Engineering Tribology Episode 2 Part 2 pdf

Computer Program for the Analysis of Vibrational Stability in a Partial Arc Journal Bearing An example of a computer program ‘STABILITY’ for analysis of vibrational stability in a partia

and a specific damping coefficient, e.g ‘C * xx’ is calculated according to:

developed by Hori [7] In this theory a simple disc of a mass ‘m’ mounted centrally on a shaft supported by two journal bearings is considered The disc tends to vibrate in the ‘x’ and ‘y’

directions which are both normal to the shaft axis The configuration is shown in Figure 5.30

m

k 2

k 2

Combined shaft stiffness =k

Oscillation

Rotating mass e.g turbine rotor

FIGURE 5.30 Hori's model for journal bearing vibration analysis

There are two sources of disc deflection in this model; the shaft can bend and the twobearings are of finite stiffness which allows translation of the shaft This system was analyzed

by Newton's second law of motion to provide a series of equations relating the acceleration

of the rotor in either the ‘x’ or the ‘y’ direction to the mass of the disc, shaft and bearing

stiffnesses, and bearing damping coefficients The description of this analysis can be found in[7] The equations of motion of the disc can be solved to produce shaft trajectory but this isnot often required since the most important information resulting from the analysis is thelimiting shaft speed at the onset of bearing vibration The limiting shaft speed is derivedfrom the Routh-Hurwitz criterion which provides the following expression for the

‘threshold speed of self-excited vibration’ or the ‘critical frequency’ as it is often called:

A 1 ,A 2 ,A 5 are the dimensionless stiffness and damping products;

ωc * is the dimensionless bearing critical frequency

The bearing critical frequency is also given by:

ωc

ω* c =

COMPUTATIONAL HYDRODYNAMICS 251

where:

ωc is the angular speed of the shaft [rad/s];

g is the acceleration due to gravity [m/s2];

c is the radial clearance of the bearing [m]

and the ‘γ’ parameter is expressed by:

W

γ =

where;

W is the weight on the shaft [N];

k is the stiffness of the shaft [N/m]

Since the ‘γ’ parameter is independent of bearing geometry is must be specified beforecommencing computing of a solution to equation (5.104)

The ‘A’ terms relate to stiffness and damping coefficients in the following manner [7]:

The analysis is completed with the calculation of the non-dimensional critical frequency ‘ωc *’

Computer Program for the Analysis of Vibrational Stability in a Partial Arc Journal Bearing

An example of a computer program ‘STABILITY’ for analysis of vibrational stability in a

partial arc journal bearing is listed and described in the Appendix and its flow chart is shown

in Figure 5.31 The program computes the limits of bearing vibrational stability

The Vogelpohl equation is solved by the same method described for the program ‘PARTIAL’ Although the program ‘STABILITY’ specifically refers to partial arc bearings a similar

program could be developed for grooved bearings since the principles applied are the same

Example of the Analysis of Vibrational Stability in a Partial Arc Journal Bearing

Comprehensive tables of a perfectly aligned bearing can be found in [7] Of considerablepractical interest, however, is the effect of shaft misalignment on bearing critical frequency

The computed results of the effect of shaft misalignment on critical frequency of a 120° partial arc bearing, L/D = 1, eccentricity ratio 0.7 and dimensionless exciter mass 0.1, are shown in Figure 5.32 A mesh density of 11 rows in both the ‘x*’ and ‘y*’ directions was applied in

computation

It can be seen from Figure 5.32 that there is a decline in critical frequency with increasingmisalignment However, at extreme values of misalignment the critical frequency rises as a

result of the sharp increase in the principal stiffness coefficient ‘K x x * ’

In practical bearing systems where misalignment is inevitable, operating the bearing atspeeds very close to the critical speed as predicted from the perfectly aligned condition is not

recommended For example, if the value of radial clearance is 0.0002 [m] and g = 9.81 [m/s2]then the conversion factor from non-dimensional to real frequency according to equation(5.105) is equal to:

(g/c) 0.5 = (9.81/0.0002) 0.5 = 221.5 [Hz]

The calculated difference between the minimum dimensionless critical speed for the bearing

with a misalignment parameter of t = 0.2 and a perfectly aligned bearing is:

Eccentricityε

L /D ratio Dimensionless exciter massγMisalignment parametert

Special settings of iteration parameters?

No

Yes Acquire

parameters

Set values of

DX, DY, WX & WY

Set initial values of arc position

as bisecting minimum film thickness

A

Use preset values

Set initial zero values of

M(I,J), P(I,J) & switch function

Assign SWITCH=1 to M(I,J) nodes

Output: print KXX, KYX, KXY, KYY, CXX, CYX, CXY, CYY and dimensionless frequency limit

Store values of M(I,J) as MSAVE(I,J);

Store load etc.

