Formulas for Motorized Linear Motion Systems

FORMULAS FOR MOTORIZED LINEAR MOTION SYSTEMS

Haydon Kerk Motion Solutions : 203 756 7441 Pittman Motors : 267 933 2105 www.HaydonKerkPittman.com 2 Symbols andSYMBOLS Units AND UNITS

Symbol Description Units Symbol Description Units

a linear acceleration m/s2 P power W 2 a angular acceleration rad/s Pavg average power W 2 ain angular acceleration rad/s PCV power required at constant velocity W angular acceleration 2 power required at system input aout rad/s Pin W

α temperature coefficient /°C Ploss dissipated power W

DV viscous damping factor Nm s/rad Pmax maximum power W

F force N Pmax(f) maximum power (final“hot”) W

Fa linear force required during acceleration (FJ +Ff +Fg) N Pmax(i) maximum power (initial “cold”) W

Ff linear force required to overcome friction N Pout output power W

Fg linear force required to overcome gravity N PPK peak power W

FJ linear force required to overcome load inertia N Rm motor regulation RPM/Nm 2 2 g gravitational constant (9.8 m/s ) m/s Rmt motor terminal resistance Ω I current A Rmt(f) motor terminal resistance (final“hot”) Ω

ILR locked rotor current A Rmt(i) motor terminal resistance (initial “cold”) Ω

IO no-load current A Rth thermal resistance °C/W

IPK peak current A r radius m

IRMS RMS current A S linear distance m J inertia Kg-m2 t time s inertia of belt Kg-m2 torque Nm JB T inertia of the gear box Kg-m2 T torque required to overcome load inertia Nm JGB a reflected inertia at system input Kg-m2 T torque required at motor shaft during acceleration Jin a (motor) Nm 2 continuous rated motor torque Jout reflected inertia at system output Kg-m TC Nm inertia of pully 1 Kg-m2 coulomb friction torque Nm JP1 TCF inertia of pully 2 Kg-m2 reverse torque required for deceleration JP2 Td Nm lead screw inertia Kg-m2 drag / preload torque Nm JS TD voltage constant torque required to overcome friction KE V/rad/s Tf Nm T torque required to overcome gravity Nm K(f) KE or KT (final“hot”) Nm/A or V/(rad/s) g torque required at system input K(i) KE or KT (initial “cold”) Nm/A or V/(rad/s) Tin Nm motor constant locked rotor torque Km Nm/ w TLR Nm

KT torque constant Nm/A Tout torque required at system output Nm

L screw lead m/rev TRMS RMS torque required over the total duty cycle Nm

m mass Kg TRMS(motor) RMS torque required at the motor shaft Nm N gear ratio n/a μ coefficient of friction n/a

η efficiency n/a VT motor terminal voltage V

n speed RPM Vbus DC drive bus voltage V no load speed nO RPM v linear velocity m/s peak speed final linear velocity m/s nPK RPM vf θ load orientation (horizontal = 0°, vertical = 90°) degrees vi initial linear velocity m/s °C peak linear velocity m/s Θa ambient temperature vPK Θf motor temperature (final“hot”) °C W energy J angular velocity Θi motor temperature (initial “cold”) °C ω rad/s Θm motor temperature °C ωin angular velocity at system input rad/s Θr motor temperature rise °C ωout angular velocity at system output rad/s Θrated rated motor temperature °C ωo no load angular velocity rad/s ωPK peak angular velocity rad/s

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Any motion system should be broken down into its individual components.

Power Controller / Mechanical Lead Screw / Supply Drive Motor Transmission Support Rails + + F x S Watts t = Out (V)(I) I I Watts (V)( ) Watts (V)( ) Watts (T)(ω) Watts (T)(ω) Watts Watts Lost Watts Lost Watts Lost Watts Lost Watts Lost 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

vpk - v a a

t 1 t 2 t 3 t dwell Time t move Total Cycle Time

Optimized for minimum power

t 1 = t 2= t 3 3S S is the total move distance vPK = t is the total move time 2t Area under profile curve represents distance moved

