motor sizing Brushed or brushless motors
Introduction to Motor Sizing
INTRODUCTION TO MOTOR SIZING
Nick Repanich Adjunct Research Professor
Department of Mechanical and Mechatronic Engineering and Sustainable Manufacturing
California State University, Chico Chico, CA 95929-0789
(530) 520-2548
Introduction to Motor Sizing 1
SEVEN STEPS OF SIZING AND SELECTION Step 1: Develop the torque and inertia equations that model the system mechanics.
1a: Draw/diagram the system to establish the relative location of the load mechanics. 1b: Develop the acceleration (Ta=), friction (Tf=), gravity (Tg=), and thrust (Tth=) torque
equations. Since Ta= (J)(α), this will also involve developing the inertial model (Jload=). Step 2: Determine the load motion profile(s) and calculate peak values.
• First find the max velocity reached (V) in a triangular move. If OK, use it. a
a
m t x
t X
V 22
==
- a simple triangular profile minimizes torque (lowers cost), but uses higher speeds • If V is too high for any reason, then find the optimized acceleration time (ta) in a trapezoidal
move using overall move time, distance & the new constrained velocity. V X
tt ma −=
- trapezoidal profiles are useful where rated motor torque drops with speed - for steppers, set maximum speed where the motor changes from the constant torque range
into the constant power range, otherwise reduction may be necessary.
• If there are multiple move profiles, find the worst-case acceleration (a), where at V
a =
Step 3: Calculate the mass moment of inertias of the load mechanics. • Anything that moves with the motor is part of the inertia (J). Step 4: Determine the peak torque (Tp) at maximum speed excluding motor inertia. • determine worst case combination of Ta , Tf , Tg , Tth ( thgfapeak TTTTT ±±±= ) • calculate power required to move the load (to find a system in the right power range) Step 5: Choose an approximate motor/drive system. • find a motor/drive with more than the required speed and about double the power • look at speed-torque curves and price list simultaneously
- with servos, use most of the speed available or you waste the power available Step 6: Determine the peak torque at speed including motor inertia. • calculate RMS torque to evaluate motor heating issues (necessary for servo systems) • give 50-100% torque margin (only use lower margins if measurable mechanics exist) • check for 10-20% velocity margin Step 7: Optimize the system. • Check the load-to-motor inertia ratio, and compare to the machine’s stiffness to the
performance desired. A ratio higher than 10:1 is an indicator of potential instability problems with non-stiff systems. (Note: It is not the cause of the problem.)
• Check the need for a power dump circuit (high inertia ratios or vertical load) • It may be necessary to adjust mechanics, add reduction, and start over. Remember, iteration
is crucial to successful design!
Introduction to Motor Sizing Direct Drive (Rotary) Formulas and Examples 2
MOTION PROFILE FORMULAS
for triangular profiles: a
a
m t x
t X
V 22
==
for trapezoidal profiles: V X
tt ma −=
Where: V = maximum velocity X = total move distance tm = total move time xa = acceleration distance ta = acceleration time Note: V is used here rather that ω, as these formulas work with either linear or rotary units.
