Engineering Guide: Sizing Linear Shaft Motors for Maximum Performance and Efficiency

Before sizing a motor, clearly define the motion requirements:

Key Parameters:

• Load mass (m): Total moving mass including payload, motor forcer, and mechanical components [kg]
• Travel distance (d): Stroke length or travel distance [mm or m]
• Maximum velocity (v_max): Peak velocity requirement [mm/s or m/s]
• Acceleration time (t_acc): Time to reach maximum velocity [seconds]
• Deceleration time (t_dec): Time to decelerate from maximum velocity [seconds]
• Settling time (t_settle): Time at destination before next move [seconds]
• Cycle time (t_cycle): Total time for complete move cycle [seconds]

Motion Profile Types:

1. Trapezoidal Profile (most common)

• Constant acceleration phase
• Constant velocity cruise phase
• Constant deceleration phase

2. Triangular Profile (short moves, no cruise)

• Constant acceleration to midpoint
• Constant deceleration to destination
• Never reaches maximum velocity

3. S-curve Profile (smooth motion)

• Gradual acceleration ramp (jerk-limited)
• Constant velocity cruise
• Gradual deceleration ramp
• Smoother but slower than trapezoidal

Example Application: Pick-and-Place

• Moving mass: 5 kg (2 kg payload + 2 kg gripper + 1 kg forcer)
• Travel distance: 300 mm
• Maximum velocity: 1 m/s
• Acceleration time: 0.1 s (10 m/s² acceleration)
• Cruise time: 0.2 s
• Deceleration time: 0.1 s
• Settling time: 0.05 s
• Cycle time: 0.45 s (2.2 cycles/second)

Multiple forces must be overcome during motion:

1. Acceleration Force (F_acc)

Force required to accelerate the mass:

F_acc = m × a

Where:

• m = moving mass [kg]
• a = acceleration [m/s²]

Note: Acceleration (a) = v_max / t_acc

Example calculation:

• m = 5 kg
• v_max = 1 m/s
• t_acc = 0.1 s
• a = 1 / 0.1 = 10 m/s²
• F_acc = 5 × 10 = 50 N

2. Friction Force (F_friction)

Bearing friction and seal friction:

F_friction = μ × N

Where:

• μ = coefficient of friction (typically 0.001-0.005 for linear bearings)
• N = normal force (= m × g for horizontal axis)

Example calculation (horizontal axis):

• μ = 0.003 (typical for ball bearing linear guides)
• N = 5 × 9.81 = 49 N
• F_friction = 0.003 × 49 = 0.15 N

3. Gravity Force (F_gravity) (vertical axis only)

F_gravity = m × g = m × 9.81

For vertical axis:

• F_gravity = 5 × 9.81 = 49 N

(Zero for horizontal axis)

4. External Process Forces (F_process)

Application-specific forces:

• Cutting forces
• Adhesive forces (pick-and-place)
• Pressure forces
• Cable drag forces

Total Force Requirement:

Horizontal axis:

F_total = F_acc + F_friction + F_process

Vertical axis:

F_total = F_acc + F_friction + F_gravity + F_process

Example (horizontal pick-and-place):

• F_acc = 50 N
• F_friction = 0.15 N
• F_process = 2 N (gripper release force)
• F_total = 50 + 0.15 + 2 = 52.2 N

Peak force determines if the motor can execute the move, but RMS (Root Mean Square) force determines if the motor can sustain continuous operation without overheating.

RMS Force Calculation:

F_RMS = sqrt[(Σ(F_i² × t_i)) / t_cycle]

Where:

• F_i = force during phase i
• t_i = duration of phase i
• t_cycle = total cycle time

Example Motion Profile Analysis:

Phase	Force (N)	Duration (s)	F² × t
Acceleration	52 N	0.1 s	270.4
Cruise (constant velocity)	2 N	0.2 s	0.8
Deceleration	48 N (slightly less)	0.1 s	230.4
Settling (stationary)	0 N	0.05 s	0

Sum: Σ(F² × t) = 270.4 + 0.8 + 230.4 + 0 = 501.6

F_RMS = sqrt(501.6 / 0.45) = sqrt(1114.7) = 33.4 N

Key Insight:

Even though peak force is 52N, RMS force is only 33.4N (64% of peak). This is because:

• High force only during brief acceleration/deceleration phases
• Low force during cruise and settling phases
• Motor spends most of cycle at low force

A motor with 70N peak force and 35N continuous force rating would be appropriate for this application.

