21.1
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
Screws
21.16
Designing a Rolling-Contact Translation
Screw
21.21
Selection of a Rigid Flange-Type Shaft
Coupling
21.30
Selection of Flexible-Coupling for a
Shaft
21.32
Selection of a Shaft Coupling for Torque
and Thrust Loads
21.34
High-Speed Power-Coupling
Characteristics
21.35

ϭ
40:1; coefficient of
slip, C
s
ϭ
0.8; and coefficient of rolling resistance, C
r
ϭ
60 lb/1000 lb (27.2 kg/
454 kg) of gross vehicle weight. The vehicle is powered by a hydrostatic trans-
mission with a 2.5-in
3
/rev (41-mL/rev) displacement pump, rated at 5000 lb / in
2
(34.5-MPa). Compare the performance produced by using a 2.5-in
3
/rev (41-mL /
rev) displacement fixed-displacement motor and a 2.5-in
3
/rev (41-mL/rev) dis-
placement variable-displacement motor with an 11-degree displacement stop. Other
pump and motor data are given on performance curves available from the pump
manufacturer.
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Source: HANDBOOK OF MECHANICAL ENGINEERING CALCULATIONS
21.2
DESIGN ENGINEERING
Calculation Procedure:

p
3
ϭ
1.31 in /rev (21.5 mL / rev)
Now we must find the pump flow from
DNE
pp
v
p
Q
ϭ
p
231
Substituting,
1.31 (2,400) (0.88)
Q
ϭ
p
231
ϭ
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, T
m
ϭ
1800 lb / in (203.3
Nm) at a motor speed of 960 rpm.
Maximum tractive effort is given by
TREn
m ƒd ƒdm

50
40
30
20
10
0
40
35
30
25
20
15
10
5
0
100
90
80
70
60
50
Volumetric efficiency, E
rv
(%)
Outlet flow, Q
p
(gpm)
Input horsepower, H
p
(hp)

5
0
100
80
60
40
20
0
Output torque, T
m
(lb-in.)
Output horsepower (hp)
Inlet flow, Q
p
(gpm)
Overall efficiency, E
moa
(%)
Motor speed, N
m
(rpm)
(b)
FIGURE 1 Continued.
SI values for Fig. 1a and 1b:
gpm L / sec lb-in. Nm
00 0 0
5 0.32 500 56.4
10 0.63 1000 112.9
15 0.95
20 1.26 psi MPa

Fig. 2 SI
lb kg mph m / sec
00
1000 454 2 0.89
2000 908 4 1.78
3000 1362 6 2.68
4000 1816 8 3.58
5000 2270 10 4.47
Nr
mL
N
ϭ
v
168 RE
ƒd ƒd
Substituting,
960 (14.5)
N
ϭ
v
168 (40)
ϭ
2.1 mi/h (0.939 m/s)
Plot these computed values as point A on the tractive-effort vs. vehicle-speed
curve, Fig. 2.
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TRANSMISSIONS, CLUTCHES, ROLLER-SCREW ACTUATORS, COUPLINGS, AND SPEED CONTROL
21.6

ϭ
4.8 mi/h (2.15 m/s)
The tractive effort at this speed is
TREn
m ƒd ƒdm
T
ϭ
emax
r
L
Substituting,
1,000 (40) (0.9)
T
ϭ
e
14.5
ϭ
2,482 lb (1127 kg)
Plot these values as point B on Fig. 2.
From the curves for the variable-displacement motor, N
m
ϭ
3580 rpm and T
m
ϭ
560 lb/in (63.2 Nm). Therefore, as before, maximum vehicle speed produced by
the variable-displacement motor is
3,580 (14.5)
N
ϭ

