Handbook of Mechanical Engineering Calculations P22 ppt

SECTION 22 BEARING DESIGN AND SELECTION Determining Stresses, Loading, Bending Moments, and Spring Rate in Spoked Journal Bearing Frictional Horsepower Loss During Operation 22.21 Journa

FIGURE 1 Shaft and 6-spoked bearing system

hav-ing three rotor masses (Product Engineerhav-ing.)

SECTION 22 BEARING DESIGN AND

SELECTION

Determining Stresses, Loading, Bending

Moments, and Spring Rate in Spoked

Journal Bearing Frictional Horsepower

Loss During Operation 22.21

Journal Bearing Operation Analysis

Load Capacity of Gas Bearings 22.46

DETERMINING STRESSES, LOADING, BENDING

MOMENTS, AND SPRING RATE IN SPOKED

BEARING SUPPORTS

Spoked bearing supports are used in gas turbines, large air-cooling fans, motor casings slotted for air circulation, and a variety of other applications For theshaft and 6-spoked bearing system in Fig 1 having three rotor masses and theseparameters and symbols,

A⫽ 7 in (for ring also)

C S⫽ distance, neutral axis to extreme fiber (of spoke), in (cm)

C R⫽ distance, neutral axis to extreme fiber (of ring), in (cm)

E R , E S⫽ elasticity moduli (ring and spoke), psi (kPa)

R R1⫽ axial loading in inclined spokes, lb; (⫹ for upper two, ⫺ for lower

two)

(N)

F R2⫽ axial loading in vertical spokes, lb; (⫹ for top, ⫺ for bottom) (N)

F T⫽ tangential load at OD of inclined spokes, lb; (clockwise on left side,

counterclockwise on right side)

M⫽ max bending moment (6-spoked support) in outer ring at OD of

vertical spokes, in-lb; ( ⫹ at inner-fiber upper point and outer-fiber

T⫽ axial loading in outer ring at OD of vertical spokes in 6-spoked

support; all spokes in 4-spoked support (⫺ at upper points, ⫹ at

bending moment in the spoke at the hub, and (f) the spring rate of the spokedbearing support Use the free-body diagram, Fig 3, to analyze this bearing support

FIGURE 2 Typical 6-spoke bearing support having the mount

load at the top for an aircraft gas turbine; in a stationary plant, mount

load would be at the bottom of the support (Product Engineering.)

2. Determine the axial loading in the outer ring at the outside diameter (OD)

Find T, the axial loading from Fig 4 for the 144 parameter as

10T

⫽1.55

P

Substituting,

FIGURE 3 Free-body diagram for 6- and 4-spoke bearing

supports (Product Engineering.)

1.55P (1.55)(10,000)

T⫽ ⫽ ⫽1550 lb (6894 N)

3. Compute the total stress in the outer ring at the top vertical spoke

Use the relation

A (0.3) 7

4. Find the total axial loading in one of the inclined spokes

Using the F R1curve in Fig 4 for the same parameter, 144,

FIGURE 4 Six-spoke bearing-support parameter curves (Product Engineering.)

5. Determine the bending moment in the spoke at the hub

The bending moment in the spoke at the hub is (F T )(L)⫽(1470)(10)⫽14,700

6. Find the spring rate of the spoked bearing support

For the 144 parameter,

Related Calculations. Figure 5 gives values for 4-spoke bearing supports Use

it in the same way that Fig 4 was used in this procedure

With more jet aircraft being built, an increase in the use of aero-derivative gasturbines in central-station and industrial power plants, wider use of air conditioningthroughout the world, and construction of larger and larger electric motors, the

FIGURE 5 Four-spoke bearing-support parameter curves (Product Engineering.)

spoked bearing support is gaining greater attention The procedure presented herecan be used for any of these applications, plus many related ones

In calculations for spoked bearings the rings are assumed supported by idally varying tangential skin-shear reactions Spokes are also assumed pinned tothe ring but rigidly attached to the hub

sinuso-This procedure is the work of Lawrence Berko, Supervising Design Engineer,

