Calculator Directory
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Calculator 1
TOYO Sizing Software
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Calculator 2
Motion Profile Calculator
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Calculator 3
Actuator Loading & Moments
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Calculator 4
Linear to Rotary Converter
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Calculator 5
Load Inertia & Sizing Ratio
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Sizing Motors & Drives for Linear Actuators

Selecting the optimal combination of linear actuators, mechanical drivetrains, and servo motors requires a balanced understanding of kinematic acceleration, steady-state thermal loads, and reflected dynamic inertia. Undersizing leads to positional lag, motor overheating, tracking errors, or premature bearing failure. Oversizing increases system inertia, drives up cabinet footprint, and needlessly inflates automation bill-of-materials costs.

A rigorous electric motion sizing workflow systematically addresses four foundational mechanical criteria:

  1. Kinematic Motion Profiling: Evaluating trapezoidal versus triangular velocity profiles to minimize required peak velocity, acceleration torque, and mechanical jerk during point-to-point moves.
  2. Guideway Loading & 3-Axis Moments: Quantifying normal loading (FN) and moment limits (Roll Mx, Pitch My, and Yaw Mz) induced by center-of-gravity offsets, mounting inclination angles, and dynamic deceleration.
  3. Linear-to-Rotary Conversion: Equating linear thrust and velocity to motor shaft speed (RPM) and torque (N·m or lb·in), while factoring in mechanical efficiency and gearbox reduction ratios.
  4. Reflected Load Inertia & Mismatch Tuning: Calculating reflected load inertia (Jload) at the motor shaft and maintaining an ideal inertia mismatch ratio (JL : JM ≤ 10:1) to guarantee rapid servo loop tuning and zero mechanical resonance.

Below is the TOYO Engineering Motion Suite, developed to provide automation integrators, machine builders, and mechanical design engineers with instantaneous, field-tested sizing calculations across both Metric (SI) and American Imperial units.

Calculator 1

Official TOYO Sizing Software

Cloud Actuator & Linear Motor Sizing Engine

TOYO's official cloud sizing software provides an interactive, comprehensive environment for sizing single-axis and multi-axis Cartesian systems across TOYO's complete portfolio of ball screw actuators, belt-driven modules, rack and pinion drives, and ironcore direct-drive linear motors.

The web application models your application's complete duty cycle, stroke profiles, payload masses, orientation angles, and operating environments. It evaluates critical ball screw whip speeds, guideway dynamic moments, thermal motor ratings, and expected actuator service life (km or hours), outputting verified engineering selection reports and recommended motor/drive packages.

Notice: Leaving TOYO Robotics Americas Clicking "Launch TOYO Sizing Software" will direct you to TOYO's factory web sizing application hosted on toyorobot.com/Select. The tool opens in a secure new browser tab so you can retain your session on this engineering portal.
Launch TOYO Sizing Software
TOYO Sizing Engine
Cloud-Based Sizing Highlights
  • Actuators & Linear Motors: Multi-axis Cartesian and single-axis evaluation
  • Duty Cycle Profiling: Acceleration, dwell times, and payload CoG offsets
  • Life & Safety Factors: Dynamic bearing life, critical speed, and thermal checks
  • PDF Engineering Report: Exportable sizing reports for project archives
Calculator 2

Motion Profile Calculator

Velocity & Acceleration Dynamics

Calculate the required linear acceleration, deceleration, and peak velocity for point-to-point moves. Compare a 1/3 Trapezoidal move, a Triangular move (zero flat dwell), and an optional Custom Accel/Decel move with results presented in dedicated tabs with copyable summaries.

Total point-to-point travel distance.
seconds (s)
Total allowable move execution time from start to stop.
Enable to input specific ramp times for asymmetric or custom constant-velocity motion.
Calculator 3

Actuator Loading & Moment Calculator

Thrust, Normal Load & Moments

Evaluate worst-case kinematic and static loads on your linear actuator. Calculate maximum thrust required, normal loading, and 3-axis moment loads (−Roll \(M_x\), Pitch \(M_y\), and Yaw \(M_z\)) based on mounting inclination angle, payload mass, acceleration, process forces, and center of gravity (CoG) offsets.

Sets payload, dimension, thrust, and moment measurement units.
° (0° to 90°)
0° (Pure Horizontal) 45° (Incline) 90° (Pure Vertical)
kg
Total mass of moving payload tooling, parts, and brackets.
m/s²
Equivalent G-force: 0.00 G
N
External continuous processing load (e.g. cutting, dispensing tip drag, insertion).
μ (0 to 1)
Profile rail recirculating linear bearings typically μ ≈ 0.03. Bushings ≈ 0.15 to 0.30.
Payload Center of Gravity (CoG) Lever Arm Offsets
Reference datum: Carriage Center Point
Rodless Actuator Coordinate Reference Hover or focus an input below to highlight its axis
Along travel stroke direction (+ forward / − rear)
mm
Generates Pitch Moment (My) with normal load.
Perpendicular horizontal offset (+ right / − left)
mm
Generates Roll (Mx) & Yaw (Mz) twisting.
Height / elevation above carriage mounting plane
mm
Increases Pitch (My) during acceleration & deceleration.
Accounts for friction surges, lubrication variations, and drive wear.
Calculator 4

Linear to Rotary Motion Calculator

Kinematic & Dynamic Conversion

Determine the required rotary motor shaft speed and output torque based on your actuator's linear travel velocity, axial thrust requirement, drive mechanism lead / pulley travel, and optional gear reduction.

