Servo Motor Selection Flow Chart START Selection Has the machine Been Selected? NO Explanation References • Determine the size, mass, coefficient of friction, and external forces of all the moving part of the Servo Motor the rotation of which affects. --- YES Has the Operating Pattern Been Selected? YES Calculating the Load Inertia For Motor Shaft Conversion Value Calculating the Added Load Torque For Motor Shaft Conversion Value Select a motor temporarily Calculate Acceleration/ Deceleration Torque Confirm Maximum Momentary Torque and Calculate Effective Torque 2 1 NO • Determine the operating pattern (relationship between time and speed) of each part that must be controlled. • Convert the operating pattern of each controlled element into the motor shaft operating pattern. • Operation Pattern Formula • The elements of the machine can be separated so that inertia can be calculated for each part that moves as the Servo Motor rotates. • Calculate the inertia applied to each element to calculate the total load inertia of the motor shaft conversion value. • Inertia Formulas • Calculation of Friction Torque Calculates the frictional force for each element, where necessary, and converts it to friction torque for a motor shaft. • Calculation of External Torque Calculates the external force for each element, where necessary, and converts it to external torque of a motor shaft. • Calculates the total load torque for the motor shaft conversion value. • Load Torque Formulas • Select a motor temporarily based upon the motor shaft converted load inertia, friction torque, external torque and r.p.m of a motor. --- • Calculate the Acceleration/Deceleration Torque from the Load Inertia or Operating Pattern. • Acceleration/Deceleration Torque Formulas • Calculate the necessary torque for each part of the Operating Pattern from the Friction Torque, External Torque and Acceleration/ Deceleration Torque. • Confirm that the maximum value for the Torque for each operating part (Maximum Momentary Torque) is less than the Maximum Momentary Torque of the motor. • Calculate the Effective Torque from the Torque for each Operating part, and confirm that it is less than the Rated Torque for the motor. • Calculation of Maximum Momentary Torque, Effective Torque 3 1 2 Explanation Calculate Regenerative Energy NO Is the Resolution OK? References • Calculate Regenerative Energy from the Torque of all the moving parts. • Please see the user manual of each product for the details on calculation of the regenerative energy. • Check if the the number of encoder pulses meets the system specified resolution. • Accuracy of Positioning • Check if the calculation meets the specifications of the temporarily selected motor. If not, change the temporarily selected motor and re-calculate it. • The following table YES NO Are the Check Items on Characteristics All OK? YES Specialized Check Items Check Items Load Inertia Load Inertia ≤ Motor Rotor Inertia x Applicable Inertia Ratio Effective Torque Effective Torque < Motor Rated Torque • Please allow a margin of about 20%. * Maximum Momentary Torque Maximum Momentary Torque < Motor Maximum Momentary Torque • Please allow a margin of about 20%. * • For the motor Maximum Momentary Torque, use the value that is combined with a driver and the one of the motor itself. Maximum Rotation Speed Maximum Rotation Speed ≤ Rated Rotation Speed of a motor • Try to get as close to the motor's rated rotations as possible. It will increase the operating efficiency of a motor. • For the formula, please see "Straight-line Speed and Motor Rotation Speed" on Page 11. Regenerative Energy Regenerative Energy ≤ Regenerative Energy Absorption of a motor • When the Regenerative Energy is large, connect a Regenerative