Walter Kide & Co., as reported in Product Engineering magazine SI values were

added by the handbook editor

HYDRODYNAMIC EQUATIONS FOR BEARING

DESIGN CALCULATIONS

Bearing design also requires a number of hydrodynamic formulas involving draulics, fluid flow, power, pressure head, torque, fluid viscosity, and fluid density.Table 1 presents useful formulas for the hydrodynamic design of bearings in bothUSCS and SI units

FIGURE 6 Gear loads on a typical gear (Product neering.)

Engi-GRAPHIC COMPUTATION OF BEARING LOADS

ON GEARED SHAFTS

Geared shafts having loads and reactions as shown in Fig 6 are arranged as shown

in Fig 7 The physical characteristics of the gears are:

Pressure angle of all gears, ␾ ⫽ 20 deg

Determine the bearing loads resulting from gear action using the total force directly.Use a combined numerical and graphical solution

Calculation Procedure:

1. Find the load on each gear in the set

The tangential forces, F T on the driver, Fig 6, is F T ⫽ 2T / D, where T⫽torque

transmitted by the gear, lb / in (Nm); D⫽gear pitch diameter, in (cm) Substituting

SI Values

1148 lb-in (129.7 Nm)

608 lb-in (68.7 Nm)

621 lb-in (70.2 Nm) 145.5 lb-in (16.4 Nm)

FIGURE 8 Couple vector diagram for the gear set in Fig 7 (Product Engineering.)

TABLE 2 Bearing Loads on Geared Shafts

with the given torque on the driver of 100 lb / in (11.3 Nm), we have, F T ⫽

2(100) / 1.75⫽ 114 lb (508 N) Then, the torque on the center shaft ⫽ D(F T) ⫽

2(114)⫽228 lb / in (25.8 Nm)

The gear loads are computed from F A⫽(percent torque on shaft)(2)(torque on

center shaft, lb / in)(sec of pressure angle on the gear) / D, where F A⫽gear load, lb

(N), on gear A, Fig 7 Substituting using the given and computed values, F A ⫽

0.4 (2)(228)(sec 20⬚) / 2 ⫽ 97 lb (431.5 N) Likewise, for gear B using a similar

relation, F B⫽0.6 (2)(228)(sec 20⬚) / 1.5⫽195 lb (867.4 N) Further, F C, for gear

C⫽114 (sec 20⬚)⫽121.5 lb (540 N)

2. Collect the needed data to prepare the graphical solution

Assemble the data as shown in Table 2

3. Prepare the couple diagram, Fig 8

When all the data are collected, draw the couple diagram, Fig 8 When drawingthis couple diagram it is important to note that: (a) Vectors representing negativecouples are drawn in the same direction but in opposite sense to the forces causingthem; (b) The direction of the closing should be such as to make the sum of all

FIGURE 9 Force vector diagram for the gear set in Fig 7 (Product Engineering.)

couples equal to zero Thus, the direction of P IIis the direction of bearing reaction.The bearing load has the same direction but is of opposite sense

In Fig 8, the vector P II measures 1149 lb / in (129.7 Nm) to scale Therefore,

the reaction on the bearing II is P II⫽1148 / 7⫽164 lb (729.7 N) at 11.5⬚

4. Construct the force vector diagram

The value of P IIfound in step 3 is now used to construct the force vector diagram

of forces acting at point X, Fig 9 Drawing the closing line gives the value and

direction of the reaction on bearing I The force vectors are drawn in the usual way

in their respective directions and sense Then, the loading on bearing I is P I⫽184

lb (818.4 N) at 27.5⬚

Related Calculations. The principles used in this procedure to obtain loads onbearings supporting the center shaft in Fig 7 can be applied to obtain loads onbearings carrying shafts with any number of gears It is limited, however, to thosecases which are statistically determinate

