Haydon Kerk Motion SolutionsPittman Motors
203 756 7441 267 933 2105
wwwHaydonKerkPittmancom
FORMULAS FOR MOTORIZED
LINEAR MOTION SYSTEMS
Symbol Description Units Symbol Description Units
a ms2linear acceleration
a rads2angular acceleration
α degCtemperature coefficient
Td Nmreverse torque required for deceleration
Tf Nmtorque required to overcome friction
Tg Nmtorque required to overcome gravity
TLR Nmlocked rotor torque
TRMS NmRMS torque required over the total duty cycle
TRMS (motor) NmRMS torque required at the motor shaft
μ nacoefficient of friction
VT Vmotor terminal voltage
Vbus VDC drive bus voltage
v mslinear velocity
vf msfinal linear velocity
msinitial linear velocity
mspeak linear velocity
W Jenergy
ω radsangular velocity
radsno load angular velocity
radspeak angular velocity
vi
vPK
ωo
ωPK
TD Nmdrag preload torque
ain rads2angular acceleration
aout rads2angular acceleration
DV Nm sradviscous damping factor
F Nforce
Fa Nlinear force required during acceleration (FJ +Ff +Fg)
Ff Nlinear force required to overcome friction
Fg Nlinear force required to overcome gravity
FJ Nlinear force required to overcome load inertia
g ms2gravitational constant (98 ms2)
I Acurrent
ILR Alocked rotor current
IO Ano-load current
IPK Apeak current
IRMS ARMS current
J Kg-m2inertia
JBKg-m2inertia of belt
JGBKg-m2inertia of the gear box
JinKg-m2reflected inertia at system input
Jout Kg-m2reflected inertia at system output
JP1Kg-m2inertia of pully 1
JP2Kg-m2inertia of pully 2
JSKg-m2lead screw inertia
K(i)
Vradsvoltage constant
K(f) NmA or V(rads)KE or KT (finalldquohotrdquo)
KE
KE or KT (initial ldquocoldrdquo)
Km Nm wmotor constant
NmA or V(rads)
KT NmAtorque constant
L mrevscrew lead
m Kgmass
N nagear ratio
RPM
ηRPMspeed
nO no load speed
nnaefficiency
RPMnPK peak speed
load orientation (horizontal = 0deg vertical = 90deg)
degCθ degrees
Θa ambient temperature
Θf degCmotor temperature (finalldquohotrdquo)
Θi degCmotor temperature (initial ldquocoldrdquo)
Θm degCmotor temperature
Θr degCmotor temperature rise
Θrated degCrated motor temperature
P Wpower
Pavg Waverage power
PCV Wpower required at constant velocity
Pin Wpower required at system input
Ploss Wdissipated power
Pmax Wmaximum power
Pmax(f) Wmaximum power (finalldquohotrdquo)
Pmax(i) Wmaximum power (initial ldquocoldrdquo)
Pout Woutput power
PPK Wpeak power
Rm RPMNmmotor regulation
Rmt Ωmotor terminal resistance
Rmt(f) Ωmotor terminal resistance (finalldquohotrdquo)
Rmt(i) Ωmotor terminal resistance (initial ldquocoldrdquo)
Rth degCWthermal resistance
r mradius
S mlinear distance
T Nmtorque
Ta Nmtorque required to overcome load inertia
Ta (motor) Nmtorque required at motor shaft during acceleration
TC Nmcontinuous rated motor torque
TCF Nmcoulomb friction torque
Tin Nmtorque required at system input
Tout Nmtorque required at system output
radsangular velocity at system inputωin
radsangular velocity at system outputωout
t stime
Symbols and Units
2 SYMBOLS AND UNITS
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1
(V)(I)Watts
F x St =
+ +MotorController
DrivePowerSupply
Mechanical Transmission
Lead Screw Support Rails
WattsLost
(V)(I)Watts
WattsLost
WattsLost
WattsLost
(T)(ω)Watts
(T)(ω)Watts
WattsLost
All analysis begins with the linear motion profile of the output and the force required
to move the load
t 1 t 2 t 3t move
Total Cycle Time
t dwell
vTime
vpk
a -a
13 13 13 Trapazoidal Motion Profile
Optimized for minimum power
t 1 = t 2= t 3
vPK =3S2t
S is the total move distancet is the total move timeArea under profile curve represents distance moved
Watts Out
Motion System Formulas
Any motion system should be broken down into its individual components
Motion Profiles
3
(V)(I)Watts
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2
MOTION SYSTEMFORMULAS
Triangular Motion Profiles
Optimized for minimum acceleration slope
t 1 = t 2
vPK =2St
S is the total move distancet is the total move time 13 13 13
Triangular
Move Profile Peak Velocity (VPK) Acceleration (a )
t 1 t 2t move
Total Cycle Time
t dwell
vTime
a -avPK
Complex Motion Profile
Linear to Rotary Formulas through a Lead Screw
(1) s = ndashndashndashndashvf - vi
2t
(2) vf = vi + a t
(3) s = vi t + ndash at 212
(4) 2 as = vf 2
- vi 2
(5) a = ( vf - vi) ( tf - ti )
= ΔvΔt
a = = rads 2 2 π a
L
ω = = rads 2 π v
L
n = = RPM 60 v
L
3 known quantities are needed to solve for the other 2 Each segment calculated individually
Linear Motion Formulas
t 1 t 2 t 3t move
Total Cycle Time
t dwell
vTime
v1
a 1
-a3
a 2
t 4 t 5
v2
Motion System Formulas4
Area under profile curve represents distance moved
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3
MOTION SYSTEMFORMULAS
Inertia Acceleration Velocity and Required Torque
Jin mout
ain aout ωin vout
Tin Fout
radsec 2
radsec
ain = 2 πa
L
ωin = 2 π v
L
Jin = m 2 πL 2
X η1 + JS Kg-m2
Τin= Τa+Τf +Τg+ΤD N-m
Τa= Jin ain
Τf =cosOslashmgμL
2 π ηN-m
N-m
Τg =sinOslashmgL
2 π ηN-m
ΤD = dragpreloadrefer to manufacturerrsquos data
N-m
Lead Screw System
Belt and Pulley System
ain = radsec 2
ωin = radsec Jin Jout
ain aout ωin ωout
Tin Tout
Jin = NJout
2X η
1 + JP1 Kg-m2
N-mFor a 1st approximation analysis JP1 JP2 and JB can be disregarded For higher precision pulley and belt inertias should be included as well as the reflected inertia of these components
+ JP2+ JB
(aout)(N) (ωout)(N)
Τin = NΤout X η
1
Motion System Formulas5
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4
MOTION SYSTEMFORMULAS
Motion System Formulas6
Gearbox System
radsec 2
ωin = radsec
Jin = NJout
2X η
1
+
Kg-m2
N-m
Drag torque can be significant (TD) depending on the viscosity of the lubricant For a first approximation analysis this can be left outGearbox inertia may be found in manufacturerrsquos data sheets or calculated using gear dimensions and materials mass
JGB
(ωout)(N) Τin = N
Τout X η1
+
ΤD
ΤRMS = Τ12t 1 + Τ2
2t2 + Τ32t 3 + Τn
2t n
t 1 + t 2 + t 3 + t n + tdwell
TωLinear Power =
FSt
= watts
Rotary Power = = watts
Mechanical Power and RMS Torque
Jin Jout
ain aout ωin ωout
Tin Tout
ain = (aout)(N)
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5
MOTION SYSTEMFORMULAS
Assuming IO is very small it can be ignored for a quick approximation of motor no-load speed If more accurate results are needed IO will also need to be adjuisted based on the motor Viscous Damping Factor (Dv) As the no-load speed increases with increased terminal voltage on a given motor IO will also increase based on the motorrsquos Dv value found on the manufacturerrsquos motor data sheet
Motor No Load SpeedVT ndash (IO x Rmt)no = 95493 x
KE
ndash ndash ndash ndash ndash ndash ndash ndash ndash ndash ndash ndash ndash
ndash ndash ndash
ndash ndash ndash
BreakawayTorque
Viscous Damping Torque (Dv)
Coulomb Friction
Τ
ωο
Motion System Formulas7
DC Motor Formulas
Motor Locked Rotor Current Locked Rotor Torque
ILR =VT
Rmt
TLR = ILR x KT
Motor Regulation
RmtRm = 95493 x
KE x KT
= x KT
VT
Rmt
ndash Theoretical using motor constants
nORm = TLR
ndash Regulation calculated using test data
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6
MOTION SYSTEMFORMULAS
Motor Power Relationships
ndash Watts lost due to winding resistance
Ploss = I2 x Rmt
ndash Output power at any point on the motor curve
Pout = ω x Tndash Motor maximum output power
Pmax = 025 x ωo x TLR
ndash Motor maximum output power (Theoretical)
Pmax = 025 x VT2
Rmt
Temperature Effects on Motor Constants
ndash Change in terminal resistance
Rmt(f) = Rmt(i) x [ 1 + αconductor (Θf ndash Θi) ]αconductor = 00040degC (copper)
K(f) = K(i) x [ 1 + αmagnet (Θf ndash Θi) ]ndash Change in torque constant and voltage constant ( KT = KE )
Magnetic Material
CeramicSmCoAlNiCoNdFeB
Θmax (degC)300degC300degC540degC150degC
The magnetic material values above represent average figures for particular material classes and estimating performance If exact values are needed consult data sheets for a specific magnet grade
Motion System Formulas8
αmagnet (degC)-00020degC-00004degC-00002degC-00012degC
In the case of a mechanically commutated motor with graphite brushes this analysis will result in a slight error because of the negative temperature coefficient of carbon For estimation purposes this can be ignored but for a more rigorous analysis the behavior of the brush material must be taken into account This is not an issue when evaluating a brushless dc motor
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
7
MOTION SYSTEMFORMULAS
Motion System Formulas9
For PMDC motors (brush type) brush drop can be approximated as a constant resistance connected in series with the armature winding This series resistance is included in the Rmt factor found on manufacturerrsquos data sheet
Copper graphiteSilver graphiteElectrographite
02 to 04 Ω 02 to 04 Ω 08 to 10 Ω
Motor Brush Resistance
Brush Material Resistance
DC Motor Thermal Resistance and Temperature Rise
ndash Thermal resistance
ndash Motor temperature rise
Θm = Θr + Θa
ndash OR ndash
Temperature Rise degCRth =
Machine Losses W
Θr = Rth x I2 x Rmt
Θr = Rth x I
2 x Rmt
1 ndash (Rth x I2 x Rmt x α)
ndash OR ndash
Θr = Rth x(TC
Km)
2
ndash Final motor temperature
Use caution when applying these formulas Depending on the conditions used to determine motor specifications only one formula should be used Contact Appications Engineering for questions
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
8
MOTION SYSTEMFORMULAS
Motion System Formulas9
For PMDC motors (brush type) brush drop can be approximated as a constant resistance connected in series with the armature winding This series resistance is included in the Rmt factor found on manufacturerrsquos data sheet
Copper graphiteSilver graphiteElectrographite
02 to 04 Ω 02 to 04 Ω 08 to 10 Ω
Motor Brush Resistance
Brush Material Resistance
DC Motor Thermal Resistance and Temperature Rise
ndash Thermal resistance
ndash Motor temperature rise
Θm = Θr + Θa
ndash OR ndash
Temperature Rise degCRth =
Machine Losses W
Θr = Rth x I2 x Rmt
Θr = Rth x I
2 x Rmt
1 ndash (Rth x I2 x Rmt x α)
ndash OR ndash
Θr = Rth x(TC
Km)
2
ndash Final motor temperature
Use caution when applying these formulas Depending on the conditions used to determine motor specifications only one formula should be used Contact Appications Engineering for questions
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
9
MOTION SYSTEMFORMULAS
Stepper Motor Formulas
Required ldquoTime Offrdquo using an LR drive (constant voltage drive)
180Step Oslash = (phases)(rotor teeth)
ndash Motor step angle
ndash RPM (revolutionsminute) to SPS (stepssecond)
Maximum accuracy of a step motor system has nothing to do with maximum resolution of a step motor system Maximum accuracy is always based on the accuracy of 1 full step but maximum resolution is based on all system components including lead screw encoder stepper motor and step mode on the controller
ndash SPS (stepssecond) to RPM (revolutionsminute)(SPS) x (Step Oslash)
RPM = 6
(RPM) x 6 SPS = (Step Oslash)
ndash Travel per step
mstep =L
positionsrev
ndash Overdriving capability
Required ldquoTime Offrdquo using a chopper drive (constant current drive)
Time Off = (Time On) (Applied Voltage)2
(Rated Voltage)ndash Time On
Time Off = (Time On) (Applied Current)2
(Rated Current)ndash Time On
It is not unusual for a customer to drive Haydon Kerk stepper motors beyond their rated power to obtain the most force in the smallest package size Precautions must be taken to prevent the motor from exceeding its maximum temperature The ldquoonrdquo time should not exceed 2 - 3 minutes As general rule of thumb driving a motor at 25 to 3x its rated voltage or current may result in wasted energy and erratic behavior due to saturation
Special notes for can-stack motors bull more easily saturated due to less active material (steel) bull more iron losses due to unlaminated steel
Motion System Formulas10
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10
MOTION SYSTEMFORMULAS
Lead Screw Performance Characteristics
End Mounting Rigidity
Impact Impact
ScrewLength
Increase In Affects Increase In Affects
Critical speed
Compression load
Critical speed
Inertia
Compression load
Stiffness
Spring Rate
Load capacity
Drive torque
Angular velocity
Load cpacity
Positioning accuracy
ScrewDiameter
Lead
Load
Preload
Nut length
SI Unit Systems
LengthMassTimeElectric currentThermodynamic temperatureLuminous intensityAmount of substance
meterkilogramsecondamperekelvincandelamole
mKgsAKcdmol
Base Units
Derived Unitsareavolumefrequencymass density (density)speed velocityangular velocityaccelerationangular accelerationforcepressure (mechanical stress)kinematic viscositydynamic viscositywork energy quantity of heatpowerentropyspecific heat capacitythermal conductivity
square metercubic meterhertzkilogram per cubic metermeter per secondradian per secondmeter per second squaredradian per second squarednewtonpascalsquare meter per secondnewton-second per square meter joulewattjoule per kelvinjoule per kilogram kelvinwatt per meter kelvin
m2
m3
HzKgm3
msradsms2
rads2
NPam2sN-sm2
JWJKJKg-K)W(m-K)
Quantity Name of Unit Symbol
Motion System Reference Tables11
Critical speed
Compression load
System stiffness
Life
Positioning accuracy
System stiffness
Drag torque
Load capacity
Stiffness
sndash1
Kg-ms2
Nm2
N-mJs or N-ms
ExamplesAn INCREASE ( ) in screw length results in a DECREASE ( ) in critical speed
An INCREASE in screw diameter results in an INCREASE in critical speed
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11
MOTION SYSTEMREFERENCE TABLES
t 1 t 2 t 3 t dwell
vTime
vpk
a -a
MassMove distanceMove timeDwell timeMotion profileScrew speed limitLoad supportOrientationRotary to linear conversionAmbient temperature
9 Kg200 mm10 s05 s13 13 13 Trapezoidal1000 RPMμ = 001VerticalTFE lead screw 8 mm diameter 275 mm long30degC
ControlDrive Power Supply
4 quadrant with encoder32 VDC 35 Arms50 A peak
0333 s 0333 s 0333 s 05 s
vPK =3S2t =
(3)(02m)(2)(10 s) =
06m2 s =
03ms
Linear Velocity
Linear Acceleration
a = vf - vi
t = 03ms - 0 ms 0333 s 0901ms2 =
1st step is to thoroughly understand system constraints and motion profile requirements
Linear Motion ExampleData
Motion System Application Example12
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12
MOTION SYSTEMAPPLICATION EXAMPLE
Lmin =60vPK
n =(60)(03ms)1000 revmin = 0018mrev = 18mmrev
Refer to Kerk screw chart ndash closest lead in an 8mm screw diameter is
2032mmrev = 002032 mrevη = 086 free wheeling nut TFE coated screw
JS = 388 x 10ndash 7 Kg-m2 Other critical lead screw considerations bull Column loading bull Critical speed
Jm aa
ωPKvPK
TF
= ωPK = 2 π vPK
L=
(2π)(03ms)002032 mrev
9276 rads
a = =2πaL
=(2π)(0901ms2)002032 mrev
2786 rads2
Linear to Rotary Conversion ndash Velocity Acceleration Inertia Torque
Minimum screw lead to maintain lt 1000 RPM shaft speed
Linear to Rotary Conversion Velocity
Linear to Rotary Conversion Acceleration
Motion System Application Example13
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
13
MOTION SYSTEMFORMULAS
Jin= m 2 πL 2
X η1 + JS= 9Kg
2 π002032 m 2
X 0861 + 388x10ndash 7Kg-m2
= 1095 X 10ndash 5 Kg-m2 + 388x10ndash 7 Kg-m2
= 1134 x 10ndash 5 Kg-m2
Τin= Τa+Τf +Τg+ΤD
Τa = Jin ain
Τg =sinOslashmgL
2 π η
= (1134 x 10ndash 5 Kg-m2)(2786 rads2)
= 00316 Nm
Τf =cosOslashmgμL
2 π η= (cos90)(9 Kg)(98ms2)(001)(002032m)
2 π (086)= 0 Nm
= (sin90)(9 Kg)(98ms2)(002032m)2 π (086)
=1792 Nm
5404 = 03316 Nm
ΤDSince a free-wheeling lead screw nut was used there is no preload force If an anti-backlash nut was used ΤD would be gt0 due to preload force Always reference manufacturerrsquos data for more information
= 0 Nm
Linear to Rotary Conversion Torque
Linear to Rotary Conversion Inertia
Motion System Application Example14
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
14
MOTION SYSTEMFORMULAS
t 1 t 2 t 3 t dwell Time0333 s 0333 s 0333 s 05 s
Rotary Motion Profile
ωPK
ω
= 9276 rads a = 2786 rads2
Τ1= Τa+Τf +Τg+ΤD
= (00316 Nm) + (0 Nm) + (03316 Nm) + (0 Nm)
= 03632 Nm
Τ2= Τa+Τf +Τg+ΤD
= (0 Nm) + (0 Nm) + (03316 Nm) + (0 Nm)
= 03316 Nm
Τ3= Τa+Τf +Τg+ΤD
= (00316 Nm) + (0 Nm) + (03316 Nm) + (0 Nm)
= 03632 Nm
Τ1
Τ2
Τ3
Motion System Application Example15
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
15
MOTION SYSTEMAPPLICATION EXAMPLE
Lead Screw Input Requirements
= 9276 rads= 2786 rads2
= 03632 Nm= 03316 Nm= ndash 03632 Nm= 0333 s= 0333 s= 0333 s= 05 s= 3369 W= 3076 W
ωPKaΤ1Τ2Τ3
t1t2t3tdwell
PPKPCV
PPK = (Τ1)(ωPK)
= (03632 Nm)(9276 rads)= 3369 W
PCV = (Τ2)(ωPK)
= (03316 Nm)(9276 rads)= 3076 W
RMS Torque Requirement Lead Screw Input
=
0333 s + 0333 s + 0333 s + 05 s
(03632 Nm)2(0333 s)+(03316 Nm)2(0333 s)+(- 03632 Nm)2(0333 s)
ΤRMS = Τ12t 1 + Τ2
2t2 + Τ32t 3
t 1 + t 2 + t 3 + tdwell
15
00439 + 00366 + 00439 = 00829 =
ΤRMS = 02879 Nm
Motion System Application Example16
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
16
MOTION SYSTEMAPPLICATION EXAMPLE
Load Parameters at the Lead Screw Shaft
= 9276 rads= 2786 rads2
= 03632 Nm= 03316 Nm= ndash 03632 Nm= 02879 Nm= 3369 W= 3076 W
ωPKaΤ1Τ2Τ3ΤRMSPPKPCV
Motor SelectionA direct-drive motor can be used however it would be large and expensive For example a DC057B-3 brush motor
would easily meet continuous torque and continuous speed as well as a 181 load-to-rotor inertia This motor however has significantly more output power than needed at 128 watts
A smaller motor with a transmission would optimize the system Look for a motor starting at a rated power around 40 watts Pittman DC040B-6
DC040B-6Reference VoltageContinuous TorqueRated CurrentRated PowerTorque ConstantVoltage ConstantTerminal ResistanceMax Winding TemperatureRotor Inertia
240 V00812 Nm236 A37 W0042 NmA0042 Vrads185 Ω155degC847 x 10ndash6 Kg-m2
Motor TransmissionLead Screw Support Rails
ΤRMS
PPK
ωPK
= 02879 Nm= 3369 W= 9276 rads
M
Other Motor Considerationsndash Type stepper brush BLDCndash Input speed (high input speed will result in high noise levels)ndash Total motor footprintndash Ambient temperaturendash Environmental conditionsndash Load-to-rotor inertiandash Encoder ready needed
Motion System Application Example17
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
17
MOTION SYSTEMAPPLICATION EXAMPLE
Gearbox Transmission Selection
When selecting a reduction ratio keep in mind that the RMS required torque at the motor shaft needs to fall below the continuous rated output torque of the motor
G30 A - Planetary GearboxMaximum output load 247 Nm
TRMS( Lead Screw)
02879 Nm
02879 Nm02879 Nm
Reduction
41
5161
Efficiency
090
090090
TRMS( Motor)
00798 Nm
00640 Nm00533 Nm
A 51 gearbox would allow about a 27 safety margin between what the system requires and the continuous torque output of the motor
System Summary 51η = 090
M
= 00807 Nm= 00737 Nm= ndash 00807 Nm= 00640 Nm= 4638 rads= 504 x 10 ndash 6 Kg-m2
= 3743 W
Τ1Τ2Τ3ΤRMS
PPK
ωPK
J
= 03632 Nm= 03316 Nm= ndash 03632 Nm= 02879 Nm= 9276 rads= 1134 x 10 ndash 5 Kg-m2
= 3369 W
Τ1Τ2Τ3ΤRMS
PPK
ωPK
J
Motion System Application Example18
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
18
MOTION SYSTEMAPPLICATION EXAMPLE
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
19
Motion Profile at the Motor
t 1 t 2 t 3 t dwell Time0333 s 0333 s 0333 s 05 s
ωΤ1
Τ2
Τ3= 4638 rads= 4429 RPM= ΤPK = 00807 Nm
ωPK
Τ1
nPK
Drive and Power Supply Requirements
Peak Current Required
IPK =TPK
KT+ IO =
00807 Nm0042 NmA
+ 0180 A = 210 A
RMS Current Required
IRMS =TRMS
KT+ IO =
00640 Nm0042 NmA
+ 0180 A = 170 A
Minimum bus Voltage Required
Vbus= IPKRmt+ ωPKKE = (210A)(185Ω) + (4638 rads)(0042 Vrads)= 389 V + 1948 V= 2337 V
0100 Nm 0200 Nm 0300 Nm 0400 Nm 0500 Nm 0600 Nm 0700 Nm
30 VDC
0100 Nm 0200 Nm 0300 Nm 0500 Nm 0600 Nm 0700 Nm
24 VDC
700060005000
RPM 4000300020001000
0400 Nm
TRMS
TPK
700060005000
RPM 4000300020001000
TRMS
TPK
Performance using 24V reference voltage
Performance using a 30V bus voltage
A 30 VDC bus will meet the application requirements as wells supply a margin of safety
MOTION SYSTEMAPPLICATION EXAMPLE
0100 Nm 0200 Nm 0300 Nm 0400 Nm 0500 Nm 0600 Nm 0700 Nm
30 VDC
0100 Nm 0200 Nm 0300 Nm 0500 Nm 0600 Nm 0700 Nm
24 VDC
700060005000
RPM 4000300020001000
0400 Nm
TRMS
TPK
700060005000
RPM 4000300020001000
TRMS
TPK
Ambient temperature Rated motor temperature Temperature rise Motor temperature
= 30degC= 155degC= 7638degC= 10638degC
Θa
Θm
ΘrΘrated
=Rth x IRMS
2 x Rmt
1 ndash (Rth x IRMS2 x Rmt x 000392degC)
Θr
=11degCω x 170A2
x 185Ω 1 ndash (11degCω x 170A2
x 185Ω x 000392degC)
= 5881 1 ndash (023)
= 5881 077
=
Θm = Θr + Θa = 7638degC + 30degC = 10638degC
7638degC
Performance using 24V reference voltage
Performance using a 30V bus voltage
Will the motor meet the temperature rise caused by the application
Motion System Application Example20
A 30 VDC bus will meet the application requirements as wells supply a margin of safety
203 756 7441 1500 Meriden Road Waterbury CT USA 06705
267 933 2105 534 Godshall DriveHarleysville PA USA 19438
wwwHaydonKerkPittmancom
copy All rights reserved AMETEK Advanced Motion Solutions No part of this doc-ument or technical information can be used reproduced or altered in any form without approval or proper authorization from AMETEK Inc and global affiliates
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MOTION SYSTEMAPPLICATION EXAMPLE
Symbol Description Units Symbol Description Units
a ms2linear acceleration
a rads2angular acceleration
α degCtemperature coefficient
Td Nmreverse torque required for deceleration
Tf Nmtorque required to overcome friction
Tg Nmtorque required to overcome gravity
TLR Nmlocked rotor torque
TRMS NmRMS torque required over the total duty cycle
TRMS (motor) NmRMS torque required at the motor shaft
μ nacoefficient of friction
VT Vmotor terminal voltage
Vbus VDC drive bus voltage
v mslinear velocity
vf msfinal linear velocity
msinitial linear velocity
mspeak linear velocity
W Jenergy
ω radsangular velocity
radsno load angular velocity
radspeak angular velocity
vi
vPK
ωo
ωPK
TD Nmdrag preload torque
ain rads2angular acceleration
aout rads2angular acceleration
DV Nm sradviscous damping factor
F Nforce
Fa Nlinear force required during acceleration (FJ +Ff +Fg)
Ff Nlinear force required to overcome friction
Fg Nlinear force required to overcome gravity
FJ Nlinear force required to overcome load inertia
g ms2gravitational constant (98 ms2)
I Acurrent
ILR Alocked rotor current
IO Ano-load current
IPK Apeak current
IRMS ARMS current
J Kg-m2inertia
JBKg-m2inertia of belt
JGBKg-m2inertia of the gear box
JinKg-m2reflected inertia at system input
Jout Kg-m2reflected inertia at system output
JP1Kg-m2inertia of pully 1
JP2Kg-m2inertia of pully 2
JSKg-m2lead screw inertia
K(i)
Vradsvoltage constant
K(f) NmA or V(rads)KE or KT (finalldquohotrdquo)
KE
KE or KT (initial ldquocoldrdquo)
Km Nm wmotor constant
NmA or V(rads)
KT NmAtorque constant
L mrevscrew lead
m Kgmass
N nagear ratio
RPM
ηRPMspeed
nO no load speed
nnaefficiency
RPMnPK peak speed
load orientation (horizontal = 0deg vertical = 90deg)
degCθ degrees
Θa ambient temperature
Θf degCmotor temperature (finalldquohotrdquo)
Θi degCmotor temperature (initial ldquocoldrdquo)
Θm degCmotor temperature
Θr degCmotor temperature rise
Θrated degCrated motor temperature
P Wpower
Pavg Waverage power
PCV Wpower required at constant velocity
Pin Wpower required at system input
Ploss Wdissipated power
Pmax Wmaximum power
Pmax(f) Wmaximum power (finalldquohotrdquo)
Pmax(i) Wmaximum power (initial ldquocoldrdquo)
Pout Woutput power
PPK Wpeak power
Rm RPMNmmotor regulation
Rmt Ωmotor terminal resistance
Rmt(f) Ωmotor terminal resistance (finalldquohotrdquo)
Rmt(i) Ωmotor terminal resistance (initial ldquocoldrdquo)
Rth degCWthermal resistance
r mradius
S mlinear distance
T Nmtorque
Ta Nmtorque required to overcome load inertia
Ta (motor) Nmtorque required at motor shaft during acceleration
TC Nmcontinuous rated motor torque
TCF Nmcoulomb friction torque
Tin Nmtorque required at system input
Tout Nmtorque required at system output
radsangular velocity at system inputωin
radsangular velocity at system outputωout
t stime
Symbols and Units
2 SYMBOLS AND UNITS
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1
(V)(I)Watts
F x St =
+ +MotorController
DrivePowerSupply
Mechanical Transmission
Lead Screw Support Rails
WattsLost
(V)(I)Watts
WattsLost
WattsLost
WattsLost
(T)(ω)Watts
(T)(ω)Watts
WattsLost
All analysis begins with the linear motion profile of the output and the force required
to move the load
t 1 t 2 t 3t move
Total Cycle Time
t dwell
vTime
vpk
a -a
13 13 13 Trapazoidal Motion Profile
Optimized for minimum power
t 1 = t 2= t 3
vPK =3S2t
S is the total move distancet is the total move timeArea under profile curve represents distance moved
Watts Out
Motion System Formulas
Any motion system should be broken down into its individual components
Motion Profiles
3
(V)(I)Watts
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
2
MOTION SYSTEMFORMULAS
Triangular Motion Profiles
Optimized for minimum acceleration slope
t 1 = t 2
vPK =2St
S is the total move distancet is the total move time 13 13 13
Triangular
Move Profile Peak Velocity (VPK) Acceleration (a )
t 1 t 2t move
Total Cycle Time
t dwell
vTime
a -avPK
Complex Motion Profile
Linear to Rotary Formulas through a Lead Screw
(1) s = ndashndashndashndashvf - vi
2t
(2) vf = vi + a t
(3) s = vi t + ndash at 212
(4) 2 as = vf 2
- vi 2
(5) a = ( vf - vi) ( tf - ti )
= ΔvΔt
a = = rads 2 2 π a
L
ω = = rads 2 π v
L
n = = RPM 60 v
L
3 known quantities are needed to solve for the other 2 Each segment calculated individually
Linear Motion Formulas
t 1 t 2 t 3t move
Total Cycle Time
t dwell
vTime
v1
a 1
-a3
a 2
t 4 t 5
v2
Motion System Formulas4
Area under profile curve represents distance moved
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3
MOTION SYSTEMFORMULAS
Inertia Acceleration Velocity and Required Torque
Jin mout
ain aout ωin vout
Tin Fout
radsec 2
radsec
ain = 2 πa
L
ωin = 2 π v
L
Jin = m 2 πL 2
X η1 + JS Kg-m2
Τin= Τa+Τf +Τg+ΤD N-m
Τa= Jin ain
Τf =cosOslashmgμL
2 π ηN-m
N-m
Τg =sinOslashmgL
2 π ηN-m
ΤD = dragpreloadrefer to manufacturerrsquos data
N-m
Lead Screw System
Belt and Pulley System
ain = radsec 2
ωin = radsec Jin Jout
ain aout ωin ωout
Tin Tout
Jin = NJout
2X η
1 + JP1 Kg-m2
N-mFor a 1st approximation analysis JP1 JP2 and JB can be disregarded For higher precision pulley and belt inertias should be included as well as the reflected inertia of these components
+ JP2+ JB
(aout)(N) (ωout)(N)
Τin = NΤout X η
1
Motion System Formulas5
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4
MOTION SYSTEMFORMULAS
Motion System Formulas6
Gearbox System
radsec 2
ωin = radsec
Jin = NJout
2X η
1
+
Kg-m2
N-m
Drag torque can be significant (TD) depending on the viscosity of the lubricant For a first approximation analysis this can be left outGearbox inertia may be found in manufacturerrsquos data sheets or calculated using gear dimensions and materials mass
JGB
(ωout)(N) Τin = N
Τout X η1
+
ΤD
ΤRMS = Τ12t 1 + Τ2
2t2 + Τ32t 3 + Τn
2t n
t 1 + t 2 + t 3 + t n + tdwell
TωLinear Power =
FSt
= watts
Rotary Power = = watts
Mechanical Power and RMS Torque
Jin Jout
ain aout ωin ωout
Tin Tout
ain = (aout)(N)
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5
MOTION SYSTEMFORMULAS
Assuming IO is very small it can be ignored for a quick approximation of motor no-load speed If more accurate results are needed IO will also need to be adjuisted based on the motor Viscous Damping Factor (Dv) As the no-load speed increases with increased terminal voltage on a given motor IO will also increase based on the motorrsquos Dv value found on the manufacturerrsquos motor data sheet
Motor No Load SpeedVT ndash (IO x Rmt)no = 95493 x
KE
ndash ndash ndash ndash ndash ndash ndash ndash ndash ndash ndash ndash ndash
ndash ndash ndash
ndash ndash ndash
BreakawayTorque
Viscous Damping Torque (Dv)
Coulomb Friction
Τ
ωο
Motion System Formulas7
DC Motor Formulas
Motor Locked Rotor Current Locked Rotor Torque
ILR =VT
Rmt
TLR = ILR x KT
Motor Regulation
RmtRm = 95493 x
KE x KT
= x KT
VT
Rmt
ndash Theoretical using motor constants
nORm = TLR
ndash Regulation calculated using test data
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6
MOTION SYSTEMFORMULAS
Motor Power Relationships
ndash Watts lost due to winding resistance
Ploss = I2 x Rmt
ndash Output power at any point on the motor curve
Pout = ω x Tndash Motor maximum output power
Pmax = 025 x ωo x TLR
ndash Motor maximum output power (Theoretical)
