# Motor Selection and Sizing

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1 Motor Selection and Sizing Motor Selection With each application, the drive system requirements greatly vary. In order to accommodate this variety of needs, Aerotech offers five types of motors. Motors Highest acceleration, highest speed No backlash, windup or wear Slotless Motors Ultra-smooth operation Zero cogging Complex amplifier design Brushless Motors High acceleration High torque More complex amplifier design Motor Sizing Process The following sections describe how to choose a motor using speed, torque, and inertia selection criteria. The basic procedure for sizing a motor is: 1. Determine move parameters 2. Calculate load inertia (or mass) 3. Calculate peak and rms torque (or force) requirements While the steps for motor sizing remain constant, different mechanical systems require different formulas to calculate the first three steps. The selection of the motor is determined by the general characteristics of the motor desired and the ability of the motor to meet the calculated requirements. Motor Sizing Software Aerotech produces a Motor Sizing Workbook that is available on Aerotech s web site. This file can be downloaded HERE. The free-of-charge motor sizing sheet features the following application support: Motor Direct Drive Leadscrew Beltdrive Rack and Pinion DC Servomotors Smooth operation, low-velocity ripple High torque Brushes limit ability to continuously start and stop Brushes require maintenance Microstepping Motors Simple operation High torque at low speeds Open loop Low acceleration capability More heat generation than servomotors 26

2 Motor Sizing and Selection Engineering Reference Motor Selection and Sizing CONTINUED Motion Velocity Profile The most common and efficient velocity profile for pointto-point moves is the 1/3-1/3-1/3 trapezoid. This profile breaks the time of the acceleration, traverse, and deceleration into three equal segments. The end result is that the profile provides the optimal move by minimizing the power required to complete the move. If you know the move distance and time, you can quickly calculate the acceleration and velocity as follows: α = acceleration in rad/s 2 θ = distance in radians t = total move time in ω = peak velocity in rad/s a = acceleration in m/s 2 x = distance in meters t = total move time in v = peak velocity in m/s Inertia and Mass In a rotary motor system, the best load-to-motor inertia match is 1:1 because it minimizes power consumption and increases system stability. Typically, systems will not achieve a 1:1 ratio; ratios as high as 10:1 can exist without adversely affecting system stability. For low bandwidth systems, higher ratios are acceptable but should be avoided due to the effect of decreased system stability. In a linear motor system, the mass is direct-coupled to the moving coil (forcer). System stability is directly dependent on the stiffness of the mechanics and bandwidth of the servo controller. Note: The inertia of a rotary system (J system ) is dependent on the mechanics of the system. The load in a linear motor system (m Load ) is the sum of all weights (kg used as a force) directly connected to the moving forcer coil. Peak Torque and Peak Force The total torque that a motor must produce to move itself and the load is: T t = total peak torque in T a = acceleration torque in T f = friction torque in T w = viscous torque in F t = total peak force in N F a = acceleration force N F f = friction force in N F g = gravity force in N A simplified expression for acceleration torque and force that provides a reasonable estimation value is: T α = acceleration torque J t = total inertia in kg-m 2 α = acceleration in rad/s 2 F a = acceleration force m t = total mass in kg a = acceleration in m/s 2 27

3 Motor Selection and Sizing CONTINUED A simplified expression for gravity torque and force that provides a reasonable estimation value is: W = weight of load in kg P = pitch of transmission (rev/mm) F g = total force in N m t = total mass in kg g = acceleration from gravity rms Torque Obtaining the root-mean-square (rms) value of the required torque in an application is important because the heating of the motor is related to the square of the rms torque output of the motor. Motor/Ball Screw Example The specifications of the system are: Ball-Screw Data Diameter: 13 mm Length: 500 mm Pitch: Efficiency: 0.5 rev/mm 90% (ball screw) 40% (lead screw) Mechanical Data Friction coefficient (µ): 0.05 Load: 4.5 kg Orientation: horizontal Move Profile Type: 1/3-1/3-1/3 trapezoid Distance: 8 mm Move time: 0.1 s Dwell time: 0.1 s Motor Brushless: BM75E STEP 1. Determine move parameters The peak speed (ω) and acceleration (α) of the motor required can be determined using the formulas: Trms = rms torque in oz-in (using 1/3-1/3-1/3 profile) T a = acceleration torque in N-n T f = friction torque in T w = viscous torque in t on = total move time in t off = dwell time between moves F rms = rms force in N (using 1/3-1/3-1/3 profile) F a = acceleration force in N F f = friction force in N F g = gravity force in N t on = total move time in t off = dwell time between moves in Note: Verify that both values are within the specifications of the motor. STEP 2. Determine motor shaft load inertia The calculation of the inertia is dependent on the mechanics of the system. For the ball screw the inertia can be calculated by: J screw = screw inertia in kg-m 2 D = screw diameter in mm L = screw length in mm 28

4 Motor Sizing and Selection Engineering Reference Motor Selection and Sizing CONTINUED The inertia of the load using a ballscrew mechanism can be calculated using: J load = payload inertia (reflected) in kg-m 2 m load = payload in kg P = screw pitch in rev/mm From the results above, the inertia at the motor shaft is: The BM75E can provide 0.53 of continuous torque and 1.41 peak torque, which is much greater than the system requires. The element of friction torque, T f, can now be derived using the friction coefficient (μ): Total peak torque required of the motor is estimated by: Knowing the total torque, the formula for rms torque is applied and determined to be: A motor that gives a good inertia match and meets the required motor speed of 3600 rpm is the BM75E, with an inertia of: This is well within the motor s rating of Checking the inertia ratio (J system /J motor ) = 2.2:1. The total inertia of the complete system is given by: Step 3. Calculate peak and rms torque The acceleration (peak) torque, T α, and friction torque, T f, are now calculated using the system inertia values. The acceleration torque required can be estimated at: 29

5 Motor Selection and Sizing CONTINUED Motor Sizing Example The following specifications summarize the system. Machine Details Friction coefficient (µ): Load: 10 kg Orientation: horizontal Air cooling: 1.36 bar (20 psi) Carriage/motor mass: 5 kg Move Profile Type: 1/3-1/3-1/3 trapezoid Distance: 350 mm Move time: 250 ms Dwell time: 275 ms STEP 1. Determine move parameters Using the 1/3-1/3-1/3 model: Where: F a = force to accelerate the load F trav = force during traverse motion F d = force required to decelerate the load F f = force due to friction ma = mass x acceleration The friction force is determined by: To compute rms or continuous force, dwell time must be known. Applying the rms force equation from the previous example, F rms is: Note: Most systems utilize a sinusoidal acceleration profile instead of the more easily modeled trapezoidal for system considerations. Sinusoidal acceleration is modeled as: Note: Verify that all values are within the specifications of the system. STEP 2. Determine moving mass STEP 3. Calculate peak and rms force The rms force of N is used to select a linear motor. In this case, a BLM-203-A can provide up to 232 N of continuous force and 902 N peak force with 1.36 bar (20 psi) forced air cooling. Breaking up the move profile into four segments, the fundamental equations for calculating forces during a trapezoidal move are: 1. F a = ma + F f 2. F trav = F f 3. F d = ma F f 4. Cycle dwell time in 30

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