GENERAL TORQUE & POWER FORMULAS
thgfapeak TTTTT ±±±= Torque Equations:
4321
4 2 43
2 32
2 21
2 1
tttt tTtTtTtT
Trms +++
+++ =
Where: Ta = Motor torque required to accelerate the inertial load Tf = Motor torque required to overcome the frictional forces Tg = Motor torque required to overcome the gravitational forces load Tth = Motor torque required to overcome any additional thrust forces T1 = Torque to accelerate the load from zero speed to max speed (Tf + Ta) T2 = Torque to keep the motor moving once it reaches max speed (Ta = 0) T3 = Torque required to decelerate from max speed to a stop (Ta - Tf) T4 = Torque required while motor is sitting still at zero speed t1 = time spent accelerating the load t2 = time spent while motor is turning at constant speed t3 = time spent decelerating the load t4 = time spent while motor is at rest Torque Unit Conversions:
mN cm
m s cmkg
s
mkg N ⋅=⎟
⎠
⎞ ⎜ ⎝
⎛⋅ ∴
⋅ =
2
2
2
2 100 1
Power Unit Conversions:
( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )
W746 1
1173.0038.745.8452.2260 2
2 Watts
=
⋅ =
⋅ =
⋅ =
⋅ =
⋅ =⋅=
hp
rpslbftrpmlbftrpmlbinrpsinozrpmmN rpsmN
π π
Introduction to Motor Sizing Direct Drive (Rotary) Formulas and Examples 3
DIRECT DRIVE (ROTARY) FORMULAS Inertia (Step 3):
Solid Cylinder Hollow Cylinder Where ρ, the density is known, use Where ρ, the density, is known use
2
4RL Jload
ρπ = ( )41422 RR
L Jload −=
ρπ
Where mass and radius are known use Where m, the mass, is known use
2
2mR Jload = ( )22212 RR
m Jload +=
2RLm ρπ= ( )2122 RRLm −= ρπ Torque (Step 2, 4-7)
( ) ⎟⎟ ⎠
⎞ ⎜⎜ ⎝
⎛ ++=
a
m mcploada t JJJT
ω
( )( )RFTf =
Where: Jcp = inertia of the coupler (g·cm
2) Jm = inertia of the motor's rotor (g·cm
2) ta = acceleration time (s) ωm = maximum motor velocity reached (rad/s) F = frictional force (N) R = radius at which the frictional force acts (cm) Note 1: When using Imperial units, specifically inertia in convenient units of oz·in2, the acceleration torque formula must be modified to correct the units. The units of oz·in2 for inertia are not proper inertial units, yet they are quite convenient to calculate inertia based on real world data. To use the acceleration torque formula with inertia in oz·in2, divide by the gravitational constant g (1/386 in/s2), which converts oz·in2 to proper units of oz·in·s2 as follows:
( ) ⎟⎟ ⎠
⎞ ⎜⎜ ⎝
⎛ ++=
a
m mcploada t JJJT
ω
2s in
386
1
Note 2: Though the correct symbol for mass moment of inertia is “I”, typically in mechatronics we (incorrectly) use the symbol “J” for inertia, as “I” is also used for electrical current.
Introduction to Motor Sizing Direct Drive (Rotary) Formulas and Examples 4
DIRECT DRIVE WITH REDUCTION
Gear Drive Inertia Formulas:
2
2
2N Rm
J loadloadload =
or
2
4
2N RL
J loadloadloadload ρπ
=
2
2 11
1 2N Rm
J geargeargear =
2
2 22
2 geargear
gear
Rm J =
Where:
Jx = inertia "as seen by the motor" m = mass R = radius N = reduction ratio (R1/R2) L = length ρ = density
Note: Most gearhead manufacturers provide the reflected inertia of the reducer in their user guide. This will help eliminate the need to calculate the individual gear inertias.
⎟⎟ ⎠
⎞ ⎜⎜ ⎝
⎛ ⎟ ⎠
⎞ ⎜ ⎝
⎛ ++=
a mr
load a t
N JJ
e J
T m ω
( )( ) eN RF
Tf =
Where: e = efficiency of transmission (reduction) N = reduction ratio Jr = effective inertia of reducer, gear box, belts and pulleys, or gears Jm = inertia of the motor's rotor ωm = maximum motor velocity reached ta = acceleration time F = frictional force R = radius at which the frictional force acts
Introduction to Motor Sizing Direct Drive (Rotary) Formulas and Examples 5
