Tài liệu Handbook of Mechanical Engineering Calculations P21 docx

SECTION 21 TRANSMISSIONS, CLUTCHES, ROLLER-SCREW ACTUATORS, COUPLINGS, AND SPEEDCONTROL Constructing Mathematical Models for Analyzing Hydrostatic Transmissions 21.1 Selecting a Clutch f

SECTION 21 TRANSMISSIONS, CLUTCHES, ROLLER-SCREW ACTUATORS, COUPLINGS, AND SPEED

CONTROL

Constructing Mathematical Models for

Analyzing Hydrostatic Transmissions

21.1

Selecting a Clutch for a Given Load

21.12

Clutch Selection for Shaft Drive 21.13

Sizing Planetary Linear-Actuator Roller

Cam Clutch Selection and Analysis 21.42

Timing-Belt Drive Selection and Analysis 21.43

Geared Speed Reducer Selection and Application 21.47

Power Transmission for a Speed Drive 21.48

Variable-CONSTRUCTING MATHEMATICAL MODELS FOR

ANALYZING HYDROSTATIC TRANSMISSIONS

Construct a mathematical model of vehicle performance for a construction vehiclepowered by a hydrostatic transmission when the vehicle is driven by a 45-hp (33.6-kW) engine at 2400 rpm, with a high idle-speed of 2600 rpm The vehicle has a

supercharge pump rated at 2 hp (1.5 kW) Other vehicle data are: loaded radius, rL

⫽ 14.5 in (36.8 cm); gross vehicle weight, W g ⫽ 8500 lb (3825 kg); weight on

drive wheels, Ww⫽5150 lb (2338 kg); final drive ratio, Rfd⫽40:1; coefficient of

slip, Cs⫽0.8; and coefficient of rolling resistance, Cr⫽60 lb / 1000 lb (27.2 kg /

454 kg) of gross vehicle weight The vehicle is powered by a hydrostatic mission with a 2.5-in3/ rev (41-mL / rev) displacement pump, rated at 5000 lb / in2

trans-(34.5-MPa) Compare the performance produced by using a 2.5-in3/ rev (41-mL /rev) displacement fixed-displacement motor and a 2.5-in3/ rev (41-mL / rev) dis-placement variable-displacement motor with an 11-degree displacement stop Otherpump and motor data are given on performance curves available from the pumpmanufacturer

Calculation Procedure:

1. Determine the vehicle speed at maximum tractive effort

The theoretical pump displacement required to absorb the input horsepower fromthe engine, using the nomenclature at the end of this procedure, is:

3

⫽1.42 in / rev (23.3 mL / rev)Next, find the horsepower-limited displacement from

D N E p p vp

Q p⫽231Substituting,

1.31 (2,400) (0.88)

Q p⫽

231

⫽12 gal / min (0.76 L / s)Using the motor torque curve from the manufacturer for the pump being con-

sidered, similar to Fig 1, these data give a motor torque, Tm⫽ 1800 lb / in (203.3Nm) at a motor speed of 960 rpm

Maximum tractive effort is given by

FIGURE 1 (a) Typical pump performance curves relate input horsepower, fluid outlet flow

rate, speed, and pressure to volumetric and overall efficiency (b) Motor performance curves

relate output horsepower, fluid inlet flow rate, speed, and pressure to volumetric and overall

FIGURE 2 Performance curve for vehicle analyzed in calculation procedure

2. Determine the tractive effort at the maximum vehicle speed

From the pump performance curve obtained from the manufacturer, the maximumpump flow is 25.2 gal / min (1.59 L / s) at 2700 lb / in218.6 (MPa) and 2.5 in3/ rev(41 mL / rev) From these data, the motor torque curves for the fixed-displacement

motor give Nm⫽2240 rpm and Tm⫽1000 lb / in (112.9 Nm)

The maximum vehicle speed produced by the fixed-displacement motor is

From the curves for the variable-displacement motor, Nm ⫽ 3580 rpm and Tm

⫽560 lb / in (63.2 Nm) Therefore, as before, maximum vehicle speed produced bythe variable-displacement motor is

3,580 (14.5)

N v⫽ 168 (40)

⫽7.7 mi / h (3.44 m / s)And the tractive effort, as before, is:

560 (40) (0.9)