Real-world conditions introduce uncertainties that require safety margins:

Recommended Safety Factors:

1. Peak Force Safety Factor: 1.5-2.0×

• Accounts for unexpected loads
• Provides headroom for future changes
• Ensures responsive control (servo loop margin)

2. RMS Force Safety Factor: 1.2-1.5×

• Accounts for thermal variations
• Provides margin for higher ambient temperatures
• Compensates for optimistic duty cycle estimates

Example Application:

Required forces:

• Peak force required: 52 N
• RMS force required: 33.4 N

With safety factors:

• Peak force with margin: 52 × 1.5 = 78 N
• RMS force with margin: 33.4 × 1.3 = 43.4 N

Motor Selection Criteria:

Select motor where:

• Motor peak force ≥ 78 N
• Motor continuous force ≥ 43.4 N

Candidate: Linear shaft motor with 100N peak, 50N continuous

• Peak margin: 100 / 52 = 1.92× ✓
• Continuous margin: 50 / 33.4 = 1.50× ✓

Even if RMS force is within motor continuous rating, verify actual thermal performance for your specific duty cycle.

Duty Cycle Definition:

Duty Cycle = (t_on / t_cycle) × 100%

Where t_on = time motor is generating force

Example calculation:

• Acceleration phase: 0.1 s (52 N)
• Cruise phase: 0.2 s (2 N)
• Deceleration phase: 0.1 s (48 N)
• Settling phase: 0.05 s (0 N)
• t_on = 0.4 s (excluding settling)
• t_cycle = 0.45 s
• Duty cycle = (0.4 / 0.45) × 100% = 89%

Thermal Derating:

For duty cycles exceeding motor's rated continuous duty cycle (typically 100% for linear shaft motors), apply derating:

F_derated = F_continuous × sqrt(DC_rated / DC_actual)

Linear shaft motors typically rate continuous force at 100% duty cycle, so no derating needed for this example.

Ambient Temperature Considerations:

Motor continuous force ratings assume 40°C ambient. For higher ambient temperatures:

• 40°C ambient: Full continuous force rating
• 50°C ambient: Derate by ~10%
• 60°C ambient: Derate by ~20% or add forced cooling

Example 1: Semiconductor Wafer Transport (Horizontal)

Requirements:

• Moving mass: 8 kg (wafer + carrier + forcer)
• Travel: 500 mm
• Velocity: 0.5 m/s
• Acceleration: 2 m/s² (clean room vibration limits)
• Cycle time: 1.2 s (continuous operation)

Force calculations:

• F_acc = 8 × 2 = 16 N
• F_friction = 8 × 9.81 × 0.002 = 0.16 N (ultra-low friction air bearings)
• F_total = 16.16 N

RMS force analysis (trapezoidal profile):

• Acceleration: 16 N for 0.25 s
• Cruise: 0.16 N for 0.5 s
• Deceleration: 16 N for 0.25 s
• Settling: 0 N for 0.2 s
• F_RMS = 11.7 N

With safety factors (1.5× peak, 1.3× continuous):

• Peak required: 16 × 1.5 = 24 N
• Continuous required: 11.7 × 1.3 = 15.2 N

Motor selection: Linear shaft motor 50N peak / 30N continuous

Example 2: Medical Device Pick-and-Place (Vertical)

Requirements:

• Moving mass: 3 kg
• Travel: 200 mm (vertical)
• Velocity: 0.3 m/s
• Acceleration: 3 m/s²
• Gripper force: 5 N
• Cycle time: 2 s

Force calculations (upward motion):

• F_acc = 3 × 3 = 9 N
• F_gravity = 3 × 9.81 = 29.4 N
• F_friction = 0.1 N
• F_gripper = 5 N
• F_total_up = 43.5 N

Force calculations (downward motion):

• Gravity assists, only need force to control deceleration: 9 N

RMS analysis:

• Upward acceleration: 43.5 N for 0.1 s
• Upward cruise: 34.5 N for 0.57 s
• Upward deceleration: 34.5 N for 0.1 s
• Grip hold: 34.4 N for 0.2 s
• Downward acceleration: 0 N (gravity)
• Downward cruise: 0 N (gravity)
• Downward deceleration: 9 N for 0.1 s
• Return settling: 0 N for 0.93 s
• F_RMS = 23.8 N

With safety factors:

• Peak required: 43.5 × 1.5 = 65 N
• Continuous required: 23.8 × 1.3 = 31 N

Motor selection: Linear shaft motor 80N peak / 48N continuous (includes brake for vertical safety)