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Formulas for Motorized Linear Motion Systems

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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 1 www.HaydonKerkPittman.com Haydon Kerk Motion Solutions: 203 756 7441/Pittman Motors: 267 933 2105 MOTION SYSTEM 3 Motion System FormulasFORMULAS 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 www.HaydonKerkPittman.com 2 Haydon Kerk Motion Solutions: 203 756 7441/Pittman Motors: 267 933 2105 MOTION SYSTEM 4 Motion System FormulasFORMULAS 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 v2 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 = t – at 2 vi + 2 πv ω = = rad/s 2 2 L (4) 2 as = vf - vi ( v - vi) 60 v f = Δv = = RPM (5) a = n ( tf - ti ) Δt L 3 known quantities are needed to solve for the other 2. Each segment calculated individually. 3 www.HaydonKerkPittman.com Haydon Kerk Motion Solutions: 203 756 7441/Pittman Motors: 267 933 2105 MOTION SYSTEM 5 Motion System FormulasFORMULAS 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. www.HaydonKerkPittman.com 4 Haydon Kerk Motion Solutions: 203 756 7441/Pittman Motors: 267 933 2105 MOTION SYSTEM 6 Motion System FormulasFORMULAS 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 5 www.HaydonKerkPittman.com Haydon Kerk Motion Solutions: 203 756 7441/Pittman Motors: 267 933 2105 MOTION SYSTEM 7 Motion System FormulasFORMULAS 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 nO Rm = TLR www.HaydonKerkPittman.com 6 Haydon Kerk Motion Solutions: 203 756 7441/Pittman Motors: 267 933 2105 MOTION SYSTEM 8 Motion System FormulasFORMULAS 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.
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