T e⫽

14.5

⫽1,390 lb (631 kg)Plot these values as point C on Fig 2

3. Find intermediate points on the tractive-effort vs vehicle-speed curve

To plot an intermediate point on the curve, Fig 2, a pump flow of 21 gal / min(1.325 L / s) is chosen arbitrarily For the fixed-displacement motor, this flow gives

N m⫽ 1800 rpm and Tm⫽ 1200 lb / in (135.5 Nm) Therefore, vehicle speed andtractive effort are

For the variable-speed motor, N m⫽2600 rpm and T m⫽ 800 lb / in (90.3 Nm).Therefore, the vehicle speed and tractive effort are:

4. Find the maximum theoretical speed for each motor type

The final point needed to construct the performance curve is the maximum retical speed of the vehicle for each type of motor For the fixed-displacementmotor, maximum motor speed is given by

For the variable-displacement motor, maximum motor speed is

2,600 (2.5) (0.95) (0.95)

N mmax⫽

1.5

⫽3,911 rpmand the maximum theoretical vehicle speed is

3,911 (14.5)

N vmax⫽ 169 (40)

⫽8.4 mi / h (3.75 m / s)which is plotted as point G on Fig 2

5. Refine the curve with rolling resistance and tractive effort at wheel slip

To refine the curve, rolling resistance, tractive effort at wheel slip, and gradabilitymust be determined Rolling resistance is found from

R r⫽W C gv r

Substituting,

8,900 (60)

R r⫽1,000

⫽510 lb (231.5 kg)Tractive effort at wheel slip is given by

T e⫽W C w s

Single-path

T e⫽0.6 W C w s

Dual-pathSubstituting,

T e⫽5,150 (0.8)

⫽4,120 lb (1870 kg)The gradability at slip is given by

T e⫺R r

⫺1

Substituting,

⫺1

G⫽tan冋 冉sin 8,500 冊册100

⫽47%

for the fixed-displacement motor

The maximum gradability for the variable-displacement motor, limited by 5000

hy-The vehicle application value expresses vehicle requirements and depends onrequired vehicle speed and maximum tractive effort For single-path applicationsonly one transmission is used For dual-path applications, where two transmissionsare used, the transmission on each side of the vehicle must be treated as if it were

a single-path system In such a case, a normal assumption is that 60 percent of thetotal weight on the drive wheels transfers to one side of the vehicle when it ne-gotiates a slope or turn The transmission application value expresses transmissioncapabilities and depends on motor torque and speed

If the transmission application value is greater than that for the vehicle, theproposed transmission is viable, and calculations to size properly the transmissioncan be made If the vehicle application value is greater, consideration must be given

to increasing pump speed, using variable-displacement motors, increasing pumpdisplacement, lowering maximum vehicle speed, or accepting a lower vehicle trac-tive effort

If a comparison of application values indicates that a proposed transmission isadequate, a more refined procedure must be used to size the transmission Thisensures that the transmission is applied within its horsepower rating, and that itmeets vehicle power requirements

The input power available to the transmission is the net engine flywheel power (kW) at full-load governed speed, less the horsepower (kW) required forsupercharge and auxiliary no-load losses For a transmission to operate satisfacto-rily, input horsepower (kW) must not exceed its rated horsepower (kW) In dual-path applications, power is split between two pumps and 80 percent of the totalinput horsepower (kW) should be used in this calculation Thus, up to 80 percent

horse-of the total input horsepower (kW), is assumed to be directed to one pump

Next, the transmission’s ability to produce the required tractive effort and pelling speed must be checked For these calculations, the pump and motor per-

pro-TABLE 1 Coefficients of Rolling Resistance

Units are lb / 1,000 lb (kg / 454 kg) gross vehicle weight.

formance curve for the proposed transmission must be available, usually from themanufacturer

The maximum tractive effort is limited by the pump relief-valve setting or wheelslip For multiple-path systems, maximum tractive effort must be divided by thenumber of motors and multiplied by 0.6 before the comparison is made

The final calculation required to determine whether a transmission meets vehiclerequirements is to check maximum vehicle speed If the transmission can producethe required tractive effort and speed, it is sized properly However, if speed is toolow and tractive effort acceptable, consideration should be given to increasing pumpspeed, using a variable-displacement motor, or decreasing the ratio of the final drive