Calculate new film thickness h and derivatives with # change imposed

Call subroutine to solve Vogelpohl equation

Calculate pressure field difference based on

M(I,J)/W(I,J)^1.5 old − M(I,J)/W(I,J)^1.5 new

1

Call subroutine 2 to find load components 2

Calculate stiffness coefficients

Call subroutine to solve Vogelpohl equation

Apply Routh Hurwitz criterion

Print warning

Take modulus of solution

Subroutine 1 from program Partial

Subroutine 2 from program Partial

FIGURE 5.31 Flow chart of program for the analysis of vibration stability in partial arc

bearings

1

3 4

on load capacity is shown, together with possible methods of controlling the negative effectsthese have on bearing performance The scope of numerical analysis is continually beingextended With increases in the speed of computing it may become possible to perform thesimultaneous analysis of several different effects on bearing performance, e.g the combinedeffect of heating, deformation and misalignment The finite difference method used innumerical analysis is versatile and simple to apply, but is also relatively inaccurate Newermethods of devising numerical equivalents of differential equations are being increasinglyadopted However, the fundamental principles of numerical analysis outlined in this chapterremain unaltered

REFERENCES

1 G Vogelpohl, Beitrage zur Kenntnis der Gleitlagerreibung, Ver Deutsch Ing., Forschungsheft, 1937, pp 386.

2 M.M Reddi and T.Y Chu, Finite Element Solution of the Steady-State Incompressible Lubrication Problem,

Transactions ASME, Journal of Lubrication Technology, Vol 92, 1970, pp 495-503.

3 J.F Booker and K.K Huebner, Application of Finite Element Methods to Lubrication, an Engineering

Approach, Transactions ASME, Journal of Lubrication Technology, Vol 94, 1972, pp 313-323.

4 A Cameron, Principles of Lubrication, Chapter by M.R Osborne on Computation of Reynolds' Equation, Longmans, London, 1966, pp 426-439.

5 A.A Raimondi and J Boyd, A Solution for the Finite Journal Bearing and its Application to Analysis and

Design, ASLE Transactions, Vol 1, 1958, pp 159-209.

6 A Cameron, Principles of Lubrication, Longmans, London, 1966, pp 305-340.

7 T Someya (editor), Journal-Bearing Data-Book, Springer Verlag, Berlin, Heidelberg, 1989.

COMPUTATIONAL HYDRODYNAMICS 255

8 A.J Colynuck and J.B Medley, Comparison of Two Finite Difference Methods for the Numerical Analysis of

Thermohydrodynamic Lubrication, Tribology Transactions, Vol 32, 1989, pp 346-356.

9 C.M.Mc Ettles, Transient Thermoelastic Effects in Fluid Film Bearings, Wear, Vol 79, 1982, pp 53-71.

10 J.H Vohr, Prediction of the Operating Temperature of Thrust Bearings, Transactions ASME, Journal of

Lubrication Technology, Vol 103, 1981, pp 97-106.

11 S.M Rohde and K.P Oh, A Thermoelastohydrodynamic Analysis of a Finite Slider Bearing, Transactions

ASME, Journal of Lubrication Technology, Vol 97, 1975, pp 450-460.

12 H.G Elrod, A Cavitation Algorithm, Transactions ASME, Journal of Lubrication Technology, Vol 103, 1981,

Hydrostatic bearings have a wide range of characteristics and need to be carefully controlledfor optimum effect The following questions summarize the potential problems that anengineer or tribologist might confront If it is possible to generate films similar tohydrodynamic films, how can these films be controlled and produced when needed? Whatare the practical applications of this type of lubrication? What are the critical designparameters of hydrostatic bearings? What is the bearing stiffness and how can it becontrolled? The engineer or tribologist should know how to find the answer to all thesequestions.

According to available records, the first hydrostatic bearing was invented in 1851 by Girard[1,2] who employed a bearing fed by high pressure water for a system of railway propulsion.Since then there have been a number of patents and publications dealing with differentdesign aspects and incorporating various features Some of these designs introduced genuineimprovements but the majority merely introduced complexity rather than simplicity and aredestined to be forgotten

As well as the true hydrostatic bearing, hybrid bearings have also been developed These arehydrodynamic bearings assisted by an externally pressurized lubricant supply

In this chapter, the mechanism of film generation in hydrostatic bearings together withmethods of calculating basic bearing operational and design parameters are discussed.Commonly used methods of controlling the bearing stiffness are also outlined.