EXAMPLE #1 - Fluted-Bit Cutting Machine
Parameters for Axis 4:
• Move distance 0.2 rev • Move time 0.15 s • Maximum velocity allowed 20 rps • Length of chuck 4.0 cm • Diameter of steel chuck 3.0 cm • Length of bit 14 cm • Diameter of steel bit 0.8 cm • Torque required during cutting 0.050 N·m
Step 1: rps 67.215.0 )2.0)(2(2 ===
mt X
V
(OK, since not above "knee" on most stepper curves)
Step 2:
( ) ⎟⎟
⎠
⎞ ⎜⎜ ⎝
⎛ ++=
a
m motorchuckbita t JJJT
ω
Step 3: 2
4
3 cmg 36.4
2 cm 8.0
) cm
g 75.7)(cm 14(
2 ⋅=⎟
⎠
⎞ ⎜ ⎝
⎛ = π
bitJ
2
4
3 cmg 5.246
2 cm 0.3
) cm
g 75.7)(cm 4(
2 ⋅=⎟
⎠
⎞ ⎜ ⎝
⎛ = π
chuckJ
Introduction to Motor Sizing Direct Drive (Rotary) Formulas and Examples 6
Step 4:
( ) mN 0056.0 s m
kg 0.0056 s
cm g 112,56
2/s 15.0 rps 67.2
)2(cmg 5.246cmg 36.4 2
2
2
2 22 ⋅=⋅=⋅=⎟
⎠
⎞ ⎜ ⎝
⎛ ⋅+⋅= πaT
Tf = 0.050 N·m
Tpeak = Ta + Tf = 0.0056 + 0.050 N·m = 0.0556 N·m @ 2.67 rps
( )( ) W93.0rps 67.2mN 0556.02 =⋅= πP
Jload = 4.36 + 246.5 = 250.86 g·cm 2
Step 5: Try Nema 23 half-stack step motor, Jm = 0.070 kg·cm
2 = 70 g·cm2
Step 6:
( ) mN 0072.0 s
cm g 770,71
2/15.0 67.2
)2(705.24636.4 2
2 ⋅=⋅=⎟
⎠
⎞ ⎜ ⎝
⎛ ++= πaT
Tpeak = Ta + Tf = 0.0072 + 0.050 N·m = 0.0572 N·m @ 2.67 rps
Step 7: (fine) 6.3
cmg 70 cmg 86.250
Ratio Inertia 2
2
= ⋅
⋅ ==
m
l
J J
)fine(% 355 mN 00572.0
mN 0572.0-mN 26.0 (@speed)Margin Torque =
⋅
⋅⋅ =
− =
peak
peakavailable
T TT
Introduction to Motor Sizing Direct Drive (Rotary) Formulas and Examples 7
PROBLEM #1 - Grinding Of Bicycle Rim Braking Surface
Parameters of Axis 3:
• Outside diameter of steel bicycle rim 33.0 cm • Move distance 1/36 rev • Move time 0.10 s • Inertia of rim-holding fixture 1686 kg·cm2 • Width of rim 4.5 cm • Inside diameter of rim 31.5 cm • Friction torque during grinding 1.6 N·m • Dwell time between moves 0.05 s • Velocity is not limited in any way • This is not the final grinding move, but is the worst case move, making
repeated passes over the weld.
Answer: approximately 20 N·m @ 0.56 rps
Introduction to Motor Sizing Leadscrew Formulas and Examples 8
LEADSCREW FORMULAS
Inertia (Step 3) ( ) 2
2
4
screw2
RL J
p m
Jload ρπ
π ==
Where: m = mass (kg) L = length (cm) R = radius (cm) ρ = density (g/cm3) p = pitch of screw (revs/unit length) (inverse of lead) The formula for Jload converts linear mass into the rotational inertia as reflected to the motor shaft by the lead screw.
Torque (Step 2, 4-7) ⎟⎟ ⎠
⎞ ⎜⎜ ⎝
⎛ ⎟ ⎠
⎞ ⎜ ⎝
⎛ +++=
a mcpscrew
load a t
JJJ e J
T m ω
pe F
TT brf π2 +=
pe mg
Tg π2 =
mgF dµ=
Where: Jscrew = inertia of lead screw or ball screw (kg·cm2) Tbr = breakaway torque of nut on screw (N·m) F = force or thrust required p = pitch of screw (revs/unit length) e = efficiency of nut and screw ωm = maximum motor speed (rev/s). (watch critical speed of screw) g = acceleration due to gravity Jcp = inertia of the coupling (kg·cm2) Jm = inertia of the motor's rotor (kg·cm2) ta = acceleration time (s) Tg = torque required to overcome gravity (vertical) µd = coefficient of dynamic friction
Introduction to Motor Sizing Leadscrew Formulas and Examples 9
EXAMPLE #3 - "Smart" Battery Inspection
Parameters for Axis 1:
• Length of steel leadscrew 36 in • Pitch of leadscrew 5 • Efficiency of leadscrew 0.65 • Radius of leadscrew 0.5 in • Move distance 1.5 in • Move time 0.5 s • Maximum allowed velocity 5 in/s • Weight of load 50 lbs. • Coefficient of friction 0.01 • Breakaway Torque 25 oz·in
Introduction to Motor Sizing Leadscrew Formulas and Examples 10
Step 1: Step 2:
Step 3:
Step 4:
h p = (6 0. 0 8)25
1 6, 800 = 0. 0894 746
watts h p
!