If speed is acceptable but tractive effort too low, give consideration to increasingthe final drive ratio The resultant loss in maximum speed can be recovered byincreasing pump speed or by using a variable-displacement motor

Once the transmission is sized to meet vehicle requirements, a mathematicalmodel can be generated to predict system performance The calculations necessary

to produce the model take into account such factors as pump speed, pump andmotor displacement, and pump and motor efficiency The expected vehicle perform-ance is represented by a tractive-effort vs speed curve

The first two steps in generating the math model are to define the upper andlower limits on the curve The upper limit is the vehicle speed produced at themaximum tractive effort; the lower limit is the tractive effort produced at maximumvehicle speed

Typically, four intermediate points on the performance curve are sufficient toprovide a rough approximation of vehicle performance Six to eight points may berequired for a complete analysis These points are calculated as shown here, exceptthat motor torque and speed are determined for pump displacements between max-imum displacement and displacement at maximum tractive effort

To complete the analysis, a number of factors must be calculated to determinehow they affect vehicle performance One factor that must be considered is rollingresistance, the portion of tractive effort required to overcome friction and move thevehicle For the vehicle to move, available tractive effort must be greater than therolling resistance If the actual coefficient of rolling resistance is not known, thevalues in Table 1 can be used

Few vehicles operate only on level ground; so the slope or grade it can climbmust be determined This factor, called gradability, is calculated as shown above.Gradability can be determined for any point along the tractive-effort vs speed curve,

up to the slip-limited effort Gradability at wheel slip is usually the upper limit

TABLE 2 Coefficients of Slip

The final factor to be considered is the tractive effort required to slip the wheels

If the actual coefficient of slip is not known, the values in Table 2 can be used

This procedure is valid for a variety of off-the-road vehicles: tractors, draggers,bulldozers, rippers, scrapers, excavators, loaders, trenchers, hauling units, etc It is

the work of Charles Griesel, Sperry Vickers, as reported in Machine Design

Mag-azine SI values were added by the handbook editor

Nomenclature

A t⫽Transmission application value

A r⫽Vehicle application value

C r⫽Coefficient of rolling resistance, lb / 1,000 lb (kg / 454 kg)

C s⫽Coefficient of slip

D m⫽Motor displacement, in3/ rev (mL / rev)

D p⫽Pump displacement in3/ rev (mL / rev)

D pt ⫽Theoretical pump displacement, in3/ rev (mL / rev)

E ƒd ⫽Final drive efficiency, %

E poa ⫽Overall pump efficiency, %

E tm , E tp ⫽Pump or motor torque efficiency, %

E vm , E vp ⫽Pump or motor volumetric efficiency, %

G ⫽Gradability, %

H p⫽Pump input horsepower, hp (kW)

H r⫽Pump rated horsepower, hp (kW)

N mmax⫽Maximum motor speed, rpm

N pmax ⫽Maximum pump speed, rpm

N pr ⫽Rated pump speed, rpm

N v⫽Vehicle speed, mi / h (m / s)

N vmax⫽Maximum vehicle speed, mi / h (m / s)

n m⫽Number of motors

P p⫽Pump pressure, lb / in2(kPa)

Q p⫽Pump output flow, gal / min (L / s)

R ƒd ⫽Final drive output ratio

W gv ⫽Gross vehicle weight, lb (kg)

W w ⫽Weight on drive wheels, lb (kg)

FIGURE 3 Basic friction clutch Adjustable spring tension holds the two friction surfaces together and sets the overload limit As soon as the overload is removed the clutch reengages.

(Product Engineering.)

SELECTING A CLUTCH FOR A GIVEN LOAD

Choose a clutch for a lathe designed for automatic operation There will be no gearshifting in the headstock All speed changes will be made using hydraulically op-erated clutches to connect the proper gear train to the output shaft Determine thenumber of plates and the operating force required for the clutch if it is to transmit

a torque of 300 lb / in (33.9 Nm) under normal operating conditions Design theclutch to slip under 300 percent of rated torque to protect the gears and other parts

of the drive Space limitations dictate an upper limit of 4 in (10.2 cm) and a lowerlimit of 2.5 in (6.35 cm) for the diameters of the friction surfaces The clutch willoperate in an oil atmosphere

Calculation Procedure:

1. Choose the type of clutch to use

Based on the proposed application, choose a wet clutch with hardened-steel plates.(Since the clutch is operating in an oil atmosphere, use of a dry clutch could lead

to operational problems.)