Energy Absorption Resistance to increase the Absorption capacity of the driver. Encoder Resolution Ensure that the Encoder Resolution meets the system specifications. Characteristics of a Positioner Check if the Pulse Frequency does not exceed the Maximum Response Frequency or Maximum Command Frequency of a Positioner. Operating Conditions Ensure that values of the ambient operating temperature/ humidity, operating atmosphere, shock and vibrations meet the product specifications. END Selection * When handling vertical loads and a load affected by the external torque, allow for about 30% of capacity. 4 Formulas ■Formulas for Operating Patterns speed v0 Maximum Speed v0 = X0 tA X0: Distance Moved in t0 Time (mm) v0: Maximum Speed (mm/s) Triangular Acceleration/Deceleration Time tA = X0 v0 t0: Positioning Time (s) tA: Acceleration/ Deceleration Time (s) tA time tA Travel Distance X0 = v0·tA Maximum Speed v0 = t0 X0 speed X0 t0 – tA Acceleration/Deceleration Time tA = t0 – X0 v0 v0 Total Travel Time t0 = tA + X0 v0 Constant-velocity travel time tB = t0 – 2 · tA = 2 2 · X0 – t0 = X0 – tA Trapezoid tA tB tA v0 v0 time Total Travel Distance X0 = v0 (t0 – tA) t0 XA XB Acceleration/Deceleration Travel Distance XA = v0 ·tA = v0 ·t0 – X0 2 XA X0 2 Constant-velocity travel distance XB = v0 ·tB = 2·X0 – v0 ·t0 speed Ascending Time tA = v0 – v1 α v0 Ascending Time (tA) including distance moved v1 Speed and Slope When Ascending vg time tg tA XA = 1 α·tA2 + v1 ·tA 2 XA = 1 (v0 – v1) + v1 ·tA α 2 2 Speed after Ascending v0 = v1 + α·tA Speed Gradient vg tg 5 Conditions for Trapezoidal Operating Pattern speed v0 X0 < t02·α 4 X0: Positioning Distance (mm) Maximum Speed Speed and Slope Trapezoid pattern v0 = tA tA time t0·α 4X0 ) (1– 1– t0·α 2 tA = tA: Acceleration/Deceleration Time (s) v0: Maximum Speed (mm/s) α: Speed Gradient Ascending Time t0 speed t0: Positioning Time (s) t 4X0 v0 ) = 0 (1 – 1 – 2 t0 ·α α Conditions for Triangular Operating Pattern v0 X0 ≥ t02 · α 4 Maximum Speed Speed and Slope Triangular Pattern v0 = tA time tA t0 v [mm/s] Ascending Time tA = X0 α·X0 X0 α Linear Movement Perform the following unitary conversions X [mm] Rotating Part Linear Movement Rotating Movement X: Distance (mm) θ: Angle (rad) v: Speed (mm/s) ω: Angular Velocity (rad/s) θ [rad] ω= 2π·N 60 N: Rotating Speed (r/min) ω [rad/s] N [r/min] 6 ■Inertia Formulas D2: Cylinder Inner Diameter (mm) D1: Cylinder Outer Diameter (mm) 2 2 JW = M (D1 + D2 ) × 10– 6 (kg·m2) 8 Cylindrical Inertia M: Cylinder Mass (kg) JW: Cylinder Inertia (kg·m2) M: Cylinder Mass (kg) M Eccentric Disc Inertia (Cylinder which rotates off the center axis) C JC: Inertia around the center axis of Cylinder JW: Inertia (kg·m2) JW = JC + M·re2 × 10–6 (kg·m2) re: Rotational Radius (mm) Center of rotation M: Square Cylinder Mass (kg) M b: Height (mm) Inertia of Rotating Square Cylinder 2 2 JW = M (a + b ) × 10–6 (kg·m2) 12 JW: Inertia (kg·m2) L: Length (mm) a: Width (mm) M: Load Mass (kg) Inertia of Linear Movement JB: Ball Screw Inertia (kg·m2) 2 JW = M ( 2πP ) × 10 –6 + JB (kg·m2) P: Ball Screw Pitch (mm) JW: Inertia (kg·m2) D: Diameter (mm) JW M1: Mass of Cylinder (kg) J1: Cylinder Inertia (kg·m2) Inertia of Lifting Object by Pulley JW = J1 + J2 2 J2: Inertia due to the Object (kg·m ) ·D2 M2 ·D2 = M1 × 10–6 (kg·m2) + 8 4 ( ) M2: Mass of Object (kg) JW: Inertia (kg·m2) 7 M Rack Inertia of Rack and Pinion Movement JW = M·D2 × 10–6 (kg · m2) 4 JW = D2 (M1 + M2) × 10–6 (kg·m2) 4 JW JW: Inertia (kg·m2) D M: Mass (kg) D: Pinion Diameter (mm) D (mm) JW Inertia of Suspended Counterbalance JW: Inertia (kg·m2) M2 M1: Mass (kg) M2: Mass (kg) M1 M3 : Mass of Object (kg) D1 : Cylinder 1 Diameter (mm) M4 : Mass of Belt (kg) Inertia when Carrying Object via Conveyor