Bending-moment and shear diagrams can now be constructed since the tude and direction of all forces acting on the shaft are known To calculate thebearing loads resulting from gear action, both the magnitude and direction of thetooth reaction must be known This reaction is the force at the pitch circle exerted

magni-by the tooth in the direction perpendicular to, and away from the tooth surface.Thus, the tooth reaction of a gear is always in the same general direction as itsmotion

Most techniques for evaluating bearing loads separate the total force acting onthe gear into tangential and separating components This tends to complicate thesolution The method given in this procedure uses the total force directly

Since a force can be replaced by an equal force acting at a different point, plus

a couple, the total gear force can be considered as acting at the intersection of theshaft centerline and a line passing through the mid-face of the gear, if the appro-

SI Values

1600 lb-in (180.8 Nm) 1.625 in (4.128 cm

8 in (20.3 cm) 1.00 in (2.54 cm) 3.00 in (7.62 cm) 4.00 in (10.16 cm) 3.50 in (8.89 cm)

2100 lb (9341 N)

1970 lb (8763 N)

FIGURE 10 Sheave drive having bearing side loads (Product Engineering.)

priate couple is included For example, in Fig 7, the total force on gear B is

equivalent to a force F B applied at point X plus the couple b⫻F B In establishingthe couples for the other gears, a sign convention must be adopted to distinguishclockwise and counterclockwise moments

If a vector diagram is now drawn for all couples acting on the shaft, the closingline will be equal (to scale) to the couple resulting from the reaction at bearing II.Knowing the distance between the two bearings, the load on bearing II can befound, the direction being the same as that of the couple caused by it

The load on bearing I is found in the same manner by drawing force vector

diagram for all the forces acting at X, including the load on bearing II found from

the couple diagram

This procedure is the work of Zbigniew Jania, Project Engineer, Ford Motor

Company, as reported in Product Engineering SI values were added by the

FIGURE 11 Load diagram for sheave drive shaft (Product Engineering.)

1600 lb / in (181 Nm) The tangential and separating forces at the point of toothcontact are 1970 lb (8763 N) and 718 lb (3194 N) respectively Use numerical andgraphical analysis methods

Calculation Procedure:

1. Construct a vector diagram to determine the side forces on the bearings

Draw the vector diagram in Fig 10 to combine the tangential and separating forces

of 1970 lb (8763 N) and 718 lb (3195 N) vectorially to give a resultant R of 2100

lb (9341 N) This resultant, R, can be replaced by a parallel force, R⬘, of equalmagnitude at the shaft centerline and a moment of 1970⫻4 in⫽7880 lb / in (890Nm) in a plane normal to the shaft centerline

Construct a load diagram, of the shaft, Fig 11 Taking moments, we find thatthe resultant loads at the bearings are: Bearing A⫽1575 lb (7006 N); Bearing B

⫽525 lb (2335 N) Next, each bearing load condition must be treated separately

2. Use the fictitious-force approach to find the bearing load conditions

The key to the solution is to replace R⬘ with fictitious forces in the plane of theoutput sheave, Fig 10, of such magnitudes that the bearing loads are unchanged

Figures 12a and 12b show the procedure and the necessary forces with their

directions of application These fictitious forces, if plotted to some convenient scale

on polar coordinate paper with the leading edge at the origin, will represent thebearing loads resulting from gear action

If the side load is applied in the same direction as fictitious force R⬘A, it will belimited by the allowable bearing capacity of bearing A Since the maximum per-missible radial load is given as 2100 lb (9341 N) by the bearing manufacturer, the

resultant radial force, R⬘A, acting in the plane of the output sheave must not exceed

2400 lb (10,675 N), since from Fig 12a: (1575⫹ 525)4⫽(1800⫹ 600)3.5 Ofthis 2400 lb (10,675 N), the fictitious force of 1800 lb (8006 N) must be considered

a definite component The actual applied load would then be limited to 600 lb(2669 N) (⫽ 2400⫺ 1800), in the direction of R⬘A

FIGURE 12 Load diagram for sheave dive shaft using fictitious forces (Product Engineering.)