1,800 (14.5)
N
ϭ
v
168 (40)
ϭ
3.8 mi/h (1.698 m/s)
800 (40) (0.9)
T
ϭ
e
14.5
ϭ
2,979 lb (1352 kg)
which are plotted as point D on Fig. 2.
For the variable-speed motor, N
m
ϭ
2600 rpm and T
m
ϭ
800 lb / in (90.3 Nm).
Therefore, the vehicle speed and tractive effort are:
2,600 (14.5)
N
ϭ
v
168 (40)
ϭ
5.6 mi/h (2.5 m/s)

2,346 rpm
The maximum theoretical vehicle speed is found from
Nr
mmax L
N
ϭ
v
max
168 R
jd
Substituting,
2,346 (14.5)
N
ϭ
v
max
168 (40)
ϭ
5.1 mi/h (2.5 m/s)
which is plotted as point F on Fig. 2.
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TRANSMISSIONS, CLUTCHES, ROLLER-SCREW ACTUATORS, COUPLINGS, AND SPEED CONTROL
21.8
DESIGN ENGINEERING
For the variable-displacement motor, maximum motor speed is
2,600 (2.5) (0.95) (0.95)
N
ϭ

510 lb (231.5 kg)
Tractive effort at wheel slip is given by
T
ϭ
WC
ews
Single-path
T
ϭ
0.6 WC
ews
Dual-path
Substituting,
T
ϭ
5,150 (0.8)
e
ϭ
4,120 lb (1870 kg)
The gradability at slip is given by
T
Ϫ
R
er
Ϫ
1
G
ϭ
tan sin 100
ͫͩ ͪͬ

Ϫ
1
G
ϭ
tan sin 100
ͫͩ ͪͬ
8,500
ϭ
24%
These data are also shown on the tractive-effort curve, Fig. 2, which now gives a
complete picture of vehicle performance.
Related Calculations. The analytical technique presented here allows the hy-
drostatic transmission to be evaluated on paper, and necessary changes made before
the unit is actually built. The procedure uses a series of calculations that gradually
define transmission and vehicle data. With these data, a curve can be constructed,
Fig. 2, so that vehicle performance can be predicted for the entire operating range.
The first step in the analysis is to compare the ‘‘application values’’ of the vehicle
and the available transmissions. This comparison provides a simple way to match
vehicle requirements to transmission capabilities.
The vehicle application value expresses vehicle requirements and depends on
required vehicle speed and maximum tractive effort. For single-path applications
only one transmission is used. For dual-path applications, where two transmissions
are 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 the
total 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 transmission
capabilities and depends on motor torque and speed.
If the transmission application value is greater than that for the vehicle, the
proposed transmission is viable, and calculations to size properly the transmission
can be made. If the vehicle application value is greater, consideration must be given

Loose sand 60–150 27.2–68.1 100 45.4
Snow 25–50 9.1–22.7 — —
Units are lb / 1,000 lb (kg / 454 kg) gross vehicle weight.
formance curve for the proposed transmission must be available, usually from the
manufacturer.
The maximum tractive effort is limited by the pump relief-valve setting or wheel
slip. For multiple-path systems, maximum tractive effort must be divided by the
number of motors and multiplied by 0.6 before the comparison is made.
The final calculation required to determine whether a transmission meets vehicle
requirements is to check maximum vehicle speed. If the transmission can produce
the required tractive effort and speed, it is sized properly. However, if speed is too
low and tractive effort acceptable, consideration should be given to increasing pump
speed, 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 increasing
the final drive ratio. The resultant loss in maximum speed can be recovered by
increasing pump speed or by using a variable-displacement motor.
Once the transmission is sized to meet vehicle requirements, a mathematical
model can be generated to predict system performance. The calculations necessary
to produce the model take into account such factors as pump speed, pump and
motor 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 and
lower limits on the curve. The upper limit is the vehicle speed produced at the
maximum tractive effort; the lower limit is the tractive effort produced at maximum
vehicle speed.
Typically, four intermediate points on the performance curve are sufficient to
provide a rough approximation of vehicle performance. Six to eight points may be
required for a complete analysis. These points are calculated as shown here, except
that motor torque and speed are determined for pump displacements between max-
imum displacement and displacement at maximum tractive effort.