Belt JW: Inertia (kg·m2) M1 : Mass of Cylinder 1 (kg) 2 JW : Inertia (kg·m ) D2 : Cylinder 2 Diameter (mm) J1 : Cylinder 1 Inertia (kg·m2) M2 : Mass of Cylinder 2 (kg) 2 J2 : Inertia due to Cylinder 2 (kg·m ) JW = J1 + J2 + J3 + J4 2 ·D 2 ·D 2 JW = M1 1 + M2 2 · D12 + 8 8 D2 M3·D12 + M4·D12 × 10–6 4 4 (kg·m2) ( ) J3 : Inertia due to the Object (kg·m2) J4 : Inertia due to the Belt (kg·m2) JW : System Inertia (kg·m2) J1 : Roller 1 Inertia (kg·m2) J2 : Roller 2 Inertia (kg·m2) D1 : Roller 1 Diameter (mm) D2 : Roller 2 Diameter (mm) Inertia where Work is Placed between Rollers M : Equivalent Mass of Work (kg) J1 2 ( ) J + M·D4 JW = J1 + D1 D2 Roller 1 D1 2 2 1 × 10–6 (kg·m2) JW D2 M Roller 2 J2 Load Gears Inertia of a Load Value Converted to Motor Shaft JW: Load Inertia (kg·m2) Z2: Number of Gear Teeth on Load Side J2: Gear Inertia on Load Side (kg·m2) Motor JL = J1 + G2 (J2 + JW) (kg·m2) Z1: Number of Gear Teeth on Motor Side J1: Gear Inertia on Motor Side (kg·m2) JL: Motor Shaft Conversion Load Inertia Gear Ratio G = Z1/Z2 (kg·m2) 8 ■Load Torque Formulas F: External Force (N) Torque against external force TW = F·P × 10– 3 (N·m) 2π TW: Torque due to External Forces (N·m) P: Ball Screw Pitch (mm) M: Load Mass (kg) Torque against frictional force μ: Ball Screw Friction Coefficient TW = μMg· P × 10– 3 (N·m) 2π TW: Frictional Forces Torque (N·m) P: Ball Screw Pitch (mm) 2 g: Acceleration due to Gravity (9.8m/s ) D: Diameter (mm) Torque when external force is applied to a rotating object F: External Force (N) TW: Torque due to External Forces (N·m) TW = F· D × 10– 3 (N·m) 2 D: Diameter (mm) Torque of an object on the conveyer belt to which the external force is applied Torque of an object to which the external force is applied by Rack and Pinion TW = F· D × 10– 3 (N·m) 2 F: External Force (N) TW: Torque due to External Forces (N·m) F: External Force (N) D: Diameter (mm) TW: External Torque (N·m) M Pinion g: Acceleration due to Gravity (9.8m/s2) Torque of a Load Value Converted to Motor Shaft D × 10– 3 (N·m) 2 TW: Torque due to External Forces (N·m) Rack Torque when work is lifted at an angle. TW = F· TW: Load Torque (N·m) Plumb Line M: Mass (kg) TW = Mg·cosθ · D × 10– 3 (N·m) 2 D: Diameter (mm) Z2: Number of Gear Teeth on Load Side η: Gear Transmission Efficiency TL = TW · G (N·m) η Z1: Number of Gear Teeth on Motor Side Gear (Deceleration) Ratio G = Z1/Z2 TL: Motor Shaft Conversion Load Torque (N·m) 9 ■Acceleration/Deceleration Torque Formula Acceleration/Deceleration Torque (TA) TA = 2πN JM + JL (N·m) η 60tA ( ) η: Gear Transmission Efficiency M N: Motor Rotation Speed (r/min) JM: Motor Inertia (kg·m2) JL: Motor Shaft Conversion Load Inertia (kg·m2) Speed (Rotation Speed) N: Rotation Speed (r/min) TA: Acceleration/Deceleration Torque (N·m) N time tA Acceleration Time (s) ■Calculation of Maximum Momentary Torque, Effective Torque Maximum Momentary Torque (T1) Rotation Speed (rpm) N (r/min) T1 = TA + TL (N ·m) Effective Torque (Trms) T12 ·t1 + T22 ·t2 + T32 ·t3 t1 + t2 + t3 + t4 (N·m) T2 = TL (N·m) Trms = 0 tA Torque T1 time Acceleration Time (s) T3 = TL – TA (N·m) t1 = tA (N·m) TA T2 TL 0 time T3 t1 t2 t3 t4 Single Cycle TA: Acceleration/Deceleration Torque (N·m) TL: Servomotor Shaft Converted Load Torque (N·m) T1: Maximum Momentary Torque (N·m) Trms: Effective Torque (N·m) 10 ■Positioning Accuracy G = Z1/Z2 Gear (Deceleration) Ratio Z2: Number of Gear Teeth on Load Side S: Positioner Multiplier P: Ball Screw Pitch (mm) Z1: Number of Gear Teeth on Motor Side M Positioning Accuracy (AP) Ap = P· G (mm) R·S R: Encoder Resolution (Pulses/Rotation) Ap: Positioning Accuracy (mm) ■Straight Line Speed and Motor Rotation Speed V: Velocity (mm/s) P: Ball Screw Pitch (mm) Motor Rotations Z2: Number of Gear