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Triangular Motion Profiles vPK

a -a Optimized for minimum acceleration slope v

t 1 = t 2 t 1 t 2 Time t dwell t 2S move = Total Cycle Time vPK t is the total move distance S Move Profile Peak Velocity (VPK) Acceleration (a ) t is the total move time 1/3 1/3 1/3 Area under profile curve represents distance moved Triangular

Complex Motion Profile

2 a - v1 a 3 v a 1

t 1 t 2 t 3 t 4 t 5 t dwell Time t move Total Cycle Time

Linear Motion Formulas Linear to Rotary Formulas through a Lead Screw vf - vi (1) s = –––– t 2 2 π a 2 a = = rad/s (2) vf = vi + a t L

1 2 (3) s = vi t + – at 2 π v 2 ω = = rad/s 2 2 L (4) 2 as = vf - vi

( - ) 60 v vf vi Δv = = RPM (5) a = = n ( tf - ti ) Δt L

3 known quantities are needed to solve for the other 2. Each segment calculated individually.

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Inertia, Acceleration, Velocity, and Required Torque

Lead Screw System 2 L 1 J 2 Jin = m X η + S Kg-m 2 π 2 2 πa rad/sec in = a L 2 π v rad/sec ωin = Jin mout L

ain a out Τin= Τa+Τf +Τg+ΤD N-m

N-m ωin vout Τa= Jin ain Tin Fout cosØmgμL Τf = N-m 2 π η sinØmgL Τg = N-m 2 π η ΤD = drag/preload N-m refer to manufacturer’s data

Belt and Pulley System Jout 1 2 Jin = X + JP1+ JP2+ JB Kg-m N 2 η 2 = rad/sec ain (aout)(N)

ωin = ωout N rad/sec Jin Jout ( )( ) Τout 1 ain aout Τ = X N-m in N η ωin ωout For a 1st approximation analysis, JP1, JP2, and JB can be disregarded. For higher precision, pulley and Tin Tout belt inertias should be included, as well as the reflected inertia of these components.

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Gearbox System Jout 1 2 J = X J Kg-m in 2 η + GB N

2 = rad/sec ain (aout)(N)

ωin = ωout N rad/sec Jin Jout ( )( ) Τout 1 ain aout Τ = X ΤD N-m in N η + ωin ωout Drag torque can be significant (TD) depending on the Tin Tout 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.

Mechanical Power and RMS Torque

FS Linear Power = t = watts

Rotary Power = Tω = watts

2 2 2 2 ΤRMS = Τ1 t 1 + Τ2 t2 + Τ3 t 3 ...+ Τn t n

t 1 + t 2 + t 3 ... + t n + tdwell

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DC Motor Formulas

Motor No Load Speed

VT – (IO x Rmt) no = 9.5493 x KE 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 motor’s Dv value found on the manufacturer’s motor data sheet. Τ Viscous Damping – – – Torque (Dv)

Breakaway Torque – – – – – – – – – – – – – Coulomb Friction – – – ο ω Motor Locked Rotor Current / Locked Rotor Torque

VT ILR = Rmt

TLR = ILR x KT

VT = x KT Rmt Motor Regulation – Theoretical using motor constants Rmt Rm = 9.5493 x KE x KT – Regulation calculated using test data

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Motor Power Relationships

– Watts lost due to winding resistance 2 Ploss = I x Rmt – Output power at any point on the motor curve

Pout = ω x T – Motor maximum output power

Pmax = 0.25 x ωo x TLR – Motor maximum output power (Theoretical) 2 VT Pmax = 0.25 x Rmt Temperature Effects on Motor Constants

– Change in terminal resistance

Rmt(f) = Rmt(i) x [ 1 + αconductor (Θf – Θi) ]

αconductor = 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 ( KT = KE )

K(f) = K(i) x [ 1 + αmagnet (Θf – Θi) ]

Magnetic Material αmagnet (/°C) Θmax (°C) Ceramic -0.0020/°C 300°C SmCo -0.0004/°C 300°C AlNiCo -0.0002/°C 540°C NdFeB -0.0012/°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.

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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 Rmt factor found on manufacturer’s data sheet.