If applied in the opposite direction, the limiting side load becomes 4200 lb(18,682 N) (⫽2400⫹1800), since it acts to nullify rather than to complement thefictitious force in producing bearing load Recalling the principles of polar dia-grams, it is now apparent that a circle of 2400 lb (10,675 N) equivalent radius,

R⬘AL, would result in a polar diagram, Fig 13, of permissible side loading for thedesign life of Bearing A The validity of this solution and operation is understood

by noting that the equivalent radius is always the resultant of the fictitious force,

R⬘Aand the rotating vector which represents the applied side load

3. Analyze the other bearing in the system using a similar procedure

We will repeat the same procedure for Bearing B, Fig 10, which has a design life(either assumed or given by the manufacturer) compatible with the 3750 lb (16,680N) maximum rated load When superimposed on one another, these diagrams give

a confined realm of loading (k-l-m-n), Fig 13.

Using the chosen scale, it is now possible to conveniently measure load tations in any direction When dividing the torque transmitted at the sheave, namely

limi-7880 lb / in (890 Nm), Fig 10, by any maximum indicated side load, the sponding minimum sheave diameter is obtained

corre-4. Analyze the shaft stresses based on desired bearing life

Now that the side-load limitations, based on desired bearing life, have been lished, the imposed shaft stresses can be analyzed Consider the stress to be criticalunder Bearing B For safe loading limit, the maximum equivalent bending will betaken as 7200 lb / in (814 Nm) The equation for combined bending and torsion is:

SI Values

2400 lb (10,675 N)

280 lb (1245 N)

1800 lb (8006 N) (2.54 cm = 5338 N)

FIGURE 13 Polar diagram of sheave drive showing side forces (Product Engineering.)

FIGURE 14 Operating condition of a loaded journal bearing.

7,200⫽ ⁄2M⫹ ⁄2兹M ⫹7,880The maximum allowable bending moment is then 5000 lb / in (565 Nm) At the

specified distance of overhang, the side load, Q, caused by chain or belt pull is

therefore limited by the shaft strength of 1430 lb (6361 N) in any direction Thislimitation is shown on the polar diagram, Fig 13, by a circle with its center at theorigin and a radius equivalent to 1430 lb (6361 N) Study of the polar diagram,Fig 10, shows that Bearing B is no longer a limiting factor since it completelyencloses the now smaller region of safe loading

Related Calculations. When applying belt and chain drives to geared powertransmitting devices, excessive side loads are a frequent source of trouble in thebearings Improper selection of sheaves and arrangement of connections can result

in poor bearing life and early shaft breakage A thorough analysis of bearing loadsand shaft stresses is necessary if these troubles are to be avoided

Use of polar diagrams provides a unique method of determining the maximumpermissible side loading and the minimum size sheaves on an overhung shaft wheregear loading is present This type of analysis gives a graphical representation oflimiting side loads with relation to their corresponding directions of pull In essence

it becomes a permanent data sheet for an individually designed unit to which futurereference is readily available The approach given here is valid for geared drives infactories, waterworks installations, marine applications, air conditioning and ven-tilation, etc., wherever an overhung shaft is used

This procedure is the work of Richard J Derks, Assistant Professor of ical Engineering, University of Notre Dame, formerly Design Engineer, Twin Disc

Mechan-Clutch Company, as reported in Product Engineering.

JOURNAL BEARING FRICTIONAL HORSEPOWER

LOSS DURING OPERATION

An 8-in (20.3-cm) diameter journal bearing is designed for 140-degree optimumoil-film pressure distribution, Fig 14, when the journal length is 9 in (22.9 cm),shaft rotative speed is 1800 rpm, and the total load is 20,000 lb (9080 kg) What

is the frictional horsepower loss when this journal bearing operates under statedconditions with oil of optimum viscosity?

