66.455

30 101.304

40 138.187

50 177.932

60 221.489

70 269.948

80 324.502

90 386.324

100 456.197

110 533.668

120 615.268

130 691.232

140 740.635

150 727.552

160 607.012

170 353.042

180 0.1976

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3.3 Determination of synchronous whirl.

Considering partial arc bearing 60º to 150 º (β = 90º)

Taking values of η and ε from Table Appendix D

Using

= 6.94 A η (14 )

= 16.374 A η (15)

= 1.848 *10-4

(1- ε) (16)

Taking values of A and η from Table (Appendix D), applied load (W = 150 N) will

produce an eccentricity of 0.52 with a minimum film thickness ho = 9.24 * 10-5

m.

Performance of bearing is investigated by taking ε from 0.1 to 0.9,corresponding values

of A and η from Table (Appendix D) , and determining values of Pavg, W and ho by

using equations 14 ,15 and 16[1]. These values are tabulated in Table 3.7

3.3.1 Performance of Journal Bearing at 800 rpm

Table 3.7 Performance of Journal Bearing at 800 rpm

Ε η A Pavg W ho

m psi kPa Lb N

0.1 0.40 0.8 2.22 15.29 5.23 23.29 1.6632*10-4

0.2 0.42 1.4 4.08 28.11 9.62 43.03 1.478*10-4

0.3 0.46 2.2 7.02 48.36 16.57 74.12 1.2936*10-4

0.4 0.47 3.2 10.43 71.86 24.16 110.13 1.1088*10-4

0.5 0.50 4.2 14.57 100.38 34.38 153.79 9.24*10-5

0.6 0.52 5.3 19.12 131.73 45.12 201.15 7.392*10-5

0.7 0.56 7.3 28.37 195.45 66.93 299.40 5.55*10-5

0.8 0.60 10.8 44.97 309.84 106.10 474.62 3.69*10-5

0.9 0.72 18.0 89.94 619.68 212.20 948.35 1.848*10-5

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Fig 3.7 Bearing Performance factors from table at 800 rpm

Stiffness:

Although the stiffness of the film in the journal bearing is sharply non-linear, it may be

taken as linear for small displacement about equilibrium position. Tangent is drawn to the

curve at the operating eccentricity ratio 0.52 and its slope is determined.

From Fig. 3.7

(17)

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= (345-0) / (9.24*10-5

- 1.848*10-5

)

= 466090 N / M

Assuming rigid body conditions

√

(18)

= 27.78 cycles /sec

=1666.8 cycles /min

From above calculations it is seen that journal bearing operating at 800 rpm and 150N

load is stable up to 1666.8 rpm.

3.3.2 Performance of Journal Bearing at 900 rpm.

= 7.182 A η (19)

=19.354 A η (20)

= 1.848 *10-4

(1- ε) (21)

Table 3.8Performance of Bearing at 900 rpm and 150 N load

Ε η A Pavg Load (W ) ho

m Psi kPa Ib N

0.1 0.40 0.8 2.501 17.23 6.194 27.07 1.6632*10-4

0.2 0.42 1.4 4.596 31.66 11.38 50.90 1.478*10-4

0.3 0.46 2.2 7.911 54.50 19.58 87.58 1.2936*10-4

0.4 0.47 3.2 11.75 80.95 29.11 130.21 1.1088*10-4

0.5 0.50 4.2 16.41 113.06 40.65 181.84 9.24*10-5

0.6 0.52 5.3 21.54 148.41 53.54 239.50 7.392*10-5

0.7 0.56 7.3 31.95 220.13 79.13 353.97 5.55*10-5

0.8 0.60 10.8 50.65 348.97 128.43 574.51 3.69*10-5

0.9 0.72 18.0 101.31 698.02 250.86 1122.1 1.848*10-5

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Fig 3.8 Bearing Performance factors from table at900 rpm

Stiffness of journal bearing is determined by following equation, From Fig 3.8

= 2661527.166 N/M


√

= 66.40 cycles/sec



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3.3.3 Performance of Bearing under consideration at 1000 rpm.

= 8.685 A η (22)

= 21.51 A η (23)

= 1.848 *10-4

(1- ε) (24)

Table 3.9 Performance of Bearing under consideration at 1000 rpm

Ε η A Pavg W ho

m Psi kPa Ib N

0.1 0.40 0.8 2.7792 19.08 6.8832 30.79 1.6632*10-4

0.2 0.42 1.4 5.1067 35.18 12.647 56.57 1.478*10-4

0.3 0.46 2.2 8.789 60.55 21.768 97.37 1.2936*10-4

0.4 0.47 3.2 13.06 89.98 32.35 144.71 1.1088*10-4

0.5 0.50 4.2 18.28 125.94 45.17 202.06 9.24*10-5

0.6 0.52 5.3 23.93 164.87 59.281 265.18 7.392*10-5

0.7 0.56 7.3 35.50 244.45 87.93 393.34 5.55*10-5

0.8 0.60 10.8 56.27 387.70 139.38 623.49 3.69*10-5

0.9 0.72 18.0 112.55 775.46 278.76 1246.9 1.848*10-5

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Fig 3.9. Bearing Performance factors from table at 1000 rpm

From Fig. 3.9

= 3138140.965 N/M


√

= 72.10 cycles/sec



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3.4 Experimental Setup

Introduction to Journal Bearing Test Rig.[18]

The Journal Bearing Test Rig is used to demonstrate the pressure distribution in a

lubricant under load condition. It also measures frictional torque.

