Mass inertia calculator

Geometry Solid cylinder (shaft, roller) Hollow cylinder (tube, ring) Disc (flat cylinder) Cuboid (rectangular block) Ball Point mass / eccentric mass Spindle (linear → rotary)

Mass m

Radius r

Outer radius r_a

Inner radius r_i

Width a

Depth b

Spindle pitch P

Mass m = translationally moved mass (carriage + load)

Mass moment of inertia J

Active formula

Calculation example

Solid cylinder, steel shaft: m = 10 kg, r = 100 mm
J = ½ × 10 × 0.1² = 0.0500 kg-m²

Transmission ratio i

i = n_motor / n_load (e.g. 5 for 5:1 gearbox)

Reduced inertia J_red

J_red = J_Load / i²

J_red = Inertia reduced to motor shaft [kg-m²]
J_Load = Inertia of the load (from section 1) [kg-m²]
i = gear ratio (i = n_Motor / n_Load)

Example: J_Load = 0.050 kg-m², i = 5
J_red = 0.050 / 25 = 0.0020 kg-m²

Motor moment of inertia J_Motor

From motor catalog (J_Rotor in the data sheet)

Traffic light criteria (servo rule of thumb)

Tip: A gearbox with a higher ratio i reduces J_Load by i² - the most effective lever for improving the ratio.

The mass moment of inertia J (also called moment of inertia) describes the resistance of a body to a change in its rotational motion — analogous to mass in translational motion. The larger J, the more torque is required to accelerate the body. The unit is kg·m². J depends not only on the mass but also on the mass distribution relative to the axis of rotation: mass far from the axis increases J disproportionately.

Servo motors must dynamically accelerate and decelerate loads. The ratio J_Load/J_Motor (load inertia to motor inertia) determines how well the controller can handle the system. An excessively high ratio leads to resonance, overshoot, and unstable control behavior. For precise positioning tasks, the rule of thumb is: J_Load/J_Motor ≤ 3:1. Up to 10:1 is still tolerable in less dynamic applications.

The established rule of thumb in servo drive technology is: J_Load/J_Motor ≤ 3:1 for highly dynamic positioning tasks (e.g. material handling, pick-and-place). Up to 10:1 is acceptable for less dynamic applications with moderate accelerations. Above 10:1, special measures are required: use a gearbox with a higher gear ratio, select a motor with a larger rotor moment of inertia, or adjust the controller parameters.

With a gearbox of gear ratio i, the load inertia is reduced quadratically: J_red = J_Load / i². This means: a gearbox with i = 5 reduces the inertia acting on the motor shaft by a factor of 25. This is the main reason why gearboxes in servo drives are used not only for speed/torque adaptation but specifically for inertia reduction.

A ball screw converts rotary motion into linear motion. The inertia of the translationally moving mass m, reduced to the motor shaft, is derived from the kinetic energy equation: ½·J·ω² = ½·m·v². With v = ω·P/(2π) (P = lead in meters) it follows that J = m·(P/2π)². The smaller the lead, the lower the reduced inertia — a fine lead thus acts like a gearbox.

The mass moment of inertia J (unit: kg·m²) describes the dynamic behavior of rotating bodies and is relevant for drive design, motor selection, and controller dimensioning. The second moment of area I (unit: m⁴), on the other hand, is a purely geometric property of the cross-section and describes resistance to bending or torsion in structural mechanics. Unfortunately, both quantities share the symbol I in various standards — in drive technology, J always refers to the mass moment of inertia.