Pmax = 025 x VT2
Rmt
Temperature Effects on Motor Constants
ndash Change in terminal resistance
Rmt(f) = Rmt(i) x [ 1 + αconductor (Θf ndash Θi) ]αconductor = 00040degC (copper)
K(f) = K(i) x [ 1 + αmagnet (Θf ndash Θi) ]ndash Change in torque constant and voltage constant ( KT = KE )
Magnetic Material
CeramicSmCoAlNiCoNdFeB
Θmax (degC)300degC300degC540degC150degC
The magnetic material values above represent average figures for particular material classes and estimating performance If exact values are needed consult data sheets for a specific magnet grade
Motion System Formulas8
αmagnet (degC)-00020degC-00004degC-00002degC-00012degC
In the case of a mechanically commutated motor with graphite brushes this analysis will result in a slight error because of the negative temperature coefficient of carbon For estimation purposes this can be ignored but for a more rigorous analysis the behavior of the brush material must be taken into account This is not an issue when evaluating a brushless dc motor
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
7
MOTION SYSTEMFORMULAS
Motion System Formulas9
For PMDC motors (brush type) brush drop can be approximated as a constant resistance connected in series with the armature winding This series resistance is included in the Rmt factor found on manufacturerrsquos data sheet
Copper graphiteSilver graphiteElectrographite
02 to 04 Ω 02 to 04 Ω 08 to 10 Ω
Motor Brush Resistance
Brush Material Resistance
DC Motor Thermal Resistance and Temperature Rise
ndash Thermal resistance
ndash Motor temperature rise
Θm = Θr + Θa
ndash OR ndash
Temperature Rise degCRth =
Machine Losses W
Θr = Rth x I2 x Rmt
Θr = Rth x I
2 x Rmt
1 ndash (Rth x I2 x Rmt x α)
ndash OR ndash
Θr = Rth x(TC
Km)
2
ndash Final motor temperature
Use caution when applying these formulas Depending on the conditions used to determine motor specifications only one formula should be used Contact Appications Engineering for questions
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
8
MOTION SYSTEMFORMULAS
Motion System Formulas9
For PMDC motors (brush type) brush drop can be approximated as a constant resistance connected in series with the armature winding This series resistance is included in the Rmt factor found on manufacturerrsquos data sheet
Copper graphiteSilver graphiteElectrographite
02 to 04 Ω 02 to 04 Ω 08 to 10 Ω
Motor Brush Resistance
Brush Material Resistance
DC Motor Thermal Resistance and Temperature Rise
ndash Thermal resistance
ndash Motor temperature rise
Θm = Θr + Θa
ndash OR ndash
Temperature Rise degCRth =
Machine Losses W
Θr = Rth x I2 x Rmt
Θr = Rth x I
2 x Rmt
1 ndash (Rth x I2 x Rmt x α)
ndash OR ndash
Θr = Rth x(TC
Km)
2
ndash Final motor temperature
Use caution when applying these formulas Depending on the conditions used to determine motor specifications only one formula should be used Contact Appications Engineering for questions
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
9
MOTION SYSTEMFORMULAS
Stepper Motor Formulas
Required ldquoTime Offrdquo using an LR drive (constant voltage drive)
180Step Oslash = (phases)(rotor teeth)
ndash Motor step angle
ndash RPM (revolutionsminute) to SPS (stepssecond)
Maximum accuracy of a step motor system has nothing to do with maximum resolution of a step motor system Maximum accuracy is always based on the accuracy of 1 full step but maximum resolution is based on all system components including lead screw encoder stepper motor and step mode on the controller
ndash SPS (stepssecond) to RPM (revolutionsminute)(SPS) x (Step Oslash)
RPM = 6
(RPM) x 6 SPS = (Step Oslash)
ndash Travel per step
mstep =L
positionsrev
ndash Overdriving capability
Required ldquoTime Offrdquo using a chopper drive (constant current drive)
Time Off = (Time On) (Applied Voltage)2
(Rated Voltage)ndash Time On
Time Off = (Time On) (Applied Current)2
(Rated Current)ndash Time On
It is not unusual for a customer to drive Haydon Kerk stepper motors beyond their rated power to obtain the most force in the smallest package size Precautions must be taken to prevent the motor from exceeding its maximum temperature The ldquoonrdquo time should not exceed 2 - 3 minutes As general rule of thumb driving a motor at 25 to 3x its rated voltage or current may result in wasted energy and erratic behavior due to saturation
Special notes for can-stack motors bull more easily saturated due to less active material (steel) bull more iron losses due to unlaminated steel
Motion System Formulas10
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10
MOTION SYSTEMFORMULAS
Lead Screw Performance Characteristics
End Mounting Rigidity
Impact Impact
ScrewLength
Increase In Affects Increase In Affects
Critical speed
Compression load
Critical speed
Inertia
Compression load
Stiffness
Spring Rate
Load capacity
Drive torque
Angular velocity
Load cpacity
Positioning accuracy
ScrewDiameter
Lead
Load
Preload
Nut length
SI Unit Systems
LengthMassTimeElectric currentThermodynamic temperatureLuminous intensityAmount of substance
meterkilogramsecondamperekelvincandelamole
mKgsAKcdmol
Base Units
Derived Unitsareavolumefrequencymass density (density)speed velocityangular velocityaccelerationangular accelerationforcepressure (mechanical stress)kinematic viscositydynamic viscositywork energy quantity of heatpowerentropyspecific heat capacitythermal conductivity
square metercubic meterhertzkilogram per cubic metermeter per secondradian per secondmeter per second squaredradian per second squarednewtonpascalsquare meter per secondnewton-second per square meter joulewattjoule per kelvinjoule per kilogram kelvinwatt per meter kelvin
m2
m3
HzKgm3
msradsms2
rads2
NPam2sN-sm2
JWJKJKg-K)W(m-K)
Quantity Name of Unit Symbol
Motion System Reference Tables11
Critical speed
Compression load
System stiffness
Life
Positioning accuracy
System stiffness
Drag torque
Load capacity
Stiffness
sndash1
Kg-ms2
Nm2
N-mJs or N-ms
ExamplesAn INCREASE ( ) in screw length results in a DECREASE ( ) in critical speed
An INCREASE in screw diameter results in an INCREASE in critical speed
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11
MOTION SYSTEMREFERENCE TABLES
t 1 t 2 t 3 t dwell
vTime
vpk
a -a
MassMove distanceMove timeDwell timeMotion profileScrew speed limitLoad supportOrientationRotary to linear conversionAmbient temperature
9 Kg200 mm10 s05 s13 13 13 Trapezoidal1000 RPMμ = 001VerticalTFE lead screw 8 mm diameter 275 mm long30degC
ControlDrive Power Supply
4 quadrant with encoder32 VDC 35 Arms50 A peak
0333 s 0333 s 0333 s 05 s
vPK =3S2t =
(3)(02m)(2)(10 s) =
06m2 s =
03ms
Linear Velocity
Linear Acceleration
a = vf - vi
t = 03ms - 0 ms 0333 s 0901ms2 =
1st step is to thoroughly understand system constraints and motion profile requirements
Linear Motion ExampleData
Motion System Application Example12
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12
MOTION SYSTEMAPPLICATION EXAMPLE
Lmin =60vPK
n =(60)(03ms)1000 revmin = 0018mrev = 18mmrev
Refer to Kerk screw chart ndash closest lead in an 8mm screw diameter is
2032mmrev = 002032 mrevη = 086 free wheeling nut TFE coated screw
JS = 388 x 10ndash 7 Kg-m2 Other critical lead screw considerations bull Column loading bull Critical speed
Jm aa
ωPKvPK
TF
= ωPK = 2 π vPK
L=
(2π)(03ms)002032 mrev
9276 rads
a = =2πaL
=(2π)(0901ms2)002032 mrev
2786 rads2
Linear to Rotary Conversion ndash Velocity Acceleration Inertia Torque
Minimum screw lead to maintain lt 1000 RPM shaft speed
Linear to Rotary Conversion Velocity
Linear to Rotary Conversion Acceleration
Motion System Application Example13
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
13
MOTION SYSTEMFORMULAS
Jin= m 2 πL 2
X η1 + JS= 9Kg
2 π002032 m 2
X 0861 + 388x10ndash 7Kg-m2
= 1095 X 10ndash 5 Kg-m2 + 388x10ndash 7 Kg-m2
= 1134 x 10ndash 5 Kg-m2
Τin= Τa+Τf +Τg+ΤD
Τa = Jin ain
Τg =sinOslashmgL
2 π η
= (1134 x 10ndash 5 Kg-m2)(2786 rads2)
= 00316 Nm
Τf =cosOslashmgμL
2 π η= (cos90)(9 Kg)(98ms2)(001)(002032m)
2 π (086)= 0 Nm
= (sin90)(9 Kg)(98ms2)(002032m)2 π (086)
=1792 Nm
5404 = 03316 Nm
ΤDSince a free-wheeling lead screw nut was used there is no preload force If an anti-backlash nut was used ΤD would be gt0 due to preload force Always reference manufacturerrsquos data for more information
= 0 Nm
Linear to Rotary Conversion Torque
Linear to Rotary Conversion Inertia
Motion System Application Example14
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
14
MOTION SYSTEMFORMULAS
t 1 t 2 t 3 t dwell Time0333 s 0333 s 0333 s 05 s
Rotary Motion Profile
ωPK
ω
= 9276 rads a = 2786 rads2
Τ1= Τa+Τf +Τg+ΤD
= (00316 Nm) + (0 Nm) + (03316 Nm) + (0 Nm)
= 03632 Nm
Τ2= Τa+Τf +Τg+ΤD
= (0 Nm) + (0 Nm) + (03316 Nm) + (0 Nm)
= 03316 Nm
Τ3= Τa+Τf +Τg+ΤD
= (00316 Nm) + (0 Nm) + (03316 Nm) + (0 Nm)
= 03632 Nm
Τ1
Τ2
Τ3
Motion System Application Example15
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
15
MOTION SYSTEMAPPLICATION EXAMPLE
Lead Screw Input Requirements
= 9276 rads= 2786 rads2
= 03632 Nm= 03316 Nm= ndash 03632 Nm= 0333 s= 0333 s= 0333 s= 05 s= 3369 W= 3076 W
ωPKaΤ1Τ2Τ3
t1t2t3tdwell
PPKPCV
PPK = (Τ1)(ωPK)
= (03632 Nm)(9276 rads)= 3369 W
PCV = (Τ2)(ωPK)
= (03316 Nm)(9276 rads)= 3076 W
RMS Torque Requirement Lead Screw Input
=
0333 s + 0333 s + 0333 s + 05 s
(03632 Nm)2(0333 s)+(03316 Nm)2(0333 s)+(- 03632 Nm)2(0333 s)
ΤRMS = Τ12t 1 + Τ2
2t2 + Τ32t 3
t 1 + t 2 + t 3 + tdwell
15
00439 + 00366 + 00439 = 00829 =
ΤRMS = 02879 Nm
Motion System Application Example16
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
16
MOTION SYSTEMAPPLICATION EXAMPLE
Load Parameters at the Lead Screw Shaft
= 9276 rads= 2786 rads2
= 03632 Nm= 03316 Nm= ndash 03632 Nm= 02879 Nm= 3369 W= 3076 W
ωPKaΤ1Τ2Τ3ΤRMSPPKPCV
Motor SelectionA direct-drive motor can be used however it would be large and expensive For example a DC057B-3 brush motor
would easily meet continuous torque and continuous speed as well as a 181 load-to-rotor inertia This motor however has significantly more output power than needed at 128 watts
A smaller motor with a transmission would optimize the system Look for a motor starting at a rated power around 40 watts Pittman DC040B-6
DC040B-6Reference VoltageContinuous TorqueRated CurrentRated PowerTorque ConstantVoltage ConstantTerminal ResistanceMax Winding TemperatureRotor Inertia
240 V00812 Nm236 A37 W0042 NmA0042 Vrads185 Ω155degC847 x 10ndash6 Kg-m2
Motor TransmissionLead Screw Support Rails
ΤRMS
PPK
ωPK
= 02879 Nm= 3369 W= 9276 rads
M
Other Motor Considerationsndash Type stepper brush BLDCndash Input speed (high input speed will result in high noise levels)ndash Total motor footprintndash Ambient temperaturendash Environmental conditionsndash Load-to-rotor inertiandash Encoder ready needed
Motion System Application Example17
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
17
MOTION SYSTEMAPPLICATION EXAMPLE
Gearbox Transmission Selection
When selecting a reduction ratio keep in mind that the RMS required torque at the motor shaft needs to fall below the continuous rated output torque of the motor
G30 A - Planetary GearboxMaximum output load 247 Nm
TRMS( Lead Screw)
02879 Nm
02879 Nm02879 Nm
Reduction
41
5161
Efficiency
090
090090
TRMS( Motor)
00798 Nm
00640 Nm00533 Nm
A 51 gearbox would allow about a 27 safety margin between what the system requires and the continuous torque output of the motor
System Summary 51η = 090
M
= 00807 Nm= 00737 Nm= ndash 00807 Nm= 00640 Nm= 4638 rads= 504 x 10 ndash 6 Kg-m2
= 3743 W
Τ1Τ2Τ3ΤRMS
PPK
ωPK
J
= 03632 Nm= 03316 Nm= ndash 03632 Nm= 02879 Nm= 9276 rads= 1134 x 10 ndash 5 Kg-m2
= 3369 W
Τ1Τ2Τ3ΤRMS
PPK
ωPK
J
Motion System Application Example18
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
18
MOTION SYSTEMAPPLICATION EXAMPLE
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
19
Motion Profile at the Motor
t 1 t 2 t 3 t dwell Time0333 s 0333 s 0333 s 05 s
ωΤ1
Τ2
Τ3= 4638 rads= 4429 RPM= ΤPK = 00807 Nm
ωPK
Τ1
nPK
Drive and Power Supply Requirements
Peak Current Required
IPK =TPK
KT+ IO =
00807 Nm0042 NmA
+ 0180 A = 210 A
RMS Current Required
IRMS =TRMS
KT+ IO =
00640 Nm0042 NmA
+ 0180 A = 170 A
Minimum bus Voltage Required
Vbus= IPKRmt+ ωPKKE = (210A)(185Ω) + (4638 rads)(0042 Vrads)= 389 V + 1948 V= 2337 V
0100 Nm 0200 Nm 0300 Nm 0400 Nm 0500 Nm 0600 Nm 0700 Nm
30 VDC
0100 Nm 0200 Nm 0300 Nm 0500 Nm 0600 Nm 0700 Nm
24 VDC
700060005000
RPM 4000300020001000
0400 Nm
TRMS
TPK
700060005000
RPM 4000300020001000
TRMS
TPK
Performance using 24V reference voltage
Performance using a 30V bus voltage
A 30 VDC bus will meet the application requirements as wells supply a margin of safety
MOTION SYSTEMAPPLICATION EXAMPLE
0100 Nm 0200 Nm 0300 Nm 0400 Nm 0500 Nm 0600 Nm 0700 Nm
30 VDC
0100 Nm 0200 Nm 0300 Nm 0500 Nm 0600 Nm 0700 Nm
24 VDC
700060005000
RPM 4000300020001000
0400 Nm
TRMS
TPK
700060005000
RPM 4000300020001000
TRMS
TPK
Ambient temperature Rated motor temperature Temperature rise Motor temperature
= 30degC= 155degC= 7638degC= 10638degC
Θa
Θm
ΘrΘrated
=Rth x IRMS
2 x Rmt
1 ndash (Rth x IRMS2 x Rmt x 000392degC)
Θr
=11degCω x 170A2
x 185Ω 1 ndash (11degCω x 170A2
x 185Ω x 000392degC)
= 5881 1 ndash (023)
= 5881 077
=
Θm = Θr + Θa = 7638degC + 30degC = 10638degC
7638degC
Performance using 24V reference voltage
Performance using a 30V bus voltage
Will the motor meet the temperature rise caused by the application
Motion System Application Example20
A 30 VDC bus will meet the application requirements as wells supply a margin of safety
203 756 7441 1500 Meriden Road Waterbury CT USA 06705
267 933 2105 534 Godshall DriveHarleysville PA USA 19438
wwwHaydonKerkPittmancom
copy All rights reserved AMETEK Advanced Motion Solutions No part of this doc-ument or technical information can be used reproduced or altered in any form without approval or proper authorization from AMETEK Inc and global affiliates
NOW ASK us the tough motion control questions ONLINE
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MOTION SYSTEMAPPLICATION EXAMPLE
(V)(I)Watts
F x St =
+ +MotorController
DrivePowerSupply
Mechanical Transmission
Lead Screw Support Rails
WattsLost
(V)(I)Watts
WattsLost
WattsLost
WattsLost
(T)(ω)Watts
(T)(ω)Watts
WattsLost
All analysis begins with the linear motion profile of the output and the force required
to move the load
t 1 t 2 t 3t move
Total Cycle Time
t dwell
vTime
vpk
a -a
13 13 13 Trapazoidal Motion Profile
Optimized for minimum power
t 1 = t 2= t 3
vPK =3S2t
S is the total move distancet is the total move timeArea under profile curve represents distance moved
Watts Out
Motion System Formulas
Any motion system should be broken down into its individual components
Motion Profiles
3
(V)(I)Watts
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
2
MOTION SYSTEMFORMULAS
Triangular Motion Profiles
Optimized for minimum acceleration slope
t 1 = t 2
vPK =2St
S is the total move distancet is the total move time 13 13 13
Triangular
Move Profile Peak Velocity (VPK) Acceleration (a )
t 1 t 2t move
Total Cycle Time
t dwell
vTime
a -avPK
Complex Motion Profile
Linear to Rotary Formulas through a Lead Screw
(1) s = ndashndashndashndashvf - vi
2t
(2) vf = vi + a t
(3) s = vi t + ndash at 212
(4) 2 as = vf 2
- vi 2
(5) a = ( vf - vi) ( tf - ti )
= ΔvΔt
a = = rads 2 2 π a
L
ω = = rads 2 π v
L
n = = RPM 60 v
L
3 known quantities are needed to solve for the other 2 Each segment calculated individually
Linear Motion Formulas
t 1 t 2 t 3t move
Total Cycle Time
t dwell
vTime
v1
a 1
-a3
a 2
t 4 t 5
v2
Motion System Formulas4
Area under profile curve represents distance moved
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3
MOTION SYSTEMFORMULAS
Inertia Acceleration Velocity and Required Torque
Jin mout
ain aout ωin vout
Tin Fout
radsec 2
radsec
ain = 2 πa
L
ωin = 2 π v
L
Jin = m 2 πL 2
X η1 + JS Kg-m2
Τin= Τa+Τf +Τg+ΤD N-m
Τa= Jin ain
Τf =cosOslashmgμL
2 π ηN-m
N-m
Τg =sinOslashmgL
2 π ηN-m
ΤD = dragpreloadrefer to manufacturerrsquos data
N-m
Lead Screw System
Belt and Pulley System
ain = radsec 2
ωin = radsec Jin Jout
ain aout ωin ωout
Tin Tout
Jin = NJout
2X η
1 + JP1 Kg-m2
N-mFor a 1st approximation analysis JP1 JP2 and JB can be disregarded For higher precision pulley and belt inertias should be included as well as the reflected inertia of these components
+ JP2+ JB
(aout)(N) (ωout)(N)
Τin = NΤout X η
1
Motion System Formulas5
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4
MOTION SYSTEMFORMULAS
Motion System Formulas6
Gearbox System
radsec 2
ωin = radsec
Jin = NJout
2X η
1
+
Kg-m2
N-m
Drag torque can be significant (TD) depending on the viscosity of the lubricant For a first approximation analysis this can be left outGearbox inertia may be found in manufacturerrsquos data sheets or calculated using gear dimensions and materials mass
JGB
(ωout)(N) Τin = N
Τout X η1
+
ΤD
ΤRMS = Τ12t 1 + Τ2
2t2 + Τ32t 3 + Τn
2t n
t 1 + t 2 + t 3 + t n + tdwell
TωLinear Power =
FSt
= watts
Rotary Power = = watts
Mechanical Power and RMS Torque
Jin Jout
ain aout ωin ωout
Tin Tout
ain = (aout)(N)
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5
MOTION SYSTEMFORMULAS
Assuming IO is very small it can be ignored for a quick approximation of motor no-load speed If more accurate results are needed IO will also need to be adjuisted based on the motor Viscous Damping Factor (Dv) As the no-load speed increases with increased terminal voltage on a given motor IO will also increase based on the motorrsquos Dv value found on the manufacturerrsquos motor data sheet
Motor No Load SpeedVT ndash (IO x Rmt)no = 95493 x
KE
ndash ndash ndash ndash ndash ndash ndash ndash ndash ndash ndash ndash ndash
ndash ndash ndash
ndash ndash ndash
BreakawayTorque
Viscous Damping Torque (Dv)
Coulomb Friction
Τ
ωο
Motion System Formulas7
DC Motor Formulas
Motor Locked Rotor Current Locked Rotor Torque
ILR =VT
Rmt
TLR = ILR x KT
Motor Regulation
RmtRm = 95493 x
KE x KT
= x KT
VT
Rmt
ndash Theoretical using motor constants
nORm = TLR
ndash Regulation calculated using test data
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6
MOTION SYSTEMFORMULAS
Motor Power Relationships
ndash Watts lost due to winding resistance
Ploss = I2 x Rmt
ndash Output power at any point on the motor curve
Pout = ω x Tndash Motor maximum output power
Pmax = 025 x ωo x TLR
ndash Motor maximum output power (Theoretical)
Pmax = 025 x VT2
Rmt
Temperature Effects on Motor Constants
ndash Change in terminal resistance
Rmt(f) = Rmt(i) x [ 1 + αconductor (Θf ndash Θi) ]αconductor = 00040degC (copper)
K(f) = K(i) x [ 1 + αmagnet (Θf ndash Θi) ]ndash Change in torque constant and voltage constant ( KT = KE )
Magnetic Material
CeramicSmCoAlNiCoNdFeB
Θmax (degC)300degC300degC540degC150degC
The magnetic material values above represent average figures for particular material classes and estimating performance If exact values are needed consult data sheets for a specific magnet grade
Motion System Formulas8
αmagnet (degC)-00020degC-00004degC-00002degC-00012degC
In the case of a mechanically commutated motor with graphite brushes this analysis will result in a slight error because of the negative temperature coefficient of carbon For estimation purposes this can be ignored but for a more rigorous analysis the behavior of the brush material must be taken into account This is not an issue when evaluating a brushless dc motor
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7
MOTION SYSTEMFORMULAS
Motion System Formulas9
For PMDC motors (brush type) brush drop can be approximated as a constant resistance connected in series with the armature winding This series resistance is included in the Rmt factor found on manufacturerrsquos data sheet
Copper graphiteSilver graphiteElectrographite
02 to 04 Ω 02 to 04 Ω 08 to 10 Ω
Motor Brush Resistance
Brush Material Resistance
DC Motor Thermal Resistance and Temperature Rise
ndash Thermal resistance
ndash Motor temperature rise
Θm = Θr + Θa
ndash OR ndash
Temperature Rise degCRth =
Machine Losses W
Θr = Rth x I2 x Rmt
Θr = Rth x I
2 x Rmt
1 ndash (Rth x I2 x Rmt x α)
ndash OR ndash
Θr = Rth x(TC
Km)
2
ndash Final motor temperature
Use caution when applying these formulas Depending on the conditions used to determine motor specifications only one formula should be used Contact Appications Engineering for questions
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
8
MOTION SYSTEMFORMULAS
Motion System Formulas9
For PMDC motors (brush type) brush drop can be approximated as a constant resistance connected in series with the armature winding This series resistance is included in the Rmt factor found on manufacturerrsquos data sheet
Copper graphiteSilver graphiteElectrographite
02 to 04 Ω 02 to 04 Ω 08 to 10 Ω
Motor Brush Resistance
Brush Material Resistance
DC Motor Thermal Resistance and Temperature Rise
ndash Thermal resistance
ndash Motor temperature rise
Θm = Θr + Θa
ndash OR ndash
Temperature Rise degCRth =
Machine Losses W
Θr = Rth x I2 x Rmt
Θr = Rth x I
2 x Rmt
1 ndash (Rth x I2 x Rmt x α)
ndash OR ndash
Θr = Rth x(TC
Km)
2
ndash Final motor temperature
Use caution when applying these formulas Depending on the conditions used to determine motor specifications only one formula should be used Contact Appications Engineering for questions
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
9
MOTION SYSTEMFORMULAS
Stepper Motor Formulas
Required ldquoTime Offrdquo using an LR drive (constant voltage drive)
180Step Oslash = (phases)(rotor teeth)
ndash Motor step angle
ndash RPM (revolutionsminute) to SPS (stepssecond)
Maximum accuracy of a step motor system has nothing to do with maximum resolution of a step motor system Maximum accuracy is always based on the accuracy of 1 full step but maximum resolution is based on all system components including lead screw encoder stepper motor and step mode on the controller
ndash SPS (stepssecond) to RPM (revolutionsminute)(SPS) x (Step Oslash)
RPM = 6
(RPM) x 6 SPS = (Step Oslash)
ndash Travel per step
mstep =L
positionsrev
ndash Overdriving capability
Required ldquoTime Offrdquo using a chopper drive (constant current drive)
Time Off = (Time On) (Applied Voltage)2
(Rated Voltage)ndash Time On
Time Off = (Time On) (Applied Current)2
(Rated Current)ndash Time On
It is not unusual for a customer to drive Haydon Kerk stepper motors beyond their rated power to obtain the most force in the smallest package size Precautions must be taken to prevent the motor from exceeding its maximum temperature The ldquoonrdquo time should not exceed 2 - 3 minutes As general rule of thumb driving a motor at 25 to 3x its rated voltage or current may result in wasted energy and erratic behavior due to saturation
Special notes for can-stack motors bull more easily saturated due to less active material (steel) bull more iron losses due to unlaminated steel
Motion System Formulas10
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10
MOTION SYSTEMFORMULAS
Lead Screw Performance Characteristics
End Mounting Rigidity
Impact Impact
ScrewLength
Increase In Affects Increase In Affects
Critical speed
Compression load
Critical speed
Inertia
Compression load
Stiffness
Spring Rate
Load capacity
Drive torque
Angular velocity
Load cpacity
Positioning accuracy
ScrewDiameter
Lead
Load
Preload
Nut length
SI Unit Systems
LengthMassTimeElectric currentThermodynamic temperatureLuminous intensityAmount of substance
meterkilogramsecondamperekelvincandelamole
mKgsAKcdmol
Base Units
Derived Unitsareavolumefrequencymass density (density)speed velocityangular velocityaccelerationangular accelerationforcepressure (mechanical stress)kinematic viscositydynamic viscositywork energy quantity of heatpowerentropyspecific heat capacitythermal conductivity
square metercubic meterhertzkilogram per cubic metermeter per secondradian per secondmeter per second squaredradian per second squarednewtonpascalsquare meter per secondnewton-second per square meter joulewattjoule per kelvinjoule per kilogram kelvinwatt per meter kelvin
m2
m3
HzKgm3
msradsms2
rads2
NPam2sN-sm2
JWJKJKg-K)W(m-K)
Quantity Name of Unit Symbol
Motion System Reference Tables11
Critical speed
Compression load
System stiffness
Life
Positioning accuracy
System stiffness
Drag torque
Load capacity
Stiffness
sndash1
Kg-ms2
Nm2
N-mJs or N-ms
ExamplesAn INCREASE ( ) in screw length results in a DECREASE ( ) in critical speed
An INCREASE in screw diameter results in an INCREASE in critical speed
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11
MOTION SYSTEMREFERENCE TABLES
t 1 t 2 t 3 t dwell
vTime
vpk
a -a
MassMove distanceMove timeDwell timeMotion profileScrew speed limitLoad supportOrientationRotary to linear conversionAmbient temperature
9 Kg200 mm10 s05 s13 13 13 Trapezoidal1000 RPMμ = 001VerticalTFE lead screw 8 mm diameter 275 mm long30degC
ControlDrive Power Supply
4 quadrant with encoder32 VDC 35 Arms50 A peak
0333 s 0333 s 0333 s 05 s
vPK =3S2t =
(3)(02m)(2)(10 s) =
06m2 s =
03ms
Linear Velocity
Linear Acceleration
a = vf - vi
t = 03ms - 0 ms 0333 s 0901ms2 =
1st step is to thoroughly understand system constraints and motion profile requirements
Linear Motion ExampleData
Motion System Application Example12
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12
MOTION SYSTEMAPPLICATION EXAMPLE
Lmin =60vPK
n =(60)(03ms)1000 revmin = 0018mrev = 18mmrev
Refer to Kerk screw chart ndash closest lead in an 8mm screw diameter is
2032mmrev = 002032 mrevη = 086 free wheeling nut TFE coated screw
JS = 388 x 10ndash 7 Kg-m2 Other critical lead screw considerations bull Column loading bull Critical speed
Jm aa
ωPKvPK
TF
= ωPK = 2 π vPK
L=
(2π)(03ms)002032 mrev
9276 rads
a = =2πaL
=(2π)(0901ms2)002032 mrev
2786 rads2
Linear to Rotary Conversion ndash Velocity Acceleration Inertia Torque
Minimum screw lead to maintain lt 1000 RPM shaft speed
Linear to Rotary Conversion Velocity
Linear to Rotary Conversion Acceleration
Motion System Application Example13
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13
MOTION SYSTEMFORMULAS
Jin= m 2 πL 2
X η1 + JS= 9Kg
2 π002032 m 2
X 0861 + 388x10ndash 7Kg-m2
= 1095 X 10ndash 5 Kg-m2 + 388x10ndash 7 Kg-m2
= 1134 x 10ndash 5 Kg-m2
Τin= Τa+Τf +Τg+ΤD
Τa = Jin ain
Τg =sinOslashmgL
2 π η
= (1134 x 10ndash 5 Kg-m2)(2786 rads2)
= 00316 Nm
Τf =cosOslashmgμL
2 π η= (cos90)(9 Kg)(98ms2)(001)(002032m)
2 π (086)= 0 Nm
= (sin90)(9 Kg)(98ms2)(002032m)2 π (086)
=1792 Nm
5404 = 03316 Nm
ΤDSince a free-wheeling lead screw nut was used there is no preload force If an anti-backlash nut was used ΤD would be gt0 due to preload force Always reference manufacturerrsquos data for more information
= 0 Nm
Linear to Rotary Conversion Torque
Linear to Rotary Conversion Inertia
Motion System Application Example14
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14
MOTION SYSTEMFORMULAS
t 1 t 2 t 3 t dwell Time0333 s 0333 s 0333 s 05 s
Rotary Motion Profile
ωPK
ω
= 9276 rads a = 2786 rads2
Τ1= Τa+Τf +Τg+ΤD
= (00316 Nm) + (0 Nm) + (03316 Nm) + (0 Nm)
= 03632 Nm
Τ2= Τa+Τf +Τg+ΤD
= (0 Nm) + (0 Nm) + (03316 Nm) + (0 Nm)
= 03316 Nm
Τ3= Τa+Τf +Τg+ΤD
= (00316 Nm) + (0 Nm) + (03316 Nm) + (0 Nm)
= 03632 Nm
Τ1
Τ2
Τ3
Motion System Application Example15
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15
MOTION SYSTEMAPPLICATION EXAMPLE
Lead Screw Input Requirements
= 9276 rads= 2786 rads2
= 03632 Nm= 03316 Nm= ndash 03632 Nm= 0333 s= 0333 s= 0333 s= 05 s= 3369 W= 3076 W
ωPKaΤ1Τ2Τ3
t1t2t3tdwell
PPKPCV
PPK = (Τ1)(ωPK)
= (03632 Nm)(9276 rads)= 3369 W
PCV = (Τ2)(ωPK)
= (03316 Nm)(9276 rads)= 3076 W
RMS Torque Requirement Lead Screw Input
=
0333 s + 0333 s + 0333 s + 05 s
(03632 Nm)2(0333 s)+(03316 Nm)2(0333 s)+(- 03632 Nm)2(0333 s)
ΤRMS = Τ12t 1 + Τ2
2t2 + Τ32t 3
t 1 + t 2 + t 3 + tdwell
15
00439 + 00366 + 00439 = 00829 =
ΤRMS = 02879 Nm
Motion System Application Example16
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16
MOTION SYSTEMAPPLICATION EXAMPLE
Load Parameters at the Lead Screw Shaft
= 9276 rads= 2786 rads2
= 03632 Nm= 03316 Nm= ndash 03632 Nm= 02879 Nm= 3369 W= 3076 W
ωPKaΤ1Τ2Τ3ΤRMSPPKPCV
Motor SelectionA direct-drive motor can be used however it would be large and expensive For example a DC057B-3 brush motor
would easily meet continuous torque and continuous speed as well as a 181 load-to-rotor inertia This motor however has significantly more output power than needed at 128 watts
A smaller motor with a transmission would optimize the system Look for a motor starting at a rated power around 40 watts Pittman DC040B-6