2. Compute the number of friction plates needed for this clutch

Use the general clutch relation

D⫹d

T⫽N␮P

4

where T⫽torque transmitted by clutch, lb / in (Nm); N⫽number of friction plates

in the clutch;␮ ⫽coefficient of friction for the clutch; P ⫽total operating force

on the clutch, lb (kg); D ⫽ maximum space limitation, in (cm); d ⫽ minimumspace limitation, in (cm)

Using a pressure valve, p, of 100 lb / in2(689 kPa) for long wear of this clutch

and its plates, the total operating force for this clutch will be P⫽pA⫽100⫻␲

⫻[(D2⫺d2) / 4]⫽100⫻ ␲ ⫻[(42⫺2.52) / 4)]⫽766 lb (347.8 kg)

Substituting in the general clutch relation, with the torque at 300 percent ofnormal operating conditions, or 3 ⫻ 300 lb / in ⫽ 900 lb / in (101.7 Nm), and acoefficient of friction of 0.1, 900⫽N⫻0.1⫻ 766⫻[(4⫹2.5) / 4] Solving for

N, we find N⫽ 7.23 The next larger even whole number of friction plates is 8.Therefore, eight friction planes and nine plates will be specified

3. Determine if the chosen number of plates is optimum for this clutch

Once we’ve chosen the number of plates we have the option of either reducing the

operating force, P, and thus the pressure on the plates, by the ratio 7.23 / 8 or

keeping the pressure between the plates at 100 lb / in2(689 kPa) and reducing theouter diameter of the plates Since space is important in the design of this clutch,

we will determine the outer diameter required when p⫽100 lb / in2(689 kPa) and

hard-Related Calculations. Use this general procedure to choose either wet or dryclutches The relations given here can be applied to either type of clutch A wetclutch is chosen wherever the atmosphere in the clutch operating area is such thatoil or moisture are present and cannot be conveniently removed Using a wet clutchsaves the cost of seals and other devices needed to seal the clutch from the atmos-pheric moisture

Dry-plate clutches are used where there is no danger of oil or moisture getting

on the plates Most such clutches use either natural or forced convection for cooling.The drive material is a manmade composition in contact with cast iron, bronze, orsteel plates

This procedure is the work of Richard M Phelan, Associate Professor of chanical Engineering, Cornell University SI values were added by the handbookeditor

Me-CLUTCH SELECTION FOR SHAFT DRIVE

Choose a clutch to connect a 50-hp (37.30 kW) internal-combustion engine to a300-r / min single-acting reciprocating pump Determine the general dimensions ofthe clutch

TABLE 3 Clutch Characteristics

Calculation Procedure:

1. Choose the type of clutch for the load

Table 3 shows a typical applications for the major types of clutches Where omy is the prime consideration, a positive-engagement or a cone-type friction clutchwould be chosen Since a reciprocating pump runs at a slightly varying speed, acentrifugal clutch is not suitable For greater dependability, a disk or plate frictionclutch is more desirable than a cone clutch Assume that dependability is moreimportant than economy, and choose a disk-type friction clutch

econ-2. Determine the required clutch torque at starting capacity

A clutch must start its load from a stopped condition Under these circumstancesthe instantaneous torque may be two, three, or four times the running torque There-fore, the usual clutch is chosen so it has a torque capacity of at least twice therunning torque For internal-combustion engine drives, a starting torque of three tofour times the running torque is generally used Assume 3.5 time is used for this

engine and pump combination This is termed the clutch starting factor.

