Motion Control Formulas
Terms:
A
=acceleration rate {in/sec 2}
Tc
=constant velocity torque
CT
=carriage thrust force
{oz}
Td
=deceleration torque
{in/sec 2}
t a
=acceleration time {sec}
{in/rev}
t c
=constant time
{sec}
td
=decelerate time
{sec}
D
=deceleration rate
d
=lead of screw
e
=lead screw efficiency
ball screw
90%
FT
=total frictional force
{oz}
GR
=gear ratio
JBELT
{oz-in2}
=belt/rack inertia
JLOAD =load inertia
JLS
{oz}
=lead screw inertia
JMOTOR =motor inertia
{oz-in2}
{oz-in2}
JPULLEY=pulley/pinion inertia {oz-in2}
L
=lead screw length
{inch}
LRMS
=RMS value of load
L1
=load during time period 1
L 2
=load during time period 2
t 1
=Duration of time period 1
t 2
=Duration of time period 2
ϕ
=angle from
horizontal
t e
=constant velocity
time
{sec}
TACC
=torque to
accelerate load {oz-in}
TBREAKAWAY = breakaway torque{oz-in}
TFRICTION =required torque to
overcome friction{oz-in}
TGRAVITY =required torque to
overcome gravity{oz-in}
TMOTOR
=motor output torque
at desired speed {oz-in}
TOTHER
=function of mechanical
system selected
TSYS
=required torque to
move the load {oz-in}
μ
{degrees}
=coefficient of friction
VLOAD
=maximum linear
velocity
{in/sec}
ω
=angular
velocity
ρ
=material density (lead
screw) steel 4.48 {oz/in3}
N 1
=number of teeth in gear 1
N 2
=number of teeth in gear 2
Vmax
=maximum
velocity
r
=radius of lead screw
{in}
WGEAR1
=weight of gear 1
{oz}
r 1
=radius of gear 1
{in}
WGEAR2
=weight of gear 2
{oz}
r 2
=radius of gear 2
{in}
WLOAD
=weight of load
{oz}
SF
=safety factor 1.25-2.5
recommended
T
=total move time
Ta
=acceleration torque
Vavg
=average velocity{in/sec}
WPULLEY =weight of
pulley/pinion
{sec}
WOTHER
{in/sec}
{oz}
=weight of carriage {oz}
Note: 16 oz-in = 1 lb-in
Duty Cycle Calculation
LRMS =
{rad/sec}
√
L12 × t1 + L22 × t2
t1 + t2
564
Motion Control Formulas
Torque Equations
Horizontal Applications
Ta = TACC + TFRICTION + TOTHER
Tc = TFRICTION + TOTHER
Td = TACC + TFRICTION + TOTHER
Vertical Applications
Upward Move
Ta = TACC + TFRICTION + TOTHER + TGRAVITY
Tc = TFRICTION + TOTHER + TGRAVITY
Td = TACC – TFRICTION – TOTHER – TGRAVITY
Downward Move
Ta = TACC + TFRICTION + TOTHER – TGRAVITY
Tc = TFRICTION + TOTHER – TGRAVITY
Td = TACC – TFRICTION – TOTHER + TGRAVITY
Lead Screw Equations (Linear Motion)
TSYS = [TACC + TFRICTION + TBREAKAWAY + TGRAVITY]
TACC =
ω
1 JLOAD
+ JLS + JMOTOR
ta
e
386
JLOAD =
d2(WLOAD + WOTHER)
(2π)2
πρLr 4
JLS =
2
CT = 2πTSYSd
JMOTOR = See Motor Data
ω=
2πVLOAD
d
TFRICTION =
dFT
2πe
FT = (WLOAD + WOTHER)μ
TGRAVITY =
d(WLOAD + WOTHER)Sinφ
2πe
NOTES:
1. TSYS is the torque the motor must deliver at a velocity of ω (radians/second). This usually
occurs during the acceleration portion of a move profile for horizontal applications and
an upward move for vertical applications. During the deceleration portion of a move
profile, TFRICTION and TBREAKAWAY are subtractions from TSYS . For horizontal applications
TGRAVITY has a zero value.
2. The divisor 386 in the TACC equation represents acceleration due to gravity (386 in/sec 2
or 32.2 ft/sec 2) and converts inertia from units of oz-in 2 to oz-in-sec 2 .
3. Verify that the angular velocity of the lead screw is below the manufacturer’s
recommended rating for critical speed.