DC040B-6Reference VoltageContinuous TorqueRated CurrentRated PowerTorque ConstantVoltage ConstantTerminal ResistanceMax Winding TemperatureRotor Inertia
240 V00812 Nm236 A37 W0042 NmA0042 Vrads185 Ω155degC847 x 10ndash6 Kg-m2
Motor TransmissionLead Screw Support Rails
ΤRMS
PPK
ωPK
= 02879 Nm= 3369 W= 9276 rads
M
Other Motor Considerationsndash Type stepper brush BLDCndash Input speed (high input speed will result in high noise levels)ndash Total motor footprintndash Ambient temperaturendash Environmental conditionsndash Load-to-rotor inertiandash Encoder ready needed
Motion System Application Example17
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17
MOTION SYSTEMAPPLICATION EXAMPLE
Gearbox Transmission Selection
When selecting a reduction ratio keep in mind that the RMS required torque at the motor shaft needs to fall below the continuous rated output torque of the motor
G30 A - Planetary GearboxMaximum output load 247 Nm
TRMS( Lead Screw)
02879 Nm
02879 Nm02879 Nm
Reduction
41
5161
Efficiency
090
090090
TRMS( Motor)
00798 Nm
00640 Nm00533 Nm
A 51 gearbox would allow about a 27 safety margin between what the system requires and the continuous torque output of the motor
System Summary 51η = 090
M
= 00807 Nm= 00737 Nm= ndash 00807 Nm= 00640 Nm= 4638 rads= 504 x 10 ndash 6 Kg-m2
= 3743 W
Τ1Τ2Τ3ΤRMS
PPK
ωPK
J
= 03632 Nm= 03316 Nm= ndash 03632 Nm= 02879 Nm= 9276 rads= 1134 x 10 ndash 5 Kg-m2
= 3369 W
Τ1Τ2Τ3ΤRMS
PPK
ωPK
J
Motion System Application Example18
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18
MOTION SYSTEMAPPLICATION EXAMPLE
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19
Motion Profile at the Motor
t 1 t 2 t 3 t dwell Time0333 s 0333 s 0333 s 05 s
ωΤ1
Τ2
Τ3= 4638 rads= 4429 RPM= ΤPK = 00807 Nm
ωPK
Τ1
nPK
Drive and Power Supply Requirements
Peak Current Required
IPK =TPK
KT+ IO =
00807 Nm0042 NmA
+ 0180 A = 210 A
RMS Current Required
IRMS =TRMS
KT+ IO =
00640 Nm0042 NmA
+ 0180 A = 170 A
Minimum bus Voltage Required
Vbus= IPKRmt+ ωPKKE = (210A)(185Ω) + (4638 rads)(0042 Vrads)= 389 V + 1948 V= 2337 V
0100 Nm 0200 Nm 0300 Nm 0400 Nm 0500 Nm 0600 Nm 0700 Nm
30 VDC
0100 Nm 0200 Nm 0300 Nm 0500 Nm 0600 Nm 0700 Nm
24 VDC
700060005000
RPM 4000300020001000
0400 Nm
TRMS
TPK
700060005000
RPM 4000300020001000
TRMS
TPK
Performance using 24V reference voltage
Performance using a 30V bus voltage
A 30 VDC bus will meet the application requirements as wells supply a margin of safety
MOTION SYSTEMAPPLICATION EXAMPLE
0100 Nm 0200 Nm 0300 Nm 0400 Nm 0500 Nm 0600 Nm 0700 Nm
30 VDC
0100 Nm 0200 Nm 0300 Nm 0500 Nm 0600 Nm 0700 Nm
24 VDC
700060005000
RPM 4000300020001000
0400 Nm
TRMS
TPK
700060005000
RPM 4000300020001000
TRMS
TPK
Ambient temperature Rated motor temperature Temperature rise Motor temperature
= 30degC= 155degC= 7638degC= 10638degC
Θa
Θm
ΘrΘrated
=Rth x IRMS
2 x Rmt
1 ndash (Rth x IRMS2 x Rmt x 000392degC)
Θr
=11degCω x 170A2
x 185Ω 1 ndash (11degCω x 170A2
x 185Ω x 000392degC)
= 5881 1 ndash (023)
= 5881 077
=
Θm = Θr + Θa = 7638degC + 30degC = 10638degC
7638degC
Performance using 24V reference voltage
Performance using a 30V bus voltage
Will the motor meet the temperature rise caused by the application
Motion System Application Example20
A 30 VDC bus will meet the application requirements as wells supply a margin of safety
203 756 7441 1500 Meriden Road Waterbury CT USA 06705
267 933 2105 534 Godshall DriveHarleysville PA USA 19438
wwwHaydonKerkPittmancom
copy All rights reserved AMETEK Advanced Motion Solutions No part of this doc-ument or technical information can be used reproduced or altered in any form without approval or proper authorization from AMETEK Inc and global affiliates
NOW ASK us the tough motion control questions ONLINE
ASK THE EXPERTS
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MOTION SYSTEMAPPLICATION EXAMPLE
Triangular Motion Profiles
Optimized for minimum acceleration slope
t 1 = t 2
vPK =2St
S is the total move distancet is the total move time 13 13 13
Triangular
Move Profile Peak Velocity (VPK) Acceleration (a )
t 1 t 2t move
Total Cycle Time
t dwell
vTime
a -avPK
Complex Motion Profile
Linear to Rotary Formulas through a Lead Screw
(1) s = ndashndashndashndashvf - vi
2t
(2) vf = vi + a t
(3) s = vi t + ndash at 212
(4) 2 as = vf 2
- vi 2
(5) a = ( vf - vi) ( tf - ti )
= ΔvΔt
a = = rads 2 2 π a
L
ω = = rads 2 π v
L
n = = RPM 60 v
L
3 known quantities are needed to solve for the other 2 Each segment calculated individually
Linear Motion Formulas
t 1 t 2 t 3t move
Total Cycle Time
t dwell
vTime
v1
a 1
-a3
a 2
t 4 t 5
v2
Motion System Formulas4
Area under profile curve represents distance moved
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
3
MOTION SYSTEMFORMULAS
Inertia Acceleration Velocity and Required Torque
Jin mout
ain aout ωin vout
Tin Fout
radsec 2
radsec
ain = 2 πa
L
ωin = 2 π v
L
Jin = m 2 πL 2
X η1 + JS Kg-m2
Τin= Τa+Τf +Τg+ΤD N-m
Τa= Jin ain
Τf =cosOslashmgμL
2 π ηN-m
N-m
Τg =sinOslashmgL
2 π ηN-m
ΤD = dragpreloadrefer to manufacturerrsquos data
N-m
Lead Screw System
Belt and Pulley System
ain = radsec 2
ωin = radsec Jin Jout
ain aout ωin ωout
Tin Tout
Jin = NJout
2X η
1 + JP1 Kg-m2
N-mFor a 1st approximation analysis JP1 JP2 and JB can be disregarded For higher precision pulley and belt inertias should be included as well as the reflected inertia of these components
+ JP2+ JB
(aout)(N) (ωout)(N)
Τin = NΤout X η
1
Motion System Formulas5
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
4
MOTION SYSTEMFORMULAS
Motion System Formulas6
Gearbox System
radsec 2
ωin = radsec
Jin = NJout
2X η
1
+
Kg-m2
N-m
Drag torque can be significant (TD) depending on the viscosity of the lubricant For a first approximation analysis this can be left outGearbox inertia may be found in manufacturerrsquos data sheets or calculated using gear dimensions and materials mass
JGB
(ωout)(N) Τin = N
Τout X η1
+
ΤD
ΤRMS = Τ12t 1 + Τ2
2t2 + Τ32t 3 + Τn
2t n
t 1 + t 2 + t 3 + t n + tdwell
TωLinear Power =
FSt
= watts
Rotary Power = = watts
Mechanical Power and RMS Torque
Jin Jout
ain aout ωin ωout
Tin Tout
ain = (aout)(N)
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
5
MOTION SYSTEMFORMULAS
Assuming IO is very small it can be ignored for a quick approximation of motor no-load speed If more accurate results are needed IO will also need to be adjuisted based on the motor Viscous Damping Factor (Dv) As the no-load speed increases with increased terminal voltage on a given motor IO will also increase based on the motorrsquos Dv value found on the manufacturerrsquos motor data sheet
Motor No Load SpeedVT ndash (IO x Rmt)no = 95493 x
KE
ndash ndash ndash ndash ndash ndash ndash ndash ndash ndash ndash ndash ndash
ndash ndash ndash
ndash ndash ndash
BreakawayTorque
Viscous Damping Torque (Dv)
Coulomb Friction
Τ
ωο
Motion System Formulas7
DC Motor Formulas
Motor Locked Rotor Current Locked Rotor Torque
ILR =VT
Rmt
TLR = ILR x KT
Motor Regulation
RmtRm = 95493 x
KE x KT
= x KT
VT
Rmt
ndash Theoretical using motor constants
nORm = TLR
ndash Regulation calculated using test data
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
6
MOTION SYSTEMFORMULAS
Motor Power Relationships
ndash Watts lost due to winding resistance
Ploss = I2 x Rmt
ndash Output power at any point on the motor curve
Pout = ω x Tndash Motor maximum output power
Pmax = 025 x ωo x TLR
ndash Motor maximum output power (Theoretical)
Pmax = 025 x VT2
Rmt
Temperature Effects on Motor Constants
ndash Change in terminal resistance
Rmt(f) = Rmt(i) x [ 1 + αconductor (Θf ndash Θi) ]αconductor = 00040degC (copper)
K(f) = K(i) x [ 1 + αmagnet (Θf ndash Θi) ]ndash Change in torque constant and voltage constant ( KT = KE )
Magnetic Material
CeramicSmCoAlNiCoNdFeB
Θmax (degC)300degC300degC540degC150degC
The magnetic material values above represent average figures for particular material classes and estimating performance If exact values are needed consult data sheets for a specific magnet grade
Motion System Formulas8
αmagnet (degC)-00020degC-00004degC-00002degC-00012degC
In the case of a mechanically commutated motor with graphite brushes this analysis will result in a slight error because of the negative temperature coefficient of carbon For estimation purposes this can be ignored but for a more rigorous analysis the behavior of the brush material must be taken into account This is not an issue when evaluating a brushless dc motor
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
7
MOTION SYSTEMFORMULAS
Motion System Formulas9
For PMDC motors (brush type) brush drop can be approximated as a constant resistance connected in series with the armature winding This series resistance is included in the Rmt factor found on manufacturerrsquos data sheet
Copper graphiteSilver graphiteElectrographite
02 to 04 Ω 02 to 04 Ω 08 to 10 Ω
Motor Brush Resistance
Brush Material Resistance
DC Motor Thermal Resistance and Temperature Rise
ndash Thermal resistance
ndash Motor temperature rise
Θm = Θr + Θa
ndash OR ndash
Temperature Rise degCRth =
Machine Losses W
Θr = Rth x I2 x Rmt
Θr = Rth x I
2 x Rmt
1 ndash (Rth x I2 x Rmt x α)
ndash OR ndash
Θr = Rth x(TC
Km)
2
ndash Final motor temperature
Use caution when applying these formulas Depending on the conditions used to determine motor specifications only one formula should be used Contact Appications Engineering for questions
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
8
MOTION SYSTEMFORMULAS
Motion System Formulas9
For PMDC motors (brush type) brush drop can be approximated as a constant resistance connected in series with the armature winding This series resistance is included in the Rmt factor found on manufacturerrsquos data sheet
Copper graphiteSilver graphiteElectrographite
02 to 04 Ω 02 to 04 Ω 08 to 10 Ω
Motor Brush Resistance
Brush Material Resistance
DC Motor Thermal Resistance and Temperature Rise
ndash Thermal resistance
ndash Motor temperature rise
Θm = Θr + Θa
ndash OR ndash
Temperature Rise degCRth =
Machine Losses W
Θr = Rth x I2 x Rmt
Θr = Rth x I
2 x Rmt
1 ndash (Rth x I2 x Rmt x α)
ndash OR ndash
Θr = Rth x(TC
Km)
2
ndash Final motor temperature
Use caution when applying these formulas Depending on the conditions used to determine motor specifications only one formula should be used Contact Appications Engineering for questions
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
9
MOTION SYSTEMFORMULAS
Stepper Motor Formulas
Required ldquoTime Offrdquo using an LR drive (constant voltage drive)
180Step Oslash = (phases)(rotor teeth)
ndash Motor step angle
ndash RPM (revolutionsminute) to SPS (stepssecond)
Maximum accuracy of a step motor system has nothing to do with maximum resolution of a step motor system Maximum accuracy is always based on the accuracy of 1 full step but maximum resolution is based on all system components including lead screw encoder stepper motor and step mode on the controller
ndash SPS (stepssecond) to RPM (revolutionsminute)(SPS) x (Step Oslash)
RPM = 6
(RPM) x 6 SPS = (Step Oslash)
ndash Travel per step
mstep =L
positionsrev
ndash Overdriving capability
Required ldquoTime Offrdquo using a chopper drive (constant current drive)
Time Off = (Time On) (Applied Voltage)2
(Rated Voltage)ndash Time On
Time Off = (Time On) (Applied Current)2
(Rated Current)ndash Time On
It is not unusual for a customer to drive Haydon Kerk stepper motors beyond their rated power to obtain the most force in the smallest package size Precautions must be taken to prevent the motor from exceeding its maximum temperature The ldquoonrdquo time should not exceed 2 - 3 minutes As general rule of thumb driving a motor at 25 to 3x its rated voltage or current may result in wasted energy and erratic behavior due to saturation
Special notes for can-stack motors bull more easily saturated due to less active material (steel) bull more iron losses due to unlaminated steel
Motion System Formulas10
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10
MOTION SYSTEMFORMULAS
Lead Screw Performance Characteristics
End Mounting Rigidity
Impact Impact
ScrewLength
Increase In Affects Increase In Affects
Critical speed
Compression load
Critical speed
Inertia
Compression load
Stiffness
Spring Rate
Load capacity
Drive torque
Angular velocity
Load cpacity
Positioning accuracy
ScrewDiameter
Lead
Load
Preload
Nut length
SI Unit Systems
LengthMassTimeElectric currentThermodynamic temperatureLuminous intensityAmount of substance
meterkilogramsecondamperekelvincandelamole
mKgsAKcdmol
Base Units
Derived Unitsareavolumefrequencymass density (density)speed velocityangular velocityaccelerationangular accelerationforcepressure (mechanical stress)kinematic viscositydynamic viscositywork energy quantity of heatpowerentropyspecific heat capacitythermal conductivity
square metercubic meterhertzkilogram per cubic metermeter per secondradian per secondmeter per second squaredradian per second squarednewtonpascalsquare meter per secondnewton-second per square meter joulewattjoule per kelvinjoule per kilogram kelvinwatt per meter kelvin
m2
m3
HzKgm3
msradsms2
rads2
NPam2sN-sm2
JWJKJKg-K)W(m-K)
Quantity Name of Unit Symbol
Motion System Reference Tables11
Critical speed
Compression load
System stiffness
Life
Positioning accuracy
System stiffness
Drag torque
Load capacity
Stiffness
sndash1
Kg-ms2
Nm2
N-mJs or N-ms
ExamplesAn INCREASE ( ) in screw length results in a DECREASE ( ) in critical speed
An INCREASE in screw diameter results in an INCREASE in critical speed
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
11
MOTION SYSTEMREFERENCE TABLES
t 1 t 2 t 3 t dwell
vTime
vpk
a -a
MassMove distanceMove timeDwell timeMotion profileScrew speed limitLoad supportOrientationRotary to linear conversionAmbient temperature
9 Kg200 mm10 s05 s13 13 13 Trapezoidal1000 RPMμ = 001VerticalTFE lead screw 8 mm diameter 275 mm long30degC
ControlDrive Power Supply
4 quadrant with encoder32 VDC 35 Arms50 A peak
0333 s 0333 s 0333 s 05 s
vPK =3S2t =
(3)(02m)(2)(10 s) =
06m2 s =
03ms
Linear Velocity
Linear Acceleration
a = vf - vi
t = 03ms - 0 ms 0333 s 0901ms2 =
1st step is to thoroughly understand system constraints and motion profile requirements
Linear Motion ExampleData
Motion System Application Example12
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
12
MOTION SYSTEMAPPLICATION EXAMPLE
Lmin =60vPK
n =(60)(03ms)1000 revmin = 0018mrev = 18mmrev
Refer to Kerk screw chart ndash closest lead in an 8mm screw diameter is
2032mmrev = 002032 mrevη = 086 free wheeling nut TFE coated screw
JS = 388 x 10ndash 7 Kg-m2 Other critical lead screw considerations bull Column loading bull Critical speed
Jm aa
ωPKvPK
TF
= ωPK = 2 π vPK
L=
(2π)(03ms)002032 mrev
9276 rads
a = =2πaL
=(2π)(0901ms2)002032 mrev
2786 rads2
Linear to Rotary Conversion ndash Velocity Acceleration Inertia Torque
Minimum screw lead to maintain lt 1000 RPM shaft speed
Linear to Rotary Conversion Velocity
Linear to Rotary Conversion Acceleration
Motion System Application Example13
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
13
MOTION SYSTEMFORMULAS
Jin= m 2 πL 2
X η1 + JS= 9Kg
2 π002032 m 2
X 0861 + 388x10ndash 7Kg-m2
= 1095 X 10ndash 5 Kg-m2 + 388x10ndash 7 Kg-m2
= 1134 x 10ndash 5 Kg-m2
Τin= Τa+Τf +Τg+ΤD
Τa = Jin ain
Τg =sinOslashmgL
2 π η
= (1134 x 10ndash 5 Kg-m2)(2786 rads2)
= 00316 Nm
Τf =cosOslashmgμL
2 π η= (cos90)(9 Kg)(98ms2)(001)(002032m)
2 π (086)= 0 Nm
= (sin90)(9 Kg)(98ms2)(002032m)2 π (086)
=1792 Nm
5404 = 03316 Nm
ΤDSince a free-wheeling lead screw nut was used there is no preload force If an anti-backlash nut was used ΤD would be gt0 due to preload force Always reference manufacturerrsquos data for more information
= 0 Nm
Linear to Rotary Conversion Torque
Linear to Rotary Conversion Inertia
Motion System Application Example14
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
14
MOTION SYSTEMFORMULAS
t 1 t 2 t 3 t dwell Time0333 s 0333 s 0333 s 05 s
Rotary Motion Profile
ωPK
ω
= 9276 rads a = 2786 rads2
Τ1= Τa+Τf +Τg+ΤD
= (00316 Nm) + (0 Nm) + (03316 Nm) + (0 Nm)
= 03632 Nm
Τ2= Τa+Τf +Τg+ΤD
= (0 Nm) + (0 Nm) + (03316 Nm) + (0 Nm)
= 03316 Nm
Τ3= Τa+Τf +Τg+ΤD
= (00316 Nm) + (0 Nm) + (03316 Nm) + (0 Nm)
= 03632 Nm
Τ1
Τ2
Τ3
Motion System Application Example15
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
15
MOTION SYSTEMAPPLICATION EXAMPLE
Lead Screw Input Requirements
= 9276 rads= 2786 rads2
= 03632 Nm= 03316 Nm= ndash 03632 Nm= 0333 s= 0333 s= 0333 s= 05 s= 3369 W= 3076 W
ωPKaΤ1Τ2Τ3
t1t2t3tdwell
PPKPCV
PPK = (Τ1)(ωPK)
= (03632 Nm)(9276 rads)= 3369 W
PCV = (Τ2)(ωPK)
= (03316 Nm)(9276 rads)= 3076 W
RMS Torque Requirement Lead Screw Input
=
0333 s + 0333 s + 0333 s + 05 s
(03632 Nm)2(0333 s)+(03316 Nm)2(0333 s)+(- 03632 Nm)2(0333 s)
ΤRMS = Τ12t 1 + Τ2
2t2 + Τ32t 3
t 1 + t 2 + t 3 + tdwell
15
00439 + 00366 + 00439 = 00829 =
ΤRMS = 02879 Nm
Motion System Application Example16
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
16
MOTION SYSTEMAPPLICATION EXAMPLE
Load Parameters at the Lead Screw Shaft
= 9276 rads= 2786 rads2
= 03632 Nm= 03316 Nm= ndash 03632 Nm= 02879 Nm= 3369 W= 3076 W
ωPKaΤ1Τ2Τ3ΤRMSPPKPCV
Motor SelectionA direct-drive motor can be used however it would be large and expensive For example a DC057B-3 brush motor
would easily meet continuous torque and continuous speed as well as a 181 load-to-rotor inertia This motor however has significantly more output power than needed at 128 watts
A smaller motor with a transmission would optimize the system Look for a motor starting at a rated power around 40 watts Pittman DC040B-6
DC040B-6Reference VoltageContinuous TorqueRated CurrentRated PowerTorque ConstantVoltage ConstantTerminal ResistanceMax Winding TemperatureRotor Inertia
240 V00812 Nm236 A37 W0042 NmA0042 Vrads185 Ω155degC847 x 10ndash6 Kg-m2
Motor TransmissionLead Screw Support Rails
ΤRMS
PPK
ωPK
= 02879 Nm= 3369 W= 9276 rads
M
Other Motor Considerationsndash Type stepper brush BLDCndash Input speed (high input speed will result in high noise levels)ndash Total motor footprintndash Ambient temperaturendash Environmental conditionsndash Load-to-rotor inertiandash Encoder ready needed
Motion System Application Example17
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
17
MOTION SYSTEMAPPLICATION EXAMPLE
Gearbox Transmission Selection
When selecting a reduction ratio keep in mind that the RMS required torque at the motor shaft needs to fall below the continuous rated output torque of the motor
G30 A - Planetary GearboxMaximum output load 247 Nm
TRMS( Lead Screw)
02879 Nm
02879 Nm02879 Nm
Reduction
41
5161
Efficiency
090
090090
TRMS( Motor)
00798 Nm
00640 Nm00533 Nm
A 51 gearbox would allow about a 27 safety margin between what the system requires and the continuous torque output of the motor
System Summary 51η = 090
M
= 00807 Nm= 00737 Nm= ndash 00807 Nm= 00640 Nm= 4638 rads= 504 x 10 ndash 6 Kg-m2
= 3743 W
Τ1Τ2Τ3ΤRMS
PPK
ωPK
J
= 03632 Nm= 03316 Nm= ndash 03632 Nm= 02879 Nm= 9276 rads= 1134 x 10 ndash 5 Kg-m2
= 3369 W
Τ1Τ2Τ3ΤRMS
PPK
ωPK
J
Motion System Application Example18
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
18
MOTION SYSTEMAPPLICATION EXAMPLE
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
19
Motion Profile at the Motor
t 1 t 2 t 3 t dwell Time0333 s 0333 s 0333 s 05 s
ωΤ1
Τ2
Τ3= 4638 rads= 4429 RPM= ΤPK = 00807 Nm
ωPK
Τ1
nPK
Drive and Power Supply Requirements
Peak Current Required
IPK =TPK
KT+ IO =
00807 Nm0042 NmA
+ 0180 A = 210 A
RMS Current Required
IRMS =TRMS
KT+ IO =
00640 Nm0042 NmA
+ 0180 A = 170 A
Minimum bus Voltage Required
Vbus= IPKRmt+ ωPKKE = (210A)(185Ω) + (4638 rads)(0042 Vrads)= 389 V + 1948 V= 2337 V
0100 Nm 0200 Nm 0300 Nm 0400 Nm 0500 Nm 0600 Nm 0700 Nm
30 VDC
0100 Nm 0200 Nm 0300 Nm 0500 Nm 0600 Nm 0700 Nm
24 VDC
700060005000
RPM 4000300020001000
0400 Nm
TRMS
TPK
700060005000
RPM 4000300020001000
TRMS
TPK
Performance using 24V reference voltage
Performance using a 30V bus voltage
A 30 VDC bus will meet the application requirements as wells supply a margin of safety
MOTION SYSTEMAPPLICATION EXAMPLE
0100 Nm 0200 Nm 0300 Nm 0400 Nm 0500 Nm 0600 Nm 0700 Nm
30 VDC
0100 Nm 0200 Nm 0300 Nm 0500 Nm 0600 Nm 0700 Nm
24 VDC
700060005000
RPM 4000300020001000
0400 Nm
TRMS
TPK
700060005000
RPM 4000300020001000
TRMS
TPK
Ambient temperature Rated motor temperature Temperature rise Motor temperature
= 30degC= 155degC= 7638degC= 10638degC
Θa
Θm
ΘrΘrated
=Rth x IRMS
2 x Rmt
1 ndash (Rth x IRMS2 x Rmt x 000392degC)
Θr
=11degCω x 170A2
x 185Ω 1 ndash (11degCω x 170A2
x 185Ω x 000392degC)
= 5881 1 ndash (023)
= 5881 077
=
Θm = Θr + Θa = 7638degC + 30degC = 10638degC
7638degC
Performance using 24V reference voltage
Performance using a 30V bus voltage
Will the motor meet the temperature rise caused by the application
Motion System Application Example20
A 30 VDC bus will meet the application requirements as wells supply a margin of safety
203 756 7441 1500 Meriden Road Waterbury CT USA 06705
267 933 2105 534 Godshall DriveHarleysville PA USA 19438
wwwHaydonKerkPittmancom
copy All rights reserved AMETEK Advanced Motion Solutions No part of this doc-ument or technical information can be used reproduced or altered in any form without approval or proper authorization from AMETEK Inc and global affiliates
NOW ASK us the tough motion control questions ONLINE
ASK THE EXPERTS
HaydonKerkPittmancomwww
MOTION SYSTEMAPPLICATION EXAMPLE
Inertia Acceleration Velocity and Required Torque
Jin mout
ain aout ωin vout
Tin Fout
radsec 2
radsec
ain = 2 πa
L
ωin = 2 π v
L
Jin = m 2 πL 2
X η1 + JS Kg-m2
Τin= Τa+Τf +Τg+ΤD N-m
Τa= Jin ain
Τf =cosOslashmgμL
2 π ηN-m
N-m
Τg =sinOslashmgL
2 π ηN-m
ΤD = dragpreloadrefer to manufacturerrsquos data
N-m
Lead Screw System
Belt and Pulley System
ain = radsec 2
ωin = radsec Jin Jout
ain aout ωin ωout
Tin Tout
Jin = NJout
2X η
1 + JP1 Kg-m2
N-mFor a 1st approximation analysis JP1 JP2 and JB can be disregarded For higher precision pulley and belt inertias should be included as well as the reflected inertia of these components
+ JP2+ JB
(aout)(N) (ωout)(N)
Τin = NΤout X η
1
Motion System Formulas5
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
4
MOTION SYSTEMFORMULAS
Motion System Formulas6
Gearbox System
radsec 2
ωin = radsec
Jin = NJout
2X η
1
+
Kg-m2
N-m
Drag torque can be significant (TD) depending on the viscosity of the lubricant For a first approximation analysis this can be left outGearbox inertia may be found in manufacturerrsquos data sheets or calculated using gear dimensions and materials mass
JGB
(ωout)(N) Τin = N
Τout X η1
+
ΤD
ΤRMS = Τ12t 1 + Τ2
2t2 + Τ32t 3 + Τn
2t n
t 1 + t 2 + t 3 + t n + tdwell
TωLinear Power =
FSt
= watts
Rotary Power = = watts
Mechanical Power and RMS Torque
Jin Jout
ain aout ωin ωout
Tin Tout
ain = (aout)(N)
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
5
MOTION SYSTEMFORMULAS
Assuming IO is very small it can be ignored for a quick approximation of motor no-load speed If more accurate results are needed IO will also need to be adjuisted based on the motor Viscous Damping Factor (Dv) As the no-load speed increases with increased terminal voltage on a given motor IO will also increase based on the motorrsquos Dv value found on the manufacturerrsquos motor data sheet
Motor No Load SpeedVT ndash (IO x Rmt)no = 95493 x
KE
ndash ndash ndash ndash ndash ndash ndash ndash ndash ndash ndash ndash ndash
ndash ndash ndash
ndash ndash ndash
BreakawayTorque
Viscous Damping Torque (Dv)
Coulomb Friction
Τ
ωο
Motion System Formulas7
DC Motor Formulas
Motor Locked Rotor Current Locked Rotor Torque
ILR =VT
Rmt
TLR = ILR x KT
Motor Regulation
RmtRm = 95493 x
KE x KT
= x KT
VT
Rmt
ndash Theoretical using motor constants
nORm = TLR
ndash Regulation calculated using test data
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
6
MOTION SYSTEMFORMULAS
Motor Power Relationships
ndash Watts lost due to winding resistance
Ploss = I2 x Rmt
ndash Output power at any point on the motor curve
Pout = ω x Tndash Motor maximum output power
Pmax = 025 x ωo x TLR
ndash Motor maximum output power (Theoretical)
Pmax = 025 x VT2
Rmt
Temperature Effects on Motor Constants
ndash Change in terminal resistance
Rmt(f) = Rmt(i) x [ 1 + αconductor (Θf ndash Θi) ]αconductor = 00040degC (copper)
K(f) = K(i) x [ 1 + αmagnet (Θf ndash Θi) ]ndash Change in torque constant and voltage constant ( KT = KE )
Magnetic Material
CeramicSmCoAlNiCoNdFeB
Θmax (degC)300degC300degC540degC150degC
The magnetic material values above represent average figures for particular material classes and estimating performance If exact values are needed consult data sheets for a specific magnet grade
Motion System Formulas8
αmagnet (degC)-00020degC-00004degC-00002degC-00012degC
In the case of a mechanically commutated motor with graphite brushes this analysis will result in a slight error because of the negative temperature coefficient of carbon For estimation purposes this can be ignored but for a more rigorous analysis the behavior of the brush material must be taken into account This is not an issue when evaluating a brushless dc motor
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
7
MOTION SYSTEMFORMULAS
Motion System Formulas9
For PMDC motors (brush type) brush drop can be approximated as a constant resistance connected in series with the armature winding This series resistance is included in the Rmt factor found on manufacturerrsquos data sheet
Copper graphiteSilver graphiteElectrographite
02 to 04 Ω 02 to 04 Ω 08 to 10 Ω
Motor Brush Resistance
Brush Material Resistance
DC Motor Thermal Resistance and Temperature Rise
ndash Thermal resistance
ndash Motor temperature rise
Θm = Θr + Θa
ndash OR ndash
Temperature Rise degCRth =
Machine Losses W
Θr = Rth x I2 x Rmt
Θr = Rth x I
2 x Rmt
1 ndash (Rth x I2 x Rmt x α)
ndash OR ndash
Θr = Rth x(TC
Km)
2
ndash Final motor temperature
Use caution when applying these formulas Depending on the conditions used to determine motor specifications only one formula should be used Contact Appications Engineering for questions
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
8
MOTION SYSTEMFORMULAS
Motion System Formulas9
For PMDC motors (brush type) brush drop can be approximated as a constant resistance connected in series with the armature winding This series resistance is included in the Rmt factor found on manufacturerrsquos data sheet
Copper graphiteSilver graphiteElectrographite
02 to 04 Ω 02 to 04 Ω 08 to 10 Ω
Motor Brush Resistance
Brush Material Resistance
DC Motor Thermal Resistance and Temperature Rise
ndash Thermal resistance
ndash Motor temperature rise
Θm = Θr + Θa
ndash OR ndash
Temperature Rise degCRth =
Machine Losses W
Θr = Rth x I2 x Rmt
Θr = Rth x I
2 x Rmt
1 ndash (Rth x I2 x Rmt x α)
ndash OR ndash
Θr = Rth x(TC
Km)
2
ndash Final motor temperature
Use caution when applying these formulas Depending on the conditions used to determine motor specifications only one formula should be used Contact Appications Engineering for questions
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
9
MOTION SYSTEMFORMULAS
Stepper Motor Formulas
Required ldquoTime Offrdquo using an LR drive (constant voltage drive)
180Step Oslash = (phases)(rotor teeth)
ndash Motor step angle
ndash RPM (revolutionsminute) to SPS (stepssecond)
Maximum accuracy of a step motor system has nothing to do with maximum resolution of a step motor system Maximum accuracy is always based on the accuracy of 1 full step but maximum resolution is based on all system components including lead screw encoder stepper motor and step mode on the controller
ndash SPS (stepssecond) to RPM (revolutionsminute)(SPS) x (Step Oslash)
RPM = 6
(RPM) x 6 SPS = (Step Oslash)
ndash Travel per step
mstep =L
positionsrev
ndash Overdriving capability
Required ldquoTime Offrdquo using a chopper drive (constant current drive)
Time Off = (Time On) (Applied Voltage)2
(Rated Voltage)ndash Time On
Time Off = (Time On) (Applied Current)2
(Rated Current)ndash Time On
It is not unusual for a customer to drive Haydon Kerk stepper motors beyond their rated power to obtain the most force in the smallest package size Precautions must be taken to prevent the motor from exceeding its maximum temperature The ldquoonrdquo time should not exceed 2 - 3 minutes As general rule of thumb driving a motor at 25 to 3x its rated voltage or current may result in wasted energy and erratic behavior due to saturation
Special notes for can-stack motors bull more easily saturated due to less active material (steel) bull more iron losses due to unlaminated steel
Motion System Formulas10