Brush Material Resistance

Copper graphite 0.2 to 0.4 Ω Silver graphite 0.2 to 0.4 Ω Electrographite 0.8 to 1.0 Ω

DC Motor Thermal Resistance and Temperature Rise

– Thermal resistance Temperature Rise °C R = th Machine Losses W

– Final motor temperature

Θm = Θr + Θa

– Motor temperature rise* 2 Θr = Rth x I x Rmt – OR – 2 Rth x I x Rmt Θr = 2 1 – (Rth x I x Rmt x α) – OR – 2 TC Θr = Rth x (Km) *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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Motor Brush Resistance

Brush Material Resistance

Copper graphite 0.2 to 0.4 Ω Silver graphite 0.2 to 0.4 Ω Electrographite 0.8 to 1.0 Ω

DC Motor Thermal Resistance and Temperature Rise

– Thermal resistance Temperature Rise °C R = th Machine Losses W

– Final motor temperature

Θm = Θr + Θa

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Stepper Motor Formulas

– Motor step angle 180 Step Ø = (phases)(rotor teeth) – SPS (steps/second) to RPM (revolutions/minute) (SPS) x (Step Ø) RPM = 6 – RPM (revolutions/minute) to SPS (steps/second) (RPM) x 6 SPS = (Step Ø) – Travel per step L m/step = 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) (Applied Voltage)2 Time Off = (Time On) – Time On (Rated Voltage) Required “Time Off” using a chopper drive (constant current drive) (Applied Current)2 Time Off = (Time On) – 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

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Lead Screw Performance Characteristics

Increase In Affects Impact Increase In Affects Impact Screw Critical speed Critical speed End Mounting Length Compression load Rigidity Compression load Critical speed System stiffness Inertia Load Life Screw Compression load Positioning accuracy Diameter Stiffness Preload System stiffness Spring Rate Drag torque Load capacity Load capacity Nut length Drive torque Stiffness Angular velocity Examples: Lead Load cpacity An INCREASE ( ) in screw length results in a DECREASE ( ) in critical speed. Positioning accuracy An INCREASE in screw diameter results in an INCREASE in critical speed.

SI Unit Systems Quantity Name of Unit Symbol Base Units Length meter m Mass kilogram Kg Time second s Electric current ampere A Thermodynamic temperature kelvin K Luminous intensity candela cd Amount of substance mole mol Derived Units area square meter m2 volume cubic meter m3 frequency hertz Hz s–1 mass density (density) kilogram per cubic meter Kg/m3 speed, velocity meter per second m/s angular velocity radian per second rad/s acceleration meter per second squared m/s2 angular acceleration radian per second squared rad/s2 force newton N Kg-m/s2 pressure (mechanical stress) pascal Pa N/m2 kinematic viscosity square meter per second m2/s dynamic viscosity newton-second per square meter N-s/m2 work, energy, quantity of heat joule J N-m power watt W J/s or N-m/s entropy joule per kelvin J/K specific heat capacity joule per kilogram kelvin J/Kg-K) thermal conductivity watt per meter kelvin W/(m-K)

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Linear Motion Example Data Mass 9 Kg Control 4 quadrant with encoder Move distance 200 mm Drive Power Supply: 32 VDC, 3.5 Arms, 5.0 A peak Move time 1.0 s Dwell time 0.5 s Motion profile 1/3 1/3 1/3 Trapezoidal Screw speed limit 1000 RPM Load support μ = 0.01 Orientation Vertical Rotary to linear conversion TFE lead screw 8 mm diameter 275 mm long Ambient temperature 30°C

vpk

-a v a

t 1 t 2 t 3 t dwell Time 0.333 s 0.333 s 0.333 s 0.5 s

Linear Velocity

3S (3)(0.2m) 0.6m 0.3m = = = = vPK 2t (2)(1.0 s) 2 s s

Linear Acceleration

vf - vi 0.3m/s - 0 m/s 2 a = = = 0.901m/s t 0.333 s

1st step is to thoroughly understand system constraints and motion profile requirements.