Since T⫽63,000hp / R, where T⫽torque, lb䡠in; hp⫽horsepower transmitted;

R ⫽ shaft rpm; T ⫽ 63,000(50) / 300 ⫽ 10,500 lb䡠in (1186.3 Nm) This is therequired starting torque capacity of the clutch

TABLE 4 Clutch Service Factors

3. Determine the total required clutch torque capacity

In addition to the clutch starting factor, a service factor is also usually applied.Table 4 lists typical clutch service factors This tabulation shows that the servicefactor for a single-reciprocating pump is 2.0 Hence, the total required clutch torquecapacity ⫽ required starting torque capacity⫻ service factor ⫽ 10,500⫻ 2.0 ⫽21,000-lb䡠in (2372.7 Nm) torque capacity

4. Choose a suitable clutch for the load

Consult a manufacturer’s engineering data sheet listing clutch torque capacities forclutches of the type chosen in step 1 of this procedure Choose a clutch having arated torque equal to or greater than that computed in step 3 Table 5 shows aportion of a typical engineering data sheet A size 6 clutch would be chosen forthis drive

Related Calculations. Use the general method given here to select clutches forindustrial, commercial, marine, automotive, tractor, and similar applications Notethat engineering data sheets often list the clutch rating in terms of torque, lb䡠in,and hp / (100 r / min)

Friction clutches depend, for their load-carrying ability, on the friction and sure between two mating surfaces Usual coefficients of friction for friction clutches

pres-TABLE 5 Clutch Ratings

FIGURE 4 Recirculating roller screw which has high positioning accuracy is

well-suited for precise work, such as refocusing lenses for laser beams (Machine Design.)

range between 0.15 and 0.50 for dry surfaces, 0.05 and 0.30 for greasy surfaces,and 0.05 and 0.25 for lubricated surfaces The allowable pressure between thesurfaces ranges from a low of 8 lb / in2(55.2 kPa) to a high of 300 lb / in2(2068.5kPa)

SIZING PLANETARY LINEAR-ACTUATOR ROLLER

SCREWS

A high-speed industrial robot requires a linear actuator with a 1.2-m (3.94-ft) stroke

to advance a load averaging 5700 N (1281 lb) at 20 m / min (65.6 ft / min) The load

to reposition the arm is 1000 N (225 lb) Positioning should be within 1 mm(0.03937 in) Find the mean load, expected life, life in million revolutions, andmaximum speed of a roller screw for this application Suggest a type of lubrication,estimate screw efficiency, and calculate the power required to drive the roller screw,Fig 4

Calculation Procedure:

1. Choose the type of screw to use

Roller screws are best suited for loads exceeding 100,000 N (22,482 lb) or speedsover 6 m / min (19.7 ft / min) A roller screw without preload has a backlash of about0.02 to 0.04 mm (0.00079 to 0.00157 in) This backlash adds to the thread inac-curacy of the roller screw Depending on the accuracy class, positioning is usuallywithin 0.1 to 0.25 mm (0.00384 to 0.0098 in) over a travel of 4000 mm (157.5 in).Preloading the screw eliminates backlash, and overall positioning accuracy is notsignificantly affected by a variable external load Because more than 18 mm / min(0.709 in) velocity is required, a roller screw will be used for this application, withsingle support bearings at each end The machine served by this actuator will make

300 load cycles / h, operate 16 h / day, 240 days / yr, and function for 5 years

2. Determine the life expectancy of this roller screw actuator

Because length and load are not excessive, a medium-duty roller screw will bechosen To accommodate linear speed, shaft speed, and life, an initial selection of

a roller screw has a 44-mm (1.73-in) diameter, a 12-mm (0.472-in) lead, and a108,200 N (24,326-lb) dynamic nut capacity, based on manufacturer’s catalog data

Manufacturers often list the dynamic load of a nut or screw for an L10 life of 1million revolutions Total life of the screw is:

l

Lrev⫽冉 冊l t t t h h d y s

where L⫽ life in 106 revolutions; I⫽stroke, mm (in); s⫽ screw lead, mm / rev

(in / rev); Ih⫽strokes / h; th⫽operating hours / day; ty⫽years of service Anotherfactor may be included to account for variations in load alignment, acceleration,and lubrication

Substituting for this roller screw,

3. Find the mean load on the screw

For the advance and retract loads, the mean load on the screw is found from:

4. Determine the L 10 life of this roller-screw actuator

The L10 value for a roller screw is found from

5. Find the rotational speed and type of lubrication needed

With a linear speed of 20 m / mm (65.6 ft / min), the rotational speed in rpm will be

l / s, where I ⫽ stroke length, mm (in); s⫽ screw lead, mm / rev (in / rev) tuting, rpm⫽n ⫽(20 m⫻1000 mm / m) / 12) ⫽1667 rpm is required