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
10
MOTION SYSTEMFORMULAS
Lead Screw Performance Characteristics
End Mounting Rigidity
Impact Impact
ScrewLength
Increase In Affects Increase In Affects
Critical speed
Compression load
Critical speed
Inertia
Compression load
Stiffness
Spring Rate
Load capacity
Drive torque
Angular velocity
Load cpacity
Positioning accuracy
ScrewDiameter
Lead
Load
Preload
Nut length
SI Unit Systems
LengthMassTimeElectric currentThermodynamic temperatureLuminous intensityAmount of substance
meterkilogramsecondamperekelvincandelamole
mKgsAKcdmol
Base Units
Derived Unitsareavolumefrequencymass density (density)speed velocityangular velocityaccelerationangular accelerationforcepressure (mechanical stress)kinematic viscositydynamic viscositywork energy quantity of heatpowerentropyspecific heat capacitythermal conductivity
square metercubic meterhertzkilogram per cubic metermeter per secondradian per secondmeter per second squaredradian per second squarednewtonpascalsquare meter per secondnewton-second per square meter joulewattjoule per kelvinjoule per kilogram kelvinwatt per meter kelvin
m2
m3
HzKgm3
msradsms2
rads2
NPam2sN-sm2
JWJKJKg-K)W(m-K)
Quantity Name of Unit Symbol
Motion System Reference Tables11
Critical speed
Compression load
System stiffness
Life
Positioning accuracy
System stiffness
Drag torque
Load capacity
Stiffness
sndash1
Kg-ms2
Nm2
N-mJs or N-ms
ExamplesAn INCREASE ( ) in screw length results in a DECREASE ( ) in critical speed
An INCREASE in screw diameter results in an INCREASE in critical speed
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
11
MOTION SYSTEMREFERENCE TABLES
t 1 t 2 t 3 t dwell
vTime
vpk
a -a
MassMove distanceMove timeDwell timeMotion profileScrew speed limitLoad supportOrientationRotary to linear conversionAmbient temperature
9 Kg200 mm10 s05 s13 13 13 Trapezoidal1000 RPMμ = 001VerticalTFE lead screw 8 mm diameter 275 mm long30degC
ControlDrive Power Supply
4 quadrant with encoder32 VDC 35 Arms50 A peak
0333 s 0333 s 0333 s 05 s
vPK =3S2t =
(3)(02m)(2)(10 s) =
06m2 s =
03ms
Linear Velocity
Linear Acceleration
a = vf - vi
t = 03ms - 0 ms 0333 s 0901ms2 =
1st step is to thoroughly understand system constraints and motion profile requirements
Linear Motion ExampleData
Motion System Application Example12
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
12
MOTION SYSTEMAPPLICATION EXAMPLE
Lmin =60vPK
n =(60)(03ms)1000 revmin = 0018mrev = 18mmrev
Refer to Kerk screw chart ndash closest lead in an 8mm screw diameter is
2032mmrev = 002032 mrevη = 086 free wheeling nut TFE coated screw
JS = 388 x 10ndash 7 Kg-m2 Other critical lead screw considerations bull Column loading bull Critical speed
Jm aa
ωPKvPK
TF
= ωPK = 2 π vPK
L=
(2π)(03ms)002032 mrev
9276 rads
a = =2πaL
=(2π)(0901ms2)002032 mrev
2786 rads2
Linear to Rotary Conversion ndash Velocity Acceleration Inertia Torque
Minimum screw lead to maintain lt 1000 RPM shaft speed
Linear to Rotary Conversion Velocity
Linear to Rotary Conversion Acceleration
Motion System Application Example13
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13
MOTION SYSTEMFORMULAS
Jin= m 2 πL 2
X η1 + JS= 9Kg
2 π002032 m 2
X 0861 + 388x10ndash 7Kg-m2
= 1095 X 10ndash 5 Kg-m2 + 388x10ndash 7 Kg-m2
= 1134 x 10ndash 5 Kg-m2
Τin= Τa+Τf +Τg+ΤD
Τa = Jin ain
Τg =sinOslashmgL
2 π η
= (1134 x 10ndash 5 Kg-m2)(2786 rads2)
= 00316 Nm
Τf =cosOslashmgμL
2 π η= (cos90)(9 Kg)(98ms2)(001)(002032m)
2 π (086)= 0 Nm
= (sin90)(9 Kg)(98ms2)(002032m)2 π (086)
=1792 Nm
5404 = 03316 Nm
ΤDSince a free-wheeling lead screw nut was used there is no preload force If an anti-backlash nut was used ΤD would be gt0 due to preload force Always reference manufacturerrsquos data for more information
= 0 Nm
Linear to Rotary Conversion Torque
Linear to Rotary Conversion Inertia
Motion System Application Example14
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14
MOTION SYSTEMFORMULAS
t 1 t 2 t 3 t dwell Time0333 s 0333 s 0333 s 05 s
Rotary Motion Profile
ωPK
ω
= 9276 rads a = 2786 rads2
Τ1= Τa+Τf +Τg+ΤD
= (00316 Nm) + (0 Nm) + (03316 Nm) + (0 Nm)
= 03632 Nm
Τ2= Τa+Τf +Τg+ΤD
= (0 Nm) + (0 Nm) + (03316 Nm) + (0 Nm)
= 03316 Nm
Τ3= Τa+Τf +Τg+ΤD
= (00316 Nm) + (0 Nm) + (03316 Nm) + (0 Nm)
= 03632 Nm
Τ1
Τ2
Τ3
Motion System Application Example15
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15
MOTION SYSTEMAPPLICATION EXAMPLE
Lead Screw Input Requirements
= 9276 rads= 2786 rads2
= 03632 Nm= 03316 Nm= ndash 03632 Nm= 0333 s= 0333 s= 0333 s= 05 s= 3369 W= 3076 W
ωPKaΤ1Τ2Τ3
t1t2t3tdwell
PPKPCV
PPK = (Τ1)(ωPK)
= (03632 Nm)(9276 rads)= 3369 W
PCV = (Τ2)(ωPK)
= (03316 Nm)(9276 rads)= 3076 W
RMS Torque Requirement Lead Screw Input
=
0333 s + 0333 s + 0333 s + 05 s
(03632 Nm)2(0333 s)+(03316 Nm)2(0333 s)+(- 03632 Nm)2(0333 s)
ΤRMS = Τ12t 1 + Τ2
2t2 + Τ32t 3
t 1 + t 2 + t 3 + tdwell
15
00439 + 00366 + 00439 = 00829 =
ΤRMS = 02879 Nm
Motion System Application Example16
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16
MOTION SYSTEMAPPLICATION EXAMPLE
Load Parameters at the Lead Screw Shaft
= 9276 rads= 2786 rads2
= 03632 Nm= 03316 Nm= ndash 03632 Nm= 02879 Nm= 3369 W= 3076 W
ωPKaΤ1Τ2Τ3ΤRMSPPKPCV
Motor SelectionA direct-drive motor can be used however it would be large and expensive For example a DC057B-3 brush motor
would easily meet continuous torque and continuous speed as well as a 181 load-to-rotor inertia This motor however has significantly more output power than needed at 128 watts
A smaller motor with a transmission would optimize the system Look for a motor starting at a rated power around 40 watts Pittman DC040B-6
DC040B-6Reference VoltageContinuous TorqueRated CurrentRated PowerTorque ConstantVoltage ConstantTerminal ResistanceMax Winding TemperatureRotor Inertia
240 V00812 Nm236 A37 W0042 NmA0042 Vrads185 Ω155degC847 x 10ndash6 Kg-m2
Motor TransmissionLead Screw Support Rails
ΤRMS
PPK
ωPK
= 02879 Nm= 3369 W= 9276 rads
M
Other Motor Considerationsndash Type stepper brush BLDCndash Input speed (high input speed will result in high noise levels)ndash Total motor footprintndash Ambient temperaturendash Environmental conditionsndash Load-to-rotor inertiandash Encoder ready needed
Motion System Application Example17
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17
MOTION SYSTEMAPPLICATION EXAMPLE
Gearbox Transmission Selection
When selecting a reduction ratio keep in mind that the RMS required torque at the motor shaft needs to fall below the continuous rated output torque of the motor
G30 A - Planetary GearboxMaximum output load 247 Nm
TRMS( Lead Screw)
02879 Nm
02879 Nm02879 Nm
Reduction
41
5161
Efficiency
090
090090
TRMS( Motor)
00798 Nm
00640 Nm00533 Nm
A 51 gearbox would allow about a 27 safety margin between what the system requires and the continuous torque output of the motor
System Summary 51η = 090
M
= 00807 Nm= 00737 Nm= ndash 00807 Nm= 00640 Nm= 4638 rads= 504 x 10 ndash 6 Kg-m2
= 3743 W
Τ1Τ2Τ3ΤRMS
PPK
ωPK
J
= 03632 Nm= 03316 Nm= ndash 03632 Nm= 02879 Nm= 9276 rads= 1134 x 10 ndash 5 Kg-m2
= 3369 W
Τ1Τ2Τ3ΤRMS
PPK
ωPK
J
Motion System Application Example18
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18
MOTION SYSTEMAPPLICATION EXAMPLE
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19
Motion Profile at the Motor
t 1 t 2 t 3 t dwell Time0333 s 0333 s 0333 s 05 s
ωΤ1
Τ2
Τ3= 4638 rads= 4429 RPM= ΤPK = 00807 Nm
ωPK
Τ1
nPK
Drive and Power Supply Requirements
Peak Current Required
IPK =TPK
KT+ IO =
00807 Nm0042 NmA
+ 0180 A = 210 A
RMS Current Required
IRMS =TRMS
KT+ IO =
00640 Nm0042 NmA
+ 0180 A = 170 A
Minimum bus Voltage Required
Vbus= IPKRmt+ ωPKKE = (210A)(185Ω) + (4638 rads)(0042 Vrads)= 389 V + 1948 V= 2337 V
0100 Nm 0200 Nm 0300 Nm 0400 Nm 0500 Nm 0600 Nm 0700 Nm
30 VDC
0100 Nm 0200 Nm 0300 Nm 0500 Nm 0600 Nm 0700 Nm
24 VDC
700060005000
RPM 4000300020001000
0400 Nm
TRMS
TPK
700060005000
RPM 4000300020001000
TRMS
TPK
Performance using 24V reference voltage
Performance using a 30V bus voltage
A 30 VDC bus will meet the application requirements as wells supply a margin of safety
MOTION SYSTEMAPPLICATION EXAMPLE
0100 Nm 0200 Nm 0300 Nm 0400 Nm 0500 Nm 0600 Nm 0700 Nm
30 VDC
0100 Nm 0200 Nm 0300 Nm 0500 Nm 0600 Nm 0700 Nm
24 VDC
700060005000
RPM 4000300020001000
0400 Nm
TRMS
TPK
700060005000
RPM 4000300020001000
TRMS
TPK
Ambient temperature Rated motor temperature Temperature rise Motor temperature
= 30degC= 155degC= 7638degC= 10638degC
Θa
Θm
ΘrΘrated
=Rth x IRMS
2 x Rmt
1 ndash (Rth x IRMS2 x Rmt x 000392degC)
Θr
=11degCω x 170A2
x 185Ω 1 ndash (11degCω x 170A2
x 185Ω x 000392degC)
= 5881 1 ndash (023)
= 5881 077
=
Θm = Θr + Θa = 7638degC + 30degC = 10638degC
7638degC
Performance using 24V reference voltage
Performance using a 30V bus voltage
Will the motor meet the temperature rise caused by the application
Motion System Application Example20
A 30 VDC bus will meet the application requirements as wells supply a margin of safety
203 756 7441 1500 Meriden Road Waterbury CT USA 06705
267 933 2105 534 Godshall DriveHarleysville PA USA 19438
wwwHaydonKerkPittmancom
copy All rights reserved AMETEK Advanced Motion Solutions No part of this doc-ument or technical information can be used reproduced or altered in any form without approval or proper authorization from AMETEK Inc and global affiliates
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MOTION SYSTEMAPPLICATION EXAMPLE
Motion System Formulas6
Gearbox System
radsec 2
ωin = radsec
Jin = NJout
2X η
1
+
Kg-m2
N-m
Drag torque can be significant (TD) depending on the viscosity of the lubricant For a first approximation analysis this can be left outGearbox inertia may be found in manufacturerrsquos data sheets or calculated using gear dimensions and materials mass
JGB
(ωout)(N) Τin = N
Τout X η1
+
ΤD
ΤRMS = Τ12t 1 + Τ2
2t2 + Τ32t 3 + Τn
2t n
t 1 + t 2 + t 3 + t n + tdwell
TωLinear Power =
FSt
= watts
Rotary Power = = watts
Mechanical Power and RMS Torque
Jin Jout
ain aout ωin ωout
Tin Tout
ain = (aout)(N)
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5
MOTION SYSTEMFORMULAS
Assuming IO is very small it can be ignored for a quick approximation of motor no-load speed If more accurate results are needed IO will also need to be adjuisted based on the motor Viscous Damping Factor (Dv) As the no-load speed increases with increased terminal voltage on a given motor IO will also increase based on the motorrsquos Dv value found on the manufacturerrsquos motor data sheet
Motor No Load SpeedVT ndash (IO x Rmt)no = 95493 x
KE
ndash ndash ndash ndash ndash ndash ndash ndash ndash ndash ndash ndash ndash
ndash ndash ndash
ndash ndash ndash
BreakawayTorque
Viscous Damping Torque (Dv)
Coulomb Friction
Τ
ωο
Motion System Formulas7
DC Motor Formulas
Motor Locked Rotor Current Locked Rotor Torque
ILR =VT
Rmt
TLR = ILR x KT
Motor Regulation
RmtRm = 95493 x
KE x KT
= x KT
VT
Rmt
ndash Theoretical using motor constants
nORm = TLR
ndash Regulation calculated using test data
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6
MOTION SYSTEMFORMULAS
Motor Power Relationships
ndash Watts lost due to winding resistance
Ploss = I2 x Rmt
ndash Output power at any point on the motor curve
Pout = ω x Tndash Motor maximum output power
Pmax = 025 x ωo x TLR
ndash Motor maximum output power (Theoretical)
Pmax = 025 x VT2
Rmt
Temperature Effects on Motor Constants
ndash Change in terminal resistance
Rmt(f) = Rmt(i) x [ 1 + αconductor (Θf ndash Θi) ]αconductor = 00040degC (copper)
K(f) = K(i) x [ 1 + αmagnet (Θf ndash Θi) ]ndash Change in torque constant and voltage constant ( KT = KE )
Magnetic Material
CeramicSmCoAlNiCoNdFeB
Θmax (degC)300degC300degC540degC150degC
The magnetic material values above represent average figures for particular material classes and estimating performance If exact values are needed consult data sheets for a specific magnet grade
Motion System Formulas8
αmagnet (degC)-00020degC-00004degC-00002degC-00012degC
In the case of a mechanically commutated motor with graphite brushes this analysis will result in a slight error because of the negative temperature coefficient of carbon For estimation purposes this can be ignored but for a more rigorous analysis the behavior of the brush material must be taken into account This is not an issue when evaluating a brushless dc motor
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7
MOTION SYSTEMFORMULAS
Motion System Formulas9
For PMDC motors (brush type) brush drop can be approximated as a constant resistance connected in series with the armature winding This series resistance is included in the Rmt factor found on manufacturerrsquos data sheet
Copper graphiteSilver graphiteElectrographite
02 to 04 Ω 02 to 04 Ω 08 to 10 Ω
Motor Brush Resistance
Brush Material Resistance
DC Motor Thermal Resistance and Temperature Rise
ndash Thermal resistance
ndash Motor temperature rise
Θm = Θr + Θa
ndash OR ndash
Temperature Rise degCRth =
Machine Losses W
Θr = Rth x I2 x Rmt
Θr = Rth x I
2 x Rmt
1 ndash (Rth x I2 x Rmt x α)
ndash OR ndash
Θr = Rth x(TC
Km)
2
ndash Final motor temperature
Use caution when applying these formulas Depending on the conditions used to determine motor specifications only one formula should be used Contact Appications Engineering for questions
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8
MOTION SYSTEMFORMULAS
Motion System Formulas9
For PMDC motors (brush type) brush drop can be approximated as a constant resistance connected in series with the armature winding This series resistance is included in the Rmt factor found on manufacturerrsquos data sheet
Copper graphiteSilver graphiteElectrographite
02 to 04 Ω 02 to 04 Ω 08 to 10 Ω
Motor Brush Resistance
Brush Material Resistance
DC Motor Thermal Resistance and Temperature Rise
ndash Thermal resistance
ndash Motor temperature rise
Θm = Θr + Θa
ndash OR ndash
Temperature Rise degCRth =
Machine Losses W
Θr = Rth x I2 x Rmt
Θr = Rth x I
2 x Rmt
1 ndash (Rth x I2 x Rmt x α)
ndash OR ndash
Θr = Rth x(TC
Km)
2
ndash Final motor temperature
Use caution when applying these formulas Depending on the conditions used to determine motor specifications only one formula should be used Contact Appications Engineering for questions
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9
MOTION SYSTEMFORMULAS
Stepper Motor Formulas
Required ldquoTime Offrdquo using an LR drive (constant voltage drive)
180Step Oslash = (phases)(rotor teeth)
ndash Motor step angle
ndash RPM (revolutionsminute) to SPS (stepssecond)
Maximum accuracy of a step motor system has nothing to do with maximum resolution of a step motor system Maximum accuracy is always based on the accuracy of 1 full step but maximum resolution is based on all system components including lead screw encoder stepper motor and step mode on the controller
ndash SPS (stepssecond) to RPM (revolutionsminute)(SPS) x (Step Oslash)
RPM = 6
(RPM) x 6 SPS = (Step Oslash)
ndash Travel per step
mstep =L
positionsrev
ndash Overdriving capability
Required ldquoTime Offrdquo using a chopper drive (constant current drive)
Time Off = (Time On) (Applied Voltage)2
(Rated Voltage)ndash Time On
Time Off = (Time On) (Applied Current)2
(Rated Current)ndash Time On
It is not unusual for a customer to drive Haydon Kerk stepper motors beyond their rated power to obtain the most force in the smallest package size Precautions must be taken to prevent the motor from exceeding its maximum temperature The ldquoonrdquo time should not exceed 2 - 3 minutes As general rule of thumb driving a motor at 25 to 3x its rated voltage or current may result in wasted energy and erratic behavior due to saturation
Special notes for can-stack motors bull more easily saturated due to less active material (steel) bull more iron losses due to unlaminated steel
Motion System Formulas10
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10
MOTION SYSTEMFORMULAS
Lead Screw Performance Characteristics
End Mounting Rigidity
Impact Impact
ScrewLength
Increase In Affects Increase In Affects
Critical speed
Compression load
Critical speed
Inertia
Compression load
Stiffness
Spring Rate
Load capacity
Drive torque
Angular velocity
Load cpacity
Positioning accuracy
ScrewDiameter
Lead
Load
Preload
Nut length
SI Unit Systems
LengthMassTimeElectric currentThermodynamic temperatureLuminous intensityAmount of substance
meterkilogramsecondamperekelvincandelamole
mKgsAKcdmol
Base Units
Derived Unitsareavolumefrequencymass density (density)speed velocityangular velocityaccelerationangular accelerationforcepressure (mechanical stress)kinematic viscositydynamic viscositywork energy quantity of heatpowerentropyspecific heat capacitythermal conductivity
square metercubic meterhertzkilogram per cubic metermeter per secondradian per secondmeter per second squaredradian per second squarednewtonpascalsquare meter per secondnewton-second per square meter joulewattjoule per kelvinjoule per kilogram kelvinwatt per meter kelvin
m2
m3
HzKgm3
msradsms2
rads2
NPam2sN-sm2
JWJKJKg-K)W(m-K)
Quantity Name of Unit Symbol
Motion System Reference Tables11
Critical speed
Compression load
System stiffness
Life
Positioning accuracy
System stiffness
Drag torque
Load capacity
Stiffness
sndash1
Kg-ms2
Nm2
N-mJs or N-ms
ExamplesAn INCREASE ( ) in screw length results in a DECREASE ( ) in critical speed
An INCREASE in screw diameter results in an INCREASE in critical speed
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11
MOTION SYSTEMREFERENCE TABLES
t 1 t 2 t 3 t dwell
vTime
vpk
a -a
MassMove distanceMove timeDwell timeMotion profileScrew speed limitLoad supportOrientationRotary to linear conversionAmbient temperature
9 Kg200 mm10 s05 s13 13 13 Trapezoidal1000 RPMμ = 001VerticalTFE lead screw 8 mm diameter 275 mm long30degC
ControlDrive Power Supply
4 quadrant with encoder32 VDC 35 Arms50 A peak
0333 s 0333 s 0333 s 05 s
vPK =3S2t =
(3)(02m)(2)(10 s) =
06m2 s =
03ms
Linear Velocity
Linear Acceleration
a = vf - vi
t = 03ms - 0 ms 0333 s 0901ms2 =
1st step is to thoroughly understand system constraints and motion profile requirements
Linear Motion ExampleData
Motion System Application Example12
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
12
MOTION SYSTEMAPPLICATION EXAMPLE
Lmin =60vPK
n =(60)(03ms)1000 revmin = 0018mrev = 18mmrev
Refer to Kerk screw chart ndash closest lead in an 8mm screw diameter is
2032mmrev = 002032 mrevη = 086 free wheeling nut TFE coated screw
JS = 388 x 10ndash 7 Kg-m2 Other critical lead screw considerations bull Column loading bull Critical speed
Jm aa
ωPKvPK
TF
= ωPK = 2 π vPK
L=
(2π)(03ms)002032 mrev
9276 rads
a = =2πaL
=(2π)(0901ms2)002032 mrev
2786 rads2
Linear to Rotary Conversion ndash Velocity Acceleration Inertia Torque
Minimum screw lead to maintain lt 1000 RPM shaft speed
Linear to Rotary Conversion Velocity
Linear to Rotary Conversion Acceleration
Motion System Application Example13
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
13
MOTION SYSTEMFORMULAS
Jin= m 2 πL 2
X η1 + JS= 9Kg
2 π002032 m 2
X 0861 + 388x10ndash 7Kg-m2
= 1095 X 10ndash 5 Kg-m2 + 388x10ndash 7 Kg-m2
= 1134 x 10ndash 5 Kg-m2
Τin= Τa+Τf +Τg+ΤD
Τa = Jin ain
Τg =sinOslashmgL
2 π η
= (1134 x 10ndash 5 Kg-m2)(2786 rads2)
= 00316 Nm
Τf =cosOslashmgμL
2 π η= (cos90)(9 Kg)(98ms2)(001)(002032m)
2 π (086)= 0 Nm
= (sin90)(9 Kg)(98ms2)(002032m)2 π (086)
=1792 Nm
5404 = 03316 Nm
ΤDSince a free-wheeling lead screw nut was used there is no preload force If an anti-backlash nut was used ΤD would be gt0 due to preload force Always reference manufacturerrsquos data for more information
= 0 Nm
Linear to Rotary Conversion Torque
Linear to Rotary Conversion Inertia
Motion System Application Example14
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
14
MOTION SYSTEMFORMULAS
t 1 t 2 t 3 t dwell Time0333 s 0333 s 0333 s 05 s
Rotary Motion Profile
ωPK
ω
= 9276 rads a = 2786 rads2
Τ1= Τa+Τf +Τg+ΤD
= (00316 Nm) + (0 Nm) + (03316 Nm) + (0 Nm)
= 03632 Nm
Τ2= Τa+Τf +Τg+ΤD
= (0 Nm) + (0 Nm) + (03316 Nm) + (0 Nm)
= 03316 Nm
Τ3= Τa+Τf +Τg+ΤD
= (00316 Nm) + (0 Nm) + (03316 Nm) + (0 Nm)
= 03632 Nm
Τ1
Τ2
Τ3
Motion System Application Example15
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
15
MOTION SYSTEMAPPLICATION EXAMPLE
Lead Screw Input Requirements
= 9276 rads= 2786 rads2
= 03632 Nm= 03316 Nm= ndash 03632 Nm= 0333 s= 0333 s= 0333 s= 05 s= 3369 W= 3076 W
ωPKaΤ1Τ2Τ3
t1t2t3tdwell
PPKPCV
PPK = (Τ1)(ωPK)
= (03632 Nm)(9276 rads)= 3369 W
PCV = (Τ2)(ωPK)
= (03316 Nm)(9276 rads)= 3076 W
RMS Torque Requirement Lead Screw Input
=
0333 s + 0333 s + 0333 s + 05 s
(03632 Nm)2(0333 s)+(03316 Nm)2(0333 s)+(- 03632 Nm)2(0333 s)
ΤRMS = Τ12t 1 + Τ2
2t2 + Τ32t 3
t 1 + t 2 + t 3 + tdwell
15
00439 + 00366 + 00439 = 00829 =
ΤRMS = 02879 Nm
Motion System Application Example16
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
16
MOTION SYSTEMAPPLICATION EXAMPLE
Load Parameters at the Lead Screw Shaft
= 9276 rads= 2786 rads2
= 03632 Nm= 03316 Nm= ndash 03632 Nm= 02879 Nm= 3369 W= 3076 W
ωPKaΤ1Τ2Τ3ΤRMSPPKPCV
Motor SelectionA direct-drive motor can be used however it would be large and expensive For example a DC057B-3 brush motor
would easily meet continuous torque and continuous speed as well as a 181 load-to-rotor inertia This motor however has significantly more output power than needed at 128 watts
A smaller motor with a transmission would optimize the system Look for a motor starting at a rated power around 40 watts Pittman DC040B-6
DC040B-6Reference VoltageContinuous TorqueRated CurrentRated PowerTorque ConstantVoltage ConstantTerminal ResistanceMax Winding TemperatureRotor Inertia
240 V00812 Nm236 A37 W0042 NmA0042 Vrads185 Ω155degC847 x 10ndash6 Kg-m2
Motor TransmissionLead Screw Support Rails
ΤRMS
PPK
ωPK
= 02879 Nm= 3369 W= 9276 rads
M
Other Motor Considerationsndash Type stepper brush BLDCndash Input speed (high input speed will result in high noise levels)ndash Total motor footprintndash Ambient temperaturendash Environmental conditionsndash Load-to-rotor inertiandash Encoder ready needed
Motion System Application Example17
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
17
MOTION SYSTEMAPPLICATION EXAMPLE
Gearbox Transmission Selection
When selecting a reduction ratio keep in mind that the RMS required torque at the motor shaft needs to fall below the continuous rated output torque of the motor
G30 A - Planetary GearboxMaximum output load 247 Nm
TRMS( Lead Screw)
02879 Nm
02879 Nm02879 Nm
Reduction
41
5161
Efficiency
090
090090
TRMS( Motor)
00798 Nm
00640 Nm00533 Nm
A 51 gearbox would allow about a 27 safety margin between what the system requires and the continuous torque output of the motor
System Summary 51η = 090
M
= 00807 Nm= 00737 Nm= ndash 00807 Nm= 00640 Nm= 4638 rads= 504 x 10 ndash 6 Kg-m2
= 3743 W
Τ1Τ2Τ3ΤRMS
PPK
ωPK
J
= 03632 Nm= 03316 Nm= ndash 03632 Nm= 02879 Nm= 9276 rads= 1134 x 10 ndash 5 Kg-m2
= 3369 W
Τ1Τ2Τ3ΤRMS
PPK
ωPK
J
Motion System Application Example18
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
18
MOTION SYSTEMAPPLICATION EXAMPLE
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
19
Motion Profile at the Motor
t 1 t 2 t 3 t dwell Time0333 s 0333 s 0333 s 05 s
ωΤ1
Τ2
Τ3= 4638 rads= 4429 RPM= ΤPK = 00807 Nm
ωPK
Τ1
nPK
Drive and Power Supply Requirements
Peak Current Required
IPK =TPK
KT+ IO =
00807 Nm0042 NmA
+ 0180 A = 210 A
RMS Current Required
IRMS =TRMS
KT+ IO =
00640 Nm0042 NmA
+ 0180 A = 170 A
Minimum bus Voltage Required
Vbus= IPKRmt+ ωPKKE = (210A)(185Ω) + (4638 rads)(0042 Vrads)= 389 V + 1948 V= 2337 V
0100 Nm 0200 Nm 0300 Nm 0400 Nm 0500 Nm 0600 Nm 0700 Nm
30 VDC
0100 Nm 0200 Nm 0300 Nm 0500 Nm 0600 Nm 0700 Nm
24 VDC
700060005000
RPM 4000300020001000
0400 Nm
TRMS
TPK
700060005000
RPM 4000300020001000
TRMS
TPK
Performance using 24V reference voltage
Performance using a 30V bus voltage
A 30 VDC bus will meet the application requirements as wells supply a margin of safety
MOTION SYSTEMAPPLICATION EXAMPLE
0100 Nm 0200 Nm 0300 Nm 0400 Nm 0500 Nm 0600 Nm 0700 Nm
30 VDC
0100 Nm 0200 Nm 0300 Nm 0500 Nm 0600 Nm 0700 Nm
24 VDC
700060005000
RPM 4000300020001000
0400 Nm
TRMS
TPK
700060005000
RPM 4000300020001000
TRMS
TPK
Ambient temperature Rated motor temperature Temperature rise Motor temperature
= 30degC= 155degC= 7638degC= 10638degC
Θa
Θm
ΘrΘrated
=Rth x IRMS
2 x Rmt
1 ndash (Rth x IRMS2 x Rmt x 000392degC)
Θr
=11degCω x 170A2
x 185Ω 1 ndash (11degCω x 170A2
x 185Ω x 000392degC)
= 5881 1 ndash (023)
= 5881 077
=
Θm = Θr + Θa = 7638degC + 30degC = 10638degC
7638degC
Performance using 24V reference voltage
Performance using a 30V bus voltage
Will the motor meet the temperature rise caused by the application
Motion System Application Example20
A 30 VDC bus will meet the application requirements as wells supply a margin of safety
203 756 7441 1500 Meriden Road Waterbury CT USA 06705
267 933 2105 534 Godshall DriveHarleysville PA USA 19438
wwwHaydonKerkPittmancom
copy All rights reserved AMETEK Advanced Motion Solutions No part of this doc-ument or technical information can be used reproduced or altered in any form without approval or proper authorization from AMETEK Inc and global affiliates
NOW ASK us the tough motion control questions ONLINE
ASK THE EXPERTS
HaydonKerkPittmancomwww
MOTION SYSTEMAPPLICATION EXAMPLE
Assuming IO is very small it can be ignored for a quick approximation of motor no-load speed If more accurate results are needed IO will also need to be adjuisted based on the motor Viscous Damping Factor (Dv) As the no-load speed increases with increased terminal voltage on a given motor IO will also increase based on the motorrsquos Dv value found on the manufacturerrsquos motor data sheet
Motor No Load SpeedVT ndash (IO x Rmt)no = 95493 x
KE
ndash ndash ndash ndash ndash ndash ndash ndash ndash ndash ndash ndash ndash
ndash ndash ndash
ndash ndash ndash
BreakawayTorque
Viscous Damping Torque (Dv)
Coulomb Friction
Τ
ωο
Motion System Formulas7
DC Motor Formulas
Motor Locked Rotor Current Locked Rotor Torque
ILR =VT
Rmt
TLR = ILR x KT
Motor Regulation
RmtRm = 95493 x
KE x KT
= x KT
VT
Rmt
ndash Theoretical using motor constants
nORm = TLR
ndash Regulation calculated using test data
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
6
MOTION SYSTEMFORMULAS
Motor Power Relationships
ndash Watts lost due to winding resistance
Ploss = I2 x Rmt
ndash Output power at any point on the motor curve
Pout = ω x Tndash Motor maximum output power
Pmax = 025 x ωo x TLR
ndash Motor maximum output power (Theoretical)
Pmax = 025 x VT2
Rmt
Temperature Effects on Motor Constants
ndash Change in terminal resistance
Rmt(f) = Rmt(i) x [ 1 + αconductor (Θf ndash Θi) ]αconductor = 00040degC (copper)
K(f) = K(i) x [ 1 + αmagnet (Θf ndash Θi) ]ndash Change in torque constant and voltage constant ( KT = KE )
Magnetic Material
CeramicSmCoAlNiCoNdFeB
Θmax (degC)300degC300degC540degC150degC
The magnetic material values above represent average figures for particular material classes and estimating performance If exact values are needed consult data sheets for a specific magnet grade
Motion System Formulas8
αmagnet (degC)-00020degC-00004degC-00002degC-00012degC
In the case of a mechanically commutated motor with graphite brushes this analysis will result in a slight error because of the negative temperature coefficient of carbon For estimation purposes this can be ignored but for a more rigorous analysis the behavior of the brush material must be taken into account This is not an issue when evaluating a brushless dc motor
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
7
MOTION SYSTEMFORMULAS
Motion System Formulas9
For PMDC motors (brush type) brush drop can be approximated as a constant resistance connected in series with the armature winding This series resistance is included in the Rmt factor found on manufacturerrsquos data sheet
Copper graphiteSilver graphiteElectrographite
02 to 04 Ω 02 to 04 Ω 08 to 10 Ω
Motor Brush Resistance
Brush Material Resistance
DC Motor Thermal Resistance and Temperature Rise
ndash Thermal resistance
ndash Motor temperature rise
Θm = Θr + Θa
ndash OR ndash
Temperature Rise degCRth =
Machine Losses W
Θr = Rth x I2 x Rmt
Θr = Rth x I
2 x Rmt
1 ndash (Rth x I2 x Rmt x α)
ndash OR ndash
Θr = Rth x(TC
Km)
2
ndash Final motor temperature
Use caution when applying these formulas Depending on the conditions used to determine motor specifications only one formula should be used Contact Appications Engineering for questions
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
8
MOTION SYSTEMFORMULAS
Motion System Formulas9
For PMDC motors (brush type) brush drop can be approximated as a constant resistance connected in series with the armature winding This series resistance is included in the Rmt factor found on manufacturerrsquos data sheet
Copper graphiteSilver graphiteElectrographite
02 to 04 Ω 02 to 04 Ω 08 to 10 Ω
Motor Brush Resistance
Brush Material Resistance
DC Motor Thermal Resistance and Temperature Rise
ndash Thermal resistance
ndash Motor temperature rise
Θm = Θr + Θa
ndash OR ndash
Temperature Rise degCRth =
Machine Losses W
Θr = Rth x I2 x Rmt
Θr = Rth x I
2 x Rmt
1 ndash (Rth x I2 x Rmt x α)
ndash OR ndash
Θr = Rth x(TC
Km)
2
ndash Final motor temperature
Use caution when applying these formulas Depending on the conditions used to determine motor specifications only one formula should be used Contact Appications Engineering for questions
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
9
MOTION SYSTEMFORMULAS
Stepper Motor Formulas
Required ldquoTime Offrdquo using an LR drive (constant voltage drive)
180Step Oslash = (phases)(rotor teeth)
ndash Motor step angle
ndash RPM (revolutionsminute) to SPS (stepssecond)
Maximum accuracy of a step motor system has nothing to do with maximum resolution of a step motor system Maximum accuracy is always based on the accuracy of 1 full step but maximum resolution is based on all system components including lead screw encoder stepper motor and step mode on the controller
ndash SPS (stepssecond) to RPM (revolutionsminute)(SPS) x (Step Oslash)
RPM = 6
(RPM) x 6 SPS = (Step Oslash)
ndash Travel per step
mstep =L
positionsrev
ndash Overdriving capability
Required ldquoTime Offrdquo using a chopper drive (constant current drive)
Time Off = (Time On) (Applied Voltage)2
(Rated Voltage)ndash Time On
Time Off = (Time On) (Applied Current)2
(Rated Current)ndash Time On
It is not unusual for a customer to drive Haydon Kerk stepper motors beyond their rated power to obtain the most force in the smallest package size Precautions must be taken to prevent the motor from exceeding its maximum temperature The ldquoonrdquo time should not exceed 2 - 3 minutes As general rule of thumb driving a motor at 25 to 3x its rated voltage or current may result in wasted energy and erratic behavior due to saturation
Special notes for can-stack motors bull more easily saturated due to less active material (steel) bull more iron losses due to unlaminated steel
Motion System Formulas10
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
10
MOTION SYSTEMFORMULAS
Lead Screw Performance Characteristics
End Mounting Rigidity
Impact Impact
ScrewLength
Increase In Affects Increase In Affects
Critical speed
Compression load
Critical speed
Inertia
Compression load
Stiffness
Spring Rate
Load capacity
Drive torque
Angular velocity
Load cpacity
Positioning accuracy
ScrewDiameter
Lead
Load
Preload
Nut length
SI Unit Systems
LengthMassTimeElectric currentThermodynamic temperatureLuminous intensityAmount of substance
meterkilogramsecondamperekelvincandelamole
mKgsAKcdmol
Base Units
Derived Unitsareavolumefrequencymass density (density)speed velocityangular velocityaccelerationangular accelerationforcepressure (mechanical stress)kinematic viscositydynamic viscositywork energy quantity of heatpowerentropyspecific heat capacitythermal conductivity
square metercubic meterhertzkilogram per cubic metermeter per secondradian per secondmeter per second squaredradian per second squarednewtonpascalsquare meter per secondnewton-second per square meter joulewattjoule per kelvinjoule per kilogram kelvinwatt per meter kelvin
m2
m3
HzKgm3
msradsms2
rads2
NPam2sN-sm2
JWJKJKg-K)W(m-K)
Quantity Name of Unit Symbol
Motion System Reference Tables11
Critical speed
Compression load
System stiffness
Life
Positioning accuracy
System stiffness
Drag torque
Load capacity
Stiffness
sndash1
Kg-ms2
Nm2
N-mJs or N-ms
ExamplesAn INCREASE ( ) in screw length results in a DECREASE ( ) in critical speed
An INCREASE in screw diameter results in an INCREASE in critical speed
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
11
MOTION SYSTEMREFERENCE TABLES
t 1 t 2 t 3 t dwell
vTime
vpk
a -a
MassMove distanceMove timeDwell timeMotion profileScrew speed limitLoad supportOrientationRotary to linear conversionAmbient temperature
9 Kg200 mm10 s05 s13 13 13 Trapezoidal1000 RPMμ = 001VerticalTFE lead screw 8 mm diameter 275 mm long30degC
ControlDrive Power Supply
4 quadrant with encoder32 VDC 35 Arms50 A peak
0333 s 0333 s 0333 s 05 s
vPK =3S2t =
(3)(02m)(2)(10 s) =
06m2 s =
03ms
Linear Velocity
Linear Acceleration
a = vf - vi
t = 03ms - 0 ms 0333 s 0901ms2 =
1st step is to thoroughly understand system constraints and motion profile requirements
Linear Motion ExampleData
Motion System Application Example12
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
12
MOTION SYSTEMAPPLICATION EXAMPLE
Lmin =60vPK
n =(60)(03ms)1000 revmin = 0018mrev = 18mmrev
Refer to Kerk screw chart ndash closest lead in an 8mm screw diameter is
2032mmrev = 002032 mrevη = 086 free wheeling nut TFE coated screw
JS = 388 x 10ndash 7 Kg-m2 Other critical lead screw considerations bull Column loading bull Critical speed
Jm aa
ωPKvPK
TF
= ωPK = 2 π vPK
L=
(2π)(03ms)002032 mrev
9276 rads
a = =2πaL
=(2π)(0901ms2)002032 mrev
2786 rads2
Linear to Rotary Conversion ndash Velocity Acceleration Inertia Torque
Minimum screw lead to maintain lt 1000 RPM shaft speed
Linear to Rotary Conversion Velocity
Linear to Rotary Conversion Acceleration
Motion System Application Example13
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
13
MOTION SYSTEMFORMULAS
Jin= m 2 πL 2
X η1 + JS= 9Kg
2 π002032 m 2
X 0861 + 388x10ndash 7Kg-m2
= 1095 X 10ndash 5 Kg-m2 + 388x10ndash 7 Kg-m2
= 1134 x 10ndash 5 Kg-m2
Τin= Τa+Τf +Τg+ΤD
Τa = Jin ain
Τg =sinOslashmgL
2 π η
= (1134 x 10ndash 5 Kg-m2)(2786 rads2)
= 00316 Nm
Τf =cosOslashmgμL
2 π η= (cos90)(9 Kg)(98ms2)(001)(002032m)
2 π (086)= 0 Nm
= (sin90)(9 Kg)(98ms2)(002032m)2 π (086)
=1792 Nm
5404 = 03316 Nm
ΤDSince a free-wheeling lead screw nut was used there is no preload force If an anti-backlash nut was used ΤD would be gt0 due to preload force Always reference manufacturerrsquos data for more information
= 0 Nm
Linear to Rotary Conversion Torque
Linear to Rotary Conversion Inertia
Motion System Application Example14
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
14
MOTION SYSTEMFORMULAS
t 1 t 2 t 3 t dwell Time0333 s 0333 s 0333 s 05 s
Rotary Motion Profile
ωPK
ω
= 9276 rads a = 2786 rads2
Τ1= Τa+Τf +Τg+ΤD
= (00316 Nm) + (0 Nm) + (03316 Nm) + (0 Nm)
= 03632 Nm
Τ2= Τa+Τf +Τg+ΤD
= (0 Nm) + (0 Nm) + (03316 Nm) + (0 Nm)
= 03316 Nm
Τ3= Τa+Τf +Τg+ΤD
= (00316 Nm) + (0 Nm) + (03316 Nm) + (0 Nm)
= 03632 Nm
Τ1
Τ2
Τ3
Motion System Application Example15
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
15
MOTION SYSTEMAPPLICATION EXAMPLE
Lead Screw Input Requirements
= 9276 rads= 2786 rads2
= 03632 Nm= 03316 Nm= ndash 03632 Nm= 0333 s= 0333 s= 0333 s= 05 s= 3369 W= 3076 W
ωPKaΤ1Τ2Τ3
t1t2t3tdwell
PPKPCV
PPK = (Τ1)(ωPK)
= (03632 Nm)(9276 rads)= 3369 W
PCV = (Τ2)(ωPK)
= (03316 Nm)(9276 rads)= 3076 W
RMS Torque Requirement Lead Screw Input
=
0333 s + 0333 s + 0333 s + 05 s
(03632 Nm)2(0333 s)+(03316 Nm)2(0333 s)+(- 03632 Nm)2(0333 s)
ΤRMS = Τ12t 1 + Τ2
2t2 + Τ32t 3
t 1 + t 2 + t 3 + tdwell
15
00439 + 00366 + 00439 = 00829 =
ΤRMS = 02879 Nm
Motion System Application Example16
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
16
MOTION SYSTEMAPPLICATION EXAMPLE
Load Parameters at the Lead Screw Shaft
= 9276 rads= 2786 rads2
= 03632 Nm= 03316 Nm= ndash 03632 Nm= 02879 Nm= 3369 W= 3076 W
ωPKaΤ1Τ2Τ3ΤRMSPPKPCV
Motor SelectionA direct-drive motor can be used however it would be large and expensive For example a DC057B-3 brush motor
would easily meet continuous torque and continuous speed as well as a 181 load-to-rotor inertia This motor however has significantly more output power than needed at 128 watts
A smaller motor with a transmission would optimize the system Look for a motor starting at a rated power around 40 watts Pittman DC040B-6
DC040B-6Reference VoltageContinuous TorqueRated CurrentRated PowerTorque ConstantVoltage ConstantTerminal ResistanceMax Winding TemperatureRotor Inertia
240 V00812 Nm236 A37 W0042 NmA0042 Vrads185 Ω155degC847 x 10ndash6 Kg-m2
Motor TransmissionLead Screw Support Rails
ΤRMS
PPK
ωPK
= 02879 Nm= 3369 W= 9276 rads
M
Other Motor Considerationsndash Type stepper brush BLDCndash Input speed (high input speed will result in high noise levels)ndash Total motor footprintndash Ambient temperaturendash Environmental conditionsndash Load-to-rotor inertiandash Encoder ready needed
Motion System Application Example17
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
17
MOTION SYSTEMAPPLICATION EXAMPLE
Gearbox Transmission Selection
When selecting a reduction ratio keep in mind that the RMS required torque at the motor shaft needs to fall below the continuous rated output torque of the motor
G30 A - Planetary GearboxMaximum output load 247 Nm
TRMS( Lead Screw)
02879 Nm
02879 Nm02879 Nm
Reduction
41
5161
Efficiency
090
090090
TRMS( Motor)
00798 Nm
00640 Nm00533 Nm
A 51 gearbox would allow about a 27 safety margin between what the system requires and the continuous torque output of the motor
System Summary 51η = 090
M
= 00807 Nm= 00737 Nm= ndash 00807 Nm= 00640 Nm= 4638 rads= 504 x 10 ndash 6 Kg-m2
= 3743 W
Τ1Τ2Τ3ΤRMS
PPK
ωPK
J
= 03632 Nm= 03316 Nm= ndash 03632 Nm= 02879 Nm= 9276 rads= 1134 x 10 ndash 5 Kg-m2
= 3369 W
Τ1Τ2Τ3ΤRMS
PPK
ωPK
J
Motion System Application Example18
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
18
MOTION SYSTEMAPPLICATION EXAMPLE
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
19
Motion Profile at the Motor
t 1 t 2 t 3 t dwell Time0333 s 0333 s 0333 s 05 s
ωΤ1
Τ2
Τ3= 4638 rads= 4429 RPM= ΤPK = 00807 Nm
ωPK
Τ1
nPK
Drive and Power Supply Requirements
Peak Current Required
IPK =TPK
KT+ IO =
00807 Nm0042 NmA
+ 0180 A = 210 A
RMS Current Required
IRMS =TRMS
KT+ IO =
00640 Nm0042 NmA
+ 0180 A = 170 A
Minimum bus Voltage Required
Vbus= IPKRmt+ ωPKKE = (210A)(185Ω) + (4638 rads)(0042 Vrads)= 389 V + 1948 V= 2337 V
0100 Nm 0200 Nm 0300 Nm 0400 Nm 0500 Nm 0600 Nm 0700 Nm
30 VDC
0100 Nm 0200 Nm 0300 Nm 0500 Nm 0600 Nm 0700 Nm
24 VDC
700060005000
RPM 4000300020001000
0400 Nm
TRMS
TPK
700060005000
RPM 4000300020001000
TRMS
TPK
Performance using 24V reference voltage
Performance using a 30V bus voltage
A 30 VDC bus will meet the application requirements as wells supply a margin of safety
MOTION SYSTEMAPPLICATION EXAMPLE
0100 Nm 0200 Nm 0300 Nm 0400 Nm 0500 Nm 0600 Nm 0700 Nm
30 VDC
0100 Nm 0200 Nm 0300 Nm 0500 Nm 0600 Nm 0700 Nm
24 VDC
700060005000
RPM 4000300020001000
0400 Nm
TRMS
TPK
700060005000
RPM 4000300020001000
TRMS
TPK
Ambient temperature Rated motor temperature Temperature rise Motor temperature
= 30degC= 155degC= 7638degC= 10638degC
Θa
Θm
ΘrΘrated
=Rth x IRMS
2 x Rmt
1 ndash (Rth x IRMS2 x Rmt x 000392degC)
Θr
=11degCω x 170A2
x 185Ω 1 ndash (11degCω x 170A2
x 185Ω x 000392degC)
= 5881 1 ndash (023)
= 5881 077
=
Θm = Θr + Θa = 7638degC + 30degC = 10638degC
7638degC
Performance using 24V reference voltage
Performance using a 30V bus voltage
Will the motor meet the temperature rise caused by the application
Motion System Application Example20
A 30 VDC bus will meet the application requirements as wells supply a margin of safety
203 756 7441 1500 Meriden Road Waterbury CT USA 06705
267 933 2105 534 Godshall DriveHarleysville PA USA 19438
wwwHaydonKerkPittmancom
copy All rights reserved AMETEK Advanced Motion Solutions No part of this doc-ument or technical information can be used reproduced or altered in any form without approval or proper authorization from AMETEK Inc and global affiliates
NOW ASK us the tough motion control questions ONLINE
ASK THE EXPERTS
HaydonKerkPittmancomwww
MOTION SYSTEMAPPLICATION EXAMPLE
Motor Power Relationships
ndash Watts lost due to winding resistance
Ploss = I2 x Rmt
ndash Output power at any point on the motor curve
Pout = ω x Tndash Motor maximum output power
Pmax = 025 x ωo x TLR
ndash Motor maximum output power (Theoretical)
Pmax = 025 x VT2
Rmt
Temperature Effects on Motor Constants
ndash Change in terminal resistance
Rmt(f) = Rmt(i) x [ 1 + αconductor (Θf ndash Θi) ]αconductor = 00040degC (copper)
K(f) = K(i) x [ 1 + αmagnet (Θf ndash Θi) ]ndash Change in torque constant and voltage constant ( KT = KE )
Magnetic Material
CeramicSmCoAlNiCoNdFeB
Θmax (degC)300degC300degC540degC150degC
The magnetic material values above represent average figures for particular material classes and estimating performance If exact values are needed consult data sheets for a specific magnet grade
Motion System Formulas8
αmagnet (degC)-00020degC-00004degC-00002degC-00012degC
In the case of a mechanically commutated motor with graphite brushes this analysis will result in a slight error because of the negative temperature coefficient of carbon For estimation purposes this can be ignored but for a more rigorous analysis the behavior of the brush material must be taken into account This is not an issue when evaluating a brushless dc motor
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
7
MOTION SYSTEMFORMULAS
Motion System Formulas9
For PMDC motors (brush type) brush drop can be approximated as a constant resistance connected in series with the armature winding This series resistance is included in the Rmt factor found on manufacturerrsquos data sheet
Copper graphiteSilver graphiteElectrographite
02 to 04 Ω 02 to 04 Ω 08 to 10 Ω
Motor Brush Resistance
Brush Material Resistance
DC Motor Thermal Resistance and Temperature Rise
ndash Thermal resistance
ndash Motor temperature rise
Θm = Θr + Θa
ndash OR ndash
Temperature Rise degCRth =
Machine Losses W
Θr = Rth x I2 x Rmt
Θr = Rth x I
2 x Rmt
1 ndash (Rth x I2 x Rmt x α)
ndash OR ndash
Θr = Rth x(TC
Km)
2
ndash Final motor temperature
Use caution when applying these formulas Depending on the conditions used to determine motor specifications only one formula should be used Contact Appications Engineering for questions
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
8
MOTION SYSTEMFORMULAS
Motion System Formulas9
For PMDC motors (brush type) brush drop can be approximated as a constant resistance connected in series with the armature winding This series resistance is included in the Rmt factor found on manufacturerrsquos data sheet
Copper graphiteSilver graphiteElectrographite
02 to 04 Ω 02 to 04 Ω 08 to 10 Ω
Motor Brush Resistance
Brush Material Resistance
DC Motor Thermal Resistance and Temperature Rise
ndash Thermal resistance
ndash Motor temperature rise
Θm = Θr + Θa
ndash OR ndash
Temperature Rise degCRth =
Machine Losses W
Θr = Rth x I2 x Rmt
Θr = Rth x I
2 x Rmt
1 ndash (Rth x I2 x Rmt x α)
ndash OR ndash
Θr = Rth x(TC
Km)
2
ndash Final motor temperature
Use caution when applying these formulas Depending on the conditions used to determine motor specifications only one formula should be used Contact Appications Engineering for questions
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
9
MOTION SYSTEMFORMULAS
Stepper Motor Formulas
Required ldquoTime Offrdquo using an LR drive (constant voltage drive)
180Step Oslash = (phases)(rotor teeth)
ndash Motor step angle
ndash RPM (revolutionsminute) to SPS (stepssecond)
Maximum accuracy of a step motor system has nothing to do with maximum resolution of a step motor system Maximum accuracy is always based on the accuracy of 1 full step but maximum resolution is based on all system components including lead screw encoder stepper motor and step mode on the controller
ndash SPS (stepssecond) to RPM (revolutionsminute)(SPS) x (Step Oslash)
RPM = 6
(RPM) x 6 SPS = (Step Oslash)
ndash Travel per step
mstep =L
positionsrev
ndash Overdriving capability
Required ldquoTime Offrdquo using a chopper drive (constant current drive)
Time Off = (Time On) (Applied Voltage)2
(Rated Voltage)ndash Time On
Time Off = (Time On) (Applied Current)2
(Rated Current)ndash Time On
It is not unusual for a customer to drive Haydon Kerk stepper motors beyond their rated power to obtain the most force in the smallest package size Precautions must be taken to prevent the motor from exceeding its maximum temperature The ldquoonrdquo time should not exceed 2 - 3 minutes As general rule of thumb driving a motor at 25 to 3x its rated voltage or current may result in wasted energy and erratic behavior due to saturation
Special notes for can-stack motors bull more easily saturated due to less active material (steel) bull more iron losses due to unlaminated steel
Motion System Formulas10
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
10
MOTION SYSTEMFORMULAS
Lead Screw Performance Characteristics
End Mounting Rigidity
Impact Impact
ScrewLength
Increase In Affects Increase In Affects
Critical speed
Compression load
Critical speed
Inertia
Compression load
Stiffness
Spring Rate
Load capacity
Drive torque
Angular velocity
Load cpacity
Positioning accuracy
ScrewDiameter
Lead
Load
Preload
Nut length
SI Unit Systems
LengthMassTimeElectric currentThermodynamic temperatureLuminous intensityAmount of substance
meterkilogramsecondamperekelvincandelamole
mKgsAKcdmol
Base Units
Derived Unitsareavolumefrequencymass density (density)speed velocityangular velocityaccelerationangular accelerationforcepressure (mechanical stress)kinematic viscositydynamic viscositywork energy quantity of heatpowerentropyspecific heat capacitythermal conductivity
square metercubic meterhertzkilogram per cubic metermeter per secondradian per secondmeter per second squaredradian per second squarednewtonpascalsquare meter per secondnewton-second per square meter joulewattjoule per kelvinjoule per kilogram kelvinwatt per meter kelvin
m2
m3
HzKgm3
msradsms2
rads2
NPam2sN-sm2
JWJKJKg-K)W(m-K)
Quantity Name of Unit Symbol
Motion System Reference Tables11
Critical speed
Compression load
System stiffness
Life
Positioning accuracy
System stiffness
Drag torque
Load capacity
Stiffness
sndash1
Kg-ms2
Nm2
N-mJs or N-ms
ExamplesAn INCREASE ( ) in screw length results in a DECREASE ( ) in critical speed
An INCREASE in screw diameter results in an INCREASE in critical speed
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11
MOTION SYSTEMREFERENCE TABLES
t 1 t 2 t 3 t dwell
vTime
vpk
a -a
MassMove distanceMove timeDwell timeMotion profileScrew speed limitLoad supportOrientationRotary to linear conversionAmbient temperature
9 Kg200 mm10 s05 s13 13 13 Trapezoidal1000 RPMμ = 001VerticalTFE lead screw 8 mm diameter 275 mm long30degC
ControlDrive Power Supply
4 quadrant with encoder32 VDC 35 Arms50 A peak
0333 s 0333 s 0333 s 05 s
vPK =3S2t =
(3)(02m)(2)(10 s) =
06m2 s =
03ms
Linear Velocity
Linear Acceleration
a = vf - vi
t = 03ms - 0 ms 0333 s 0901ms2 =
1st step is to thoroughly understand system constraints and motion profile requirements
Linear Motion ExampleData
Motion System Application Example12
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12
MOTION SYSTEMAPPLICATION EXAMPLE
Lmin =60vPK
n =(60)(03ms)1000 revmin = 0018mrev = 18mmrev
Refer to Kerk screw chart ndash closest lead in an 8mm screw diameter is
2032mmrev = 002032 mrevη = 086 free wheeling nut TFE coated screw
JS = 388 x 10ndash 7 Kg-m2 Other critical lead screw considerations bull Column loading bull Critical speed
Jm aa
ωPKvPK
TF
= ωPK = 2 π vPK
L=
(2π)(03ms)002032 mrev
9276 rads
a = =2πaL
=(2π)(0901ms2)002032 mrev
2786 rads2
Linear to Rotary Conversion ndash Velocity Acceleration Inertia Torque
Minimum screw lead to maintain lt 1000 RPM shaft speed
Linear to Rotary Conversion Velocity
Linear to Rotary Conversion Acceleration
Motion System Application Example13
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13
MOTION SYSTEMFORMULAS
Jin= m 2 πL 2
X η1 + JS= 9Kg
2 π002032 m 2
X 0861 + 388x10ndash 7Kg-m2
= 1095 X 10ndash 5 Kg-m2 + 388x10ndash 7 Kg-m2
= 1134 x 10ndash 5 Kg-m2
Τin= Τa+Τf +Τg+ΤD
Τa = Jin ain
Τg =sinOslashmgL
2 π η
= (1134 x 10ndash 5 Kg-m2)(2786 rads2)
= 00316 Nm
Τf =cosOslashmgμL
2 π η= (cos90)(9 Kg)(98ms2)(001)(002032m)
2 π (086)= 0 Nm
= (sin90)(9 Kg)(98ms2)(002032m)2 π (086)
=1792 Nm
5404 = 03316 Nm
ΤDSince a free-wheeling lead screw nut was used there is no preload force If an anti-backlash nut was used ΤD would be gt0 due to preload force Always reference manufacturerrsquos data for more information
= 0 Nm
Linear to Rotary Conversion Torque
Linear to Rotary Conversion Inertia
Motion System Application Example14
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14
MOTION SYSTEMFORMULAS
t 1 t 2 t 3 t dwell Time0333 s 0333 s 0333 s 05 s
Rotary Motion Profile
ωPK
ω
= 9276 rads a = 2786 rads2
Τ1= Τa+Τf +Τg+ΤD
= (00316 Nm) + (0 Nm) + (03316 Nm) + (0 Nm)
= 03632 Nm
Τ2= Τa+Τf +Τg+ΤD
= (0 Nm) + (0 Nm) + (03316 Nm) + (0 Nm)
= 03316 Nm
Τ3= Τa+Τf +Τg+ΤD
= (00316 Nm) + (0 Nm) + (03316 Nm) + (0 Nm)
= 03632 Nm
Τ1
Τ2
Τ3
Motion System Application Example15
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15
MOTION SYSTEMAPPLICATION EXAMPLE
Lead Screw Input Requirements
= 9276 rads= 2786 rads2
= 03632 Nm= 03316 Nm= ndash 03632 Nm= 0333 s= 0333 s= 0333 s= 05 s= 3369 W= 3076 W
ωPKaΤ1Τ2Τ3
t1t2t3tdwell
PPKPCV
PPK = (Τ1)(ωPK)
= (03632 Nm)(9276 rads)= 3369 W
PCV = (Τ2)(ωPK)
= (03316 Nm)(9276 rads)= 3076 W
RMS Torque Requirement Lead Screw Input
=
0333 s + 0333 s + 0333 s + 05 s
(03632 Nm)2(0333 s)+(03316 Nm)2(0333 s)+(- 03632 Nm)2(0333 s)
ΤRMS = Τ12t 1 + Τ2
2t2 + Τ32t 3
t 1 + t 2 + t 3 + tdwell
15
00439 + 00366 + 00439 = 00829 =
ΤRMS = 02879 Nm
Motion System Application Example16
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16
MOTION SYSTEMAPPLICATION EXAMPLE
Load Parameters at the Lead Screw Shaft
= 9276 rads= 2786 rads2
= 03632 Nm= 03316 Nm= ndash 03632 Nm= 02879 Nm= 3369 W= 3076 W
ωPKaΤ1Τ2Τ3ΤRMSPPKPCV
Motor SelectionA direct-drive motor can be used however it would be large and expensive For example a DC057B-3 brush motor
would easily meet continuous torque and continuous speed as well as a 181 load-to-rotor inertia This motor however has significantly more output power than needed at 128 watts
A smaller motor with a transmission would optimize the system Look for a motor starting at a rated power around 40 watts Pittman DC040B-6
DC040B-6Reference VoltageContinuous TorqueRated CurrentRated PowerTorque ConstantVoltage ConstantTerminal ResistanceMax Winding TemperatureRotor Inertia
240 V00812 Nm236 A37 W0042 NmA0042 Vrads185 Ω155degC847 x 10ndash6 Kg-m2
Motor TransmissionLead Screw Support Rails
ΤRMS
PPK
ωPK
= 02879 Nm= 3369 W= 9276 rads
M
Other Motor Considerationsndash Type stepper brush BLDCndash Input speed (high input speed will result in high noise levels)ndash Total motor footprintndash Ambient temperaturendash Environmental conditionsndash Load-to-rotor inertiandash Encoder ready needed
Motion System Application Example17
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17
MOTION SYSTEMAPPLICATION EXAMPLE
Gearbox Transmission Selection
When selecting a reduction ratio keep in mind that the RMS required torque at the motor shaft needs to fall below the continuous rated output torque of the motor
G30 A - Planetary GearboxMaximum output load 247 Nm
TRMS( Lead Screw)
02879 Nm
02879 Nm02879 Nm
Reduction
41
5161
Efficiency
090
090090
TRMS( Motor)
00798 Nm
00640 Nm00533 Nm
A 51 gearbox would allow about a 27 safety margin between what the system requires and the continuous torque output of the motor
System Summary 51η = 090
M
= 00807 Nm= 00737 Nm= ndash 00807 Nm= 00640 Nm= 4638 rads= 504 x 10 ndash 6 Kg-m2
= 3743 W
Τ1Τ2Τ3ΤRMS
PPK
ωPK
J
= 03632 Nm= 03316 Nm= ndash 03632 Nm= 02879 Nm= 9276 rads= 1134 x 10 ndash 5 Kg-m2
= 3369 W
Τ1Τ2Τ3ΤRMS
PPK
ωPK
J
Motion System Application Example18
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MOTION SYSTEMAPPLICATION EXAMPLE
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19
Motion Profile at the Motor
t 1 t 2 t 3 t dwell Time0333 s 0333 s 0333 s 05 s
ωΤ1
Τ2
Τ3= 4638 rads= 4429 RPM= ΤPK = 00807 Nm
ωPK
Τ1
nPK
Drive and Power Supply Requirements
Peak Current Required
IPK =TPK
KT+ IO =
00807 Nm0042 NmA
+ 0180 A = 210 A
RMS Current Required
IRMS =TRMS
KT+ IO =
00640 Nm0042 NmA
+ 0180 A = 170 A
Minimum bus Voltage Required
Vbus= IPKRmt+ ωPKKE = (210A)(185Ω) + (4638 rads)(0042 Vrads)= 389 V + 1948 V= 2337 V
0100 Nm 0200 Nm 0300 Nm 0400 Nm 0500 Nm 0600 Nm 0700 Nm
30 VDC
0100 Nm 0200 Nm 0300 Nm 0500 Nm 0600 Nm 0700 Nm
24 VDC
700060005000
RPM 4000300020001000
0400 Nm
TRMS
TPK
700060005000
RPM 4000300020001000
TRMS
TPK
Performance using 24V reference voltage
Performance using a 30V bus voltage
A 30 VDC bus will meet the application requirements as wells supply a margin of safety
MOTION SYSTEMAPPLICATION EXAMPLE
0100 Nm 0200 Nm 0300 Nm 0400 Nm 0500 Nm 0600 Nm 0700 Nm
30 VDC
0100 Nm 0200 Nm 0300 Nm 0500 Nm 0600 Nm 0700 Nm
24 VDC
700060005000
RPM 4000300020001000
0400 Nm
TRMS
TPK
700060005000
RPM 4000300020001000
TRMS
TPK
Ambient temperature Rated motor temperature Temperature rise Motor temperature
= 30degC= 155degC= 7638degC= 10638degC
Θa
Θm
ΘrΘrated
=Rth x IRMS
2 x Rmt
1 ndash (Rth x IRMS2 x Rmt x 000392degC)
Θr
=11degCω x 170A2
x 185Ω 1 ndash (11degCω x 170A2
x 185Ω x 000392degC)
= 5881 1 ndash (023)
= 5881 077
=
Θm = Θr + Θa = 7638degC + 30degC = 10638degC
7638degC
Performance using 24V reference voltage
Performance using a 30V bus voltage
Will the motor meet the temperature rise caused by the application
Motion System Application Example20
A 30 VDC bus will meet the application requirements as wells supply a margin of safety
203 756 7441 1500 Meriden Road Waterbury CT USA 06705
267 933 2105 534 Godshall DriveHarleysville PA USA 19438
wwwHaydonKerkPittmancom
copy All rights reserved AMETEK Advanced Motion Solutions No part of this doc-ument or technical information can be used reproduced or altered in any form without approval or proper authorization from AMETEK Inc and global affiliates
NOW ASK us the tough motion control questions ONLINE
ASK THE EXPERTS
HaydonKerkPittmancomwww
MOTION SYSTEMAPPLICATION EXAMPLE
Motion System Formulas9
For PMDC motors (brush type) brush drop can be approximated as a constant resistance connected in series with the armature winding This series resistance is included in the Rmt factor found on manufacturerrsquos data sheet
Copper graphiteSilver graphiteElectrographite
02 to 04 Ω 02 to 04 Ω 08 to 10 Ω
Motor Brush Resistance
Brush Material Resistance
DC Motor Thermal Resistance and Temperature Rise
ndash Thermal resistance
ndash Motor temperature rise
Θm = Θr + Θa
ndash OR ndash
Temperature Rise degCRth =
Machine Losses W
Θr = Rth x I2 x Rmt
Θr = Rth x I
2 x Rmt
1 ndash (Rth x I2 x Rmt x α)
ndash OR ndash
Θr = Rth x(TC
Km)
2
ndash Final motor temperature
Use caution when applying these formulas Depending on the conditions used to determine motor specifications only one formula should be used Contact Appications Engineering for questions
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8
MOTION SYSTEMFORMULAS
Motion System Formulas9
For PMDC motors (brush type) brush drop can be approximated as a constant resistance connected in series with the armature winding This series resistance is included in the Rmt factor found on manufacturerrsquos data sheet
Copper graphiteSilver graphiteElectrographite
02 to 04 Ω 02 to 04 Ω 08 to 10 Ω
Motor Brush Resistance
Brush Material Resistance
DC Motor Thermal Resistance and Temperature Rise
ndash Thermal resistance
ndash Motor temperature rise
Θm = Θr + Θa
ndash OR ndash
Temperature Rise degCRth =
Machine Losses W
Θr = Rth x I2 x Rmt
Θr = Rth x I
2 x Rmt
1 ndash (Rth x I2 x Rmt x α)
ndash OR ndash
Θr = Rth x(TC
Km)
2
ndash Final motor temperature
Use caution when applying these formulas Depending on the conditions used to determine motor specifications only one formula should be used Contact Appications Engineering for questions
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9
MOTION SYSTEMFORMULAS
Stepper Motor Formulas
Required ldquoTime Offrdquo using an LR drive (constant voltage drive)
180Step Oslash = (phases)(rotor teeth)
ndash Motor step angle
ndash RPM (revolutionsminute) to SPS (stepssecond)
Maximum accuracy of a step motor system has nothing to do with maximum resolution of a step motor system Maximum accuracy is always based on the accuracy of 1 full step but maximum resolution is based on all system components including lead screw encoder stepper motor and step mode on the controller
ndash SPS (stepssecond) to RPM (revolutionsminute)(SPS) x (Step Oslash)
RPM = 6
(RPM) x 6 SPS = (Step Oslash)
ndash Travel per step
mstep =L
positionsrev
ndash Overdriving capability
Required ldquoTime Offrdquo using a chopper drive (constant current drive)
Time Off = (Time On) (Applied Voltage)2
(Rated Voltage)ndash Time On
Time Off = (Time On) (Applied Current)2
(Rated Current)ndash Time On
It is not unusual for a customer to drive Haydon Kerk stepper motors beyond their rated power to obtain the most force in the smallest package size Precautions must be taken to prevent the motor from exceeding its maximum temperature The ldquoonrdquo time should not exceed 2 - 3 minutes As general rule of thumb driving a motor at 25 to 3x its rated voltage or current may result in wasted energy and erratic behavior due to saturation
Special notes for can-stack motors bull more easily saturated due to less active material (steel) bull more iron losses due to unlaminated steel
Motion System Formulas10
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10
MOTION SYSTEMFORMULAS
Lead Screw Performance Characteristics
End Mounting Rigidity
Impact Impact
ScrewLength
Increase In Affects Increase In Affects
Critical speed
Compression load
Critical speed
Inertia
Compression load
Stiffness
Spring Rate
Load capacity
Drive torque
Angular velocity
Load cpacity
Positioning accuracy
ScrewDiameter
Lead
Load
Preload
Nut length
SI Unit Systems
LengthMassTimeElectric currentThermodynamic temperatureLuminous intensityAmount of substance
meterkilogramsecondamperekelvincandelamole
mKgsAKcdmol
Base Units
Derived Unitsareavolumefrequencymass density (density)speed velocityangular velocityaccelerationangular accelerationforcepressure (mechanical stress)kinematic viscositydynamic viscositywork energy quantity of heatpowerentropyspecific heat capacitythermal conductivity
square metercubic meterhertzkilogram per cubic metermeter per secondradian per secondmeter per second squaredradian per second squarednewtonpascalsquare meter per secondnewton-second per square meter joulewattjoule per kelvinjoule per kilogram kelvinwatt per meter kelvin
m2
m3
HzKgm3
msradsms2
rads2
NPam2sN-sm2
JWJKJKg-K)W(m-K)
Quantity Name of Unit Symbol
Motion System Reference Tables11
Critical speed
Compression load
System stiffness
Life
Positioning accuracy
System stiffness
Drag torque
Load capacity
Stiffness
sndash1
Kg-ms2
Nm2
N-mJs or N-ms
ExamplesAn INCREASE ( ) in screw length results in a DECREASE ( ) in critical speed
An INCREASE in screw diameter results in an INCREASE in critical speed
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
11
MOTION SYSTEMREFERENCE TABLES
t 1 t 2 t 3 t dwell
vTime
vpk
a -a
MassMove distanceMove timeDwell timeMotion profileScrew speed limitLoad supportOrientationRotary to linear conversionAmbient temperature
9 Kg200 mm10 s05 s13 13 13 Trapezoidal1000 RPMμ = 001VerticalTFE lead screw 8 mm diameter 275 mm long30degC
ControlDrive Power Supply
4 quadrant with encoder32 VDC 35 Arms50 A peak
0333 s 0333 s 0333 s 05 s
vPK =3S2t =
(3)(02m)(2)(10 s) =
06m2 s =
03ms
Linear Velocity
Linear Acceleration
a = vf - vi
t = 03ms - 0 ms 0333 s 0901ms2 =
1st step is to thoroughly understand system constraints and motion profile requirements
Linear Motion ExampleData
Motion System Application Example12
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
12
MOTION SYSTEMAPPLICATION EXAMPLE
Lmin =60vPK
n =(60)(03ms)1000 revmin = 0018mrev = 18mmrev
Refer to Kerk screw chart ndash closest lead in an 8mm screw diameter is
2032mmrev = 002032 mrevη = 086 free wheeling nut TFE coated screw
JS = 388 x 10ndash 7 Kg-m2 Other critical lead screw considerations bull Column loading bull Critical speed
Jm aa
ωPKvPK
TF
= ωPK = 2 π vPK
L=
(2π)(03ms)002032 mrev
9276 rads
a = =2πaL
=(2π)(0901ms2)002032 mrev
2786 rads2
Linear to Rotary Conversion ndash Velocity Acceleration Inertia Torque
Minimum screw lead to maintain lt 1000 RPM shaft speed
Linear to Rotary Conversion Velocity
Linear to Rotary Conversion Acceleration
Motion System Application Example13
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13
MOTION SYSTEMFORMULAS
Jin= m 2 πL 2
X η1 + JS= 9Kg
2 π002032 m 2
X 0861 + 388x10ndash 7Kg-m2
= 1095 X 10ndash 5 Kg-m2 + 388x10ndash 7 Kg-m2
= 1134 x 10ndash 5 Kg-m2
Τin= Τa+Τf +Τg+ΤD
Τa = Jin ain
Τg =sinOslashmgL
2 π η
= (1134 x 10ndash 5 Kg-m2)(2786 rads2)
= 00316 Nm
Τf =cosOslashmgμL
2 π η= (cos90)(9 Kg)(98ms2)(001)(002032m)
2 π (086)= 0 Nm
= (sin90)(9 Kg)(98ms2)(002032m)2 π (086)
=1792 Nm
5404 = 03316 Nm
ΤDSince a free-wheeling lead screw nut was used there is no preload force If an anti-backlash nut was used ΤD would be gt0 due to preload force Always reference manufacturerrsquos data for more information
= 0 Nm
Linear to Rotary Conversion Torque
Linear to Rotary Conversion Inertia
Motion System Application Example14
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14
MOTION SYSTEMFORMULAS
t 1 t 2 t 3 t dwell Time0333 s 0333 s 0333 s 05 s
Rotary Motion Profile
ωPK
ω
= 9276 rads a = 2786 rads2
Τ1= Τa+Τf +Τg+ΤD
= (00316 Nm) + (0 Nm) + (03316 Nm) + (0 Nm)
= 03632 Nm
Τ2= Τa+Τf +Τg+ΤD
= (0 Nm) + (0 Nm) + (03316 Nm) + (0 Nm)
= 03316 Nm
Τ3= Τa+Τf +Τg+ΤD
= (00316 Nm) + (0 Nm) + (03316 Nm) + (0 Nm)
= 03632 Nm
Τ1
Τ2
Τ3
Motion System Application Example15
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
15
MOTION SYSTEMAPPLICATION EXAMPLE
Lead Screw Input Requirements
= 9276 rads= 2786 rads2
= 03632 Nm= 03316 Nm= ndash 03632 Nm= 0333 s= 0333 s= 0333 s= 05 s= 3369 W= 3076 W
ωPKaΤ1Τ2Τ3
t1t2t3tdwell
PPKPCV
PPK = (Τ1)(ωPK)
= (03632 Nm)(9276 rads)= 3369 W
PCV = (Τ2)(ωPK)
= (03316 Nm)(9276 rads)= 3076 W
RMS Torque Requirement Lead Screw Input
=
0333 s + 0333 s + 0333 s + 05 s
(03632 Nm)2(0333 s)+(03316 Nm)2(0333 s)+(- 03632 Nm)2(0333 s)
ΤRMS = Τ12t 1 + Τ2
2t2 + Τ32t 3
t 1 + t 2 + t 3 + tdwell
15
00439 + 00366 + 00439 = 00829 =
ΤRMS = 02879 Nm
Motion System Application Example16
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
16
MOTION SYSTEMAPPLICATION EXAMPLE
Load Parameters at the Lead Screw Shaft
= 9276 rads= 2786 rads2
= 03632 Nm= 03316 Nm= ndash 03632 Nm= 02879 Nm= 3369 W= 3076 W
ωPKaΤ1Τ2Τ3ΤRMSPPKPCV
Motor SelectionA direct-drive motor can be used however it would be large and expensive For example a DC057B-3 brush motor
would easily meet continuous torque and continuous speed as well as a 181 load-to-rotor inertia This motor however has significantly more output power than needed at 128 watts
A smaller motor with a transmission would optimize the system Look for a motor starting at a rated power around 40 watts Pittman DC040B-6
DC040B-6Reference VoltageContinuous TorqueRated CurrentRated PowerTorque ConstantVoltage ConstantTerminal ResistanceMax Winding TemperatureRotor Inertia
240 V00812 Nm236 A37 W0042 NmA0042 Vrads185 Ω155degC847 x 10ndash6 Kg-m2
Motor TransmissionLead Screw Support Rails
ΤRMS
PPK
ωPK
= 02879 Nm= 3369 W= 9276 rads
M
Other Motor Considerationsndash Type stepper brush BLDCndash Input speed (high input speed will result in high noise levels)ndash Total motor footprintndash Ambient temperaturendash Environmental conditionsndash Load-to-rotor inertiandash Encoder ready needed
Motion System Application Example17
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
17
MOTION SYSTEMAPPLICATION EXAMPLE
Gearbox Transmission Selection
When selecting a reduction ratio keep in mind that the RMS required torque at the motor shaft needs to fall below the continuous rated output torque of the motor
G30 A - Planetary GearboxMaximum output load 247 Nm
TRMS( Lead Screw)
02879 Nm
02879 Nm02879 Nm
Reduction
41
5161
Efficiency
090
090090
TRMS( Motor)
00798 Nm
00640 Nm00533 Nm
A 51 gearbox would allow about a 27 safety margin between what the system requires and the continuous torque output of the motor
System Summary 51η = 090
M
= 00807 Nm= 00737 Nm= ndash 00807 Nm= 00640 Nm= 4638 rads= 504 x 10 ndash 6 Kg-m2
= 3743 W
Τ1Τ2Τ3ΤRMS
PPK
ωPK
J
= 03632 Nm= 03316 Nm= ndash 03632 Nm= 02879 Nm= 9276 rads= 1134 x 10 ndash 5 Kg-m2
= 3369 W
Τ1Τ2Τ3ΤRMS
PPK
ωPK
J
Motion System Application Example18
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
18
MOTION SYSTEMAPPLICATION EXAMPLE
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
19
Motion Profile at the Motor
t 1 t 2 t 3 t dwell Time0333 s 0333 s 0333 s 05 s
ωΤ1
Τ2
Τ3= 4638 rads= 4429 RPM= ΤPK = 00807 Nm
ωPK
Τ1
nPK
Drive and Power Supply Requirements
Peak Current Required
IPK =TPK
KT+ IO =
00807 Nm0042 NmA
+ 0180 A = 210 A
RMS Current Required
IRMS =TRMS
KT+ IO =
00640 Nm0042 NmA
+ 0180 A = 170 A
Minimum bus Voltage Required
Vbus= IPKRmt+ ωPKKE = (210A)(185Ω) + (4638 rads)(0042 Vrads)= 389 V + 1948 V= 2337 V
0100 Nm 0200 Nm 0300 Nm 0400 Nm 0500 Nm 0600 Nm 0700 Nm
30 VDC
0100 Nm 0200 Nm 0300 Nm 0500 Nm 0600 Nm 0700 Nm
24 VDC
700060005000
RPM 4000300020001000
0400 Nm
TRMS
TPK
700060005000
RPM 4000300020001000
TRMS
TPK
Performance using 24V reference voltage
Performance using a 30V bus voltage
A 30 VDC bus will meet the application requirements as wells supply a margin of safety
MOTION SYSTEMAPPLICATION EXAMPLE
0100 Nm 0200 Nm 0300 Nm 0400 Nm 0500 Nm 0600 Nm 0700 Nm
30 VDC
0100 Nm 0200 Nm 0300 Nm 0500 Nm 0600 Nm 0700 Nm
24 VDC
700060005000
RPM 4000300020001000
0400 Nm
TRMS
TPK
700060005000
RPM 4000300020001000
TRMS
TPK
Ambient temperature Rated motor temperature Temperature rise Motor temperature
= 30degC= 155degC= 7638degC= 10638degC
Θa
Θm
ΘrΘrated
=Rth x IRMS
2 x Rmt
1 ndash (Rth x IRMS2 x Rmt x 000392degC)
Θr
=11degCω x 170A2
x 185Ω 1 ndash (11degCω x 170A2
x 185Ω x 000392degC)
= 5881 1 ndash (023)
= 5881 077
=
Θm = Θr + Θa = 7638degC + 30degC = 10638degC
7638degC
Performance using 24V reference voltage
Performance using a 30V bus voltage
Will the motor meet the temperature rise caused by the application
Motion System Application Example20
A 30 VDC bus will meet the application requirements as wells supply a margin of safety
203 756 7441 1500 Meriden Road Waterbury CT USA 06705
267 933 2105 534 Godshall DriveHarleysville PA USA 19438
wwwHaydonKerkPittmancom
copy All rights reserved AMETEK Advanced Motion Solutions No part of this doc-ument or technical information can be used reproduced or altered in any form without approval or proper authorization from AMETEK Inc and global affiliates
NOW ASK us the tough motion control questions ONLINE
ASK THE EXPERTS
HaydonKerkPittmancomwww
MOTION SYSTEMAPPLICATION EXAMPLE
Motion System Formulas9
For PMDC motors (brush type) brush drop can be approximated as a constant resistance connected in series with the armature winding This series resistance is included in the Rmt factor found on manufacturerrsquos data sheet
Copper graphiteSilver graphiteElectrographite
02 to 04 Ω 02 to 04 Ω 08 to 10 Ω
Motor Brush Resistance
Brush Material Resistance
DC Motor Thermal Resistance and Temperature Rise
ndash Thermal resistance
ndash Motor temperature rise
Θm = Θr + Θa
ndash OR ndash
Temperature Rise degCRth =
Machine Losses W
Θr = Rth x I2 x Rmt
Θr = Rth x I
2 x Rmt
1 ndash (Rth x I2 x Rmt x α)
ndash OR ndash
Θr = Rth x(TC
Km)
2
ndash Final motor temperature
Use caution when applying these formulas Depending on the conditions used to determine motor specifications only one formula should be used Contact Appications Engineering for questions
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
9
MOTION SYSTEMFORMULAS
Stepper Motor Formulas
Required ldquoTime Offrdquo using an LR drive (constant voltage drive)
180Step Oslash = (phases)(rotor teeth)
ndash Motor step angle
ndash RPM (revolutionsminute) to SPS (stepssecond)
Maximum accuracy of a step motor system has nothing to do with maximum resolution of a step motor system Maximum accuracy is always based on the accuracy of 1 full step but maximum resolution is based on all system components including lead screw encoder stepper motor and step mode on the controller
ndash SPS (stepssecond) to RPM (revolutionsminute)(SPS) x (Step Oslash)
RPM = 6
(RPM) x 6 SPS = (Step Oslash)
ndash Travel per step
mstep =L
positionsrev
ndash Overdriving capability
Required ldquoTime Offrdquo using a chopper drive (constant current drive)
Time Off = (Time On) (Applied Voltage)2
(Rated Voltage)ndash Time On
Time Off = (Time On) (Applied Current)2
(Rated Current)ndash Time On
It is not unusual for a customer to drive Haydon Kerk stepper motors beyond their rated power to obtain the most force in the smallest package size Precautions must be taken to prevent the motor from exceeding its maximum temperature The ldquoonrdquo time should not exceed 2 - 3 minutes As general rule of thumb driving a motor at 25 to 3x its rated voltage or current may result in wasted energy and erratic behavior due to saturation
Special notes for can-stack motors bull more easily saturated due to less active material (steel) bull more iron losses due to unlaminated steel
Motion System Formulas10
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10
MOTION SYSTEMFORMULAS
Lead Screw Performance Characteristics
End Mounting Rigidity
Impact Impact
ScrewLength
Increase In Affects Increase In Affects
Critical speed
Compression load
Critical speed
Inertia
Compression load
Stiffness
Spring Rate
Load capacity
Drive torque
Angular velocity
Load cpacity
Positioning accuracy
ScrewDiameter
Lead
Load
Preload
Nut length
SI Unit Systems
LengthMassTimeElectric currentThermodynamic temperatureLuminous intensityAmount of substance
meterkilogramsecondamperekelvincandelamole
mKgsAKcdmol
Base Units
Derived Unitsareavolumefrequencymass density (density)speed velocityangular velocityaccelerationangular accelerationforcepressure (mechanical stress)kinematic viscositydynamic viscositywork energy quantity of heatpowerentropyspecific heat capacitythermal conductivity
square metercubic meterhertzkilogram per cubic metermeter per secondradian per secondmeter per second squaredradian per second squarednewtonpascalsquare meter per secondnewton-second per square meter joulewattjoule per kelvinjoule per kilogram kelvinwatt per meter kelvin
m2
m3
HzKgm3
msradsms2
rads2
NPam2sN-sm2
JWJKJKg-K)W(m-K)
Quantity Name of Unit Symbol
Motion System Reference Tables11
Critical speed
Compression load
System stiffness
Life
Positioning accuracy
System stiffness
Drag torque
Load capacity
Stiffness
sndash1
Kg-ms2
Nm2
N-mJs or N-ms
ExamplesAn INCREASE ( ) in screw length results in a DECREASE ( ) in critical speed
An INCREASE in screw diameter results in an INCREASE in critical speed
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11
MOTION SYSTEMREFERENCE TABLES
t 1 t 2 t 3 t dwell
vTime
vpk
a -a
MassMove distanceMove timeDwell timeMotion profileScrew speed limitLoad supportOrientationRotary to linear conversionAmbient temperature
9 Kg200 mm10 s05 s13 13 13 Trapezoidal1000 RPMμ = 001VerticalTFE lead screw 8 mm diameter 275 mm long30degC
ControlDrive Power Supply
4 quadrant with encoder32 VDC 35 Arms50 A peak
0333 s 0333 s 0333 s 05 s
vPK =3S2t =
(3)(02m)(2)(10 s) =
06m2 s =
03ms
Linear Velocity
Linear Acceleration
a = vf - vi
t = 03ms - 0 ms 0333 s 0901ms2 =
1st step is to thoroughly understand system constraints and motion profile requirements
Linear Motion ExampleData
Motion System Application Example12
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
12
MOTION SYSTEMAPPLICATION EXAMPLE
Lmin =60vPK
n =(60)(03ms)1000 revmin = 0018mrev = 18mmrev
Refer to Kerk screw chart ndash closest lead in an 8mm screw diameter is
2032mmrev = 002032 mrevη = 086 free wheeling nut TFE coated screw
JS = 388 x 10ndash 7 Kg-m2 Other critical lead screw considerations bull Column loading bull Critical speed
Jm aa
ωPKvPK
TF
= ωPK = 2 π vPK
L=
(2π)(03ms)002032 mrev
9276 rads
a = =2πaL
=(2π)(0901ms2)002032 mrev
2786 rads2
Linear to Rotary Conversion ndash Velocity Acceleration Inertia Torque
Minimum screw lead to maintain lt 1000 RPM shaft speed
Linear to Rotary Conversion Velocity
Linear to Rotary Conversion Acceleration
Motion System Application Example13
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
13
MOTION SYSTEMFORMULAS
Jin= m 2 πL 2
X η1 + JS= 9Kg
2 π002032 m 2
X 0861 + 388x10ndash 7Kg-m2
= 1095 X 10ndash 5 Kg-m2 + 388x10ndash 7 Kg-m2
= 1134 x 10ndash 5 Kg-m2
Τin= Τa+Τf +Τg+ΤD
Τa = Jin ain
Τg =sinOslashmgL
2 π η
= (1134 x 10ndash 5 Kg-m2)(2786 rads2)
= 00316 Nm
Τf =cosOslashmgμL
2 π η= (cos90)(9 Kg)(98ms2)(001)(002032m)
2 π (086)= 0 Nm
= (sin90)(9 Kg)(98ms2)(002032m)2 π (086)
=1792 Nm
5404 = 03316 Nm
ΤDSince a free-wheeling lead screw nut was used there is no preload force If an anti-backlash nut was used ΤD would be gt0 due to preload force Always reference manufacturerrsquos data for more information
= 0 Nm
Linear to Rotary Conversion Torque
Linear to Rotary Conversion Inertia
Motion System Application Example14
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
14
MOTION SYSTEMFORMULAS
t 1 t 2 t 3 t dwell Time0333 s 0333 s 0333 s 05 s
Rotary Motion Profile
ωPK
ω
= 9276 rads a = 2786 rads2
Τ1= Τa+Τf +Τg+ΤD
= (00316 Nm) + (0 Nm) + (03316 Nm) + (0 Nm)
= 03632 Nm
Τ2= Τa+Τf +Τg+ΤD
= (0 Nm) + (0 Nm) + (03316 Nm) + (0 Nm)
= 03316 Nm
Τ3= Τa+Τf +Τg+ΤD
= (00316 Nm) + (0 Nm) + (03316 Nm) + (0 Nm)
= 03632 Nm
Τ1
Τ2
Τ3
Motion System Application Example15
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
15
MOTION SYSTEMAPPLICATION EXAMPLE
Lead Screw Input Requirements
= 9276 rads= 2786 rads2
= 03632 Nm= 03316 Nm= ndash 03632 Nm= 0333 s= 0333 s= 0333 s= 05 s= 3369 W= 3076 W
ωPKaΤ1Τ2Τ3
t1t2t3tdwell
PPKPCV
PPK = (Τ1)(ωPK)
= (03632 Nm)(9276 rads)= 3369 W
PCV = (Τ2)(ωPK)
= (03316 Nm)(9276 rads)= 3076 W
RMS Torque Requirement Lead Screw Input
=
0333 s + 0333 s + 0333 s + 05 s
(03632 Nm)2(0333 s)+(03316 Nm)2(0333 s)+(- 03632 Nm)2(0333 s)
ΤRMS = Τ12t 1 + Τ2
2t2 + Τ32t 3
t 1 + t 2 + t 3 + tdwell
15
00439 + 00366 + 00439 = 00829 =
ΤRMS = 02879 Nm
Motion System Application Example16
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
16
MOTION SYSTEMAPPLICATION EXAMPLE
Load Parameters at the Lead Screw Shaft
= 9276 rads= 2786 rads2
= 03632 Nm= 03316 Nm= ndash 03632 Nm= 02879 Nm= 3369 W= 3076 W
ωPKaΤ1Τ2Τ3ΤRMSPPKPCV
Motor SelectionA direct-drive motor can be used however it would be large and expensive For example a DC057B-3 brush motor
would easily meet continuous torque and continuous speed as well as a 181 load-to-rotor inertia This motor however has significantly more output power than needed at 128 watts
A smaller motor with a transmission would optimize the system Look for a motor starting at a rated power around 40 watts Pittman DC040B-6
DC040B-6Reference VoltageContinuous TorqueRated CurrentRated PowerTorque ConstantVoltage ConstantTerminal ResistanceMax Winding TemperatureRotor Inertia
240 V00812 Nm236 A37 W0042 NmA0042 Vrads185 Ω155degC847 x 10ndash6 Kg-m2
Motor TransmissionLead Screw Support Rails
ΤRMS
PPK
ωPK
= 02879 Nm= 3369 W= 9276 rads
M
Other Motor Considerationsndash Type stepper brush BLDCndash Input speed (high input speed will result in high noise levels)ndash Total motor footprintndash Ambient temperaturendash Environmental conditionsndash Load-to-rotor inertiandash Encoder ready needed
Motion System Application Example17
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
17
MOTION SYSTEMAPPLICATION EXAMPLE
Gearbox Transmission Selection
When selecting a reduction ratio keep in mind that the RMS required torque at the motor shaft needs to fall below the continuous rated output torque of the motor
G30 A - Planetary GearboxMaximum output load 247 Nm
TRMS( Lead Screw)
02879 Nm
02879 Nm02879 Nm
Reduction
41
5161
Efficiency
090
090090
TRMS( Motor)
00798 Nm
00640 Nm00533 Nm
A 51 gearbox would allow about a 27 safety margin between what the system requires and the continuous torque output of the motor
System Summary 51η = 090
M
= 00807 Nm= 00737 Nm= ndash 00807 Nm= 00640 Nm= 4638 rads= 504 x 10 ndash 6 Kg-m2
= 3743 W
Τ1Τ2Τ3ΤRMS
PPK
ωPK
J
= 03632 Nm= 03316 Nm= ndash 03632 Nm= 02879 Nm= 9276 rads= 1134 x 10 ndash 5 Kg-m2
= 3369 W
Τ1Τ2Τ3ΤRMS
PPK
ωPK
J
Motion System Application Example18
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
18
MOTION SYSTEMAPPLICATION EXAMPLE
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
19
Motion Profile at the Motor
t 1 t 2 t 3 t dwell Time0333 s 0333 s 0333 s 05 s
ωΤ1
Τ2
Τ3= 4638 rads= 4429 RPM= ΤPK = 00807 Nm
ωPK
Τ1
nPK
Drive and Power Supply Requirements
Peak Current Required
IPK =TPK
KT+ IO =
00807 Nm0042 NmA
+ 0180 A = 210 A
RMS Current Required
IRMS =TRMS
KT+ IO =
00640 Nm0042 NmA
+ 0180 A = 170 A
Minimum bus Voltage Required
Vbus= IPKRmt+ ωPKKE = (210A)(185Ω) + (4638 rads)(0042 Vrads)= 389 V + 1948 V= 2337 V
0100 Nm 0200 Nm 0300 Nm 0400 Nm 0500 Nm 0600 Nm 0700 Nm
30 VDC
0100 Nm 0200 Nm 0300 Nm 0500 Nm 0600 Nm 0700 Nm
24 VDC
700060005000
RPM 4000300020001000
0400 Nm
TRMS
TPK
700060005000
RPM 4000300020001000
TRMS
TPK
Performance using 24V reference voltage
Performance using a 30V bus voltage
A 30 VDC bus will meet the application requirements as wells supply a margin of safety
MOTION SYSTEMAPPLICATION EXAMPLE
0100 Nm 0200 Nm 0300 Nm 0400 Nm 0500 Nm 0600 Nm 0700 Nm
30 VDC
0100 Nm 0200 Nm 0300 Nm 0500 Nm 0600 Nm 0700 Nm
24 VDC
700060005000
RPM 4000300020001000
0400 Nm
TRMS
TPK
700060005000
RPM 4000300020001000
TRMS
TPK
Ambient temperature Rated motor temperature Temperature rise Motor temperature
= 30degC= 155degC= 7638degC= 10638degC
Θa
Θm
ΘrΘrated
=Rth x IRMS
2 x Rmt
1 ndash (Rth x IRMS2 x Rmt x 000392degC)
Θr
=11degCω x 170A2
x 185Ω 1 ndash (11degCω x 170A2
x 185Ω x 000392degC)
= 5881 1 ndash (023)
= 5881 077
=
Θm = Θr + Θa = 7638degC + 30degC = 10638degC
7638degC
Performance using 24V reference voltage
Performance using a 30V bus voltage
Will the motor meet the temperature rise caused by the application
Motion System Application Example20
A 30 VDC bus will meet the application requirements as wells supply a margin of safety
203 756 7441 1500 Meriden Road Waterbury CT USA 06705
267 933 2105 534 Godshall DriveHarleysville PA USA 19438
wwwHaydonKerkPittmancom
copy All rights reserved AMETEK Advanced Motion Solutions No part of this doc-ument or technical information can be used reproduced or altered in any form without approval or proper authorization from AMETEK Inc and global affiliates
NOW ASK us the tough motion control questions ONLINE
ASK THE EXPERTS
HaydonKerkPittmancomwww
MOTION SYSTEMAPPLICATION EXAMPLE
Stepper Motor Formulas
Required ldquoTime Offrdquo using an LR drive (constant voltage drive)
180Step Oslash = (phases)(rotor teeth)
ndash Motor step angle
ndash RPM (revolutionsminute) to SPS (stepssecond)
Maximum accuracy of a step motor system has nothing to do with maximum resolution of a step motor system Maximum accuracy is always based on the accuracy of 1 full step but maximum resolution is based on all system components including lead screw encoder stepper motor and step mode on the controller
ndash SPS (stepssecond) to RPM (revolutionsminute)(SPS) x (Step Oslash)
RPM = 6
(RPM) x 6 SPS = (Step Oslash)
ndash Travel per step
mstep =L
positionsrev
ndash Overdriving capability
Required ldquoTime Offrdquo using a chopper drive (constant current drive)
Time Off = (Time On) (Applied Voltage)2
(Rated Voltage)ndash Time On
Time Off = (Time On) (Applied Current)2
(Rated Current)ndash Time On
It is not unusual for a customer to drive Haydon Kerk stepper motors beyond their rated power to obtain the most force in the smallest package size Precautions must be taken to prevent the motor from exceeding its maximum temperature The ldquoonrdquo time should not exceed 2 - 3 minutes As general rule of thumb driving a motor at 25 to 3x its rated voltage or current may result in wasted energy and erratic behavior due to saturation
Special notes for can-stack motors bull more easily saturated due to less active material (steel) bull more iron losses due to unlaminated steel
Motion System Formulas10
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
10
MOTION SYSTEMFORMULAS
Lead Screw Performance Characteristics
End Mounting Rigidity
Impact Impact
ScrewLength
Increase In Affects Increase In Affects
Critical speed
Compression load
Critical speed
Inertia
Compression load
Stiffness
Spring Rate
Load capacity
Drive torque
Angular velocity
Load cpacity
Positioning accuracy
ScrewDiameter
Lead
Load
Preload
Nut length
SI Unit Systems
LengthMassTimeElectric currentThermodynamic temperatureLuminous intensityAmount of substance
meterkilogramsecondamperekelvincandelamole
mKgsAKcdmol
Base Units
Derived Unitsareavolumefrequencymass density (density)speed velocityangular velocityaccelerationangular accelerationforcepressure (mechanical stress)kinematic viscositydynamic viscositywork energy quantity of heatpowerentropyspecific heat capacitythermal conductivity
square metercubic meterhertzkilogram per cubic metermeter per secondradian per secondmeter per second squaredradian per second squarednewtonpascalsquare meter per secondnewton-second per square meter joulewattjoule per kelvinjoule per kilogram kelvinwatt per meter kelvin
m2
m3
HzKgm3
msradsms2
rads2
NPam2sN-sm2
JWJKJKg-K)W(m-K)
Quantity Name of Unit Symbol
Motion System Reference Tables11
Critical speed
Compression load
System stiffness
Life
Positioning accuracy
System stiffness
Drag torque
Load capacity
Stiffness
sndash1
Kg-ms2
Nm2
N-mJs or N-ms
ExamplesAn INCREASE ( ) in screw length results in a DECREASE ( ) in critical speed
An INCREASE in screw diameter results in an INCREASE in critical speed
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
11
MOTION SYSTEMREFERENCE TABLES
t 1 t 2 t 3 t dwell
vTime
vpk
a -a
MassMove distanceMove timeDwell timeMotion profileScrew speed limitLoad supportOrientationRotary to linear conversionAmbient temperature
9 Kg200 mm10 s05 s13 13 13 Trapezoidal1000 RPMμ = 001VerticalTFE lead screw 8 mm diameter 275 mm long30degC
ControlDrive Power Supply
4 quadrant with encoder32 VDC 35 Arms50 A peak
0333 s 0333 s 0333 s 05 s
vPK =3S2t =
(3)(02m)(2)(10 s) =
06m2 s =
03ms
Linear Velocity
Linear Acceleration
a = vf - vi
t = 03ms - 0 ms 0333 s 0901ms2 =
1st step is to thoroughly understand system constraints and motion profile requirements
Linear Motion ExampleData
Motion System Application Example12
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
12
MOTION SYSTEMAPPLICATION EXAMPLE
Lmin =60vPK
n =(60)(03ms)1000 revmin = 0018mrev = 18mmrev
Refer to Kerk screw chart ndash closest lead in an 8mm screw diameter is
2032mmrev = 002032 mrevη = 086 free wheeling nut TFE coated screw
JS = 388 x 10ndash 7 Kg-m2 Other critical lead screw considerations bull Column loading bull Critical speed
Jm aa
ωPKvPK
TF
= ωPK = 2 π vPK
L=
(2π)(03ms)002032 mrev
9276 rads
a = =2πaL
=(2π)(0901ms2)002032 mrev
2786 rads2
Linear to Rotary Conversion ndash Velocity Acceleration Inertia Torque
Minimum screw lead to maintain lt 1000 RPM shaft speed
Linear to Rotary Conversion Velocity
Linear to Rotary Conversion Acceleration
Motion System Application Example13
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
13
MOTION SYSTEMFORMULAS
Jin= m 2 πL 2
X η1 + JS= 9Kg
2 π002032 m 2
X 0861 + 388x10ndash 7Kg-m2
= 1095 X 10ndash 5 Kg-m2 + 388x10ndash 7 Kg-m2
= 1134 x 10ndash 5 Kg-m2
Τin= Τa+Τf +Τg+ΤD
Τa = Jin ain
Τg =sinOslashmgL
2 π η
= (1134 x 10ndash 5 Kg-m2)(2786 rads2)
= 00316 Nm
Τf =cosOslashmgμL
2 π η= (cos90)(9 Kg)(98ms2)(001)(002032m)
2 π (086)= 0 Nm
= (sin90)(9 Kg)(98ms2)(002032m)2 π (086)
=1792 Nm
5404 = 03316 Nm
ΤDSince a free-wheeling lead screw nut was used there is no preload force If an anti-backlash nut was used ΤD would be gt0 due to preload force Always reference manufacturerrsquos data for more information
= 0 Nm
Linear to Rotary Conversion Torque
Linear to Rotary Conversion Inertia
Motion System Application Example14
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
14
MOTION SYSTEMFORMULAS
t 1 t 2 t 3 t dwell Time0333 s 0333 s 0333 s 05 s
Rotary Motion Profile
ωPK
ω
= 9276 rads a = 2786 rads2
Τ1= Τa+Τf +Τg+ΤD
= (00316 Nm) + (0 Nm) + (03316 Nm) + (0 Nm)
= 03632 Nm
Τ2= Τa+Τf +Τg+ΤD
= (0 Nm) + (0 Nm) + (03316 Nm) + (0 Nm)
= 03316 Nm
Τ3= Τa+Τf +Τg+ΤD
= (00316 Nm) + (0 Nm) + (03316 Nm) + (0 Nm)
= 03632 Nm
Τ1
Τ2
Τ3
Motion System Application Example15
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
15
MOTION SYSTEMAPPLICATION EXAMPLE
Lead Screw Input Requirements
= 9276 rads= 2786 rads2
= 03632 Nm= 03316 Nm= ndash 03632 Nm= 0333 s= 0333 s= 0333 s= 05 s= 3369 W= 3076 W