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Minimum screw lead to maintain < 1000 RPM shaft speed

60vPK (60)(0.3m/s) L = = = 0.018m/rev = 18mm/rev min n 1000 rev/min

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 – 7 2 JS = 38.8 x 10 Kg-m Other critical lead screw considerations • Column loading • Critical speed

Linear to Rotary Conversion – Velocity, Acceleration, Inertia, Torque

vPK ωPK m J a a F T Linear to Rotary Conversion: Velocity

2 π vPK (2π)(0.3m/s) ωPK = = = 92.76 rad/s L 0.02032 m/rev

Linear to Rotary Conversion: Acceleration

2 2πa (2π)(0.901m/s ) 2 = = 278.6 rad/s a = L 0.02032 m/rev

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Linear to Rotary Conversion: Inertia

2 2 L 1 0.02032 m 1 – 7 2 J 38.8x10 Kg-m Jin= m X η + S= 9Kg X + 2 π 2 π 0.86 – 5 2 – 7 2 = 10.95 X 10 Kg-m + 38.8x10 Kg-m – 5 2 = 11.34 x 10 Kg-m

Linear to Rotary Conversion: Torque

Τin= Τa+Τf +Τg+ΤD

– 5 2 2 Τa = Jin ain = (11.34 x 10 Kg-m )(278.6 rad/s ) = 0.0316 Nm

2 cosØmgμL (cos90)(9 Kg)(9.8m/s )(0.01)(0.02032m) Τf = =

2 π η 2 π (0.86) = 0 Nm

2 sinØmgL (sin90)(9 Kg)(9.8m/s )(0.02032m) Τg = = 2 η π 2 π (0.86) 1.792 Nm = 0.3316 Nm = 5.404

Since a free-wheeling lead screw nut was used, there is no preload force. ΤD = 0 Nm If an anti-backlash nut was used, ΤD would be >0 due to preload force. Always reference manufacturer’s data for more information.

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Rotary Motion Profile

Τ2 ω Τ1 Τ3

t 1 t 2 t 3 t dwell Time 0.333 s 0.333 s 0.333 s 0.5 s

2 ωPK = 92.76 rad/s a = 278.6 rad/s

Τ1= Τa+Τf +Τg+ΤD = (0.0316 Nm) + (0 Nm) + (0.3316 Nm) + (0 Nm) = 0.3632 Nm

Τ2= Τa+Τf +Τg+ΤD = (0 Nm) + (0 Nm) + (0.3316 Nm) + (0 Nm) = 0.3316 Nm

Τ3= Τa+Τf +Τg+ΤD = (0.0316 Nm) + (0 Nm) + (0.3316 Nm) + (0 Nm) = 0.3632 Nm

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Lead Screw Input Requirements

ωPK = 92.76 rad/s a = 278.6 rad/s2 = 0.3632 Nm P = Τ1 PK (Τ1)(ωPK) Τ = 0.3316 Nm 2 = (0.3632 Nm)(92.76 rad/s) Τ3 = – 0.3632 Nm = 33.69 W t1 = 0.333 s t2 = 0.333 s PCV = (Τ2)( ) t3 = 0.333 s ωPK tdwell = 0.5 s = (0.3316 Nm)(92.76 rad/s) P = 33.69 W PK = 30.76 W PCV = 30.76 W

RMS Torque Requirement @ Lead Screw Input

2 2 2 ΤRMS = Τ1 t 1 + Τ2 t2 + Τ3 t 3

t 1 + t 2 + t 3 + tdwell

= 2 2 2 (0.3632 Nm) (0.333 s)+(0.3316 Nm) (0.333 s)+(- 0.3632 Nm) (0.333 s)

0.333 s + 0.333 s + 0.333 s + 0.5 s

= 0.0439 + 0.0366 + 0.0439 = 0.0829 1.5

ΤRMS = 0.2879 Nm

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Load Parameters at the Lead Screw Shaft ω PK = 92.76 rad/s Lead Screw / 2 a = 278.6 rad/s Motor Transmission Support Rails Τ1 = 0.3632 Nm M Τ2 = 0.3316 Nm Τ = – 0.3632 Nm = 0.2879 Nm 3 ΤRMS ΤRMS = 0.2879 Nm = 33.69 W PPK PPK = 33.69 W ωPK = 92.76 rad/s PCV = 30.76 W 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