565
Motion Control Formulas
Thrust Equations
CT =
2πeTMOTOR
d
Direct Drive Equations (Rotary Motion)
TSYS = [TACC + TFRICTION]SF
TACC =
ω
1
[J
+ JMOTOR]
ta
386 LOAD
Solid Cylinders
JLOAD =
WLOAD*r2
2
=
πρLr 4
2
WLOAD = πρLr 2
L
r
JMOTOR = See Motor Data
Hollow Cylinders
JLOAD =
WLOAD
2
[r 22 – r12] =
πρL 4
[r 2 – r14]
2
r2
WLOAD = πρL[r 22 – r12]
L
r1
JMOTOR = See Motor Data
Gear Drive Equations (Rotary Motion)
TSYS = TACC + TBREAKAWAY +
TACC =
TLOAD
e(GR)
SF
ω
1 JLOAD
+ JGEARS + JMOTOR
ta
386 e(GR)2
N1
r1
ω = VLOAD2πGR
Gear2
N1
GR =
N2
Gear1
r2
N2
NOTES:
1. TSYS is the torque the motor must deliver at a velocity of ω (radians/second). The
divisor 386 in the TACC equation represents acceleration due to gravity (386 in/sec 2 or
32.2 ft/sec 2) and converts inertia from units of oz-in 2 to oz-in-sec 2 .
566
Motion Control Formulas
Gear Drive Equations (Rotary Motion) (Continued)
JGEARS = JGEAR1 + JGEAR2
WGEAR1*r12
JGEAR1 =
JGEAR2 =
2
r1
1
(GR)2
N1
WGEAR2*r22
2
r2
N2
Worm Gear
Belt/Tangential Drive Equations (Linear Motion)
TSYS = [TACC + TFRICTION + TBREAKAWAY + TGRAVITY]SF
ω
1
[J
+ JPULLEY + JBELT + JMOTOR]
ta
386 LOAD
TACC =
JLOAD = WLOAD*r 2
JPULLEY =
WPULLEY
2
r2
Multiply by
(Number
of Pulleys)
JBELT = WBELT*r 2
JMOTOR = See Motor Data
W
TFRICTION = WLOADr μCosφ
TGRAVITY = WLOADr μSinφ
ω=
VLOAD
r
r
NOTES:
1. TSYS is the torque the motor must deliver at a velocity of ω (radians/second). This usually
occurs during the acceleration portion of a move profile for horizontal applications and
an upward move for vertical applications. During the deceleration portion of a move
profile, TFRICTION and TBREAKAWAY are subtractions from TSYS . For horizontal applications
TGRAVITY has a zero value.
2. The divisor 386 in the TACC equation represents acceleration due to gravity (386 in/sec 2
or 32.2 ft/sec 2) and converts inertia from units of oz-in 2 to oz-in-sec 2 .
567
Motion Control Formulas
Move Profiles
Triangular Move
Vmax
Vavg
Velocity
(in/sec)
Vmax = 2Vavg = 2
T
2
ta = td =
ta
td
A = −D =
T (time in sec)
X
T
2Vmax
T
=
4X
T2
X (distance in inches)
Trapezoidal Move
Vmax
Velocity
(in/sec)
ta
td
tc
T (time in sec)
X (distance in inches)
T
3
Assume: ta = tc = td =
Then: Vmax =
1.5X
T
A = −D =
Selected Coefficients of Friction
Materials
Conditions
Hard Steel on Hard Steel Dry – Static
Coefficient
of Friction
0.78
Greasy – Sliding
0.15
Mild Steel on Cast Iron
Greasy – Sliding
0.183
Mild Steel on Mild Steel
Dry – Static
0.74
Greasy – Sliding
0.16
Dry – Sliding
0.42
Mild Steel on Babbit
Teflon on Teflon
Greasy – Sliding
0.17
Dry – Static
0.04
0.04
Teflon on Steel
Dry – Static
Brass on Steel
Dry – Static
0.51
Brass on Cast Iron
Dry – Static
0.35
Cast Iron on Oak
Dry – Static
0.49
568
4.5X
T2
Leadscrew Efficiencies
Type
Ball-Nut
Acme with Metal Nut
Acme Plastic Nut
Efficiency (%)
High
Median
Low
95
55
85
90
35
65
85
20
50
Since viscous lubricant is required,
the coefficient of friction is both speed and
temperature dependent.