Fig. 3.10 Journal Bearing Test Rig 60

This is a sturdy versatile machine, which facilitates study of pressure at corresponding

angular position of the pressure sensor with the load line. The JBTR equipment consists

of a vertically mounted shaft and driven by a variable speed motor. A metallic bellow

connects brass bearing at bottom and top is fixed to frictional torque load cell. Bearing

made of brass material encloses the shaft at the lower end and is immersed in an oil

sump. An rpm sensor disc is mounted on the driven pulley to measure the revolution of

the shaft per minute. A stepper motor moves the bearing in the direction of the rotation of

the shaft unto 180o in steps of 9

o. A pressure sensor is fixed on the bearing, which

Frictional force load cell

Loading lever

Oil tank

Base plate

Timer belt assembly

Pressure sensor

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measures the film pressure distributed in the oil film. Radial load is applied by dead

weights through a lever mechanism. The assembly of the shaft and the bearing is

immersed in oil so as to provide continuous lubrication at all times. The equipment is

connected to the controller, which displays the values of the angular position of pressure

sensor with reference to the load line and the corresponding pressure values. Normal

load, rotational speed can be varied to suit the test conditions. Frictional torque value can

also be displayed on the controller. Data obtained are transmitted to PC through data

acquisition cable.

The JBTR Equipment consists of Journal assembly, Bearing assembly, Loading

assembly, Lubrication system, Sensors, Controller, Data Acquisition software and cables.

Journal assembly:

Journal is housed in spindle housing supported on bearings. Spindle housing is fixed

firmly to base plate Journal is rotated by a AC motor through pulley arrangement. On top

of the spindle an rpm sensor disc is mounted to measure speed.

Bearing assembly:

Diameter and length of the brass bearing is made equal to get ratio L/D = 1. A narrow

orifice inside bearing lets oil into pressure sensor mounted area. Bearing is `mounted on

bottom of a metallic bellow and on top of bellow a frictional torque load cell is mounted.

Load cell unit is mounted on to indexing pulley. Indexing pulley is driven by stepper

motor (Fig 3.10). Angular range for stepper motor is set unto 180o, in steps of 9

o.

Loading assembly:

A lever arrangement is provided with a lever ratio of 1:5. To one end of the lever a

loading pan is attached on which the weights can be placed in the range of 150N to 750N.

To the other end is mounted a ball bearing through which the load is applied to the brass

bearing. Lever is pivoted to get 1:5 loading ratio.

Lubrication system:

An oil filling arrangement is provided on the top of base plate to gravity feed the brass

bearing by means of a flexible tube before start of test to remove any trapped air. An oil

sump is provided to ensure continuous lubrication for the journal bearing (Fig3.10). The

oil sump is placed on bottom base plat, which is fixed on to the supporting structure.

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Sensors:

Pressure sensor: A calibrated pressure sensor to measure unto 1000psi is fixed to the

bearing to measure pressure at various points around the journal (Fig. 3.10). Output is

100mv for 1000 psi.

Proximity sensor: A sensor disc rotates with spindle at the top and a proximity sensor is

fixed on base plate near it to measure the rpm of the journal

Frictional torque load cell: Frictional torque is measured by load cell mounted below

the indexing plate. During indexing the load cell also moves along. Torque increases

with increase in rpm.

Controller:

Test parameter such as disc speed can be set with the front panel settings on the

controller. The load is applied by dead weights through loading lever assembly. The

pressure and Torque readings are displayed along with indexing angle during test.

Pressure and Angle readings are processed and serially transmitted through data

acquisition cable.

The front panel of controller has:

Power ON : Switches ON cotroller

Pressure : Display pressure value in psi

Frictional torque : Display frictional torque value in Nmm

Angle : Display position of pressure switch in degrees

Speed : Display speed on journal in rpm

Test start : Starting of stepper motor

Motor start – stop : Motor switch ON & OFF

Zero buttons : Initialize pressure and frictional torque value to zero.

The back panel has :

Port’s for signal input cable & data acquisition cable MS connector for control cable and

fuse .

Data acquisition software includes

i. Data acquisition cable.

ii. One CD containing Winducom 2004 software

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Specifications of Experimental set up.

Journal Bearing test rig was used for experimentation. Refer Appendix G for

specifications.

3.4.1Experimental pressure distribution at synchronous speed

Bearing under consideration discussed in section 3.1 From theoretical calculations of

synchronous whirl for 800 rpm and 150 N load the stability speed is 1666 rpm, at this

speed the experimental analysis is carried out to plot the pressure distribution . From the

plot we can see that the pressure distribution is normal and regular up to this speed of

1666 rpm. i.e. the bearing is stable up to 1666 rpm. If the pressure distribution is not

normal it means that bearing is operating above the stability speed.

Graph 3.1 Pressure distributions at 1666 rpm and 150 N load.

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3.4.2 Theoretical pressure distribution at 1666 rpm and 150 N

Fig. 3.11 Pressure Distribution plot for 1666 rpm and 150N (Theoretical)

Experimental pressure distribution on Journal bearing test rig for bearing under

consideration (Graph 3.1) shows that maximum pressure is 52 Psi at 82.5 degrees (172.5

degrees in graph as plot starts from 90 degrees). This is in good agreement with the

theoretical value of maximum pressure. So we can conclude that the bearing is stable up to

the speed of 1666 rpm.

From theoretical calculations of synchronous whirl for 900 rpm and 150 N load the

stability speed is 3984.05 rpm, at this speed the experimental analysis on set up is not

possible as maximum operating speed of set up is limited to 2000 rpm. Similarly for 1000

rpm and 150 N load the experimental analysis on available set up is not possible.

chapter 3 stability analysis of hydrodynamic journal...

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