Calculation Procedure:

1. Find the journal rubbing speed

The rubbing speed of a journal bearing is given by V r⫽␲(D)(rpm), where D⫽

journal diameter, ft (m) Substituting, V r ⫽ 3.1416(8.12)(1800) ⫽ 3770 ft / min(1149 m / min)

2. Compute the rubbing surface area

The rubbing surface area of a journal bearing, A r ⫽ (d / 2)(␣)(␲)(L), where d ⫽

bearing diameter, in;␣ ⫽optimum bearing angle, degrees; L⫽bearing length, in

Substituting, A r⫽ (8 / 2)(140 / 180)(␲)(9)⫽87.96 in2(567.5 cm2)

3. Calculate the bearing pressure

With a total load of 20,000 lb (9080 kg), the bearing pressure⫽total load / bearingrubbing surface area, or 20,000 / 87.96⫽ 227.4 lb / in2(1567 kPa)

4. Find the frictional horsepower (kW) loss

Use the relation ƒhl ⫽␮(P)(V r) / 33,000, where ƒhl⫽frictional hp (kW) loss;␮ ⫽

coefficient of friction for the journal bearing; P⫽total load on the bearing, lb (kg);

V r⫽ rubbing speed, ft / min

Using standard engineering handbooks, the coefficient of friction can be found

to be 0.002 for a journal bearing having the computed rubbing speed and the givenbearing pressure Substituting,

ƒhl⫽(0.002)(20,000)(377) / 33,000⫽4.569 hp (3.4 kW)

Related Calculations. Use this general procedure to find the frictional powerloss in journal bearings serving motors, engines, pumps, compressors, and similarequipment Data on the coefficient of friction is available in standard handbooks.Figure 15 shows a typical plot of the coefficient of friction for journal bearings.The coefficient of friction chose for a well-designed journal bearing is that for fullfluid film lubrication

JOURNAL BEARING OPERATION ANALYSIS

A 2.5-in (6.36-cm) diameter journal bearing is subjected to a load of 1000 lb (454kg) while the shaft it supports is rotating at 200 rpm If the coefficient of friction

is 0.02 and L / D⫽3.0, find (a) bearing projected area, (b) pressure on bearing, (c)total work, (d) work of friction, (e) total heat generated, and (f) heat generated perminute per unit area

Calculation Procedure:

1. Determine the bearing dimensions and projected area

The length / diameter ratio, L / D⫽3.0; hence L⫽3D⫽3⫻2.25⫽6.75 in (17.15cm) (a) Then the projected area⫽L⫻D⫽6.75⫻2.5⫽15.19 in2(97.99 cm2)

2. Find the pressure on the bearing and the total work

(b) The pressure on the bearing⫽total load on bearing, P lb / (projected area, in2)

⫽ 1000 / 15.19 ⫽ 65.8 lb / in2 (453.6 kPa) (c) The total friction work transmitted

FIGURE 15 Various zones of possible lubrication for a journal bearing.

by the bearing, W ⫽ friction factor (P)(␲)(bearing diameter, in / 12 in / ft)(rpm)

Substituting, W⫽0.02(1000)(␲)(2.25 / 12)(200)⫽2356 ft-lb / min (53.2 W)

3. Compute the work of friction and total heat generated

(d) The work of friction, w ⫽ W / LD, where the symbols are as defined earlier

Substituting, w⫽ 2356 / 15.19⫽ 155.1 ft-lb / min (3.5 W) (e) The total heat erated⫽Q ⫽W / 778⫽ 155.1 / 778⫽3.03 Btu / min (3.19 kJ / min)

gen-4. Find the heat generated per unit area

The heat generated per unit area, q⫽ w / 778 ⫽155.1 / 778 ⫽ 0.199 Btu / in2min(0.0325 kJ / cm2min)