Motor Selection 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

DC040B-6 Reference Voltage 24.0 V Continuous Torque 0.0812 Nm Rated Current 2.36 A Rated Power 37 W Torque Constant 0.042 Nm/A Voltage Constant 0.042 V/rad/s Terminal Resistance 1.85 Ω Max. Winding Temperature 155°C Rotor Inertia 8.47 x 10–6 Kg-m2

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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 Gearbox Maximum output load 2.47 Nm

TRMS(@ Lead Screw) Reduction Efficiency TRMS(@ Motor)

0.2879 Nm 4:1 0.90 0.0798 Nm 0.2879 Nm 5:1 0.90 0.0640 Nm 0.2879 Nm 6:1 0.90 0.0533 Nm

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

Τ1 = 0.0807 Nm Τ1 = 0.3632 Nm Τ2 = 0.0737 Nm Τ2 = 0.3316 Nm Τ3 = – 0.0807 Nm Τ3 = – 0.3632 Nm ΤRMS= 0.0640 Nm ΤRMS= 0.2879 Nm

ωPK = 463.8 rad/s ωPK = 92.76 rad/s J = 5.04 x 10 – 6 Kg-m2 J = 11.34 x 10 – 5 Kg-m2

PPK = 37.43 W PPK = 33.69 W

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Motion Profile at the Motor

Τ2 ω ωPK = 463.8 rad/s Τ1 Τ3 nPK = 4429 RPM

Τ1 = ΤPK = 0.0807 Nm t t t t 1 2 3 dwell Time 0.333 s 0.333 s 0.333 s 0.5 s Drive and Power Supply Requirements Peak Current Required

TPK 0.0807 Nm I PK = + IO = + 0.180 A = 2.10 A KT 0.042 Nm/A RMS Current Required

TRMS 0.0640 Nm I RMS = + IO = + 0.180 A = 1.70 A KT 0.042 Nm/A Minimum bus Voltage Required

Vbus= IPKRmt+ ωPKKE = (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 TRMS

TPK 5000 24 RPM 4000 VDC 3000 2000 1000

0.100 Nm 0.200 Nm 0.300 Nm 0.400 Nm 0.500 Nm 0.600 Nm 0.700 Nm Performance using a 30V bus voltage 7000 6000 TRMS 30 T 5000 VDC PK RPM 4000 3000 2000 1000

0.100 Nm 0.200 Nm 0.300 Nm 0.400 Nm 0.500 Nm 0.600 Nm 0.700 Nm A 30 VDC bus will meet the application requirements as wells supply a margin of safety.

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Performance using 24V reference voltage 7000

6000 TRMS

TPK 5000 24 RPM 4000 VDC 3000 2000 1000

0.100 Nm 0.200 Nm 0.300 Nm 0.400 Nm 0.500 Nm 0.600 Nm 0.700 Nm Performance using a 30V bus voltage 7000 6000 TRMS 30 T 5000 VDC PK RPM 4000 3000 2000 1000 MOTION SYSTEM 0.100 Nm 0.200 Nm 0.300 Nm 0.400APPLICATION Nm 0.500 Nm 0.600 Nm 0.700EXAMPLE Nm A 30 VDC bus will meet the application requirements as wells supply a margin of safety. Will the motor meet the temperature rise caused by the application?

Ambient temperature ...... Θa = 30°C Rated motor temperature ... Θrated = 155°C Temperature rise ...... Θr = 76.38°C Motor temperature ...... Θm = 106.38°C 2 Rth x IRMS x Rmt = Θr 2 1 – (Rth x IRMS x Rmt x 0.00392/°C)

2 11°C/ω x 1.70A x 1.85Ω = 2 1 – (11°C/ω x 1.70A x 1.85Ω x 0.00392/°C) 58.81 58.81 = = = 76.38°C 1 – (0.23) 0.77

Θm = Θr + Θa = 76.38°C + 30°C = 106.38°C

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