" #
$
% & = 6 6. 7 watts
Step 5: Choose motor with about 0.2 hp Try S83-93, Jm = 6.70
V = (2) (1. 5in)
0.5 sec = 6 in / sec, too high
ta = tm − X V
= 0. 5 − 1.5 5
= 0.2 sec
V = 5 in sec
× 5 rev in
= 2 5 rev sec
T a = 1
386 Jload
e + Jleadscrew + Jmotor + Jc p
# $ %
& ' (
2πV ta
Jload =
W 2πp( )
2 = 5 0(16) 2π5( )
2 = 0. 8 1 oz ⋅in 2
Jleadscrew = πLρR4
2 = π 2
(3 6)( 4. 4 8) . 5( )4 = 1 5.8 oz ⋅ in2
Jc p = 0
T a =
1 386
. 81 . 65
+ 15 .8! " #
$ % &
2π(25 ) . 2
= 34. 69 oz⋅ in @ 25 rps
T f = T br + F
2πp e = 25 + 0.01(50 )(16 )
2π 5(. 65 ) = 25. 39 oz⋅ in
Tpeak = 34.69 + 25.39 = 60.08 oz·in
Introduction to Motor Sizing Leadscrew Formulas and Examples 11
Step 6: Step 7: Pick S83-93
T a =
1 386
.81 .65
+ 1 5. 8 + 6.70! " #
$ % &
2π( 2 5) . 2
= 4 8. 3 oz⋅ in
T peak = 4 8. 3 + 2 5. 3 9 = 7 3. 7 oz⋅ in @ 25 rps
Jl Jm
= 1 6.61 6. 7
= 2. 4 8, O.K.
Introduction to Motor Sizing Tangential Formulas and Examples 12
TANGENTIAL DRIVES FORMULAS
Inertia Formulas:
2 dplload RmJ =
22
42 pppp
fpdp
RLRm JJ
ρπ ===
2 dpbb RmJ =
Where:
Jdp = inertia of the driving pulley Jfp = inertia of the follow pulley Jb = inertia of the belt Rdp = radius of the driving pulley ml = mass of the load mp = mass of the pulley mb = mass of the belt
Introduction to Motor Sizing Appendix A 13
Belt Drive Torque Formulas:
Direct Drive: ⎟⎟ ⎠
⎞ ⎜⎜ ⎝
⎛ ⎟⎟ ⎠
⎞ ⎜⎜ ⎝
⎛ ++
++ =
a
m mdp
b
mechanicsbeltload a t
JJ e JJJ
T ω
dpf FRT =
With a gearhead: ⎟⎟ ⎠
⎞ ⎜⎜ ⎝
⎛ ⎟ ⎟ ⎠
⎞ ⎜ ⎜ ⎝
⎛ +++
++ =
a
ghm mgh
ghgh
dp
ghghb
mechanicsbeltload a t
N JJ
Ne J
Nee JJJ
T ω
22
ghgh
dp f Ne
FR T =
Where: ta = acceleration time eb = efficiency of belt over pulleys egh = efficiency of gearhead Ngh = reduction ratio of gearhead Jmechanics = inertia of other moving masses Rdp = radius of driving pulley Jgh = effective inertia of reducer ωm = maximum motor speed Jm = inertia of the motor's rotor F = force or thrust or friction forces at the drive pulley
Introduction to Motor Sizing Appendix A 14
Rack & Pinion Torque Formulas: Note: In the following formulas, the rack is stationary and the motor moves with the load. If the rack moves and the motor is stationary, then use the belt and pulley formulas, where the rack acts essentially like the belt.
