327 10. Driving Torque 10-1. Friction and Efficiency When the friction coefficient is , and lead angle is , ball screw's efficiency is indicated by the following formulas. 10-2. Load Torque The load torque (constant velocity torque) required for the drive power source (motor, etc.) selection is as follows. (1) Forward Action Torque required when converting rotational force into axial force (2) Reverse Action External axial load when converting axial force into rotational (3) Friction Torque Caused by Preloading This is a torque generated by preloading. As external loads increase, the preload of the nut is released and therefore the friction torque by preloading also decreases. This selection provides a guide for selecting ball screw frictional properties and the driving motor. When rotational force is converted into axial force (Forward Action) 1−μ tan = 1+μ/tan When axial force is converted into rotational force (Reverse Action) 1−μ/tan ’= 1+μ tan PL T= (N · cm) 2π Where: T: Load Torque (N·cm) P: External Axial Load (N) L: Ball Screw Lead (cm) : Ball Screw Efficiency (0.9) 2π T P= (N) 'L Where: P: External Axial Load (N) T: Load Torque (N·cm) L: Ball Screw Lead (cm) ': Ball Screw Efficiency (0.9) Under No load Where: PL: Preload (N) L: Ball Screw Lead (cm) K: Coefficient of Internal Friction : Lead Angle D: Thread Outer Diameter PLL TP= K (N · cm) 2π K=0.05(tan )− L ≈tan−1 πD F W Motors Gear J2 Work Piece Pinion J1 Ball Screw J3 Table 2Z 1Z 11. Selecting the Driving Motors When selecting a driving motor, it is necessary to satisfy the following conditions: 1.Ensure a marginal force sufficient to counter the load torque exerted on the motor's output thread. 2.Enable starting, stopping at prescribed pulse speeds, sufficiently powered to counter the moment of inertia exerted on the motor's output thread. 3.Obtain the prescribed acceleration and deceleration constants, sufficient to counter the moment of inertia exerted on the motor's output thread. (1) Constant Speed Torque Exerted on the Motor Output Thread This is the amount of torque required to drive the output thread against the applied external load, at a constant speed. Where: P3PL T1: Driving Torque at Constant Speed (N·cm) P: External Axial Load (N) P=F+μMg F: Thrust Reaction Produced in Cutting Force (N) M: Masses of Table and Work Piece (kg) μ: Coefficient of Friction on Sliding Surfaces g: Gravitational Acceleration (9.8m/s2) L: Ball Screw Lead (cm) : Mechanical Efficiency of Ball Screw or Gear TP: Friction Torque Caused by Preloading (N·cm) Referto Formula 10-2-(3) PL: Preload (N) Z1: Number of Pinion's Teeth Z2: No. of Gear's Teeth PL (3PL−P) Z1 T1= +TP (N · cm) 2π 3PL Z2 (2) Acceleration Torque Exerted on the Motor Output Thread This is the amount of torque required to drive the output shaft against the external load during acceleration. 2πN T2= JM = JM ×10−3 (N · cm) 60t Where: T2: Driving Torque in Acceleration (N·cm) : Motor Thread Angular Acceleration (rad/s2) N: Motor Thread Rotational Speed (rpm) t: Acceleration Time (s) JM: Moment of Inertia Exerted on the Motor (kg·cm2) J1: Moment of Inertia Exerted on Pinion (kg·cm2) J2: Moment of Inertia Exerted on Gear (kg·cm2) J3: Moment of Inertia Exerted on Ball Screw (kg·cm2) J4: Moment of Inertia Exerted on Motor's Rotor (kg·cm2) J5: Moment of Inertia of Moving Body (kg·cm2) J6: Moment of Inertia of Coupling (kg·cm2) M: Masses of Table and Work Piece (kg) L: Ball Screw Lead (cm) Moment of inertia exerted on cylinders as screws and cylinders such as Gears (Calculation of J1~J4, J6) Z1 2 JM= J1+J4 + (J2+J3+J5+J6) (kg · cm2) Z2 π J= D4ℓ (kg · cm2) 32 Where: D: Cylinder Outer Diameter (cm) ℓ: Cylinder Length (cm) : Material Specific Gravity =7.8×10−3 (kg/cm3) L J5=M 2 (kg · cm2) 2π (3) Total Torque Exerted on the Motor Output Thread Overall torque can be obtained by adding results from formulas (1) and (2). Once you have temporarily found the type of motor you need, check 1. effective torque, 2. acceleration constant and 3. motor overload properties and heat tolerance during repeated starting, stopping. It is necessary to ensure a sufficient margin for these parameters. PL (3PL−P) Z1 2π N TM= T1+T2= +TP +JM ×10−3(N·cm) 2π 3PL Z2 60t Where: TM: Total Torque Exerted on the Motor Output Thread (N·cm) T1: Driving Torque at Constant Speed (N·cm) T2: Driving Torque at In Acceleration (N·cm) Ball Screws Selection of Ball Screws 4
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