6.2 HYDROSTATIC BEARING ANALYSIS

The analysis of hydrostatic bearings is much simpler than the analysis of hydrodynamicbearings It is greatly simplified by the condition that the surfaces of these bearings areparallel

Flat Circular Hydrostatic Pad Bearings

Consider, as an example, a flat circular hydrostatic pad bearing with a central recess as shown

The pressure distribution can be calculated by considering the lubricant flow in a bearing For

a bearing supplied with lubricant under pressure, the flow rate given by equation (4.18)becomes:

⌠

6 ηQ

By rearranging equation (6.4), the lubricant flow, i.e the minimum amount of lubricant

required from the pump to maintain film thickness ‘h’ in a bearing, is obtained:

p r is the recess pressure [Pa];

h is the lubricant film thickness [m];

η is the lubricant dynamic viscosity [Pas];

R is the outer radius of the bearing [m];

R 0 is the radius of the recess [m];

Q is the lubricant flow [m3/s]

It can be seen that by merely substituting for flow (eq 6.5), the pressure distribution (eq 6.4) isexpressed only in terms of the recess pressure and bearing geometry, i.e.:

r dr

Since the recess pressure is constant the

expression is reduced to:

HYDROSTATIC LUBRICATION 261

T = T r + T l

where:

T is the total friction torque [Nm];

T r is the friction torque acting on the recess area [Nm];

T l is the friction torque acting on the bearing load area [Nm]

The bearing surfaces are parallel which ensures a uniform film thickness, h = constant, hence

the velocity equation (4.11) becomes:

u = U z

h

It should be noted that in this bearing the upper surface is moving while the bottom surfaceremains stationary This is the opposite situation from the hydrodynamic pad bearingdiscussed previously Thus the following conditions for velocity apply:

recess area and the other to the bearing load (land) area, i.e.:

dF = η rdθdr + η U

h r U r d θdr

where:

h is the hydrostatic film thickness in the bearing load area [m];

h r is the hydrostatic film thickness in the recess area [m]

Substituting (6.10) into (6.9) gives:

and substituting gives:

T is the torque needed to rotate the bearing [Nm]

Since in practical applications the recess depth ‘h r’ is at least 16 to 20 times the bearing film

thickness ‘h’ [2], the first term of equation (6.11) which is related to the recess area is very

small and may be neglected

· Friction Power Loss

The friction power loss which is transmitted through the operating surfaces can be calculatedfrom:

H f = T ω = 2Tπn

where:

ω is the bearing angular velocity, ω = 2πn, [rad/s];

H f is the friction power loss in the bearing [W]

Substituting for torque (eq 6.11) yields:

It has been suggested that in practice this equation gives slightly underestimated results due

to the neglect of the recess effects such as flow recirculation [2,7] To allow for these effects ithas been proposed to increase the recess component in equation (6.12) by a factor of four [2,7].Very often, however, in practical applications an allowance for these effects is made bytreating the whole bearing area as the bearing load area [2] and hence equation (6.12) becomes:

H f= 2π3 ηn 2 R 4

h

Non-Flat Circular Hydrostatic Pad Bearings

Flat hydrostatic bearings are only suitable for supporting a load normal to the plane ofcontact In some mechanical systems it is, however, very convenient to support oblique loadswhile allowing rotation and non-flat circular pad bearings are suitable for this purpose.Examples of non-flat bearings used in mechanical equipment are bearings based on a conical

or hemi-spherical shape The typical geometries of these bearings are shown in Figure 6.2

HYDROSTATIC LUBRICATION 263

Non-flat circular pad bearings can be analysed in the same manner as already discussed forflat circular pads For example, the geometry of the conical bearing is shown in Figure 6.3 andthe following analysis is applicable

FIGURE 6.2 Typical geometries of non-flat hydrostatic bearings; a) footstep, b) spherical, c)

conical, d) hydrostatic screw thread

Over the flat part of the bearing surface, hydrostatic pressure is equal to the supply pressurewhile a nearly linear decrease in pressure prevails in the conical bearing region The pressureprofile is then very similar to the flat pad bearing pressure profile already discussed In theconical bearing, pressure does show a fully asymptotic profile outside the constant pressure

region because the bearing radius ‘r’ increases at a slower rate with respect to distance

travelled by the escaping fluid The calculation procedure of bearing operating parameterssuch as pressure, load, flow and friction force is the same as for flat circular pad bearings Anallowance has to be made, however, for the bearing geometry as shown in the followingsections

· Pressure Distribution

In this bearing, the lubricant flow through an elemental ring at radius ‘r’ is given by the

equation:

p = p r ln(R /R 0 )

ln(R /r)

ln(R /R 0 )

· Friction Torque

The friction torque is calculated from equation (6.9), according to the same principles as

outlined for flat circular pad bearings The friction force in its differential form ‘dF’, also has

recess and bearing load components, i.e.:

dF = η rdθdr + η U

h

dr cosφwhere:

h is the hydrostatic film thickness in the bearing load area [m];

h r is the hydrostatic film thickness in the recess area [m]

Expressing ‘U’ in terms of revolutions per second ‘n’ and substituting into (6.9) yields:

R

r 3

cosφAssuming no change in viscosity ‘η’ and rotational velocity ‘n’ and integrating gives:

T is the torque needed to rotate the bearing [Nm]

Similarly as with flat circular pad bearings, the recess component in these bearings is usuallyvery small and may be omitted for elementary estimates of friction torque

· Friction Power Loss

The friction power loss which is transmitted through the operating surfaces for a conicalhydrostatic bearing is given by: