FORMULAS FOR MOTORIZED LINEAR MOTION SYSTEMS

FOR MOTORIZED LINEAR MOTION SYSTEMS Haydon Kerk Motion Solutions Pittman Motors : 203 756 7441 : 267 933 2105

SYMBOLS AND UNITS Symbol Description Units Symbol Description Units a linear acceleration m/s 2 a angular acceleration rad/s 2 a in angular acceleration rad/s 2 a out angular acceleration rad/s 2 α D V F temperature coefficient viscous damping factor force / C s/rad F a linear force required during acceleration (F J +F f +F g ) N F f linear force required to overcome friction N F g linear force required to overcome gravity N F J linear force required to overcome load inertia N g gravitational constant (9.8 m/s 2 ) m/s 2 I I LR I O I PK I RMS current locked rotor current no-load current peak current RMS current J inertia Kg-m 2 J B inertia of belt Kg-m 2 J GB inertia of the gear box Kg-m 2 J in reflected inertia at system input Kg-m 2 J out reflected inertia at system output Kg-m 2 J P1 inertia of pully 1 Kg-m 2 J P2 inertia of pully 2 Kg-m 2 J S lead screw inertia Kg-m 2 K E K (f) K (i) K m K T L m N η n n O n PK θ voltage constant K E or K T (final hot ) K E or K T (initial cold ) motor constant torque constant screw lead mass gear ratio efficiency speed no load speed peak speed load orientation (horizontal 0, vertical 90 ) N A A A A A V/rad/s /A or V/(rad/s) /A or V/(rad/s) / w /A m/rev Kg n/a n/a RPM RPM RPM degrees Θ a ambient temperature C Θ f motor temperature (final hot ) C Θ i motor temperature (initial cold ) C Θ m motor temperature C Θ r motor temperature rise C Θ rated rated motor temperature C P P avg P CV P in P loss P max P max(f) P max(i) P out P PK R m R mt R mt(f) R mt(i) R th r S t T T a T a (motor) T C T CF T d T D T f T g T in T LR T out T RMS T RMS (motor) μ V T V bus v v f v i v PK ω ω in ω out ω o ω PK power average power power required at constant velocity power required at system input dissipated power maximum power maximum power (final hot ) maximum power (initial cold ) output power peak power motor regulation motor terminal resistance motor terminal resistance (final hot ) motor terminal resistance (initial cold ) thermal resistance radius linear distance time torque torque required to overcome load inertia torque required at motor shaft during acceleration continuous rated motor torque coulomb friction torque reverse torque required for deceleration drag / preload torque torque required to overcome friction torque required to overcome gravity torque required at system input locked rotor torque torque required at system output RMS torque required over the total duty cycle RMS torque required at the motor shaft coefficient of friction motor terminal voltage DC drive bus voltage linear velocity final linear velocity initial linear velocity peak linear velocity energy angular velocity angular velocity at system input angular velocity at system output no load angular velocity peak angular velocity RPM/ Ω Ω Ω C/ m m s n/a V V m/s m/s m/s m/s J rad/s rad/s rad/s rad/s rad/s 1

Any motion system should be broken down into its individual components. Power Supply Controller / Drive + + Motor Mechanical Transmission Lead Screw / Support Rails (V)(I) atts atts Lost (V)(I) atts atts Lost (V)(I) atts atts Lost (T)(ω) atts atts Lost (T)(ω) atts atts Lost F x S t atts Out All analysis begins with the linear motion profile of the output and the force required to move the load. Motion Profiles 1 /3 1 /3 1 /3 Trapazoidal Motion Profile v pk v a -a t 1 t 2 t 3 t move Total Cycle Time Optimized for minimum power t dwell Time t 1 t 2 t 3 v PK 3S 2t S is the total move distance t is the total move time Area under profile curve represents distance moved 2

Triangular Motion Profiles Optimized for minimum acceleration slope t 1 t 2 v PK 2S t v a v PK -a t 1 t 2 t t dwell move Total Cycle Time Time S is the total move distance t is the total move time Area under profile curve represents distance moved Move Profile Peak Velocity (V PK ) Acceleration (a ) 1 /3 1 /3 1 /3 Triangular Complex Motion Profile v 2 v a 1 v 1 a 2 -a 3 t 1 t 2 t 3 t 4 t 5 t move Total Cycle Time t dwell Time Linear Motion Formulas v f - v i t (1) s 2 (2) v f v i + a t (3) s v i t + 1 at 2 2 (4) 2 as v f 2 - v i 2 (5) a ( v f - v i ) ( t f - t i ) Δv Δt Linear to Rotary Formulas through a Lead Screw a ω n 2 π a rad/s 2 L 2 π v L 60 v L rad/s RPM 3 known quantities are needed to solve for the other 2. Each segment calculated individually. 3

Inertia, Acceleration, Velocity, and Required Torque Lead Screw System J in m out ain a out ω in v out T in F out J in m 2 π L 2 X η 1 + J S Kg-m 2 a in 2 πa L 2 π v ω in L Τ in Τ a +Τ f +Τ g +Τ D Τ a J in a in Τ f cosøm gμl 2 π η Τ g sinøm gl 2 π η Τ D drag/preload refer to manufacturer s data rad/sec 2 rad/sec N-m N-m N-m N-m N-m Belt and Pulley System J in J out 2 N + J P2 + J B X η 1 + J P1 Kg-m 2 a in rad/sec 2 (a out)(n) J in J out ain a out ω in ω out T in T out ω in Τ in (ω out)(n) Τ out 1 X N η rad/sec N-m For a 1st approximation analysis, J P1, J P2, and J B can be disregarded. For higher precision, pulley and belt inertias should be included, as well as the reflected inertia of these components. 4

Gearbox System J in J out ain a out ω in ω out T in T out J in J out 2 N ω in Τ in X η 1 a in (a out)(n) (ω out)(n) Τ out 1 X N η + J GB Kg-m 2 rad/sec 2 rad/sec N-m Drag torque can be significant (T D ) depending on the viscosity of the lubricant. For a first approximation analysis, this can be left out. Gearbox inertia may be found in manufacturer s data sheets, or calculated using gear dimensions and materials / mass. + Τ D Mechanical Power and RMS Torque Linear Power FS t watts Rotary Power Tω watts Τ RMS Τ 1 2 t 1 + Τ 2 2 t 2 + Τ 3 2 t 3...+ Τ n 2 t n t 1 + t 2 + t 3... + t n + t dwell 5

Motor No Load Speed n o 9.5493 x V T (I O x R mt ) K E Assuming I O is very small, it can be ignored for a quick approximation of motor no-load speed. If more accurate results are needed, I O will also need to be adjuisted based on the motor Viscous Damping Factor (D v ). As the no-load speed increases with increased terminal voltage on a given motor, I O will also increase based on the motor s D v value found on the manufacturer s motor data sheet. Τ Breakaway Torque DC Motor Formulas Viscous Damping Torque (D v ) ο Coulomb Friction Motor Locked Rotor Current / Locked Rotor Torque MOTION SYSTEM ω I LR V T R mt Motor Regulation T LR I LR x K T V T x K T R mt Theoretical using motor constants R mt R m 9.5493 x KE x K T Regulation calculated using test data n O R m TLR 6

Motor Power Relationships atts lost due to winding resistance P loss I 2 x R mt Output power at any point on the motor curve P out ω x T Motor maximum output power P max 0.25 x ω o x T LR Motor maximum output power (Theoretical) 2 P max 0.25 x V T R mt Temperature Effects on Motor Constants Change in terminal resistance R mt(f) R mt(i) x [ 1 + α conductor (Θ f Θ i ) ] α conductor 0.0040/ C (copper) 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. Change in torque constant and voltage constant ( K T K E ) K (f) K (i) x [ 1 + α magnet (Θ f Θ i ) ] Magnetic Material Ceramic SmCo AlNiCo NdFeB α magnet (/ C) -0.0020/ C -0.0004/ C -0.0002/ C -0.0012/ C Θ max ( C) 300 C 300 C 540 C 150 C 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. 7

Motor Brush Resistance 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 R mt factor found on manufacturer s data sheet. Brush Material Copper graphite Silver graphite Electrographite Resistance 0.2 to 0.4 Ω 0.2 to 0.4 Ω 0.8 to 1.0 Ω DC Motor Thermal Resistance and Temperature Rise Thermal resistance Temperature Rise C R th Machine Losses Final motor temperature Motor temperature rise* OR Θ m Θ r + Θ a Θ r R th x I 2 x R mt Θ r OR Θ r R th x( R th x I 2 x R mt 1 (R th x I 2 x R mt x α) T C K m)2 *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. 8

Motor Brush Resistance 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 R mt factor found on manufacturer s data sheet. Brush Material Copper graphite Silver graphite Electrographite Resistance 0.2 to 0.4 Ω 0.2 to 0.4 Ω 0.8 to 1.0 Ω DC Motor Thermal Resistance and Temperature Rise Thermal resistance Temperature Rise C R th Machine Losses Final motor temperature Motor temperature rise* OR Θ m Θ r + Θ a Θ r R th x I 2 x R mt Θ r OR Θ r R th x( R th x I 2 x R mt 1 (R th x I 2 x R mt x α) T C K m)2 *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. 9

Motor step angle Stepper Motor Formulas 180 Step Ø (phases)(rotor teeth) SPS (steps/second) to RPM (revolutions/minute) RPM RPM (revolutions/minute) to SPS (steps/second) SPS (SPS) x (Step Ø) 6 (RPM) x 6 (Step Ø) Travel per step m/step L positions/rev 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. Overdriving capability Required Time Off using an L/R drive (constant voltage drive) Time Off (Time On) (Applied Voltage) 2 Time On (Rated Voltage) Required Time Off using a chopper drive (constant current drive) Time Off (Time On) (Applied Current) 2 Time On (Rated Current) 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 on time should not exceed 2-3 minutes. As general rule of thumb, driving a motor at 2.5 to 3x its rated voltage or current may result in wasted energy and erratic behavior due to saturation. Special notes for can-stack motors: more easily saturated due to less active material (steel) more iron losses due to unlaminated steel 10

REFERENCE TABLES Lead Screw Performance Characteristics Increase In Affects Impact Increase In Affects Impact Screw Length Screw Diameter Lead Critical speed Compression load Critical speed Inertia Compression load Stiffness Spring Rate Load capacity Drive torque Angular velocity Load cpacity Positioning accuracy End Mounting Rigidity Load Preload Nut length Critical speed Compression load System stiffness Life Positioning accuracy System stiffness Drag torque Load capacity Stiffness Examples: An INCREASE ( ) in screw length results in a DECREASE ( ) in critical speed. An INCREASE in screw diameter results in an INCREASE in critical speed. SI Unit Systems Quantity Name of Unit Symbol Base Units Length Mass Time Electric current Thermodynamic temperature Luminous intensity Amount of substance Derived Units area volume frequency mass density (density) speed, velocity angular velocity acceleration angular acceleration force pressure (mechanical stress) kinematic viscosity dynamic viscosity work, energy, quantity of heat power entropy specific heat capacity thermal conductivity meter kilogram second ampere kelvin candela mole square meter cubic meter hertz kilogram per cubic meter meter per second radian per second meter per second squared radian per second squared newton pascal square meter per second newton-second per square meter joule watt joule per kelvin joule per kilogram kelvin watt per meter kelvin m Kg s A K cd mol m 2 m 3 Hz Kg/m 3 m/s rad/s m/s 2 rad/s 2 N Pa m 2 /s N-s/m 2 s 1 J J/K J/Kg-K) /(m-k) Kg-m/s 2 N/m 2 N-m J/s or N-m/s 11

Linear Motion Example MOTION SYSTEM APPLICATION EXAMPLE Data Mass Move distance Move time Dwell time Motion profile Screw speed limit Load support Orientation Rotary to linear conversion Ambient temperature 9 Kg 200 mm 1.0 s 0.5 s Control Drive Power Supply: 1/3 1/3 1/3 Trapezoidal 1000 RPM μ 0.01 Vertical TFE lead screw 8 mm diameter 275 mm long 30 C 4 quadrant with encoder 32 VDC, 3.5 Arms, 5.0 A peak v pk v a -a t 1 t 2 t 3 t dwell 0.333 s 0.333 s 0.333 s 0.5 s Time Linear Velocity v PK 3S 2t (3)(0.2m) (2)(1.0 s) 0.6m 2 s 0.3m s Linear Acceleration a v f - v i t 0.3m/s - 0 m/s 0.333 s 0.901m/s 2 1st step is to thoroughly understand system constraints and motion profile requirements. 12

Minimum screw lead to maintain < 1000 RPM shaft speed L min 60v PK n (60)(0.3m/s) 1000 rev/min 0.018m/rev 18mm/rev Refer to Kerk screw chart closest lead in an 8mm screw diameter is 20.32mm/rev 0.02032 m/rev η 0.86 free wheeling nut, TFE coated screw J S 38.8 x 10 7 Kg-m 2 Other critical lead screw considerations Column loading Critical speed Linear to Rotary Conversion Velocity, Acceleration, Inertia, Torque v PK a ω PK a Linear to Rotary Conversion: Velocity m F J T ω PK 2 π v PK L (2π)(0.3m/s) 0.02032 m/rev 92.76 rad/s Linear to Rotary Conversion: Acceleration a 2πa L (2π)(0.901m/s 2 ) 0.02032 m/rev 278.6 rad/s 2 13

Linear to Rotary Conversion: Inertia J in m L 2X 1 2 π η + J S 9Kg 0.02032 m 2 X 1 2 π 0.86 + 38.8x10 7 Kg-m 2 10.95 X 10 5 Kg-m 2 + 38.8x10 7 Kg-m 2 11.34 x 10 5 Kg-m 2 Linear to Rotary Conversion: Torque Τ in Τ a +Τ f +Τ g +Τ D Τ a J in a in (11.34 x 10 5 Kg-m 2 )(278.6 rad/s 2 ) 0.0316 Τ f cosøm gμl 2 π η (cos90)(9 Kg)(9.8m/s2 )(0.01)(0.02032m) 2 π (0.86) 0 Τ g sinøm gl 2 π η (sin90)(9 Kg)(9.8m/s2 )(0.02032m) 2 π (0.86) 1.792 5.404 0.3316 Τ D 0 Since a free-wheeling lead screw nut was used, there is no preload force. If an anti-backlash nut was used, Τ D would be >0 due to preload force. Always reference manufacturer s data for more information. 14

APPLICATION EXAMPLE Rotary Motion Profile ω Τ 1 Τ 2 Τ 3 t 1 t 2 t 3 t dwell Time 0.333 s 0.333 s 0.333 s 0.5 s ω PK 92.76 rad/s a 278.6 rad/s 2 Τ 1 Τ a +Τ f +Τ g +Τ D (0.0316 ) + (0 ) + (0.3316 ) + (0 ) 0.3632 Τ 2 Τ a +Τ f +Τ g +Τ D (0 ) + (0 ) + (0.3316 ) + (0 ) 0.3316 Τ 3 Τ a +Τ f +Τ g +Τ D (0.0316 ) + (0 ) + (0.3316 ) + (0 ) 0.3632 15

APPLICATION EXAMPLE Lead Screw Input Requirements ω PK a Τ 1 Τ 2 Τ 3 t 1 t 2 t 3 t dwell P PK P CV 92.76 rad/s 278.6 rad/s 2 0.3632 0.3316 0.3632 0.333 s 0.333 s 0.333 s 0.5 s 33.69 30.76 P PK (Τ 1 )(ω PK ) (0.3632 )(92.76 rad/s) 33.69 P CV (Τ 2 )(ω PK ) (0.3316 )(92.76 rad/s) 30.76 RMS Torque Requirement @ Lead Screw Input Τ RMS Τ 1 2 t 1 + Τ 2 2 t 2 + Τ 3 2 t 3 t 1 + t 2 + t 3 + t dwell (0.3632 ) 2 (0.333 s)+(0.3316 ) 2 (0.333 s)+(- 0.3632 ) 2 (0.333 s) 0.333 s + 0.333 s + 0.333 s + 0.5 s 0.0439 + 0.0366 + 0.0439 1.5 0.0829 Τ RMS 0.2879 16

APPLICATION EXAMPLE Load Parameters at the Lead Screw Shaft ω PK a Τ 1 Τ 2 Τ 3 Τ RMS P PK P CV 92.76 rad/s 278.6 rad/s 2 0.3632 0.3316 0.3632 0.2879 33.69 30.76 Motor Selection Motor Transmission Τ RMS P PK ω PK Lead Screw / Support Rails 0.2879 33.69 92.76 rad/s A 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 1.8:1 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 M Other Motor Considerations Type: stepper, brush, BLDC Input speed (high input speed will result in high noise levels) Total motor footprint Ambient temperature Environmental conditions Load-to-rotor inertia Encoder ready needed DC040B-6 Reference Voltage Continuous Torque Rated Current Rated Power Torque Constant Voltage Constant Terminal Resistance Max. inding Temperature Rotor Inertia 24.0 V 0.0812 2.36 A 37 0.042 /A 0.042 V/rad/s 1.85 Ω 155 C 8.47 x 10 6 Kg-m 2 17

APPLICATION EXAMPLE Gearbox / Transmission Selection hen 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 Gearbox Maximum output load 2.47 T RMS(@ Lead Screw) Reduction Efficiency T RMS(@ Motor) 0.2879 4:1 0.90 0.0798 0.2879 5:1 0.90 0.0640 0.2879 6:1 0.90 0.0533 A 5:1 gearbox would allow about a 27% safety margin between what the system requires and the continuous torque output of the motor. System Summary 5:1 η 0.90 M Τ 1 Τ 2 Τ 3 Τ RMS ω PK J P PK 0.0807 0.0737 0.0807 0.0640 463.8 rad/s 5.04 x 10 6 Kg-m 2 37.43 Τ 1 Τ 2 Τ 3 Τ RMS ω PK J P PK 0.3632 0.3316 0.3632 0.2879 92.76 rad/s 11.34 x 10 5 Kg-m 2 33.69 18

APPLICATION EXAMPLE Motion Profile at the Motor ω Τ 2 Τ1 Τ ω PK 463.8 rad/s 3 t 1 t 2 t 3 t dwell Time 0.333 s 0.333 s 0.333 s 0.5 s Drive and Power Supply Requirements Peak Current Required I PK T PK K T + I O RMS Current Required I T RMS RMS + I K O T Minimum bus Voltage Required n PK Τ 1 0.0807 0.042 /A + 0.180 A 2.10 A 4429 RPM Τ PK 0.0807 0.0640 0.042 /A + 0.180 A 1.70 A V bus I PK R mt + ω PK K E (2.10A)(1.85Ω) + (463.8 rad/s)(0.042 V/rad/s) 3.89 V + 19.48 V 23.37 V Performance using 24V reference voltage 7000 6000 5000 RPM 4000 3000 2000 1000 24 VDC T RMS T PK 0.100 0.200 0.300 0.400 0.500 0.600 0.700 Performance using a 30V bus voltage 7000 6000 5000 RPM 4000 3000 2000 1000 30 VDC T RMS T PK 0.100 0.200 0.300 0.400 0.500 0.600 0.700 A 30 VDC bus will meet the application requirements as wells supply a margin of safety. 19

APPLICATION EXAMPLE ill the motor meet the temperature rise caused by the application? Ambient temperature... Rated motor temperature... Temperature rise... Motor temperature... Θ a Θ rated Θ r Θ m 30 C 155 C 76.38 C 106.38 C Θ r 2 R th x I RMS x R mt 2 1 (R th x I RMS x R mt x 0.00392/ C) 11 C/ω x 1.70A 2 x 1.85Ω 1 (11 C/ω x 1.70A 2 x 1.85Ω x 0.00392/ C) 58.81 1 (0.23) 58.81 76.38 C 0.77 Θ m Θ r + Θ a 76.38 C + 30 C 106.38 C NO... ASK us the tough motion control questions ONLINE! ASK THE EXPERTS www..com HaydonKerkPittman 203 756 7441 1500 Meriden Road aterbury, CT USA 06705 267 933 2105 534 Godshall Drive Harleysville, PA USA 19438 All rights reserved. AMETEK Advanced Motion Solutions. No part of this document or technical information can be used, reproduced or altered in any form without approval or proper authorization from AMETEK, Inc and global affiliates.