ωPKaΤ1Τ2Τ3
t1t2t3tdwell
PPKPCV
PPK = (Τ1)(ωPK)
= (03632 Nm)(9276 rads)= 3369 W
PCV = (Τ2)(ωPK)
= (03316 Nm)(9276 rads)= 3076 W
RMS Torque Requirement Lead Screw Input
=
0333 s + 0333 s + 0333 s + 05 s
(03632 Nm)2(0333 s)+(03316 Nm)2(0333 s)+(- 03632 Nm)2(0333 s)
ΤRMS = Τ12t 1 + Τ2
2t2 + Τ32t 3
t 1 + t 2 + t 3 + tdwell
15
00439 + 00366 + 00439 = 00829 =
ΤRMS = 02879 Nm
Motion System Application Example16
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
16
MOTION SYSTEMAPPLICATION EXAMPLE
Load Parameters at the Lead Screw Shaft
= 9276 rads= 2786 rads2
= 03632 Nm= 03316 Nm= ndash 03632 Nm= 02879 Nm= 3369 W= 3076 W
ωPKaΤ1Τ2Τ3ΤRMSPPKPCV
Motor SelectionA direct-drive motor can be used however it would be large and expensive For example a DC057B-3 brush motor
would easily meet continuous torque and continuous speed as well as a 181 load-to-rotor inertia This motor however has significantly more output power than needed at 128 watts
A smaller motor with a transmission would optimize the system Look for a motor starting at a rated power around 40 watts Pittman DC040B-6
DC040B-6Reference VoltageContinuous TorqueRated CurrentRated PowerTorque ConstantVoltage ConstantTerminal ResistanceMax Winding TemperatureRotor Inertia
240 V00812 Nm236 A37 W0042 NmA0042 Vrads185 Ω155degC847 x 10ndash6 Kg-m2
Motor TransmissionLead Screw Support Rails
ΤRMS
PPK
ωPK
= 02879 Nm= 3369 W= 9276 rads
M
Other Motor Considerationsndash Type stepper brush BLDCndash Input speed (high input speed will result in high noise levels)ndash Total motor footprintndash Ambient temperaturendash Environmental conditionsndash Load-to-rotor inertiandash Encoder ready needed
Motion System Application Example17
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
17
MOTION SYSTEMAPPLICATION EXAMPLE
Gearbox Transmission Selection
When selecting a reduction ratio keep in mind that the RMS required torque at the motor shaft needs to fall below the continuous rated output torque of the motor
G30 A - Planetary GearboxMaximum output load 247 Nm
TRMS( Lead Screw)
02879 Nm
02879 Nm02879 Nm
Reduction
41
5161
Efficiency
090
090090
TRMS( Motor)
00798 Nm
00640 Nm00533 Nm
A 51 gearbox would allow about a 27 safety margin between what the system requires and the continuous torque output of the motor
System Summary 51η = 090
M
= 00807 Nm= 00737 Nm= ndash 00807 Nm= 00640 Nm= 4638 rads= 504 x 10 ndash 6 Kg-m2
= 3743 W
Τ1Τ2Τ3ΤRMS
PPK
ωPK
J
= 03632 Nm= 03316 Nm= ndash 03632 Nm= 02879 Nm= 9276 rads= 1134 x 10 ndash 5 Kg-m2
= 3369 W
Τ1Τ2Τ3ΤRMS
PPK
ωPK
J
Motion System Application Example18
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
18
MOTION SYSTEMAPPLICATION EXAMPLE
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
19
Motion Profile at the Motor
t 1 t 2 t 3 t dwell Time0333 s 0333 s 0333 s 05 s
ωΤ1
Τ2
Τ3= 4638 rads= 4429 RPM= ΤPK = 00807 Nm
ωPK
Τ1
nPK
Drive and Power Supply Requirements
Peak Current Required
IPK =TPK
KT+ IO =
00807 Nm0042 NmA
+ 0180 A = 210 A
RMS Current Required
IRMS =TRMS
KT+ IO =
00640 Nm0042 NmA
+ 0180 A = 170 A
Minimum bus Voltage Required
Vbus= IPKRmt+ ωPKKE = (210A)(185Ω) + (4638 rads)(0042 Vrads)= 389 V + 1948 V= 2337 V
0100 Nm 0200 Nm 0300 Nm 0400 Nm 0500 Nm 0600 Nm 0700 Nm
30 VDC
0100 Nm 0200 Nm 0300 Nm 0500 Nm 0600 Nm 0700 Nm
24 VDC
700060005000
RPM 4000300020001000
0400 Nm
TRMS
TPK
700060005000
RPM 4000300020001000
TRMS
TPK
Performance using 24V reference voltage
Performance using a 30V bus voltage
A 30 VDC bus will meet the application requirements as wells supply a margin of safety
MOTION SYSTEMAPPLICATION EXAMPLE
0100 Nm 0200 Nm 0300 Nm 0400 Nm 0500 Nm 0600 Nm 0700 Nm
30 VDC
0100 Nm 0200 Nm 0300 Nm 0500 Nm 0600 Nm 0700 Nm
24 VDC
700060005000
RPM 4000300020001000
0400 Nm
TRMS
TPK
700060005000
RPM 4000300020001000
TRMS
TPK
Ambient temperature Rated motor temperature Temperature rise Motor temperature
= 30degC= 155degC= 7638degC= 10638degC
Θa
Θm
ΘrΘrated
=Rth x IRMS
2 x Rmt
1 ndash (Rth x IRMS2 x Rmt x 000392degC)
Θr
=11degCω x 170A2
x 185Ω 1 ndash (11degCω x 170A2
x 185Ω x 000392degC)
= 5881 1 ndash (023)
= 5881 077
=
Θm = Θr + Θa = 7638degC + 30degC = 10638degC
7638degC
Performance using 24V reference voltage
Performance using a 30V bus voltage
Will the motor meet the temperature rise caused by the application
Motion System Application Example20
A 30 VDC bus will meet the application requirements as wells supply a margin of safety
203 756 7441 1500 Meriden Road Waterbury CT USA 06705
267 933 2105 534 Godshall DriveHarleysville PA USA 19438
wwwHaydonKerkPittmancom
copy All rights reserved AMETEK Advanced Motion Solutions No part of this doc-ument or technical information can be used reproduced or altered in any form without approval or proper authorization from AMETEK Inc and global affiliates
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MOTION SYSTEMAPPLICATION EXAMPLE
Lead Screw Performance Characteristics
End Mounting Rigidity
Impact Impact
ScrewLength
Increase In Affects Increase In Affects
Critical speed
Compression load
Critical speed
Inertia
Compression load
Stiffness
Spring Rate
Load capacity
Drive torque
Angular velocity
Load cpacity
Positioning accuracy
ScrewDiameter
Lead
Load
Preload
Nut length
SI Unit Systems
LengthMassTimeElectric currentThermodynamic temperatureLuminous intensityAmount of substance
meterkilogramsecondamperekelvincandelamole
mKgsAKcdmol
Base Units
Derived Unitsareavolumefrequencymass density (density)speed velocityangular velocityaccelerationangular accelerationforcepressure (mechanical stress)kinematic viscositydynamic viscositywork energy quantity of heatpowerentropyspecific heat capacitythermal conductivity
square metercubic meterhertzkilogram per cubic metermeter per secondradian per secondmeter per second squaredradian per second squarednewtonpascalsquare meter per secondnewton-second per square meter joulewattjoule per kelvinjoule per kilogram kelvinwatt per meter kelvin
m2
m3
HzKgm3
msradsms2
rads2
NPam2sN-sm2
JWJKJKg-K)W(m-K)
Quantity Name of Unit Symbol
Motion System Reference Tables11
Critical speed
Compression load
System stiffness
Life
Positioning accuracy
System stiffness
Drag torque
Load capacity
Stiffness
sndash1
Kg-ms2
Nm2
N-mJs or N-ms
ExamplesAn INCREASE ( ) in screw length results in a DECREASE ( ) in critical speed
An INCREASE in screw diameter results in an INCREASE in critical speed
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MOTION SYSTEMREFERENCE TABLES
t 1 t 2 t 3 t dwell
vTime
vpk
a -a
MassMove distanceMove timeDwell timeMotion profileScrew speed limitLoad supportOrientationRotary to linear conversionAmbient temperature
9 Kg200 mm10 s05 s13 13 13 Trapezoidal1000 RPMμ = 001VerticalTFE lead screw 8 mm diameter 275 mm long30degC
ControlDrive Power Supply
4 quadrant with encoder32 VDC 35 Arms50 A peak
0333 s 0333 s 0333 s 05 s
vPK =3S2t =
(3)(02m)(2)(10 s) =
06m2 s =
03ms
Linear Velocity
Linear Acceleration
a = vf - vi
t = 03ms - 0 ms 0333 s 0901ms2 =
1st step is to thoroughly understand system constraints and motion profile requirements
Linear Motion ExampleData
Motion System Application Example12
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12
MOTION SYSTEMAPPLICATION EXAMPLE
Lmin =60vPK
n =(60)(03ms)1000 revmin = 0018mrev = 18mmrev
Refer to Kerk screw chart ndash closest lead in an 8mm screw diameter is
2032mmrev = 002032 mrevη = 086 free wheeling nut TFE coated screw
JS = 388 x 10ndash 7 Kg-m2 Other critical lead screw considerations bull Column loading bull Critical speed
Jm aa
ωPKvPK
TF
= ωPK = 2 π vPK
L=
(2π)(03ms)002032 mrev
9276 rads
a = =2πaL
=(2π)(0901ms2)002032 mrev
2786 rads2
Linear to Rotary Conversion ndash Velocity Acceleration Inertia Torque
Minimum screw lead to maintain lt 1000 RPM shaft speed
Linear to Rotary Conversion Velocity
Linear to Rotary Conversion Acceleration
Motion System Application Example13
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MOTION SYSTEMFORMULAS
Jin= m 2 πL 2
X η1 + JS= 9Kg
2 π002032 m 2
X 0861 + 388x10ndash 7Kg-m2
= 1095 X 10ndash 5 Kg-m2 + 388x10ndash 7 Kg-m2
= 1134 x 10ndash 5 Kg-m2
Τin= Τa+Τf +Τg+ΤD
Τa = Jin ain
Τg =sinOslashmgL
2 π η
= (1134 x 10ndash 5 Kg-m2)(2786 rads2)
= 00316 Nm
Τf =cosOslashmgμL
2 π η= (cos90)(9 Kg)(98ms2)(001)(002032m)
2 π (086)= 0 Nm
= (sin90)(9 Kg)(98ms2)(002032m)2 π (086)
=1792 Nm
5404 = 03316 Nm
ΤDSince a free-wheeling lead screw nut was used there is no preload force If an anti-backlash nut was used ΤD would be gt0 due to preload force Always reference manufacturerrsquos data for more information
= 0 Nm
Linear to Rotary Conversion Torque
Linear to Rotary Conversion Inertia
Motion System Application Example14
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MOTION SYSTEMFORMULAS
t 1 t 2 t 3 t dwell Time0333 s 0333 s 0333 s 05 s
Rotary Motion Profile
ωPK
ω
= 9276 rads a = 2786 rads2
Τ1= Τa+Τf +Τg+ΤD
= (00316 Nm) + (0 Nm) + (03316 Nm) + (0 Nm)
= 03632 Nm
Τ2= Τa+Τf +Τg+ΤD
= (0 Nm) + (0 Nm) + (03316 Nm) + (0 Nm)
= 03316 Nm
Τ3= Τa+Τf +Τg+ΤD
= (00316 Nm) + (0 Nm) + (03316 Nm) + (0 Nm)
= 03632 Nm
Τ1
Τ2
Τ3
Motion System Application Example15
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MOTION SYSTEMAPPLICATION EXAMPLE
Lead Screw Input Requirements
= 9276 rads= 2786 rads2
= 03632 Nm= 03316 Nm= ndash 03632 Nm= 0333 s= 0333 s= 0333 s= 05 s= 3369 W= 3076 W
ωPKaΤ1Τ2Τ3
t1t2t3tdwell
PPKPCV
PPK = (Τ1)(ωPK)
= (03632 Nm)(9276 rads)= 3369 W
PCV = (Τ2)(ωPK)
= (03316 Nm)(9276 rads)= 3076 W
RMS Torque Requirement Lead Screw Input
=
0333 s + 0333 s + 0333 s + 05 s
(03632 Nm)2(0333 s)+(03316 Nm)2(0333 s)+(- 03632 Nm)2(0333 s)
ΤRMS = Τ12t 1 + Τ2
2t2 + Τ32t 3
t 1 + t 2 + t 3 + tdwell
15
00439 + 00366 + 00439 = 00829 =
ΤRMS = 02879 Nm
Motion System Application Example16
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MOTION SYSTEMAPPLICATION EXAMPLE
Load Parameters at the Lead Screw Shaft
= 9276 rads= 2786 rads2
= 03632 Nm= 03316 Nm= ndash 03632 Nm= 02879 Nm= 3369 W= 3076 W
ωPKaΤ1Τ2Τ3ΤRMSPPKPCV
Motor SelectionA direct-drive motor can be used however it would be large and expensive For example a DC057B-3 brush motor
would easily meet continuous torque and continuous speed as well as a 181 load-to-rotor inertia This motor however has significantly more output power than needed at 128 watts
A smaller motor with a transmission would optimize the system Look for a motor starting at a rated power around 40 watts Pittman DC040B-6
DC040B-6Reference VoltageContinuous TorqueRated CurrentRated PowerTorque ConstantVoltage ConstantTerminal ResistanceMax Winding TemperatureRotor Inertia
240 V00812 Nm236 A37 W0042 NmA0042 Vrads185 Ω155degC847 x 10ndash6 Kg-m2
Motor TransmissionLead Screw Support Rails
ΤRMS
PPK
ωPK
= 02879 Nm= 3369 W= 9276 rads
M
Other Motor Considerationsndash Type stepper brush BLDCndash Input speed (high input speed will result in high noise levels)ndash Total motor footprintndash Ambient temperaturendash Environmental conditionsndash Load-to-rotor inertiandash Encoder ready needed
Motion System Application Example17
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MOTION SYSTEMAPPLICATION EXAMPLE
Gearbox Transmission Selection
When selecting a reduction ratio keep in mind that the RMS required torque at the motor shaft needs to fall below the continuous rated output torque of the motor
G30 A - Planetary GearboxMaximum output load 247 Nm
TRMS( Lead Screw)
02879 Nm
02879 Nm02879 Nm
Reduction
41
5161
Efficiency
090
090090
TRMS( Motor)
00798 Nm
00640 Nm00533 Nm
A 51 gearbox would allow about a 27 safety margin between what the system requires and the continuous torque output of the motor
System Summary 51η = 090
M
= 00807 Nm= 00737 Nm= ndash 00807 Nm= 00640 Nm= 4638 rads= 504 x 10 ndash 6 Kg-m2
= 3743 W
Τ1Τ2Τ3ΤRMS
PPK
ωPK
J
= 03632 Nm= 03316 Nm= ndash 03632 Nm= 02879 Nm= 9276 rads= 1134 x 10 ndash 5 Kg-m2
= 3369 W
Τ1Τ2Τ3ΤRMS
PPK
ωPK
J
Motion System Application Example18
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MOTION SYSTEMAPPLICATION EXAMPLE
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Motion Profile at the Motor
t 1 t 2 t 3 t dwell Time0333 s 0333 s 0333 s 05 s
ωΤ1
Τ2
Τ3= 4638 rads= 4429 RPM= ΤPK = 00807 Nm
ωPK
Τ1
nPK
Drive and Power Supply Requirements
Peak Current Required
IPK =TPK
KT+ IO =
00807 Nm0042 NmA
+ 0180 A = 210 A
RMS Current Required
IRMS =TRMS
KT+ IO =
00640 Nm0042 NmA
+ 0180 A = 170 A
Minimum bus Voltage Required
Vbus= IPKRmt+ ωPKKE = (210A)(185Ω) + (4638 rads)(0042 Vrads)= 389 V + 1948 V= 2337 V
0100 Nm 0200 Nm 0300 Nm 0400 Nm 0500 Nm 0600 Nm 0700 Nm
30 VDC
0100 Nm 0200 Nm 0300 Nm 0500 Nm 0600 Nm 0700 Nm
24 VDC
700060005000
RPM 4000300020001000
0400 Nm
TRMS
TPK
700060005000
RPM 4000300020001000
TRMS
TPK
Performance using 24V reference voltage
Performance using a 30V bus voltage
A 30 VDC bus will meet the application requirements as wells supply a margin of safety
MOTION SYSTEMAPPLICATION EXAMPLE
0100 Nm 0200 Nm 0300 Nm 0400 Nm 0500 Nm 0600 Nm 0700 Nm
30 VDC
0100 Nm 0200 Nm 0300 Nm 0500 Nm 0600 Nm 0700 Nm
24 VDC
700060005000
RPM 4000300020001000
0400 Nm
TRMS
TPK
700060005000
RPM 4000300020001000
TRMS
TPK
Ambient temperature Rated motor temperature Temperature rise Motor temperature
= 30degC= 155degC= 7638degC= 10638degC
Θa
Θm
ΘrΘrated
=Rth x IRMS
2 x Rmt
1 ndash (Rth x IRMS2 x Rmt x 000392degC)
Θr
=11degCω x 170A2
x 185Ω 1 ndash (11degCω x 170A2
x 185Ω x 000392degC)
= 5881 1 ndash (023)
= 5881 077
=
Θm = Θr + Θa = 7638degC + 30degC = 10638degC
7638degC
Performance using 24V reference voltage
Performance using a 30V bus voltage
Will the motor meet the temperature rise caused by the application
Motion System Application Example20
A 30 VDC bus will meet the application requirements as wells supply a margin of safety
203 756 7441 1500 Meriden Road Waterbury CT USA 06705
267 933 2105 534 Godshall DriveHarleysville PA USA 19438
wwwHaydonKerkPittmancom
copy All rights reserved AMETEK Advanced Motion Solutions No part of this doc-ument or technical information can be used reproduced or altered in any form without approval or proper authorization from AMETEK Inc and global affiliates
NOW ASK us the tough motion control questions ONLINE
ASK THE EXPERTS
HaydonKerkPittmancomwww
MOTION SYSTEMAPPLICATION EXAMPLE
t 1 t 2 t 3 t dwell
vTime
vpk
a -a
MassMove distanceMove timeDwell timeMotion profileScrew speed limitLoad supportOrientationRotary to linear conversionAmbient temperature
9 Kg200 mm10 s05 s13 13 13 Trapezoidal1000 RPMμ = 001VerticalTFE lead screw 8 mm diameter 275 mm long30degC
ControlDrive Power Supply
4 quadrant with encoder32 VDC 35 Arms50 A peak
0333 s 0333 s 0333 s 05 s
vPK =3S2t =
(3)(02m)(2)(10 s) =
06m2 s =
03ms
Linear Velocity
Linear Acceleration
a = vf - vi
t = 03ms - 0 ms 0333 s 0901ms2 =
1st step is to thoroughly understand system constraints and motion profile requirements
Linear Motion ExampleData
Motion System Application Example12
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12
MOTION SYSTEMAPPLICATION EXAMPLE
Lmin =60vPK
n =(60)(03ms)1000 revmin = 0018mrev = 18mmrev
Refer to Kerk screw chart ndash closest lead in an 8mm screw diameter is
2032mmrev = 002032 mrevη = 086 free wheeling nut TFE coated screw
JS = 388 x 10ndash 7 Kg-m2 Other critical lead screw considerations bull Column loading bull Critical speed
Jm aa
ωPKvPK
TF
= ωPK = 2 π vPK
L=
(2π)(03ms)002032 mrev
9276 rads
a = =2πaL
=(2π)(0901ms2)002032 mrev
2786 rads2
Linear to Rotary Conversion ndash Velocity Acceleration Inertia Torque
Minimum screw lead to maintain lt 1000 RPM shaft speed
Linear to Rotary Conversion Velocity
Linear to Rotary Conversion Acceleration
Motion System Application Example13
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MOTION SYSTEMFORMULAS
Jin= m 2 πL 2
X η1 + JS= 9Kg
2 π002032 m 2
X 0861 + 388x10ndash 7Kg-m2
= 1095 X 10ndash 5 Kg-m2 + 388x10ndash 7 Kg-m2
= 1134 x 10ndash 5 Kg-m2
Τin= Τa+Τf +Τg+ΤD
Τa = Jin ain
Τg =sinOslashmgL
2 π η
= (1134 x 10ndash 5 Kg-m2)(2786 rads2)
= 00316 Nm
Τf =cosOslashmgμL
2 π η= (cos90)(9 Kg)(98ms2)(001)(002032m)
2 π (086)= 0 Nm
= (sin90)(9 Kg)(98ms2)(002032m)2 π (086)
=1792 Nm
5404 = 03316 Nm
ΤDSince a free-wheeling lead screw nut was used there is no preload force If an anti-backlash nut was used ΤD would be gt0 due to preload force Always reference manufacturerrsquos data for more information
= 0 Nm
Linear to Rotary Conversion Torque
Linear to Rotary Conversion Inertia
Motion System Application Example14
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MOTION SYSTEMFORMULAS
t 1 t 2 t 3 t dwell Time0333 s 0333 s 0333 s 05 s
Rotary Motion Profile
ωPK
ω
= 9276 rads a = 2786 rads2
Τ1= Τa+Τf +Τg+ΤD
= (00316 Nm) + (0 Nm) + (03316 Nm) + (0 Nm)
= 03632 Nm
Τ2= Τa+Τf +Τg+ΤD
= (0 Nm) + (0 Nm) + (03316 Nm) + (0 Nm)
= 03316 Nm
Τ3= Τa+Τf +Τg+ΤD
= (00316 Nm) + (0 Nm) + (03316 Nm) + (0 Nm)
= 03632 Nm
Τ1
Τ2
Τ3
Motion System Application Example15
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MOTION SYSTEMAPPLICATION EXAMPLE
Lead Screw Input Requirements
= 9276 rads= 2786 rads2
= 03632 Nm= 03316 Nm= ndash 03632 Nm= 0333 s= 0333 s= 0333 s= 05 s= 3369 W= 3076 W
ωPKaΤ1Τ2Τ3
t1t2t3tdwell
PPKPCV
PPK = (Τ1)(ωPK)
= (03632 Nm)(9276 rads)= 3369 W
PCV = (Τ2)(ωPK)
= (03316 Nm)(9276 rads)= 3076 W
RMS Torque Requirement Lead Screw Input
=
0333 s + 0333 s + 0333 s + 05 s
(03632 Nm)2(0333 s)+(03316 Nm)2(0333 s)+(- 03632 Nm)2(0333 s)
ΤRMS = Τ12t 1 + Τ2
2t2 + Τ32t 3
t 1 + t 2 + t 3 + tdwell
15
00439 + 00366 + 00439 = 00829 =
ΤRMS = 02879 Nm
Motion System Application Example16
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16
MOTION SYSTEMAPPLICATION EXAMPLE
Load Parameters at the Lead Screw Shaft
= 9276 rads= 2786 rads2
= 03632 Nm= 03316 Nm= ndash 03632 Nm= 02879 Nm= 3369 W= 3076 W
ωPKaΤ1Τ2Τ3ΤRMSPPKPCV
Motor SelectionA direct-drive motor can be used however it would be large and expensive For example a DC057B-3 brush motor
would easily meet continuous torque and continuous speed as well as a 181 load-to-rotor inertia This motor however has significantly more output power than needed at 128 watts
A smaller motor with a transmission would optimize the system Look for a motor starting at a rated power around 40 watts Pittman DC040B-6
DC040B-6Reference VoltageContinuous TorqueRated CurrentRated PowerTorque ConstantVoltage ConstantTerminal ResistanceMax Winding TemperatureRotor Inertia
240 V00812 Nm236 A37 W0042 NmA0042 Vrads185 Ω155degC847 x 10ndash6 Kg-m2
Motor TransmissionLead Screw Support Rails
ΤRMS
PPK
ωPK
= 02879 Nm= 3369 W= 9276 rads
M
Other Motor Considerationsndash Type stepper brush BLDCndash Input speed (high input speed will result in high noise levels)ndash Total motor footprintndash Ambient temperaturendash Environmental conditionsndash Load-to-rotor inertiandash Encoder ready needed
Motion System Application Example17
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17
MOTION SYSTEMAPPLICATION EXAMPLE
Gearbox Transmission Selection
When selecting a reduction ratio keep in mind that the RMS required torque at the motor shaft needs to fall below the continuous rated output torque of the motor
G30 A - Planetary GearboxMaximum output load 247 Nm
TRMS( Lead Screw)
02879 Nm
02879 Nm02879 Nm
Reduction
41
5161
Efficiency
090
090090
TRMS( Motor)
00798 Nm
00640 Nm00533 Nm
A 51 gearbox would allow about a 27 safety margin between what the system requires and the continuous torque output of the motor
System Summary 51η = 090
M
= 00807 Nm= 00737 Nm= ndash 00807 Nm= 00640 Nm= 4638 rads= 504 x 10 ndash 6 Kg-m2
= 3743 W
Τ1Τ2Τ3ΤRMS
PPK
ωPK
J
= 03632 Nm= 03316 Nm= ndash 03632 Nm= 02879 Nm= 9276 rads= 1134 x 10 ndash 5 Kg-m2
= 3369 W
Τ1Τ2Τ3ΤRMS
PPK
ωPK
J
Motion System Application Example18
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MOTION SYSTEMAPPLICATION EXAMPLE
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Motion Profile at the Motor
t 1 t 2 t 3 t dwell Time0333 s 0333 s 0333 s 05 s
ωΤ1
Τ2
Τ3= 4638 rads= 4429 RPM= ΤPK = 00807 Nm
ωPK
Τ1
nPK
Drive and Power Supply Requirements
Peak Current Required
IPK =TPK
KT+ IO =
00807 Nm0042 NmA
+ 0180 A = 210 A
RMS Current Required
IRMS =TRMS
KT+ IO =
00640 Nm0042 NmA
+ 0180 A = 170 A
Minimum bus Voltage Required
Vbus= IPKRmt+ ωPKKE = (210A)(185Ω) + (4638 rads)(0042 Vrads)= 389 V + 1948 V= 2337 V
0100 Nm 0200 Nm 0300 Nm 0400 Nm 0500 Nm 0600 Nm 0700 Nm
30 VDC
0100 Nm 0200 Nm 0300 Nm 0500 Nm 0600 Nm 0700 Nm
24 VDC
700060005000
RPM 4000300020001000
0400 Nm
TRMS
TPK
700060005000
RPM 4000300020001000
TRMS
TPK
Performance using 24V reference voltage
Performance using a 30V bus voltage
A 30 VDC bus will meet the application requirements as wells supply a margin of safety
MOTION SYSTEMAPPLICATION EXAMPLE
0100 Nm 0200 Nm 0300 Nm 0400 Nm 0500 Nm 0600 Nm 0700 Nm
30 VDC
0100 Nm 0200 Nm 0300 Nm 0500 Nm 0600 Nm 0700 Nm
24 VDC
700060005000
RPM 4000300020001000
0400 Nm
TRMS
TPK
700060005000
RPM 4000300020001000
TRMS
TPK
Ambient temperature Rated motor temperature Temperature rise Motor temperature
= 30degC= 155degC= 7638degC= 10638degC
Θa
Θm
ΘrΘrated
=Rth x IRMS
2 x Rmt
1 ndash (Rth x IRMS2 x Rmt x 000392degC)
Θr
=11degCω x 170A2
x 185Ω 1 ndash (11degCω x 170A2
x 185Ω x 000392degC)
= 5881 1 ndash (023)
= 5881 077
=
Θm = Θr + Θa = 7638degC + 30degC = 10638degC
7638degC
Performance using 24V reference voltage
Performance using a 30V bus voltage
Will the motor meet the temperature rise caused by the application
Motion System Application Example20
A 30 VDC bus will meet the application requirements as wells supply a margin of safety
203 756 7441 1500 Meriden Road Waterbury CT USA 06705
267 933 2105 534 Godshall DriveHarleysville PA USA 19438
wwwHaydonKerkPittmancom
copy All rights reserved AMETEK Advanced Motion Solutions No part of this doc-ument or technical information can be used reproduced or altered in any form without approval or proper authorization from AMETEK Inc and global affiliates
NOW ASK us the tough motion control questions ONLINE
ASK THE EXPERTS
HaydonKerkPittmancomwww
MOTION SYSTEMAPPLICATION EXAMPLE
Lmin =60vPK
n =(60)(03ms)1000 revmin = 0018mrev = 18mmrev
Refer to Kerk screw chart ndash closest lead in an 8mm screw diameter is
2032mmrev = 002032 mrevη = 086 free wheeling nut TFE coated screw
JS = 388 x 10ndash 7 Kg-m2 Other critical lead screw considerations bull Column loading bull Critical speed
Jm aa
ωPKvPK
TF
= ωPK = 2 π vPK
L=
(2π)(03ms)002032 mrev
9276 rads
a = =2πaL
=(2π)(0901ms2)002032 mrev
2786 rads2
Linear to Rotary Conversion ndash Velocity Acceleration Inertia Torque
Minimum screw lead to maintain lt 1000 RPM shaft speed
Linear to Rotary Conversion Velocity
Linear to Rotary Conversion Acceleration
Motion System Application Example13
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13
MOTION SYSTEMFORMULAS
Jin= m 2 πL 2
X η1 + JS= 9Kg
2 π002032 m 2
X 0861 + 388x10ndash 7Kg-m2
= 1095 X 10ndash 5 Kg-m2 + 388x10ndash 7 Kg-m2
= 1134 x 10ndash 5 Kg-m2
Τin= Τa+Τf +Τg+ΤD
Τa = Jin ain
Τg =sinOslashmgL
2 π η
= (1134 x 10ndash 5 Kg-m2)(2786 rads2)
= 00316 Nm
Τf =cosOslashmgμL
2 π η= (cos90)(9 Kg)(98ms2)(001)(002032m)
2 π (086)= 0 Nm
= (sin90)(9 Kg)(98ms2)(002032m)2 π (086)
=1792 Nm
5404 = 03316 Nm
ΤDSince a free-wheeling lead screw nut was used there is no preload force If an anti-backlash nut was used ΤD would be gt0 due to preload force Always reference manufacturerrsquos data for more information
= 0 Nm
Linear to Rotary Conversion Torque
Linear to Rotary Conversion Inertia
Motion System Application Example14
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14
MOTION SYSTEMFORMULAS
t 1 t 2 t 3 t dwell Time0333 s 0333 s 0333 s 05 s
Rotary Motion Profile
ωPK
ω
= 9276 rads a = 2786 rads2
Τ1= Τa+Τf +Τg+ΤD
= (00316 Nm) + (0 Nm) + (03316 Nm) + (0 Nm)
= 03632 Nm
Τ2= Τa+Τf +Τg+ΤD
= (0 Nm) + (0 Nm) + (03316 Nm) + (0 Nm)
= 03316 Nm
Τ3= Τa+Τf +Τg+ΤD
= (00316 Nm) + (0 Nm) + (03316 Nm) + (0 Nm)
= 03632 Nm
Τ1
Τ2
Τ3
Motion System Application Example15
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MOTION SYSTEMAPPLICATION EXAMPLE
Lead Screw Input Requirements
= 9276 rads= 2786 rads2
= 03632 Nm= 03316 Nm= ndash 03632 Nm= 0333 s= 0333 s= 0333 s= 05 s= 3369 W= 3076 W
ωPKaΤ1Τ2Τ3
t1t2t3tdwell
PPKPCV
PPK = (Τ1)(ωPK)
= (03632 Nm)(9276 rads)= 3369 W
PCV = (Τ2)(ωPK)
= (03316 Nm)(9276 rads)= 3076 W
RMS Torque Requirement Lead Screw Input
=
0333 s + 0333 s + 0333 s + 05 s
(03632 Nm)2(0333 s)+(03316 Nm)2(0333 s)+(- 03632 Nm)2(0333 s)
ΤRMS = Τ12t 1 + Τ2
2t2 + Τ32t 3
t 1 + t 2 + t 3 + tdwell
15
00439 + 00366 + 00439 = 00829 =
ΤRMS = 02879 Nm
Motion System Application Example16
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MOTION SYSTEMAPPLICATION EXAMPLE
Load Parameters at the Lead Screw Shaft
= 9276 rads= 2786 rads2
= 03632 Nm= 03316 Nm= ndash 03632 Nm= 02879 Nm= 3369 W= 3076 W
ωPKaΤ1Τ2Τ3ΤRMSPPKPCV
Motor SelectionA direct-drive motor can be used however it would be large and expensive For example a DC057B-3 brush motor
would easily meet continuous torque and continuous speed as well as a 181 load-to-rotor inertia This motor however has significantly more output power than needed at 128 watts
A smaller motor with a transmission would optimize the system Look for a motor starting at a rated power around 40 watts Pittman DC040B-6
DC040B-6Reference VoltageContinuous TorqueRated CurrentRated PowerTorque ConstantVoltage ConstantTerminal ResistanceMax Winding TemperatureRotor Inertia
240 V00812 Nm236 A37 W0042 NmA0042 Vrads185 Ω155degC847 x 10ndash6 Kg-m2
Motor TransmissionLead Screw Support Rails
ΤRMS
PPK
ωPK
= 02879 Nm= 3369 W= 9276 rads
M
Other Motor Considerationsndash Type stepper brush BLDCndash Input speed (high input speed will result in high noise levels)ndash Total motor footprintndash Ambient temperaturendash Environmental conditionsndash Load-to-rotor inertiandash Encoder ready needed
Motion System Application Example17
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17
MOTION SYSTEMAPPLICATION EXAMPLE
Gearbox Transmission Selection
When selecting a reduction ratio keep in mind that the RMS required torque at the motor shaft needs to fall below the continuous rated output torque of the motor
G30 A - Planetary GearboxMaximum output load 247 Nm
TRMS( Lead Screw)
02879 Nm
02879 Nm02879 Nm
Reduction
41
5161
Efficiency
090
090090
TRMS( Motor)
00798 Nm
00640 Nm00533 Nm
A 51 gearbox would allow about a 27 safety margin between what the system requires and the continuous torque output of the motor
System Summary 51η = 090
M
= 00807 Nm= 00737 Nm= ndash 00807 Nm= 00640 Nm= 4638 rads= 504 x 10 ndash 6 Kg-m2
= 3743 W
Τ1Τ2Τ3ΤRMS
PPK
ωPK
J
= 03632 Nm= 03316 Nm= ndash 03632 Nm= 02879 Nm= 9276 rads= 1134 x 10 ndash 5 Kg-m2
= 3369 W
Τ1Τ2Τ3ΤRMS
PPK
ωPK
J
Motion System Application Example18
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MOTION SYSTEMAPPLICATION EXAMPLE
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Motion Profile at the Motor
t 1 t 2 t 3 t dwell Time0333 s 0333 s 0333 s 05 s
ωΤ1
Τ2
Τ3= 4638 rads= 4429 RPM= ΤPK = 00807 Nm
ωPK
Τ1
nPK
Drive and Power Supply Requirements
Peak Current Required
IPK =TPK
KT+ IO =
00807 Nm0042 NmA
+ 0180 A = 210 A
RMS Current Required
IRMS =TRMS
KT+ IO =
00640 Nm0042 NmA
+ 0180 A = 170 A
Minimum bus Voltage Required
Vbus= IPKRmt+ ωPKKE = (210A)(185Ω) + (4638 rads)(0042 Vrads)= 389 V + 1948 V= 2337 V
0100 Nm 0200 Nm 0300 Nm 0400 Nm 0500 Nm 0600 Nm 0700 Nm
30 VDC
0100 Nm 0200 Nm 0300 Nm 0500 Nm 0600 Nm 0700 Nm
24 VDC
700060005000
RPM 4000300020001000
0400 Nm
TRMS
TPK
700060005000
RPM 4000300020001000
TRMS
TPK
Performance using 24V reference voltage
Performance using a 30V bus voltage
A 30 VDC bus will meet the application requirements as wells supply a margin of safety
MOTION SYSTEMAPPLICATION EXAMPLE
0100 Nm 0200 Nm 0300 Nm 0400 Nm 0500 Nm 0600 Nm 0700 Nm
30 VDC
0100 Nm 0200 Nm 0300 Nm 0500 Nm 0600 Nm 0700 Nm
24 VDC
700060005000
RPM 4000300020001000
0400 Nm
TRMS
TPK
700060005000
RPM 4000300020001000
TRMS
TPK
Ambient temperature Rated motor temperature Temperature rise Motor temperature
= 30degC= 155degC= 7638degC= 10638degC
Θa
Θm
ΘrΘrated
=Rth x IRMS
2 x Rmt
1 ndash (Rth x IRMS2 x Rmt x 000392degC)
Θr
=11degCω x 170A2
x 185Ω 1 ndash (11degCω x 170A2
x 185Ω x 000392degC)
= 5881 1 ndash (023)
= 5881 077
=
Θm = Θr + Θa = 7638degC + 30degC = 10638degC
7638degC
Performance using 24V reference voltage
Performance using a 30V bus voltage
Will the motor meet the temperature rise caused by the application
Motion System Application Example20
A 30 VDC bus will meet the application requirements as wells supply a margin of safety