Substi-Knowing the rotational speed, we can compute the nD value, where D⫽

nom-inal screw diameter, mm Substituting, nD⫽ 1667(44)⫽ 73,348

Lubrication type defines the roller screw speed limit This limit is given by the

nD value computed above, or v D / s, where v⫽linear nut speed, mm / s (in / s); other

symbols as before Oil lubrication allows nD values as high as 140,000, while grease permits nD values to 93,000 For rolled-thread ball screws, nD values are about 64 percent of these Since this roller screw has an nD value of 73,348, grease

lubrication is acceptable

If lubrication of the roller screw is not regular or old lubricant is used, the lifefigure can be modified by a factor of 0.5 to 0.66 Further, if the lubricant is likely

to be contaminated, an adjustment factor of 0.33 to 0.5 can be used

6. Compute the maximum speed of the screw shaft

The maximum permissible speed of the screw shaft is 80 percent of the first criticalspeed and is given by:

5

0.8(392) (10 ) ad o

␻max⫽ l2

where ␻max ⫽ maximum permissible speed, rpm; a ⫽ screw support factor from

Table 3; do⫽screw shaft root diameter, mm (in); l⫽distance between centers ofscrew shaft support bearings, mm (in) Substituting,

5

0.8(392⫻10 ) 2.47(42)

␻max⫽ 1,2002

⫽2,259 rpm

7. Calculate the theoretical efficiency of this roller screw

Efficiency of converting rotary motion to linear motion is estimated with

for medium-duty designs At low loads, less than 10 percent of dynamic capacity,

e is within 10 percent of the calculated value As load increases to the dynamic

capacity, efficiency estimates are less certain, usually within 25 percent of the culated value Substituting,

cal-12

e⫽12⫹0.0325(42)

⫽0.90

8. Find the input power to the roller screw

Input power to drive the load at a given speed is found from

⫽1,679 W (2.25 hp)

9. Calculate the screw root diameter required to avoid buckling

Buckling of roller screws becomes a problem when the screw shaft is loaded incompression To avoid buckling, the screw root diameter must exceed the critical

or minimum screw diameter for the load, or

F l

4 m

d0⬎冪34,000b where b⫽screw support factor from Table 6 Substituting,

The limiting factor in this application appears to be the maximum shaft speed

If the bearings are encased at one end of the shaft, allowing a⫽3.85 and b⫽2,

a 36-mm (1.41-in) diameter screw with 66,000 N (14,838 lb) dynamic capacity andthe same lead will also perform adequately

Related Calculations. Roller screws are cost-effective alternatives to ballscrews in applications requiring high speed, long life, and high load capacity Theload-bearing advantage of roller screws lies in their contact area Ball screws trans-fer load through point contact on the balls Consequently, load is carried through

a discrete number of points Roller screws, by contrast, transmit load through tacts on each roller Unlike a ball, contact on a roller can be ground precisely for

con-a duty requirement

TABLE 6 Screw Speed and Compression Load Support Factors***

Support bearings on shaft Speed factor, a Compression factor, b

The absence of recirculation has the added benefit of allowing greater speedsthan ball screws Because the rollers are in constant contact with the screw, asopposed to being lifted and repositioned, roller screws can be driven at higherrotational linear speeds than ball screws Nonrecirculation also means the rollersare not subjected to cyclic stressing This further improves fatigue life

Load is a good parameter for starting the sizing process for a roller screw Whilethe load often fluctuates and reverses with each cycle, once a mean load is calcu-lated, a unit can be selected by using the nut’s dynamic capacity Constant meanload is given by

F L ⫹F L ⫹ 䡠 䡠 䡠 ⫹F L

3 1 1 2 2 n n

F m⫽冪 L1⫹L2⫹ 䡠 䡠 䡠L n where Fm⫽constant mean load, N (lb); F1through Fn⫽constant loads encountered

during operation, N (lb) These loads correspond to L1through Ln, which are the

number of revolutions under a particular constant load

If the loads are fairly constant as the screw advances or retracts, mean load isfound from

3 3

F ⫹F

3 a r

F m⫽冪 2

where Fa⫽load during advance, N (lb); Fr⫽load during retraction N (lb)

