Formulas and Units
Transmission technical calculations – Main Formulas
Size designations and units according to the SI-units
Linear movement:
s
v=
t
s = v× t
sa =
a=
1
2
× a × ta
2
Rotation
ω = 2× π ×f
rad/s
m/s
v = ω × r = 2× π ×f × r
m/s
m
M = F× r
Nm
P = M×ω
W
&
M = J×ω
Nm
J × ω2
W=
2
Ws or J
J = m× r2
kgm2
m
v
ta
m/s2
P = F× v
W
F = m×a
N
W = F×s
Ws
W=
m × v2
2
Ws
Units used
v = velocity in m/s
F= force in N
M = mass in kg
W = work in Ws = J = Nm
P= power in W
= ang. velocity in rad./sec.
s = length in m
f = frequency in rev./sec.
t = time in sec.
r = radius in m
a = acc. in m/s
2
J = Rotational mass
moment of inertia in
kgm2
& = angular acc. in rad/s2
ω
Ma= acceleration torque in
Nm
M = torque in Nm
ta = acc. time in sec.
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Formulae for the transmission technique
Power
Rotational Movement:
Ps = M × ω
ω =
π ×n
30
Linear Movement:
W (without loss)
rad/s
P=
P=
M×n π
×
η
30
W
P=
M×n π
1
× ×
η
30 1000
kW
Torque
M = F× r
MA =
P × 9550
×η
n
P = F× v×
1
η
F× v 1
×
1000 η
W
kW
Lead screw:
M=
F× p
2000 × π × η
Nm
Toothed belt:
Nm
v = π × D× n
Nm
m=
D
z
D=
z×t
π
m/min
Units used:
M = torque in Nm
= angular velocity in rad/s
n = revolutions/min.
= efficiency (motor)
F = force in N
z = number of teeth
t = distance between teeth in mm
v = velocity in m/min
MA = delivered torque in Nm
r = radius in m
P= power in kW or W
D= diameter in m
m = module
p = pitch in mm/rev
Ps = transmitted shaft power
P= necessary motor power
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Acceleration torque
J×n π
Ma =
×
ta
30
Nm
For operation of electrical motors with gear transmission:
Ma =
J red × n π
×
ta
30
Nm
Reduction of rotational mass moment of inertia
J red
n2
1
= 2 ×J = 2 ×J
n mot
i
kgm2
Linearly moveable masses is reduced
to the number of revolutions of the
motor according to:
J red = 91.2 × m ×
v2
n
2
kgm2
mot
Rotational mass moment of inertia of a
solid cylinder:
1
J = × m × r2y
2
kgm2
Units used:
Ma = acceleration torque in Nm
J = rotational mass moment of inertia in
kgm2
Jred = reduced rotational mass moment
referred to the motor shaft in kgm2
nmot = number of revolutions of motor in
rev/min.
n = number of revolutions in rev./min.
m = mass in kg
ta = acceleration time in s
ry = outer radius of solid cylinder
v = velocity in m/s
i=
n mot
gear ratio
n
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Acceleration and deceleration time
ta =
J×n
9.55 × M a
s
Braking work
A=
Mb
J × n 2 mot
× red
Mb + ML
182.4
Ws
Necessary power for linear movement
P=
F× v
1000 × η
kW
Force at sliding friction
F = m×g×µ
N
Units used:
Ma = acceleration torque in Nm
= friction coefficient
Mb = braking torque in Nm
ML = load torque reduced to the
motorshaft in Nm
ta = acceleration time in s
J = rotational mass moment of inertia in
kgm2
Jred = reduced rotational mass moment of
inertia referred to the motor shaft
nmot = number of revolutions of motor in
rev/min.
P = power in kW
F = force in N
n = Number of revolutions in rev./min.
v = linear velocity in m/s
W = work in Ws or J
m = load in kg
g = gravity (9.81 m/s2)
= efficiency of linear movement
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Frictional force during linear movement using wheels or rails
F=
2× m×g
d
× (µ1 × + f) × µ 2 N
D
2
By approximate calculations it is often simple to use the specific running resistance R in
N/ton carriage weight by calculation of the required power.