203 756 7441 1500 Meriden Road Waterbury CT USA 06705
267 933 2105 534 Godshall DriveHarleysville PA USA 19438
wwwHaydonKerkPittmancom
copy All rights reserved AMETEK Advanced Motion Solutions No part of this doc-ument or technical information can be used reproduced or altered in any form without approval or proper authorization from AMETEK Inc and global affiliates
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MOTION SYSTEMAPPLICATION EXAMPLE
Jin= m 2 πL 2
X η1 + JS= 9Kg
2 π002032 m 2
X 0861 + 388x10ndash 7Kg-m2
= 1095 X 10ndash 5 Kg-m2 + 388x10ndash 7 Kg-m2
= 1134 x 10ndash 5 Kg-m2
Τin= Τa+Τf +Τg+ΤD
Τa = Jin ain
Τg =sinOslashmgL
2 π η
= (1134 x 10ndash 5 Kg-m2)(2786 rads2)
= 00316 Nm
Τf =cosOslashmgμL
2 π η= (cos90)(9 Kg)(98ms2)(001)(002032m)
2 π (086)= 0 Nm
= (sin90)(9 Kg)(98ms2)(002032m)2 π (086)
=1792 Nm
5404 = 03316 Nm
ΤDSince a free-wheeling lead screw nut was used there is no preload force If an anti-backlash nut was used ΤD would be gt0 due to preload force Always reference manufacturerrsquos data for more information
= 0 Nm
Linear to Rotary Conversion Torque
Linear to Rotary Conversion Inertia
Motion System Application Example14
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14
MOTION SYSTEMFORMULAS
t 1 t 2 t 3 t dwell Time0333 s 0333 s 0333 s 05 s
Rotary Motion Profile
ωPK
ω
= 9276 rads a = 2786 rads2
Τ1= Τa+Τf +Τg+ΤD
= (00316 Nm) + (0 Nm) + (03316 Nm) + (0 Nm)
= 03632 Nm
Τ2= Τa+Τf +Τg+ΤD
= (0 Nm) + (0 Nm) + (03316 Nm) + (0 Nm)
= 03316 Nm
Τ3= Τa+Τf +Τg+ΤD
= (00316 Nm) + (0 Nm) + (03316 Nm) + (0 Nm)
= 03632 Nm
Τ1
Τ2
Τ3
Motion System Application Example15
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
15
MOTION SYSTEMAPPLICATION EXAMPLE
Lead Screw Input Requirements
= 9276 rads= 2786 rads2
= 03632 Nm= 03316 Nm= ndash 03632 Nm= 0333 s= 0333 s= 0333 s= 05 s= 3369 W= 3076 W
ωPKaΤ1Τ2Τ3
t1t2t3tdwell
PPKPCV
PPK = (Τ1)(ωPK)
= (03632 Nm)(9276 rads)= 3369 W
PCV = (Τ2)(ωPK)
= (03316 Nm)(9276 rads)= 3076 W
RMS Torque Requirement Lead Screw Input
=
0333 s + 0333 s + 0333 s + 05 s
(03632 Nm)2(0333 s)+(03316 Nm)2(0333 s)+(- 03632 Nm)2(0333 s)
ΤRMS = Τ12t 1 + Τ2
2t2 + Τ32t 3
t 1 + t 2 + t 3 + tdwell
15
00439 + 00366 + 00439 = 00829 =
ΤRMS = 02879 Nm
Motion System Application Example16
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16
MOTION SYSTEMAPPLICATION EXAMPLE
Load Parameters at the Lead Screw Shaft
= 9276 rads= 2786 rads2
= 03632 Nm= 03316 Nm= ndash 03632 Nm= 02879 Nm= 3369 W= 3076 W
ωPKaΤ1Τ2Τ3ΤRMSPPKPCV
Motor SelectionA direct-drive motor can be used however it would be large and expensive For example a DC057B-3 brush motor
would easily meet continuous torque and continuous speed as well as a 181 load-to-rotor inertia This motor however has significantly more output power than needed at 128 watts
A smaller motor with a transmission would optimize the system Look for a motor starting at a rated power around 40 watts Pittman DC040B-6
DC040B-6Reference VoltageContinuous TorqueRated CurrentRated PowerTorque ConstantVoltage ConstantTerminal ResistanceMax Winding TemperatureRotor Inertia
240 V00812 Nm236 A37 W0042 NmA0042 Vrads185 Ω155degC847 x 10ndash6 Kg-m2
Motor TransmissionLead Screw Support Rails
ΤRMS
PPK
ωPK
= 02879 Nm= 3369 W= 9276 rads
M
Other Motor Considerationsndash Type stepper brush BLDCndash Input speed (high input speed will result in high noise levels)ndash Total motor footprintndash Ambient temperaturendash Environmental conditionsndash Load-to-rotor inertiandash Encoder ready needed
Motion System Application Example17
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17
MOTION SYSTEMAPPLICATION EXAMPLE
Gearbox Transmission Selection
When selecting a reduction ratio keep in mind that the RMS required torque at the motor shaft needs to fall below the continuous rated output torque of the motor
G30 A - Planetary GearboxMaximum output load 247 Nm
TRMS( Lead Screw)
02879 Nm
02879 Nm02879 Nm
Reduction
41
5161
Efficiency
090
090090
TRMS( Motor)
00798 Nm
00640 Nm00533 Nm
A 51 gearbox would allow about a 27 safety margin between what the system requires and the continuous torque output of the motor
System Summary 51η = 090
M
= 00807 Nm= 00737 Nm= ndash 00807 Nm= 00640 Nm= 4638 rads= 504 x 10 ndash 6 Kg-m2
= 3743 W
Τ1Τ2Τ3ΤRMS
PPK
ωPK
J
= 03632 Nm= 03316 Nm= ndash 03632 Nm= 02879 Nm= 9276 rads= 1134 x 10 ndash 5 Kg-m2
= 3369 W
Τ1Τ2Τ3ΤRMS
PPK
ωPK
J
Motion System Application Example18
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MOTION SYSTEMAPPLICATION EXAMPLE
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Motion Profile at the Motor
t 1 t 2 t 3 t dwell Time0333 s 0333 s 0333 s 05 s
ωΤ1
Τ2
Τ3= 4638 rads= 4429 RPM= ΤPK = 00807 Nm
ωPK
Τ1
nPK
Drive and Power Supply Requirements
Peak Current Required
IPK =TPK
KT+ IO =
00807 Nm0042 NmA
+ 0180 A = 210 A
RMS Current Required
IRMS =TRMS
KT+ IO =
00640 Nm0042 NmA
+ 0180 A = 170 A
Minimum bus Voltage Required
Vbus= IPKRmt+ ωPKKE = (210A)(185Ω) + (4638 rads)(0042 Vrads)= 389 V + 1948 V= 2337 V
0100 Nm 0200 Nm 0300 Nm 0400 Nm 0500 Nm 0600 Nm 0700 Nm
30 VDC
0100 Nm 0200 Nm 0300 Nm 0500 Nm 0600 Nm 0700 Nm
24 VDC
700060005000
RPM 4000300020001000
0400 Nm
TRMS
TPK
700060005000
RPM 4000300020001000
TRMS
TPK
Performance using 24V reference voltage
Performance using a 30V bus voltage
A 30 VDC bus will meet the application requirements as wells supply a margin of safety
MOTION SYSTEMAPPLICATION EXAMPLE
0100 Nm 0200 Nm 0300 Nm 0400 Nm 0500 Nm 0600 Nm 0700 Nm
30 VDC
0100 Nm 0200 Nm 0300 Nm 0500 Nm 0600 Nm 0700 Nm
24 VDC
700060005000
RPM 4000300020001000
0400 Nm
TRMS
TPK
700060005000
RPM 4000300020001000
TRMS
TPK
Ambient temperature Rated motor temperature Temperature rise Motor temperature
= 30degC= 155degC= 7638degC= 10638degC
Θa
Θm
ΘrΘrated
=Rth x IRMS
2 x Rmt
1 ndash (Rth x IRMS2 x Rmt x 000392degC)
Θr
=11degCω x 170A2
x 185Ω 1 ndash (11degCω x 170A2
x 185Ω x 000392degC)
= 5881 1 ndash (023)
= 5881 077
=
Θm = Θr + Θa = 7638degC + 30degC = 10638degC
7638degC
Performance using 24V reference voltage
Performance using a 30V bus voltage
Will the motor meet the temperature rise caused by the application
Motion System Application Example20
A 30 VDC bus will meet the application requirements as wells supply a margin of safety
203 756 7441 1500 Meriden Road Waterbury CT USA 06705
267 933 2105 534 Godshall DriveHarleysville PA USA 19438
wwwHaydonKerkPittmancom
copy All rights reserved AMETEK Advanced Motion Solutions No part of this doc-ument or technical information can be used reproduced or altered in any form without approval or proper authorization from AMETEK Inc and global affiliates
NOW ASK us the tough motion control questions ONLINE
ASK THE EXPERTS
HaydonKerkPittmancomwww
MOTION SYSTEMAPPLICATION EXAMPLE
t 1 t 2 t 3 t dwell Time0333 s 0333 s 0333 s 05 s
Rotary Motion Profile
ωPK
ω
= 9276 rads a = 2786 rads2
Τ1= Τa+Τf +Τg+ΤD
= (00316 Nm) + (0 Nm) + (03316 Nm) + (0 Nm)
= 03632 Nm
Τ2= Τa+Τf +Τg+ΤD
= (0 Nm) + (0 Nm) + (03316 Nm) + (0 Nm)
= 03316 Nm
Τ3= Τa+Τf +Τg+ΤD
= (00316 Nm) + (0 Nm) + (03316 Nm) + (0 Nm)
= 03632 Nm
Τ1
Τ2
Τ3
Motion System Application Example15
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MOTION SYSTEMAPPLICATION EXAMPLE
Lead Screw Input Requirements
= 9276 rads= 2786 rads2
= 03632 Nm= 03316 Nm= ndash 03632 Nm= 0333 s= 0333 s= 0333 s= 05 s= 3369 W= 3076 W
ωPKaΤ1Τ2Τ3
t1t2t3tdwell
PPKPCV
PPK = (Τ1)(ωPK)
= (03632 Nm)(9276 rads)= 3369 W
PCV = (Τ2)(ωPK)
= (03316 Nm)(9276 rads)= 3076 W
RMS Torque Requirement Lead Screw Input
=
0333 s + 0333 s + 0333 s + 05 s
(03632 Nm)2(0333 s)+(03316 Nm)2(0333 s)+(- 03632 Nm)2(0333 s)
ΤRMS = Τ12t 1 + Τ2
2t2 + Τ32t 3
t 1 + t 2 + t 3 + tdwell
15
00439 + 00366 + 00439 = 00829 =
ΤRMS = 02879 Nm
Motion System Application Example16
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16
MOTION SYSTEMAPPLICATION EXAMPLE
Load Parameters at the Lead Screw Shaft
= 9276 rads= 2786 rads2
= 03632 Nm= 03316 Nm= ndash 03632 Nm= 02879 Nm= 3369 W= 3076 W
ωPKaΤ1Τ2Τ3ΤRMSPPKPCV
Motor SelectionA direct-drive motor can be used however it would be large and expensive For example a DC057B-3 brush motor
would easily meet continuous torque and continuous speed as well as a 181 load-to-rotor inertia This motor however has significantly more output power than needed at 128 watts
A smaller motor with a transmission would optimize the system Look for a motor starting at a rated power around 40 watts Pittman DC040B-6
DC040B-6Reference VoltageContinuous TorqueRated CurrentRated PowerTorque ConstantVoltage ConstantTerminal ResistanceMax Winding TemperatureRotor Inertia
240 V00812 Nm236 A37 W0042 NmA0042 Vrads185 Ω155degC847 x 10ndash6 Kg-m2
Motor TransmissionLead Screw Support Rails
ΤRMS
PPK
ωPK
= 02879 Nm= 3369 W= 9276 rads
M
Other Motor Considerationsndash Type stepper brush BLDCndash Input speed (high input speed will result in high noise levels)ndash Total motor footprintndash Ambient temperaturendash Environmental conditionsndash Load-to-rotor inertiandash Encoder ready needed
Motion System Application Example17
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
17
MOTION SYSTEMAPPLICATION EXAMPLE
Gearbox Transmission Selection
When selecting a reduction ratio keep in mind that the RMS required torque at the motor shaft needs to fall below the continuous rated output torque of the motor
G30 A - Planetary GearboxMaximum output load 247 Nm
TRMS( Lead Screw)
02879 Nm
02879 Nm02879 Nm
Reduction
41
5161
Efficiency
090
090090
TRMS( Motor)
00798 Nm
00640 Nm00533 Nm
A 51 gearbox would allow about a 27 safety margin between what the system requires and the continuous torque output of the motor
System Summary 51η = 090
M
= 00807 Nm= 00737 Nm= ndash 00807 Nm= 00640 Nm= 4638 rads= 504 x 10 ndash 6 Kg-m2
= 3743 W
Τ1Τ2Τ3ΤRMS
PPK
ωPK
J
= 03632 Nm= 03316 Nm= ndash 03632 Nm= 02879 Nm= 9276 rads= 1134 x 10 ndash 5 Kg-m2
= 3369 W
Τ1Τ2Τ3ΤRMS
PPK
ωPK
J
Motion System Application Example18
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MOTION SYSTEMAPPLICATION EXAMPLE
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
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Motion Profile at the Motor
t 1 t 2 t 3 t dwell Time0333 s 0333 s 0333 s 05 s
ωΤ1
Τ2
Τ3= 4638 rads= 4429 RPM= ΤPK = 00807 Nm
ωPK
Τ1
nPK
Drive and Power Supply Requirements
Peak Current Required
IPK =TPK
KT+ IO =
00807 Nm0042 NmA
+ 0180 A = 210 A
RMS Current Required
IRMS =TRMS
KT+ IO =
00640 Nm0042 NmA
+ 0180 A = 170 A
Minimum bus Voltage Required
Vbus= IPKRmt+ ωPKKE = (210A)(185Ω) + (4638 rads)(0042 Vrads)= 389 V + 1948 V= 2337 V
0100 Nm 0200 Nm 0300 Nm 0400 Nm 0500 Nm 0600 Nm 0700 Nm
30 VDC
0100 Nm 0200 Nm 0300 Nm 0500 Nm 0600 Nm 0700 Nm
24 VDC
700060005000
RPM 4000300020001000
0400 Nm
TRMS
TPK
700060005000
RPM 4000300020001000
TRMS
TPK
Performance using 24V reference voltage
Performance using a 30V bus voltage
A 30 VDC bus will meet the application requirements as wells supply a margin of safety
MOTION SYSTEMAPPLICATION EXAMPLE
0100 Nm 0200 Nm 0300 Nm 0400 Nm 0500 Nm 0600 Nm 0700 Nm
30 VDC
0100 Nm 0200 Nm 0300 Nm 0500 Nm 0600 Nm 0700 Nm
24 VDC
700060005000
RPM 4000300020001000
0400 Nm
TRMS
TPK
700060005000
RPM 4000300020001000
TRMS
TPK
Ambient temperature Rated motor temperature Temperature rise Motor temperature
= 30degC= 155degC= 7638degC= 10638degC
Θa
Θm
ΘrΘrated
=Rth x IRMS
2 x Rmt
1 ndash (Rth x IRMS2 x Rmt x 000392degC)
Θr
=11degCω x 170A2
x 185Ω 1 ndash (11degCω x 170A2
x 185Ω x 000392degC)
= 5881 1 ndash (023)
= 5881 077
=
Θm = Θr + Θa = 7638degC + 30degC = 10638degC
7638degC
Performance using 24V reference voltage
Performance using a 30V bus voltage
Will the motor meet the temperature rise caused by the application
Motion System Application Example20
A 30 VDC bus will meet the application requirements as wells supply a margin of safety
203 756 7441 1500 Meriden Road Waterbury CT USA 06705
267 933 2105 534 Godshall DriveHarleysville PA USA 19438
wwwHaydonKerkPittmancom
copy All rights reserved AMETEK Advanced Motion Solutions No part of this doc-ument or technical information can be used reproduced or altered in any form without approval or proper authorization from AMETEK Inc and global affiliates
NOW ASK us the tough motion control questions ONLINE
ASK THE EXPERTS
HaydonKerkPittmancomwww
MOTION SYSTEMAPPLICATION EXAMPLE
Lead Screw Input Requirements
= 9276 rads= 2786 rads2
= 03632 Nm= 03316 Nm= ndash 03632 Nm= 0333 s= 0333 s= 0333 s= 05 s= 3369 W= 3076 W
ωPKaΤ1Τ2Τ3
t1t2t3tdwell
PPKPCV
PPK = (Τ1)(ωPK)
= (03632 Nm)(9276 rads)= 3369 W
PCV = (Τ2)(ωPK)
= (03316 Nm)(9276 rads)= 3076 W
RMS Torque Requirement Lead Screw Input
=
0333 s + 0333 s + 0333 s + 05 s
(03632 Nm)2(0333 s)+(03316 Nm)2(0333 s)+(- 03632 Nm)2(0333 s)
ΤRMS = Τ12t 1 + Τ2
2t2 + Τ32t 3
t 1 + t 2 + t 3 + tdwell
15
00439 + 00366 + 00439 = 00829 =
ΤRMS = 02879 Nm
Motion System Application Example16
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
16
MOTION SYSTEMAPPLICATION EXAMPLE
Load Parameters at the Lead Screw Shaft
= 9276 rads= 2786 rads2
= 03632 Nm= 03316 Nm= ndash 03632 Nm= 02879 Nm= 3369 W= 3076 W
ωPKaΤ1Τ2Τ3ΤRMSPPKPCV
Motor SelectionA direct-drive motor can be used however it would be large and expensive For example a DC057B-3 brush motor
would easily meet continuous torque and continuous speed as well as a 181 load-to-rotor inertia This motor however has significantly more output power than needed at 128 watts
A smaller motor with a transmission would optimize the system Look for a motor starting at a rated power around 40 watts Pittman DC040B-6
DC040B-6Reference VoltageContinuous TorqueRated CurrentRated PowerTorque ConstantVoltage ConstantTerminal ResistanceMax Winding TemperatureRotor Inertia
240 V00812 Nm236 A37 W0042 NmA0042 Vrads185 Ω155degC847 x 10ndash6 Kg-m2
Motor TransmissionLead Screw Support Rails
ΤRMS
PPK
ωPK
= 02879 Nm= 3369 W= 9276 rads
M
Other Motor Considerationsndash Type stepper brush BLDCndash Input speed (high input speed will result in high noise levels)ndash Total motor footprintndash Ambient temperaturendash Environmental conditionsndash Load-to-rotor inertiandash Encoder ready needed
Motion System Application Example17
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
17
MOTION SYSTEMAPPLICATION EXAMPLE
Gearbox Transmission Selection
When selecting a reduction ratio keep in mind that the RMS required torque at the motor shaft needs to fall below the continuous rated output torque of the motor
G30 A - Planetary GearboxMaximum output load 247 Nm
TRMS( Lead Screw)
02879 Nm
02879 Nm02879 Nm
Reduction
41
5161
Efficiency
090
090090
TRMS( Motor)
00798 Nm
00640 Nm00533 Nm
A 51 gearbox would allow about a 27 safety margin between what the system requires and the continuous torque output of the motor
System Summary 51η = 090
M
= 00807 Nm= 00737 Nm= ndash 00807 Nm= 00640 Nm= 4638 rads= 504 x 10 ndash 6 Kg-m2
= 3743 W
Τ1Τ2Τ3ΤRMS
PPK
ωPK
J
= 03632 Nm= 03316 Nm= ndash 03632 Nm= 02879 Nm= 9276 rads= 1134 x 10 ndash 5 Kg-m2
= 3369 W
Τ1Τ2Τ3ΤRMS
PPK
ωPK
J
Motion System Application Example18
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MOTION SYSTEMAPPLICATION EXAMPLE
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Motion Profile at the Motor
t 1 t 2 t 3 t dwell Time0333 s 0333 s 0333 s 05 s
ωΤ1
Τ2
Τ3= 4638 rads= 4429 RPM= ΤPK = 00807 Nm
ωPK
Τ1
nPK
Drive and Power Supply Requirements
Peak Current Required
IPK =TPK
KT+ IO =
00807 Nm0042 NmA
+ 0180 A = 210 A
RMS Current Required
IRMS =TRMS
KT+ IO =
00640 Nm0042 NmA
+ 0180 A = 170 A
Minimum bus Voltage Required
Vbus= IPKRmt+ ωPKKE = (210A)(185Ω) + (4638 rads)(0042 Vrads)= 389 V + 1948 V= 2337 V
0100 Nm 0200 Nm 0300 Nm 0400 Nm 0500 Nm 0600 Nm 0700 Nm
30 VDC
0100 Nm 0200 Nm 0300 Nm 0500 Nm 0600 Nm 0700 Nm
24 VDC
700060005000
RPM 4000300020001000
0400 Nm
TRMS
TPK
700060005000
RPM 4000300020001000
TRMS
TPK
Performance using 24V reference voltage
Performance using a 30V bus voltage
A 30 VDC bus will meet the application requirements as wells supply a margin of safety
MOTION SYSTEMAPPLICATION EXAMPLE
0100 Nm 0200 Nm 0300 Nm 0400 Nm 0500 Nm 0600 Nm 0700 Nm
30 VDC
0100 Nm 0200 Nm 0300 Nm 0500 Nm 0600 Nm 0700 Nm
24 VDC
700060005000
RPM 4000300020001000
0400 Nm
TRMS
TPK
700060005000
RPM 4000300020001000
TRMS
TPK
Ambient temperature Rated motor temperature Temperature rise Motor temperature
= 30degC= 155degC= 7638degC= 10638degC
Θa
Θm
ΘrΘrated
=Rth x IRMS
2 x Rmt
1 ndash (Rth x IRMS2 x Rmt x 000392degC)
Θr
=11degCω x 170A2
x 185Ω 1 ndash (11degCω x 170A2
x 185Ω x 000392degC)
= 5881 1 ndash (023)
= 5881 077
=
Θm = Θr + Θa = 7638degC + 30degC = 10638degC
7638degC
Performance using 24V reference voltage
Performance using a 30V bus voltage
Will the motor meet the temperature rise caused by the application
Motion System Application Example20
A 30 VDC bus will meet the application requirements as wells supply a margin of safety
203 756 7441 1500 Meriden Road Waterbury CT USA 06705
267 933 2105 534 Godshall DriveHarleysville PA USA 19438
wwwHaydonKerkPittmancom
copy All rights reserved AMETEK Advanced Motion Solutions No part of this doc-ument or technical information can be used reproduced or altered in any form without approval or proper authorization from AMETEK Inc and global affiliates
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MOTION SYSTEMAPPLICATION EXAMPLE
Load Parameters at the Lead Screw Shaft
= 9276 rads= 2786 rads2
= 03632 Nm= 03316 Nm= ndash 03632 Nm= 02879 Nm= 3369 W= 3076 W
ωPKaΤ1Τ2Τ3ΤRMSPPKPCV
Motor SelectionA direct-drive motor can be used however it would be large and expensive For example a DC057B-3 brush motor
would easily meet continuous torque and continuous speed as well as a 181 load-to-rotor inertia This motor however has significantly more output power than needed at 128 watts
A smaller motor with a transmission would optimize the system Look for a motor starting at a rated power around 40 watts Pittman DC040B-6
DC040B-6Reference VoltageContinuous TorqueRated CurrentRated PowerTorque ConstantVoltage ConstantTerminal ResistanceMax Winding TemperatureRotor Inertia
240 V00812 Nm236 A37 W0042 NmA0042 Vrads185 Ω155degC847 x 10ndash6 Kg-m2
Motor TransmissionLead Screw Support Rails
ΤRMS
PPK
ωPK
= 02879 Nm= 3369 W= 9276 rads
M
Other Motor Considerationsndash Type stepper brush BLDCndash Input speed (high input speed will result in high noise levels)ndash Total motor footprintndash Ambient temperaturendash Environmental conditionsndash Load-to-rotor inertiandash Encoder ready needed
Motion System Application Example17
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MOTION SYSTEMAPPLICATION EXAMPLE
Gearbox Transmission Selection
When selecting a reduction ratio keep in mind that the RMS required torque at the motor shaft needs to fall below the continuous rated output torque of the motor
G30 A - Planetary GearboxMaximum output load 247 Nm
TRMS( Lead Screw)
02879 Nm
02879 Nm02879 Nm
Reduction
41
5161
Efficiency
090
090090
TRMS( Motor)
00798 Nm
00640 Nm00533 Nm
A 51 gearbox would allow about a 27 safety margin between what the system requires and the continuous torque output of the motor
System Summary 51η = 090
M
= 00807 Nm= 00737 Nm= ndash 00807 Nm= 00640 Nm= 4638 rads= 504 x 10 ndash 6 Kg-m2
= 3743 W
Τ1Τ2Τ3ΤRMS
PPK
ωPK
J
= 03632 Nm= 03316 Nm= ndash 03632 Nm= 02879 Nm= 9276 rads= 1134 x 10 ndash 5 Kg-m2
= 3369 W
Τ1Τ2Τ3ΤRMS
PPK
ωPK
J
Motion System Application Example18
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MOTION SYSTEMAPPLICATION EXAMPLE
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
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Motion Profile at the Motor
t 1 t 2 t 3 t dwell Time0333 s 0333 s 0333 s 05 s
ωΤ1
Τ2
Τ3= 4638 rads= 4429 RPM= ΤPK = 00807 Nm
ωPK
Τ1
nPK
Drive and Power Supply Requirements
Peak Current Required
IPK =TPK
KT+ IO =
00807 Nm0042 NmA
+ 0180 A = 210 A
RMS Current Required
IRMS =TRMS
KT+ IO =
00640 Nm0042 NmA
+ 0180 A = 170 A
Minimum bus Voltage Required
Vbus= IPKRmt+ ωPKKE = (210A)(185Ω) + (4638 rads)(0042 Vrads)= 389 V + 1948 V= 2337 V
0100 Nm 0200 Nm 0300 Nm 0400 Nm 0500 Nm 0600 Nm 0700 Nm
30 VDC
0100 Nm 0200 Nm 0300 Nm 0500 Nm 0600 Nm 0700 Nm
24 VDC
700060005000
RPM 4000300020001000
0400 Nm
TRMS
TPK
700060005000
RPM 4000300020001000
TRMS
TPK
Performance using 24V reference voltage
Performance using a 30V bus voltage
A 30 VDC bus will meet the application requirements as wells supply a margin of safety
MOTION SYSTEMAPPLICATION EXAMPLE
0100 Nm 0200 Nm 0300 Nm 0400 Nm 0500 Nm 0600 Nm 0700 Nm
30 VDC
0100 Nm 0200 Nm 0300 Nm 0500 Nm 0600 Nm 0700 Nm
24 VDC
700060005000
RPM 4000300020001000
0400 Nm
TRMS
TPK
700060005000
RPM 4000300020001000
TRMS
TPK
Ambient temperature Rated motor temperature Temperature rise Motor temperature
= 30degC= 155degC= 7638degC= 10638degC
Θa
Θm
ΘrΘrated
=Rth x IRMS
2 x Rmt
1 ndash (Rth x IRMS2 x Rmt x 000392degC)
Θr
=11degCω x 170A2
x 185Ω 1 ndash (11degCω x 170A2
x 185Ω x 000392degC)
= 5881 1 ndash (023)
= 5881 077
=
Θm = Θr + Θa = 7638degC + 30degC = 10638degC
7638degC
Performance using 24V reference voltage
Performance using a 30V bus voltage
Will the motor meet the temperature rise caused by the application
Motion System Application Example20
A 30 VDC bus will meet the application requirements as wells supply a margin of safety
203 756 7441 1500 Meriden Road Waterbury CT USA 06705
267 933 2105 534 Godshall DriveHarleysville PA USA 19438
wwwHaydonKerkPittmancom
copy All rights reserved AMETEK Advanced Motion Solutions No part of this doc-ument or technical information can be used reproduced or altered in any form without approval or proper authorization from AMETEK Inc and global affiliates
NOW ASK us the tough motion control questions ONLINE
ASK THE EXPERTS
HaydonKerkPittmancomwww
MOTION SYSTEMAPPLICATION EXAMPLE
Gearbox Transmission Selection
When selecting a reduction ratio keep in mind that the RMS required torque at the motor shaft needs to fall below the continuous rated output torque of the motor
G30 A - Planetary GearboxMaximum output load 247 Nm
TRMS( Lead Screw)
02879 Nm
02879 Nm02879 Nm
Reduction
41
5161
Efficiency
090
090090
TRMS( Motor)
00798 Nm
00640 Nm00533 Nm
A 51 gearbox would allow about a 27 safety margin between what the system requires and the continuous torque output of the motor
System Summary 51η = 090
M
= 00807 Nm= 00737 Nm= ndash 00807 Nm= 00640 Nm= 4638 rads= 504 x 10 ndash 6 Kg-m2
= 3743 W
Τ1Τ2Τ3ΤRMS
PPK
ωPK
J
= 03632 Nm= 03316 Nm= ndash 03632 Nm= 02879 Nm= 9276 rads= 1134 x 10 ndash 5 Kg-m2
= 3369 W
Τ1Τ2Τ3ΤRMS
PPK
ωPK
J
Motion System Application Example18
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
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MOTION SYSTEMAPPLICATION EXAMPLE
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
19
Motion Profile at the Motor
t 1 t 2 t 3 t dwell Time0333 s 0333 s 0333 s 05 s
ωΤ1
Τ2
Τ3= 4638 rads= 4429 RPM= ΤPK = 00807 Nm
ωPK
Τ1
nPK
Drive and Power Supply Requirements
Peak Current Required
IPK =TPK
KT+ IO =
00807 Nm0042 NmA
+ 0180 A = 210 A
RMS Current Required
IRMS =TRMS
KT+ IO =
00640 Nm0042 NmA
+ 0180 A = 170 A
Minimum bus Voltage Required
Vbus= IPKRmt+ ωPKKE = (210A)(185Ω) + (4638 rads)(0042 Vrads)= 389 V + 1948 V= 2337 V
0100 Nm 0200 Nm 0300 Nm 0400 Nm 0500 Nm 0600 Nm 0700 Nm
30 VDC
0100 Nm 0200 Nm 0300 Nm 0500 Nm 0600 Nm 0700 Nm
24 VDC
700060005000
RPM 4000300020001000
0400 Nm
TRMS
TPK
700060005000
RPM 4000300020001000
TRMS
TPK
Performance using 24V reference voltage
Performance using a 30V bus voltage
A 30 VDC bus will meet the application requirements as wells supply a margin of safety
MOTION SYSTEMAPPLICATION EXAMPLE
0100 Nm 0200 Nm 0300 Nm 0400 Nm 0500 Nm 0600 Nm 0700 Nm
30 VDC
0100 Nm 0200 Nm 0300 Nm 0500 Nm 0600 Nm 0700 Nm
24 VDC
700060005000
RPM 4000300020001000
0400 Nm
TRMS
TPK
700060005000
RPM 4000300020001000
TRMS
TPK
Ambient temperature Rated motor temperature Temperature rise Motor temperature
= 30degC= 155degC= 7638degC= 10638degC
Θa
Θm
ΘrΘrated
=Rth x IRMS
2 x Rmt
1 ndash (Rth x IRMS2 x Rmt x 000392degC)
Θr
=11degCω x 170A2
x 185Ω 1 ndash (11degCω x 170A2
x 185Ω x 000392degC)
= 5881 1 ndash (023)
= 5881 077
=
Θm = Θr + Θa = 7638degC + 30degC = 10638degC
7638degC
Performance using 24V reference voltage
Performance using a 30V bus voltage
Will the motor meet the temperature rise caused by the application
Motion System Application Example20
A 30 VDC bus will meet the application requirements as wells supply a margin of safety
203 756 7441 1500 Meriden Road Waterbury CT USA 06705
267 933 2105 534 Godshall DriveHarleysville PA USA 19438
wwwHaydonKerkPittmancom
copy All rights reserved AMETEK Advanced Motion Solutions No part of this doc-ument or technical information can be used reproduced or altered in any form without approval or proper authorization from AMETEK Inc and global affiliates
NOW ASK us the tough motion control questions ONLINE
ASK THE EXPERTS
HaydonKerkPittmancomwww
MOTION SYSTEMAPPLICATION EXAMPLE
wwwHaydonKerkPittmancomHaydon Kerk Motion Solutions 203 756 7441Pittman Motors 267 933 2105
19
Motion Profile at the Motor
t 1 t 2 t 3 t dwell Time0333 s 0333 s 0333 s 05 s
ωΤ1
Τ2
Τ3= 4638 rads= 4429 RPM= ΤPK = 00807 Nm
ωPK
Τ1
nPK
Drive and Power Supply Requirements
Peak Current Required
IPK =TPK
KT+ IO =
00807 Nm0042 NmA
+ 0180 A = 210 A
RMS Current Required
IRMS =TRMS
KT+ IO =
00640 Nm0042 NmA
+ 0180 A = 170 A
Minimum bus Voltage Required
Vbus= IPKRmt+ ωPKKE = (210A)(185Ω) + (4638 rads)(0042 Vrads)= 389 V + 1948 V= 2337 V
0100 Nm 0200 Nm 0300 Nm 0400 Nm 0500 Nm 0600 Nm 0700 Nm
30 VDC
0100 Nm 0200 Nm 0300 Nm 0500 Nm 0600 Nm 0700 Nm
24 VDC
700060005000
RPM 4000300020001000
0400 Nm
TRMS
TPK
700060005000
RPM 4000300020001000
TRMS
TPK
Performance using 24V reference voltage
Performance using a 30V bus voltage
A 30 VDC bus will meet the application requirements as wells supply a margin of safety
MOTION SYSTEMAPPLICATION EXAMPLE
0100 Nm 0200 Nm 0300 Nm 0400 Nm 0500 Nm 0600 Nm 0700 Nm
30 VDC
0100 Nm 0200 Nm 0300 Nm 0500 Nm 0600 Nm 0700 Nm
24 VDC
700060005000
RPM 4000300020001000
0400 Nm
TRMS
TPK
700060005000
RPM 4000300020001000
TRMS
TPK
Ambient temperature Rated motor temperature Temperature rise Motor temperature
= 30degC= 155degC= 7638degC= 10638degC
Θa
Θm
ΘrΘrated
=Rth x IRMS
2 x Rmt
1 ndash (Rth x IRMS2 x Rmt x 000392degC)
Θr
=11degCω x 170A2
x 185Ω 1 ndash (11degCω x 170A2
x 185Ω x 000392degC)
= 5881 1 ndash (023)
= 5881 077
=
Θm = Θr + Θa = 7638degC + 30degC = 10638degC
7638degC
Performance using 24V reference voltage
Performance using a 30V bus voltage
Will the motor meet the temperature rise caused by the application
Motion System Application Example20
A 30 VDC bus will meet the application requirements as wells supply a margin of safety
203 756 7441 1500 Meriden Road Waterbury CT USA 06705
267 933 2105 534 Godshall DriveHarleysville PA USA 19438
wwwHaydonKerkPittmancom
copy All rights reserved AMETEK Advanced Motion Solutions No part of this doc-ument or technical information can be used reproduced or altered in any form without approval or proper authorization from AMETEK Inc and global affiliates
NOW ASK us the tough motion control questions ONLINE
ASK THE EXPERTS
HaydonKerkPittmancomwww
MOTION SYSTEMAPPLICATION EXAMPLE
0100 Nm 0200 Nm 0300 Nm 0400 Nm 0500 Nm 0600 Nm 0700 Nm
30 VDC
0100 Nm 0200 Nm 0300 Nm 0500 Nm 0600 Nm 0700 Nm
24 VDC
700060005000
RPM 4000300020001000
0400 Nm
TRMS
TPK
700060005000
RPM 4000300020001000
TRMS
TPK
Ambient temperature Rated motor temperature Temperature rise Motor temperature
= 30degC= 155degC= 7638degC= 10638degC
Θa
Θm
ΘrΘrated
=Rth x IRMS
2 x Rmt
1 ndash (Rth x IRMS2 x Rmt x 000392degC)
Θr
=11degCω x 170A2
x 185Ω 1 ndash (11degCω x 170A2
x 185Ω x 000392degC)
= 5881 1 ndash (023)
= 5881 077
=
Θm = Θr + Θa = 7638degC + 30degC = 10638degC
7638degC
Performance using 24V reference voltage
Performance using a 30V bus voltage
Will the motor meet the temperature rise caused by the application
Motion System Application Example20
A 30 VDC bus will meet the application requirements as wells supply a margin of safety
203 756 7441 1500 Meriden Road Waterbury CT USA 06705
267 933 2105 534 Godshall DriveHarleysville PA USA 19438
wwwHaydonKerkPittmancom
copy All rights reserved AMETEK Advanced Motion Solutions No part of this doc-ument or technical information can be used reproduced or altered in any form without approval or proper authorization from AMETEK Inc and global affiliates
NOW ASK us the tough motion control questions ONLINE
ASK THE EXPERTS
HaydonKerkPittmancomwww
MOTION SYSTEMAPPLICATION EXAMPLE