Nomenclature
A
t
ϭ
Transmission application value
A
r
ϭ
Vehicle application value
C
r
ϭ
Coefficient of rolling resistance, lb /1,000 lb (kg / 454 kg)
C
s
ϭ
Coefficient of slip
D
m
ϭ
Motor displacement, in
3
/rev (mL/rev)
D
p
ϭ
Pump displacement in
3
/rev (mL/rev)
D

H
p
ϭ
Pump input horsepower, hp (kW)
H
r
ϭ
Pump rated horsepower, hp (kW)
N
m
, N
p
ϭ
Motor or pump speed, rpm
N
mmax
ϭ
Maximum motor speed, rpm
N
pmax
ϭ
Maximum pump speed, rpm
N
pr
ϭ
Rated pump speed, rpm
N
v
ϭ
Vehicle speed, mi/ h (m / s)

Loaded radius, in (cm)
T
e
ϭ
Tractive effort, lb (kg)
T
emax
ϭ
Maximum tractive effort, lb (kg)
T
m
ϭ
Motor torque, lb/in (Nm)
W
g
v
ϭ
Gross vehicle weight, lb (kg)
W
w
ϭ
Weight on drive wheels, lb (kg)
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TRANSMISSIONS, CLUTCHES, ROLLER-SCREW ACTUATORS, COUPLINGS, AND SPEED CONTROL
21.12
DESIGN ENGINEERING
FIGURE 3 Basic friction clutch. Adjustable spring tension
holds the two friction surfaces together and sets the overload

torque transmitted by clutch, lb / in (Nm); N
ϭ
number of friction plates
in the clutch;
␮
ϭ
coefficient of friction for the clutch; P
ϭ
total operating force
on the clutch, lb (kg); D
ϭ
maximum space limitation, in (cm); d
ϭ
minimum
space limitation, in (cm).
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TRANSMISSIONS, CLUTCHES, ROLLER-SCREW ACTUATORS, COUPLINGS, AND SPEED CONTROL
TRANSMISSIONS, CLUTCHES, ETC.
21.13
Using a pressure valve, p, of 100 lb /in
2
(689 kPa) for long wear of this clutch
and its plates, the total operating force for this clutch will be P
ϭ
pA
ϭ
100
ϫ

ϫ
0.1
ϫ
766
ϫ
[(4
ϩ
2.5)/ 4]. Solving for
N,wefindN
ϭ
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/ in
2
(689 kPa) and reducing the
outer diameter of the plates. Since space is important in the design of this clutch,
we will determine the outer diameter required when p
ϭ
100 lb/in
2
(689 kPa) and
N
ϭ
8.
Substituting pA
ϭ
p(

0.1
ϫ
100
44
Solving the resulting cubic equation for D,wefindD
ϭ
3.90 in (9.906 cm). Solving
for P,wefindP
ϭ
704 lb (319.6 kg).
Our specifications for this clutch will be: Plates 9 (eight friction planes), hard-
ened steel, outer diameter of friction surface
ϭ
3.90 in (9.906 cm); inner diameter
friction surface
ϭ
2.50 in (6.35 cm); operating force
ϭ
704 lb (319.6 kg).
Related Calculations. Use this general procedure to choose either wet or dry
clutches. The relations given here can be applied to either type of clutch. A wet
clutch is chosen wherever the atmosphere in the clutch operating area is such that
oil or moisture are present and cannot be conveniently removed. Using a wet clutch
saves 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, or
steel plates.
This procedure is the work of Richard M. Phelan, Associate Professor of Me-

ϭ
63,000hp/R, where T
ϭ
torque, lb
⅐
in; hp
ϭ
horsepower transmitted;
R
ϭ
shaft rpm; T
ϭ
63,000(50)/ 300
ϭ
10,500 lb
⅐
in (1186.3 Nm). This is the
required starting torque capacity of the clutch.
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TRANSMISSIONS, CLUTCHES, ROLLER-SCREW ACTUATORS, COUPLINGS, AND SPEED CONTROL
TRANSMISSIONS, CLUTCHES, ETC.
21.15
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 service
factor for a single-reciprocating pump is 2.0. Hence, the total required clutch torque