Related Calculations. This procedure shows that the analysis of journal ings is a simple task requiring knowledge only of the physical dimensions or ratios

bear-of the bearing, the coefficient bear-of friction and the shaft rpm Using these data, anyjournal bearing can be designed for the mechanical requirements of the machine orstructure

BEARING-TYPE SELECTION FOR A KNOWN

LOAD

Choose a suitable bearing for a 3-in (7.6-cm) diameter 100-r / min shaft carrying atotal radial load of 12,000 lb (53,379 N) A reasonable degree of shaft misalignmentmust be allowed by the bearing Quiet operation of the shaft is desired Lubricationwill be intermittent

Calculation Procedure:

1. Analyze the desired characteristics of the bearing

Two major types of bearings are available to the designer, rolling and sliding Rolling bearings are of two types, ball and roller Sliding bearings are also of two types, journal for radial loads and thrust for axial loads only or for combined axial

and radial loads Table 3 shows the principal characteristics of rolling and slidingbearings Based on the data in Table 3, a sliding bearing would be suitable for this

application because it has a fair misalignment tolerance and a quiet noise level.

Both factors are key considerations in the bearing choice

2. Choose the bearing materials

Table 4 shows that a porous-bronze bearing, suitable for intermittent lubrication,can carry a maximum pressure load of 4000 lb / in2(27,580.0 kPa) at a maximum

shaft speed of 1500 ft / min (7.62 m / s) By using the relation l⫽L / (Pd), where l

⫽ bearing length, in, L⫽ load, lb, d⫽ shaft diameter, in, the required length of

this sleeve bearing is l⫽L / (Pd)⫽ 12,000 / [(4000)(3)]⫽1 in (2.5 cm)

Compute the shaft surface speed V ft / min from V⫽␲dR / 12, where d⫽shaft

diameter, in; R⫽shaft rpm Thus, V⫽␲(3)(100) / 12⫽78.4 ft / min (0.4 m / s).With the shaft speed known, the PV, or pressure-velocity, value of the bearingcan be computed For this bearing, with an operating pressure of 4000 lb / in2

(27,580.0 kPa), PV⫽4000⫻78.4 ⫽313,600 (lb / in2) (ft / min) (10,984.3 kPa䡠m/ s) This is considerably in excess of the PV limit of 50,000 (lb / in2) (ft / min)(1751.3 kPa䡠m / s) listed in Table 4 To come within the recommended PV limit,the operating pressure of the bearing must be reduced

Assume an operating pressure of 600 lb / in2(4137.0 kPa) Then l ⫽L / (Pd)⫽

12,000 / [(600)(3)] ⫽6.67 in (16.9 cm), say 7 in (17.8 cm) The PV value of thebearing then is (600)(78.4)⫽47,000 (lb / in2) (ft / min) (1646.3 kPa䡠m / s) This is

a satisfactory value for a porous-bronze bearing because the recommend limit is50,000 (lb / in2) (ft / min) (1751.3 kPa䡠m / s)

3. Check the selected bearing size

The sliding bearing chosen will have a diameter somewhat in excess of 3 in (7.6cm) and a length of 7 in (17.8 cm) If this length is too great to fit in the allowablespace, another bearing material will have to be studied, by using the same proce-dure Figure 16 shows the space occupied by rolling and sliding bearings of varioustypes

Table 5 shows the load-carrying capacity and maximum operating temperaturefor oil-film journal sliding bearings that are regularly lubricated These bearings are

termed full film because they receive a supply of lubricant at regular intervals.

Surface speeds of 20,000 to 25,000 ft / min (101.6 to 127.0 m / s) are common forindustrial machines fitted with these bearings This corresponds closely to the sur-face speed for ball and roller bearings

4. Evaluate oil-film bearings

Oil-film sliding bearings are chosen by the method of the next calculation dure The bearing size is made large enough that the maximum operating temper-ature listed in Table 5 is not exceeded Table 6 lists typical design load limits foroil-film bearings in various services Figure 17 shows the typical temperature limitsfor rolling and sliding bearings made of various materials