Pre-sets mechanical efficiency ηmech for common linear transmissions.
%
Mechanical efficiency of the actuator transmission.
Maximum or rated operational travel speed.
Total axial thrust force required (including payload friction & acceleration).
mm / rev
Ball screw lead (lead per 360° revolution) or belt pulley linear travel per revolution (π × pitch diameter). Length unit automatically tracks with linear speed.
Check to include an inline speed reducer, planetary gearbox, or timing belt wrap ratio.
Calculator 5

Load Inertia & Motor Sizing Ratio Calculator

Reflected Inertia & Servo Sizing Ratio

Calculate the total reflected load inertia at the motor shaft and evaluate the critical inertia mismatch ratio (Jload : Jmotor) for ball screw and belt-driven linear actuators. Maintain servo control loop stability, prevent mechanical resonance, and optimize dynamic acceleration rates.

Mass of tooling, parts, or workpieces carried by the actuator.
Tare mass of the linear stage moving carriage block.
Ball Screw Material Preset: Carbon Steel (ρ = 7,850 kg/m³) Rotary screw inertia calculated automatically from solid cylinder geometry.
Linear travel per full turn of the screw.
Nominal outer diameter of the ball screw shaft.
Total shaft length (or stroke + ~150 mm end margins).
Check to include an inline planetary speed reducer, belt wrap ratio, or flexible coupling. Defaults to 1:1 direct coupling if unchecked.
Leave blank if unknown to view reflected load inertia only (Jload). Enter a value to evaluate the load-to-rotor inertia mismatch ratio (JL : JM).

Engineering Formulas & Conversion Principles

Rotary Speed from Linear Velocity

The relationship between linear velocity v and mechanism rotational speed Nmech is determined by the linear distance per revolution p (lead or pulley pitch circumference):

N_mech (RPM) = (v / p) × 60

When a speed reduction gear ratio i is present (e.g. i = 4 for a 4:1 reducer), motor shaft speed becomes:
N_motor = N_mech × i

Rotary Torque from Linear Thrust

Required input torque is derived by equating linear work to rotary work while accounting for system mechanical efficiency η:

T_mech = (F × p) / (2 × π × η_mech)

Factoring in a gearbox with ratio i and gear efficiency ηgear, the required motor shaft torque is:
T_motor = T_mech / (i × η_gear)

Trapezoidal Move Profile (1/3 - 1/3 - 1/3)

Equally divides total move time T into acceleration, constant velocity, and deceleration segments (tacc = tdec = tflat = T / 3):

V_max = 1.5 × (s / T) = 1.5 × V_avg
Accel = Decel = V_max / (T / 3) = 4.5 × s / T²
Triangular Move Profile (Zero Dwell)

Smooth acceleration and deceleration with zero constant speed cruise (tacc = tdec = T / 2), yielding minimum acceleration for a given time but higher peak speed:

V_max = 2.0 × (s / T) = 2.0 × V_avg
Accel = Decel = V_max / (T / 2) = 4.0 × s / T²
Actuator Axial Thrust Mechanics

Net axial drive force sums gravitational weight along stroke, guideway friction, and inertial acceleration:

F_N = m × g × cos(θ)
F_thrust = (m × g × sin(θ)) + (μ × F_N) + (m × a) + F_ext
F_thrust,max = F_thrust × Safety_Factor
3-Axis Dynamic Moment Loading

Moments are evaluated around carriage guideway centers based on payload Center of Gravity (CoG) lever arms:

Pitch (M_y) = |F_N × L_x| + |(F_accel + F_g + F_ext) × L_z|
Roll (M_x) = |F_N × L_y|
Yaw (M_z) = |(F_accel + F_ext) × L_y|
Reflected Linear Mass & Screw Inertia

Linear mass reflects to rotational inertia via the radius conversion constant R = Lead / (2π) or Dp / 2. Ball screw solid cylinder inertia is governed by the 4th power of radius:

J_linear = m_total × R²
J_screw = (π × ρ_steel × L × r&sup4;) / 2
Gearbox Reflected Ratio & Mismatch Sizing

A speed reducer attenuates total mechanism inertia by the square of its ratio (i²), bringing the final mismatch ratio into the ideal control band (≤ 10:1):

J_load,reflected = (J_shaft / i²) + J_gearbox + J_coupling
Inertia_Ratio = J_load,reflected / J_motor