Direct Drive: ⎟⎟ ⎠
⎞ ⎜⎜ ⎝
⎛ ⎟ ⎟ ⎠
⎞ ⎜ ⎜ ⎝
⎛ ++ =
a
m
g
pmload a te
JJJ T
ω
g
p f e
FR T =
With gearhead on motor: ⎟⎟ ⎠
⎞ ⎜⎜ ⎝
⎛ ⎟ ⎟ ⎠
⎞ ⎜ ⎜ ⎝
⎛ + +
+ =
a
m
g
mgh
ghg
pload a t
N e JJ
eNe JJ
T ω
2
ghg
p f
eNe
FR T
2 =
Where: Jload = inertia of moving load eg = efficiency of transmission between rack and pinion Jp = inertia of pinion gear and shafts and couplers and outboard bearings (if any) Rp = effective radius of pinion gear F = force, thrust or friction Jm = inertia of the motor's rotor ωm = maximum motor speed ta = acceleration time Jgh = effective inertia of reducer egh = efficiency of reducer N = reduction ratio
Introduction to Motor Sizing Appendix A 15
APPENDIX A STANDARD END FIXTURING METHODS
The method of end fixturing has a direct effect on critical speed, column load bearing capacity, and system stiffness. Four common methods are shown below.
Fixed-Free Fixed-Simple One end held in a duplex (preloaded) bearing, the other end is free
One end held in a duplex (preloaded) bearing, one end held in a single bearing.
Simple-Simple Fixed-Fixed Both ends held in a single bearing. Both ends held in duplex (preloaded)
bearings.
DENSITIES OF COMMON
MATERIALS Material oz/in3 gm/cm3
Aluminum Alloys 1.54 2.8 Brass, Bronze 4.80 8.6 Copper 5.15 8.9 Plastics 0.64 1.1 Steel (carbon,alloys,stainless) 4.48 7.8 Hard Wood 0.46 0.80
KINEMATIC EQUATIONS
(straight-line motion w/ constant acceleration) Equation Contains
x xv xa t tavv xxx += 0 ü ü ü
( )tvvxx xx 02 1
0 ++= ü ü ü 2
2 1
0 0 tatvxx xx ++= ü ü ü
( )0 22 2
0 xxavv xxx −+= ü ü ü
COEFFICIENTS OF FRICTION
Materials (dry contact unless noted) µs µd Steel on Steel Steel on Steel (lubricated) Aluminum on Steel Copper on Steel Brass on Steel Teflon on Steel Round rails w/ball bearings Linear guides - radius groove Linear guides - gothic arch (depends on load, pre-load & type)
0.74 0.23 0.61
0.04 0.002 0.003 0.008 to .05
0.58 0.15 0.45 0.22 0.19 0.04 0.002 0.002 0.004 to .02
Introduction to Motor Sizing Appendix A 16
BALL SCREW FORMULAS Critical Speed
⎟⎟ ⎠
⎞ ⎜⎜ ⎝
⎛
⋅
×⋅ =
2
610 Lf
dS N
s c
Nc = critical speed (rpm) d = root diameter of screw (cm) L = distance between bearings (cm) fs = Safety factor (recommend 1.25) S = end support conditions factor = 4.22 fixed - free = 18.85 fixed - simple = 27.31 fixed – fixed Note: The ‘S’ term carries units cmrev ⋅ /min. This equation is based on the natural frequency of the rod, and acquires its time component from that derivation. Column Load
⎟⎟ ⎠
⎞ ⎜⎜ ⎝
⎛ ⋅⋅× =
2
461003.14 L
dS Fcl
Fcl = maximum load (lb) S = end fixity factor = .250 one end fixed one free = 1.0 both ends supported
(simple) = 2.0 one fixed one simple = 4.0 both fixed d = root diameter of screw L = distance between ball nut and
load-carrying bearing Note: The ‘S’ term carries hidden units that make the formula simpler and easier to use. Backdriving Torque (mainly used to determine holding brake torque)
π2 elF
Tb ××
=
Tb = torque required to backdrive F = axial load l = lead of screw e = efficiency of screw
Life Expectancy
For Other then Rated (Dynamic) Load
( )
( )3m
6
3
6
load dynamic/F inches 10
load icload/dynam operating inches 10
Life
=
=
For Equivalent Load
( ) ( )( )3 3311 nnm FYFYF …+=
Fm = equivalent load Fn = a particular increment of load Yn = the portion of a cycle (sub
cycles) of a particular increment of load expressed as a decimal, i.e. the sum of the sub cycles must equal one. Example if L1 is applied for 20% of the cycle, L2 is applied for 30% of the cycle and L3 is applied for 50% of the cycle, then the associated Y values are Y1 =.2, Y2 = .3, Y3 = .5.