P=
R ×q×v
1000 × η
kW
Heavier carriages on rails, roller bearings R = 70 – 100N/ton
Lighter carriages on rails, roller bearings
R = 100 – 150N/ton
Units used:
F = force in N
m = load in kg
g = gravity
D = wheel- or roller diameter in m
f = rolling friction radius
1
= bearing friction
2
= rail- or side friction
v = velocity in m/s
= efficiency
q = load in ton
d = shaft diameter in m
Rolling friction radius, f (m):
Steel against steel
Steel against wood
Hard rubber against steel
Hard rubber against concrete
Inflated rubber tire against concrete
f = 0.0003 – 0.0008
f = 0.0012
f = 0.007 – 0.02
f = 0.01 – 0.02
f = 0.004 – 0.025
Bearing, rail- and side friction:
Roller bearings 1 = 0.005
Sliding bearings 1 = 0.08 – 0.1
Roller bearings 2 = 1.6
Slide bearings 2 = 1.15
Sideguides with rollerbearings 2 = 1.1
Roller guides side friction 2 = 1.8
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SI - Units
SI basic units
For Motion Control
Symbol
Measure
Unit
m
kg
s
A
K
length
mass
time
electrical current
temperature
metre
kilogram
second
ampere
Kelvin
Designation
Measure
Unit
Symbol
a
distance
angle
angle
diameter
height
length
radius
distance
volume
linear acceleration
angular acc.
frequency
gravity
revolutions per unit
angular velocity
time constant
time
linear velocity
metre
radian
degree
metre
metre
metre
metre
metre
cubic-metre
m
rad
Newton
Newton
N
N
2
kgm
Newtonmetre
kilogram
Watt
Joule
Nm
kg
W
J
i
force
weight force
Rotational mass
moment of inertia
torque
mass
power
energy
efficiency
friction coefficient
gear ratio
I
P
R
S,Ps
U
current
active power
resistance
appearent power
voltage
Ampere
Watt
Ohm
Voltampere
Volt
A
W
,
d
h
l
r
s
V
a
&
ω
f
g
n
w
T
t
v
Mechanical
F
G
J
M
m
P
W
Electrical
Hertz
rev./min.
second
second
m
m
m
m
m
3
m
2
m/s
2
rad/s
Hz
2
m/s
1/s
rad/s
s
s
m/s
W, VA
V
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The rotating mass moment of inertia of rotating bodies
Body
Rotation
Symbol
Rotational mass
moment of inertia,
J in kgm2
m × r2
Hollow cylinder
Around own axis
Homogeneous
cylinder
Around own axis
m 2
×r
2
Thickwalled cylinder
Around own axis
m
× (r 21 + r 2 2 )
2
Disc
Around own axis
Disc
Around own plane
Sphere
Around own center
m 2
×r
2
m 2
×r
4
2× m 2
×r
5
Thinwalled sphere
Around own center
2× m 2
×r
3
Thin rod
Perpendicular around
own axis
m 2
×l
12
Steiners Equation
Rotational mass moment of inertia relative to a parallel shaft in the distance a
J = J0 + m × r 2
J0 = rotational mass
moment of inertia
relative to the center of
gravity axis
kgm2
kgm2
m = mass of the body
kg
r = shaft distance
m
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Correlation between rotational mass moment of inertia and rotating mass
J = m × rJ
2
kgm2
Units used:
J = rotational mass moment of inertia in
kgm2
m = mass in kg
rJ = inertial radius in m
The efficiency for different types of drives are often values obtained by experience:
Some normal values for parts with rollerbearings:
Drive belt with 180 force transmitting angle
Chain with 180 force transmitting angle
Toothed rod
Transporting belt with 180 transmitting angle
Wire with 180 transmitting angle
= 0.9 – 0.95
= 0.9 – 0.96
= 0.8 – 0.9
= 0.8 – 0.85
= 0.9 – 0.95
The friction values are difficult to give correctly and are dependant on surface conditions
and lubrication.
Some normal values:
Steel against steel
static friction, dry
dynamic friction, dry
static friction, viscous
dynamic friction, viscous
= 0.12 – 0.6
= 0.08 – 0.5
= 0.12 – 0.35
= 0.04 – 0.25
Wood against steel
static friction, dry
Dynamic friction, viscous
= 0.45 – 0.75
= 0.3 – 0.6
Wood against wood
Static friction, dry
Dynamic friction, viscous
= 0.4 – 0.75
= 0.3 – 0.5
Plastic against steel
static friction, dry
Dynamic friction, viscous
= 0.2 – 0.45
= 0.18 – 0.35
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The twisting torque on a shaft from a pulling force is according to the following:
M = F× r × η = F×
d0
×η
2
Nm
Efficiency,
Toothed wheel
= 0.95
Chain wheel
= 0.95
Toothed belt
= 0.80
Flat belt
= 0.40
Flat belt, pre-tensed
= 0.20
Units used:
F = cross force
M = twisting torque
do = effective diameter on
toothed wheel or chain wheel
= efficiency
r = radius in m
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