TRANSMISSIONS, CLUTCHES, ROLLER-SCREW ACTUATORS, COUPLINGS, AND SPEED CONTROL
21.16
DESIGN ENGINEERING
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 the
surfaces ranges from a low of 8 lb / in
2
(55.2 kPa) to a high of 300 lb/ in
2
(2068.5
kPa).
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, and
maximum 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.
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TRANSMISSIONS, CLUTCHES, ROLLER-SCREW ACTUATORS, COUPLINGS, AND SPEED CONTROL
TRANSMISSIONS, CLUTCHES, ETC.
21.17

life in 10
6
revolutions; I
ϭ
stroke, mm (in); s
ϭ
screw lead, mm/ rev
(in/ rev); I
h
ϭ
strokes/ h; t
h
ϭ
operating hours /day; t
y
ϭ
years of service. Another
factor may be included to account for variations in load alignment, acceleration,
and lubrication.
Substituting for this roller screw,
1,200
ϫ
2
L
ϭ
(300) 16 (240) 5
rev
12
6
ϭ

ϭ
m
Ί
2
ϭ
4,532 N (1019 lb)
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TRANSMISSIONS, CLUTCHES, ROLLER-SCREW ACTUATORS, COUPLINGS, AND SPEED CONTROL
21.18
DESIGN ENGINEERING
4. Determine the L
10
life of this roller-screw actuator
The L
10
value for a roller screw is found from
3
C
6
L
ϭϫ
10 rev
ͩͪ
10
F
m
where L
10

ϫ
1000 mm /m)/ 12)
ϭ
1667 rpm is required.
Knowing the rotational speed, we can compute the nD value, where D
ϭ
nom-
inal screw diameter, mm. Substituting, nD
ϭ
1667(44)
ϭ
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 ϭ
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 life
figure 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 critical
speed and is given by:
5
0.8(392) (10 ) ad

2,259 rpm
7. Calculate the theoretical efficiency of this roller screw
Efficiency of converting rotary motion to linear motion is estimated with
s
e
ϭ
s
ϩ
Kd
o
where e
ϭ
efficiency; K
ϭ
friction angle factor, 0.0375 for heavy duty and 0.0325
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TRANSMISSIONS, CLUTCHES, ROLLER-SCREW ACTUATORS, COUPLINGS, AND SPEED CONTROL
TRANSMISSIONS, CLUTCHES, ETC.
21.19
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 cal-
culated value. Substituting,
12
e
ϭ
12
ϩ

Ί
34,000b
where b
ϭ
screw support factor from Table 6. Substituting,
4,532(1,200)
4
d
Ͼ
0
Ί
34,000(1)
d
Ͼ
3.56 mm (0.14 in.)
0
The root diameter of the selected shaft is d
o
ϭ
44 mm (1.74 in). Since this is greater
than the 3.56 mm (0.14 in), the shaft is unlikely to buckle.
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
ϭ
3.85 and b
ϭ
2,
a 36-mm (1.41-in) diameter screw with 66,000 N (14,838 lb) dynamic capacity and
the same lead will also perform adequately.
Related Calculations. Roller screws are cost-effective alternatives to ball

opposed to being lifted and repositioned, roller screws can be driven at higher
rotational linear speeds than ball screws. Nonrecirculation also means the rollers
are not subjected to cyclic stressing. This further improves fatigue life.
Load is a good parameter for starting the sizing process for a roller screw. While
the 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 mean
load is given by
33 3
FL
ϩ
FL
ϩ ⅐⅐⅐ ϩ
FL
3
11 22 nn
F
ϭ
m
Ί
L
ϩ
L
ϩ ⅐⅐⅐
L
12 n
where F
m
ϭ
constant mean load, N (lb); F
1