Teeth on Load Side M Z1: Number of Gear Teeth on Motor Side N: Motor Rotation Speed (r/min) N = 60V (r/min) P· G G = Z1/Z2 Gear (Deceleration) Ratio 11 Sample Calculations 1Machinery Selection • Load Mass M = 5 (kg) • Ball Screw Pitch P = 10 (mm) P • Ball Screw Diameter D = 20 (mm) • Ball Screw Mass MB = 3 (kg) M MB Direct Connection • Ball Screw Friction Coefficient μ = 0.1 • Since there is no decelerator, G = 1, η = 1 D 2Determining Operating Pattern (mm/s) speed • One Speed Change • Velocity for a Load Travel V = 300 (mm/s) 300 • Strokes L = 360 (mm) • Stroke Travel Time tS = 1.4 (s) • Acceleration/Deceleration Time tA = 0.2 (s) 0 • Positioning Accuracy AP = 0.01 (mm) Time (s) 0.2 1.0 0.2 0.2 3Calculation of Motor Shaft Conversion Load Inertia Ball screw Inertia JB 2 JB = MBD × 10– 6 8 Load Inertia JW JW = M Motor Shaft Conversion Load Inertia JL JL = G2 × (JW + J2) + J1 2 JB = ( 2πP ) × 10 –6 3 × 202 × 10– 6 = 1.5 × 10– 4 (kg·m2) 8 2 + JB JW = 5 × ( 2 ×103.14 ) × 10 –6 + 1.5 × 10– 4 = 1.63 × 10– 4 (kg·m2) JL = JW = 1.63 × 10– 4 (kg·m2) 4Load Torque Calculation Torque against Friction Torque TW TW = μMg P × 10– 3 2π TW = 0.1 × 5 × 9.8 × Motor Shaft Conversion Load Torque TL TL = G ·TW η TL = TW = 7.8 × 10–3 (N·m) 10 × 10– 3 = 7.8 × 10– 3 (N·m) 2 × 3.14 5Calculation of Rotation Speed Rotations N N = 60V P·G N= 60 × 300 = 1800 (r/min) 10 × 1 6Motor Temporary Selection [In case OMNUC U Series Servo Motor is temporarily selected] The Rotor/Inertia of the selected servo motor is JM ≥ JL 30 more than 1/30* of a load 80% of the Rated Torque of the selected servo motor is more than the load torque of the servomotor shaft conversion value TM × 0.8 > TL JL 1.63 × 10–4 = = 5.43 × 10–6 (kg·m2) 30 30 Temporarily selected Model R88M-U20030 (JM = 1.23 × 10–5). Rated Torque for R88M – U20030 Model from TM = 0.637 (N·m) TM = 0.637 (N·m) × 0.8 > TL = 7.8 × 10-3 (N·m) * Note that this value changes according to the Series. 12 7Calculation of Acceleration/Deceleration Torque Acceleration/ Deceleration Torque TA TA = 2π·N JM + JL η 60tA ( ) TA = 2π × 1800 1.63 × 10– 4 × 1.23 × 10– 5 + = 0.165 (N·m) 60 × 0.2 1.0 ) ( 8Calculation of Maximum Momentary Torque, Effective Torque Required Max. Momentary Torque is T1 = TA + TL = 0.165 + 0.0078 = 0.173 (mm/s) 300 (N·m) T2 = TL = 0.0078 (N·m) =– 0.157 (N·m) speed T3 = TL – TA = 0.0078 – 0.165 0 Effective Torque Trms is Trms = 0.1732 × 0.2 + 0.00782 × 1.0 + 0.1572 × 0.2 0.2 + 1.0 + 0.2 + 0.2 Trms = 0.0828 (N·m) t1 0.2 Acceleration/ Decceleration Torque Trms = T12·t1 + T22·t2 + T32·t3 t1 + t2 + t3 + t4 Time (s) (N·m) t2 1.0 t3 0.2 t4 0.2 0.2 Single Cycle 0.165 TA 0 Time (s) -0.165 Load Torque of Servomotor Shaft Conversion (N·m) TL 0.0078 Time (s) 0.173 Total Torque T1 T2 0.0078 T3 Time (s) -0.157 9Result of Examination Load Inertia [Load Inertia JL = 1.63 × 10–4 (kg·m2)] ≤ [Motor Rotor Inertia JM = 1.23 × 10–5] × [Applied Inertia = 30] Conditions Satisfied Effective Torque [Effective Torque Trms = 0.0828 (N·m)] < [Servomotor Rated Torque 0.637 (N·m) × 0.8] Conditions Satisfied Maximum Momentary Torque [Maximum Momentary Torque T1 = 0.173 N·m < [Servomotor Maximum Momentary Torque 1.91 (N·m) × 0.8] Conditions Satisfied Maximum Rotation Speed [Maximum Rotations Required N = 1800 (r/min)] ≤ [Servomotor Rated Rotation Speed 3000 (r/min)] Conditions Satisfied The encoder resolution when the positioner multiplication factor is set to 1 is Encoder Resolution R= P·G = 10 × 1 = 1000 (Pulses/Rotations) Ap·S 0.01 × 1 Conditions Satisfied The encoder specification of U Series 2048 (pulses/rotation) should be set 1000 with the Encoder Dividing Rate Setting. Note.This example omits calculations for the regenerative energy, operating conditions, or positioner characteristics. 13