Lubrication of a roller screw is similar to that for a roller bearing A circulatinglubrication system is preferred, especially in hot or dirty environments Here, cooloil reduces frictional heat buildup during operation and flushes debris from thethreads Oil should have a viscosity of 100 ISO or about SAE 30 during operation.With heavy load or at low speed, an EP additive is recommended to improve filmstrength

Grease lubrication is acceptable when oil is impractical Wipers should be used

on the nut to prevent dirty grease from entering the threads The bearing greasesmost often used are NLGI 2 Lithium-base greases are generally suitable from⫺30

to 110⬚C (⫺20⬚F to 230⬚F)

Alignment and shock can severely alter roller-screw bearing life If the load is

not completely axial, the constant mean load, Fm, may be adjusted by a factor of 1.05 to 1.1 If traverse loads are applied to the screw, Fmis adjusted by a factor of1.1 to 1.5

If accelerations are low and speed variations in the application are smooth, noadjustment to the screw capacity factor is necessary However, if the speed variesrapidly, or vibrations or high-frequency oscillations are present, an adjustment factor

of 1.05 to 1.2 should be applied to Fm.

Roller screws find many applications including catapult reset after aircraft launchfrom aircraft carriers, automated arms for industrial presses, rudder control on largeaircraft, robot actuators, etc

The data and calculation in this procedure are the work of Pierre C Lemor,

Manager, Linear Components, SKF Component Systems, as reported in Machine Design magazine SI units were added by the handbook editor.

DESIGNING A ROLLING-CONTACT TRANSLATION

SCREW

A rolling-contact screw-operated translation device is being designed to raise a10,000-lb (44,480-N) load a distance of 15 in (38.1 cm) as rapidly as possible Theproposed design is shown schematically in Fig 5 Modified square threads havebeen proposed for this power screw The nut will be made of bronze; the screw ofAISI 3140 steel oil-quenched and tempered at 1000⬚F (537.8⬚C) All thrust and

guide bearings will be rolling-contact types with negligible friction Determine (a) the dimensions of the screw and nut for a factor of safety of 2; (b) the time required

to raise the load; (c) the required horsepower (kW) of the electric drive motor.

Calculation Procedure:

1. Determine the required screw diameter

Consider the screw as a column Then, the J B Johnson formula applies Checkingthread-property tables, it is found that a 1-in (2.54-cm) four-threads-per-inch (2.54cm) is the smallest modified square thread that will be satisfactory for this design

FIGURE 5 Translation device to raise load.

However, the unsupported length of this screw must also carry the screw torque,and screw strength must be checked under the combined stress of the column andtorque loads

2. Find the screw torque

10,000⫻0.875 0.9962⫻0.0909⫹0.14

2 0.9962⫺0.14⫻0.0909

3. Check screw strength under combined stress

Make the check by comparing the factor of safety under combined stress with thespecified factor of safety of 2 Or,

s / 2 y

兹(s / 2) ⫹(s ) s where sy ⫽ yield point, lb / in2 (kPa) from an AISI-3140 steel properties plot ⫽132,000 lb / in2(909.5 MPa); s⫽equivalent normal stress for the column load, lb/ in2(kPa); ss⫽ shear stress produced by the torque, lb / in2(kPa)

FIGURE 7 Translation screw geometric relations.

To compute an equivalent normal stress for the column load we will use

E⫽ 30,000 lb / in2(206,700 MPa)

Thus,

TABLE 7 Coefficients of Friction for Screw Threads and Thrust Collars*

Steel screw and bronze or cast-iron nut Thrust-collar friction

Average coefficient

of friction, ␮

Average coefficient

0.10 0.08

0.08 0.06

*After C W Ham and D G Ryan, An Experimental Investigation of the Friction of Screw Threads, Univ.

Illinois Eng Expt Sta Bull 247.

J / c⫽␲d / 16 i ⫽[␲ ⫻(0.750) ] / 16⫽0.0828Thus,

s s⫽ ⫽14,000 lb / in (96.5 MPa)0.0828

Solving for the factor of safety we find:

132,000 / 2

兹(29,700 / 2) ⫹(14,000)Therefore, since 3.23 is greater than 2, the screw has more than adequate strength

as a column